A multiplicative Tate spectral sequence for compact Lie group actions
aa r X i v : . [ m a t h . A T ] A ug A MULTIPLICATIVE TATE SPECTRAL SEQUENCEFOR COMPACT LIE GROUP ACTIONS
ALICE HEDENLUND AND JOHN ROGNES
Abstract.
Given a compact Lie group G and a commutative orthogonal ringspectrum R such that R [ G ] ∗ = π ∗ ( R ∧ G + ) is finitely generated and projectiveover π ∗ ( R ), we construct a multiplicative G -Tate spectral sequence for each R -module X in orthogonal G -spectra, with E -page given by the Hopf algebraTate cohomology of R [ G ] ∗ with coefficients in π ∗ ( X ). Under mild hypotheses,such as X being bounded below and the derived page RE ∞ vanishing, thisspectral sequence converges strongly to the homotopy π ∗ ( X tG ) of the G -Tateconstruction X tG = [ g EG ∧ F ( EG + , X )] G . Contents
1. Introduction 12. Tate cohomology for Hopf algebras 93. Homotopy groups of orthogonal G -spectra 414. Sequences of spectra and spectral sequences 485. The G -homotopy fixed point spectral sequence 746. The G -Tate spectral sequence 88References 1141. Introduction
This paper grew out of an attempt to spell out the details for the Tate spectralsequence for the circle group T . The construction of a multiplicative Tate spectralsequence for finite groups has been around for a while now: the first constructioncan be found in [GM95], and another one, which makes the multiplicative propertiesof the spectral sequence more transparent, can be found in [HM03]. However, whilemultiplicativity of the T -Tate spectral sequences has been used in computations,the authors of this paper have found references discussing the details for how sucha spectral sequence is constructed surprisingly lacking. We hope that this paperwill fill that gap in the literature.The authors’ motivation for considering the T -Tate spectral sequence comesfrom the study of topological Hochschild homology and its refinements, such astopological cyclic homology. Given an E -ring spectrum B , topological Hochschildhomology THH( B ), first defined in the unpublished manuscript [B¨ok85], is a gen-uine T -equivariant spectrum. The study of the Tate construction on this spectrumusing the entire circle action goes back to [BM94] and [AR02], and was put in thespotlight by Hesselholt in [Hes18] under the name of topological periodic cyclichomology: TP( B ) = THH( B ) t T . Date : August 21, 2020.
Recently, Bhatt–Morrow–Scholze showed that there is a tight connection betweentopological periodic cyclic homology and crystalline cohomology [BMS19].
Background and aim.
Classically, Tate cohomology is a way to combine grouphomology and group cohomology into a single multiplicative cohomology theory,and was first introduced by Tate in his study of class field theory [Tat52]. We sketchthe main ideas involved following [CE56, Section XII.3] and [Bro82]. Given a finitegroup G , the main observation of Tate cohomology is this: if we dualise a projectiveresolution of Z as a trivial module over Z [ G ], we end up with a “coresolution” of Z by projective Z [ G ]-modules. This “coresolution” Hom Z ( P ∗ , Z ) can be spliced withthe original projective resolution P ∗ , and we so obtain a bi-infinite resolution ˆ P ∗ of Z called a complete resolution. Tate cohomology of G with coefficients in a G -module M is defined as ˆ H n ( G, M ) = H n (Hom G ( ˆ P ∗ , M )) . The Tate construction in the category of G -spectra can be seen as a generalisationof Tate cohomology in the context of higher algebra. Given a compact Lie group G and orthogonal G -spectrum X , we define the G -homotopy orbits and G -homotopyfixed points of X as X hG = EG + ∧ G X and X hG = F ( EG + , X ) G , respectively. Here EG denotes a free contractible G -space. These can be regardedas generalisations of group homology and group cohomology. Indeed, if G is a finitegroup and X = HM is the Eilenberg–Mac Lane spectrum on the G -module M ,then the homotopy groups of the G -homotopy orbits and G -homotopy fixed pointsof HM recover group homology and group cohomology of G with coefficients in M ,respectively. We define the G -Tate construction on X as the G -fixed point spectrum X tG = (cid:16)g EG ∧ F ( EG + , X ) (cid:17) G with respect to the diagonal G -action. Here, g EG denotes the mapping cone of thecollapse map c : EG + → S . This is a generalisation of Tate cohomology in thesense that the homotopy groups of the Tate construction on HM for a G -module M recover the Tate cohomology groups of the finite group G with coefficients in M .One important property of the Tate construction is that it is multiplicative inthe sense that any pairing X ∧ Y → Z of orthogonal G -spectra gives rise to apairing X tG ∧ Y tG −→ Z tG of their Tate constructions. This relies on the existenceof G -maps EG + → EG + ∧ EG + and g EG ∧ g EG → g EG . It is well-known that thediagonal map EG + → EG + ∧ EG + induces a pairing X hG ∧ Y hG −→ Z hG , making the G -homotopy fixed points construction a lax symmetric monoidal func-tor. The inclusion S → g EG and the canonical identifications S ∧ g EG ∼ = g EG ∼ = g EG ∧ S induce a natural map X hG −→ X tG and pairings X hG ∧ Y tG → Z tG and X tG ∧ Y hG → Z tG . There is a G -map N : g EG ∧ g EG → g EG extending the canonical identifications, and any two suchextensions are homotopic. Any choice of extension then induces a pairing X tG ∧ Y tG −→ Z tG MULTIPLICATIVE TATE SPECTRAL SEQUENCE 3 compatible with the above-mentioned map and pairings . In general, the ex-tension N will only be commutative and associative up to (coherent) homotopy,so X X tG is not a lax symmetric monoidal functor to the category of orthogo-nal spectra, but only satisfies a homotopy coherent version of this property, whichcould be made precise using operad actions. For our purposes it suffices to notethat it is lax symmetric monoidal as a functor to the stable homotopy category.Given an orthogonal G -spectrum X , the aim of the present paper is to constructa G -Tate spectral sequence ˆ E rs,t ( X ) = ⇒ π s + t ( X tG ) , with an algebraically specified E -page, converging, in some suitable sense, to thehomotopy groups of the G -Tate construction on X . Moreover, we would like thisspectral sequence to be multiplicative, in the sense that a pairing X ∧ Y → Z oforthogonal G -spectra should induce a pairing( ˆ E r ( X ) , ˆ E r ( Y )) −→ ˆ E r ( Z )of G -Tate spectral sequences. Finally, we want the pairing of E ∞ -pages to becompatible with the pairing π ∗ ( X tG ) ⊗ π ∗ ( Y tG ) −→ π ∗ ( Z tG )of abutments. In particular, if X is an orthogonal G -ring spectrum, then the G -Tate spectral sequence of X should be an algebra spectral sequence convergingmultiplicatively to π ∗ ( X tG ). As already mentioned, how to construct such spectralsequences is well–known in the situation of G being a finite group. Our goal is togeneralise this to higher dimensional compact Lie groups. Main results.
Let us start by describing roughly, without going into too muchdetail, what we will do in this paper. We will carry out the construction of mul-tiplicative and conditionally convergent Tate spectral sequences for compact Liegroups G such that S [ G ] ∗ = π ∗ ( S [ G ]) is finitely generated projective as a moduleover S ∗ = π ∗ ( S ). Here S denotes the sphere spectrum and S [ G ] = S ∧ G + is the unreduced suspension spectrum of G . Under these assumptions, S [ G ] ∗ is afinitely generated and projective cocommutative Hopf algebra over S ∗ , and we willshow that we have access to a multiplicative G -Tate spectral sequence with E -pagegiven by the complete Ext-groupsˆ E s, ∗ ( X ) = d Ext − s S [ G ] ∗ ( S ∗ , π ∗ ( X ))of S ∗ over S [ G ] ∗ with coefficients in the S [ G ] ∗ -module π ∗ ( X ). The multiplicativestructure in complete Ext is given by a graded commutative and associative cupproduct, and this will serve as a substitute for the failure of X X tG to belax symmetric monoidal. This spectral sequence will be strongly convergent undermild hypotheses, such as for instance in the case when the derived E ∞ -page RE ∞ vanishes and the spectrum X is bounded below.We note that this generality includes the case where G = T is the circle group, ourmain interest, but does not cover cases such as G = SO (3). We therefore broadenour scope by considering a commutative ‘ground’ orthogonal ring spectrum R Work by Nikolaus–Scholze shows that this multiplicative structure is actually unique, in ahomotopy theoretical sense; see [NS18, Theorem I.3.1]. This will not be important for our work,though. For somewhat technical reasons, it is not sufficient for us to assume that R is homotopycommutative. We analyse the product in the filtered R -module G -spectrum g EG ∧ F ( EG + , R ∧ X ) ∼ = L ∧ R M ,
ALICE HEDENLUND AND JOHN ROGNES and a compact Lie group G such that R [ G ] ∗ = π ∗ ( R [ G ]) is finitely generated andprojective over R ∗ = π ∗ ( R ), where R [ G ] = R ∧ G + . This then includes cases such as R = S [1 /
2] and R = H F , with G = SO (3). Givenan R -module X in orthogonal G -spectra we shall construct a multiplicative G -Tatespectral sequence ˆ E s, ∗ ( X ) = d Ext − sR [ G ] ∗ ( R ∗ , π ∗ ( X )) = ⇒ π s + ∗ ( X tG )where the E -page is now given as complete Ext of R ∗ over R [ G ] ∗ with coefficientsin π ∗ ( X ). This will be strongly convergent under the same conditions as before. Tate cohomology of Hopf algebras.
In Section 2 we develop a theory of Tate co-homology of a finitely generated and projective Hopf algebra Γ over a (possiblygraded) commutative ring k , with the aim being to algebraically describe the E -page of a suitable Tate spectral sequence. Our approach will be different from thecomplete resolution approach, and we instead rely on the so–called Tate complex.Given a projective Γ-resolution P ∗ of k , we will denote the mapping cone of theaugmentation map ǫ : P ∗ → k as e P ∗ . The Tate complex of a Γ-module M , firstdefined in [Gre95], is the Γ-chain complexhm ∗ ( M ) = e P ∗ ⊗ k Hom k ( P, M ) ∗ where Γ acts diagonally on the tensor product and by conjugation on Hom( P, M ) ∗ .In the aforementioned paper, the author shows that in the classical case, meaning k = Z and Γ = Z [ G ] for a finite group G , there is a zig-zag of maps e P ∗ ⊗ k Hom k ( P ∗ , M ) / / e P ∗ ⊗ k Hom k ( ˆ P ∗ , M ) Hom k ( ˆ P ∗ , M ) o o which become quasi-isomorphisms after taking G -invariants. The conclusion isthat Tate cohomology can also be computed as the (co)homology groups of the G -invariants of the Tate complex. We show that a similar result holds true in oursetting: under the assumption that Γ is a finitely generated and projective Hopfalgebra over k , the homology of the Γ-invariants of hm ∗ ( M ), which we can reason-ably refer to as the Tate cohomology of Γ with coefficients in M , is isomorphic tothe complete Ext of k over Γ with coefficients in M . Theorem 1.1. If Γ is a finitely generated and projective Hopf algebra over k , then d Ext n Γ ( k, M ) ∼ = H − n (Hom Γ ( k, hm ∗ ( M ))) . The above result, which in the text corresponds to Theorem 2.26 and Re-mark 2.27, relies crucially on a result by Pareigis which exhibits the k -dual ofa Hopf algebra Γ as an induced Γ-module. Theorem 1.2 (Pareigis) . Let Γ be a finitely generated projective Hopf algebraover k . Then there is an isomorphism Hom k (Γ , k ) ∼ = Ind Γ k P (Hom k (Γ , k )) of right Γ -modules, where P (Hom k (Γ , k )) is a finitely generated projective k -moduleof constant rank , given as the primitives for the right Γ -coaction on Hom k (Γ , k ) . with L = R ∧ g EG and M = F ( EG + , R ∧ X ), as a composition L ∧ R M ∧ R L ∧ R M (23) −→ L ∧ R L ∧ R M ∧ R M φ ∧ ψ −→ L ∧ R M for filtered products φ : L ∧ R L → L and ψ : M ∧ R M → M . Homotopy commutativity is notsufficient to ensure that the twist map τ : M ∧ R L → L ∧ R M implicit in the definition of (23) isan R - R -bimodule map. MULTIPLICATIVE TATE SPECTRAL SEQUENCE 5
Our main reason for working primarily with Tate complexes, as opposed tocomplete resolutions, has to do with multiplicative structures. Recall that the cupproduct ⌣ : Ext ∗ Γ ( k, M ) ⊗ k Ext ∗ Γ ( k, N ) −→ Ext ∗ Γ ( k, M ⊗ k N )relies on the existence of a Γ-linear chain map Ψ : P ∗ → P ∗ ⊗ k P ∗ covering theidentity map id : k → k ⊗ k k . Such a chain map exists and is unique up to chainhomotopy, by elementary homological algebra. One can extend this cup productto a product on Hopf algebra Tate cohomology by the existence of a Γ-linear chainmap Φ : e P ∗ ⊗ k e P ∗ → e P ∗ extending the fold map e P ∗ ⊕ k e P ∗ → e P ∗ . For Γ-modules M and N the composite pairing e P ∗ ⊗ k Hom k ( P ∗ , M ) ⊗ k e P ∗ ⊗ k Hom k ( P ∗ , N ) ⊗ τ ⊗ −→ e P ∗ ⊗ k e P ∗ ⊗ k Hom k ( P ∗ , M ) ⊗ Hom k ( P ∗ , N ) ⊗ ⊗ α −→ e P ∗ ⊗ k e P ∗ ⊗ k Hom k ( P ∗ ⊗ k P ∗ , M ⊗ k N ) Φ ⊗ Ψ ∗ −→ e P ∗ ⊗ k Hom k ( P ∗ , M ⊗ k N )is Γ-linear, and it induces an associative, unital, and graded commutative pairing ⌣ : d Ext ∗ Γ ( k, M ) ⊗ k d Ext ∗ Γ ( k, N ) −→ d Ext ∗ Γ ( k, M ⊗ k N )after passing to homology, which we refer to as the cup product on Tate coho-mology. This extends the cup product on ordinary Ext, in a suitable sense. SeeProposition 2.31.Finally, in Section 2.6, we do a full computation of the Tate cohomology, togetherwith the cup product, of the Hopf algebraΓ = k [ s ] / ( s = ηs ) , | s | = 1 . where s is a primitive element and k is a graded commutative ring with an element η in internal degree 1 satisfying 2 η = 0. This has relevance in the situation G = T ,which is our main case of interest. Indeed, we have π ∗ ( S [ T ]) ∼ = π ∗ ( S )[ s ] / ( s = ηs )where η is the image of the complex Hopf map in π ( S ) ∼ = Z /
2. See Proposition 3.3.The conclusion of the computation is the following theorem, which in the text isTheorem 2.52 and Remark 2.54.
Theorem 1.3.
Tate cohomology of
Γ = k [ s ] / ( s = ηs ) with coefficients in the Γ -module M is isomorphic to the homology of the differential graded Γ -module M [ t, t − ] with differential d ( m ) = tms and d ( t ) = t η , where m is an element of M and t has homological degree − , internal degree | t | = − and total degree k t k = − . If µ : M ⊗ N → L is a pairing of Γ -modules, thenthe cup product ⌣ : d Ext c Γ ( k, M ) ⊗ d Ext c Γ ( k, N ) −→ d Ext c + c Γ ( k, M ⊗ N ) −→ d Ext c + c Γ ( k, L ) is precisely the one induced by the obvious pairing M [ t, t − ] ⊗ N [ t, t − ] −→ L [ t, t − ] on homology. ALICE HEDENLUND AND JOHN ROGNES
Sequences of spectra and spectral sequences.
The main difficulty of the paper liesin verifying that there is a construction of the Tate spectral sequence that is mul-tiplicative. To deal with multiplicative structures on spectral sequences we havedecided to employ Cartan–Eilenberg systems. These are mathematical gadgets,first introduced in [CE56], which determine a spectral sequence. For us, the ad-vantage is that there is a useful notion of pairings of Cartan–Eilenberg systems,and that one can prove that a pairing of Cartan–Eilenberg systems gives rise toa pairing of the associated spectral sequences. Our contribution is a detailed andexplicit proof that a pairing of sequences of orthogonal G -spectra gives rise to apairing of Cartan–Eilenberg systems. Here, sequence simply means a sequentialdiagram · · · −→ X i − −→ X i −→ X i +1 −→ · · · of maps of orthogonal G -spectra, and pairing φ : ( X ⋆ , Y ⋆ ) → Z ⋆ refers to a collectionof G -maps φ i,j : X i ∧ Y j −→ Z i + j for all integers i and j , making the squares X i − ∧ Y j Z i + j − X i ∧ Y j − X i ∧ Y j Z i,j X i ∧ Y jφ i − ,j φ i,j − φ i,j φ i,j commute strictly. It is well-known that a sequence of orthogonal G -spectra givesrise to an unrolled exact couple on equivariant homotopy groups, which in turngives rise to a spectral sequence. That a pairing of sequences gives rise of a pairingof the corresponding spectral sequences can also reasonably be regarded as folklore,but as the authors feel that an explicit reference for this is not available at the timeof writing, we have decided to give a complete proof of this fact.For homotopical control in the proofs, some sort of ‘cofibrant replacement’ of thesequence X ⋆ is needed. In this paper we have chosen to use the classical telescopeconstruction to deal with these sorts of issues. See Section 4.3. Our main reasonfor this is that these ‘cofibrant replacements’ behave well with respect to monoidalproperties. This allows us to always approximate a sequence X ⋆ with an equivalentsequence T ⋆ ( X ) in a way that will make our analysis of multiplicative structuresmore manageable.The main result of Section 4 of the paper is the following, which in the textcorresponds to Theorem 4.27. Theorem 1.4.
A pairing φ : ( X ⋆ , Y ⋆ ) → Z ⋆ of sequences of orthogonal G -spectragives rise to a pairing φ : ( E ∗ ( X ⋆ ) , E ∗ ( Y ⋆ )) → E ∗ ( Z ⋆ ) . Explicitly, we have accessto a collection of homomorphisms φ r : E r ( X ⋆ ) ⊗ E r ( Y ⋆ ) −→ E r ( Z ⋆ ) for all r ≥ , such that: (1) The Leibniz rule d r φ r = φ r ( d r ⊗
1) + φ r (1 ⊗ d r ) holds as an equality of homomorphisms E ri ( X ⋆ ) ⊗ E rj ( Y ⋆ ) −→ E ri + j − r ( Z ⋆ ) for all i, j ∈ Z and r ≥ . MULTIPLICATIVE TATE SPECTRAL SEQUENCE 7 (2)
The diagram E r +1 ( X ⋆ ) ⊗ E r +1 ( Y ⋆ ) E r +1 ( Z ⋆ ) H ( E r ( X ⋆ ) ⊗ E r ( Y ⋆ )) H ( E r ( Z ⋆ )) φ r +1 ∼ = H ( φ r ) commutes for all r ≥ .Moreover, the induced pairing φ ∗ on filtered abutments is compatible with thepairing φ ∞ of E ∞ -pages in the sense of Proposition 4.12. Explicitly, the diagram F i A ∞ ( X ⋆ ) F i − A ∞ ( X ⋆ ) ⊗ F j A ∞ ( Y ⋆ ) F j − A ∞ ( Y ⋆ ) ¯ φ ∗ / / β ⊗ β (cid:15) (cid:15) F i + j A ∞ ( Z ⋆ ) F i + j − A ∞ ( Z ⋆ ) (cid:15) (cid:15) β (cid:15) (cid:15) E ∞ i ( X ⋆ ) ⊗ E ∞ j ( Y ⋆ ) φ ∞ / / E ∞ i + j ( Z ⋆ ) commutes, for all i, j ∈ Z . Here the abutments are given as A ∞ ( X ⋆ ) ∼ = π G ∗ Tel( X ⋆ ) A ∞ ( Y ⋆ ) ∼ = π G ∗ Tel( Y ⋆ ) A ∞ ( Z ⋆ ) ∼ = π G ∗ Tel( Z ⋆ ) with filtrations by the images F i A ∞ ( X ⋆ ) = im( π G ∗ ( X i ) −→ A ∞ ( X ⋆ )) F j A ∞ ( Y ⋆ ) = im( π G ∗ ( Y j ) −→ A ∞ ( Y ⋆ )) F k A ∞ ( Z ⋆ ) = im( π G ∗ ( Z k ) −→ A ∞ ( Z ⋆ )) , respectively.The G -Tate spectral sequence. Given an R -module X in orthogonal G -spectra, thereare a number of ways of constructing Tate spectral sequences additively; as men-tioned, the difficulty lies in establishing multiplicative properties of the construc-tions. The standard way of constructing a Tate spectral sequence seems to be byfiltering the Tate construction X tG = (cid:16)g EG ∧ F ( EG + , X ) (cid:17) G ≃ (cid:16) ( R ∧ g EG ) ∧ R F R ( R ∧ EG + , X ) (cid:17) G by filtering g EG , in some suitable sense, dualising this filtration, and splicing, inanalogy with the construction of complete resolutions by dualising and splicingprojective resolutions. This is far from ideal if one aims to prove any multiplicativeproperties of the Tate spectral sequence. We will instead prove multiplicativityof the Tate spectral sequence using a construction along the lines of [HM03]. Inthis construction, we filter F ( EG + , X ) and g EG separately, and totalise to get afiltration on the Tate construction.In more detail, we proceed as follows in Section 6. We start by giving the free G -space EG the simplicial skeletal filtration F ⋆ EG coming from the constructionof EG using the simplicial bar construction. This induces a filtration E ⋆ = R ∧ F ⋆ EG on R ∧ EG + , which in turn induces a filtration M ⋆ ( X ) = F R ( E/E − ⋆ − , X ) ALICE HEDENLUND AND JOHN ROGNES on F R ( R ∧ EG + , X ), and a filtration e E ⋆ = cone( E ⋆ − −→ R )on R ∧ g EG . The convolution filtration HM ⋆ ( X ) = ( e E ∧ T ( M ( X ))) ⋆ = colim i + j ≤ ⋆ e E i ∧ R T ( M ( X )) j is referred to as the Hesselholt–Madsen filtration. For homotopical control wehave ‘cofibrantly replaced’ the filtration M ⋆ ( X ) with its telescopic approxima-tion T ⋆ ( M ( X )). Under our projectivity assumptions, we show that the E -pageof the spectral sequence arising from the Hesselholt–Madsen filtration is given byˆ E c, ∗ ∼ = Hom R [ G ] ∗ ( R ∗ , hm c ( π ∗ ( X ))) , so that the E -page is given as the Hopf algebra Tate cohomology groupsˆ E c, ∗ ∼ = d Ext − cR [ G ] ∗ ( R ∗ , π ∗ ( X )) , as defined in Section 2. See Proposition 6.16 and Theorem 6.17. We note that theHesselholt–Madsen G -Tate spectral sequence is not obviously conditionally conver-gent, so for convergence issues we need to do some additional work.The existence of a multiplicative structure on the Hesselholt–Madsen G -Tatespectral sequence relies on the existence of filtration-preserving maps EG + −→ EG + ∧ EG + and g EG ∧ g EG −→ g EG .
The first is known to exist, and we prove by obstruction theory that the second oneexists under the assumption that R [ G ] ∗ is projective over R ∗ . See Proposition 6.9.This guarantees that a pairing X ∧ R Y → Z of R -modules in orthogonal spectrainduces a pairing ( HM ⋆ ( X ) , HM ⋆ ( Y )) −→ HM ⋆ ( Z )of the corresponding Hesselholt–Madsen filtrations. The work done in Section 4then guarantees that the G -Tate spectral sequence constructed from the Hesselholt–Madsen filtration has a multiplicative structure. Moreover, we show that the mul-tiplicative structure on the E -page agrees with the one given by cup product onTate cohomology. See Theorem 6.18 and Theorem 6.21.To settle questions about convergence we compare the Hesselholt–Madsen filtra-tion to another possible filtration of the Tate construction. The filtration we arereferring to is the filtration GM ⋆ ( X ) given in each degree as GM k ( X ) = ( e E k ∧ R T ( M ( X )) for k ≥ e E ∧ R T k ( M ( X )) for k ≤ GM k − ( X ) → GM k ( X ) for k ≥ e E k − → e E k in the filtration e E ⋆ , while the maps for k ≤ T ⋆ ( M ( X )). This filtration is referred to as the Greenlees–May filtration . It isstraight-forward to show that the spectral sequence arising from the Greenlees–Mayfiltration is conditionally convergent; see Lemma 6.37. Moreover, in Lemma 6.25we show that there is a map of filtrations α : GM ⋆ ( X ) −→ HM ⋆ ( X ) , which induces an isomorphism of spectral sequences from the E -page and on.See Proposition 6.31. We can then deduce convergence results for the Hesselholt–Madsen G -Tate spectral sequence in certain favourable situations, such as in thecase when the spectrum X is bounded below and the derived E ∞ -page RE ∞ van-ishing. In particular, we have the following result, which in the text corresponds toTheorem 6.43. MULTIPLICATIVE TATE SPECTRAL SEQUENCE 9
Theorem 1.5.
If the Greenlees–May G -Tate spectral sequence for X is stronglyconvergent, then so is the Hesselholt–Madsen G -Tate spectral sequence for X . Organisation of the paper.
Let us discuss the various sections contained in thispaper, and how they relate to one another.
Section 2:
In this section we develop a theory of Tate cohomology for finitelygenerated projective Hopf algebras, with a view toward being able to sat-isfactorily describe the E -page of a G -Tate spectral sequence for compactLie groups. Section 3:
In this section we do a quick review of orthogonal G -spectra.Most of this section can be regarded as well–known to people working ingenuine equivariant stable homotopy theory. However, we want to highlightProposition 3.6, for which we have not found a reference, and which will beimportant in later parts of the paper. Section 4:
In this section we discuss sequences of orthogonal G -spectra, Cartan–Eilenberg systems, and spectral sequences, with a special focus on multi-plicative structures. This section may well be read separately from the restof the paper, possibly in addition to Section 3.1, which contains a quickrecap on orthogonal G -spectra. We hope it can be of use as a reference formultiplicative structures on spectral sequences coming from sequences ofspectra. Section 5:
In this section we discuss the G -homotopy fixed point spectralsequence for an orthogonal G -spectrum. This is meant as a warm-up tothe G -Tate spectral sequence, but can absolutely be read in its own right. Section 6:
In this section we discuss various constructions of the G -Tatespectral sequence of an orthogonal G -spectrum. The reader who only caresfor the T -Tate spectral sequence will find a summary of the relevant resultsat the very end of the paper, in Section 6.7.2. Tate cohomology for Hopf algebras
The algebraic objects that we are led to work with when constructing the Tatespectral sequence are Hopf algebras and chain complexes of modules over these. Thetopological context we will discuss later in the paper allows for Hopf algebras overfairly complicated rings, which forces us to work in the generality of Hopf algebrasover arbitrary, possibly graded, commutative rings. We give a brief account ofthis in Section 2.1 and Section 2.2. We go on to give a suitable definition of Tatecohomology of Hopf algebras via the so-called Tate complex in Section 2.3. InSection 2.4 we relate this definition to the ordinary definition of Tate cohomologyin terms of complete resolutions. In particular, we show in Theorem 2.26 thatour definition agrees with what is traditionally referred to as Tate cohomologyor complete Ext, in the case when our Hopf algebra Γ is finitely generated andprojective over its base ring k . The crucial point that allows us to do this is aresult of Pareigis, which in particular forces the k -dual of Γ to be finitely generatedand projective over Γ, under the same hypotheses. We discuss the multiplicativestructure of Tate cohomology in Section 2.5, and finish with an explicit computationin Section 2.6.2.1. Modules over Hopf algebras.
Let k be a graded commutative ring, wherewe mean commutative in the graded sense. All unlabelled tensors and homs areto be taken over k . We denote the closed symmetric monoidal category of right k -modules by Mod( k ). Note that such modules are implicitly graded, and that mor-phisms of such modules, which we will refer to as k -linear homomorphisms, aredegree-preserving. Definition 2.1. A Hopf algebra Γ over k is a k -module equipped with five k -linearhomomorphisms: multiplication φ : Γ ⊗ Γ → Γ, comultiplication ψ : Γ → Γ ⊗ Γ, unit η : k → Γ, counit ǫ : Γ → k , and antipode χ : Γ → Γ. These are subject to thefollowing conditions:(1) Multiplication and unit provide Γ with the structure of a k -algebra.(2) Comultiplication and counit provide Γ with the structure of a k -coalgebra.(3) Comultiplication and counit are k -algebra morphisms, or equivalently, mul-tiplication and unit are k -coalgebra morphisms.(4) The antipode satisfies the formulae φ (1 ⊗ χ ) ψ = ηǫ = φ ( χ ⊗ ψ .We say that a Hopf algebra is cocommutative if the comultiplication satisfies τ ψ = ψ , where τ denotes the twist in Mod( k ). We are going to assume that all Hopf al-gebras we work with are cocommutative in this paper.A module over a Hopf algebra is just a module over the underlying k -algebra.For a right Γ-module M we denote the right action by ρ M : M ⊗ Γ → M . We denotethe category of right Γ-modules by Mod(Γ). This is a closed symmetric monoidalcategory if we endow the category with the tensor products and internal homsover k together with appropriate Γ-actions on these objects. Here let M , N , and L be Γ-modules. The tensor product M ⊗ N is endowed with the diagonal Γ-action.This is the composition M ⊗ N ⊗ Γ ⊗ ⊗ ψ / / M ⊗ N ⊗ Γ ⊗ Γ ⊗ τ ⊗ / / M ⊗ Γ ⊗ N ⊗ Γ ρ M ⊗ ρ N / / M ⊗ N .
The unit of the tensor product is k regarded as a trivial Γ-module via the counit: k ⊗ Γ ⊗ ǫ / / k ⊗ k = k . The internal hom Hom(
N, L ) becomes a Γ-module by giving it the conjugate Γ-action. This is the Γ-action that needs to be on the internal hom to make surethat Hom( N, − ) is right adjoint to ( − ) ⊗ N : Mod(Γ) → Mod(Γ). In otherwords, the characterising feature of the conjugate Γ-action is that it is the Γ-action on Hom(
N, L ) that makes the counit Hom(
N, L ) ⊗ N → L and the unit M → Hom( M ⊗ N, N ) into Γ-linear maps. Explicitly, the conjugate action isadjoint to the compositionHom(
N, L ) ⊗ Γ ⊗ N ⊗ τ −−→ Hom(
N, L ) ⊗ N ⊗ Γ ⊗ ⊗ ψ −−−−−→ Hom(
N, L ) ⊗ N ⊗ Γ ⊗ Γ ⊗ ⊗ χ ⊗ −−−−−−→ Hom(
N, L ) ⊗ N ⊗ Γ ⊗ Γ ⊗ ρ N ⊗ −−−−−→ Hom(
N, L ) ⊗ N ⊗ Γ ev ⊗ −−−→ L ⊗ Γ ρ L −−→ L .
These actions on tensor and hom-objects ensure that the forgetful functor U : Mod(Γ) → Mod( k ) is strict closed monoidal. Lemma 2.2.
Let M and N be Γ -modules, where we assume that M is projectiveover Γ and N is projective over k . Then M ⊗ N is projective over Γ .Proof. By the tensor-hom adjunction we have a natural isomorphismHom Γ ( M ⊗ N, − ) ∼ = Hom Γ ( M, Hom( N, − ))of functors. Since N is projective over k the functor Hom( N, − ) is exact, andsince M is projective over Γ the functor Hom Γ ( M, − ) is exact. The functor Hom Γ ( M ⊗ N, − ) is then also exact, being naturally isomorphic to the composition of two exactfunctors. This is equivalent to the assertion that M ⊗ N is projective over Γ. (cid:3) Symmetry uses that Γ is cocommutative.
MULTIPLICATIVE TATE SPECTRAL SEQUENCE 11
The forgetful functor U admits a left adjointInd Γ k : Mod( k ) −→ Mod(Γ) , which we refer to as induction . This functor sends a k -module C to C ⊗ Γ withthe Γ-action given by C ⊗ Γ ⊗ Γ ⊗ φ −−−→ C ⊗ Γ . The forgetful functor U also admits a right adjointCoind Γ k : Mod( k ) −→ Mod(Γ) , which is referred to as coinduction . This functor sends a k -module C to Hom(Γ , C )with Γ-action given as the adjoint ofHom(Γ , C ) ⊗ Γ ⊗ Γ ⊗ τ −−→ Hom(Γ , C ) ⊗ Γ ⊗ Γ ⊗ ⊗ χ −−−−→ Hom(Γ , C ) ⊗ Γ ⊗ Γ ⊗ φ −−−→ Hom(Γ , C ) ⊗ Γ ev −→ C .
The fact that the forgetful functor is strict monoidal makes sure that induc-tion and coinduction interact with the forgetful functor in various useful ways.In [LMSM86, Section 2.4] the following formulae, in the context of equivariant sta-ble homotopy theory, are called untwisting isomorphisms , and we will refer to themas such also in this paper.
Proposition 2.3.
Let M be a Γ -module and let C be a k -module. There arenatural Γ -module isomorphisms: (1) Ind Γ k ( C ⊗ U ( M )) ∼ = Ind Γ k ( C ) ⊗ M (2) Hom( M, Coind Γ k ( C )) ∼ = Coind Γ k (Hom( U ( M ) , C ))(3) Hom(Ind Γ k ( C ) , M ) ∼ = Coind Γ k (Hom( C, U ( M ))) . Proof.
The result follows formally from the Yoneda lemma together with the factthat Mod(Γ) → Mod( k ) is strict closed monoidal. Let us show the first isomor-phism, the other two are proven in a similar manner.Consider the functor corepresented by Ind Γ k ( C ⊗ U ( M )). By adjunctions we havenatural isomorphismsHom Γ (Ind Γ k ( C ⊗ U ( M )) , − ) ∼ = Hom( C ⊗ U ( M ) , U ( − )) ∼ = Hom( C, Hom( U ( M ) , U ( − ))) . Since the forgetful functor is strict closed monoidal we have the identityHom( C, Hom( U ( M ) , U ( − ))) = Hom( C, U (Hom( M, − )))and by adjunctions againHom( C, U (Hom( M, − ))) ∼ = Hom Γ (Ind Γ k ( C ) , Hom( M, − )) ∼ = Hom Γ (Ind Γ k ( C ) ⊗ M, − ) . The Yoneda lemma now asserts that we have a natural isomorphism, as wanted. (cid:3)
Corollary 2.4.
Let M be a Γ -module. There are natural isomorphisms Ind Γ k ( U ( M )) ∼ = Γ ⊗ M and Coind Γ k ( U ( M )) ∼ = Hom(Γ , M ) where the Γ -action on the right hand sides are the ordinary diagonal and conjugateactions, respectively.Proof. Use that Γ = Ind Γ k ( k ). (cid:3) We will also deal a lot with functional duals of modules over Hopf algebras, solet us now recall this story.
Definition 2.5.
For each Γ-module M let DM = Hom( M, k )be its functional dual . This is a Γ-module by using the usual conjugate Γ-action.Note that the evaluation pairing ev : Hom(
N, L ) ⊗ N → L gives rise to a natu-ral Γ-linear pairing α : Hom( N, L ) ⊗ Hom( N ′ , L ′ ) −→ Hom( N ⊗ N ′ , L ⊗ L ′ )adjoint to the compositionHom( N, L ) ⊗ Hom( N ′ , L ′ ) ⊗ N ⊗ N ′ ⊗ τ ⊗ −−−−→ Hom(
N, L ) ⊗ L ⊗ Hom( N ′ , L ′ ) ⊗ N ′ ev ⊗ ev −−−−→ L ⊗ L ′ . In the case N ′ = L = k this specialises to a natural Γ-linear homomorphism ν : DN ⊗ L ′ −→ Hom(
N, L ′ ) . This map is an isomorphism when N is finitely generated and projective over k .So far we have only discussed Γ-modules, but we can also talk about (right)comodules over Γ, by which we mean comodules over the underlying coalgebrastructure of Γ. If Γ is finitely generated projective over k then we can endow itsfunctional dual D Γ with such a Γ-coaction. Moreover, this Γ-coaction is compatiblewith the Γ-action in a suitable way. See [Par71, Prop. 2]. This allows us to concludethe following.
Theorem 2.6 ([Par71, Lem. 2, Prop. 3]) . Let Γ be a finitely generated projectiveHopf algebra over k . Then there is an isomorphism D Γ ∼ = Ind Γ k P ( D Γ) of right Γ -modules, where P ( D Γ) is a finitely generated projective k -module of con-stant rank , given as the primitives for the right Γ -coaction on D Γ . Note in particular that a direct consequence of Γ being finitely generated andprojective over k is that D Γ is itself finitely generated and projective over Γ. Thisresult will be crucial in our treatment of Tate cohomology of Hopf algebras. For now,let us simply note that the result implies that we have a ‘Wirthm¨uller isomorphism’.
Corollary 2.7.
Let Γ be a finitely generated and projective Hopf algebra over k and let C be a k -module. There is a natural isomorphism Ind Γ k ( P ( D Γ) ⊗ C ) ∼ = Coind Γ k ( C ) of Γ -modules.Proof. We can assume that C is obtained from a Γ-module M by forgetting the Γ-action, as in C = U M . By the first untwisting isomorphism of Proposition 2.3 wehave Ind Γ k ( P ( D Γ) ⊗ U ( M )) ∼ = Ind Γ k ( P ( D Γ)) ⊗ M .
By Pareigis’ result it follows thatInd Γ k ( P ( D Γ)) ⊗ M ∼ = D Γ ⊗ M .
Finally, by Γ being finitely generated projective over k and untwisting, more specif-ically Corollary 2.4, we have D Γ ⊗ M ∼ = Hom(Γ , M ) ∼ = Coind Γ k ( U ( M )) . (cid:3) MULTIPLICATIVE TATE SPECTRAL SEQUENCE 13
Since P ( D Γ) is tensor-invertible over k the above tells us that induced modulesare the same things as coinduced modules. We also note that that duals of finitelygenerated projectives over Γ are themselves finitely generated projective over Γ. Corollary 2.8.
Let Γ be a finitely generated projective Hopf algebra over k andlet M be a finitely generated projective Γ -module. Then its dual DM is also finitelygenerated projective over Γ .Proof. Since M is finitely generated, we can find a short exact sequence0 / / ker( r ) / / L i ∈ I Σ n i Γ r / / M / / I is a finite indexing set. Since M is projective, this shortexact sequence splits, so that we can find a Γ-linear map u : M → L i ∈ I Σ n i Γ suchthat ru = id M . Consider the k -dual picture. Applying Hom( − , k ) gives us a longexact sequence of Γ-modules: · · · Ext k ( M, k ) o o D ker( r ) o o D (cid:0)L i ∈ I Σ n i Γ (cid:1) o o DM r ∗ o o . o o Since M is projective over Γ, which is in turn projective over k , it follows that M isprojective over k , from which we conclude that Ext k ( M, k ) ∼ = 0. We are left with ashort exact sequence, which is also split since u ∗ r ∗ = id DM . We conclude that DM is finitely generated projective over Γ since it is a retract of the finitely generatedprojective Γ-module D M i ∈ I Σ n i Γ ! ∼ = M i ∈ I Σ − n i D Γ ∼ = M i ∈ I Σ − n i Ind Γ k P ( D Γ) . (cid:3) Chain complexes of Γ -modules. In this section we give the conventions forchain complexes of Γ-modules. A lot is standard; the category Mod(Γ) is an abeliancategory and what we mean by chain complexes of Γ-modules is nothing more thanthe ordinary category of chain complexes in this abelian category. However, we wantto make a point of clarifying certain subtle points, especially related to grading andsigns.
Definition 2.9. A chain complex X ∗ of Γ -modules is a family ( X n ) n ∈ Z of Γ-modules together with morphisms of Γ-modules ∂ : X n → X n − , called boundaries ,such that ∂ = 0. A chain map f : X ∗ → Y ∗ is a family of Γ-module homomor-phisms f n : X n → Y n that commute with the boundaries.The Γ-module X n in the chain complex is of course implicitly graded: X n = M ℓ ∈ Z X n,ℓ , and if we want to emphasize the bigrading we will write X ∗ , ∗ for the complex. Weremark on the following point: as the boundaries are morphisms of Γ-modules theypreserve the Γ-module grading in the sense that they are maps ∂ : X n,ℓ → X n − ,ℓ .We use the following terminology for the different degrees. Definition 2.10.
Let X ∗ be a chain complex of Γ-modules. If x is an elementin X n,ℓ we say that x has homological degree n , internal degree | x | = ℓ , and totaldegree k x k = n + ℓ .As indicated, we will use the notations k x k and | x | for the total and internaldegree of x , respectively. The category of chain complexes of Γ-modules is closed symmetric monoidal. If X ∗ , Y ∗ , and Z ∗ are chain complexes of Γ-modules then thetensor product X ∗ ⊗ Y ∗ is defined in each degree as( X ⊗ Y ) n = M i + j = n X i ⊗ Y j , ∂ ( x ⊗ y ) = ∂ ( x ) ⊗ y + ( − || x || x ⊗ ∂ ( y ) . The unit for the tensor product is k concentrated in homological degree 0. Notethat the twist isomorphism is given as τ : ( X ⊗ Y ) ∗ −→ ( Y ⊗ X ) ∗ , x ⊗ y ( − k x kk y k y ⊗ x . The hom complex Hom(
Y, Z ) ∗ is defined asHom( Y, Z ) n = Y i + j = n Hom( Y − i , Z j ) , ( ∂f )( x ) = ∂ ( f ( x )) − ( − || f || f ( ∂ ( x )) . We will in particular be interested in the case when Z ∗ is a Γ-module M , regardedas a chain complex concentrated in homological degree 0. In this case, we willoften denote the differential in the resulting function complex as ∂ ∗ = Hom( ∂, Y, M ) n = Hom( Y − n , M ) , ( ∂ ∗ f )( x ) = − ( − || f || f ( ∂ ( x )) . As before, the evaluation pairing ev : Hom(
Y, Z ) ∗ ⊗ Y ∗ → Z ∗ gives rise to a natu-ral Γ-chain map α : Hom( Y, Z ) ∗ ⊗ Hom( Y ′ , Z ′ ) ∗ −→ Hom( Y ∗ ⊗ Y ′∗ , Z ∗ ⊗ Z ′∗ )adjoint to the compositionHom( Y, Z ) ∗ ⊗ Hom( Y ′ , Z ′ ) ∗ ⊗ Y ∗ ⊗ Y ′∗ ⊗ τ ⊗ −−−−→ Hom(
Y, Z ) ∗ ⊗ Y ∗ ⊗ Hom( Y ′ , Z ′ ) ∗ ⊗ Y ′∗ ev ⊗ ev −−−−→ Z ∗ ⊗ Z ′∗ . Note that this introduces a sign in the formula for α coming from the twist τ .Explicitly, α ( f ⊗ g )( x ⊗ y ) = ( − k g kk x k f ( x ) ⊗ g ( y ) . (2.1)We now turn to suspensions and mapping cones. These are determined by specifyinga ‘circle chain complex’ and an ‘interval chain complex’. The interval object is thechain complex I ∗ given as0 −→ k { i } ∂ −→ k { i } −→ , ∂ ( i ) = i . Both of the generators i and i are regarded as a having internal degree 0 andthe subscripts indicate the homological degrees. The circle object is the chaincomplex C ∗ given as 0 −→ k { c } −→ , again with c regarded as having internal degree 0 and with the subscript indicatingthe homological degree.The convention we will use in this paper is that chain complexes are suspendedon the left. In more precise terms, the suspension of a chain complex X ∗ is thechain complex X [1] ∗ = C ∗ ⊗ X ∗ . From the definition of the symmetric monoidalstructure and the appropriate identifications we get X [1] n ∼ = X n − , ∂ X [1] ( x ) = − ∂ X ( x ) . MULTIPLICATIVE TATE SPECTRAL SEQUENCE 15
Definition 2.11.
The mapping cone of a chain map f : X ∗ → Y ∗ is the chaincomplex cone( f ) ∗ given as the pushout in the diagram X ∗ f / / i (cid:15) (cid:15) Y ∗ (cid:15) (cid:15) I ∗ ⊗ X ∗ / / cone( f ) ∗ where i ( x ) = i ⊗ x .Explicitly, we havecone( f ) n ∼ = X n − ⊕ Y n , ∂ ( x, y ) = ( − ∂ ( x ) , ∂ ( y ) + f ( x )) . We have a short exact sequence of chain complexes0 / / Y ∗ / / cone( f ) ∗ / / X [1] ∗ / / y (0 , y ) and the second one is ( x, y ) x . We leave it tothe reader to convince themself that these are indeed chain maps.2.3. Tate complexes.
Consider a projective Γ-resolution ǫ : P ∗ → k of the triv-ial Γ-module k . We will denote the mapping cone of the map ǫ as e P ∗ . With theconventions from before we hence have e P n ∼ = ( k if n = 0 P n − otherwisewith boundary ˜ ∂ : e P n → e P n − given as˜ ∂ ( x ) = ( − ∂ ( x ) if n ≥ ǫ ( x ) if n = 1.Let us use the notation i : k → e P ∗ for the inclusion. We now define the so-calledTate complex. Definition 2.12.
For each Γ-module M lethm ∗ ( M ) = e P ∗ ⊗ Hom(
P, M ) ∗ be the Tate complex of [Gre95, § G -spectrum that we call the Hesselholt–Madsen filtration, which is adapted from [HM03],and this explains the notation “hm”. See Section 6.3. Explicitly, the Tate complexis given in each homological degree byhm n ( M ) = M i + j = n e P i ⊗ Hom( P − j , M )with boundary given as ∂ hm ( x ⊗ f ) = ˜ ∂ ( x ) ⊗ f + ( − || x || x ⊗ ∂ ∗ ( f ) . Definition 2.13.
For an integer n let d Ext n Γ ( k, M ) = H − n (Hom Γ ( k, hm ∗ ( M )))be the k -module given by the ( − n )th homology of the chain complexHom Γ ( k, hm ∗ ( M )) = Hom Γ ( k, e P ∗ ⊗ Hom(
P, M ) ∗ ) . We call this the n th Tate cohomology group of Γ with coefficients in the Γ-module M . To be able to compare this definition to the standard definition of Tate cohomol-ogy in terms of complete resolutions, it is convenient to introduce an alternative,quasi-isomorphic chain complex.
Definition 2.14.
For each Γ-module M , let gm ∗ ( M ) be the pushout in the diagram M Hom(
P, M ) ∗ e P ∗ ⊗ M gm ∗ ( M ) . ǫ ∗ i ⊗ Here the top horizontal morphism is the map ǫ ∗ = Hom( ǫ,
1) : M ∼ = Hom( k, M ) −→ Hom(
P, M ) ∗ contravariantly induced by the augmentation, and the left hand vertical morphismis the map i ⊗ M ∼ = k ⊗ M −→ e P ∗ ⊗ M induced by the inclusion of k into the mapping cone e P ∗ = cone( ǫ ).Complexes of this type arise from a filtration of the Tate construction on a G -spectrum that we call the Greenlees–May filtration, which is adapted from [GM95],and this explains the notation “gm”. See Section 6.5. Proposition 2.15.
Explicitly, the complex gm ∗ ( M ) is given in each homologicaldegree as gm n ( M ) ∼ = ( e P n ⊗ M if n ≥ P − n , M ) if n ≤ and under these identifications the boundary ∂ gm : gm n ( M ) → gm n − ( M ) is givenas ∂ gm = ˜ ∂ ⊗ if n ≥ ∂ ∗ if n ≤ e P ⊗ M ǫ ⊗ −−→ M ǫ ∗ −→ Hom( P , M ) if n = 1 .Proof. The only non-trivial case happens when the homological degree is n = 0. Inthis case we have a pushout square M Hom( P , M ) e P ⊗ M gm ( M ) . ǫ ∗ i ⊗ Since i ⊗ P , M ) → gm ( M ).It is straight-forward to see that the boundary ∂ : gm n ( M ) → gm n − ( M ) is givenby ˜ ∂ ⊗ ∂ ∗ when n ≥ n ≤
0, respectively. For the remaining boundary,note that the element 1 ⊗ m in e P ⊗ M is identified with the element f : y mǫ ( y )in Hom( P , M ) when both are viewed as elements of the pushout gm ( M ). Theboundary wants to take the element x ⊗ m in e P ⊗ M ∼ = gm ( M ) to ǫ ( x ) ⊗ m in e P ⊗ M . This is identified with the map y ( − | m || x | mǫ ( x ) ǫ ( y ) in Hom( P , M ).Schematically, we are taking the composite e P ⊗ M ǫ ⊗ −→ e P ⊗ M ∼ = M ∼ = Hom( k, M ) ǫ ∗ −→ Hom( P , M ) . (cid:3) MULTIPLICATIVE TATE SPECTRAL SEQUENCE 17
Visually, gm ∗ ( M ) is the complex · · · e P ⊗ M e P ⊗ M Hom( P , M ) Hom( P , M ) · · · .M e ∂ ⊗ ǫ ⊗ ∂ ∗ ǫ ∗ We will often refer to the complex gm ∗ ( M ) as being obtained by ‘splicing’ e P ∗ ⊗ M and Hom( P, M ) ∗ together.Note that the universal property of the pushout ensures that we have an in-duced Γ-chain map θ : gm ∗ ( M ) → hm ∗ ( M ) in the commutative diagram M Hom(
P, M ) ∗ e P ∗ ⊗ M gm ∗ ( M ) hm ∗ ( M ) . ǫ ∗ i ⊗ i ⊗ ⊗ ǫ ∗ θ Here the ‘bendy’ map1 ⊗ ǫ ∗ : e P ∗ ⊗ M ∼ = e P ∗ ⊗ Hom( k, M ) −→ e P ∗ ⊗ Hom( P ∗ , M )is again the map contravariantly induced by the augmentation, and the other‘bendy’ map i ⊗ P, M ) ∗ ∼ = k ⊗ Hom(
P, M ) ∗ −→ e P ∗ ⊗ Hom(
P, M ) ∗ is induced by the inclusion i : k → e P ∗ . Proposition 2.16.
The k -linear chain map Hom(1 , θ ) : Hom Γ ( k, gm ∗ ( M )) −→ Hom Γ ( k, hm ∗ ( M )) is a quasi-isomorphism, inducing isomorphisms H n (Hom Γ ( k, gm ∗ ( M ))) ∼ = −→ d Ext − n Γ ( k, M ) for all integers n .Proof. We compatibly filter gm ∗ ( M ) and hm ∗ ( M ), setting F s gm k ( M ) = ( k > s, gm k ( M ) for k ≤ s and F s hm k ( M ) = M i + j = ki ≤ s e P i ⊗ Hom( P − j , M ) . We obtain a vertical map of short exact sequences0 / / F s − gm ∗ ( M ) / / θ s − (cid:15) (cid:15) F s gm ∗ ( M ) / / θ s (cid:15) (cid:15) F s gm ∗ ( M ) F s − gm ∗ ( M ) / / ¯ θ s (cid:15) (cid:15) / / F s − hm ∗ ( M ) / / F s hm ∗ ( M ) / / F s hm ∗ ( M ) F s − hm ∗ ( M ) / / Each horizontal short exact sequence is degree-wise split as an extension of Γ-modules, hence remains short exact after applying Hom Γ ( k, − ).For s = 0, the map F gm ∗ ( M ) → F hm ∗ ( M ) is an isomorphism. We claim foreach s ≥ θ s : F s gm ∗ ( M ) F s − gm ∗ ( M ) −→ F s hm ∗ ( M ) F s − hm ∗ ( M )induces a quasi-isomorphism Hom(1 , ¯ θ s ) after applying Hom Γ ( k, − ). It follows byinduction that Hom(1 , θ s ) is a quasi-isomorphism for each s ≥
0. Passing to colimitsover s it follows that Hom(1 , θ ) is a quasi-isomorphism.It remains to prove the claim. We can rewrite ¯ θ s for s ≥ ⊗ ǫ ∗ : e P s ⊗ M −→ e P s ⊗ Hom(
P, M ) ∗ . Here e P s is Γ-projective, so it suffices to prove thatHom Γ (1 , ⊗ ǫ ∗ ) : Hom Γ ( k, L ⊗ M ) −→ Hom Γ ( k, L ⊗ Hom(
P, M ) ∗ )is a quasi-isomorphism for any projective Γ-module L . By preservation of quasi-isomorphisms under passage to retracts, we may assume that L is free. Since Hom Γ ( k, − )commutes with direct sums, it suffices to consider the case L = Γ. Using the Hopfalgebra structure of Γ, there is a natural untwisting isomorphismΓ ⊗ N ∼ = Ind Γ k ( N )for any Γ-module N , where Γ acts diagonally on the left hand side and we usethe induced Γ-action on the right hand side. See Corollary 2.4. The augmentation ǫ : P ∗ → k admits a k -linear chain homotopy inverse. Hence ǫ ∗ : M → Hom(
P, M ) ∗ also admits such a chain homotopy inverse, andInd Γ k ( ǫ ∗ ) : Ind Γ k ( M ) −→ Ind Γ k (Hom( P, M ) ∗ )admits a Γ-linear chain homotopy inverse. By naturality of the untwisting isomor-phism, 1 ⊗ ǫ ∗ : Γ ⊗ M −→ Γ ⊗ Hom(
P, M ) ∗ admits a Γ-module chain homotopy inverse, and therefore induces a k -modulechain homotopy equivalence after applying Hom Γ ( k, − ). This proves the claimthat Hom Γ (1 , ⊗ ǫ ∗ ) is a quasi-isomorphism. (cid:3) Corollary 2.17.
The inclusion
Hom(
P, M ) ∗ → gm ∗ ( M ) induces an isomorphism γ : Ext n Γ ( k, M ) −→ d Ext n Γ ( k, M ) for each n ≥ , and a surjection for n = 0 . Complete resolutions.
In this section, we make the standing assumptionthat the cocommutative Hopf algebra Γ is finitely generated and projective over k .Let us now relate the complex gm ∗ ( M ) to the complete resolutions often used whendefining Tate cohomology. Definition 2.18.
Let ˆ P ∗ be the pullback in the diagramˆ P ∗ / / (cid:15) (cid:15) P ∗ ǫ (cid:15) (cid:15) D e P ∗ i ∗ / / k where D e P ∗ = Hom( e P ∗ , k ). MULTIPLICATIVE TATE SPECTRAL SEQUENCE 19
Proposition 2.19.
Explicitly, the chain complex ˆ P ∗ is given in each homologicaldegree as ˆ P n ∼ = ( P n for n ≥ , D ( e P − n ) for n < with boundary given as ˆ ∂ n = ∂ n for n > , ǫ ∗ ◦ ǫ for n = 0 , D ( ˜ ∂ − n ) for n < under these identifications.Proof. The only non-trivial case is when we are dealing with something involvinghomological degree 0. Since D e P → k is an isomorphism, it follows that the pro-jection ˆ P → P is one, as well. This shows that the chain complex is given in eachhomological degree as asserted. The only thing left to prove is that the bound-aries are given as claimed. To do so, note that the inverse to the projection is themap P → ˆ P given by P −→ ˆ P = P × k D e P , x ( x, ǫ ( x )) . It is clear that boundary ˆ ∂ : ˆ P n → ˆ P n − is given by ∂ n and D ( ˜ ∂ − n ) when n > n <
0, respectively. When n = 0, we are looking at the boundary P × k D e P −→ D e P , ( x, f ) ǫ ∗ ( f )which under the identifications made above corresponds to the composition P −→ P × k D e P −→ D e P , x ( ǫ ∗ ◦ ǫ )( x ) . (cid:3) Diagrammatically, we can visualise ˆ P ∗ as the “spliced” complex · · · / / ˆ P / / ˆ P / / ǫ (cid:26) (cid:26) ✹✹✹✹✹✹ ˆ P − / / ˆ P − / / · · · k ǫ ∗ C C ✞✞✞✞✞✞✞ . We will show that if P ∗ is assumed to be a projective resolution of finite type , thenthis is a complete resolution. See Remark 2.27. First we need a lemma. Lemma 2.20.
Let Q ∗ A ∗ C ∗ B ∗ g ′ f ′ fg be a pullback diagram of chain complexes. Assume that there is some chain map φ : B ∗ → A ∗ such that f φ = id B ∗ and φf ≃ id A ∗ witnessed by a chain homotopy H : A n → A n +1 satisfying f H = 0 . Then there is a chain map φ ′ : C ∗ → Q ∗ suchthat f ′ φ ′ = id C ∗ and φ ′ f ′ ≃ id Q ∗ . Recall that a chain complex C ∗ of projective modules is said to be of finite type if it is finitelygenerated in each homological degree. Proof.
Consider the diagram C ∗ Q ∗ A ∗ C ∗ B ∗ φg id C ∗ φ ′ g ′ f ′ fg in which we have an induced chain map φ ′ : C ∗ → Q ∗ by the universal property ofa pullback. This shows that f ′ φ ′ = id C ∗ . Let us show that this constitutes a lefthomotopy inverse, as well.Let H n : A n → A n +1 be the chain homotopy between φf and id A ∗ . That isid A ∗ − φf = ∂H n + H n − ∂ . We want to use this data to build a chain homotopy between id Q ∗ and φ ′ f ′ . To dothis, consider the diagram A n Q n +1 A n +1 C n +1 B n +1 H n h g ′ f ′ fg which commutes since f H n = 0, by assumption. Again, by the universal propertyof a pullback we have induced maps h : A n → Q n +1 . Let us set H ′ n = hg ′ : Q n −→ Q n +1 . We claim that these maps constitute a chain homotopy between id Q ∗ and φ ′ f ′ .That is, we claim that they satisfyid Q ∗ − φ ′ f ′ = ∂H ′ n + H ′ n − ∂ . To show this, we appeal to the uniqueness of maps induced from pullbacks. Considerthe diagram Q n Q n A n C n B n∂H n g ′ + H n − ∂g ′ g ′ f ′ fg . This diagram commutes, since f ∂H n g ′ + f H n − ∂g ′ = ∂f H n g ′ + f H n − ∂g ′ = ∂f g ′ hg ′ + f g ′ h∂g ′ = ∂gf ′ hg ′ + gf ′ h∂g ′ = 0 , so we do indeed have a unique induced map in the diagram. We claim that thequestion-mark in the diagram can be filled by both id Q ∗ − φ ′ f ′ and ∂H ′ n + H ′ n − ∂ , MULTIPLICATIVE TATE SPECTRAL SEQUENCE 21 so they must agree by uniqueness of the induced map. Checking this claim isstraight-forward. The checks g ′ (id Q ∗ − φ ′ f ′ ) = g ′ − g ′ φ ′ f ′ = g ′ − φgf ′ = g ′ − φf g ′ = (id A ∗ − φf ) g ′ = ( ∂H n + H n − ∂ ) g ′ , and f ′ (id Q ∗ − φ ′ f ′ ) = f ′ − f ′ φ ′ f ′ = f ′ − f ′ = 0show that the map id Q ∗ − φ ′ f ′ fits into the diagram. The checks g ′ ( ∂H ′ n + H ′ n − ∂ ) = g ′ ∂hg ′ + g ′ hg ′ ∂ = ∂g ′ hg ′ + g ′ hg ′ ∂ = ∂H n g ′ + H n − g ′ ∂ = ∂H n g ′ + H n − ∂g ′ = ( ∂H n + H n − ∂ ) g ′ , and f ′ ( ∂H ′ n + H ′ n − ∂ ) = f ′ ∂H ′ n + f ′ H ′ n − ∂ = ∂ ′ f ′ H ′ n + f ′ H ′ n − ∂ = ∂ ′ f ′ hg ′ + f ′ hg ′ ∂ = 0show that the map ∂H ′ n + H ′ n − ∂ also fits into the diagram, which concludes theproof. (cid:3) Proposition 2.21.
Assume that P ∗ is of finite type over Γ . Then ˆ P ∗ is an acycliccomplex of projective Γ -modules such that Hom Γ ( ˆ P ∗ , Q ) is acyclic for every coin-duced Γ -module Q .Proof. Since P n is finitely generated and projective over Γ in each homologicaldegree n , it follows that ˆ P n must be finitely generated and projective over Γ, aswell, by Corollary 2.8.To show that ˆ P ∗ is acyclic, we show that it is k -linearly contractible. Since ǫ : P ∗ → k is a chain homotopy equivalence, we can find a homotopy inverse f : k → P ∗ . In this case, we can pick η : k → P ∗ so that ǫη = id k on thenose. Since k is concentrated in homological degree 0 we know that the chainhomotopy ηǫ ≃ id P ∗ is zero after post-composition with ǫ , so that Lemma 2.20applies. This shows that the map ˆ P ∗ → D e P ∗ is a chain homotopy equivalence.Since e P ∗ is chain contractible, we conclude that so is its dual D e P ∗ and hencealso ˆ P ∗ .If Q = Coind Γ k ( C ) for some k -module C , thenHom Γ ( ˆ P ∗ , Coind Γ k ( C )) ∼ = Hom( ˆ P ∗ , C ) . Since ˆ P ∗ is k -linearly contractible it follows that Hom( ˆ P ∗ , C ) is contractible, andtherefore acyclic. (cid:3) Let M be a Γ-module. The chain map ˆ P ∗ → P ∗ induces a chain mapHom( P, M ) ∗ −→ Hom( ˆ
P , M ) ∗ which is an isomorphism in homological degrees ∗ ≤
0. In addition to this map, wealso have a chain map composition e P ∗ ⊗ M −→ DD e P ∗ ⊗ M ν −→ Hom( D e P , M ) ∗ −→ Hom( ˆ
P , M ) ∗ . In particular, this is an isomorphism in homological degrees ∗ ≥ e P ∗ is of finite type over k . Note that the chain maps described abovefit into the commutative diagram M Hom(
P, M ) ∗ e P ∗ ⊗ M gm ∗ ( M ) Hom( ˆ P , M ) ∗ , ǫ ∗ i ⊗ β so that we have an induced chain map β by the universal property of gm ∗ ( M ). Proposition 2.22.
Suppose that e P ∗ is of finite type over k . Then the map β : gm ∗ ( M ) ∼ = −→ Hom( ˆ
P , M ) ∗ is a natural isomorphism of Γ -chain complexes.Proof. The assertion is clear in homological degrees n > n <
0. If n = 0 weare looking at the diagram M Hom(
P, M ) e P ⊗ M gm ( M ) Hom( ˆ P , M ) , ∼ = ∼ = ∼ = β in which we have marked the obvious isomorphisms. It is then clear that β is anisomorphism, as well. (cid:3) Corollary 2.23.
Suppose that the projective Γ -resolution P ∗ is of finite type over k .Then there is a natural isomorphism d Ext n Γ ( k, M ) ∼ = H n (Hom Γ ( ˆ P , M ) ∗ ) for all n ∈ Z .Proof. Combine Proposition 2.16 and Proposition 2.22. (cid:3)
It turns out that, under the assumption that Γ is finitely generated projectiveover k , we can always construct the projective resolution P ∗ so that it is of finitetype over Γ. It is then necessarily also of finite type over k . This can be done via thebar construction, which we now review. See ([May72, § § §
11] and [GM74, App.A]) for more details.
MULTIPLICATIVE TATE SPECTRAL SEQUENCE 23
Construction 2.24 (The bar construction) . Let Γ be a k -algebra, M a right Γ-module, and N a left Γ-module. We form a simplicial object B • ( M, Γ , N ) : ∆ op → Mod( k ) as follows. In simplicial degree q we let B q ( M, Γ , N ) = M ⊗ Γ ⊗ q ⊗ N. It is customary to write m [ γ | · · · | γ q ] n = m ⊗ γ ⊗ · · · ⊗ γ q ⊗ n for an element in the q th simplicial degree; hence the terminology bar construction.In this notation, the face maps are given as d i ( m [ γ | · · · | γ q ] n ) = mγ [ γ | · · · | γ q ] n i = 0 m [ γ | · · · | γ i γ i +1 | · · · | γ q ] n < i < qm [ γ | · · · | γ q − ] γ q n i = q and the degeneracy maps are given by s i ( m [ γ | · · · | γ q ] n ) = m [ γ | · · · | γ i | | γ i +1 | · · · | γ q ] n . The simplicial k -module B • ( M, Γ , N ) can be turned into a non-negative k -complex in essentially two ways. • The most straight-forward way to turn B • ( M, Γ , N ) into a Γ-chain complex B ∗ ( M, Γ , N ) is by taking the Γ-module in homological degree n to be equalto the n -simplices of B • ( M, Γ , N ) and to let the boundary in the chaincomplex be the alternating sum of the face maps: B n = B n ( M, Γ , N ) and ∂ = n X i =0 ( − i d i : B n −→ B n − . This is referred to as the bar complex . • To get a smaller quasi-isomorphic chain complex, more convenient for com-putations, we can turn B • ( M, Γ , N ) into a chain complex N B ∗ = N B ∗ ( M, Γ , N )quotienting out by the degenerate simplices. Explicitly, in homological de-gree n we have N B n = B n / ( s B n − + · · · + s n − B n − ) ∼ = M ⊗ Γ ⊗ n ⊗ N , where Γ = coker( η ) ∼ = ker( ǫ ) . The boundary ∂ : N B n → N B n − is given by the same formula as before,which makes sense because ∂ ( s B n − + · · · + s n − B n − ) ⊂ ( s B n − + · · · + s n − B n − ). We refer to ( N B ∗ , ∂ ) as the normalised bar complex .There is a natural Γ-action on the simplicial k -module B • ( M, Γ , Γ) arising fromviewing N = Γ as a Γ-Γ-bimodule. Explicitly, in each simplicial degree we have theright Γ-action B q ( M, Γ , Γ) ⊗ Γ → B q ( M, Γ , Γ) given by m [ γ | · · · | γ q ] γ q +1 ⊗ γ m [ γ | · · · | γ q ] γ q +1 γ and this Γ-action commutes with the simplicial structure maps of B • ( M, Γ , Γ),so that B • ( M, Γ , Γ) extends to a simplicial Γ-module. It is a standard exercisein simplicial homotopy theory to check that B • ( M, Γ , Γ) is simplicially homotopyequivalent to M viewed as a constant simplicial Γ-module. As a consequence, thecomplexes B ∗ ( M, Γ , Γ) and
N B ∗ ( M, Γ , Γ) This isomorphism follows from ǫη = id k . are resolutions of M as a Γ-module. We refer to these as the bar resolution and normalised bar resolution of M as a Γ-module, respectively. See [May72, Prop. 9.9]and [GM74, Lem. A.8]. Proposition 2.25.
Assume that Γ is finitely generated and projective over k . If M is finitely generated projective over k , then the bar resolution B ∗ ( M, Γ , Γ) and thenormalised bar resolution N B ∗ ( M, Γ , Γ) are Γ -projective resolutions of M of finitetype.Proof. Since B • ( M, Γ , Γ) is simplicially homotopy equivalent to M , the bar resolu-tion is a resolution of M . It is finitely generated, and projective in each degree byan application of Lemma 2.2. The proof in the normalised case is very similar. (cid:3) Theorem 2.26.
When Γ is finitely generated and projective over k , each shortexact sequence −→ M ′ −→ M −→ M ′′ −→ of Γ -modules induces a long exact sequence . . . −→ d Ext n Γ ( k, M ′ ) −→ d Ext n Γ ( k, M ) −→ d Ext n Γ ( k, M ′′ ) δ −→ d Ext n +1Γ ( k, M ′ ) −→ . . . . Furthermore, if M is an induced or coinduced Γ -module , then d Ext n Γ ( k, M ) = 0 forall n ∈ Z .Proof. If Γ is finitely generated projective over k , then the bar complex B ∗ ( k, Γ , Γ)constitutes a projective Γ-resolution of k of finite type, so that Proposition 2.21applies. The long exact sequence is then induced by the short exact sequence0 −→ Hom Γ ( ˆ P ∗ , M ′ ) −→ Hom Γ ( ˆ P ∗ , M ) −→ Hom Γ ( ˆ P ∗ , M ′′ ) −→ k -module chain complexes. Here we are using Corollary 2.23 to identify theterms in the long exact homology sequence. That Tate cohomology vanishes oninduced/coinduced modules is a direct consequence of condition (2) in Proposi-tion 2.21. (cid:3) Remark . Note that if Γ is finitely generated projective, the chain complex ˆ P ∗ constructed from P ∗ = B ∗ ( k, Γ , Γ) is indeed a complete Γ-resolution of k in the senseof [CK97, Definition 1.1]. This uses Proposition 2.21 and the fact that all projec-tive Γ-modules are retracts of induced Γ-modules, which are coinduced Γ-modulesby Corollary 2.7. In particular, the results of [CK97] apply and we can concludethat our d Ext Γ ( k, − ) agrees with what is traditionally referred to as “complete Ext”,in this case.2.5. Multiplicative structure of Tate cohomology.
We will now define a suit-able pairing on Hopf algebra Tate cohomology. As before, we will assume that Γ isfinitely generated and projective over k , so that this theory coincides with completeExt. Proposition 2.28.
There is a unique, up to chain homotopy, Γ -linear chain map Ψ : P ∗ → P ∗ ⊗ P ∗ covering the identity map id : k → k ⊗ k .Proof. We first note that the chain complex P ∗ ⊗ P ∗ , with diagonal Γ-action, is a Γ-resolution of k ⊗ k = k . To see this, use the spectral sequence associated to P ∗ ⊗ P ∗ viewed as a double complex. This converges strongly to the homology of P ∗ ⊗ P ∗ .The first page of the spectral sequence is given by E s,t ∼ = H t ( P s ⊗ P ∗ ) ∼ = P s ⊗ H t ( P ∗ ) , Due to the assumption that Γ is finitely generated projective induced modules are coinducedby Corollary 2.7, and vice versa, so these are actually equivalent conditions.
MULTIPLICATIVE TATE SPECTRAL SEQUENCE 25 since P s is projective over Γ, and hence over k . This is zero unless t = 0, whereit is P s . The d -differential is induced by the horizontal differential in the doublecomplex, so that the E -page is k concentrated at the origin.Classical homological algebra then asserts that there is a unique (up to chainhomotopy) Γ-linear chain map as asserted; see [ML95, Chapter III Thm. 6.1]. (cid:3) The Γ-linear chain map Ψ : P ∗ → P ∗ ⊗ P ∗ described above induces a producton Ext ∗ Γ ( k, − ) via the pairingHom Γ ( P ∗ , M ) ⊗ Hom Γ ( P ∗ , N ) α −→ Hom Γ ( P ∗ ⊗ P ∗ , M ⊗ N ) Ψ ∗ −→ Hom Γ ( P ∗ , M ⊗ N )of k -module complexes. By cocommutativity of Γ and uniqueness (up to chainhomotopy) of Ψ, we have that Ψ ≃ τ ◦ Ψ. Passing to homology, this gives us anassociative, unital, and graded commutative multiplication ⌣ : Ext ∗ Γ ( k, M ) ⊗ Ext ∗ Γ ( k, N ) −→ Ext ∗ Γ ( k, M ⊗ N )that we will refer to as the cup product . In particular, Ext ∗ Γ ( k, k ) is a k -algebra,and Ext ∗ Γ ( k, M ) is an Ext Γ ( k, k )-module for each Γ-module M . If M is a Γ-modulealgebra, then Ext ∗ Γ ( k, M ) is an Ext ∗ Γ ( k, k )-algebra.We proceed to define the cup product in Tate cohomology for Hopf algebras.For this, we need a unique (up to chain homotopy) Γ-linear extension of the foldmap in the category of chain complexes of Γ-modules under k . Explicitly, the foldmap ∇ is the induced map in the commutative diagram k ⊗ k e P ∗ e P ∗ e P ∗ ⊕ k e P ∗ e P ∗ , i ⊗ ⊗ i idid ∇ where the inner square is a pushout diagram. Let us start with a more generalresult. Lemma 2.29.
Let A ∗ , B ∗ , and C ∗ be chain complexes of Γ -modules, where weassume that C ∗ is non-negative and exact. Let i : A ∗ → B ∗ be an injective chainmap and assume that Q ∗ = coker( i ) is projective over Γ in each homological degree.Then, for each chain map f : A ∗ → C ∗ there is a chain map g : B ∗ → C ∗ suchthat gi = f . Moreover, this chain map is unique up to a chain homotopy that iszero on the image of i .Proof. Consider the diagram0 A ∗ B ∗ Q ∗ C ∗ if rg where the top sequence is short exact in each homological degree. Since C ∗ is non-negative, we must have g n = 0 for n <
0. To construct the rest of the chain map weproceed by induction. Assume inductively that we have constructed g m satisfying g m i m = f m and g m − ∂ = ∂g m for all m < n . Since 0 = g n − ∂ = ∂g n − ∂ and C ∗ is exact we know that g n − ∂ lands in ∂ ( C n ). Consider the diagram0 A n B n Q n C n ∂ ( C n ) i n f n r n g n g n − ∂ in which we want to find a dashed map g n : B n → C n that makes both trianglescommute. Since Q n is projective, the short exact sequence at the top of the diagramsplits, and we can find s n : Q n → B n and t n : B n → A n such that i n t n + s n r n = id B n . Moreover, we can find a map h n : Q n → C n such that g n − ∂s n = ∂h n . We define g n : B n → C n by setting g n = f n t n + h n r n . This map satisfies g n i n = f n t n i n + h n r n i n = f n + 0 = f n and ∂g n = ∂f n t n + ∂h n r n = f n − ∂t n + ∂h n r n = g n − i n − ∂t n + g n − ∂s n r n = g n − ∂ ( i n t n + s n r n )= g n − ∂ , which concludes the construction of g .We now show that the map g : B ∗ → C ∗ is unique up to a chain homotopy that iszero on i ( A ∗ ). Let g ′ : B ∗ → C ∗ be another chain map satisfying f = g ′ i . We wantto show that we can find a chain homotopy between k = g − g ′ and 0 that is zeroon the image of i . That is, we want to find a collection of maps H n : B n → C n +1 such that k n = H n − ∂ + ∂H n and H n i n = 0for all n . Again, we use induction. Since C ∗ is non-negative we must have H n = 0for n <
0. Assume inductively that we have constructed H m : B m → C m +1 satisfying k m = H m − ∂ + ∂H m and H m i m = 0for all m < n . Consider the map k n − H n − ∂ : B n → C n . Since ∂ ( k n − H n − ∂ ) = ∂k n − ∂H n − ∂ = ∂k n − ( k n − − H n − ∂ ) ∂ = ∂k n − k n − ∂ = 0and C ∗ is exact we know that k n − H n − ∂ lands in ∂ ( C n +1 ). Consider the diagram Q n C n +1 ∂ ( C n +1 ) 0 k n s n − H n − ∂s n β n in which we have a map β n since Q n is projective. We define H n = β n r n : B n −→ C n +1 , MULTIPLICATIVE TATE SPECTRAL SEQUENCE 27 which vanishes on the image of i n since r n i n = 0. Furthermore, we have H n − ∂ + ∂H n = H n − ∂ + ∂β n r n = H n − ∂ + k n s n r n − H n − ∂s n r n = H n − ∂ + k n − H n − ∂ = k n where the penultimate equality sign follows from the fact that k n and H n − vanishon the image of i n and i n − , respectively, so that k n = k n ( i n t n + s n r n ) = k n s n r n and H n − ∂ = H n − ∂ ( i n t n + s n r n )= H n − ∂i n t n + H n − ∂s n r n = H n − i n ∂t n + H n − ∂s n r n = H n − ∂s n r n . (cid:3) Proposition 2.30.
There is a unique, up to chain homotopy, Γ -linear chain map Φ : e P ∗ ⊗ e P ∗ → e P ∗ that makes the diagram e P ∗ ⊕ k e P ∗ e P ∗ e P ∗ ⊗ e P ∗∇ Φ commute.Proof. This is an application of Lemma 2.29. The diagram we are considering is0 e P ∗ ⊕ k e P ∗ e P ∗ ⊗ e P ∗ Q ∗ e P ∗ i ∇ r Φ We only need to check that the cokernel of the map i is projective over Γ in eachhomological degree. By construction, the cokernel is the total cokernel of the com-mutative diagram k ⊗ k e P ∗ ⊗ kk ⊗ e P ∗ e P ∗ ⊗ e P ∗ . i ⊗ ⊗ i ⊗ ii ⊗ This can be calculated by computing the cokernels of the two horizontal mapsfollowed by the cokernel of induced vertical map. Explicitly:coker( i ) ∼ = coker(coker( i ⊗ −→ coker( i ⊗ ∼ = coker(1 ⊗ i : P [1] ∗ ⊗ k −→ P [1] ∗ ⊗ e P ∗ ) ∼ = P [1] ∗ ⊗ P [1] ∗ . In particular, we note that the cokernel is a complex of projective Γ-modules. (cid:3)
We can now define a pairing on hm ∗ ( − ) using Φ and Ψ. For Γ-modules M and N the composite pairing e P ∗ ⊗ Hom( P ∗ , M ) ⊗ e P ∗ ⊗ Hom( P ∗ , N ) ⊗ τ ⊗ −→ e P ∗ ⊗ e P ∗ ⊗ Hom( P ∗ , M ) ⊗ Hom( P ∗ , N ) ⊗ ⊗ α −→ e P ∗ ⊗ e P ∗ ⊗ Hom( P ∗ ⊗ P ∗ , M ⊗ N ) Φ ⊗ Ψ ∗ −→ e P ∗ ⊗ Hom( P ∗ , M ⊗ N )is Γ-linear, so it induces a pairingHom Γ ( k, hm ∗ ( M )) ⊗ Hom Γ ( k, hm ∗ ( N )) −→ Hom Γ ( k, hm ∗ ( M ⊗ N ))of k -module complexes. Note that the uniqueness of Φ up to chain homotopyguarantees that Φ ◦ τ ≃ Φ, and we have already observed that τ ◦ Ψ ≃ Ψ, whichensures that we get an associative, unital, and graded commutative pairing ⌣ : d Ext ∗ Γ ( k, M ) ⊗ d Ext ∗ Γ ( k, N ) −→ d Ext ∗ Γ ( k, M ⊗ N )after passing to homology. The inclusion Hom( P, M ) ∗ → hm ∗ ( M ) provides us witha map Ext ∗ Γ ( k, M ) −→ d Ext ∗ Γ ( k, M ) . This map is compatible with the multiplicative structures we have defined above,in the sense of the following proposition.
Proposition 2.31.
The two diagrams
Ext b Γ ( k, M ) ⊗ Ext b Γ ( k, N ) ⌣ / / (cid:15) (cid:15) Ext b + b Γ ( k, M ⊗ N ) (cid:15) (cid:15) Ext b Γ ( k, M ) ⊗ d Ext b Γ ( k, N ) / / (cid:15) (cid:15) d Ext b + b Γ ( k, M ⊗ N ) d Ext b Γ ( k, M ) ⊗ d Ext b Γ ( k, N ) ⌣ / / d Ext b + b Γ ( k, M ⊗ N ) and Ext b Γ ( k, M ) ⊗ Ext b Γ ( k, N ) ⌣ / / (cid:15) (cid:15) Ext b + b Γ ( k, M ⊗ N ) (cid:15) (cid:15) d Ext b Γ ( k, M ) ⊗ Ext b Γ ( k, N ) / / (cid:15) (cid:15) d Ext b + b Γ ( k, M ⊗ N ) d Ext b Γ ( k, M ) ⊗ d Ext b Γ ( k, N ) ⌣ / / d Ext b + b Γ ( k, M ⊗ N ) commute. In particular, it follows that d Ext ∗ Γ ( k, k ) is an Ext ∗ Γ ( k, k ) -algebra, and Ext ∗ Γ ( k, M ) → d Ext ∗ Γ ( k, M ) is an Ext ∗ Γ ( k, k ) -module homomorphism. If M is a Γ -module algebra, then Ext ∗ Γ ( k, M ) → d Ext ∗ Γ ( k, M ) is an Ext ∗ Γ ( k, k ) -algebra homomor-phism. MULTIPLICATIVE TATE SPECTRAL SEQUENCE 29
Proof.
This follows from the commutative diagrams k ⊗ k kk ⊗ e P ∗ e P ∗ e P ∗ ⊗ e P ∗ e P ∗ Φ and k ⊗ k k e P ∗ ⊗ k e P ∗ e P ∗ ⊗ e P ∗ e P ∗ . Φ (cid:3) Computation.
In this section we look at a sample computation of the Tatecohomology of a Hopf algebra. Let k be a graded commutative ring with an ele-ment η in degree 1 such that 2 η = 0. We will consider the Hopf algebraΓ = k [ s ] / ( s = ηs ) , | s | = 1 . Here s is a primitive element, so that comultiplication is given by ψ = s ⊗ ⊗ s ,counit by ǫ ( s ) = 0, and antipode by χ ( s ) = − s . To clarify: our goal is to compute d Ext ∗ Γ ( k, M ) where M is a Γ-module. This situation naturally appears when weconsider the Tate construction on a spectrum X with an action of the circle T . Inthis situation, we will haveΓ = π ∗ ( S [ T ]) k = π ∗ ( S ) M = π ∗ ( X ) . See Proposition 3.3 and Section 6.A projective resolution P ∗ of k as a trivial Γ-module is · · · / / Γ { p } ∂ / / Γ { p } ∂ / / Γ { p } ∂ / / Γ { p } / / p b being | p b | = b and the total de-gree being k p b k = 2 b . As (right) k -modules we have P b = Γ { p b } = k { p b , p b s } where k p b s k = 2 b + 1. The boundary of the complex is given by ∂ b +1 ( p b +1 ) = ( p b s b ≥ p b ( s + η ) b ≥ ǫ : P ∗ → k is given by ǫ ( p ) = 1.By definition, the mapping cone e P ∗ = cone( P ∗ → k ) is isomorphic to the complex · · · / / Γ { ˜ p } ˜ ∂ / / Γ { ˜ p } ˜ ∂ / / Γ { ˜ p } ˜ ∂ / / k { ˜ p } / / | ˜ p | = 0 and | ˜ p a | = a − a ≥ k ˜ p k = 0 and k ˜ p a k = 2 a − a ≥
1. As before, wecan also write this as a complex of (right) k -modules Γ { ˜ p a } = k { ˜ p a , ˜ p a s } for a ≥ | ˜ p a s | = a and k ˜ p a s k = 2 a . The boundary is given by˜ ∂ a (˜ p a ) = ˜ p a = 1 − ˜ p a − s a ≥ − ˜ p a − ( s + η ) a ≥ ∗ ( M ), or rather,its Γ-invariants. Recall that the Tate complex hm ∗ ( M ) is given in each homologicaldegree by hm c ( M ) = M a + b = c e P a ⊗ Hom( P − b , M ) with boundary given as ∂ hm ( x, f ) = ˜ ∂ ( x ) ⊗ f + ( − k x k x ⊗ ∂ ∗ ( f ) . When calculating the second term we also remember that( ∂ ∗ f )( v ) = − ( − k f k f ( ∂ ( v ))for an element f ∈ Hom( P ∗ , M ) since M is concentrated in homological degree 0.It will also be useful to consider the Tate complex in its bicomplex version. Inthis case, we will write (hm ∗ , ∗ ( M ) , ∂ h , ∂ v ) wherehm a,b ( M ) = e P a ⊗ Hom( P − b , M )and the horizontal and vertical boundaries are the first and the second term in theformula for the boundary in hm ∗ , respectively. The total complex of this bicomplexis equal to the Tate complex, by definition. Moreover, let us write ( U ∗ , ∗ , ∂ h , ∂ v ) forthe restriction of the bicomplex to the Γ-invariants U a,b = Hom Γ ( k, e P a ⊗ Hom( P − b , M )) . We refer to the total complex of this bicomplex as ( U ∗ , ∂ h + ∂ v ); it is isomorphicto Hom Γ ( k, hm ∗ ( M )). Let us introduce some notation for the different elements inthe bicomplex U ∗ , ∗ to keep our computations from becoming too messy. Notation 2.32.
Let x be an element of M and write f b · x = ˜ p ⊗ (cid:18) p b xp b s xs (cid:19) for an element in Hom Γ ( k, f P ⊗ Hom( P b , M )) for b ≥ Notation 2.33.
Let y be an element of M and write g a,b · y = ˜ p a ⊗ (cid:18) p b yp b s ys (cid:19) + ˜ p a s ⊗ (cid:18) p b p b s ( − | y | y (cid:19) for an element in Hom Γ ( k, f P a ⊗ Hom( P b , M )) for a ≥ b ≥ Notation 2.34.
Let z be an element of M and write h a,b · z = ˜ p a s ⊗ (cid:18) p b zp b s z ( s + η ) (cid:19) for an element in Hom Γ ( k, f P a ⊗ Hom( P b , M )) for a ≥ b ≥ g a,b · y ) · s = 0 , and analogously for f b · x and h a,b · z , using the following lemma. Lemma 2.35.
The (right) conjugate action of s on an element f of Hom(
M, N ) is given by ( f s )( m ) = ( − k m k ( f ( m ) s − f ( ms )) . Proof.
Recall that the characterising property of the conjugate action is that itis the action on a function object Hom(
M, N ) such that the evaluation pairingev : Hom(
M, N ) ⊗ M → N is Γ-linear. Explicitly, the Γ-action on Hom( M, N )makes the diagram Hom(
M, N ) ⊗ M ⊗ Γ ev ⊗ / / ρ Hom(
M,N ) ⊗ M (cid:15) (cid:15) N ⊗ Γ ρ N (cid:15) (cid:15) Hom(
M, N ) ⊗ M ev / / N MULTIPLICATIVE TATE SPECTRAL SEQUENCE 31 commute. The top composition sends a generic element to f ⊗ m ⊗ s f ( m ) ⊗ s f ( m ) s , while the bottom composition sends the generic element to f ⊗ m ⊗ s f ⊗ ms + ( − k m k f s ⊗ m f ( ms ) + ( − k m k ( f s )( m ) . These must agree, which necessarily gives us the assertion. (cid:3)
Furthermore, these form an ‘ M -basis’ of the Γ-invariants of the Tate complex inthe sense of the following proposition. Proposition 2.36.
Let b ≥ and a ≥ . There are k -module isomorphisms Σ − b M ∼ = −→ Hom Γ ( k, e P ⊗ Hom( P b , M )) x f b · x and Σ a − b − M ⊕ Σ a − b M ∼ = −→ Hom Γ ( k, e P a ⊗ Hom( P b , M ))( y, z ) g a,b · y + h a,b · z . Proof.
The maps are clearly injective, so we only need to show that they are sur-jective.A general element in e P ⊗ Hom( P b , M ) is on the form˜ p ⊗ (cid:18) p b xp b s y (cid:19) . By Lemma 2.35, the right action of the primitive element s on such an element is˜ p ⊗ (cid:18) p b xs − yp b s yη − ys (cid:19) . For our original element to be Γ-invariant this must be zero, which gives us y = xs .In other words, a Γ-invariant element in e P ⊗ Hom( P b , M ) can be written f b · x ,where we let x range throughout M . The grading suspension appearing in theisomorphism makes sure that this is actually a map of graded k -modules. Indeed,the internal degree of our element is | f b · x | = | ˜ p | + | x | − | p b | = | x | − b . A general element in e P a ⊗ Hom(
P, M ) b is on the form˜ p a ⊗ (cid:18) p b xp b s y (cid:19) + ˜ p a s ⊗ (cid:18) p b zp b s w (cid:19) . We assume that this is a homogeneous element, so that | y | = | x | + 1, | z | = | x | − | w | = | x | . Letting the primitive element s act on this element from the rightwe obtain ( − | x | ˜ p a s ⊗ (cid:18) p b xp b s y (cid:19) + ˜ p a ⊗ (cid:18) p b xs − yp b s yη − ys (cid:19) − ( − | x | ˜ p a sη ⊗ (cid:18) p b zp b s w (cid:19) + ˜ p a s ⊗ (cid:18) p b zs − wp b s wη − ws (cid:19) . For our element to be Γ-invariant we want this to add up to zero. In other words,we need to solve the following system of equations xs − y = 0 yη − ys = 0( − | x | x − ( − | x | zη + zs − w = 0( − | x | y − ( − | x | wη + wη − ws = 0 . It is straight-forward to check that the solutions are given by the two independentequations ( y = xsw = ( − | x | x − ( − | x | zη + zs, which tells us that a Γ-invariant element can be written g a,b · x + h a,b · z where we are free to vary x and z in M . The suspensions in the source of the k -isomorphism are again there to make sure that the grading is preserved by theisomorphism. Indeed, | g a,b · x | = | ˜ p a | + | x | − | p b | = a − | x | − b and | h a,b · z | = | ˜ p a s | + | z | − | p b | = a + | z | − b . (cid:3) We now need to figure out what the boundary on these generic Γ-invariantelements looks like. Keeping track of all the signs we end up with the followingdescription of the horizontal and vertical boundaries in terms of our ‘ M -basis’. Lemma 2.37.
The horizontal boundary on f b · x is given by ∂ h ( f b · x ) = 0 and the vertical boundary is given by ∂ v ( f b · x ) = ( − ( − | x | f b +1 · xs b ≥ even − ( − | x | f b +1 · x ( s + η ) b ≥ odd. Lemma 2.38.
The horizontal boundary on g a,b · y is given by ∂ h ( g a,b · y ) = f b · y for a = 1 − h a − ,b · y for a ≥ even g a − ,b · yη − h a − ,b · y for a ≥ oddand the vertical boundary is given by ∂ v ( g a,b · y ) = ( ( − | y | g a,b +1 · ys + h a,b +1 · y for b ≥ even ( − | y | g a,b +1 · y ( s + η ) + h a,b +1 · y for b ≥ odd. Lemma 2.39.
The horizontal boundary on h a,b · z is given by ∂ h ( h a,b · z ) = ( h a − ,b · zη for a ≥ even for a ≥ odd . and the vertical boundary is given by ∂ v ( h a,b · z ) = ( − ( − | z | h a,b +1 · z ( s + η ) for b ≥ even − ( − | z | h a,b +1 · zs for b ≥ odd. We calculate the homology of U ∗ by filtering the first tensor factor of U ∗ , ∗ andusing the spectral sequence for the total complex of a bicomplex: E a, − b = H − b (Hom Γ ( k, e P a ⊗ Hom( P ∗ , M ))) ⇒ H a − b (Hom Γ ( k, e P ∗ ⊗ Hom( P ∗ , M ))) . The bicomplex ( U ∗ , ∗ , ∂ h , ∂ v ) is displayed in Figure 2.1 for the convenience of thereader. MULTIPLICATIVE TATE SPECTRAL SEQUENCE 33
Remark . Let us clarify how to interpret Figure 2.1 and the matrix notationappearing in it. The horizontal and vertical boundaries are given in terms of the ‘ M -bases’ { f b } for U , − b and { g a,b , h a,b } for U a, − b . We record f b · x and g a,b · y + h a,b · z as the column vectors (cid:2) x (cid:3) and (cid:20) yz (cid:21) , respectively. The boundaries are then indicated by multiplication with the corre-sponding matrices appearing in the figure. Multiplication is done, as is usual, withthe matrix on the left hand side. So, to clarify: A vertical boundary ∂ v : U a, − b → U a, − b − recorded as a 2 × M ) (cid:20) i jk ℓ (cid:21) (cid:20) yz (cid:21) = (cid:20) iy + jzky + ℓz (cid:21) indicates that this boundary is given as g a,b · y + h a,b · z g a,b +1 · ( iy + jz ) + h a,b +1 · ( ky + ℓz ) . Note that in this convention Γ ends up acting on M from the left, through the twistisomorphism followed by the right action. To see that the boundaries given in thematrix notation actually agree with the ones given in Lemma 2.37, Lemma 2.38,and Lemma 2.39, we have to switch the position of the Γ-values ( i , j , k , and ℓ )and the M -values ( y and z ), which typically introduces a sign. For example, theboundary ∂ v : U a, − b → U a, − b − for even b is recorded in the figure as (cid:20) s − ( s + η ) (cid:21) . Left multiplication of this matrix with the column vector corresponding to g a,b · y gives (cid:20) s − ( s + η ) (cid:21) (cid:20) y (cid:21) = (cid:20) syy (cid:21) , which tells us that this vertical boundary is given by g a,b · y g a,b +1 · sy + h a,b +1 · y = ( − | y | g a,b +1 · ys + h a,b +1 · y , which is indeed in agreement with Lemma 2.38.Before we explicitly compute the first page of the spectral sequence for the bi-complex U ∗ , ∗ , we again introduce some notation. Notation 2.41.
Let z be an element of M and write u a · z = − ( − | z | g a, · z ( s + η ) − h a, · z = − ( − | z | ˜ p a ⊗ (cid:18) p z ( s + η ) p s (cid:19) − ˜ p a s ⊗ (cid:18) p zp s (cid:19) for the specified element in Hom Γ ( k, f P a ⊗ Hom( P b , M )) for a ≥ b ≥ Proposition 2.42.
The E -page of the bicomplex spectral sequence for U ∗ , ∗ is givenas H − b (Hom Γ ( k, e P ⊗ Hom(
P, M ) ∗ )) ∼ = f · ker( s ) for b = 0 , f b · ker( s + η )im( s ) for b ≥ odd, f b · ker( s )im( s + η ) for b ≥ even, U , − s ] (cid:15) (cid:15) U ,
0[ 1 0 ] o o h s − ( s + η ) i (cid:15) (cid:15) U , h − η i o o h s − ( s + η ) i (cid:15) (cid:15) U , h η − i o o h s − ( s + η ) i (cid:15) (cid:15) . . . h − η i o o U , − − ( s + η ) ] (cid:15) (cid:15) U , −
1[ 1 0 ] o o h s + η − s i (cid:15) (cid:15) U , − h − η i o o h s + η − s i (cid:15) (cid:15) U , − h η − i o o h s + η − s i (cid:15) (cid:15) . . . h − η i o o U , − − s ] (cid:15) (cid:15) U , −
2[ 1 0 ] o o h s − ( s + η ) i (cid:15) (cid:15) U , − h − η i o o h s − ( s + η ) i (cid:15) (cid:15) U , − h η − i o o h s − ( s + η ) i (cid:15) (cid:15) . . . h − η i o o ... ... ... ... Figure 2.1.
The bicomplex ( U ∗ , ∗ , ∂ h , ∂ v ) for Γ = k [ s ] / ( s = ηs ) when a = 0 , and as H − b (Hom Γ ( k, e P a ⊗ Hom(
P, M ) ∗ )) ∼ = ( u a · M for b = 00 otherwise,when a ≥ .Proof. This is essentially an exercise in linear algebra using the matrices in Fig-ure 2.1. The kernels of the boundaries are computed by computing the nullspacesof the corresponding matrices. Similarly, the images of boundaries are computed bycomputing the column spaces of the matrices. Let us give the details for the a ≥ a = 0 case is directly visible by inspecting the figure. The nullspacesof the matrices (cid:20) s − ( s + η ) (cid:21) and (cid:20) s + η − s (cid:21) are generated by the column vectors (cid:20) s + η (cid:21) and (cid:20) s (cid:21) , respectively, and the column spaces are generated by the column vectors (cid:20) s (cid:21) and (cid:20) s + η (cid:21) , respectively. From this is follows that the homology is concentrated in homologicaldegree b = 0, in the a ≥ g a, · ( s + η ) z + h a, · z = ( − | z | g a, · z ( s + η ) + h a, · z for varying z in M . For reasons concerning the multiplicative structure, we havedecided to denote the above element by − u a · z . (cid:3) MULTIPLICATIVE TATE SPECTRAL SEQUENCE 35
In particular, note that the above result tells us that the E -page of the spectralsequence is concentrated around the boundary of the fourth quadrant. The d -differential d : E a, − b → E a − , − b in the spectral sequence is induced by the hor-izontal boundary, and is by degree reasons only non-zero on the positive a -axis.There it is given by d ( u a · z ) = − ( − | z | f · z ( s + η ) for a = 1 − ( − | z | u a − · zs for a ≥ − ( − | z | u a − · z ( s + η ) for a ≥ a - and b -axes and that E a, − b ∼ = f b · ker( s )im( s + η ) for a = 0 and b ≥ f b · ker( s + η )im( s ) for a = 0 and b ≥ u a · ker( s )im( s + η ) for b = 0 and a ≥ u a · ker( s + η )im( s ) for b = 0 and a ≥ d -cycles in the total complex U ∗ . Weconclude: Proposition 2.43. d Ext c Γ ( k, M ) ∼ = f c · ker( s )im( s + η ) for c ≥ even, f c · ker( s + η )im( s ) for c ≥ odd, u − c · ker( s )im( s + η ) for c ≤ − even, u − c · ker( s + η )im( s ) for c ≤ − odd. Now all that remains is to describe the multiplicative structure. That is, giventwo Γ-modules M and N we want to determine the cup product ⌣ : d Ext c Γ ( k, M ) ⊗ d Ext c Γ ( k, N ) −→ d Ext c + c Γ ( k, M ⊗ N ) . In order to do so we need a Γ-linear chain map Ψ : P ∗ → P ∗ ⊗ P ∗ covering theidentity of k , and a Γ-linear chain map Φ : e P ∗ ⊗ e P ∗ → e P ∗ extending the fold map,as per Section 2.5. Lemma 2.44. A Γ -linear chain map Ψ : P ∗ → P ∗ ⊗ P ∗ that covers the identity isgiven by Ψ( p b ) = X b + b = b p b ⊗ p b . By Γ -linearity we have Ψ( p b s ) = X b + b = b p b s ⊗ p b + p b ⊗ p b s. Proof.
Note that ∂ ( p b ) = p b − ( s + ( b − η )for b ≥
1. To verify that Ψ, as specified in the statement of the lemma, is a chainmap, we must show that ∂ (Ψ( p b )) = X b + b = b ∂ ( p b ) ⊗ p b + p b ⊗ ∂ ( p b )= X b + b = b p b − ( s + ( b − η ) ⊗ p b + p b ⊗ p b − ( s + ( b − η )is equal to Ψ( ∂ ( p b )) = Ψ( p b − s ) + Ψ( p b − )( b − η . Here X b + b = b p b − s ⊗ p b + p b ⊗ p b − s = Ψ( p b − ) s , so it remains to check that X b + b = b p b − ( b − η ⊗ p b + p b ⊗ p b − ( b − η = Ψ( p b − )( b − η . When b is odd the terms of the left hand side cancel in pairs, and the right handside is zero. When b is even only the terms with b and b both even contribute tothe left hand side, and these add up to Ψ( p b − ) η , as required. Finally,( ǫ ⊗ ǫ )(Ψ( p )) = 1 = ǫ ( p ) , so Ψ is indeed a chain map covering k ⊗ k = k . (cid:3) Lemma 2.45. A Γ -linear chain map Φ : e P ∗ ⊗ e P ∗ → e P ∗ that extends the fold mapis given by Φ(˜ p a ⊗ ˜ p a ) = 0Φ(˜ p a ⊗ ˜ p a s ) = − ˜ p a Φ(˜ p a s ⊗ ˜ p a ) = − ˜ p a Φ(˜ p a s ⊗ ˜ p a s ) = − ˜ p a ( s + η ) for a , a ≥ and a = a + a . Furthermore, Φ(˜ p ⊗ ˜ p a ) = ˜ p a Φ(˜ p ⊗ ˜ p a s ) = ˜ p a s Φ(˜ p a ⊗ ˜ p ) = ˜ p a Φ(˜ p a s ⊗ ˜ p ) = ˜ p a s and Φ(˜ p ⊗ ˜ p ) = ˜ p .Proof. Note that the differential in the chain complex e P ∗ can be described as˜ ∂ (˜ p a ) = ( ˜ p for a = 1 − ˜ p a − ( s + aη ) for a ≥ MULTIPLICATIVE TATE SPECTRAL SEQUENCE 37
To check that Φ, as specified in the statement of the lemma, is Γ-linear, we observethat Φ((˜ p a ⊗ ˜ p a ) s ) = Φ(˜ p a ⊗ ˜ p a s − ˜ p a s ⊗ ˜ p a )= − ˜ p a + ˜ p a = 0= Φ(˜ p a ⊗ ˜ p a ) s and that Φ((˜ p a ⊗ ˜ p a s ) s ) = Φ(˜ p a s ⊗ ˜ p a s + ˜ p a ⊗ ˜ p a ηs )= − ˜ p a ( s + η ) + ˜ p a η = − ˜ p a s = Φ(˜ p a ⊗ ˜ p a s ) s . The check that Φ is a chain map is contained in the computationsΦ( ∂ e P ∗ ⊗ e P ∗ (˜ p a ⊗ ˜ p a )) = Φ( ˜ ∂ (˜ p a ) ⊗ ˜ p a ) − Φ(˜ p a ⊗ ˜ ∂ (˜ p a ))= − Φ(˜ p a − ( s + a η ) ⊗ ˜ p a )+Φ(˜ p a ⊗ ˜ p a − ( s + a η )) for a , a ≥ p ⊗ ˜ p a )+Φ(˜ p ⊗ ˜ p a − ( s + a η )) for a = 1, a ≥ − Φ(˜ p a − ( s + a η ) ⊗ ˜ p ) − Φ(˜ p a ⊗ ˜ p ) for a ≥ a = 1= ˜ p a − − ˜ p a − = 0= ˜ ∂ (Φ(˜ p a ⊗ ˜ p a ))andΦ( ∂ e P ∗ ⊗ e P ∗ (˜ p a ⊗ ˜ p a s )) = Φ( ˜ ∂ (˜ p a ) ⊗ ˜ p a s ) − Φ(˜ p a ⊗ ˜ ∂ (˜ p a s ))= − Φ(˜ p a − ( s + a η ) ⊗ ˜ p a s )+Φ(˜ p a ⊗ ˜ p a − ( s + a η ) s ) for a , a ≥ p ⊗ ˜ p a s )+Φ(˜ p ⊗ ˜ p a − ( s + a η ) s ) for a = 1, a ≥ − Φ(˜ p a − ( s + a η ) ⊗ ˜ p s ) − Φ(˜ p a ⊗ ˜ p s ) for a ≥ a = 1= ˜ p a − ( s + aη )= − ˜ ∂ (˜ p a )= ˜ ∂ (Φ(˜ p a ⊗ ˜ p a s )) . (cid:3) Now we want to use the above chain maps to compute the multiplicative struc-ture. Recall that the cup product is induced by the composite pairing e P ∗ ⊗ Hom(
P, M ) ∗ ⊗ e P ∗ ⊗ Hom(
P, N ) ∗ ⊗ τ ⊗ −→ e P ∗ ⊗ e P ∗ ⊗ Hom(
P, M ) ∗ ⊗ Hom(
P, N ) ∗ ⊗ ⊗ α −→ e P ∗ ⊗ e P ∗ ⊗ Hom( P ⊗ P, M ⊗ N ) ∗ Φ ⊗ Ψ ∗ −→ e P ∗ ⊗ Hom(
P, M ⊗ N ) ∗ . There are two signs to be wary of here; the first one comes from twisting Hom(
P, M ) ∗ past the second e P ∗ -factor, and the second sign comes from using the canonicalmap α . Please refer to Equation 2.1. The cup product computations, which canbe found in the lemmas below, are straight-forward computations. We only includethe verification of two of the lemmas, as the other three are very similar. Lemma 2.46.
Let f b · m and f b · n be cycles with homology classes in d Ext b Γ ( k, M ) and d Ext b Γ ( k, N ) , respectively. The cup product of these is the cycle f b · m ⌣ f b · n = f b + b · m ⊗ n with homology class in d Ext b + b Γ ( k, M ⊗ N ) . Lemma 2.47.
Let u a · m and u a · n be cycles with homology classes in d Ext − a Γ ( k, M ) and d Ext − a Γ ( k, N ) , respectively. The cup product of these is the cycle u a · m ⌣ u a · n = u a + a · m ⊗ n with homology class in d Ext − a − a Γ ( k, M ⊗ N ) . Lemma 2.48.
Let f · m and u a · n be cycles with homology classes in d Ext ( k, M ) and d Ext − a Γ ( k, N ) , respectively. The cup product of these is the cycle f · m ⌣ u a · n = u a · m ⊗ n with homology class in d Ext − a Γ ( k, M ⊗ N ) .Proof. Since f · m is assumed to be a cycle in U ∗ we know that m is an elementin ker( s ). An explicit description of this cycle is then f · m = ˜ p ⊗ (cid:18) p mp s (cid:19) . The first map 1 ⊗ τ ⊗ p ⊗ (cid:18) p mp s (cid:19) ⊗ (cid:18) − ( − | n | ˜ p a ⊗ (cid:18) p n ( s + η ) p s (cid:19) − ˜ p a s ⊗ (cid:18) p np s (cid:19)(cid:19)
7→ − ( − | m | + | n | ˜ p ⊗ ˜ p a ⊗ (cid:18) p mp s (cid:19) ⊗ (cid:18) p n ( s + η ) p s (cid:19) − ˜ p ⊗ ˜ p a s ⊗ (cid:18) p mp s (cid:19) ⊗ (cid:18) p np s (cid:19) . The second map in the composite is 1 ⊗ ⊗ α , so that − ( − | m | + | n | ˜ p ⊗ ˜ p a ⊗ (cid:18) p mp s (cid:19) ⊗ (cid:18) p n ( s + η ) p s (cid:19) − ˜ p ⊗ ˜ p a s ⊗ (cid:18) p mp s (cid:19) ⊗ (cid:18) p np s (cid:19)
7→ − ( − | m | + | n | ˜ p ⊗ ˜ p a ⊗ p ⊗ p m ⊗ n ( s + η ) p s ⊗ p p ⊗ p s p s ⊗ p s − ˜ p ⊗ ˜ p a s ⊗ p ⊗ p m ⊗ np s ⊗ p p ⊗ p s p s ⊗ p s . MULTIPLICATIVE TATE SPECTRAL SEQUENCE 39
Lastly, the computations of Ψ and Φ given in Lemma 2.44 and Lemma 2.45 tell usthat the final map in the composite is such that − ( − | m | + | n | ˜ p ⊗ ˜ p a ⊗ p ⊗ p m ⊗ n ( s + η ) p s ⊗ p p ⊗ p s p s ⊗ p s − ˜ p ⊗ ˜ p a s ⊗ p ⊗ p m ⊗ np s ⊗ p p ⊗ p s p s ⊗ p s
7→ − ( − | m | + | n | ˜ p a ⊗ (cid:18) p m ⊗ n ( s + η ) p s (cid:19) − ˜ p a s ⊗ (cid:18) p m ⊗ np s (cid:19) where the target can be identified with u a · ( m ⊗ n ), as wanted. (cid:3) Lemma 2.49.
Let u a · m and f · n be cycles with homology classes in d Ext − a Γ ( k, M ) and d Ext ( k, N ) , respectively. The cup product of these is the cycle u a · m ⌣ f · n = u a · m ⊗ n with homology class in d Ext − a Γ ( k, M ⊗ N ) . Lemma 2.50.
Let f · m and u · n be cycles with homology classes in d Ext ( k, M ) and d Ext − ( k, N ) , respectively. Then the two cycles f · m ⌣ u · n ≃ f · m ⊗ n. are homologous in the complex U ∗ , so that they determine the same class in d Ext ( k, M ⊗ N ) .Proof. Since f · m and u · n are assumed to be cycles we know that m and n areelements in ker( s + η ), which directly implies that m ⊗ n is an element of ker( s )since ( m ⊗ n ) · s = m ⊗ ns + ( − | n | ms ⊗ n = m ⊗ ns + m ⊗ nη + m ⊗ nη + ( − | n | ms ⊗ n = m ⊗ n ( s + η ) + ( − | n | m ( s + η ) ⊗ n = 0 . An explicit description of the two cycles f · m and u · n is f · m = ˜ p ⊗ (cid:18) p mp s ms (cid:19) and u · n = − ˜ p s ⊗ (cid:18) p np s (cid:19) . The first map 1 ⊗ τ ⊗ − ˜ p ⊗ (cid:18) p mp s ms (cid:19) ⊗ ˜ p s ⊗ (cid:18) p np s (cid:19)
7→ − ˜ p ⊗ ˜ p s ⊗ (cid:18) p mp s ms (cid:19) ⊗ (cid:18) p np s (cid:19) . The second map in the composite is 1 ⊗ ⊗ α , so that − ˜ p ⊗ ˜ p s ⊗ (cid:18) p mp s ms (cid:19) ⊗ (cid:18) p np s (cid:19)
7→ − ˜ p ⊗ ˜ p s ⊗ p ⊗ p m ⊗ np s ⊗ p ( − | n | ms ⊗ np ⊗ p s p s ⊗ p s . Lastly, the computations of Ψ and Φ given in Lemma 2.44 and Lemma 2.45 tellsus that the final map in the composite is such that − ˜ p ⊗ ˜ p s ⊗ p ⊗ p m ⊗ np s ⊗ p ( − | n | ms ⊗ np ⊗ p s p s ⊗ p s
7→ − ˜ p s ⊗ (cid:18) p m ⊗ np s ( − | n | ms ⊗ n (cid:19) . The right hand term can be identified with − h , · m ⊗ n . We conclude that f · m ⌣ u · n = − h , · m ⊗ n. Note that the boundary of g , · m ⊗ n is ∂ ( g , · m ⊗ n ) = f · m ⊗ n + h , · m ⊗ n , which tells us that f · m ⊗ n and − h , · m ⊗ n are homologous in U ∗ , and hencerepresent the same class in d Ext ( k, M ⊗ N ). (cid:3) We decide to make a final change of notation.
Notation 2.51.
Let t b · m and t − a · n denote the homology classes t b · m = [ f b · m ] and t − a · n = [ u a · n ]in d Ext b Γ ( k, M ) for b ≥ d Ext − a Γ ( k, N ) for a ≥
1, respectively.Note that t b · m has internal and total degrees equal to those of f b · m , so that | t b · m | = | m | − b and k t b · m k = | m | − b . Similarly, t − a · n has internal and total degrees equal to that of u a · n , so that | t − a · n | = a + | n | and k t − a · n k = 2 a + | n | . We conclude that, formally, the symbol t has homological degree −
1, internal degree | t | = − k t k = −
2. Using this new notation we have the followingtheorem.
Theorem 2.52. d Ext c Γ ( k, M ) ∼ = t c · ker( s )im( s + η ) for c even, t c · ker( s + η )im( s ) for c odd.The cup product d Ext c Γ ( k, M ) ⊗ d Ext c Γ ( k, N ) −→ d Ext c + c Γ ( k, M ⊗ N ) is given by ( t c · m ) ⌣ ( t c · n ) = t c + c · m ⊗ n . Corollary 2.53. d Ext c Γ ( k, k ) ∼ = ( t c · coker( η ) for c even, t c · ker( η ) for c odd.In this case, the cup product d Ext c Γ ( k, k ) ⊗ d Ext c Γ ( k, k ) −→ d Ext c + c Γ ( k, k ) is given by ( t c · x ) ⌣ ( t c · y ) = t c + c · xy and makes d Ext ∗ Γ ( k, k ) into a k -algebra over which d Ext ∗ Γ ( k, M ) is a module for any Γ -module M . MULTIPLICATIVE TATE SPECTRAL SEQUENCE 41
Remark . Note that in Theorem 2.52 above, the answer is also the homologyof the differential graded Γ-module M [ t, t − ]with differential d ( m ) = tms and d ( t ) = t η , where m is an element of M and t has homological degree − − d ( t c m ) = d ( t c ) m + t c d ( m ) = ct c +1 ηm + t c +1 ms = ( t c +1 ms if c is even t c +1 m ( s + η ) if c is odd,which gives us the same homology groups as in Theorem 2.52. Note that this isalso true multiplicatively: if µ : M ⊗ N → L is a pairing of Γ-modules, then thecup product d Ext c Γ ( k, M ) ⊗ d Ext c Γ ( k, N ) −→ d Ext c + c Γ ( k, M ⊗ N ) −→ d Ext c + c Γ ( k, L )is precisely the one induced by the obvious pairing M [ t, t − ] ⊗ N [ t, t − ] −→ L [ t, t − ]on homology. 3. Homotopy groups of orthogonal G -spectra In this section we discuss some results regarding equivariant stable homotopygroups. Our chosen model for equivariant spectra is orthogonal G -spectra, andwe recall some basic theory about these objects and their homotopy groups inSection 3.1. In Section 3.2 we define the main Hopf algebra that we will work within this paper, namely the (non-equivariant) homotopy groups of the unreducedsuspension spectrum of a compact Lie group, also referred to as the spherical groupring S [ G ] of that group. Since our main group of interest is the circle T , we also givean explicit description of π ∗ ( S [ T ]) as an algebra over π ∗ ( S ). Lastly, in Section 3.3we show, under suitable projectivity assumptions, that we can sometimes describethe equivariant homotopy groups π G ∗ ( X ) of an orthogonal G -spectrum X as the‘ π ∗ ( S [ G ])-invariants’ of the non-equivariant homotopy groups π ∗ ( X ).3.1. Equivariant homotopy groups.
Let G be a compact Lie group, and let X be an orthogonal G -spectrum, as in [MM02, § II.2] and [Sch18, § G acts from the right. Recall that, in particular, X associates to each (finite-dimensional, orthogonal) G -representation V a based G -space X ( V ), and to each pair ( U, V ) of G -representations a G -equivariant structuremap σ : Σ U X ( V ) → X ( U ⊕ V ).We can define G -equivariant homotopy groups π G ∗ ( X ) associated to X . To dothis, one fixes a complete G -universe U containing a fixed copy of R ∞ . Notethat the set of finite-dimensional G -subrepresentations of U is partially orderedby inclusion. For non-negative integers q ≥
0, we define the q th G -equivarianthomotopy group of X as the colimit, over this directed partially ordered set, of thesets of homotopy classes [ f ] of G -maps f : Σ V S q → X ( V ): π Gq ( X ) = colim V [Σ V S q , X ( V )] G . A complete G -universe is an orthogonal representation of countably infinite dimension in whichevery finite dimensional G -representation, and their countably infinite direct sums, embeds. Similarly, to define the non-positive G -equivariant homotopy groups, we let π G − q ( X ) = colim V [Σ V − R q S , X ( V )] G where V − R q denotes the orthogonal complement of R q in V . These definitionsagree for q = 0. Each equivariant homotopy group π Gq ( X ) is naturally an abeliangroup.The category of orthogonal G -spectra is symmetric monoidal, with the symmetricmonoidal product being denoted ∧ and referred to as the smash product. The unitof this symmetric monoidal structure is the sphere spectrum S with the trivial G -action. Any pairing φ : X ∧ Y → Z of orthogonal G -spectra gives rise to a pairingof the corresponding equivariant homotopy groups. Consider classes [ f ] ∈ π Gp ( X )and [ g ] ∈ π Gq ( Y ), represented by homotopy classes of G -maps f : Σ V S p → X ( V )and g : Σ W S q → Y ( W ), respectively. The induced pairing φ ∗ : π Gp ( X ) ⊗ π Gq ( Y ) −→ π Gp + q ( Z )maps [ f ] ⊗ [ g ] to the element represented by the homotopy class of the compositeΣ V ⊕ W S p + q ∼ = −→ Σ V S p ∧ Σ W S q f ∧ g −→ X ( V ) ∧ Y ( W ) φ −→ Z ( V ⊕ W ) . Similar constructions can be carried out if p or q is negative, although this is a bittricky. In this way, we obtain a pairing φ ∗ : π G ∗ ( X ) ⊗ π G ∗ ( Y ) −→ π G ∗ ( Z )of graded abelian groups. Remark . More generally, given a group homomorphism α : G → H × K andan α -equivariant map X ∧ Y → Z , where X , Y and Z are orthogonal H -, K -and G -spectra, respectively, we obtain a pairing π H ∗ ( X ) ⊗ π K ∗ ( Y ) −→ π G ∗ ( Z ) . Here, α -equivariant means that the diagram X ∧ Y ∧ G + X ∧ Y ∧ H + ∧ K + X ∧ H + ∧ Y ∧ K + X ∧ YZ ∧ G + Z ∧ ∧ α ∧ τ ∧ commutes.If R is a commutative (non-equivariant) orthogonal ring spectrum, with multi-plication µ : R ∧ R → R , then the induced pairing µ ∗ : π ∗ ( R ) ⊗ π ∗ ( R ) −→ π ∗ ( R )on (non-equivariant) homotopy groups makes R ∗ = π ∗ ( R ) into a graded commuta-tive ring. A right R -module in orthogonal G -spectra is an orthogonal G -spectrum X with an associative and unital action ρ : X ∧ R → X , defined in the category oforthogonal G -spectra. Here R is regarded as a G -spectrum with trivial action. Inthis case, there is an induced pairing ρ ∗ : π G ∗ ( X ) ⊗ R ∗ −→ π G ∗ ( X )making π G ∗ ( X ) into a right R ∗ -module. If X and Y are two R -modules in orthog-onal G -spectra, then the canonical map X ∧ Y → X ∧ R Y induces a pairing π G ∗ ( X ) ⊗ π G ∗ ( Y ) −→ π G ∗ ( X ∧ R Y ) MULTIPLICATIVE TATE SPECTRAL SEQUENCE 43 that equalizes the two composites from π G ∗ ( X ) ⊗ R ∗ ⊗ π G ∗ ( Y ), so that we have theinduced dashed map making the diagram π G ∗ ( X ) ⊗ R ∗ ⊗ π G ∗ ( Y ) π G ∗ ( X ) ⊗ π G ∗ ( Y ) π G ∗ ( X ) ⊗ R ∗ π G ∗ ( Y ) π ∗ ( X ∧ R Y )commute.3.2. A cocommutative Hopf algebra.
Let us introduce the Hopf algebra thatwe will work with through the remainder of this paper. The right R -action on R [ G ] = R ∧ G + is given by the composite map R ∧ G + ∧ R ∧ τ −→ R ∧ R ∧ G + µ ∧ −→ R ∧ G + . Lemma 3.2. If R [ G ] ∗ = π ∗ ( R ∧ G + ) is flat as a (right) R ∗ -module, then R [ G ] ∗ isnaturally a cocommutative Hopf algebra over R ∗ = π ∗ ( R ) .Proof. We have a pairing R [ G ] ∗ ⊗ R ∗ R [ G ] ∗ = π ∗ ( R ∧ G + ) ⊗ π ∗ ( R ) π ∗ ( R ∧ G + ) · −→ π ∗ (( R ∧ G + ) ∧ R ( R ∧ G + )) ∼ = π ∗ ( R ∧ G + ∧ G + ) , where the left R ∗ -action on the right hand copy of R [ G ] ∗ is equal to that obtainedby twisting the right R ∗ -action. When R [ G ] ∗ is flat as a right R ∗ -module, it followsby a well-known induction over the cells of a CW structure on the right hand copyof G that the pairing above is an isomorphism of R ∗ -modules.The unit inclusion { e } → G , group multiplication G × G → G , collapse G → { e } ,diagonal G → G × G and group inverse G → G give us R -module maps R → R ∧ G + , R ∧ G + ∧ R R ∧ G + → R ∧ G + , R ∧ G + → R , R ∧ G + → R ∧ G + ∧ R R ∧ G + and R ∧ G + → R ∧ G + that induce R ∗ -module homomorphisms η : R ∗ −→ R [ G ] ∗ φ : R [ G ] ∗ ⊗ R ∗ R [ G ] ∗ −→ R [ G ] ∗ ǫ : R [ G ] ∗ −→ R ∗ ψ : R [ G ] ∗ −→ R [ G ] ∗ ⊗ R ∗ R [ G ] ∗ χ : R [ G ] ∗ −→ R [ G ] ∗ which make R [ G ] ∗ a Hopf algebra over R ∗ . The cocommutativity of the diagonalimplies that ψ is cocommutative. (cid:3) By the discussion in Section 2.1, the category of modules over R [ G ] ∗ is closedsymmetric monoidal. Note that if X is an R -module in orthogonal G -spectra, thenthe commuting right R - and G -actions combine to define an action γ : X ∧ R R [ G ] ∼ = X ∧ G + −→ X which makes the underlying (non-equivariant) orthogonal spectrum of X into aright R [ G ]-module in the category of (non-equivariant) R -modules. The inducedpairing γ ∗ : π ∗ ( X ) ⊗ R ∗ R [ G ] ∗ −→ π ∗ ( X )then gives the (non-equivariant) homotopy groups π ∗ ( X ) the structure of a right R [ G ] ∗ -module. If Y is a second R -module in orthogonal G -spectra, the pairing π ∗ ( X ) ⊗ R ∗ π ∗ ( Y ) · −→ π ∗ ( X ∧ R Y ) is a homomorphism of R [ G ] ∗ -modules, where the Hopf algebra R [ G ] ∗ acts diagonallyon the left hand side. Likewise, π ∗ F R ( X, Y ) −→ Hom R ∗ ( π ∗ ( X ) , π ∗ ( Y ))is a homomorphism of R [ G ] ∗ -modules, where the Hopf algebra R [ G ] ∗ acts by con-jugation on the right hand side.The case we are the most interested in is when the Lie group is the circle, so letus compute the homotopy groups of the spherical group ring of this specific group. Proposition 3.3.
When G = T = U (1) is the circle group, R [ T ] ∗ = R ∗ [ s ] / ( s = ηs ) with | s | = | η | = 1 . Here s generates the augmentation ideal R [ T ] ∗ = ker( ǫ : R [ T ] ∗ −→ R ∗ ) = R ∗ { s } , and η is the image of the complex Hopf map in π ( S ) ∼ = Z / . The generator s is primitive, so the coproduct and involution are given by ψ ( s ) = s ⊗ ⊗ s and χ ( s ) = − s .Proof. It suffices to prove the result for R = S . Proving this we would knowthat S [ T ] ∗ is free over S ∗ , so that the case of a general ground ring spectrum R follows immediately from the isomorphism R [ T ] ∗ ∼ = S [ T ] ∗ ⊗ S ∗ R ∗ .To prove the result for the sphere spectrum, we start by noting that the cofibresequence S ∼ = 1 + −→ T + −→ T ∼ = S admits a retraction T + → + . Hence the induced stable cofibre sequence S i −→ S [ T ] p −→ Σ S admits a retraction c : S [ T ] → S and a section s : Σ S → S [ T ] with ps ≃ cs ≃
0. The maps i and s represent classes in S [ T ] ∗ of total (and internal) degree 0and 1, respectively, and induce an isomorphism S ∗ { i, s } ∼ = S [ T ] ∗ . Here, i is themultiplicative unit and s generates the augmentation ideal S [ T ] ∗ = S ∗ { s } . It onlyremains to prove that we have the relation s = ηs in S [ T ] . This is the contentof formula (1.4.4) in [Hes96]. We give the following direct argument using thebar construction [Seg68, § §
7] and the bar spectral sequence [Seg68, § § T is the geometric realization B T = | B • T | ≃ C P ∞ ofthe simplicial space [ q ] B q T = T q , with the usual face and degeneracy maps. There is a standard filtration of B T bysimplicial skeleta. The associated spectral sequence in (reduced) stable homotopyhas E -page given as the normalised bar complex N B ∗ ( S ∗ , S [ T ] ∗ , S ∗ )0 ← d ←− S [ T ] ∗ d ←− S [ T ] ∗ ⊗ S ∗ S [ T ] ∗ d ←− · · · , which we reviewed in Construction 2.24. This spectral sequence converges (strongly)to π ∗ Σ ∞ ( B T ) ∼ = π ∗ Σ ∞ ( C P ∞ ). The part of the E -page that will be relevant to us MULTIPLICATIVE TATE SPECTRAL SEQUENCE 45 is pictured below, with the origin in the lower left hand corner: . . . . . . . . . . . . . . . S [ T ] S [ T ] ⊗ S [ T ] . . . . . . S [ T ] . . . d ( x ⊗ y ) = ǫ ( x ) y − xy + xǫ ( y ) = − xy for x, y ∈ S [ T ] ∗ , since x and y both augment to zero. With prior understand-ing of the stable homotopy groups of C P ∞ we can now figure out the displayeddifferential. The stable class of the inclusion S ∼ = C P → C P ∞ is well-knownto generate π Σ ∞ C P ∞ ∼ = Z . For degree reasons this must be detected by ± s in E ∞ , = E , = S [ T ] = Z { s } . The 4-cell in C P is attached by the Hopf fibration η to C P , so that π Σ ∞ C P ∞ = 0. This forces ηs ∈ E , = S [ T ] = Z / { ηs } to be aboundary in the spectral sequence. For degree reasons, the only possibility is that ηs = d ( s ⊗ s ) , so that s = ηs .Note that the coproduct ψ ( s ) must contain the terms s ⊗ ⊗ s by counitality,and cannot contain other terms since S [ T ] ∗ is connected and | s | = 1. Hence s is aprimitive element of our Hopf algebra. (cid:3) Proposition 3.4.
When G = U = Sp (1) is the -sphere group, R [ U ] ∗ = R ∗ [ t ] / ( t = ˙ νt ) with | t | = | ˙ ν | = 3 . Here t generates the augmentation ideal R [ U ] ∗ = ker( ǫ : R [ U ] ∗ −→ R ∗ ) = R ∗ { t } , and ˙ ν is the image of a generator of π ( S ) ∼ = Z / . The coproduct is given by ψ ( t ) = t ⊗ ⊗ t .Proof. Similar to the circle case. (cid:3)
Note that we cannot assert from this line of argument that ˙ ν is the image of thequaternionic Hopf map. The bar spectral sequence argument for R = S only showsthat t = ˙ νt with ˙ ν some generator of π ( S ) ∼ = Z /
24. A more geometric argumentmight link ˙ ν to the standard generator ν of π ( S ), but we will not pursue this here.3.3. A restriction homomorphism.
Any G -spectrum can be viewed as a non-equivariant spectrum via the inclusion homomorphism 1 → G . This gives rise to amap of graded abelian groupsres G : π G ∗ ( X ) −→ π ∗ ( X )taking the homotopy class of a G -map f : Σ V S q → X ( V ) to the homotopy class ofthe underlying non-equivariant map, and similarly for G -maps Σ V − R q S → X ( V ).Here 1 denotes the trivial group, and we write π ∗ ( X ) in place of π ∗ ( X ), as theyare simply the ordinary non-equivariant homotopy groups of X viewed as a non-equivariant orthogonal spectrum X . Tracing through definitions shows that res G
16 ALICE HEDENLUND AND JOHN ROGNES is R ∗ -linear if X is an R -module in orthogonal G -spectra. We are interested in thefollowing refined restriction homomorphism. Lemma 3.5.
There is a natural R ∗ -module homomorphism ω X : π G ∗ ( X ) −→ Hom R [ G ] ∗ ( R ∗ , π ∗ ( X )) making the diagram π G ∗ ( X ) ω X (cid:15) (cid:15) res G ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ Hom R [ G ] ∗ ( R ∗ , π ∗ ( X )) / / / / π ∗ ( X ) ˜ γ / / ˜ ǫ / / Hom R ∗ ( R [ G ] ∗ , π ∗ ( X )) commute. Here ˜ ǫ denotes the adjoint of the trivial R [ G ] ∗ -action ǫ ∗ on π ∗ ( X ) , whichequals the composite ǫ ∗ : π ∗ ( X ) ⊗ R ∗ R [ G ] ∗ ⊗ ǫ −→ π ∗ ( X ) ⊗ R ∗ R ∗ ∼ = π ∗ ( X ) . Similarly ˜ γ denotes the adjoint to the R [ G ] ∗ -action γ ∗ : π ∗ ( X ) ⊗ R ∗ R [ G ] ∗ −→ π ∗ ( X ) on π ∗ ( X ) .Proof. We claim that the two composite homomorphisms π G ∗ ( X ) ⊗ R ∗ R [ G ] ∗ res G ⊗ / / π ∗ ( X ) ⊗ R ∗ R [ G ] ∗ γ ∗ / / ǫ ∗ / / π ∗ ( X )are equal. This implies that the two adjoint homomorphisms π G ∗ ( X ) res G / / π ∗ ( X ) ˜ γ / / ˜ ǫ / / Hom R ∗ ( R [ G ] ∗ , π ∗ ( X ))are equal, so that res G factors uniquely through the equalizer Hom R [ G ] ∗ ( R ∗ , π ∗ ( X ))of the two right-hand arrows.By fibrant replacement we may assume that X is an Ω- G -spectrum [MM02,Def. III.3.1], meaning that each adjoint structure map X ( V ) → Ω W − V X ( W ) is aweak G -equivalence, where V ⊂ W lie in our fixed complete G -universe U . Theneach element x in π G ∗ ( X ) is represented by the homotopy class [ f ] of a G -map f : S m → X ( R n ), for suitable non-negative integers m and n . Here G acts triviallyon S m , so f factors through the fixed points X ( R n ) G , where the G -action γ is trivial.It follows that ˜ γ and ˜ ǫ agree on res G ( x ) ⊗ y for any y ∈ R [ G ] ∗ , as claimed. (cid:3) Proposition 3.6. If R [ G ] ∗ is projective as an R ∗ -module, and X ≃ F ( G + , Y ) forsome R -module Y in orthogonal G -spectra, then the natural homomorphism ω X : π G ∗ ( X ) ∼ = −→ Hom R [ G ] ∗ ( R ∗ , π ∗ ( X )) is an isomorphism.Proof. By fibrant replacement, we may assume that Y is an Ω- G -spectrum. Asusual we give F ( G + , Y ) ∼ = F R ( R [ G ] , Y ) the conjugate G -action. By naturality of ω X we may assume that X = F ( G + , Y ), in which case X is also an Ω- G -spectrum.Let us consider the commutative diagram π G ∗ ( X ) π ∗ ( X ) π ∗ ( Y ) Hom R ∗ ( R [ G ] ∗ , π ∗ ( Y )) res G ∼ = ∼ =˜ γ . MULTIPLICATIVE TATE SPECTRAL SEQUENCE 47
We make the maps involved a bit more explicit. The vertical isomorphisms are givenas follows. The left hand vertical isomorphism π G ∗ ( X ) = π G ∗ ( F ( G + , Y )) → π ∗ ( Y )takes the homotopy class of a G -map f : S m → X ( R n ) = F ( G + , Y ( R n )) bijectivelyto the homotopy class of f ′ : S m → Y ( R n ) given by f ′ ( s ) = f ( s )( e ), where e ∈ G isthe unit element of our group. The right hand vertical isomorphism is the specialcase Z = R [ G ] of the natural R [ G ] ∗ -module homomorphism π ∗ F R ( Z, Y ) −→ Hom R ∗ ( π ∗ ( Z ) , π ∗ ( Y )) . Indeed, this is an isomorphism whenever π ∗ ( Z ) is projective as an R ∗ -module. Thetop horizontal map is the restriction homomorphism we described at the beginningof this section, and the lower horizontal homomorphism ˜ γ is adjoint to the R [ G ] ∗ -module action on π ∗ ( Y ). Note that the diagram does indeed commute, since thelower and upper compositions both send the homotopy class of the G -map f : S m → F ( G + , Y ( R n )) to the homomorphism R [ G ] ∗ → π ∗ ( Y ) induced by the left adjoint S m ∧ G + → Y ( R n ) of f .The homomorphism ˜ γ identifies π ∗ ( Y ) with the R ∗ -submoduleHom R [ G ] ∗ ( R ∗ , Hom R ∗ ( R [ G ] ∗ , π ∗ ( Y ))) ∼ = Hom R [ G ] ∗ ( R [ G ] ∗ , π ∗ ( Y ))of its target, while res G factors through Hom R [ G ] ∗ ( R ∗ , π ∗ ( X )) by the previouslemma. Hence we can apply Hom R [ G ] ∗ ( R ∗ , − ) to the right hand vertical isomor-phism in the diagram above to obtain another isomorphism, and a commutativesquare π G ∗ ( X ) ω X / / ∼ = (cid:15) (cid:15) Hom R [ G ] ∗ ( R ∗ , π ∗ ( X )) ∼ = (cid:15) (cid:15) π ∗ ( Y ) ∼ = / / Hom R [ G ] ∗ ( R ∗ , Hom R ∗ ( R [ G ] ∗ , π ∗ ( Y ))) . It follows that ω X is an isomorphism, as asserted. (cid:3) To handle multiplicative structure, we need the following observation.
Lemma 3.7.
The natural transformation ω is monoidal, in the sense that thediagram π G ∗ ( X ) ⊗ R ∗ π G ∗ ( Y ) · / / ω X ⊗ ω Y (cid:15) (cid:15) π G ∗ ( X ∧ R Y ) ω X ∧ RY (cid:15) (cid:15) Hom R [ G ] ∗ ( R ∗ , π ∗ ( X )) ⊗ R ∗ Hom R [ G ] ∗ ( R ∗ , π ∗ ( Y )) α (cid:15) (cid:15) Hom R [ G ] ∗ ( R ∗ , π ∗ ( X ) ⊗ R ∗ π ∗ ( Y )) · / / Hom R [ G ] ∗ ( R ∗ , π ∗ ( X ∧ R Y )) commutes.Proof. Since Hom R [ G ] ∗ ( R ∗ , π ∗ ( X ∧ R Y )) → π ∗ ( X ∧ R Y ) is a monomorphism itsuffices to show that π G ∗ ( X ) ⊗ R ∗ π G ∗ ( Y ) · / / res G ⊗ res G (cid:15) (cid:15) π G ∗ ( X ∧ R Y ) res G (cid:15) (cid:15) π ∗ ( X ) ⊗ R ∗ π ∗ ( Y ) · / / π ∗ ( X ∧ R Y )commutes, which is clear. (cid:3) Sequences of spectra and spectral sequences
In this section we associate a Cartan–Eilenberg system, an exact couple, anda spectral sequence to any sequence of orthogonal G -spectra. We identify cer-tain well-behaved sequences called filtrations, and use these to show how pairingsof sequences induce pairings of Cartan–Eilenberg systems and spectral sequences.This is essentially the content of Section 4.5 and Section 4.6, culminating in The-orem 4.26 and Theorem 4.27. We shall rely on the classical telescope constructionto approximate general sequences by equivalent filtrations. Finally we discuss howpairings can be internalized in terms of the convolution product of two sequences.4.1. Cartan–Eilenberg systems.
A Cartan–Eilenberg system is an algebraicstructure, introduced in [CE56], which determines two exact couples [Mas52] anda spectral sequence. This structure has the advantage that one can give a usefuldefinition of a pairing of Cartan–Eilenberg systems, which determines a pairing ofthe corresponding spectral sequences. These definitions were reviewed by Douadyin [Dou59a] and [Dou59b]. As opposed to these sources, which use cohomologicalindexing, we adopt homological indexing for our Cartan–Eilenberg systems, as in[HR19].We start with some preliminary definitions. We will in particular make use ofthe posets [1] = { → } and [2] = { → → } regarded as categories. Note thatwe have three functors δ , δ , δ : [1] −→ [2] , with subscript indicating which object of the target is skipped. In addition, wehave natural transformations i : δ −→ δ and p : δ −→ δ . Let ( I , ≤ ) be a linearly ordered set. The following definitions can be found in [HR19,Def. 4.1]. Definition 4.1. • Let I [1] = Fun([1] , I ). The objects in this category are pairs ( i, j ) in I with i ≤ j , and we have a single morphism ( i, j ) → ( i ′ , j ′ ) precisely when i ≤ i ′ and j ≤ j ′ . • Let I [2] = Fun([2] , I ). The objects in this category are triples ( i, j, k ) in I with i ≤ j ≤ k , and we have a single morphism ( i, j, k ) → ( i ′ , j ′ , k ′ )precisely when i ≤ i ′ , j ≤ j ′ and k ≤ k ′ .The functors δ , δ , and δ defined above induce functors d , d , d : I [2] −→ I [1] . These map ( i, j, k ) to ( j, k ), ( i, k ) and ( i, j ), respectively. The natural transforma-tions i and p induce natural transformations ι : d −→ d and π : d −→ d with components ι : ( i, j ) → ( i, k ) and π : ( i, k ) → ( j, k ), respectively.Let k be a graded ring and let A be the graded abelian category of k -modules.The grading k x k of a homogeneous element x ∈ M in an object M of A will bereferred to as its total degree . Definition 4.2 ([HR19, Def. 4.2, Def. 6.1]) . An I -system in A is a pair ( H, ∂ )where H : I [1] → A is a functor and ∂ : Hd → Hd is a natural transformation of MULTIPLICATIVE TATE SPECTRAL SEQUENCE 49 functors I [2] → A , such that the triangle Hd Hι / / Hd Hπ (cid:15) (cid:15) Hd ∂ b b ❋❋❋❋❋❋❋❋ is exact. We assume that Hι and Hπ have total degree 0, while ∂ has total de-gree −
1. We generically write η : H ( i, j ) → H ( i ′ , j ′ ) for the total degree 0 mor-phisms in A induced by morphisms in I [1] . Definition 4.3. • A finite Cartan–Eilenberg system is a Z -system ( H, ∂ ), where Z denotes theintegers with its usual linear ordering. • An extended Cartan–Eilenberg system is a I -system ( H, ∂ ) for I = Z ∪{±∞} , with the extended linear ordering where −∞ is initial and + ∞ isterminal. • A Cartan–Eilenberg system is an extended Cartan–Eilenberg system (
H, ∂ )such that the following condition, called (SP.5), is satisfied: The canonicalhomomorphism colim j H ( i, j ) ∼ = −→ H ( i, ∞ )is an isomorphism for all i ∈ Z .An extended Cartan–Eilenberg system thus associates to each pair ( i, j ) with −∞ ≤ i ≤ j ≤ ∞ a module H ( i, j ), in a functorial way. Furthermore, it associatesto each triple ( i, j, k ) with −∞ ≤ i ≤ j ≤ k ≤ ∞ a long exact sequence . . . −→ H ( i, j ) −→ H ( i, k ) −→ H ( j, k ) ∂ −→ H ( i, j ) −→ . . . , where ∂ is a natural transformation of total degree −
1. If the homomorphism incondition (SP.5) is an isomorphism for one −∞ ≤ i < ∞ , then it is an isomorphismfor every such i . This follows by using the 5-Lemma twice in the following map ofexact sequences: . . . / / H ( −∞ , i ) / / = (cid:15) (cid:15) colim j H ( −∞ , j ) / / (cid:15) (cid:15) colim j H ( i, j ) ∂ / / (cid:15) (cid:15) H ( −∞ , i ) / / = (cid:15) (cid:15) . . .. . . / / H ( −∞ , i ) / / H ( −∞ , ∞ ) / / H ( i, ∞ ) ∂ / / H ( −∞ , i ) / / . . . . An extended Cartan–Eilenberg system determines a finite Cartan–Eilenberg systemby restriction to ( i, j ) with −∞ < i ≤ j < ∞ . Remark . Apart from the switch in variance, the definition given by Cartan andEilenberg in [CE56, § XV.7] corresponds to our Cartan–Eilenberg systems. Thisis also the definition recalled in [McC01, Ex. 2.2]. In [Dou59a, § II C], Douadyworks with data defining an Adams spectral sequence, which is concentrated innon-negative cohomological (so: non-positive homological) filtration degrees. Hetherefore assumes that H ( i,
0) = H ( i, j ) for all i ≤ ≤ j ≤ ∞ , so that condi-tion (SP.5) is trivially satisfied. Definition 4.5 ([HR19, Def. 7.1]) . Let (
H, ∂ ) be a Cartan–Eilenberg system. Letthe left couple ( A, E ) be the exact couple given by A s = H ( −∞ , s ) and E s = H ( s − , s ) fitting together in the exact triangle A s − / / A s (cid:15) (cid:15) E s∂ a a ❉❉❉❉❉❉❉❉ associated to the triple ( −∞ , s − , s ). The abutment of this exact couple is A ∞ = colim s A s ∼ = H ( −∞ , ∞ ) . This abutment is exhaustively filtered by the images F s A ∞ = im( A s −→ A ∞ ) . The Cartan–Eilenberg system and the left couple give rise to the same spectralsequence ( E r , d r ). The pages of this spectral sequence are given by E rs = Z rs /B rs and the differentials d rs : E rs → E rs − r are of total degree −
1. Here Z rs = ker( ∂ : E s −→ H ( s − r, s − B rs = im( ∂ : H ( s, s + r − −→ E s )define the r -cycles and r -boundaries in filtration degree s , respectively, and d rs ([ x ]) = [ ∂ (˜ x )] , where x ∈ Z rs and ˜ x ∈ H ( s − r, s ) satisfies η (˜ x ) = x . There are preferred isomor-phisms H ( E r , d r ) ∼ = E r +1 . We let Z ∞ s = lim r Z rs , B ∞ s = colim r B rs , and E ∞ s = Z ∞ s /B ∞ s . We refer to [CE56, § XV.1] or [HR19, Prop. 4.9] for the verification that ( E r , d r ) isindeed a spectral sequence. Note in particular that it only depends on the finite partof the Cartan–Eilenberg system ( H, ∂ ). The abutment and E ∞ -page are related asfollows. Lemma 4.6.
There is a natural monomorphism β : F s A ∞ F s − A ∞ −→ E ∞ s in each filtration degree s .Proof. See [Boa99, Lem. 5.6] or [HR19, Lem. 3.15(a)]. (cid:3)
The main purpose of reviewing the above definitions is to let us record thefollowing definitions of pairings of (finite and classical) Cartan–Eilenberg systems.We assume that k is graded commutative, and write ⊗ in place of ⊗ k . Definition 4.7.
Let ( H ′ , ∂ ), ( H ′′ , ∂ ) and ( H, ∂ ) be finite Cartan–Eilenberg systemsin A . A pairing φ : ( H ′ , H ′′ ) → H of such systems is a collection of k -modulehomomorphisms φ r : H ′ ( i − r, i ) ⊗ H ′′ ( j − r, j ) −→ H ( i + j − r, i + j )of total degree 0, for all i, j ∈ Z and r ≥
1. These are required to satisfy thefollowing two conditions:
MULTIPLICATIVE TATE SPECTRAL SEQUENCE 51
SSP I:
Each square H ′ ( i − r, i ) ⊗ H ′′ ( j − r, j ) φ r / / η ⊗ η (cid:15) (cid:15) H ( i + j − r, i + j ) η (cid:15) (cid:15) H ′ ( i ′ − r ′ , i ′ ) ⊗ H ′′ ( j ′ − r ′ , j ′ ) φ r ′ / / H ( i ′ + j ′ − r ′ , i ′ + j ′ )commutes, for all integers i, j, i ′ , j ′ and r, r ′ ≥ i ≤ i ′ , i − r ≤ i ′ − r ′ , j ≤ j ′ and j − r ≤ j ′ − r ′ . SSP II:
In the (non-commutative) diagram H ′ ( i − r, i ) ⊗ H ′′ ( j − r, j ) ∂ ⊗ η (cid:15) (cid:15) φ r ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘ η ⊗ ∂ / / H ′ ( i − , i ) ⊗ H ′′ ( j − r − , j − r ) φ (cid:15) (cid:15) H ( i + j − r, i + j ) ∂ ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘ H ′ ( i − r − , i − r ) ⊗ H ′′ ( j − , j ) φ / / H ( i + j − r − , i + j − r )the inner composition is the sum of the two outer ones: ∂φ r = φ ( ∂ ⊗ η ) + φ ( η ⊗ ∂ ) . In terms of elements, this identity in H ( i + j − r − , i + j − r ) can be written ∂φ r ( x ⊗ y ) = φ ( ∂x ⊗ ηy ) + ( − k x k φ ( ηx ⊗ ∂y )for x ∈ H ′ ( i − r, i ) of total degree k x k and y ∈ H ′′ ( j − r, j ). Remark . Apart from the switch in variance, this definition agrees with thatof Douady [Dou59b, § II A], except for the fact that we ignore r = 0, since φ carries no information, and Douady omits the cases i > j >
0, due to hisfocus on Adams spectral sequences. In the definition given in [McC01, Ex. 2.3], thehomomorphism ϕ is missing from the right hand term in his equation (2), and theconditions n ≥ q ≥ Definition 4.9.
Let ( ′ E r , d r ), ( ′′ E r , d r ) and ( E r , d r ) be k -module spectral se-quences. A pairing φ : ( ′ E ∗ , ′′ E ∗ ) → E ∗ of such spectral sequences consists of acollection of k -module homomorphisms φ r : ′ E r ⊗ ′′ E r −→ E r for all r ≥
1, such that:(1) The Leibniz rule d r φ r = φ r ( d r ⊗
1) + φ r (1 ⊗ d r )holds as an equality of homomorphisms ′ E ri ⊗ ′′ E rj −→ E ri + j − r for all i, j ∈ Z and r ≥ ′ E r +1 ⊗ ′′ E r +1 E r +1 H ( ′ E r ⊗ ′′ E r ) H ( E r ) φ r +1 ∼ = H ( φ r ) commutes for all r ≥ By a multiplicative spectral sequence , we mean a spectral sequence ( E r , d r )equipped with a pairing φ : ( E ∗ , E ∗ ) −→ E ∗ . If φ a : E a ⊗ E a → E a (usually with a = 1 or a = 2) is associative and unital, theneach pairing φ r for r ≥ a is also associative and unital, and we call ( E r , d r ) r ≥ a an algebra spectral sequence . Theorem 4.10 ([Dou59b, Thm. II A]) . Let ( H ′ , ∂ ) , ( H ′′ , ∂ ) and ( H, ∂ ) be finiteCartan–Eilenberg systems, with associated spectral sequences ( ′ E r , d r ) , ( ′′ E r , d r ) and ( E r , d r ) . Let φ : ( H ′ , H ′′ ) → H be a pairing of finite Cartan–Eilenberg systems.Then there is a pairing φ : ( ′ E ∗ , ′′ E ∗ ) → E ∗ of spectral sequences, uniquely definedby the condition φ = φ .Proof. Douady leaves the proof to the reader (“s’il existe”). Starting with setting φ : ′ E i ⊗ ′′ E j → E i + j equal to φ : ′ H ( i − , i ) ⊗ ′′ H ( j − , j ) −→ H ( i + j − , i + j ) , the point is to inductively show that d r satisfies the Leibniz rule with respectto the pairing φ r of E r -pages, so that φ r +1 can be defined to be equal to theinduced pairing in homology with respect to d r . A full proof can be found in[Hel17, Prop. 3.4.2]. (cid:3) We now move from finite Cartan–Eilenberg systems to classical ones.
Definition 4.11.
Let ( H ′ , ∂ ), ( H ′′ , ∂ ) and ( H, ∂ ) be Cartan–Eilenberg systems. A pairing φ : ( H ′ , H ′′ ) → H of such systems consists of a pairing ( φ r ) r ≥ of the re-stricted finite Cartan–Eilenberg systems, together with k -module homomorphisms φ ∞ : H ′ ( −∞ , i ) ⊗ H ′′ ( −∞ , j ) −→ H ( −∞ , i + j )of total degree 0, for all i, j ∈ Z . These are required to satisfy the followingadditional condition: SPP III:
Each square H ′ ( −∞ , i ) ⊗ H ′′ ( −∞ , j ) φ ∞ / / η ⊗ η (cid:15) (cid:15) H ( −∞ , i + j ) η (cid:15) (cid:15) H ′ ( i − r, i ) ⊗ H ′′ ( j − r, j ) φ r / / H ( i + j − r, i + j )commutes, for all integers i, j and r ≥ φ r in the definition above must satisfy (SPP I) and (SPP II),by virtue of defining a pairing of finite Cartan–Eilenberg systems. The new condi-tion (SPP III) is an analogue of (SPP I) for r = ∞ .With notation as in Definition 4.5, we can rewrite φ ∞ as compatible pairings φ i,j : A ′ i ⊗ A ′′ j −→ A i + j in the corresponding left couples, for all i, j ∈ Z . Passing to colimits, we obtain apairing of abutments φ ∗ : A ′∞ ⊗ A ′′∞ −→ A ∞ . MULTIPLICATIVE TATE SPECTRAL SEQUENCE 53
This is filtration-preserving in the sense that it sends F i A ′∞ ⊗ F j A ′′∞ to F i + j A ∞ ,by virtue of the commutative diagram A ′ i ⊗ A ′′ j φ i,j / / (cid:15) (cid:15) (cid:15) (cid:15) A i + j (cid:15) (cid:15) (cid:15) (cid:15) F i A ′∞ ⊗ F j A ′′∞ / / (cid:15) (cid:15) F i + j A ∞ (cid:15) (cid:15) (cid:15) (cid:15) A ′∞ ⊗ A ′′∞ φ ∗ / / A ∞ .Being filtration-preserving, the pairing φ ∗ then induces pairings of filtration sub-quotients ¯ φ ∗ : F i A ′∞ F i − A ′∞ ⊗ F j A ′′∞ F j − A ′′∞ −→ F i + j A ∞ F i + j − A ∞ for all i, j ∈ Z .In a similar way, the spectral sequence pairing φ = ( φ r ) r ≥ , induced by thepairing ( φ r ) r ≥ per Theorem 4.10, maps ′ Z r ⊗ ′′ Z r −→ Z r , ′ B r ⊗ ′′ Z r −→ B r , ′ Z r ⊗ ′′ B r −→ B r , hence also maps ′ Z ∞ ⊗ ′′ Z ∞ −→ Z ∞ , ′ B ∞ ⊗ ′′ Z ∞ −→ B ∞ , ′ Z ∞ ⊗ ′′ B ∞ −→ B ∞ . It follows that φ also induces k -module homomorphisms(4.1) φ ∞ : ′ E ∞ i ⊗ ′′ E ∞ j −→ E ∞ i + j , sending [ x ] ⊗ [ y ] to [ φ ( x ⊗ y )] for any pair of infinite cycles x and y . Condi-tion (SPP III) ensures that we have the following compatibility. Proposition 4.12.
Let φ = ( φ r ) : ( H ′ , H ′′ ) → H be a pairing of Cartan–Eilenbergsystems, with induced pairing φ = ( φ r ) : ( ′ E ∗ , ′′ E ∗ ) → E ∗ of spectral sequences,per Theorem 4.10. Then the pairing φ ∗ of filtered abutments is compatible with thepairing φ ∞ of E ∞ -pages, in the sense that the diagram F i A ′∞ F i − A ′∞ ⊗ F j A ′′∞ F j − A ′′∞ ¯ φ ∗ / / β ⊗ β (cid:15) (cid:15) F i + j A ∞ F i + j − A ∞ (cid:15) (cid:15) β (cid:15) (cid:15) ′ E ∞ i ⊗ ′′ E ∞ j φ ∞ / / E ∞ i + j commutes, for all i, j ∈ Z .Proof. A detailed proof is given in [Hel17, Prop. 3.4.4]. (cid:3)
Remark . As a consequence of Theorem 4.10, if (
H, ∂ ) is a multiplicativeCartan–Eilenberg system, meaning that it is equipped with a pairing φ : ( H, H ) → H , then the associated spectral sequence ( E r , d r ) is also multiplicative. Moreover,Proposition 4.12 tells us that the induced pairing on the filtered abutment A ∞ is compatible with the induced pairing on the E ∞ -page of the spectral sequence. Inthis situation, we say that A ∞ is a multiplicative abutment . When ( E r , d r ) con-verges weakly to A ∞ , meaning that the filtration ( F s A ∞ ) s is exhaustive and β isan isomorphism, multiplicativity of the abutment means that we can reconstructthe product φ ∗ on A ∞ from the product φ ∞ on E ∞ , up to the usual ambiguitycreated by extensions.4.2. Sequences.
Our Cartan–Eilenberg systems will in practice be obtained fromfiltrations and sequences. Let us first set up some terminology, so that it is clearwhat we are discussing. Again, G denotes a compact Lie group. Definition 4.14.
A sequential diagram X ⋆ of orthogonal G -spectra and G -mapson the form · · · −→ X i − −→ X i −→ X i +1 −→ · · · , indexed over i ∈ Z , is called a sequence .We can extend the sequence to be indexed over Z ∪ {±∞} by setting X −∞ = ∗ and X ∞ = Tel( X ⋆ ) , where Tel( X ⋆ ) = _ i ∈ Z [ i, i + 1] + ∧ X i / ∼ is the classical telescope construction . Here, the equivalence relation ∼ is given byidentifying { i } + ∧ X i − with { i } + ∧ X i using the G -map X i − → X i . There arestandard inclusions X i ∼ = { i } + ∧ X i ⊂ Tel( X ⋆ ) for all i ∈ Z , and each diagram X i − / / $ $ ■■■■■■■■■ X i (cid:15) (cid:15) Tel( X ⋆ )commutes up to preferred homotopy. Definition 4.15.
The Cartan–Eilenberg system ( H = H ( X ⋆ ) , ∂ ) associated to thesequence X ⋆ of orthogonal G -spectra is given by H ( i, j ) = π G ∗ ( X i −→ X j )for all −∞ ≤ i ≤ j ≤ ∞ , and ∂ : π G ∗ ( X j → X k ) −→ π G ∗− ( X i → X j )for all −∞ ≤ i ≤ j ≤ k ≤ ∞ .Let us elaborate on the definition above. In the q ≥ H ( i, j ) = π Gq ( X → Y )denotes the colimit, over the partially ordered set of finite-dimensional G -sub-representations V of the complete G -universe U , of the groups of homotopy classes [ f ′ , f ]of pairs ( f ′ , f ) of G -maps f ′ : Σ V S q − → X ( V ) and f : Σ V D q → Y ( V ) making thesquare Σ V S q − / / f ′ (cid:15) (cid:15) Σ V D qf (cid:15) (cid:15) X ( V ) / / Y ( V ) MULTIPLICATIVE TATE SPECTRAL SEQUENCE 55 commute. Similar definitions can be made for q ≤
0, but will be left to the reader.By the stability of the homotopy category of orthogonal G -spectra there is a naturalisomorphism π Gq ( X → Y ) ∼ = π Gq ( Y ∪ CX ) , where Y ∪ CX is the mapping cone of X → Y . This isomorphism takes thehomotopy class [ f ′ , f ] to (the image in the colimit over V of) the homotopy classof the composite mapΣ V S q ≃ −→ Σ V ( D q ∪ CS q − ) f ∪ Cf ′ −→ ( Y ∪ CX )( V ) , where the first map is a ( V -suspended) homotopy inverse to the collapse map D q ∪ CS q − → D q /S q − ∼ = S q .The connecting homomorphism ∂ : π Gq ( Y → Z ) → π Gq − ( X → Y ) mentioned inthe definition takes the homotopy class [ g ′ , g ] of a pair of G -maps g ′ : Σ V S q − → Y ( V ) and g : Σ V D q → Z ( V ) to the homotopy class [ ∗ , g ′ π ] of the maps ∗ : Σ V S q − −→ ∗ −→ X ( V ) and g ′ π : Σ V D q − π −→ Σ V S q − −→ Y ( V ) , where π : D q − → S q − identifies D q − /S q − with S q − . The diagramΣ V S q − / / (cid:15) (cid:15) Σ V D q − π (cid:15) (cid:15) ∗ / / (cid:15) (cid:15) Σ V S q − g ′ (cid:15) (cid:15) X ( V ) / / Y ( V )evidently commutes. Under the isomorphism π Gq − ( X → Y ) ∼ = π Gq − ( Y ∪ CX ) thehomotopy class [ ∗ , g ′ π ] corresponds to the homotopy class of the composite mapΣ V S q − g ′ −→ Y ( V ) −→ ( Y ∪ CX )( V ) . Note that the graded abelian group π G ∗ ( X i → X j ) is functorial in i ≤ j , thehomomorphism ∂ is natural in i ≤ j ≤ k , and the sequence · · · −→ π Gq ( X i → X j ) −→ π Gq ( X i → X k ) −→ π Gq ( X j → X k ) ∂ −→ π Gq − ( X i → X j ) −→ · · · is exact for all i ≤ j ≤ k and q ∈ Z . The canonical homomorphismcolim j π G ∗ ( X j ) ∼ = −→ π G ∗ Tel( X ⋆ )is an isomorphism, which implies that condition (SP.5) is satisfied. Hence ( H, ∂ ) isindeed a Cartan–Eilenberg system in the sense of Definition 4.3.We can extract two different exact couples [Mas52] from ( H ( X ⋆ ) , ∂ ), but shallonly be concerned with the ‘left’ couple of Definition 4.5. Explicitly, the exactcouple ( A, E ) = ( A ( X ⋆ ) , E ( X ⋆ )) associated to X ⋆ is given by A s = π G ∗ ( X s ) and E s = π G ∗ ( X s − → X s ) , fitting together in the exact triangle π G ∗ ( X s − ) / / π G ∗ ( X s ) (cid:15) (cid:15) π G ∗ ( X s − → X s ) ∂ h h PPPPPPPPPPPP where ∂ has total degree − Recall from [Boa99, Def. 5.10] that the spectral sequence associated to the un-rolled exact couple (
A, E ) is said to be conditionally convergent to the abutment(4.2) A ∞ = colim s π G ∗ ( X s ) ∼ = π G ∗ Tel( X ⋆ )if and only if A −∞ = lim s A s = 0 and RA −∞ = Rlim s A s = 0 . Here Rlim = lim denotes the (first right) derived limit of a sequence. In view ofthe short exact sequence0 −→ Rlim s π G ∗ +1 ( X s ) −→ π G ∗ (holim s X s ) −→ lim s π G ∗ ( X s ) −→ π G ∗ (holim s X s ) = 0 . In particular, conditional convergence holds if holim s X s ≃ G ∗ .The spectral sequence ( E r = E r ( X ⋆ ) , d r ) r ≥ associated to the sequence X ⋆ (andthe Cartan–Eilenberg system ( H ( X ⋆ ) , ∂ ), and the exact couple ( A ( X ⋆ ) , E ( X ⋆ )))has E s,t = π Gs + t ( X s − → X s )and d : E s,t → E s − ,t is equal to the composite homomorphism π Gs + t ( X s − → X s ) ∂ −→ π Gs + t − ( X s − ) −→ π Gs + t − ( X s − → X s − ) . Here s + t is the total degree , s is the filtration degree , and t will be called the internal degree . The d r -differentials have the form d r : E rs,t −→ E rs − r,t + r − and there are preferred isomorphisms H ( E r , d r ) ∼ = E r +1 for all r ≥ Filtrations.
The category of orthogonal G -spectra is based topological, mean-ing that it is enriched in the closed symmetric monoidal category of compactlygenerated weak Hausdorff spaces with base point. Definition 4.16.
Let I = [0 , ∂I = { , } . • A G -map i : A → X of orthogonal G -spectra is an h -cofibration (= Hurewiczcofibration) if it has the homotopy extension property with respect to anytarget Z : X ∪ A A ∧ I + / / (cid:15) (cid:15) ZX ∧ I + ssssss . • A G -map p : E → B of orthogonal G -spectra is an h -fibration (= Hurewiczfibration) if it has the homotopy lifting property with respect to any source X : X / / (cid:15) (cid:15) E p (cid:15) (cid:15) X ∧ I + / / ; ; ①①①①① B . MULTIPLICATIVE TATE SPECTRAL SEQUENCE 57 • Adapting [SV02, Def. 2.4], we say that i : A → X is a strong h -cofibration ifthe G -map X ∪ A A ∧ I + → X ∧ I + has the left lifting property with respectto any h -fibration: X ∪ A A ∧ I + / / (cid:15) (cid:15) E p (cid:15) (cid:15) X ∧ I + / / ssssss B .Strong h -cofibrations are closed under cobase change, retracts, arbitrary sums,and sequential colimits [SV02, Lem. 2.6]. For each map f : X → Y the inclusion i : X → Y ∪ X X ∧ I + is a strong h -cofibration [SV02, Rmk. 3.3(2)]. It followsthat each q -cofibration (= Quillen cofibration, [MM02, Def. III.2.3]) is a strong h -cofibration. Each strong h -cofibration is evidently an h -cofibration. Our mainreason for working with strong h -cofibrations is the availability of the followingtheorem. Theorem 4.17 ([SV02, Thm. 2.7]) . If i : A → X and j : B → Y are strong h -cofibrations, then the pushout-product map i ∧ ∪ ∧ j : A ∧ Y ∪ A ∧ B X ∧ B −→ X ∧ Y is a strong h -cofibration. We can now specify well-behaved sequences, called filtrations, for which we candirectly prove that pairings of sequences induce pairings of Cartan–Eilenberg sys-tems and of spectral sequences.
Definition 4.18.
Let X ⋆ be a sequence of orthogonal G -spectra. We say that X ⋆ is a filtration if each G -map X i − → X i for i ∈ Z is a strong h -cofibration.In particular, if X ⋆ is a filtration, then the G -maps are all h -cofibrations, so thecanonical maps X j ∪ CX i −→ X j /X i and Tel( X ⋆ ) −→ colim i X i = [ i X i are G -equivalences, so that H ( i, j ) ∼ = π G ∗ ( X j /X i ) and A ∞ ∼ = π G ∗ [ i X i ! in the associated Cartan–Eilenberg system.We can always approximate a sequence X ⋆ with an equivalent filtration T ⋆ ( X ).To do this, we proceed as follows. For each integer j we let T j ( X ) = { j } + ∧ X j ∨ _ i We now turn to discussing pairings of sequences andhow these behave under passage to mapping telescopes. Definition 4.20. Let X ⋆ , Y ⋆ and Z ⋆ be sequences of orthogonal G -spectra. A pairing φ : ( X ⋆ , Y ⋆ ) → Z ⋆ is a collection of G -maps φ i,j : X i ∧ Y j −→ Z i + j for all integers i and j , making the squares(4.3) X i − ∧ Y j Z i + j − X i ∧ Y j − X i ∧ Y j Z i,j X i ∧ Y jφ i − ,j φ i,j − φ i,j φ i,j commute. We say that a sequence X ⋆ is multiplicative if it comes equipped with apairing φ : ( X ⋆ , X ⋆ ) → X ⋆ .We note that, from [MM02, § II.3] and [Sch18, § X i ∧ Y j of orthogonal G -spectra is defined in such a way that φ i,j associates to each pairof G -representations U and V a G -map of based G -spaces φ i,j ( U, V ) : X i ( U ) ∧ Y j ( V ) −→ Z i + j ( U ⊕ V ) , subject to bilinearity relations for varying U and V . MULTIPLICATIVE TATE SPECTRAL SEQUENCE 59 Lemma 4.21. A pairing of sequences φ : ( X ⋆ , Y ⋆ ) → Z ⋆ induces a pairing of se-quences T ( φ ) : ( T ⋆ ( X ) , T ⋆ ( Y )) → T ⋆ ( Z ) , such that the diagram T i ( X ) ∧ T j ( Y ) T ( φ ) i,j / / ≃ G (cid:15) (cid:15) T i + j ( Z ) ≃ G (cid:15) (cid:15) X i ∧ Y j φ i,j / / Z i + j commutes for all integers i and j .Proof. Given a pairing φ of sequences, note that we can form G -maps(4.4) [ i, i + 1] + ∧ X i ∧ [ j, j + 1] + ∧ Y j −→ [ k, k + 2] + ∧ Z k −→ Tel( Z ⋆ )for any integers i and j , with k = i + j . Here [ i, i + 1] × [ j, j + 1] → [ k, k + 2] sends( x, y ) to x + y , while X i ∧ Y j → Z k is given by φ i,j . The second map factors through[ k, k + 1] + ∧ Z k ∪ [ k + 1 , k + 2] + ∧ Z k +1 , and is given by Z k → Z k +1 on [ k + 1 , k + 2] ⊂ [ k, k + 2]. The maps (4.4) for varying i and j are compatible with the identifications defining Tel( X ⋆ ) and Tel( Y ⋆ ), hencecombine to define a G -mapTel( φ ) : Tel( X ⋆ ) ∧ Tel( Y ⋆ ) −→ Tel( Z ⋆ ) . By construction, it restricts to compatible G -maps T ( φ ) i,j : T i ( X ) ∧ T j ( Y ) −→ T i + j ( Z )for all integers i and j , defining the pairing of sequences T ( φ ) : ( T ⋆ ( X ) , T ⋆ ( Y )) → T ⋆ ( Z )). It is then clear that the square in the lemma commutes, and that thevertical maps are G -equivariant deformation retractions. (cid:3) Corollary 4.22. If ( X ⋆ , φ ) is a multiplicative sequence, then so is ( T ⋆ ( X ) , T ( φ )) .Moreover, the equivalence ǫ : T ⋆ ( X ) → X ⋆ respects the multiplicative structures. Pairings of Cartan–Eilenberg systems, I. The goal of the following twosections is to show that a pairing of sequences gives rise to a pairing of the resultingCartan–Eilenberg systems. By Theorem 4.10 and Proposition 4.12 this is enoughto guarantee that we have a pairing of the associated spectral sequences in such away that the induced pairing on filtered abutments is compatible with the pairingon E ∞ -pages. Referring back to Definition 4.7 and Definition 4.11, we note thatthere are three things to check. In this section we deal with (SPP I) and (SPP III).Let φ : ( X ⋆ , Y ⋆ ) → Z ⋆ be a pairing of sequences. For integers i , j and r , with r ≥ 1, we define induced pairings φ r : H ( X ⋆ )( i − r, i ) ⊗ H ( Y ⋆ )( j − r, j ) −→ H ( Z ⋆ )( i + j − r, i + j )as homomorphisms φ r : π Gp ( X i − r → X i ) ⊗ π Gq ( Y j − r → Y j ) −→ π Gp + q ( Z i + j − r → Z i + j ) . Here p and q range over all integers, but for (relative) brevity we concentrate onthe case when p ≥ q ≥ 0. Given two pairs of vertical G -mapsΣ U S p − / / f ′ (cid:15) (cid:15) Σ U D pf (cid:15) (cid:15) X i − r ( U ) / / X i ( U ) and Σ V S q − / / g ′ (cid:15) (cid:15) Σ V D qg (cid:15) (cid:15) Y j − r ( V ) / / Y j ( V ) we first form the commutative diagramΣ U ⊕ V S p − ∧ S q − / / ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘ ∼ = (cid:15) (cid:15) Σ U ⊕ V D p ∧ S q − ( ( ◗◗◗◗◗◗◗◗◗◗◗◗ ∼ = (cid:15) (cid:15) Σ U ⊕ V S p − ∧ D q / / ∼ = (cid:15) (cid:15) Σ U ⊕ V D p ∧ D q ∼ = (cid:15) (cid:15) Σ U S p − ∧ Σ V S q − / / ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘ f ′ ∧ g ′ (cid:15) (cid:15) Σ U D p ∧ Σ V S q − ( ( ◗◗◗◗◗◗◗◗◗◗◗◗ f ∧ g ′ (cid:15) (cid:15) Σ U S p − ∧ Σ V D q / / f ′ ∧ g (cid:15) (cid:15) Σ U D p ∧ Σ V D qf ∧ g (cid:15) (cid:15) X i − r ( U ) ∧ Y j − r ( V ) / / ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ φ i − r,j − r ( U,V ) (cid:15) (cid:15) X i ( U ) ∧ Y j − r ( V ) ' ' PPPPPPPPPPPP φ i,j − r ( U,V ) (cid:15) (cid:15) X i − r ( U ) ∧ Y j ( V ) / / φ i − r,j ( U,V ) ( ( ◗◗◗◗◗◗◗◗◗◗◗◗ X i ( U ) ∧ Y j ( V ) φ i,j ( U,V ) (cid:15) (cid:15) Z i + j − r ( U ⊕ V ) / / Z i + j − r ( U ⊕ V ) / / Z i + j ( U ⊕ V ) . For typographical reasons, we will often suppress the stabilising G -representations U and V and simply display this diagram as S p − ∧ S q − / / ' ' PPPPPPPPPPPP f ′ ∧ g ′ (cid:15) (cid:15) D p ∧ S q − & & ◆◆◆◆◆◆◆◆◆◆◆ f ∧ g ′ (cid:15) (cid:15) S p − ∧ D q / / f ′ ∧ g (cid:15) (cid:15) D p ∧ D qf ∧ g (cid:15) (cid:15) X i − r ∧ Y j − r / / ' ' ❖❖❖❖❖❖❖❖❖❖❖ φ i − r,j − r (cid:15) (cid:15) X i ∧ Y j − r & & ▼▼▼▼▼▼▼▼▼▼ φ i,j − r (cid:15) (cid:15) X i − r ∧ Y j / / φ i − r,j ' ' ◆◆◆◆◆◆◆◆◆◆◆ X i ∧ Y jφ i,j (cid:15) (cid:15) Z i + j − r / / Z i + j − r / / Z i + j . Let S p + q − = S p − ∧ D q ∪ S p − ∧ S q − D p ∧ S q − and W = X i − r ∧ Y j ∪ X i − r ∧ Y j − r X i ∧ Y j − r denote the pushouts in the squares of the upper and middle layer of the diagram,respectively. In particular, S p + q − is the boundary of D p ∧ D q ∼ = D p + q . We thenhave an induced commutative diagram(4.5) S p − ∧ S q − S p + q − D p + q X i − r ∧ Y j − r W X i ∧ Y j Z i + j − r Z i + j − r Z i + j . f ′ ∧ g ′ ( f ∧ g ) ′ f ∧ gφ i − r,j − r φ W φ i,j MULTIPLICATIVE TATE SPECTRAL SEQUENCE 61 Here, ( f ∧ g ) ′ : S p + q − → W is the induced map between the pushouts in the topand bottom square of the boxed-shaped diagram appearing above, and φ W is theinduced map in the diagram X i − r ∧ Y j − r X i ∧ Y j − r X i − r ∧ Y j W Z i + j − r . φ i,j − r φ i − r,j φ W We define the homomorphism φ r : π Gp ( X i − r → X i ) ⊗ π Gq ( Y j − r → Y j ) → π Gp + q ( Z i + j − r → Z i + j )as sending [ f ′ , f ] ⊗ [ g ′ , g ] to the homotopy class of the pair φ W ( f ∧ g ) ′ : S p + q − → Z i + j − r and φ i,j ( f ∧ g ) : D p + q → Z i + j , which is an element of π Gp + q ( Z i + j − r → Z i + j ). As a diagram, this pair is visualisedas the commutative square S p + q − / / φ W ( f ∧ g ) ′ (cid:15) (cid:15) D p + qφ i,j ( f ∧ g ) (cid:15) (cid:15) Z i + j − r / / Z i + j . In symbols: φ r : [ f ′ , f ] ⊗ [ g ′ , g ] [ φ W ( f ∧ g ) ′ , φ i,j ( f ∧ g )] . Spelled out with the stabilising G -representations U and V , this diagram should beinterpreted as the commutative diagramΣ U ⊕ V S p + q − / / ∼ = (cid:15) (cid:15) Σ U ⊕ V D p + q ∼ = (cid:15) (cid:15) Σ U S p − ∧ Σ V D q ∪ Σ U D p ∧ Σ V S q − / / ( f ∧ g ) ′ (cid:15) (cid:15) Σ U D p ∧ Σ V D qf ∧ g (cid:15) (cid:15) X i − r ( U ) ∧ Y j ( V ) ∪ X i ( U ) ∧ Y j − r ( V ) / / φ W ( U,V ) (cid:15) (cid:15) X i ( U ) ∧ Y j ( V ) φ i,j ( U,V ) (cid:15) (cid:15) Z i + j − r ( U ⊕ V ) / / Z i + j ( U ⊕ V )The pushouts on the left hand side are formed along Σ U S p − ∧ Σ V S q − and X i − r ( U ) ∧ Y j − r ( V ), respectively. Remark . We note that the pushout W is not generally equivalent to the cor-responding homotopy pushout, but this will hold if X ⋆ and Y ⋆ are filtrations.We also note that if one only has a weak pairing, in the sense that the squares (4.3)commute up to homotopy, then there is in general no preferred commuting homo-topy in the diagram W / / φ W (cid:15) (cid:15) X i ∧ Y jφ i,j (cid:15) (cid:15) Z i + j − r / / Z i + j , and therefore no well-defined pairing φ r . Any construction of spectral sequencepairings that only assumes such compatibility at the level of the (stable) homotopycategory is therefore likely to contain a logical gap.The pairing φ r is evidently natural in i , j and r , in the sense that the square(4.6) H ( X ⋆ )( i − r, i ) ⊗ H ( Y ⋆ )( j − r, j ) H ( Z ⋆ )( i + j − r, i + j ) H ( X ⋆ )( i ′ − r ′ , i ′ ) ⊗ H ( Y ⋆ )( j ′ − r ′ , j ′ ) H ( Z ⋆ )( i ′ + j ′ − r ′ , i ′ + j ′ ) φ r φ r ′ commutes for all integers i , j , r ≥ i ′ , j ′ and r ′ ≥ i ≤ i ′ , i − r ≤ i ′ − r ′ , j ≤ j ′ and j − r ≤ j ′ − r ′ . As we recalled from [Dou59b, § II A] in Definition 4.7,this is the first (SPP I) of two conditions for ( φ r ) r ≥ to define a pairing of (finite)Cartan–Eilenberg systems.We now check condition (SPP III). The pairings φ r can be extended to thecase r = ∞ by letting φ ∞ : H ( X ⋆ )( −∞ , i ) ⊗ H ( Y ⋆ )( −∞ , j ) −→ H ( Z ⋆ )( −∞ , i + j )be defined by homomorphisms φ ∞ : π Gp ( X i ) ⊗ π Gq ( Y j ) −→ π Gp + q ( Z i + j ) . Given G -maps f : Σ U S p → X i ( U ) and g : Σ V S q → Y j ( V ), the homomorphism φ ∞ sends the homotopy classes [ f ] and [ g ] to the homotopy class of the compositeΣ U ⊕ V S p + q ∼ = Σ U S p ∧ Σ V S q f ∧ g −→ X i ( U ) ∧ Y j ( V ) φ i,j ( U,V ) −→ Z i + j ( U ⊕ V ) . In symbols, suppressing U and V : φ ∞ : [ f ] ⊗ [ g ] [ φ i,j ( f ∧ g )] . Recalling that our convention is such that X −∞ = Y −∞ = Z −∞ = ∗ , we note thatthe isomorphism π Gp ( X i ) ∼ = π Gp ( X −∞ → X i ) takes the homotopy class of f to thehomotopy class [ ∗ , f π ] of the pair ∗ : Σ U S p − → X −∞ ( U ) and f π : Σ U D p → X i ( U ).Here π : D p → S p identifies D p /S p − with S p . The pairing φ ∞ then corresponds tothe pairing φ r as defined in the paragraph above, for r = ∞ , with every referenceto X i − r , Y j − r , Z i + j − r and Z i + j − r replaced by ∗ . By the discussion above, it thenalso follows that the extended naturality condition(4.7) H ( X ⋆ )( −∞ , i ) ⊗ H ( Y ⋆ )( −∞ , j ) H ( Z ⋆ )( −∞ , i + j ) H ( X ⋆ )( i − r, i ) ⊗ H ( Y ⋆ )( j − r, j ) H ( Z ⋆ )( i + j − r, i + j ) φ ∞ φ r holds for the pairings φ r with 1 ≤ r ≤ ∞ . This is condition (SPP III) fromDefinition 4.11.4.6. Pairings of Cartan–Eilenberg systems, II. Having proved (SPP I) and(SPP III), we now turn to the second condition (SPP II) from [Dou59b, § II A].Recall that it says that, for ( φ r ) r ≥ to define a pairing of Cartan–Eilenberg systems,we want the Leibniz rule(4.8) ∂φ r = φ ( ∂ ⊗ η ) + φ ( η ⊗ ∂ )to hold. That is, we want the composite H ( X ⋆ )( i − r, i ) ⊗ H ( Y ⋆ )( j − r, j ) φ r −→ H ( Z ⋆ )( i + j − r, i + j ) ∂ −→ H ( Z ⋆ )( i + j − r − , i + j − r ) MULTIPLICATIVE TATE SPECTRAL SEQUENCE 63 to be equal to the sum of the composite homomorphisms H ( X ⋆ )( i − r, i ) ⊗ H ( Y ⋆ )( j − r, j ) ∂ ⊗ η −→ H ( X ⋆ )( i − r − , i − r ) ⊗ H ( Y ⋆ )( j − , j ) φ −→ H ( Z ⋆ )( i + j − r − , i + j − r )and H ( X ⋆ )( i − r, i ) ⊗ H ( Y ⋆ )( j − r, j ) η ⊗ ∂ −→ H ( X ⋆ )( i − , i ) ⊗ H ( Y ⋆ )( j − r − , j − r ) φ −→ H ( Z ⋆ )( i + j − r − , i + j − r ) . Here η : H ( X ⋆ )( i − r, i ) −→ H ( X ⋆ )( i − , i ) η : H ( Y ⋆ )( j − r, j ) −→ H ( Y ⋆ )( j − , j )denote the natural maps. Regarding signs in the Leibniz rule, we recall the con-vention that( ∂ ⊗ x ⊗ y ) = ∂x ⊗ y and (1 ⊗ ∂ )( x ⊗ y ) = ( − p x ⊗ ∂y , for x ∈ π Gp ( X i − r → X i ) in total degree p of H ( X ⋆ )( i − r, i ).To verify condition (4.8) for a given pairing φ : ( X ⋆ , Y ⋆ ) → Z ⋆ , it follows fromthe naturality of the boundary homomorphisms ∂ , and the case i = i ′ , j = j ′ , r ≥ r ′ = 1 of (4.6), that it suffices to establish the rule(4.9) ∂φ r = φ r ( ∂ ⊗ 1) + φ r (1 ⊗ ∂ ) . Here the left hand side is the composite H ( X ⋆ )( i − r, i ) ⊗ H ( Y ⋆ )( j − r, j ) φ r −→ H ( Z ⋆ )( i + j − r, i + j ) ∂ −→ H ( Z ⋆ )( i + j − r, i + j − r ) , and the right hand side is the sum of the two composite homomorphisms H ( X ⋆ )( i − r, i ) ⊗ H ( Y ⋆ )( j − r, j ) ∂ ⊗ −→ H ( X ⋆ )( i − r, i − r ) ⊗ H ( Y ⋆ )( j − r, j ) φ r −→ H ( Z ⋆ )( i + j − r, i + j − r )and H ( X ⋆ )( i − r, i ) ⊗ H ( Y ⋆ )( j − r, j ) ⊗ ∂ −→ H ( X ⋆ )( i − r, i ) ⊗ H ( Y ⋆ )( j − r, j − r ) φ r −→ H ( Z ⋆ )( i + j − r, i + j − r ) . We shall now show that the identity (4.9) holds for pairings of filtrations of orthog-onal G -spectra. Thereafter we use approximation by mapping telescopes to deducethat the identity holds for pairings of arbitrary sequences, as well. Proposition 4.24. If φ : ( X ⋆ , Y ⋆ ) → Z ⋆ is a pairing of sequences of orthogonal G -spectra, and X ⋆ and Y ⋆ are filtrations, then ∂φ r = φ r ( ∂ ⊗ 1) + φ r (1 ⊗ ∂ ) as homomorphisms H ( X ⋆ )( i − r, i ) ⊗ H ( Y ⋆ )( j − r, j ) −→ H ( Z ⋆ )( i + j − r, i + j − r ) for all integers i, j, r with r ≥ . Proof. In this proof we will, for the same typographical reasons as in Section 4.5,suppress the stabilising representations U and V implicit in the presentation ofelements of π Gp ( X i − r → X i ) and π Gq ( Y j − r → Y j ) by homotopy classes of pairs ( f ′ , f )and ( g ′ , g ) of G -maps. The reader can reconstruct how the diagrams could beembellished with these suspensions and shifts.For each map A → B we have natural maps B → B ∪ CA → B/A to thehomotopy cofibre and cofibre. The right hand map is an equivalence when A → B is an h -cofibration. Applied to the left hand maps in diagram (4.5), this gives us acommutative diagram S p + q − / / ( f ∧ g ) ′ (cid:15) (cid:15) S p + q − ∪ C ( S p − ∧ S q − ) ≃ / / ( f ∧ g ) ′ ∪ C ( f ′ ∧ g ′ ) (cid:15) (cid:15) S p − ∧ S q ∨ S p ∧ S q − f ′ ∧ g ′′ ∨ f ′′ ∧ g ′ (cid:15) (cid:15) W / / φ W (cid:15) (cid:15) W ∪ C ( X i − r ∧ Y j − r ) ( ≃ )Θ / / Φ (cid:15) (cid:15) X i − r ∧ Y j /Y j − r ∨ X i /X i − r ∧ Y j − r Z i + j − r / / Z i + j − r ∪ CZ i + j − r .Here f ′′ : S p −→ X i /X i − r and g ′′ : S q −→ Y j /Y j − r are the quotient maps induced by ( f ′ , f ) and ( g ′ , g ), respectively, and we writeΦ = φ W ∪ Cφ i − r,j − r for brevity. If X ⋆ and Y ⋆ are filtrations, as we assume, then X i − r ∧ Y j − r −→ W is a h -cofibration, so the collapse mapΘ : W ∪ C ( X i − r ∧ Y j − r ) −→ X i − r ∧ Y j /Y j − r ∨ X i /X i − r ∧ Y j − r is an equivalence.The left hand side of Equation (4.9) applied to [ f ′ , f ] ⊗ [ g ′ , g ] is ∂φ r ([ f ′ , f ] ⊗ [ g ′ , g ]) = ∂ [ φ W ( f ∧ g ) ′ , φ i,j ( f ∧ g )]= [ ∗ , φ W ( f ∧ g ) ′ π ] . (4.10)Under the isomorphism π Gp + q − ( Z i + j − r → Z i + j − r ) ∼ = π Gp + q − ( Z i + j − r ∪ CZ i + j − r )this corresponds to the homotopy class of the composite map S p + q − −→ Z i + j − r −→ Z i + j − r ∪ CZ i + j − r in the diagram above. Equivalently, by the commutativity of the diagram, we candescribe it as the homotopy class of the composite map S p + q − −→ S p + q − ∪ C ( S p − ∧ S q − ) ( f ∧ g ) ′ ∪ C ( f ′ ∧ g ′ ) −→ W ∪ C ( X i − r ∧ Y j − r ) Φ −→ Z i + j − r ∪ C ( Z i + j − r ) . Alternatively, we can describe it as the image Φ ∗ ([ a ]) of the homotopy class [ a ] ofthe composite map a : S p + q − −→ S p + q − ∪ C ( S p − ∧ S q − ) ( f ∧ g ) ′ ∪ C ( f ′ ∧ g ′ ) −→ W ∪ C ( X i − r ∧ Y j − r )under the homomorphismΦ ∗ : π Gp + q − ( W ∪ C ( X i − r ∧ Y j − r )) −→ π Gp + q − ( Z i + j − r ∪ CZ i + j − r ) . MULTIPLICATIVE TATE SPECTRAL SEQUENCE 65 We shall confirm that Equation (4.9) holds by writing it in the formΦ ∗ ([ a ]) = Φ ∗ ([ b ]) + ( − p Φ ∗ ([ c ])for some specific classes [ b ] and [ c ], to be defined later, and showing that[ a ] = [ b ] + ( − p [ c ]in π Gp + q − ( W ∪ C ( X i − r ∧ Y j − r )). The latter identity will be confirmed by showingthat Θ ∗ ([ a ]) = Θ ∗ ([ b ]) + ( − p Θ ∗ ([ c ]) , where Θ ∗ is the isomorphismΘ ∗ : π Gp + q − ( W ∪ C ( X i − r ∧ Y j − r )) ( ∼ =) −→ π Gp + q − ( X i − r ∧ Y j /Y j − r ∨ X i /X i − r ∧ Y j − r )induced by the map Θ on homotopy. With this aim in mind, we first note that Θ ∗ ([ a ])is the homotopy class of the composite S p + q − −→ S p + q − ∪ C ( S p − ∧ S q − ) ≃ −→ ( S p − ∧ S q ) ∨ ( S p ∧ S q − ) ( f ′ ∧ g ′′ ) ∨ ( f ′′ ∧ g ′ ) −→ X i − r ∧ Y j /Y j − r ∨ X i /X i − r ∧ Y j − r , again by commutativity of the above diagram. Checking orientations in the bound-ary of D p ∧ D q , the composition of all but the last map in the displayed maphas degree +1 when projected to S p − ∧ S q , and degree ( − p when projectedto S p ∧ S q − . Hence Θ ∗ ([ a ]) is the sum of the homotopy class of the composite S p − ∧ S q f ′ ∧ g ′′ −→ X i − r ∧ Y j /Y j − r in −→ X i − r ∧ Y j /Y j − r ∨ X i /X i − r ∧ Y j − r (4.11)and ( − p times the homotopy class of the composite S p ∧ S q − f ′′ ∧ g ′ −→ X i /X i − r ∧ Y j − r in −→ X i − r ∧ Y j /Y j − r ∨ X i /X i − r ∧ Y j − r . (4.12)The first term of the right hand side of Equation (4.9) applied to [ f ′ , f ] ⊗ [ g ′ , g ]is(4.13) φ r ( ∂ ⊗ f ′ , f ] ⊗ [ g ′ , g ]) = φ r ([ ∗ , f ′ π ] ⊗ [ g ′ , g ]) . Unravelling the definition of φ r , see the discussion in Section 4.5 for more details,we form the commutative diagram S p − ∧ S q − / / ( ( PPPPPPPPPPPP (cid:15) (cid:15) D p − ∧ S q − ( ( PPPPPPPPPPPP f ′ π ∧ g ′ (cid:15) (cid:15) S p − ∧ D q / / (cid:15) (cid:15) D p − ∧ D qf ′ π ∧ g (cid:15) (cid:15) ∗ / / ' ' PPPPPPPPPPPPPPPP (cid:15) (cid:15) X i − r ∧ Y j − r ' ' ❖❖❖❖❖❖❖❖❖❖❖❖ = (cid:15) (cid:15) ∗ / / (cid:15) (cid:15) X i − r ∧ Y j = (cid:15) (cid:15) X i − r ∧ Y j − r / / ' ' PPPPPPPPPPP φ i − r,j − r (cid:15) (cid:15) X i − r ∧ Y j − r ' ' ❖❖❖❖❖❖❖❖❖❖❖❖ φ i − r,j − r (cid:15) (cid:15) X i − r ∧ Y j / / φ i − r,j ' ' ❖❖❖❖❖❖❖❖❖❖❖❖ X i − r ∧ Y jφ i − r,j (cid:15) (cid:15) Z i + j − r / / Z i + j − r / / Z i + j − r . We also introduce the pushouts S p + q − = S p − ∧ D q ∪ S p − ∧ S q − D p − ∧ S q − and U = X i − r ∧ Y j ∪ X i − r ∧ Y j − r X i − r ∧ Y j − r mapping to D p − ∧ D q ∼ = D p + q − and X i − r ∧ Y j , respectively. This leads to thecommutative diagram S p + q − / / ∗∪ f ′ π ∧ g ′ (cid:15) (cid:15) D p + q − f ′ π ∧ g (cid:15) (cid:15) X i − r ∧ Y j − r / / (cid:15) (cid:15) φ i − r,j − r (cid:28) (cid:28) X i − r ∧ Y j = (cid:15) (cid:15) U / / φ U (cid:15) (cid:15) X i − r ∧ Y jφ i − r,j (cid:15) (cid:15) $ $ ■■■■■■■■■ W φ W z z ✉✉✉✉✉✉✉✉✉✉ Z i + j − r / / Z i + j − r . The class in π Gp + q − ( Z i + j − r → Z i + j − r ) described in Equation (4.13) is representedvisually as the big rectangle in the diagram, that is, by the pair of maps φ i − r,j − r ( ∗ ∪ f ′ π ∧ g ′ ) : S p + q − → Z i + j − r φ i − r,j ( f ′ π ∧ g ) : D p + q − → Z i + j − r . MULTIPLICATIVE TATE SPECTRAL SEQUENCE 67 We can extend the diagram to the right, as follows, D p + q − / / f ′ π ∧ g (cid:15) (cid:15) D p + q − ∪ CS p + q − ≃ / / f ′ π ∧ g ∪ C ( ∗∪ f ′ π ∧ g ′ ) (cid:15) (cid:15) S p + q − f ′ ∧ g ′′ (cid:15) (cid:15) X i − r ∧ Y j / / (cid:15) (cid:15) X i − r ∧ ( Y j ∪ CY j − r ) ( ≃ ) / / (cid:15) (cid:15) X i − r ∧ Y j /Y j − r in (cid:15) (cid:15) W / / φ W (cid:15) (cid:15) W ∪ C ( X i − r ∧ Y j − r ) ( ≃ )Θ / / Φ (cid:15) (cid:15) X i − r ∧ Y j /Y j − r ∨ X i /X i − r ∧ Y j − r Z i + j − r / / Z i + j − r ∪ CZ i + j − r where the maps marked ( ≃ ) are equivalences by our assumption that X i − r → X i and Y j − r → Y j are h -cofibrations. Under the isomorphism π Gp + q − ( Z i + j − r → Z i + j − r ) ∼ = π Gp + q − ( Z i + j − r ∪ CZ i + j − r ) the class described in Equation (4.13) isgiven by the composite S p + q − ≃ −→ D p + q − ∪ CS p + q − −→ X i − r ∧ ( Y j ∪ CY j − r ) −→ W ∪ C ( X i − r ∧ Y j − r ) Φ −→ Z i + j − r ∪ CZ i + j − r , where the first map is a homotopy inverse to the collapse map. This is the im-age Φ ∗ ([ b ]) of the homotopy class [ b ] of the composite map b : S p + q − ≃ −→ D p + q − ∪ CS p + q − −→ W ∪ C ( X i − r ∧ Y j − r ) . Since the composite S p + q − ≃ −→ D p + q − ∪ CS p + q − ≃ −→ S p + q − ∼ = S p − ∧ S q is homotopic to the identity, Θ ∗ ([ b ]) is the homotopy class of the map (4.11). Thatis, it is the image of [ f ′ ∧ g ′′ ] under the inclusion (in ) ∗ .The second term of the right hand side of (4.9) applied to [ f ′ , f ] ⊗ [ g ′ , g ] is(4.14) φ r (1 ⊗ ∂ )([ f ′ , f ] ⊗ [ g ′ , g ]) = ( − p φ r ([ f ′ , f ] ⊗ [ ∗ , g ′ π ]) . By a similar analysis as for the first term of the sum, the class φ r ([ f ′ , f ] ⊗ [ ∗ , g ′ π ])in π Gp + q − ( Z i + j − r → Z i + j − r ) is represented by a pair of maps φ i − r,j − r ( f ′ ∧ g ′ π ∪ ∗ ) : S p + q − −→ Z i + j − r φ i,j − r ( f ∧ g ′ π ) : D p + q − −→ Z i + j − r , where S p + q − is the boundary of D p + q − ∼ = D p ∧ D q − . The corresponding class in π Gp + q − ( Z i + j − r ∪ CZ i + j − r ) is the image Φ ∗ ([ c ]) under Φ ∗ of the homotopy class [ c ]of the composite map c : S p + q − ≃ −→ D p + q − ∪ CS p + q − −→ W ∪ C ( X i − r ∧ Y j − r ) . The class Θ ∗ ([ c ]) is then the homotopy class of the map (4.12). That is, it is theimage of [ f ′′ ∧ g ′ ] under (in ) ∗ .Summarising, we have now defined classes [ a ], [ b ], and [ c ] in π Gp + q − ( W ∪ C ( X i − r ∧ Y j − r )) satisfying Θ ∗ ([ a ]) = Θ ∗ ([ b ]) + ( − p Θ ∗ ([ c ]) . Since Θ ∗ is an isomorphism, we deduce that[ a ] = [ b ] + ( − p [ c ] and Φ ∗ ([ a ]) = Φ ∗ ([ b ]) + ( − p Φ ∗ ([ c ]) . Since Φ ∗ ([ a ]), Φ ∗ ([ b ]) and ( − p Φ ∗ ([ c ]) are the three parts in Equation (4.9) evalu-ated at [ f ′ , f ] ⊗ [ g ′ , g ], and [ f ′ , f ] and [ g ′ , g ] were arbitrarily chosen, it follows thatEquation (4.9) holds whenever X ⋆ and Y ⋆ are filtrations. (cid:3) We now extend the result above to all pairings of sequences. Proposition 4.25. If φ : ( X ⋆ , Y ⋆ ) → Z ⋆ is a pairing of sequences of orthogonal G -spectra, then ∂φ r = φ r ( ∂ ⊗ 1) + φ r (1 ⊗ ∂ ) as homomorphisms H ( X ⋆ )( i − r, i ) ⊗ H ( Y ⋆ )( j − r, j ) −→ H ( Z ⋆ )( i + j − r, i + j − r ) for all integers i, j, r with r ≥ .Proof. Let T ( φ ) : ( T ⋆ ( X ) , T ⋆ ( Y )) → T ⋆ ( Z ) be the pairing of filtrations defined as inthe proof of Lemma 4.21. The equivalence ǫ : T ⋆ ( X ) → X ⋆ and its analogues for Y ⋆ and Z ⋆ are compatible with the pairings. Hence we have commutative diagramswith vertical isomorphisms H ( T ⋆ ( X ))( i − r, i ) ∂ / / ǫ ∼ = (cid:15) (cid:15) H ( T ⋆ ( X ))( i − r, i − r ) ǫ ∼ = (cid:15) (cid:15) H ( X ⋆ )( i − r, i ) ∂ / / H ( X ⋆ )( i − r, i − r ) , together with its analogues for Y ⋆ and Z ⋆ , and H ( T ⋆ ( X ))( i − r, i ) ⊗ H ( T ⋆ ( Y ))( j − r, j ) T ( φ ) r / / ǫ ⊗ ǫ ∼ = (cid:15) (cid:15) H ( T ⋆ ( Z ))( i + j − r, i + j ) ǫ ∼ = (cid:15) (cid:15) H ( X ⋆ )( i − r, i ) ⊗ H ( Y ⋆ )( j − r, j ) φ r / / H ( Z ⋆ )( i + j − r, i + j ) ,for all r ≥ 1. By Proposition 4.24 applied to the pairing of filtrations T ( φ ) we knowthat ∂T ( φ ) r = T ( φ ) r ( ∂ ⊗ 1) + T ( φ ) r (1 ⊗ ∂ )as homomorphisms H ( T ⋆ ( X ))( i − r, i ) ⊗ H ( T ⋆ ( Y ))( j − r, j ) −→ H ( T ⋆ ( Z ))( i + j − r, i + j − r ) . In view of the vertical isomorphisms ǫ , this implies that ∂φ r = φ r ( ∂ ⊗ 1) + φ r (1 ⊗ ∂ )as homomorphisms H ( X ⋆ )( i − r, i ) ⊗ H ( Y ⋆ )( j − r, j ) → H ( Z ⋆ )( i + j − r, i + j − r ). (cid:3) This finishes the goal we set out for ourselves at the start of Section 4.5, namelyto prove that a pairing of sequences gives rise to a pairing of the resulting Cartan–Eilenberg systems. Let us phrase this conclusion in a theorem, so that we can referback to it when needed. Theorem 4.26. A pairing φ : ( X ⋆ , Y ⋆ ) → Z ⋆ of sequences gives rise to a pairing φ : ( H ( X ⋆ ) , H ( Y ⋆ )) → H ( Z ⋆ ) of the associated Cartan–Eilenberg systems, in thesense of Definition 4.11.Proof. The proof of (SPP I) and (SPP III) is the content of Section 4.5 and theproof of (SPP II) is the content of the present section. (cid:3) This directly gives us the following consequence for the associated spectral se-quences. MULTIPLICATIVE TATE SPECTRAL SEQUENCE 69 Theorem 4.27. A pairing φ : ( X ⋆ , Y ⋆ ) → Z ⋆ of sequences of orthogonal G -spectragives rise to a pairing φ : ( E ∗ ( X ⋆ ) , E ∗ ( Y ⋆ )) → E ∗ ( Z ⋆ ) , in the sense of Defini-tion 4.9. Explicitly, we have access to a collection of homomorphisms φ r : E r ( X ⋆ ) ⊗ E r ( Y ⋆ ) −→ E r ( Z ⋆ ) for all r ≥ , such that: (1) The Leibniz rule d r φ r = φ r ( d r ⊗ 1) + φ r (1 ⊗ d r ) holds as an equality of homomorphisms E ri ( X ⋆ ) ⊗ E rj ( Y ⋆ ) −→ E ri + j − r ( Z ⋆ ) for all i, j ∈ Z and r ≥ . (2) The diagram E r +1 ( X ⋆ ) ⊗ E r +1 ( Y ⋆ ) E r +1 ( Z ⋆ ) H ( E r ( X ⋆ ) ⊗ E r ( Y ⋆ )) H ( E r ( Z ⋆ )) φ r ∼ = H ( φ r +1 ) commutes for all r ≥ .Moreover, the induced pairing φ ∗ on filtered abutments is compatible with thepairing φ ∞ of E ∞ -pages in the sense of Proposition 4.12. Explicitly, the diagram F i A ∞ ( X ⋆ ) F i − A ∞ ( X ⋆ ) ⊗ F j A ∞ ( Y ⋆ ) F j − A ∞ ( Y ⋆ ) ¯ φ ∗ / / β ⊗ β (cid:15) (cid:15) F i + j A ∞ ( Z ⋆ ) F i + j − A ∞ ( Z ⋆ ) (cid:15) (cid:15) β (cid:15) (cid:15) E ∞ i ( X ⋆ ) ⊗ E ∞ j ( Y ⋆ ) φ ∞ / / E ∞ i + j ( Z ⋆ ) commutes, for all i, j ∈ Z . Here the abutments are given as A ∞ ( X ⋆ ) ∼ = π G ∗ Tel( X ⋆ ) A ∞ ( Y ⋆ ) ∼ = π G ∗ Tel( Y ⋆ ) A ∞ ( Z ⋆ ) ∼ = π G ∗ Tel( Z ⋆ ) , and are filtered by the images F i A ∞ ( X ⋆ ) = im( π G ∗ ( X i ) −→ A ∞ ( X ⋆ )) F j A ∞ ( Y ⋆ ) = im( π G ∗ ( Y j ) −→ A ∞ ( Y ⋆ )) F k A ∞ ( Z ⋆ ) = im( π G ∗ ( Z k ) −→ A ∞ ( Z ⋆ )) , respectively.Proof. This follows from combining Theorem 4.26 with Theorem 4.10 and Propo-sition 4.12. (cid:3) Corollary 4.28. If ( X ⋆ , φ ) is a multiplicative sequence of orthogonal G -spectra,then the associated spectral sequence ( E ( X ⋆ ) , d ) is multiplicative with multiplicativeabutment. The convolution product. Given two sequences X ⋆ and Y ⋆ of orthogonal G -spectra there is an initial pairing ι : ( X ⋆ , Y ⋆ ) → Z ⋆ , where the sequence Z ⋆ is givenat each level by Z k = colim i + j ≤ k X i ∧ Y j , with the canonical G -maps Z k − → Z k between them. We call this sequence Z ⋆ the (Day) convolution product of X ⋆ and Y ⋆ , and write Z ⋆ = ( X ∧ Y ) ⋆ . The universal pairing ι : ( X ⋆ , Y ⋆ ) → ( X ∧ Y ) ⋆ has components ι i,j : X i ∧ Y j −→ ( X ∧ Y ) i + j , each given by a structure map to the colimit. Per the discussion of Section 4.5, theuniversal pairing ι : ( X ⋆ , Y ⋆ ) → ( X ∧ Y ) ⋆ induces homomorphisms ι r : π Gp ( X i − r → X i ) ⊗ π Gq ( Y j − r → Y j ) −→ π Gp + q (( X ∧ Y ) i + j − r → ( X ∧ Y ) i + j )for r ≥ 1, and Theorem 4.27 shows that the pairing ι extends to a pairing ι : E ∗ ( X ⋆ ) ⊗ E ∗ ( Y ⋆ ) → E ∗ (( X ∧ Y ) ⋆ ) of spectral sequences, in such a way that the induced pair-ing on filtered abutments¯ ι ∗ : F i π G ∗ Tel( X ⋆ ) F i − π G ∗ Tel( X ⋆ ) ⊗ F j π G ∗ Tel( Y ⋆ ) F j − π G ∗ Tel( Y ⋆ ) −→ F i + j π G ∗ Tel(( X ∧ Y ) ⋆ ) F i + j − π G ∗ Tel(( X ∧ Y ) ⋆ )is compatible with the induced pairing on E ∞ -pages. Remark . The colimit defining Z k can equally well be calculated over the cofinalsubcategory of pairs ( i, j ) ∈ Z with k − ≤ i + j ≤ k , i.e., as the colimit of thezig-zag diagram: . . . / / X i − ∧ Y k − i +1 X i − ∧ Y k − i / / O O X i ∧ Y k − i X i ∧ Y k − i − / / O O . . . If ( X ⋆ , φ ) and ( Y ⋆ , ψ ) are multiplicative sequences of orthogonal G -spectra, thenthe convolution product (( X ∧ Y ) ⋆ , φ ∧ ψ ) is a multiplicative sequence as well. Here,the component ( φ ∧ ψ ) i,j : ( X ∧ Y ) i ∧ ( X ∧ Y ) j −→ ( X ∧ Y ) i + j is defined as the colimit over i + i ≤ i and j + j ≤ j of the composite maps X i ∧ Y i ∧ X j ∧ Y j ∧ τ ∧ −→ X i ∧ X j ∧ Y i ∧ Y j φ i ,j ∧ ψ i ,j −→ X i + j ∧ Y i + j ι i j ,i j −→ ( X ∧ Y ) i + j + i + j −→ ( X ∧ Y ) i + j . Lemma 4.30. If ( X ⋆ , φ ) and ( Y ⋆ , ψ ) are multiplicative sequences, then the homo-morphism ι : E ( X ⋆ ) ⊗ E ( Y ⋆ ) → E (( X ∧ Y ) ⋆ ) is multiplicative, in the sense thatthe diagram E ( X ⋆ ) ⊗ E ( Y ⋆ ) ⊗ E ( X ⋆ ) ⊗ E ( Y ⋆ ) E (( X ∧ Y ) ⋆ ) ⊗ E (( X ∧ Y ) ⋆ ) E ( X ⋆ ) ⊗ E ( X ⋆ ) ⊗ E ( Y ⋆ ) ⊗ E ( Y ⋆ ) E ( X ⋆ ) ⊗ E ( Y ⋆ ) E (( X ∧ Y ) ⋆ ) ι ⊗ ι ⊗ τ ⊗ ∼ = ( φ ∧ ψ ) φ ⊗ ψ ι commutes. MULTIPLICATIVE TATE SPECTRAL SEQUENCE 71 Proof. Let us write θ = φ ∧ ψ for brevity. The diagrams X i ∧ Y i ∧ X j ∧ Y j ι i ,i ∧ ι j ,j / / ∧ τ ∧ ∼ = (cid:15) (cid:15) ( X ∧ Y ) i + i ∧ ( X ∧ Y ) j + j θ i i ,j j (cid:15) (cid:15) X i ∧ X j ∧ Y i ∧ Y j φ i ,j ∧ ψ i ,j (cid:15) (cid:15) X i + j ∧ Y i + j ι i j ,i j / / ( X ∧ Y ) i + j + i + j commute, and are compatible, for all i , i , j and j . This implies that thecomposite homomorphism H ( X ⋆ )( i − r, i ) ⊗ H ( Y ⋆ )( i − r, i ) ⊗ H ( X ⋆ )( j − r, j ) ⊗ H ( Y ⋆ )( j − r, j ) ι r ⊗ ι r −→ H (( X ∧ Y ) ⋆ )( i + i − r, i + i ) ⊗ H (( X ∧ Y ) ⋆ )( j + j − r, j + j ) θ r −→ H (( X ∧ Y ) ⋆ )( i + i + j + j − r, i + i + j + j )is equal to the composite homomorphism H ( X ⋆ )( i − r, i ) ⊗ H ( Y ⋆ )( i − r, i ) ⊗ H ( X ⋆ )( j − r, j ) ⊗ H ( Y ⋆ )( j − r, j ) ∼ = −→ H ( X ⋆ )( i − r, i ) ⊗ H ( X ⋆ )( j − r, j ) ⊗ H ( Y ⋆ )( i − r, i ) ⊗ H ( Y ⋆ )( j − r, j ) φ r ⊗ ψ r −→ H ( X ⋆ )( i + j − r, i + j ) ⊗ H ( Y ⋆ )( i + j − r, i + j ) ι r −→ H (( X ∧ Y ) ⋆ )( i + j + i + j − r, i + j + i + j )for each r ≥ 1, where the homomorphisms ι r , φ r , ψ r and θ r are defined as inSection 4.5. For r = 1, this gives the claim of the lemma. (cid:3) Note that for general sequences X ⋆ and Y ⋆ we typically have no homotopicalcontrol of their convolution product. However, if both X ⋆ and Y ⋆ are filtrations,then we can view each X i ∧ Y j as a subspectrum ofcolim i X i ∧ colim j Y j = [ i X i ∧ [ j Y j and their colimit for i + j ≤ k can be formed as the union( X ∧ Y ) k = [ i + j = k X i ∧ Y j . Proposition 4.31. If the sequences X ⋆ and Y ⋆ are filtrations, then their convolu-tion product ( X ∧ Y ) ⋆ is a filtration.Proof. We must show that each map( X ∧ Y ) k − −→ ( X ∧ Y ) k is a strong h -cofibration. This is the colimit of a sequence of maps, each of whichis the cobase change of a pushout-product map X i − ∧ Y j ∪ X i ∧ Y j − −→ X i ∧ Y j with i + j = k , where the pushout is formed over X i − ∧ Y j − . By assumption X i − → X i and Y j − → Y j are strong h -cofibrations, so the conclusion followsimmediately from Theorem 4.17. (cid:3) In the special case when two arbitrary sequences X ⋆ and Y ⋆ are first replaced withequivalent filtrations T ⋆ ( X ) and T ⋆ ( Y ), we can give the following alternative, moreexplicit, argument for why the resulting convolution product is always a filtration. Lemma 4.32. For any two sequences X ⋆ and Y ⋆ of orthogonal G -spectra, theconvolution product ( T ( X ) ∧ T ( Y )) ⋆ is a filtration.Proof. In degree k , ( T ( X ) ∧ T ( Y )) k = [ i + j = k T i ( X ) ∧ T j ( Y ) . This is the subspectrum of Tel( X ⋆ ) ∧ Tel( Y ⋆ ) with telescope coordinates x and y satisfying ⌈ x ⌉ + ⌈ y ⌉ ≤ k . Here ⌈ x ⌉ denotes the least integer i with x ≤ i . Theinclusion ( T ( X ) ∧ T ( Y )) k − → ( T ( X ) ∧ T ( Y )) k is then the composite of a sequenceof cobase changes of maps of the form i : A −→ B ∪ A A ∧ I + , with A the double mapping cylinder of the diagram X i − ∧ Y j ←− X i − ∧ Y j − −→ X i ∧ Y j − and B = X i ∧ Y j , for i + j = k . Since each such map i is a strong h -cofibration,so is the structure map in ( T ( X ) ∧ T ( Y )) ⋆ , as claimed. (cid:3) As a consequence of Proposition 4.31, we can write first page of the spectralsequence associated to the convolution product of two filtrations X ⋆ and Y ⋆ as E k (( X ∧ Y ) ⋆ ) = π G ∗ (( X ∧ Y ) k − → ( X ∧ Y ) k ) ∼ = π G ∗ (( X ∧ Y ) k / ( X ∧ Y ) k − ) ∼ = M i + j = k π G ∗ ( X i /X i − ∧ Y j /Y j − )since ( X ∧ Y ) k ( X ∧ Y ) k − ∼ = _ i + j = k X i /X i − ∧ Y j /Y j − . Furthermore, the diagram(4.15) E ( X ⋆ ) ⊗ E ( Y ⋆ ) E (( X ∧ Y ) ⋆ ) E ( X ⋆ ) ⊗ E ( Y ⋆ ) E (( X ∧ Y ) ⋆ ) ι d ⊗ ⊗ d d ι commutes. To proceed we usually need more explicit control of the d -differentialfor ( X ∧ Y ) ⋆ , e.g., by use of (4.15) in situations where ι is surjective.Suppose now that ( X ⋆ , φ ) and ( Y ⋆ , ψ ) are multiplicative sequences, and alsoassume that the former is a filtration. This will be the situation when we filterthe G -Tate construction in Section 6. By Corollary 4.22, the telescopic replace-ment ( T ⋆ ( Y ) , T ( ψ )) is a multiplicative filtration, and by Proposition 4.31 the con-volution product ( X ∧ T ( Y )) ⋆ , φ ∧ T ( ψ )) is then also a multiplicative filtration.Lemma 4.30 shows that ι : E ( X ⋆ ) ⊗ E ( Y ⋆ ) → E (( X ∧ Y ) ⋆ ) is multiplicative, in MULTIPLICATIVE TATE SPECTRAL SEQUENCE 73 the sense that the diagram E ( X ⋆ ) ⊗ E ( T ⋆ ( Y )) ⊗ E ( X ⋆ ) ⊗ E ( T ⋆ ( Y )) E (( X ∧ T ( Y )) ⋆ ) ⊗ E (( X ∧ T ( Y )) ⋆ ) E ( X ⋆ ) ⊗ E ( X ⋆ ) ⊗ E ( T ⋆ ( Y )) ⊗ E ( T ⋆ ( Y )) E ( X ⋆ ) ⊗ E ( T ⋆ ( Y )) E (( X ∧ T ( Y )) ⋆ ) ι ⊗ ι ⊗ τ ⊗ ∼ = ( φ ∧ T ( ψ )) φ ⊗ T ( ψ ) ι commutes for all i, j ∈ Z . In situations where ι is surjective, this gives us algebraiccontrol of the product on E (( X ∧ T ( Y )) ⋆ ) in terms of the products on E ( X ⋆ ) and E ( T ⋆ ( Y )) ∼ = E ( Y ⋆ ). Remark . The results of this section readily generalise to the case of sequencesof R -modules in orthogonal G -spectra, for any fixed commutative orthogonal ringspectrum R . Letting X ⋆ denote a sequence · · · −→ X i − −→ X i −→ X i +1 → · · · of R -module G -spectra and R -module G -maps, the telescope Tel( X ⋆ ) is an R -module G -spectrum, and the Cartan–Eilenberg system ( H, ∂ ), the exact couple ( A, E ),the filtered abutment A ∞ ∼ = π G ∗ Tel( X ⋆ ) and the spectral sequence ( E r , d r ) all livein the category of R ∗ -modules. The telescope filtration and equivalence ǫ : T ⋆ ( X ) −→ X ⋆ also live in the category of R -modules.Given sequences X ⋆ , Y ⋆ and Z ⋆ of R -modules in orthogonal G -spectra, an R -bilinear pairing φ : ( X ⋆ , Y ⋆ ) −→ Z ⋆ consists of compatible R -linear G -maps φ : X i ∧ R Y j −→ Z i + j , where the usual smash product has been replaced with the smash product over R .Such pairings induce R ∗ -module homomorphisms φ r : H ( X ⋆ )( i − r, i ) ⊗ R ∗ H ( Y ⋆ )( j − r, j ) −→ H ( Z ⋆ )( i + j − r, i + j ) , where the usual tensor product has been replaced with the tensor product over R ∗ .The Leibniz rule holds for φ r , so that φ induces an R ∗ -linear pairing of R ∗ -modulespectral sequences φ r : E r ( X ⋆ ) ⊗ R ∗ E r ( Y ⋆ ) −→ E r ( Z ⋆ ) . The corresponding R ∗ -linear pairings ¯ φ ∗ and φ ∞ of the filtration subquotientsand E ∞ -pages are compatible under the R ∗ -module monomorphism β : F i A ∞ F i − A ∞ −→ E ∞ i . The universal R -bilinear pairing ι : ( X ⋆ , Y ⋆ ) → Z ⋆ is given by the R -module convo-lution product Z ⋆ = ( X ∧ R Y ) ⋆ , with( X ∧ R Y ) k = colim i + j ≤ k X i ∧ R Y j . If X ⋆ and Y ⋆ are R -module filtrations, then ( X ∧ R Y ) ⋆ is an R -module filtration.For general R -module sequences X ⋆ and Y ⋆ the diagram (4.15) commutes after replacing ⊗ and ∧ by ⊗ R ∗ and ∧ R , respectively. Finally, if ( X ⋆ , φ ) and ( Y ⋆ , ψ ) aremultiplicative R -module sequences then ι : E ( X ⋆ ) ⊗ R ∗ E ( Y ⋆ ) → E (( X ∧ R Y ) ⋆ )will be multiplicative. This depends on the existence of R -module maps(23) = 1 ∧ τ ∧ X i ∧ R Y i ∧ R X j ∧ R Y j −→ X i ∧ R X j ∧ R Y i ∧ R Y j that are strictly compatible for varying i , j , i and j . This is a point where we usethe assumption that R is strictly commutative, not just homotopy commutative,as an orthogonal ring spectrum.5. The G -homotopy fixed point spectral sequence Given an R -module X of orthogonal G -spectra we can define the G -homotopyfixed points as the genuine fixed points X hG = F ( EG + , X ) G = F R ( R ∧ EG + , X ) G . In this section we construct a spectral sequence E r ∗ , ∗ ( X ) = ⇒ π ∗ ( X hG )with abutment being the homotopy groups of the G -homotopy fixed points of X , forany compact Lie group G . This spectral sequence will be induced by the filtration,covered in Section 5.1, on the free and contractible G -space EG coming from thesimplicial bar construction. In Section 5.2, we show that this spectral sequence ismultiplicative with multiplicative abutment. See Theorem 5.6. Under the assump-tion that R [ G ] ∗ is finitely generated and projective over R ∗ we can algebraicallyidentify the E -page of the G -homotopy fixed point spectral sequence as E ∗ , ∗ ( X ) ∼ = Ext −∗ R [ G ] ∗ ( R ∗ , π ∗ ( X )) , with the multiplicative structure being identified with the cup product on the right-hand side. See Theorem 5.14. Lastly, in Section 5.4 we discuss the relationshipbetween the simplicial skeletal filtration on E T and the often used filtration comingfrom odd-dimensional spheres.5.1. The filtered G -space EG . As always, G is a compact Lie group. We let EG = B ( ∗ , G, G )be the free and contractible (right) G -space obtained by taking the geometric real-ization of the simplicial space[ q ] B q ( ∗ , G, G ) = G q × G , with the usual face and degeneracy maps [May75, § B • ( ∗ , G, G ), so EG is indeed contractible [May72, Prop. 9.8]. We let F i EG be the image of the structure map ∆ i × B i ( ∗ , G, G ) → EG to the geometric real-ization, yielding the following exhaustive filtration [May72, Def. 11.1]: ∅ = F − EG ⊂ F EG ⊂ · · · ⊂ F i − EG ⊂ F i EG ⊂ · · · ⊂ EG . Here, the group G acts freely from the right in each simplicial degree, hence alsoon each term in this filtration. The structure map induces a G -equivariant homeo-morphism Σ i ( G ∧ i ∧ G + ) ∼ = ∆ i /∂ ∆ i ∧ G ∧ i ∧ G + ∼ = F i EG/F i − EG for each i ≥ 0. Each smash power G ∧ i = G ∧ · · · ∧ G (with i copies of G ) is formedwith respect to the base point e ∈ G given by the unit element. MULTIPLICATIVE TATE SPECTRAL SEQUENCE 75 Remark . When G is finite, F i EG gives the i -skeleton of a free G -CW-structureon EG . When G = T = U (1) is the circle group, F i EG gives the 2 i - and 2 i + 1-skeleta of a G -CW structure. Similarly, when G = U = Sp (1) is the 3-sphere, F i EG gives the 4 i -, 4 i + 1-, 4 i + 2- and 4 i + 3-skeleta of a G -CW structure. Forother Lie groups the relationship is more complicated. Hence our filtration willagree with that used by Greenlees and May in [GM95, § 9] when G is finite, be adoubly accelerated version when G = T , and be a quadruply accelerated versionwhen G = U . The two filtrations might be quite different for other compact Liegroups G , though.We give the cartesian product EG × EG the product filtration: F k ( EG × EG ) = [ i + j = k F i EG × F j EG . Note that the diagonal G -map ∆ : EG → EG × EG , sending x to ∆( x ) = ( x, x ), isnot filtration-preserving. However, by [Seg68, Lem. 5.4] or [May72, Lem. 11.15] itis naturally homotopic to a filtration-preserving map D : EG → EG × EG , whichwe call a diagonal approximation for EG . By inspection, both D and the naturalhomotopy ∆ ≃ D are G -equivariant. Subject to this condition, the precise choiceof diagonal approximation will not be important, only its existence. Lemma 5.2. Any diagonal approximation D induces a commutative diagram ofbased G -spaces and G -maps F k EGF k − EG D ′ k / / (cid:15) (cid:15) (cid:15) (cid:15) D ′ i,j ' ' F k ( EG × EG ) F k − ( EG × EG ) pr i,j / / (cid:15) (cid:15) (cid:15) (cid:15) F i EGF i − EG ∧ F j EGF j − EG (cid:15) (cid:15) (cid:15) (cid:15) EGF k − EG D k / / D i,j EG × EGF k − ( EG × EG ) / / EGF i − EG ∧ EGF j − EG for all i + j = k . The G -maps D i,j are compatible for varying i and j , in the sensethat the squares EGF i − EG ∧ EGF j − EG (cid:15) (cid:15) EGF k − EG D i,j o o (cid:15) (cid:15) D i,j / / EGF i − EG ∧ EGF j − EG (cid:15) (cid:15) EGF i EG ∧ EGF j − EG EGF k EG D i +1 ,j o o D i,j +1 / / EGF i − EG ∧ EGF j EG . commute.Proof. This follows from the inclusions D ( F k − EG ) ⊂ F k − ( EG × EG ) ⊂ ( F i − EG × EG ) ∪ ( EG × F j − EG )and the splitting F k EGF k − EG ∼ = _ i + j = k F i EGF i − EG ∧ F j EGF j − EG . (cid:3) G -homotopy fixed points. Let X be an orthogonal G -spectrum. In thissection we will construct a spectral sequence computing the homotopy groups ofthe G -homotopy fixed points of X , that is, the G -fixed points of a fibrant replace-ment of the function spectrum F ( EG + , X ): X hG = F ( EG + , X ) G . To this end, note that the sequence of based G -spaces EG + = EGF − EG → EGF EG → · · · → EGF i − EG → EGF i EG → · · · → ∗ induces a sequence M ⋆ ( X ) = F ( EG/EG − ⋆ − , X )of orthogonal G -spectra. Explicitly, M ⋆ ( X ) is the sequence · · · → F (cid:18) EGF i EG , X (cid:19) → F (cid:18) EGF i − EG , X (cid:19) → . . . · · · → F (cid:18) EGF EG , X (cid:19) → F ( EG + , X ) = F ( EG + , X ) = . . . . Definition 5.3. The spectral sequence ( E r ( X ) , d r ) = ( E r ( M ⋆ ( X )) , d r ) associatedto the sequence M ⋆ ( X ) above is called the G -homotopy fixed point spectral sequence of X .Each map of function spectra F ( EG/F i EG, X ) → F ( EG/F i − EG, X ) is amonomorphism of orthogonal G -spectra, but it is unlikely in general that these mapsare h -cofibrations, so M ⋆ ( X ) need not be a filtration. Since the sequence M ⋆ ( X ) iseventually constant, there is a natural G -equivalence M ∞ ( X ) = Tel( M ⋆ ( X )) ≃ G F ( EG + , X ) . There is also a G -equivalenceholim s M s ( X ) = holim s F (cid:18) EGF − s − EG , X (cid:19) ∼ = F (cid:18) hocolim i EGF i − EG , X (cid:19) ≃ G ∗ , since hocolim i F i − EG ≃ G EG . We conclude that the G -homotopy fixed pointspectral sequence is always conditionally convergent to the abutment A ∞ ( M ⋆ ( X )) ∼ = π G ∗ Tel( M ⋆ ( X )) ∼ = π G ∗ F ( EG + , X ) = π ∗ ( X hG ) . Let us now explicitly compute the E -page of this spectral sequence. Lemma 5.4. The E -page of the G -homotopy fixed point spectral sequence of X isgiven by E − i, ∗ ( M ⋆ ( X )) ∼ = π G − i + ∗ F (cid:18) F i EGF i − EG , X (cid:19) ∼ = π G ∗ F ( G + , F ( G ∧ i , X )) and the differential d − i, ∗ : E − i, ∗ ( X ) → E − i − , ∗ ( X ) is contravariantly induced by the composite G -map F i +1 EGF i EG −→ EGF i EG ≃ EGF i − EG ∪ C (cid:16) F i EGF i − EG (cid:17) −→ Σ F i EGF i − EG . Proof. The cofibre sequence F i EGF i − EG −→ EGF i − EG −→ EGF i EG MULTIPLICATIVE TATE SPECTRAL SEQUENCE 77 of based G -spaces is a homotopy cofibre sequence, hence induces a homotopy fibresequence M − i − ( X ) −→ M − i ( X ) −→ F (cid:18) F i EGF i − EG , X (cid:19) of orthogonal G -spectra. It follows that E − i, ∗ ( X ) ∼ = π G − i + ∗ F (cid:18) F i EGF i − EG , X (cid:19) ∼ = π G − i + ∗ F (Σ i ( G ∧ i ∧ G + ) , X ) ∼ = π G ∗ F ( G + , F ( G ∧ i , X ))for i ≥ 0. The d -differential is the composite of the connecting homomorphism π G − i + ∗ ( M − i − ( X ) → M − i ( X )) ∂ −→ π G − i − ∗ ( M − i − ( X )) ∼ = π G − i − ∗ F (cid:18) EGF i EG , X (cid:19) induced by EGF i EG ≃ EGF i − EG ∪ C (cid:18) F i EGF i − EG (cid:19) −→ Σ F i EGF i − EG , and the homomorphism π G − i − ∗ ( M − i − ( X )) −→ π G − i − ∗ ( M − i − ( X ) → M − i − ( X ))induced by F i +1 EGF i EG −→ EGF i EG . (cid:3) Remark . When G is finite, the spectral sequence E r ( M ⋆ ( X )) agrees with the G -homotopy fixed point spectral sequence associated to a G -equivariant Whitehead(or Postnikov) tower for X . When G = T or U it is an accelerated version ofthe latter spectral sequence. In Theorem 5.14 we will give an algebraic descrip-tion of E ( M ⋆ ( X )) when X is an R -module and R [ G ] ∗ is finitely generated andprojective over R ∗ .The homotopy fixed point construction is a lax symmetric monoidal functor. Tosee this, let µ : X ∧ Y → Z be a pairing of orthogonal G -spectra, and recall thediagonal map ∆ : EG → EG × EG . The associated pairing X hG ∧ Y hG → Z hG isgiven by the composite F ( EG + , X ) G ∧ F ( EG + , Y ) G α −→ F ( EG + ∧ EG + , X ∧ Y ) G (∆ + ) ∗ −→ F ( EG + , X ∧ Y ) Gµ ∗ −→ F ( EG + , Z ) G . From this point of view it is hence relevant to understand how the homotopy fixedpoint spectral sequence interacts with multiplicative structures. First note that themaps D i,j from Lemma 5.2 induce G -maps F (cid:18) EGF i − EG , X (cid:19) ∧ F (cid:18) EGF j − EG , Y (cid:19) α −→ F (cid:18) EGF i − EG ∧ EGF j − EG , X ∧ Y (cid:19) D ∗ i,j −→ F (cid:18) EGF k − EG , X ∧ Y (cid:19) µ ∗ −→ F (cid:18) EGF k − EG , Z (cid:19) for k = i + j . These are compatible for varying i and j , in the sense of Definition 4.20,and so define the components ¯ µ − i, − j of a pairing¯ µ : ( M ⋆ ( X ) , M ⋆ ( Y )) → M ⋆ ( Z )of sequences of orthogonal G -spectra. Theorem 5.6. Let µ : X ∧ Y → Z be a pairing of orthogonal G -spectra. There isthen a pairing ¯ µ r : E r ( M ⋆ ( X )) ⊗ E r ( M ⋆ ( Y )) −→ E r ( M ⋆ ( Z )) of the associated G -homotopy fixed point spectral sequences, and the induced pair-ing ¯ µ ∗ on filtered abutments is compatible with the induced pairing ¯ µ ∞ : E ∞ ( M ⋆ ( X )) ⊗ E ∞ ( M ⋆ ( Y )) → E ∞ ( M ⋆ ( Z )) of E ∞ -pages.Moreover, the pairing ¯ µ of E -pages is contravariantly induced by D ′ i,j : F k EGF k − EG −→ F i EGF i − EG ∧ F j EGF j − EG under the isomorphism of Lemma 5.4, and the pairing ¯ µ ∗ : π ∗ ( X hG ) ⊗ π ∗ ( Y hG ) −→ π ∗ ( Z hG ) equals the pairing induced by X hG ∧ Y hG → Z hG .Proof. In the paragraph before this theorem we noted that a map µ : X ∧ Y → Z of orthogonal G -spectra gives rise to a pairing ¯ µ : ( M ⋆ ( X ) , M ⋆ ( Y )) → M ⋆ ( Z ) ofsequences. By Theorem 4.27 it follows that we have an induced pairing between theassociated spectral sequences, and that the induced pairing ¯ µ ∗ on filtered abutmentsis compatible with the pairing ¯ µ ∞ of E ∞ -pages.Tracing through the definitions shows that the pairing ¯ µ − i, − j of E -pages iscompatible with the pairing induced by D ′ i,j under the isomorphism E − i, ∗ ( M ⋆ ( X )) = π G − i + ∗ ( M − i − ( X ) → M − i ( X )) ∼ = π G − i + ∗ F (cid:18) F i EGF i − EG , X (cid:19) and its analogues for Y and Z .The abutment A ∞ ( M ⋆ ( X )) ∼ = π G ∗ F ( EG + , X ) is filtered by the images F s π G ∗ F ( EG + , X ) = im( π G ∗ F ( EG/EG − s − , X ) −→ π G ∗ F ( EG + , X )) . Note that this exhaustive filtration is constant for s ≥ 0. The pairing ¯ µ ∗ is inducedby the composite map¯ µ , : F ( EG + , X ) ∧ F ( EG + , Y ) α −→ F ( EG + ∧ EG + , X ∧ Y ) D ∗ , −→ F ( EG + , X ∧ Y ) µ ∗ −→ F ( EG + , Z ) . In view of the based G -homotopy ∆ + ≃ D + = D , , it is also induced by thecomposite map F ( EG + , X ) ∧ F ( EG + , Y ) α −→ F ( EG + ∧ EG + , X ∧ Y ) ∆ ∗ + −→ F ( EG + , X ∧ Y ) µ ∗ −→ F ( EG + , Z ) , where ∆ + : EG + → ( EG × EG ) + ∼ = EG + ∧ EG + . (cid:3) MULTIPLICATIVE TATE SPECTRAL SEQUENCE 79 Corollary 5.7. If X is a multiplicative orthogonal G -spectrum, then the G -homotopyfixed point spectral sequence ( E r ( M ⋆ ( X )) , d r ) is a conditionally convergent and mul-tiplicative spectral sequence, with multiplicative abutment π ∗ ( X hG ) . Algebraic description of the E - and E -pages. Under suitable flatnesshypotheses there is an algebraic description of the first two pages of the homo-topy fixed point spectral sequence. Recall from Section 3 that R is our ‘ground’commutative orthogonal ring spectrum. We write R ∗ ( X ) = π ∗ ( R ∧ X )for the associated (reduced) homology theory. We will assume that R [ G ] ∗ is flatover R ∗ , so that R [ G ] ∗ is a cocommutative Hopf algebra over R ∗ , per Lemma 3.2.Let us write E = R ∧ EG + and E i = R ∧ ( F i EG ) + . Each map E i − → E i is a q -cofibration, hence a strong h -cofibration, so that E ⋆ isa filtration · · · −→ E i − −→ E i −→ E i +1 −→ · · · of R -modules in orthogonal G -spectra. Here E i = ∗ for i < 0, and E ∞ = Tel( E ⋆ ) ≃ G E . The R - and G -equivariant collapse map 1 ∧ c : E = R ∧ EG + → R ∧ S = R is a non-equivariant R -equivalence, inducing an R [ G ] ∗ -module isomorphism π ∗ ( E ) ∼ = R ∗ . Definition 5.8. Let ( P ∗ , ∗ , ∂ ) = N B ∗ ( R ∗ , R [ G ] ∗ , R [ G ] ∗ ) denote the normalised barresolution, as defined in Construction 2.24.Explicitly, the normalised bar resolution of the R [ G ] ∗ -module R ∗ is a non-negative chain complex given in homological degree n ≥ P n, ∗ = N B n ( R ∗ , R [ G ] ∗ , R [ G ] ∗ ) = R [ G ] ⊗ n ∗ ⊗ R ∗ R [ G ] ∗ , where R [ G ] ∗ = coker( η : R ∗ → R [ G ] ∗ ) ∼ = ker( ǫ : R [ G ] ∗ → R ∗ )denotes the augmentation (co-)ideal, and R [ G ] ⊗ n ∗ is its n -th tensor power over R ∗ .The boundary ∂ n : P n, ∗ → P n − , ∗ is induced by the alternating sum of face opera-tors n X i =0 ( − i d i for n ≥ 1, with d i = ( ǫ ⊗ ⊗ n for i = 0,1 ⊗ i − ⊗ φ ⊗ ⊗ n − i for 0 < i ≤ n .Note that the simplicial contraction [May72, Prop. 9.8] of B • ( R ∗ , R [ G ] ∗ , R [ G ] ∗ )shows that the augmentation ǫ : P , ∗ = R [ G ] ∗ → R ∗ admits an R ∗ -linear chainhomotopy inverse, so that the augmented chain complex . . . −→ P q, ∗ ∂ q −→ P q − , ∗ → · · · → P , ∗ ∂ −→ P , ∗ ǫ −→ R ∗ −→ P ∗ , ∗ , ∂ ) is a flat R [ G ] ∗ -module resolution of R ∗ . Lemma 5.9. If R [ G ] ∗ is flat over R ∗ , then the ( E , d ) -page of the non-equivarianthomotopy spectral sequence E i, ∗ = π i + ∗ ( E i − → E i ) associated to E ⋆ is isomorphic to ( P ∗ , ∗ , ∂ ) . The edge homomorphism P , ∗ → π ∗ ( E ) ∼ = R ∗ is equal to the augmentation ǫ : R [ G ] ∗ → R ∗ , and makes ( P ∗ , ∗ , ∂ ) a flat R [ G ] ∗ -module resolution of R ∗ . In particular, the spectral sequence collapsesat the E -page, where is it given by E = E ∞ ∼ = R ∗ concentrated in filtration degree i = 0 .Proof. The R -module filtration E ⋆ has an associated R [ G ] ∗ -module spectral se-quence (for non-equivariant homotopy groups) with E -page E i, ∗ = π i + ∗ ( E i − → E i ) ∼ = R i + ∗ (cid:18) F i EGF i − EG (cid:19) and d -differential equal to the composite R i + ∗ (cid:18) F i EGF i − EG (cid:19) ∂ −→ R i − ∗ ( F i − EG + ) −→ R i − ∗ (cid:18) F i − EGF i − EG (cid:19) . By the proof of [Seg68, Prop. 5.1] or [May72, Thm. 11.14], ( E , d ) is the normalizedchain complex associated to the simplicial R [ G ] ∗ -module[ q ] R ∗ ( B q ( ∗ , G, G ) + ) = R ∗ (( G q × G ) + ) . The products R [ G ] ∗ ⊗ R ∗ R [ G ] ∗ ⊗ R ∗ · · · ⊗ R ∗ R [ G ] ∗· −→ π ∗ ( R ∧ G + ∧ R R ∧ G + ∧ R · · · ∧ R R ∧ G + ) ∼ = π ∗ ( R ∧ G + ∧ G + ∧ · · · ∧ G + )induce a homomorphism of simplicial R [ G ] ∗ -modules B • ( R ∗ , R [ G ] ∗ , R [ G ] ∗ ) −→ R ∗ ( B • ( ∗ , G, G )) + ) . Since R [ G ] ∗ is assumed to be flat over R ∗ the products are isomorphisms, sothat ( E , d ) is indeed isomorphic to the normalized chain complex associated tothe simplicial R [ G ] ∗ -module B • ( R ∗ , R [ G ] ∗ , R [ G ] ∗ ). (cid:3) Remark . If R [ G ] ∗ is projective over R ∗ , then R [ G ] ∗ is also R ∗ -projective,and each P q, ∗ is R [ G ] ∗ -projective by Lemma 2.2. It follows that the chain com-plex ( P ∗ , ∗ , ∂ ) is a projective R [ G ] ∗ -module resolution of R ∗ . Moreover, if R [ G ] ∗ is finitely generated over R ∗ , then so is R [ G ] ∗ , and each P q, ∗ is finitely generatedas an R [ G ] ∗ -module. We conclude that ( P ∗ , ∗ , ∂ ) is a projective resolution of finitetype, in this case.To deal with the multiplicative structure of the spectral sequence we introducethe convolution product ( E ∧ R E ) ⋆ . Explicitly, this is given by( E ∧ R E ) k = R ∧ F k ( EG × EG ) + , with filtration subquotients E ∧ R E ) k ( E ∧ R E ) k − ∼ = _ i + j = k E i E i − ∧ R E j E j − . Let in i,j denote the inclusion of the ( i, j )-th summand in this splitting. Lemma 5.11. The ( E , d ) -page of the homotopy spectral sequence associatedto ( E ∧ R E ) ⋆ is isomorphic to the tensor product ( P ∗ , ∗ ⊗ R ∗ P ∗ , ∗ , ∂ ⊗ ⊗ ∂ ) , with the same signs occurring in the boundary as specified in Section 2.2. In par-ticular, this spectral sequence collapses at the E -page, where it is given by E = E ∞ ∼ = R ∗ ⊗ R ∗ R ∗ ∼ = R ∗ MULTIPLICATIVE TATE SPECTRAL SEQUENCE 81 concentrated in filtration degree .Proof. Theorem 4.27 applied to the initial pairing ι : ( E ⋆ , E ⋆ ) → ( E ∧ R E ) ⋆ givesus a pairing ι r : E r ( E ⋆ ) ⊗ R ∗ E r ( E ⋆ ) −→ E r (( E ∧ R E ) ⋆ )of R [ G ] ∗ -module spectral sequences. Since each copy of E ⋆ is a filtration, the pairing ι i,j : P i, ∗ ⊗ R ∗ P j, ∗ = E i, ∗ ( E ⋆ ) ⊗ R ∗ E j, ∗ ( E ⋆ ) −→ E k, ∗ (( E ∧ R E ) ⋆ ) , for r = 1 and i + j = k , is induced by the product P i, ∗ ⊗ R ∗ P j, ∗ · −→ π ∗ (cid:18) E i E i − ∧ R E j E j − (cid:19) and the inclusion in i,j : E i E i − ∧ R E j E j − −→ ( E ∧ R E ) k ( E ∧ R E ) k − . Since R [ G ] ∗ is flat over R ∗ , so that each P i, ∗ is flat over R ∗ , the product is anisomorphism. Adding these together for i + j = k we obtain the degree k part ofan isomorphism of R [ G ] ∗ -module chain complexes ι : P ∗ , ∗ ⊗ R ∗ P ∗ , ∗ ∼ = −→ E ∗ , ∗ (( E ∧ R E ) ⋆ ) . In particular, Theorem 4.27 ensures that the tensor product boundary operator ∂ ⊗ ⊗ ∂ on the left hand side corresponds to the d -differential on the right handside. The calculation of the E -page then follows as in the proof of Proposition 2.28. (cid:3) Lemma 5.12. The diagonal approximation D : EG → EG × EG induces a map offiltrations ∧ D + : E ⋆ → ( E ∧ R E ) ⋆ and a chain map (1 ∧ D + ) : E ( E ⋆ ) −→ E (( E ∧ R E ) ⋆ ) , which corresponds, under the isomorphisms of Lemma 5.9 and Lemma 5.11, toan R [ G ] ∗ -module chain map Ψ : P ∗ , ∗ −→ P ∗ , ∗ ⊗ R ∗ P ∗ , ∗ . In particular, the component Ψ i,j = pr i,j ◦ Ψ k : P k, ∗ → P i, ∗ ⊗ R ∗ P j, ∗ of Ψ k , for k = i + j , is induced by the G -map D ′ i,j of Lemma 5.2 and Theorem 5.6.The chain map Ψ is characterised, uniquely up to chain homotopy equivalence,by the commutative square P ∗ , ∗ Ψ / / ǫ (cid:15) (cid:15) P ∗ , ∗ ⊗ R ∗ P ∗ , ∗ ǫ ⊗ ǫ (cid:15) (cid:15) R ∗ ∼ = / / R ∗ ⊗ R ∗ R ∗ of R [ G ] ∗ -module complexes.Proof. The map of E -pages induced by the diagonal approximation is inducedby 1 ∧ D ′ k , and the ( i, j )-th component in the direct sum splitting of its target canbe recovered by projecting to that summand, which is therefore induced by 1 ∧ D ′ i,j . By naturality of the edge homomorphism, we have a commutative square of R [ G ] ∗ -modules P , ∗ Ψ / / ǫ (cid:15) (cid:15) P , ∗ ⊗ R ∗ P , ∗ ǫ ⊗ ǫ (cid:15) (cid:15) R ∗ ∼ = / / R ∗ ⊗ R ∗ R ∗ .Hence the R [ G ] ∗ -module chain map Ψ : P ∗ , ∗ → P ∗ , ∗ ⊗ R ∗ P ∗ , ∗ is a lift of the iso-morphism R ∗ ∼ = R ∗ ⊗ R ∗ R ∗ . Since ǫ : P ∗ , ∗ → R ∗ is an R [ G ] ∗ -projective com-plex over R ∗ , and ǫ ⊗ ǫ : P ∗ , ∗ ⊗ R ∗ P ∗ , ∗ → R ∗ ⊗ R ∗ R ∗ is a resolution, it followsfrom [ML95, Thm. III.6.1] that such a chain map Ψ exists and is unique up tochain homotopy. (cid:3) We now suppose that X is an R -module in orthogonal G -spectra. There arethen compatible adjunction equivalences F R ( E/E i − , X ) ∼ = F ( EG/F i − EG, X ) = M − i ( X )for all i . The left hand side exhibits M ⋆ ( X ) as a sequence of R -modules in orthog-onal G -spectra, so that the G -homotopy fixed point spectral sequence E r ( M ⋆ ( X ))is a spectral sequence of R ∗ -modules. Theorem 5.6 readily generalizes: If Y and Z are also R -modules in orthogonal G -spectra, and µ : X ∧ R Y → Z is a map in thiscategory, then we obtain a pairing of R ∗ -module spectral sequences¯ µ r : E r ( M ⋆ ( X )) ⊗ R ∗ E r ( M ⋆ ( Y )) −→ E r ( M ⋆ ( Z ))such that the resulting pairing of E ∞ -pages is compatible with the R ∗ -linear pairing¯ µ ∗ : π G ∗ F ( EG + , X ) ⊗ R ∗ π G ∗ F ( EG + , Y ) −→ π G ∗ F ( EG + , Z )of abutments. We can now give algebraic descriptions of the ( E , d )-pages and thepairing ¯ µ , for R [ G ] ∗ projective over R ∗ . Proposition 5.13. Assume that R [ G ] ∗ is projective as an R ∗ -module. There isthen a natural isomorphism E − i, ∗ ( M ⋆ ( X )) ∼ = Hom R [ G ] ∗ ( P i, ∗ , π ∗ ( X )) of R ∗ -modules. Under this isomorphism, the d -differential d − i, ∗ : E − i, ∗ ( M ⋆ ( X )) −→ E − i − , ∗ ( M ⋆ ( X )) corresponds to the boundary in the chain complex, with signs as specified in Sec-tion 2.2. The pairing ¯ µ : E − i, ∗ ( M ⋆ ( X )) ⊗ R ∗ E − j, ∗ ( M ⋆ ( Y )) −→ E − k, ∗ ( M ⋆ ( Z )) with i + j = k is contravariantly induced by the component Ψ i,j : P k, ∗ −→ P i, ∗ ⊗ R ∗ P j, ∗ of the chain map Ψ .Proof. By Lemma 5.4 and adjunction isomorphisms E − i, ∗ ( M ⋆ ( X )) ∼ = π G − i + ∗ F ( F i EG/F i − EG, X ) ∼ = π G − i + ∗ F R ( E i /E i − , X ) . Note that the spectrum appearing in the last term can be written F R ( E i /E i − , X ) ∼ = F ( G + , X ′ ) with X ′ = F R ( R ∧ G ∧ i , X ) ∼ = F ( G ∧ i , X ) . Under our assumption that R [ G ] ∗ is projective, it follows from Proposition 3.6 thatthe natural R ∗ -module homomorphism ω : π G − i + ∗ F R ( E i /E i − , X ) ∼ = −→ Hom R [ G ] ∗ ( R ∗ , π − i + ∗ F R ( E i /E i − , X )) MULTIPLICATIVE TATE SPECTRAL SEQUENCE 83 is an isomorphism. Moreover, since P i, ∗ = π i + ∗ ( E i /E i − ) is projective over R [ G ] ∗ and hence also over R ∗ , it follows that the natural R [ G ] ∗ -module homomorphism π − i + ∗ F R ( E i /E i − , X ) ∼ = −→ Hom R ∗ ( π i + ∗ ( E i /E i − ) , π ∗ ( X )) = Hom R ∗ ( P i, ∗ , π ∗ ( X ))is an isomorphism. Applying the functor Hom R [ G ] ∗ ( R ∗ , − ) this yields an isomor-phismHom R [ G ] ∗ ( R ∗ , π − i + ∗ F R ( E i /E i − , X )) ∼ = −→ Hom R [ G ] ∗ ( R ∗ , Hom R ∗ ( P i, ∗ , π ∗ ( X ))) ∼ = Hom R [ G ] ∗ ( P i, ∗ , π ∗ ( X )) . Composing this chain of R ∗ -module isomorphisms gives the asserted natural iso-morphism.We now identify the d -differential. By Lemma 5.4 again, we have a commutativediagram E − i, ∗ ( M ⋆ ( X )) d − i, ∗ (cid:15) (cid:15) ∼ = / / π G − i + ∗ F R ( E i /E i − , X ) (cid:15) (cid:15) π G − i − ∗ F R ( E/E i , X ) (cid:15) (cid:15) E − i − , ∗ ( M ⋆ ( X )) ∼ = / / π G − i − ∗ F R ( E i +1 /E i , X )of R ∗ -modules. By the naturality of ω in Lemma 3.5 the diagram π G − i + ∗ F R ( E i /E i − , X ) (cid:15) (cid:15) ω ∼ = / / Hom R [ G ] ∗ ( R ∗ , π − i + ∗ F R ( E i /E i − , X )) (cid:15) (cid:15) π G − i − ∗ F R ( E/E i , X ) (cid:15) (cid:15) ω / / Hom R [ G ] ∗ ( R ∗ , π − i − ∗ F R ( E/E i , X )) (cid:15) (cid:15) π G − i − ∗ F R ( E i +1 /E i , X ) ω ∼ = / / Hom R [ G ] ∗ ( R ∗ , π − i − ∗ F R ( E i +1 /E i , X ))commutes. Note that these two diagrams fit together along one edge. We also havea commutative diagram of R [ G ] ∗ -modules π − i + ∗ F R ( E i /E i − , X ) (cid:15) (cid:15) ∼ = / / Hom R ∗ ( P i, ∗ , π ∗ ( X )) (cid:15) (cid:15) Hom( ∂ i +1 , t t π − i − ∗ F R ( E/E i , X ) / / (cid:15) (cid:15) Hom R ∗ ( π i +1+ ∗ ( E/E i ) , π ∗ ( X )) (cid:15) (cid:15) π − i − ∗ F R ( E i +1 /E i , X ) ∼ = / / Hom R ∗ ( P i +1 , ∗ , π ∗ ( X ))since ∂ i +1 : P i +1 , ∗ → P i, ∗ can be calculated by either composite from the left to theright in the diagram π i +1+ ∗ ( E i +1 /E i ) / / ∂ ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘ π i +1+ ∗ ( E/E i ) ∂ ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ ∂ (cid:15) (cid:15) π i + ∗ ( E i ) / / π i + ∗ ( E i /E i − ) . Applying Hom R [ G ] ∗ ( R ∗ , − ) we obtain a commutative diagram of R ∗ -modules, whichfits together with the previous one. Hence the square E − i, ∗ ( M ⋆ ( X )) ∼ = / / d − i, ∗ (cid:15) (cid:15) Hom R [ G ] ∗ ( P i, ∗ , π ∗ ( X )) Hom( ∂ i +1 , (cid:15) (cid:15) E − i − , ∗ ( M ⋆ ( X )) ∼ = / / Hom R [ G ] ∗ ( P i +1 , ∗ , π ∗ ( X ))commutes, as asserted.We now identify the multiplicative structure on the E -page. By Theorem 5.6,the diagram E − i, ∗ ( M ⋆ ( X )) ⊗ R ∗ E − j, ∗ ( M ⋆ ( Y )) ¯ µ / / ∼ = (cid:15) (cid:15) E − k, ∗ ( M ⋆ ( Z )) ∼ = (cid:15) (cid:15) π G − i + ∗ F R ( E i /E i − , X ) ⊗ R ∗ π G − j + ∗ F R ( E j /E j − , Y ) / / π G − k + ∗ F R ( E k /E k − , Z )commutes, where the lower arrow is induced by1 ∧ D ′ i,j : E k /E k − → E i /E i − ∧ R E j /E j − . Since the natural homomorphism ω is monoidal, per Lemma 3.7, the composite π G − i + ∗ F R ( E i /E i − , X ) ⊗ R ∗ π G − j + ∗ F R ( E j /E j − , Y ) −→ π G − k + ∗ F R ( E k /E k − , Z ) ω −→ Hom R [ G ] ∗ ( R ∗ , π − k + ∗ F R ( E k /E k − , Z ))is equal to the composite π G − i + ∗ F R ( E i /E i − , X ) ⊗ R ∗ π G − j + ∗ F R ( E j /E j − , Y ) ω ⊗ ω −→ Hom R [ G ] ∗ ( R ∗ , π − i + ∗ F R ( E i /E i − , X )) ⊗ R ∗ Hom R [ G ] ∗ ( R ∗ , π − j + ∗ F R ( E j /E j − , Y )) α −→ Hom R [ G ] ∗ ( R ∗ , π − i + ∗ F R ( E i /E i − , X ) ⊗ R ∗ π − j + ∗ F R ( E j /E j − , Y )) µ ∗ −→ Hom R [ G ] ∗ ( R ∗ , π − k + ∗ F R ( E k /E k − , Z )) . Note that the final arrow is also induced by 1 ∧ D ′ i,j : E k /E k − → E i /E i − ∧ R E j /E j − . Next, we use the commutative diagram π − i + ∗ F R ( E i /E i − , X ) ⊗ R ∗ π − j + ∗ F R ( E j /E j − , Y ) / / ∼ = (cid:15) (cid:15) π − k + ∗ F R ( E k /E k − , Z ) ∼ = (cid:15) (cid:15) Hom R ∗ ( P i, ∗ , π ∗ ( X )) ⊗ R ∗ Hom R ∗ ( P j, ∗ , π ∗ ( Y )) / / Hom R ∗ ( P k, ∗ , π ∗ ( Z ))of R [ G ] ∗ -modules, where the lower homomorphism is induced by 1 ∧ D ′ i,j . In viewof the isomorphism P i, ∗ ⊗ R ∗ P j, ∗ ∼ = π i + j ( E i /E i − ∧ R E j /E j − ) from the proof ofLemma 5.11, this is the same homomorphism as that induced by Ψ i,j , as definedin Lemma 5.12.Applying the monoidal functor Hom R [ G ] ∗ ( R ∗ , − ), we obtain a commutative squareof R ∗ -modules. Combining these results we have a commutative square E − i, ∗ ( M ⋆ ( X )) ⊗ R ∗ E − j, ∗ ( M ⋆ ( Y )) ¯ µ / / ∼ = (cid:15) (cid:15) E − k, ∗ ( M ⋆ ( Z )) ∼ = (cid:15) (cid:15) Hom R [ G ] ∗ ( P i, ∗ , π ∗ ( X )) ⊗ R ∗ Hom R [ G ] ∗ ( P j, ∗ , π ∗ ( Y )) / / Hom R [ G ] ∗ ( P k, ∗ , π ∗ ( Z )) MULTIPLICATIVE TATE SPECTRAL SEQUENCE 85 where the lower homomorphism is induced by Ψ i,j , meaning that it is equal to thecomposite Hom R [ G ] ∗ ( P i, ∗ , π ∗ ( X )) ⊗ R ∗ Hom R [ G ] ∗ ( P j, ∗ , π ∗ ( Y )) α −→ Hom R [ G ] ∗ ( P i, ∗ ⊗ R ∗ P j, ∗ , π ∗ ( X ) ⊗ R ∗ π ∗ ( Y )) Ψ ∗ i,j −→ Hom R [ G ] ∗ ( P k, ∗ , π ∗ ( X ) ⊗ R ∗ π ∗ ( Y )) µ ∗ −→ Hom R [ G ] ∗ ( P k, ∗ , π ∗ ( Z )) . This is the same as the ( i, j )-component of the chain mapHom R [ G ] ∗ ( P ∗ , ∗ , π ∗ ( X )) ⊗ R ∗ Hom R [ G ] ∗ ( P ∗ , ∗ , π ∗ ( Y )) α −→ Hom R [ G ] ∗ ( P ∗ , ∗ ⊗ R ∗ P ∗ , ∗ , π ∗ ( X ) ⊗ R ∗ π ∗ ( Y )) Ψ ∗ −→ Hom R [ G ] ∗ ( P ∗ , ∗ , π ∗ ( X ) ⊗ R ∗ π ∗ ( Y )) µ ∗ −→ Hom R [ G ] ∗ ( P ∗ , ∗ , π ∗ ( Z ))induced by Ψ. (cid:3) As a direct consequence, we get a description of the E -page of the homotopyfixed point spectral sequence. Theorem 5.14. Let G be a compact Lie group and let R be a commutative or-thogonal ring spectrum. Moreover, let µ : X ∧ R Y → Z be a pairing of R -modulesin orthogonal G -spectra. Assume that R [ G ] ∗ is projective as an R ∗ -module. Thenthere is a natural isomorphism E − i, ∗ ( M ⋆ ( X )) ∼ = Ext iR [ G ] ∗ ( R ∗ , π ∗ ( X )) of R ∗ -modules, for each integer i . The pairing ¯ µ : E − i, ∗ ( M ⋆ ( X )) ⊗ R ∗ E − j, ∗ ( M ⋆ ( Y )) −→ E − i − j, ∗ ( M ⋆ ( Z )) is given by the cup product ⌣ : Ext iR [ G ] ∗ ( R ∗ , π ∗ ( X )) ⊗ R ∗ Ext jR [ G ] ∗ ( R ∗ , π ∗ ( Y )) −→ Ext i + jR [ G ] ∗ ( R ∗ , π ∗ ( Z )) associated to the R [ G ] ∗ -module pairing µ ∗ : π ∗ ( X ) ⊗ R ∗ π ∗ ( Y ) → π ∗ ( Z ) , in Ext overthe Hopf algebra R [ G ] ∗ .Proof. By Proposition 5.13, the first page of the spectral sequence, together withits d -differential, is identified with the R ∗ -chain complex Hom R [ G ] ∗ ( P ∗ , ∗ , π ∗ ( X ))where ( P ∗ , ∗ , ∂ ) is a projective resolution of R ∗ . It follows that the E -page is givenby the homology of this chain complex, which by definition is the graded R ∗ -module E ∗ , ∗ ( X ) ∼ = Ext ∗ R [ G ] ∗ ( R ∗ , π ∗ ( X )) . Let us now identify the multiplication on the E -page with the cup product.Let f : P i, ∗ → π ∗ ( X ) and g : P j, ∗ → π ∗ ( Y ) be (graded) R [ G ] ∗ -module homo-morphisms with f ∂ i +1 = 0 and g∂ j +1 = 0. They correspond to i - and j -cyclesin Hom R [ G ] ∗ ( P ∗ , ∗ , π ∗ ( X )) and Hom R [ G ] ∗ ( P ∗ , ∗ , π ∗ ( Y )), respectively, with homologyclasses [ f ] ∈ E − i, ∗ ( X ) and [ g ] ∈ E − j, ∗ ( Y ). The pairing of E -pages sends [ f ] ⊗ [ g ]to the homology class in E − k, ∗ ( Z ) of the k -cycle given by the composite (graded) R [ G ] ∗ -module homomorphism P k, ∗ Ψ i,j −→ P i, ∗ ⊗ R ∗ P j, ∗ f ⊗ g −→ π ∗ ( X ) ⊗ R ∗ π ∗ ( Y ) µ ∗ −→ π ∗ ( Z ) . The verification that µ ∗ ( f ⊗ g )Ψ i,j is a k -cycle uses the fact that Ψ i,j is a componentof an R [ G ] ∗ -module chain map Ψ : P ∗ , ∗ → P ∗ , ∗ ⊗ R ∗ P ∗ , ∗ , so thatΨ i,j ∂ k +1 = ( ∂ i +1 ⊗ i +1 ,j + (1 ⊗ ∂ j +1 )Ψ i,j +1 . This is the definition of the cup product ⌣ : Ext ∗ R [ G ] ∗ ( R ∗ , π ∗ ( X )) ⊗ R ∗ Ext ∗ R [ G ] ∗ ( R ∗ , π ∗ ( Y )) −→ Ext ∗ R [ G ] ∗ ( R ∗ , π ∗ ( Z ))associated to the pairing µ ∗ . See Section 2.5. (cid:3) Remark . A well-known consequence of the comparison theorem [ML95, Thm. III.6.1]is that Ext iR [ G ] ∗ ( R ∗ , π ∗ ( X )) = H i (Hom R [ G ] ∗ ( P ∗ , ∗ , π ∗ ( X )))can be calculated with any projective R [ G ] ∗ -module resolution P ∗ , ∗ of R ∗ , notnecessarily the one introduced in Definition 5.8. Likewise, by Proposition 2.28, thecup product can be calculated with any R [ G ] ∗ -module chain mapΨ : P ∗ , ∗ −→ P ∗ , ∗ ⊗ R ∗ P ∗ , ∗ lifting R ∗ ∼ = R ∗ ⊗ R ∗ R ∗ , not necessarily the one induced by a given diagonal ap-proximation D . Example . When G is finite, R [ G ] ∗ = R ∗ [ G ] ∼ = Z [ G ] ⊗ Z R ∗ , any projective Z [ G ]-module resolution Q ∗ of Z induces up to a projective R ∗ [ G ]-module resolution P ∗ , ∗ = Q ∗ ⊗ Z R ∗ of R ∗ , and any Z [ G ]-module diagonal approxi-mation Ψ : Q ∗ → Q ∗ ⊗ Z Q ∗ induces up to an R ∗ [ G ]-module diagonal approximationΨ ⊗ P ∗ , ∗ = Q ∗ ⊗ Z R ∗ → Q ∗ ⊗ Z Q ∗ ⊗ Z R ∗ ∼ = P ∗ , ∗ ⊗ R ∗ P ∗ , ∗ . Hence there is anatural isomorphismExt iR ∗ [ G ] ( R ∗ , π ∗ ( X )) ∼ = Ext i Z [ G ] ( Z , π ∗ ( X )) = H i ( G, π ∗ ( X ))identifying the E -page of the G -homotopy fixed point spectral sequence with thegroup cohomology of the G -module π ∗ ( X ), and this identification is compatiblewith the cup product structure on both sides. Example . When G = T is the circle group, we showed in Proposition 3.3 that R [ T ] ∗ = R ∗ [ s ] / ( s = ηs ) and R [ T ] ∗ = R ∗ { s } . As we discussed in Definition 5.8, the normalized bar resolution gives a (minimal)resolution P ∗ , ∗ = N B ∗ ( R ∗ , R [ T ] ∗ , R [ T ] ∗ ) of R ∗ , with P i, ∗ = R [ T ] ⊗ i ∗ ⊗ R ∗ R [ T ] ∗ ∼ = R [ T ] ∗ { ¯ p i } . Here ¯ p i = s ⊗· · ·⊗ s ⊗ s | . . . | s ]1 has homological degree i , internal degree | ¯ p i | = i and total degree k p i k = 2 i , for i ≥ 0. The differential is given by ∂ i (¯ p i ) = ¯ p i − (( i − η + ( − i s )for i ≥ 1. This means that P ∗ , ∗ is not strictly equal to the resolution P ∗ specified atthe beginning of Section 2.6, with P i = R [ T ] ∗ { p i } and ∂ i ( p i ) = p i − ( s +( i − η ), dueto the sign ( − i before the contribution from the last face operator. However, thetwo resolutions are isomorphic, by way of the chain map sending ¯ p i to ( − i ( i +1) / p i for each i ≥ 0. Even without this isomorphism, we are free to use P ∗ to calculateExt ∗ R [ T ] ∗ ( R ∗ , π ∗ ( X )) as the homology of Hom R [ T ] ∗ ( P ∗ , π ∗ ( X )), and that calculationwas essentially done in Section 2.6. For each b ≥ x f b · x := (cid:18) p b xp b s xs (cid:19) defines a bijection Σ − b π ∗ ( X ) ∼ = Hom R ∗ [ G ] (Hom( P b , π ∗ ( X ))), and the boundary onsuch an element is given by ∂ v ( f b · x ) = ( − ( − | x | f b +1 · xs for b ≥ − ( − | x | f b +1 · x ( s + η ) for b ≥ MULTIPLICATIVE TATE SPECTRAL SEQUENCE 87 Hence we can compute the homology asExt bR [ T ] ∗ ( R ∗ , π ∗ ( X )) ∼ = f · ker( s : π ∗ ( X ) → π ∗ +1 ( X )) for b = 0, f b · ker( s + η : π ∗ ( X ) → π ∗ +1 ( X ))im( s : π ∗− ( X ) → π ∗ ( X )) for b ≥ f b · ker( s : π ∗ ( X ) → π ∗ +1 ( X ))im( s + η : π ∗− ( X ) → π ∗ ( X )) for b ≥ P ∗ → P ∗ ⊗ R ∗ P ∗ lifting the identity on R ∗ . Such a map is given in Lemma 2.44, so thatwe can compute the cup product as f b · x ⌣ f b · y = f b + b · x ⊗ y . Please compare with Lemma 2.46. Formally writing the class of f b · x as t b · x , we canthen express Ext R [ T ] ∗ ( R ∗ , π ∗ ( X )) as the homology of the differential graded R [ T ] ∗ -module π ∗ ( X )[ t ]with differential given by d ( x ) = txs and d ( t ) = t η , for x ∈ π ∗ ( X ). Here, t hashomological degree − 1, internal degree | t | = − k t k = − The odd spheres filtration. In the important case G = T , the circle actionon odd-dimensional spheres provides a pleasant alternative model for EG . Foreach i ≥ S ( i C ) = S i − be the unit sphere in i C = C i , with the standard,free T -action. We obtain an exhaustive filtration ∅ ⊂ S ( C ) ⊂ · · · ⊂ S ( i C ) ⊂ S (( i + 1) C ) ⊂ · · · ⊂ S ( ∞ C )of free T -spaces. Here S (( i + 1) C ) is obtained from S ( i C ) by attaching a free T -equivariant 2 i -cell D i × T along the group action map S i − × T ∼ = S ( i C ) × T → S ( i C ) , so that S (( i + 1) C ) is the 2 i -skeleton in a free T -CW structure on S ( ∞ C ). Thisfiltered model for a free, contractible T -CW complex was used in [BR05, § 2] todiscuss the T -homotopy fixed point spectral sequence.There are well-known T -equivariant homeomorphisms S (( i + 1) C ) ∼ = T ∗ · · · ∗ T ∗ T with ( i + 1) copies of T , where ∗ denotes the join of spaces. These homeomorphismsare compatible for varying i ≥ 0, and S ( ∞ C ) is isomorphic as a filtered space toMilnor’s infinite join construction from [Mil56], for G = T , which we denote by E G = G ∗ G ∗ G ∗ . . . . The identifications made in the iterated join are included among those made ingeometric realization. Hence the structure map ∆ i × G i × G → EG factors througha G -map q i : G ∗ · · · ∗ G ∗ G −→ F i EG with ( i + 1) copies of G , collapsing degenerate simplices. These are compatible forvarying i , yielding a G -map q : E G → EG . As explained in [Seg68, § E G ∼ = EG N of the two-sided bar construction for atopological category G N , and there is a continuous functor G N → G inducing the G -maps q i and q . It follows that the filtration-preserving diagonal approximation D N : EG N → EG N × EG N constructed in [Seg68, Lem. 5.4] is compatible with the diagonal approximation D : EG → EG × EG that we have used in the presentpaper. In particular, the T -map q ∗ : F ( E T + , X ) −→ F ( E T + , X ) ∼ = F ( S ( ∞ C ) + , X )maps our multiplicative sequence M ⋆ ( X ) to the multiplicative tower used in [BR05, § G = T the G -maps q i and q are equivalences, so that thetwo multiplicative towers of orthogonal G -spectra are equivalent. Hence they giveisomorphic T -homotopy fixed point spectral sequences, converging to the same mul-tiplicative filtration on the abutment.A similar discussion applies for the 3-sphere G = U = Sp (1) acting on the unitspheres in i H = H i , showing that S ( ∞ H ) ∼ = E U is a perfectly good alternativefiltered model for E U .6. The G -Tate spectral sequence Given an R -module X in orthogonal G -spectra we can define its G -Tate con-struction as the genuine fixed points X tG = ( g EG ∧ F ( EG + , X )) G , where g EG is the mapping cone of the collapse map EG + → S . In this section weconstruct an R ∗ -module spectral sequenceˆ E r ∗ , ∗ = ⇒ π ∗ ( X tG )with abutment the G -equivariant homotopy groups of g EG ∧ F ( EG + , X ), for anycompact Lie group G . We do this by letting the filtration E ⋆ induce a filtration e E ⋆ of R ∧ g EG and consider the so-called Hesselholt–Madsen filtration HM ⋆ ( X ) = ( e E ∧ R T ( M ( X ))) ⋆ obtained by forming a convolution product. Under the assumption that R [ G ] ∗ is finitely generated and projective over R ∗ we show that the resulting spectralsequence ˆ E r ∗ , ∗ ( X ) = E r ∗ , ∗ ( HM ⋆ ( X )) is multiplicative, as a functor of X , with mul-tiplicative abutment. With the same assumptions we also algebraically identifythe E -page as ˆ E ∗ , ∗ ( X ) ∼ = d Ext −∗ R [ G ] ∗ ( R ∗ , π ∗ ( X )) , with the multiplicative structure given by cup product on the right-hand side.See Theorem 6.18. To say something about the convergence of this spectral se-quence we compare the Hesselholt–Madsen filtration to another filtration GM ⋆ ( X )of g EG ∧ F ( EG + , X ), dubbed the Greenlees–May filtration. While the multiplica-tive properties of the Greenlees–May G -Tate spectral sequence are less clear, it iseasy to obtain convergence results for the latter spectral sequence. By the compar-ison we can then also obtain convergence results for the Hesselholt–Madsen G -Tatespectral sequence. See Section 6.6, and in particular Theorem 6.44.6.1. The filtered G -space g EG . As always, let G be any compact Lie group. Let c : EG + → S denote the based and G -equivariant collapse map, and define g EG = S ∪ C ( EG + )to be its reduced mapping cone, as in [Car84, p. 198] and [GM95, p. 2]. Non-equivariantly, c is an equivalence, so g EG is (non-equivariantly) contractible. For i ≥ F i g EG = S ∪ C ( F i − EG + ) MULTIPLICATIVE TATE SPECTRAL SEQUENCE 89 be the mapping cone of c restricted to F i − EG + , where F i − EG is defined as inSection 5.1. For i < 0, we set F i g EG = ∗ . This defines an exhaustive filtration(6.1) ∗ = F − g EG ⊂ S = F g EG ⊂ · · · ⊂ F i − g EG ⊂ F i g EG ⊂ · · · ⊂ g EG of based G -spaces. Each map F i − g EG → F i g EG is a strong h -cofibration, so this isindeed a filtration, as opposed to simply a sequence. Moreover, there are homeo-morphisms F i g EGF i − g EG ∼ = Σ F i − EGF i − EG for i ≥ 1. Per Theorem 4.17, each pushout-product map F i − g EG ∧ F j g EG ∪ F i g EG ∧ F j − g EG −→ F i g EG ∧ F j g EG is a strong h -cofibration, with cofibre F i g EGF i − g EG ∧ F j g EGF j − g EG . Remark . When G is finite, F i g EG gives the i -skeleton of a based and non-free G -CW structure on g EG . When G = T = U (1), F g EG = S is the 0-skeleton,while F i g EG for i ≥ i − i -skeleton of a G -CW structure on g EG .Similarly, when G = U = Sp (1), F i g EG gives the 4 i − i − i − 1- and 4 i -skeletaof a G -CW structure. Remark . For G = T , the G -equivalences q i − : S ( i C ) → F i − EG from Sec-tion 5.4 induce G -equivalences ˜ q i : S i C → F i g EG , where we identify the one-pointcompactification S i C with the mapping cone S ∪ C ( S ( i C ) + ). Hence we have a G -equivalence from the exhaustive filtration ∗ → S → · · · → S ( i − C → S i C → · · · → S ∞ C to (6.1), showing that we may use S ∞ C as a filtered replacement for g EG , if desired.We give g EG ∧ g EG the (convolved) smash product filtration, with F k ( g EG ∧ g EG ) = [ i + j = k F i g EG ∧ F j g EG . The identifications S ∧ g EG ∼ = g EG ∼ = g EG ∧ S agree on S ∧ S ∼ = S , hence combineto a fold map ∇ : g EG ∪ S g EG ∼ = g EG ∧ S ∪ S ∧ g EG −→ g EG . We seek a G -map N : g EG ∧ g EG → g EG extending ∇ , so that the diagram g EG ∧ S ∪ S ∧ g EG / / ∇ ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ g EG ∧ g EG N (cid:15) (cid:15) g EG commutes. For these pairings to induce pairing of spectral sequences, we mustarrange that N is filtration-preserving. We do not know how to give a directdefinition of such an extension N : g EG ∧ g EG → g EG , in analogy with the explicitdiagonal approximation D : EG → EG × EG . Instead we will use obstructiontheory to show that such a filtration-preserving extension N of ∇ exists after basechange to our ground ring spectrum R , assuming that R [ G ] ∗ is projective over R ∗ .See Proposition 6.9. Definition 6.3. Let e E = R ∧ g EG and e E i = R ∧ F i g EG . Each map e E i − → e E i is a strong h -cofibration, so that e E ⋆ is a filtration . . . −→ e E i − −→ e E i −→ e E i +1 −→ . . . of R -modules in orthogonal G -spectra. Here e E i = ∗ for i < e E = R , and e E ∞ = Tel( e E ⋆ ) ≃ G e E . Since g EG is non-equivariantly contractible, π ∗ ( e E ) = 0.Applying non-equivariant homotopy we obtain the following unrolled exact cou-ple(6.2) · · · π ∗ ( e E i − ) π ∗ ( e E i ) · · · π ∗ ( e E i − → e E i ) α β∂ with ∂ of total degree − 1. Recall the R [ G ] ∗ -module resolution ( P ∗ , ∗ , ∂ ) of R ∗ ,introduced in Definition 5.8. Definition 6.4. Let ( e P ∗ , ∗ , e ∂ ) be the mapping cone of the augmentation ǫ : P ∗ , ∗ → R ∗ , in the sense of Definition 2.11.Explicitly, we have e P i, ∗ ∼ = ( R ∗ for i = 0 P i − , ∗ for i ≥ ∂ : e P i, ∗ → e P i − , ∗ given as˜ ∂ = ( ǫ ( x ) for i = 1 − ∂ ( x ) for i ≥ e P ∗ , ∗ is an exact complex of flat R ∗ -modules, by our standing as-sumption that R [ G ] ∗ is flat. If, furthermore, R [ G ] ∗ is finitely generated projectiveover R ∗ , then so is each e P i, ∗ . Lemma 6.5. If R [ G ] ∗ is flat over R ∗ , then the ( E , d ) -page of the non-equivarianthomotopy spectral sequence e E i, ∗ = π i + ∗ ( e E i − → e E i ) associated to e E ⋆ is isomorphic to ( e P ∗ , ∗ , ˜ ∂ ) . In particular, the spectral sequencecollapses of the E -page, where it is given by e E = e E ∞ = 0 . Proof. Note that e E is the mapping cone of the collapse map 1 ∧ c : E → R and canbe viewed as the pushout E / / ∧ c (cid:15) (cid:15) I ∧ E (cid:15) (cid:15) R / / e E .Let I ⋆ be the filtration ∗ −→ { , } −→ I = −→ I = −→ I = −→ · · · MULTIPLICATIVE TATE SPECTRAL SEQUENCE 91 of the unit interval I = [0 , ∂I = { , } sits in filtration degree 0. Let L ⋆ ( R )be the non-negative filtration consisting of R ’s and identity maps between them.We then have a pushout of filtrations(6.3) E ⋆ ( I ∧ E ) ⋆ L ⋆ ( R ) e E ⋆ ∧ c with colimit being the pushout square above. That this is indeed a pushout offiltrations can be checked in each filtration degree separately, noting that( I ∧ E ) k = ∂I ∧ E k ∪ I ∧ E k − ∼ = E k ∪ CE k − . It follows as in Lemma 5.9 that we have a commutative square of associated chaincomplexes(6.4) P ∗ , ∗ ∼ = / / ǫ (cid:15) (cid:15) ∂I ∗ ⊗ P ∗ , ∗ / / / / I ∗ ⊗ P ∗ , ∗ (cid:15) (cid:15) R ∗ / / / / e E ∗ , ∗ .Here R ∗ is the chain complex consisting of R ∗ concentrated in homological degree 0,and I ∗ is the reduced cellular chain complex0 −→ Z { i } ∂ −→ Z { i } −→ I , with ∂ ( i ) = i . Both i and i have internal degree 0, and lie in homologicaldegree as indicated by their subscript. The chain complex ∂I ∗ is the subcomplexgiven by Z { i } concentrated in homological degree 0. Since the map P ∗ , ∗ ∼ = ∂I ∗ ⊗ P ∗ , ∗ −→ I ∗ ⊗ P ∗ , ∗ is injective, a Mayer–Vietoris argument for the filtration subquotients of (6.3) showsthat (6.4) is in fact a pushout of chain complexes. This proves that e E ∗ , ∗ is indeedthe algebraic mapping cone of ǫ : P ∗ , ∗ → R ∗ , by the definition of the latter chaincomplex. (cid:3) Lemma 6.6. The ( E , d ) -page of the non-equivariant homotopy spectral sequenceassociated to ( e E ∧ R e E ) ⋆ is isomorphic to ( e P ∗ , ∗ ⊗ R ∗ e P ∗ , ∗ , ˜ ∂ ⊗ ⊗ ˜ ∂ ) .Proof. This is very similar to Lemma 5.11. (cid:3) Lemma 6.7. The homomorphism π ∗ ( e E i − ) → π ∗ ( e E i ) is zero, for each i .Proof. This follows from the exactness of ( e E ∗ , ∗ , e d ) ∼ = ( e P ∗ , ∗ , ˜ ∂ ), by an induction on i in the unrolled exact couple (6.2). The claim is clear for i ≤ 0. Assume by inductionthat α : π ∗ ( e E i − ) → π ∗ ( e E i ) is zero, for some i ≥ 0. Then β : π i + ∗ ( e E i ) → e P i, ∗ isinjective. Consider any class x ∈ π i + ∗ ( e E i ). Since ˜ ∂ i ( β ( x )) = β∂β ( x ) = 0, exactnessat e P i, ∗ implies that β ( x ) = ˜ ∂ i +1 ( y ) = β∂ ( y ) for some y ∈ e P i +1 , ∗ . By injectivenessof β it follows that x = ∂ ( y ). Since x was arbitrary, ∂ : e P i +1 , ∗ → π i + ∗ ( e E i ) issurjective, so α : π ∗ ( e E i ) → π ∗ ( e E i +1 ) is zero. (cid:3) Lemma 6.8. There always exists an R -module map of orthogonal G -spectra N : e E ∧ R e E −→ e E extending ∇ : e E ∪ R e E → e E , and any two choices are homotopic. Proof. This follows by obstruction theory, since g EG ∪ g EG ∼ = g EG ∧ S ∪ S ∧ g EG ⊂ g EG ∧ g EG can be given the structure of a free relative G -CW complex, and π ∗ ( e E ) = 0. (cid:3) The above lemma, together with the map ∆ + : EG + → EG + ∧ EG + , makes surethat the Tate construction is multiplicative, in the sense that each G -equivariant R -module pairing X ∧ R Y → Z induces an R -module pairing X tG ∧ R Y tG → Z tG .See Section 6.2. To arrange that the Tate spectral sequence preserves this structurewe need to make sure that we can find a filtration-preserving approximation of N ,in the same way as we could find the filtration-preserving approximation of D . Thefollowing proposition addresses difficulties raised in Problem 11.8 and Problem 14.8of [GM95]. Proposition 6.9. Suppose that R [ G ] ∗ is projective over R ∗ . Then there exists afiltration-preserving map N : ( e E ∧ R e E ) ⋆ −→ e E ⋆ of R -modules in orthogonal G -spectra, extending the fold map ∇ : e E ⋆ ∪ R e E ⋆ ∼ = ( e E ⋆ ∧ R R ) ∪ ( R ∧ R e E ⋆ ) −→ e E ⋆ . Proof. We inductively assume that ∇ has been extended to a filtration-preservingmap N k − : ( e E ∧ R e E ) k − → e E k − , and show that N k − can be further extendedto a filtration-preserving map N k : ( e E ∧ R e E ) k → e E k . It suffices to extend N k − over e E i ∧ R e E j for i, j ≥ i + j = k . In particular, there is only something toprove for k ≥ 2. Let us consider the diagram e E i − ∧ R e E j ∪ e E i ∧ R e E j − N k − / / (cid:15) (cid:15) (cid:15) (cid:15) e E k − α (cid:15) (cid:15) e E i ∧ R e E j N i,j / / ❴❴❴❴❴❴❴ (cid:15) (cid:15) e E k e E i / e E i − ∧ R e E j / e E j − where the left hand column is a (Hurewicz) cofibre sequence. By the homotopy ex-tension property, in order to find a dashed map N i,j making the diagram commute,it suffices to find an extension up to homotopy of α ◦ N k − . Let W = E i − /E i − ∧ R E j − /E j − ∼ = R ∧ G ∧ i − ∧ G + ∧ G ∧ j − ∧ G + so that Σ W ∼ = e E i / e E i − ∧ R e E j / e E j − . There is then a (stably defined) homotopycofibre sequenceΣ W ∂ −→ e E i − ∧ R e E j ∪ e E i ∧ R e E j − −→ e E i ∧ R e E j −→ Σ W and it suffices to prove that α ◦ N k − ◦ ∂ : Σ W → e E k is null-homotopic. We confirmthis by showing that α induces the trivial homomorphism α ∗ : [Σ W, e E k − ] GR −→ [Σ W, e E k ] GR , where [ − , − ] GR denotes homotopy classes of G -maps of R -modules in orthogonal G -spectra. Note that G acts diagonally on the two copies of G + in W , so that thereis an untwisting isomorphism W ∼ = V ∧ G + where V = R ∧ G ∧ i − ∧ G ∧ j − ∧ G + MULTIPLICATIVE TATE SPECTRAL SEQUENCE 93 has trivial G -action. By adjunction we can therefore rewrite the homomorphismabove as α ∗ : [Σ V, e E k − ] R −→ [Σ V, e E k ] R where [ − , − ] R denotes homotopy classes of maps of (non-equivariant) R -modules.By our assumption that R [ G ] ∗ is R ∗ -projective, it follows that π ∗ ( V ) ∼ = R [ G ] ⊗ i − ∗ ⊗ R ∗ R [ G ] ⊗ j − ∗ ⊗ R ∗ R [ G ] ∗ is R ∗ -projective. Hence we can rewrite α ∗ as the homomorphismHom R ∗ (Σ π ∗ ( V ) , π ∗ ( e E k − )) −→ Hom R ∗ (Σ π ∗ ( V ) , π ∗ ( e E k ))given by composition with α : π ∗ ( e E k − ) → π ∗ ( e E k ). By Lemma 6.7 that homomor-phism is zero, which completes the proof. (cid:3) Definition 6.10. Suppose that R [ G ] ∗ is projective over R ∗ . LetΦ : e P ∗ , ∗ ⊗ R ∗ e P ∗ , ∗ −→ e P ∗ , ∗ be the R [ G ] ∗ -module chain map that corresponds, under the isomorphism of Lemma 6.5and Lemma 6.6, to the pairing N of ( E , d )-pages induced by the filtration-preserving map N : ( e E ∧ R e E ) ⋆ → e E ⋆ of Proposition 6.9. Lemma 6.11. Suppose that R [ G ] ∗ is projective over R ∗ . Then the map Φ : e P ∗ , ∗ ⊗ R ∗ e P ∗ , ∗ → e P ∗ , ∗ is uniquely characterized, up to R [ G ] ∗ -module chain homotopy, by being an R [ G ] ∗ -module chain map that extends the fold map ∇ .Proof. By construction, Φ extends the fold map, and it follows that this map isunique up chain homotopy equivalence by Proposition 2.30. (cid:3) The G -Tate construction. Let X be an R -module in orthogonal G -spectra .In this section, we discuss the Tate construction and its multiplicative properties. Definition 6.12. The G -Tate construction X tG is the G -fixed point spectrum of(a fibrant replacement of) g EG ∧ F ( EG + , X ): X tG = (cid:16)g EG ∧ F ( EG + , X ) (cid:17) G Note that the homotopy groups π ∗ ( X tG ) ∼ = π G ∗ ( g EG ∧ F ( EG + , X ))naturally form an R ∗ -module, and that we can write g EG ∧ F ( EG + , X ) ∼ = e E ∧ R F R ( E, X ) . The inclusion S → g EG induces a G -map F ( EG + , X ) ∼ = S ∧ F ( EG + , X ) −→ g EG ∧ F ( EG + , X )and a map of their corresponding G -fixed points X hG −→ X tG . We can write theseas maps of R -modules, using the inclusion R → e E , to obtain a G -map F R ( E, X ) ∼ = R ∧ R F R ( E, X ) −→ e E ∧ R F R ( E, X )and a canonical map γ : X hG = F R ( E, X ) G −→ (cid:16) e E ∧ R F R ( E, X ) (cid:17) G = X tG , inducing a homomorphism π ∗ ( X hG ) → π ∗ ( X tG ) of R ∗ -modules. If X is an orthogonal G -spectrum without R -action, the discussion in this section applies to R ∧ X in place of X . The Tate construction interacts well with the multiplicative structure on homo-topy fixed points we described in the paragraph preceding Remark 5.5. Note firstthat given a pairing µ : X ∧ R Y → Z of R -modules in orthogonal G -spectra, the R -module pairing X hG ∧ R Y hG → Z hG extends to R -module pairings X tG ∧ R Y hG → Z tG and X hG ∧ R Y tG → Z tG . The first is given by a composite (cid:16) e E ∧ R F R ( E, X ) (cid:17) G ∧ R F R ( E, Y ) G ∧ −→ (cid:16) e E ∧ R F R ( E, X ) ∧ R F R ( E, Y ) (cid:17) G ∧ α −→ (cid:16) e E ∧ R F R ( E ∧ R E, X ∧ R Y ) (cid:17) G ∧ (1 ∧ ∆ + ) ∗ −→ (cid:16) e E ∧ R F R ( E, X ∧ R Y ) (cid:17) G ∧ µ ∗ −→ (cid:16) e E ∧ R F R ( E, Z ) (cid:17) G . The second one is similar, and left to the reader. The two pairings induce R ∗ -modulepairings π ∗ ( X tG ) ⊗ R ∗ π ∗ ( Y hG ) → π ∗ ( Z tG ) and π ∗ ( X hG ) ⊗ R ∗ π ∗ ( Y tG ) → π ∗ ( Z tG ).These pairings are all compatible via the canonical map, meaning that the R -modulediagram X tG ∧ R Y hG (cid:15) (cid:15) X hG ∧ R Y hGγ ∧ o o (cid:15) (cid:15) ∧ γ / / X hG ∧ R Y tG (cid:15) (cid:15) Z tG Z hGγ o o γ / / Z tG , and the induced R ∗ -module diagram both commute. Per Lemma 6.8 we can choosea unique (up to homotopy) extension N : e E ∧ R e E → e E of the fold map ∇ : e E ∪ R e E → e E , in the category of R -modules in orthogonal G -spectra. We can then promotethe two R -module pairings to an R -module pairing X tG ∧ R Y tG → Z tG , given bythe composite (cid:16) e E ∧ R F R ( E, X ) (cid:17) G ∧ R (cid:16) e E ∧ R F R ( E, Y ) (cid:17) G ∧ −→ (cid:16) e E ∧ R F R ( E, X ) ∧ R e E ∧ R F R ( E, Y ) (cid:17) G ∧ τ ∧ −→ (cid:16) e E ∧ R e E ∧ R F R ( E, X ) ∧ R F R ( E, Y ) (cid:17) G ∧ ∧ α −→ (cid:16) e E ∧ R e E ∧ R F R ( E ∧ R E, X ∧ R Y ) (cid:17) GN ∧ (1 ∧ ∆ + ) ∗ −→ (cid:16) e E ∧ R F R ( E, X ∧ R Y ) (cid:17) G ∧ µ ∗ −→ (cid:16) e E ∧ R F R ( E, Z ) (cid:17) G . These pairings are also compatible via the canonical map, meaning that the R -module diagram X tG ∧ R Y hG ∧ γ / / ( ( PPPPPPPPPPPPP X tG ∧ R Y tG (cid:15) (cid:15) X hG ∧ R Y tGγ ∧ o o v v ♥♥♥♥♥♥♥♥♥♥♥♥♥ Z tG and the induced R ∗ -module diagram both commute. Taken together, these dia-grams show that γ : X hG → X tG and γ ∗ : π ∗ ( X hG ) → π ∗ ( X tG ) MULTIPLICATIVE TATE SPECTRAL SEQUENCE 95 are multiplicative. We would now like to access π ∗ ( X tG ) and the pairings abovethrough filtrations and their associated spectral sequences.6.3. The Hesselholt–Madsen filtration. We can now generalize the filtrationof X tG from [HM03, § G . Definition 6.13. Let HM ⋆ ( X ) = ( e E ∧ R T ( M ( X ))) ⋆ be the filtration · · · → HM k − ( X ) → HM k ( X ) → HM k +1 ( X ) → . . . of R -modules in orthogonal G -spectra given by the Day convolution product of thefiltrations e E ⋆ and T ⋆ ( M ( X )).Recall that we introduced the filtration e E ⋆ in Definition 6.3, the sequence M ⋆ ( X )in Section 5.2, and its telescopic approximation T ⋆ ( M ( X )) in Section 4.3. Theconvolution product of e E ⋆ and T ⋆ ( M ( X )) was defined in Section 4.7, and is afiltration by Proposition 4.31. We can realize HM k ( X ) = [ i + j = k e E i ∧ R T j ( M ( X ))as a subspectrum of e E ∧ R Tel( M ⋆ ( X )). The structure maps HM k − ( X ) → HM k ( X )are then inclusions of subspectra. These are (strong) h -cofibrations, so the canonicalmap Tel( HM ⋆ ( X )) −→ colim k HM k ( X ) = e E ∧ R Tel( M ⋆ ( X ))is an equivalence. Since M j ( X ) = F R ( E, X ) for all j ≥ M ⋆ ( X )) ≃ G −→ F R ( E, X )and a further equivalence e E ∧ R Tel( M ⋆ ( X )) ≃ G −→ e E ∧ R F R ( E, X ) ∼ = g EG ∧ F ( EG + , X ) . Definition 6.14. Let X be an R -module in orthogonal G -spectra. We define the G -Tate spectral sequence for X to be the R ∗ -module spectral sequence ( ˆ E r ( X ) , d r )associated to the filtration HM ⋆ ( X ) withˆ E r ( X ) = E r ( HM ⋆ ( X ))for each r ≥ G -Tate spectral sequence for X is the colimit A ∞ ( HM ⋆ ( X )) ∼ = π G ∗ Tel( HM ⋆ ( X )) ∼ = π G ∗ ( e E ∧ R F R ( E, X )) ∼ = π ∗ ( X tG ) , filtered by the image submodules F k π ∗ ( X tG ) = im (cid:0) π G ∗ ( HM k ( X )) → π G ∗ Tel( HM ⋆ ( X )) ∼ = π ∗ ( X tG ) (cid:1) . Remark . In general, we do not claim that the G -Tate spectral sequenceconverges to the stated abutment, neither in the conditional nor in the weaksense. As we recalled in Section 4.2, conditional convergence to the colimit holdsif holim k HM k ( X ) ≃ G ∗ . The latter condition would follow from an interchangeof homotopy colimits and homotopy limits. More precisely, for each a ≥ k , consider the subspectrum S a,k = [ i + j = ki ≤ a e E i ∧ R T j ( M ( X )) of e E ∧ R Tel( M ⋆ ( X )). Then hocolim a S a,k ≃ G HM k ( X ), and the sufficient conditionholim k HM k ( X ) ≃ G ∗ for conditional convergence is equivalent to(6.5) holim k hocolim a S a,k ≃ G ∗ . On the other hand, holim j e E i ∧ R T j ( M ( X )) ≃ G ∗ for each i , since F i g EG is afinite G -CW space. It follows by induction that holim k S a,k ≃ G ∗ for each finite a ,which implies that(6.6) hocolim a holim k S a,k ≃ G ∗ . Without further hypotheses we do not see how to deduce (6.5) from (6.6).6.4. Algebraic description of ˆ E and ˆ E . Under the assumption that R [ G ] ∗ is finitely generated projective over R ∗ , we can algebraically describe the E -and E -pages of the G -Tate spectral sequence, in the same way as we did forthe G -homotopy fixed points spectral sequence in Section 5.3. Proposition 6.16. Suppose that R [ G ] ∗ is R ∗ -projective. There is then a naturalisomorphism of R ∗ -module chain complexes E ∗ , ∗ ( HM ⋆ ( X )) ∼ = Hom R [ G ] ∗ ( R ∗ , e P ∗ , ∗ ⊗ R ∗ Hom R ∗ ( P ∗ , ∗ , π ∗ ( X ))) . In the notation of Definition 2.12 and Definition 6.14, we have ˆ E ∗ , ∗ ( X ) ∼ = Hom R [ G ] ∗ ( R ∗ , hm ∗ ( π ∗ ( X ))) , where the d -differential on the left hand side corresponds to the boundary Hom(1 , ∂ hm ) on the right hand side.Proof. We first check that the natural restriction homomorphism(6.7) ω : E ∗ , ∗ ( HM ⋆ ( X )) ∼ = −→ Hom R [ G ] ∗ ( R ∗ , E ∗ , ∗ (( e E ∧ R T ( M ( X ))) ⋆ ))from Lemma 3.5 is an isomorphism of R ∗ -module chain complexes, where HM ⋆ ( X )at the left hand side is treated as an R -module filtration in orthogonal G -spectra,while ( e E ∧ R T ( M ( X ))) ⋆ at the right hand side refers to the underlying R -modulefiltration in non-equivariant orthogonal spectra, with the residual R [ G ]-module ac-tion. We first note that we have E k, ∗ ( HM ⋆ ( X )) = π Gk + ∗ ( HM k − ( X ) → HM k ( X )) ∼ = π Gk + ∗ ( HM k ( X ) /HM k − ( X ))while E k, ∗ (( e E ∧ R T ( M ( X ))) ⋆ ) = π k + ∗ ( HM k − ( X ) → HM k ( X )) ∼ = π k + ∗ ( HM k ( X ) /HM k − ( X )) . Secondly, we note that HM k ( X ) /HM k − ( X ) ∼ = _ i + j = k e E i / e E i − ∧ R T j ( M ( X )) /T j − ( M ( X )) ∼ = _ i + j = k e E i / e E i − ∧ R ( M j ( X ) ∪ CM j − ( X )) ≃ G _ i + j = k e E i / e E i − ∧ R F R ( E − j /E − j − , X ) , which is moreover G -equivalent to F ( G + , X ′ ) for X ′ = _ i + j = k e E i / e E i − ∧ Σ j F ( G ∧− j , X ) . MULTIPLICATIVE TATE SPECTRAL SEQUENCE 97 This uses that R [ G ] is dualisable (see Definition 6.28). Proposition 3.6 thereforeimplies that the natural restriction homomorphism (6.7) is an isomorphism in everyhomological degree. We check that it is also an isomorphism of chain complexes.The d -differential in the spectral sequence appearing in the left hand side corre-sponds to the composition π Gk + ∗ ( HM k ( X ) /HM k − ( X )) ∂ −→ π Gk − ∗ ( HM k − ( X )) −→ π Gk − ∗ ( HM k − /HM k − ( X ))) , which by the naturality of ω corresponds to Hom(1 , d k, ∗ ), where d k, ∗ is the d -differential in the spectral sequence appearing in the right hand side. This is givenby the composite π k + ∗ ( HM k ( X ) /HM k − ( X )) ∂ −→ π k − ∗ ( HM k − ( X )) −→ π k − ∗ ( HM k − /HM k − ( X ))) . Hence (6.7) is indeed an isomorphism of chain complexes.We now want to identify E ∗ , ∗ (( e E ∧ R T ( M ( X ))) ⋆ ) with the Tate complex hm ∗ ( π ∗ ( X )).For this aim, we can use the canonical pairing ι : ( e E ⋆ , T ⋆ ( M ( X ))) −→ ( e E ∧ R T ( M ( X ))) ⋆ of R -module filtrations to obtain an R [ G ] ∗ -module chain map of the associated E -pages ι : E ( e E ⋆ ) ⊗ R ∗ E ( T ⋆ ( M ( X ))) −→ E (( e E ∧ R T ( M ( X ))) ⋆ )as in Theorem 4.27, but in the non-equivariant setting. This map is the direct sumof the maps π i + ∗ ( e E i / e E i − ) ⊗ R ∗ π j + ∗ ( T j ( M ( X )) /T j − ( M ( X ))) · −→ π k + ∗ ( e E i / e E i − ∧ R T j ( M ( X )) /T j − ( M ( X )))for i + j = k . Note that each one of these maps is an isomorphism, because e P i, ∗ = π i + ∗ ( e E i / e E i − ) is projective, hence flat, over R ∗ . We conclude that ι is anisomorphism of R [ G ] ∗ -chain complexes, and thus induces an isomorphism(6.8) Hom(1 , ι ) : Hom R [ G ] ∗ ( R ∗ , E ( e E ⋆ ) ⊗ R ∗ E ( T ⋆ ( M ( X )))) ∼ = −→ Hom R [ G ] ∗ ( R ∗ , E (( e E ∧ R T ( M ( X ))) ⋆ ))of R ∗ -module complexes.The equivalence ǫ : T ⋆ ( M ( X )) → M ⋆ ( X ) induces an isomorphism ǫ : E ( T ⋆ ( M ( X ))) ∼ = −→ E ( M ⋆ ( X ))of R [ G ] ∗ -module chain complexes, which in turn induces an isomorphism(6.9) Hom(1 , ⊗ ǫ ) : Hom R [ G ] ∗ ( R ∗ , E ( e E ⋆ ) ⊗ R ∗ E ( T ⋆ ( M ( X )))) ∼ = −→ Hom R [ G ] ∗ ( R ∗ , E ( e E ⋆ ) ⊗ R ∗ E ( M ⋆ ( X ))) , of R ∗ -module chain complexes. Finally, we have E j, ∗ ( M ⋆ ( X )) ∼ = π j + ∗ ( F R ( E − j /E − j − , X )) ∼ = Hom R ∗ ( P − j, ∗ , π ∗ ( X )) as R [ G ] ∗ -modules, because π − j + ∗ ( E − j /E − j − ) ∼ = P − j, ∗ is R ∗ -projective. The d -differentials correspond to ˜ ∂ and Hom( ∂, 1) by the argument in the proof of Propo-sition 5.13. Hence we have an isomorphism(6.10) Hom R [ G ] ∗ ( R ∗ , E ∗ , ∗ ( e E ⋆ ) ⊗ R ∗ E ∗ , ∗ ( M ⋆ ( X ))) ∼ = Hom R [ G ] ∗ ( R ∗ , e P ∗ , ∗ ⊗ R ∗ Hom R ∗ ( P ∗ , ∗ , π ∗ ( X )))of R ∗ -module complexes.When strung together, the numbered isomorphisms (6.7) through (6.10) estab-lish the asserted identification of the G -Tate spectral sequence ( E , d )-page forthe orthogonal G -spectrum R -module X with the Tate complex for the R [ G ] ∗ -module π ∗ ( X ). (cid:3) Theorem 6.17. Let X be an R -module in orthogonal G -spectra, and supposethat R [ G ] ∗ is R ∗ -projective. Then there is a natural R ∗ -module isomorphism ˆ E i, ∗ ( X ) = E i, ∗ ( HM ⋆ ( X )) ∼ = d Ext − iR [ G ] ∗ ( R ∗ , π ∗ ( X )) , for each integer i .Proof. This is immediate by passage to homology from Proposition 6.16. Here weare using the definition of Hopf algebra Tate cohomology given in Definition 2.13. (cid:3) We now go on to discuss the multiplicative structure of the Tate spectral se-quence. Let µ : X ∧ R Y → Z be a pairing of R -modules in orthogonal G -spectra.As discussed in the paragraph before Theorem 5.6, the diagonal approximation D and µ combine to define a pairing ¯ µ : ( M ⋆ ( X ) , M ⋆ ( Y )) → M ⋆ ( Z ) of sequences of R -modules in orthogonal G -spectra. By Lemma 4.21 we have an induced pairing T (¯ µ ) : ( T ⋆ ( M ( X )) , T ⋆ ( M ( Y ))) −→ T ⋆ ( M ( Z ))of filtrations. By Proposition 6.9 there is also a pairing of filtrations N : ( e E ⋆ , e E ⋆ ) −→ e E ⋆ which extends the fold map. Hence ( e E ⋆ , N ) is a multiplicative R -module filtrationin orthogonal G -spectra. We can now form the induced pairing of convolutionfiltrations θ = N ∧ T (¯ µ ) : ( HM ⋆ ( X ) , HM ⋆ ( Y )) −→ HM ⋆ ( Z ) . This has components θ i,j : HM i ( X ) ∧ R HM j ( Y ) −→ HM i + j ( Z )given by the union over i + i = i and j + j = j of the composite maps e E i ∧ R T i ( M ( X )) ∧ R e E j ∧ R T j ( M ( Y )) ∧ τ ∧ −→ e E i ∧ R e E j ∧ R T i ( M ( X )) ∧ R T j ( M ( Y )) N i ,j ∧ T (¯ µ ) i ,j −→ e E i + j ∧ R T i + j ( M ( Z )) ι i j ,i j −→ HM i + j ( Z ) . MULTIPLICATIVE TATE SPECTRAL SEQUENCE 99 Viewing HM i ( X ) as a subspectrum of e E ∧ R Tel( M ⋆ ( X )) (and similarly for Y and Z in place of X ) the maps θ i,j are compatible with the composite map e E ∧ R F R ( E, X ) ∧ R e E ∧ R F R ( E, Y ) ∧ τ ∧ −→ e E ∧ R e E ∧ R F R ( E, X ) ∧ R F R ( E, Y ) ∧ α −→ e E ∧ R e E ∧ R F R ( E ∧ R E, X ∧ R Y ) N ∧ −→ e E ∧ R F R ( E ∧ R E, X ∧ R Y ) (1 ∧ D + ) ∗ −→ e E ∧ R F R ( E, X ∧ R Y ) ∧ µ ∗ −→ e E ∧ R F R ( E, Z ) . This is G -homotopic to the corresponding map with ∆ in place of D , which definesthe product θ ∗ : π ∗ ( X tG ) ⊗ R ∗ π ∗ ( Y tG ) → π ∗ ( Z tG )that we introduced in Section 6.2. Hence this product is filtration-preserving, taking F i π ∗ ( X tG ) ⊗ R ∗ F j π ∗ ( Y tG ) to F i + j π ∗ ( Z tG ) for all i and j . We write ¯ θ ∗ for theinduced pairing of filtration subquotients. Theorem 6.18. Let µ : X ∧ R Y → Z be a pairing of R -modules in orthogonal G -spectra, and assume that R [ G ] ∗ is projective over R ∗ . The pairing θ = N ∧ T (¯ µ ) : ( HM ⋆ ( X ) , HM ⋆ ( Y )) → HM ⋆ ( Z ) of filtrations induces a pairing of G -Tate spectral sequences θ : ˆ E ∗ ( X ) ⊗ R ∗ ˆ E ∗ ( Y ) −→ ˆ E ∗ ( Z ) . in the sense of Definition 4.9. Moreover, the induced pairing θ ∞ of E ∞ -pages iscompatible with the pairing ¯ θ ∗ of filtration subquotients, in the sense of Proposi-tion 4.12.Proof. This is a direct consequence of Theorem 4.27. (cid:3) Corollary 6.19. If ( X, µ ) a multiplicative R -module in orthogonal G -spectra, then ( HM ⋆ ( X ) , N ∧ T (¯ µ )) is a multiplicative filtration, and the G -Tate spectral sequence ( ˆ E r ( X ) , d r ) = ( E r ( HM ⋆ ( X )) , d r ) is a multiplicative spectral sequence with multiplicative abutment π ∗ ( X tG ) .Proof. This follows from Corollary 4.28. (cid:3) Proposition 6.20. Let µ : X ∧ R Y → Z be a pairing of R -modules in orthogo-nal G -spectra and assume that R [ G ] ∗ is R ∗ -projective. Under the isomorphism ofProposition 6.16, the pairing θ : ˆ E ( X ) ⊗ R ∗ ˆ E ( Y ) −→ ˆ E ( Z ) corresponds to the pairing covariantly induced by Φ : e P ∗ , ∗ ⊗ R ∗ e P ∗ , ∗ −→ e P ∗ , ∗ andcontravariantly induced by Ψ : P ∗ , ∗ −→ P ∗ , ∗ ⊗ R ∗ P ∗ , ∗ , as in Section 2.5.Proof. For typographical reasons we will use the abbreviation M R [ G ] ∗ = Hom R [ G ] ∗ ( R ∗ , M )in what follows, for various R [ G ] ∗ -modules M . In the same way as in the proofof Proposition 6.16, the notation E ( HM ⋆ ( X )) will refer to the E -page of theassociated spectral sequence on equivariant homotopy groups, while E (( e E ∧ R T ( M ( X ))) ⋆ ) will refer to the E -page of the associated non-equivariant spectralsequence. 00 ALICE HEDENLUND AND JOHN ROGNES We first note some results regarding multiplicative compatibility. Firstly, thenatural homomorphism ω is monoidal by Lemma 3.7, so the diagram E ( HM ⋆ ( X )) ⊗ R ∗ E ( HM ⋆ ( Y )) θ / / ω ⊗ ω ∼ = (cid:15) (cid:15) E ( HM ⋆ ( Z )) ω ∼ = (cid:15) (cid:15) E (( e E ∧ R T ( M ( X ))) ⋆ ) R [ G ] ∗ ⊗ R ∗ E (( e E ∧ R T ( M ( Y ))) ⋆ ) R [ G ] ∗ α (cid:15) (cid:15) E (( e E ∧ R T ( M ( X ))) ⋆ ) ⊗ R ∗ E (( e E ∧ R T ( M ( X ))) ⋆ ) R [ G ] ∗ ( θ ) R [ G ] ∗ / / E (( e E ∧ R T ( M ( Z ))) ⋆ ) R [ G ] ∗ commutes. Secondly, by a slight generalization of Lemma 4.30, the pairing ι : E ( e E ⋆ ) ⊗ R ∗ E ( T ⋆ ( M ( X ))) −→ E (( ˜ E ∧ R T ( M ( X ))) ⋆ )and its variants for Y and Z in place of X are multiplicatively compatible in thesense that the diagram E ( e E ⋆ ) ⊗ R ∗ E ( T ⋆ ( M ( X ))) ⊗ R ∗ E ( e E ⋆ ) ⊗ R ∗ E ( T ⋆ ( M ( Y ))) ⊗ τ ⊗ / / ι ⊗ ι ∼ = (cid:15) (cid:15) E ( e E ⋆ ) ⊗ R ∗ E ( e E ⋆ ) ⊗ R ∗ E ( T ⋆ ( M ( X ))) ⊗ R ∗ E ( T ⋆ ( M ( Y ))) N ⊗ T (¯ µ ) (cid:15) (cid:15) E ( e E ⋆ ) ⊗ R ∗ E ( T ⋆ ( M ( Z ))) ι ∼ = (cid:15) (cid:15) E (( e E ∧ R T ( M ( X ))) ⋆ ) ⊗ R ∗ E (( e E ∧ R T ( M ( Y ))) ⋆ ) θ / / E (( e E ∧ R T ( M ( Z ))) ⋆ )commutes. Note that, by Definition 6.10, the map N : E ( e E ⋆ ) ⊗ R ∗ E ( e E ⋆ ) −→ E ( e E ⋆ )corresponds to Φ : e P ∗ , ∗ ⊗ R ∗ e P ∗ , ∗ → e P ∗ , ∗ under the isomorphism E ∗ , ∗ ( e E ⋆ ) ∼ = e P ∗ , ∗ . Asdiscussed in the proofs of Proposition 4.25, Theorem 5.6 and (the non-equivariantversion of) Proposition 5.13, T (¯ µ ) : E ( T ⋆ ( M ( X ))) ⊗ R ∗ E ( T ⋆ ( M ( Y ))) −→ E ( T ⋆ ( M ( Z )))corresponds to the composite homomorphismHom R ∗ ( P ∗ , ∗ , π ∗ ( X )) ⊗ R ∗ Hom R ∗ ( P ∗ , ∗ , π ∗ ( Y )) α −→ Hom R ∗ ( P ∗ , ∗ ⊗ R ∗ P ∗ , ∗ , π ∗ ( X ) ⊗ R ∗ π ∗ ( Y )) Ψ ∗ −→ Hom R ∗ ( P ∗ , ∗ , π ∗ ( X ) ⊗ R ∗ π ∗ ( Y )) µ ∗ −→ Hom R ∗ ( P ∗ , ∗ , π ∗ ( Z ))under the isomorphisms E ( T ⋆ ( M ( X ))) ∼ = E ( M ⋆ ( X )) ∼ = Hom R ∗ ( P ∗ , ∗ , π ∗ ( X )) MULTIPLICATIVE TATE SPECTRAL SEQUENCE 101 and their variants with Y and Z in place of X .Combining all of these results, we have shown that θ corresponds to the com-positehm( π ∗ ( X )) R [ G ] ∗ ⊗ R ∗ hm( π ∗ ( Y )) R [ G ] ∗ α −→ (hm ∗ ( π ∗ ( X )) ⊗ R ∗ hm( π ∗ ( Y ))) R [ G ] ∗ ⊗ τ ⊗ −→ ( e P ∗ , ∗ ⊗ R ∗ e P ∗ , ∗ ⊗ R ∗ Hom R ∗ ( P ∗ , ∗ , π ∗ ( X )) ⊗ R ∗ Hom R ∗ ( P ∗ , ∗ , π ∗ ( Y ))) R [ G ] ∗ ⊗ ⊗ α −→ ( e P ∗ , ∗ ⊗ R ∗ e P ∗ , ∗ ⊗ R ∗ Hom R ∗ ( P ∗ , ∗ ⊗ R ∗ P ∗ , ∗ , π ∗ ( X ) ⊗ R ∗ π ∗ ( Y ))) R [ G ] ∗ Φ ⊗ −→ ( e P ∗ , ∗ ⊗ R ∗ Hom R ∗ ( P ∗ , ∗ ⊗ R ∗ P ∗ , ∗ , π ∗ ( X ) ⊗ R ∗ π ∗ ( Y ))) R [ G ] ∗ ⊗ Ψ ∗ −→ hm ∗ ( π ∗ ( X ) ⊗ R ∗ π ∗ ( Y )) R [ G ] ∗ ⊗ µ ∗ −→ hm ∗ ( π ∗ ( Z )) R [ G ] ∗ , where we have abbreviatedhm ∗ ( π ∗ ( X )) = e P ∗ , ∗ ⊗ R ∗ Hom R ∗ ( P ∗ , ∗ , π ∗ ( X )) . Note that this is the pairing that induces the cup product, as in Section 2.5. (cid:3) Theorem 6.21. Let µ : X ∧ R Y → Z be a pairing of R -modules in orthogonal G -spectra, and assume that R [ G ] ∗ is R ∗ -projective. Then the pairing θ : E i, ∗ ( HM ⋆ ( X )) ⊗ R ∗ E j, ∗ ( HM ⋆ ( Y )) −→ E i + j, ∗ ( HM ⋆ ( Z )) of G -Tate spectral sequence E -pages corresponds, under the isomorphism of The-orem 6.17, to the cup product ⌣ : d Ext − i, ∗ R [ G ] ∗ ( R ∗ , π ∗ ( X )) ⊗ R ∗ d Ext − j, ∗ R [ G ] ∗ ( R ∗ , π ∗ ( Y )) −→ d Ext − i − j, ∗ R [ G ] ∗ ( R ∗ , π ∗ ( Z )) associated to µ ∗ : π ∗ ( X ) ⊗ R ∗ π ∗ ( Y ) → π ∗ ( Z ) .Proof. This is immediate by passage to homology from Proposition 6.20. See Sec-tion 2.5 for the definition of the cup product in Hopf algebra Tate cohomology. (cid:3) Corollary 6.22. If ( X, µ ) is a multiplicative R -module in orthogonal G -spectra,then the product in ˆ E ( X ) = E ( HM ⋆ ( X )) corresponds to the cup product in d Ext ∗ R [ G ] ∗ ( R ∗ , π ∗ ( X )) that is associated to the product µ ∗ in π ∗ ( X ) . Note independence of the particular choices of maps D and N , since the resultingchain homomorphisms Ψ and Φ are unique up to homotopy, per Proposition 2.28and Proposition 2.30.6.5. The Greenlees–May filtration. In [Gre87, § F ⋆ g EG with its Spanier–Whitehead dual to obtain a sequence of G -spectra · · · −→ D ( F g EG ) −→ D ( F g EG ) −→ S −→ Σ ∞ F g EG −→ Σ ∞ F g EG −→ · · · with mapping telescope equivalent to g EG . The induced sequence . . . −→ D ( F g EG ) ∧ F ( EG + , X ) −→ Σ ∞ F ( EG + , X ) −→ F g EG ∧ F ( EG + , X ) −→ · · · was used in [GM95, (9.5), Thm. 10.3] to define a spectral sequence with abutmentbeing the homotopy groups of the G -Tate construction on X . In this section, wewill define a spliced filtration GM ⋆ ( X ) with a map to the Hesselholt–Madsen filtra-tion HM ⋆ ( X ), and show that the induced map of G -homotopy spectral sequencesˇ E r ( X ) = E r ( GM ⋆ ( X )) −→ E r ( HM ⋆ ( X )) = ˆ E r ( X )is an isomorphism for r ≥ 2. Thereafter we show that GM ⋆ ( X ) is equivalent to thespliced sequence of Greenlees and May, at least for finite groups G . For other com-pact Lie groups the sequences will differ in the same way that our filtration F ⋆ g EG differs from the G -CW skeletal filtration. See Remark 6.1. 02 ALICE HEDENLUND AND JOHN ROGNES Definition 6.23. Recall the filtration e E ⋆ from Definition 6.3 and let GM ⋆ ( X ) bethe filtration of orthogonal G -spectra defined as GM k ( X ) = ( e E k ∧ R T ( M ( X )) for k ≥ e E ∧ R T k ( M ( X )) for k ≤ GM k − ( X ) → GM k ( X ) for k ≥ e E k − → e E k in the filtration e E ⋆ , while the maps for k ≤ T ⋆ ( M ( X )).We refer to the filtration GM ⋆ ( X ) as the Greenlees–May filtration . Notation 6.24. Let ˇ E r ( X ) = E r ( GM ⋆ ( X ))denote the G -homotopy spectral sequence associated to the filtration GM ⋆ ( X ).We now discuss the map of filtrations between the Greenlees–May filtration andthe Hesselholt–Madsen filtration. Lemma 6.25. The inclusions e E k ∧ R T ( M ( X )) → ( e E ∧ R T ( M ( X ))) k for k ≥ and e E ∧ R T k ( M ( X )) → ( e E ∧ R T ( M ( X ))) k for k ≤ define a map of filtrations α : GM ⋆ ( X ) −→ HM ⋆ ( X ) of R -modules in orthogonal G -spectra. The induced maps of mapping telescopes andcolimits Tel( GM ⋆ ( X )) ≃ G / / ≃ G (cid:15) (cid:15) Tel( HM ⋆ ( X )) ≃ G (cid:15) (cid:15) e E ∧ R T ( M ( X )) ≃ G / / e E ∧ R Tel( M ⋆ ( X )) are all equivalences.Proof. Recall from Section 6.3 that HM k ( X ) = [ i + j = k e E i ∧ R T j ( M ( X ))as a subspectrum of e E ∧ R Tel( M ( X )). The existence of the filtered map α is thenclear. The vertical maps from mapping telescopes to colimits are equivalences,since GM ⋆ ( X ) and HM ⋆ ( X ) are both filtrations. The lower horizontal map is alsoan equivalence, since the sequence M ⋆ ( X ) is constant for ⋆ ≥ (cid:3) As a consequence of the above lemma, there is a map α : ˇ E ∗ ( X ) → ˆ E ∗ ( X )of R ∗ -module spectral sequences. Remark . Recall from Proposition 6.9 that, under the assumption that R [ G ] ∗ is R ∗ -projective, we have a filtration-preserving pairing N : ( e E ⋆ , e E ⋆ ) → e E ⋆ . How-ever, when ( X, µ ) is multiplicative, the induced pairing N ∧ T (¯ µ ) : ( HM ⋆ ( X ) , HM ⋆ ( X )) −→ HM ⋆ ( X )does usually not restrict to a multiplication on GM ⋆ ( X ). For instance, GM a ( X ) ∧ R GM − b ( X ) with a > b > HM a ( X ) ∧ R HM − b ( X ) and e E a ∧ R T − b ( M ( X )) in HM a − b ( X ), which is hardly ever in GM a − b ( X ). Hence GM ⋆ ( X )is not a multiplicative filtration, and ˇ E r ( X ) is not evidently a multiplicative spec-tral sequence. Nonetheless, we will show that ˇ E r ( X ) is isomorphic to the G -Tatespectral sequence ˆ E r ( X ) for r ≥ 2, which we showed to be multiplicative in Theo-rem 6.18. This will then show that ( ˇ E r ( X ) , d r ) is also multiplicative, at least for r ≥ MULTIPLICATIVE TATE SPECTRAL SEQUENCE 103 Thinking only about the additive properties of the spectral sequence ˇ E r ( X ), wecan safely replace the filtration GM ⋆ ( X ) with a simpler, but equivalent, sequence. Lemma 6.27. There is an equivalence from GM ⋆ ( X ) to the sequence GM ′ ⋆ ( X ) with GM ′ k ( X ) = ( e E k ∧ R F R ( E, X ) for k ≥ , F R ( E/E − k − , X ) for k ≤ .Proof. The equivalences ǫ : T k ( M ( X )) → M k ( X ) induce the following commutativediagram. . . . / / e E ∧ R T − ( M ( X )) / / ǫ ≃ (cid:15) (cid:15) e E ∧ R T ( M ( X )) / / ǫ ≃ (cid:15) (cid:15) e E ∧ R T ( M ( X )) / / ǫ ≃ (cid:15) (cid:15) . . .. . . / / M − ( X ) / / F R ( E, X ) / / e E ∧ R F R ( E, X ) / / . . . Here M k ( X ) = F R ( E/E − k − , X ) for k ≤ (cid:3) We refer to [LMSM86, § III.1] for the basic Spanier–Whitehead duality theoryin a closed symmetric monoidal category. In the case of (the homotopy categoryof) R -modules in orthogonal G -spectra, we refer to the objects called ‘finite’ byLewis and May as ‘dualisable’. Definition 6.28. For an R -module X in orthogonal G -spectra, let D ( X ) = F R ( X, R )be its functional dual . For dualisable X we refer to D ( X ) as the Spanier–Whiteheaddual of X . There are natural maps ρ : X → D ( D ( X )) and ν : DX ∧ R Y → F R ( X, Y ) , which are equivalences when X is dualisable, essentially by definition. Lemma 6.29. Each term in the filtration e E ⋆ is dualisable.Proof. We can give G a finite CW structure, with e as a 0-cell. It follows that the barconstruction is a finite G -CW space in each simplicial degree B q ( ∗ , G, G ) = G q × G ,so that G ∧ q ∧ G + is a finite G -CW space and R ∧ G ∧ q ∧ G + is a dualisable R -modulein orthogonal G -spectra. By induction, this implies that E i − is dualisable, andtherefore the mapping cone e E i is also dualisable, for each i ≥ (cid:3) Lemma 6.30. The E -page of the spectral sequence associated to the sequence GM ′ ⋆ ( X ) is the R ∗ -module chain complex with ˇ E ∗ , ∗ ( X ) ∼ = Hom R [ G ] ∗ ( R ∗ , gm ∗ ( π ∗ ( X ))) where we use the notation of Definition 2.14.Proof. For ⋆ ≤ GM ′ ⋆ ( X ) agrees with the sequence M ⋆ ( X ) from Sec-tion 5.2, so ( ˇ E ∗ ( X ) , d ) for ∗ ≤ R [ G ] ∗ ( P −∗ , ∗ , π ∗ ( X )) by Proposi-tion 5.13.For ⋆ ≥ GM ′ ⋆ ( X ) agrees with the filtration e E ⋆ ∧ R F R ( E, X ). Itssubquotients for i ≥ e E i / e E i − ) ∧ R F R ( E, X ) with e E i / e E i − ∼ = Σ( E i − /E i − ) ∼ = R ∧ Σ i ( G ∧ i − ∧ G + ) . Let d be the dimension of G . Since G + is stably dualisable, with Spanier–Whiteheaddual D ( G + ) ≃ G Σ − d G + , each subquotient above is equivalent to F ( G + , X ′ ) forsome R -module X ′ in orthogonal G -spectra. It follows from Proposition 3.6 thatˇ E ∗ ( X ) ∼ = Hom R [ G ] ∗ ( R ∗ , E ∗ ( e E ⋆ ∧ R F R ( E, X ))) 04 ALICE HEDENLUND AND JOHN ROGNES for ∗ ≥ 1. Here E i ( e E ⋆ ∧ R F R ( E, X )) = π i + ∗ ( e E i − ∧ R F R ( E, X ) → e E i ∧ R F R ( E, X )) ∼ = π i + ∗ (( e E i / e E i − ∧ R F R ( E, X ))) ∼ = π i + ∗ ( e E i / e E i − ) ⊗ R ∗ π ∗ F R ( E, X ) ∼ = e P i, ∗ ⊗ R ∗ π ∗ ( X )for i ≥ 1, since e P i, ∗ = π i + ∗ ( e E i / e E i − ) is projective, hence flat, over R ∗ , and c : E → R induces an isomorphism π ∗ ( X ) ∼ = π ∗ F R ( E, X ) of R [ G ] ∗ -modules. This showsthat ( ˇ E ∗ ( X ) , d ) for ∗ ≥ R [ G ] ∗ ( R ∗ , e P ∗ , ∗ ⊗ R ∗ π ∗ ( X )).It remains to verify that d : ˇ E ( X ) → ˇ E ( X ) is as asserted. By definition, it isgiven by the left-to-right composite in the following diagram. π G ∗ (( e E / e E ) ∧ R X ) ∂ / / ∧ c ∗ ∼ = (cid:15) (cid:15) π G ∗ ( X ) c ∗ (cid:15) (cid:15) π G ∗ (( e E / e E ) ∧ R F R ( E, X )) ∂ / / π G ∗ ( F R ( E, X )) / / π G ∗ ( F R ( E , X ))By naturality of ω , as in Lemma 3.5, this is obtained from the left-to-right composite π ∗ (( e E / e E ) ∧ R X ) ∂ / / π ∗ ( X ) c ∗ ∼ = (cid:15) (cid:15) π ∗ ( F R ( E, X )) / / π ∗ ( F R ( E , X ))by applying Hom R [ G ] ∗ ( R ∗ , − ). Under the isomorphisms above, this is the compo-sition e P , ∗ ⊗ R ∗ π ∗ ( X ) ˜ ∂ −→ e P , ∗ ⊗ R ∗ π ∗ ( X ) ∼ = π ∗ ( X ) ∼ = Hom R ∗ ( R ∗ , π ∗ ( X )) ǫ ∗ −→ Hom R ∗ ( P , π ∗ ( X )) . As we made explicit in Proposition 2.15, this equals the boundary gm ( π ∗ ( X )) → gm ( π ∗ ( X )). (cid:3) Proposition 6.31. The filtration-preserving map α : GM ⋆ ( X ) → HM ⋆ ( X ) inducesan isomorphism of spectral sequences α r : ˇ E r ( X ) ∼ = −→ ˆ E r ( X ) for r ≥ .Proof. Comparing Proposition 6.16 and Lemma 6.30 shows that α : ˇ E ( X ) → ˆ E ( X ) is the chain map Hom(1 , α ) shown to be a quasi-isomorphism in Proposi-tion 2.16. Hence α = H ( α , d ) is an isomorphism, which implies that α r is anisomorphism for each r ≥ (cid:3) Following [Gre87, § R ∼ = e E −→ e E −→ e E −→ . . . with the Spanier–Whitehead dual sequence . . . −→ D ( e E ) −→ D ( e E ) −→ D ( e E ) ∼ = R to obtain a bi-infinite sequence(6.11) . . . −→ D ( e E ) −→ D ( e E ) −→ R −→ e E −→ e E −→ . . . MULTIPLICATIVE TATE SPECTRAL SEQUENCE 105 of dualisable R -modules in orthogonal G -spectra. This is the sequence [GM95, (9.5)]used by Greenlees and May to define their Tate spectral sequence, at least forfinite G . For G = T = U (1) or U = Sp (1) they instead repeat each term inthis sequence two or four times, respectively. For other compact Lie groups, theconnection is less direct. Proposition 6.32. There is a zig-zag of equivalences from GM ′ ⋆ ( X ) to the se-quence GM ′′ ⋆ ( X ) with GM ′′ k ( X ) = ( e E k ∧ R F R ( E, X ) for k ≥ , D ( e E − k ) ∧ R F R ( E, X ) for k ≤ .Hence the spectral sequence ˇ E r ( X ) is isomorphic to the Greenlees–May Tate spectralsequence [GM95, Thm. 10.3] for π G ∗ applied to the sequence GM ′′ ⋆ ( X ) .Proof. The zig-zag of equivalences connecting GM ′ ⋆ ( X ) to GM ′′ ⋆ ( X ) consists ofidentity maps for ⋆ ≥ 0. For ⋆ ≤ . . . / / F R ( E/E , X ) / / ≃ G (cid:15) (cid:15) F R ( E/E , X ) / / ≃ G (cid:15) (cid:15) F R ( E, X ) = (cid:15) (cid:15) . . . / / F R ( E ∪ CE , X ) / / F R ( E ∪ CE , X ) / / F R ( E, X ) . . . / / F R ( e E ∧ R E, X ) ˜∆ ∗ ≃ G O O / / F R ( e E ∧ R E, X ) ˜∆ ∗ ≃ G O O / / F R ( e E ∧ R E, X ) ˜∆ ∗ ∼ = O O . . . / / D ( e E ) ∧ R F R ( E, X ) ν ≃ G O O / / D ( e E ) ∧ R F R ( E, X ) ν ≃ G O O / / D ( e E ) ∧ R F R ( E, X ) ν ∼ = O O The two top rows are equivalent because each quotient map E ∪ CE i − → E/E i − is an equivalence, since E i − → E is a (strong) h -cofibration. The equivalencebetween the middle two rows is induced by the map e ∆ of mapping cones associatedto the diagonal equivalence ∆ : E i − → E i − ∧ R E : E i − / / ∆ ≃ G (cid:15) (cid:15) E / / = (cid:15) (cid:15) E ∪ CE i − e ∆ ≃ G (cid:15) (cid:15) E i − ∧ R E c ∧ / / R ∧ R E / / e E i ∧ R E . The lower two rows are equivalent because each e E i is dualisable by Lemma 6.29. (cid:3) Remark . Our comparison of the Hesselholt–Madsen Tate spectral sequenceˆ E r ( X ) = E r ( HM ⋆ ( X ))and the Greenlees–May Tate spectral sequenceˇ E r ( X ) = E r ( GM ⋆ ( X )) ∼ = E r ( GM ′ ⋆ ( X )) ∼ = E r ( GM ′′ ⋆ ( X ))is a little different from that of [HM03, Rmk. 4.3.6], since we obtain the Greenleessequence GM ′′ ⋆ ( X ) by splicing the two perpendicular edges ( i = 0 , j ≤ 0) and( i ≥ , j = 0) of the bifiltration HM i,j ( X ) = e E i ∧ R T j ( M ( X )) ≃ G e E i ∧ R F R ( E/E − j − , X ) , while Hesselholt and Madsen first invert a quasi-isomorphism, so as to positionboth halves of the Greenlees sequence on the line j = 0. 06 ALICE HEDENLUND AND JOHN ROGNES Remark . In the case of the circle group G = T we can work over R = S and usethe odd spheres S (( k + 1) C ) to filter E T = S ( ∞ C ), so that e E k = S k C equals (thesuspension spectrum of) a representation sphere. Then D ( e E − k ) = D ( S − k C ) = S k C is a virtual representation sphere for each k < 0. For brevity, let us also write e E k for the latter T -spectra, so that { e E k } k ∈ Z is the bi-infinite sequence (6.11), withcofibre sequences e E k − → e E k → Σ k − T + . The Greenlees–May spectral sequenceassociated to the sequence e E ⋆ ∧ F ( E T + , X )of T -spectra has E -pageˇ E k, ∗ ( X ) = π T k + ∗ (Σ k − T + ∧ F ( E T + , X )) ∼ = π ∗− k ( X )for each k ∈ Z , which we can formally write as t − k · π ∗ ( X ) with t in bidegree( − , − t in bidegree ( − , E k, ∗ ( X ) ∼ = t − k · π ∗ ( X ) and E k − , ∗ ( X ) = 0. Greenlees and May [GM95, Thm. B.8] prove that the latterspectral sequence is isomorphic to one associated to the tower of T -spectra . . . −→ g E T ∧ F ( E T + , X ℓ +1 ) −→ g E T ∧ F ( E T + , X ℓ ) −→ . . . . Here { X ℓ } ℓ denotes a T -equivariant Whitehead tower for X , with homotopy fibresequences X ℓ +1 → X ℓ → Σ ℓ Hπ ℓ ( X ). The latter spectral sequence is indexed sothat E ∗ ,ℓ ( X ) = π T ∗ + ℓ ( g E T ∧ F ( E T + , Σ ℓ Hπ ℓ ( X ))) ∼ = π ∗ ( Hπ ℓ ( X ) t T )for each integer ℓ . In particular, π k ( Hπ ℓ ( X ) t T ) ∼ = t − k · π ℓ ( X ) and π k − ( Hπ ℓ ( X ) t T ) =0, so that, formally, π ∗ ( Hπ ℓ ( X ) t T ) ∼ = π ℓ ( X )[ t, t − ]. Furthermore, Greenlees andMay argue that the latter spectral sequence is multiplicative, with respect to sometopologically defined pairings of the form π ∗ ( Hπ i ( X ) t T ) ⊗ π ∗ ( Hπ j ( Y ) t T ) −→ π ∗ ( Hπ i + j ( Z ) t T ) . However, as is implicit in [GM95, Prob. 14.8], they do not establish that thesetopological pairings agree with the evident algebraic pairings π i ( X )[ t, t − ] ⊗ π j ( Y )[ t, t − ] −→ π i + j ( Z )[ t, t − ] . Hence they do not assert that the isomorphism E ∗ , ∗ ( X ) ∼ = π ∗ ( X )[ t, t − ] takes thetopological product to the algebraic product. In particular, the higher differentialsin this spectral sequence are known to obey a Leibniz rule, but conceivably notwith respect to the most evident algebraic product.Nonetheless, we can confirm directly that the first differential in each of thesespectral sequences is a derivation with respect to the algebraic product. To expressthis, we return to the indexing used elsewhere in the paper, i.e., to the Greenlees–May spectral sequence ˇ E r ∗ , ∗ ( X ). Up to the technical issue we have pointed outabout compatibility of product structures, the following result is due to Hesselholt[Hes96, Lem. 1.4.2]. Proposition 6.35. Let X be any orthogonal T -spectrum, so that π ∗ ( X ) is a right S [ T ] ∗ -module. There is a natural isomorphism ˇ E ∗ , ∗ ( X ) ∼ = π ∗ ( X )[ t, t − ] with t in bidegree ( − , − , such that d : ˇ E k, ∗ ( X ) → ˇ E k − , ∗ ( X ) corresponds to thedifferential d : t − k · π ∗ ( X ) → t − k +1 · π ∗ ( X ) given by d ( t − k · x ) = ( t − k +1 · xs for k even, t − k +1 · x ( s + η ) for k odd. MULTIPLICATIVE TATE SPECTRAL SEQUENCE 107 Proof. By naturality of the Greenlees–May spectral sequence with respect to T -maps x : Σ ℓ S [ T ] → X , corresponding to homotopy classes x ∈ π ℓ ( X ), it suffices toprove the result in the case X = S [ T ] and x = 1 ∈ π ( S [ T ]).Consider the case X = H Z [ T ]. We have π ∗ ( H Z ) = Z { , σ } and H Z [ T ] t T ≃ ∗ since H Z , as a T -spectrum, is induced up from H Z . For bidegree reasons the T -Tatespectral sequence must collapse to zero at the E -page, which forces d ( t − k · 1) = ± t − k +1 · σ . Here, we can iteratively fix the sign of t − k implicit in the identification ˇ E k, ∗ ( X ) ∼ = t − k · π ∗ ( X ) so that each of these signs is a plus. By naturality with respect to theHurewicz homomorphism S [ T ] → H Z [ T ] it follows that d ( t − k · ≡ t − k +1 · s mod t − k +1 · η in the T -Tate spectral sequence for S [ T ], since π ( S [ T ]) = Z { s } ⊕ Z / { η } with theHurewicz homomorphism mapping s to σ .Now consider the case X = S with trivial T -action. The part k ≥ C P ∞ + . Since the 2 k -cell in Σ C P ∞ is stably attached to the 2 k − kη ,it follows that d ( t − k · 1) = t − k +1 · kη for k ≥ 2. Similarly, the part k ≤ D ( C P ∞ + ), where the − k -cell is attached to the − k − kη , so that d ( t − k · 1) = t − k +1 · kη for k ≤ 0, as well. Finally, for k = 1 the differential is induced by the composite T -mapΣ − e E / e E ∧ F ( E T + , S ) −→ e E ∧ F ( E T + , S ) −→ e E / e E − ∧ F ( E T + , S ) , which we can rewrite in terms of the counit ǫ : T + → S and its Spanier–Whiteheaddual D ( ǫ ) : S → D ( T + ) as F ( E T + , T + ) ǫ ∗ −→ F ( E T + , S ) D ( ǫ ) ∗ −→ F ( E T + , D ( T + )) . Passing to T -fixed points, this is a compositeΣ S ≃ S [ T ] h T −→ S h T −→ D ( T + ) h T ≃ S , which we claim equals η ∈ π ( S ). This can be seen using the Pontryagin–Thomcollapse S C → S C /S ∼ = Σ( T + ) associated to the embedding T ⊂ C , and theuntwisting isomorphism ζ : Σ ( T + ) ∼ = Σ C ( T + ). Then ζ (1 ∧ t ) : Σ S C → Σ C ( T + )defines a stable T -map S → T + . The composite ǫζ (1 ∧ t ) : Σ S C → S C has (non-equivariant) Hopf invariant ± 1, since the preimages of two generic points in S C arecircles in Σ S C with that linking number.This proves d ( t − · 1) = t · η in the T -Tate spectral sequence for S and, bynaturality with respect to S [ T ] → S , the asserted formulas follow. (cid:3) Convergence. In this section we use Proposition 6.31 to deduce convergenceresults for the Hesselholt–Madsen spectral sequence ˆ E r ( X ) from correspondingresults for the Greenlees–May spectral sequence ˇ E r ( X ). Lemma 6.36. The map of abutments α ∞ : A ∞ ( GM ⋆ ( X )) ∼ = −→ A ∞ ( HM ⋆ ( X )) ∼ = π ∗ ( X tG ) is an isomorphism. Hence the homomorphism α s : F s A ∞ ( GM ⋆ ( X )) −→ F s A ∞ ( HM ⋆ ( X )) 08 ALICE HEDENLUND AND JOHN ROGNES is injective, for each s ∈ Z .Proof. The first assertion follows from Lemma 6.25. The commutative diagram A s ( GM ⋆ ( X )) / / / / (cid:15) (cid:15) F s A ∞ ( GM ⋆ ( X )) / / / / α s (cid:15) (cid:15) A ∞ ( GM ⋆ ( X )) α ∞ ∼ = (cid:15) (cid:15) A s ( HM ⋆ ( X )) / / / / F s A ∞ ( HM ⋆ ( X )) / / / / A ∞ ( HM ⋆ ( X ))implies the injectivity assertion. (cid:3) Lemma 6.37. The spectral sequence ˇ E r ( X ) = E r ( GM ⋆ ( X )) = ⇒ π ∗ ( X tG ) is conditionally convergent.Proof. The spectral sequence associated to GM ⋆ ( X ) is conditionally convergent tothe colimit, in the sense of [Boa99, Def. 5.10], whenever holim s GM s ( X ) ≃ G ∗ .Since the sequences GM ⋆ ( X ) and GM ′ ⋆ ( X ) are equivalent by Lemma 6.27, we mayequally well verify that holim s GM ′ s ( X ) ≃ G ∗ . But GM ′ s ( X ) = F R ( E/E s − , X ) = M s ( X )for s ≤ 0, and we saw in Section 5.2 that holim s M s ( X ) ≃ G ∗ . Hence the Greenlees–May G -Tate spectral sequence ˇ E r ( X ) is always conditionally convergent. (cid:3) Lemma 6.38. The maps of E ∞ - and RE ∞ -pages E ∞ ( GM ⋆ ( X )) ∼ = −→ E ∞ ( HM ⋆ ( X )) RE ∞ ( GM ⋆ ( X )) ∼ = −→ RE ∞ ( HM ⋆ ( X )) are isomorphisms.Proof. Recall from [Boa99, (5.1)] that for each spectral sequence ( E r , d r ) there arefiltrations 0 = B ⊂ B ⊂ B ⊂ · · · ⊂ Z ⊂ Z ⊂ Z = E with E r ∼ = Z r /B r for r ≥ 1. We set B ∞ = colim r B r , Z ∞ = lim r Z r , E ∞ = Z ∞ /B ∞ , RE ∞ = Rlim r Z r . Letting ¯ B r = B r /B and ¯ Z r = Z r /B for r ≥ 2, we obtain a filtration0 = ¯ B ⊂ ¯ B ⊂ · · · ⊂ ¯ Z ⊂ ¯ Z ∼ = E with E r ∼ = ¯ Z r / ¯ B r for r ≥ 2. Let¯ B ∞ = colim r ¯ B r and ¯ Z ∞ = lim r ¯ Z r . Then ¯ B ∞ ∼ = B ∞ /B , while ¯ Z ∞ ∼ = Z ∞ /B and Rlim r Z r ∼ = Rlim r ¯ Z r by the lim-Rlim exact sequence. Hence E ∞ ∼ = ¯ Z ∞ / ¯ B ∞ by the Noether isomorphism, and RE ∞ ∼ = Rlim r ¯ Z r .A map of spectral sequences inducing an isomorphism of E -pages will by in-duction induce isomorphisms of ¯ B r - and ¯ Z r -pages for all r ≥ 2, and therefore alsoof ¯ B ∞ -, ¯ Z ∞ -, E ∞ - and RE ∞ -pages. (cid:3) When X is bounded below, and G is finite or equal to T = U (1) or U = Sp (1),the E -pages ˇ E ( X ) and ˆ E ( X ) are both concentrated in half-planes with enteringdifferentials [Boa99, § G the E -page ˇ E ( X )occupies a region that is only bounded by a broken line, and ˆ E ( X ) may notbe bounded in any ordinary sense. We therefore need to discuss convergence for MULTIPLICATIVE TATE SPECTRAL SEQUENCE 109 the spectral sequences ˇ E r ( X ) and ˆ E r ( X ) in the generality of whole-plane spectralsequences [Boa99, § Definition 6.39. Let ( A, E ) be the exact couple associated to a Cartan–Eilenbergsystem ( H, ∂ ). Boardman’s whole-plane obstruction group W is defined in [Boa99,Lem. 8.5] by an expression W = colim s Rlim r K ∞ im r A s . We refer to Boardman’s paper for an explanation of the notation. By [HR19,Thm. 7.5] there is an isomorphism W ∼ = ker( κ )where κ : colim j lim i H ( i, j ) −→ lim i colim j H ( i, j )is the interchange morphism, which is always surjective.While W is defined in terms of the underlying exact couple (or Cartan–Eilenbergsystem), Boardman gives the following criterion for the vanishing of W , which isinternal to the spectral sequence. Lemma 6.40 ([Boa99, Lem. 8.1]) . Suppose that for each m , there exist num-bers u ( m ) and v ( m ) such that for all u ≥ u ( m ) and v ≥ v ( m ) , the differential d u + v : E u + vu,m − u −→ E u + v − v,m + v − vanishes. Then W = 0 .Remark . If for some fixed r the E r -page of the spectral sequence is boundedfrom the side of entering differentials, in the sense that for each m there is a num-ber u ( m ) such that E ru,m − u = 0 for all u ≥ u ( m ), then Boardman’s vanishingcriterion is satisfied with v ( m ) = r − u ( m ). Hence W = 0 in these cases.Alternatively, if the spectral sequence collapses at the E r -page, so that d r andall later differentials are zero, then Boardman’s vanishing criterion is satisfiedwith u ( m ) = r and v ( m ) = 0. Thus W = 0 also in these cases. Theorem 6.42. The spectral sequence ˇ E r ( X ) = E r ( GM ⋆ ( X )) converges stronglyto A ∞ ( GM ⋆ ( X )) ∼ = π ∗ ( X tG ) if and only if RE ∞ = 0 and W = 0 for this spectralsequence.Proof. We saw that ˇ E r ( X ) is conditionally convergent in Lemma 6.37. Hence thestatements ‘ RE ∞ = 0 and W = 0’ and ‘the spectral sequence is strongly convergent’are equivalent by [Boa99, Thm. 8.10]. (cid:3) Theorem 6.43. If the Greenlees–May spectral sequence ˇ E r ( X ) = E r ( GM ⋆ ( X )) = ⇒ π ∗ ( X tG ) is strongly convergent, then the Hesselholt–Madsen spectral sequence ˆ E r ( X ) = E r ( HM ⋆ ( X )) = ⇒ π ∗ ( X tG ) is strongly convergent, as well. Moreover, F s A ∞ ( GM ⋆ ( X )) = F s A ∞ ( HM ⋆ ( X )) forall integers s .Proof. We assume ˇ E r ( X ) is strongly convergent. Explicitly, this means that theexhaustive filtration ( F s A ∞ ( GM ⋆ ( X ))) s of A ∞ ( GM ⋆ ( X )) ∼ = π ∗ ( X tG ) is complete 10 ALICE HEDENLUND AND JOHN ROGNES Hausdorff, and the left hand monomorphism β in the commutative square F s A ∞ ( GM ⋆ ( X )) F s − A ∞ ( GM ⋆ ( X )) ¯ α s / / β ∼ = (cid:15) (cid:15) F s A ∞ ( HM ⋆ ( X )) F s − A ∞ ( HM ⋆ ( X )) (cid:15) (cid:15) β (cid:15) (cid:15) E ∞ s ( GM ⋆ ( X )) α ∞ s ∼ = / / E ∞ s ( HM ⋆ ( X ))is an isomorphism. It follows that the right hand monomorphism β is also anisomorphism. Since the filtration ( F s A ∞ ( HM ⋆ ( X ))) s is exhaustive, this meansthat ˆ E r ( X ) converges weakly to π ∗ ( X tG ). It also follows that the upper homomor-phism ¯ α s is an isomorphism. By induction, this implies that the map of filtrationquotients F t A ∞ ( GM ⋆ ( X )) F s A ∞ ( GM ⋆ ( X )) ∼ = −→ F t A ∞ ( HM ⋆ ( X )) F s A ∞ ( HM ⋆ ( X ))is an isomorphism for all integers s ≤ t .Passing to colimits over t , and using the fact that α ∞ : A ∞ ( GM ⋆ ( X )) ∼ = −→ A ∞ ( HM ⋆ ( X ))is an isomorphism by Lemma 6.36, we deduce that α s : F s A ∞ ( GM ⋆ ( X )) → F s A ∞ ( HM ⋆ ( X ))is an isomorphism, for each s ∈ Z . The filtration ( F s A ∞ ( HM ⋆ ( X ))) s is thereforecomplete and Hausdorff, meaning that ˆ E r ( X ) converges strongly to π ∗ ( X tG ). (cid:3) Combining these results we obtain the following theorem, which often compen-sates for the problem that we do not a priori know when ˆ E r ( X ) = E r ( HM ⋆ ( X ))is conditionally convergent, cf. Remark 6.15. Theorem 6.44. If RE ∞ = 0 and Boardman’s vanishing criterion for W fromLemma 6.40 is satisfied for the G -Tate spectral sequence ˆ E r ( X ) = E r ( HM ⋆ ( X )) ,then this spectral sequence converges strongly and conditionally to A ∞ ( HM ⋆ ( X )) ∼ = π ∗ ( X tG ) . Note that we are not just assuming that W = 0 for ˆ E r ( X ), but that this groupvanishes for the reason given by Boardman’s criterion. Proof. Since ˇ E r ( X ) ∼ = ˆ E r ( X ) for r ≥ 2, the vanishing of RE ∞ for ˆ E r ( X ) impliesthe vanishing of RE ∞ for ˇ E r ( X ). Furthermore, the hypothesis of Boardman’s cri-terion for ˆ E r ( X ) implies the same hypothesis for ˇ E r ( X ). Hence ˇ E r ( X ) convergesstrongly by Theorem 6.42, which implies that ˆ E r ( X ) converges strongly by The-orem 6.43. By [Boa99, Thm. 8.10] strong convergence and the vanishing of RE ∞ and W imply conditional convergence. (cid:3) Summary : The T -Tate spectral sequence. The main example we had inmind when writing this paper was G = T . Note that when discussing the T -Tatespectral sequence for a T -spectrum X one could really refer to at least two differentspectral sequences: one arising from the Greenlees–May filtration and one from theHesselholt–Madsen filtration. The first has better convergence properties, whilethe latter has better multiplicative properties. Fortunately there are quite goodcomparison results between the two, as covered in Section 6.6.Let us start by summarizing the additive results regarding the Greenlees–Mayand Hesselholt–Madsen versions of the T -Tate spectral sequence. We work over There is also at least one more Tate spectral sequence, namely the one arising from a Postnikovor Whitehead tower of X ; see Remark 5.5 and Remark 6.34. MULTIPLICATIVE TATE SPECTRAL SEQUENCE 111 R = S , and write ⊗ for ⊗ S ∗ . We first note that by virtue of X being a T -spectrum,there is an action γ : X ∧ T + ∼ = X ∧ S [ T ] −→ X which makes X into a right module over the spherical group ring S [ T ]. The inducedpairing γ ∗ : π ∗ ( X ) ⊗ S [ T ] ∗ −→ π ∗ ( X )on homotopy groups then gives π ∗ ( X ) the structure of a right module over the Hopfalgebra S [ T ] ∗ ∼ = S ∗ [ s ] / ( s = ηs ) , | s | = 1 . Here η is the image of the complex Hopf map in π ( S ) ∼ = Z / 2. Note that this Hopfalgebra is finitely generated and projective over S ∗ . We denote the image γ ∗ ( x ⊗ s )by xs .There is a minimal projective S [ T ] ∗ -module resolution P ∗ of S ∗ , with P k = S [ T ] { p k } and ∂ ( p k ) = p k − ( s + ( k − η ). Let e P ∗ be the mapping cone of theaugmentation ǫ : P ∗ → S ∗ . Let the complete resolution ˆ P ∗ be the fibre product P ∗ × S ∗ D ( e P ∗ ), which is obtained by splicing P ∗ with its dual. Theorem 6.45 (Greenlees–May–Tate spectral sequence) . Given an orthogonal T -spectrum X , there is a filtration GM ⋆ ( X ) of orthogonal T -spectra, and an associ-ated S ∗ -module spectral sequence ˇ E r ( X ) = E r ( GM ⋆ ( X )) with abutment A ∞ ( GM ⋆ ( X )) ∼ = π ∗ ( X t T ) filtered by the images im( π ∗ GM ( X s ) → π ∗ ( X t T )) . We refer to this spectral sequenceas the Greenlees–May T -Tate spectral sequence for X . The following hold: E -page: The E -page of the Greenlees–May T -Tate spectral sequence can bewritten E ∗ , ∗ ( GM ⋆ ( X )) ∼ = Hom S [ T ] ∗ ( ˆ P ∗ , π ∗ ( X )) where ˆ P ∗ is a complete resolution of S ∗ as a trivial S [ T ] ∗ -module. For theminimal such resolution we can write E ∗ , ∗ ( GM ⋆ ( X )) ∼ = π ∗ ( X )[ t, t − ] with t in bidegree ( − , − , and then d ( t c · x ) = t c +1 · x ( s + cη ) for all c ∈ Z and x ∈ π ∗ ( X ) . Convergence: The Greenlees–May spectral sequence converges conditionallyto the abutment. It converges strongly to the abutment if and only if thederived E ∞ -page RE ∞ and Boardman’s whole plane obstruction group W are both trivial.Proof. The first statement is Lemma 6.30 combined with Proposition 2.22 and Propo-sition 6.35. The second statement is Lemma 6.37 combined with Theorem 6.42. (cid:3) Theorem 6.46 (Hesselholt–Madsen–Tate spectral sequence) . Given an orthogo-nal T -spectrum X , there is a filtration HM ⋆ ( X ) of orthogonal T -spectra, and anassociated S ∗ -module spectral sequence ˆ E r ( X ) = E r ( HM ⋆ ( X )) with abutment A ∞ ( HM ⋆ ( X )) ∼ = π ∗ ( X t T ) filtered by the images im( π ∗ HM ( X s ) → π ∗ ( X t T )) . We refer to this spectral sequenceas the Hesselholt–Madsen T -Tate spectral sequence for X . The following hold: 12 ALICE HEDENLUND AND JOHN ROGNES E -page: The E -page of the Hesselholt–Madsen T -Tate spectral sequence canbe written E ∗ , ∗ ( HM ⋆ ( X )) ∼ = Hom S [ T ] ∗ ( S ∗ , e P ∗ ⊗ Hom( P ∗ , π ∗ ( X ))) where P ∗ is a projective resolution of S ∗ as a trivial S [ T ] ∗ -module and e P ∗ denotes the mapping cone of the augmentation ǫ : P ∗ → S ∗ . E -page: The E -page is given in terms of Hopf algebra Tate cohomology,alias complete Ext , as E ∗ , ∗ ( HM ⋆ ( X )) ∼ = d Ext −∗ S [ T ] ∗ ( S ∗ , π ∗ ( X )) . Convergence: If the Greenlees–May T -Tate spectral sequence for X con-verges strongly, then the Hesselholt–Madsen T -Tate spectral sequence for X converges strongly, and the two associated filtrations of π ∗ ( X t T ) agree.Proof. The first statement is Proposition 6.16, the second is Theorem 6.17, and thethird statement is Theorem 6.43. (cid:3) Worth pointing out is that the Greenlees–May and the Hesselholt–Madsen ver-sions of the T -Tate spectral sequence are isomorphic from the E -page and on, perProposition 6.31. In particular, the E -page of both spectral sequences is given byˆ E − c, ∗ ( X ) ∼ = ˇ E − c, ∗ ( X ) ∼ = d Ext c S [ T ] ∗ ( S ∗ , π ∗ ( X )) ∼ = ker( s : π ∗ ( X ) → π ∗ +1 ( X ))im( s + η : π ∗− ( X ) → π ∗ ( X )) for c even,ker( s + η : π ∗ ( X ) → π ∗ +1 ( X ))im( s : π ∗− ( X ) → π ∗ ( X )) for c odd,where the last isomorphism is the result of the computation of Section 2.6.Regarding convergence, we note that Lemma 6.40 gives a criterion, internal tothe spectral sequence itself, for when Boardman’s whole-plane obstruction vanishes.In particular, if X is bounded below, either version of the Tate spectral sequenceis a half-plane spectral sequence with entering differentials (at least from the E -page), which guarantees this. In the applications we have in mind, we are in thissituation if we consider topological Hochschild homology X = THH( B ) for someconnective orthogonal ring spectrum B .Let us now summarise the multiplicative structure of the two spectral sequencesdiscussed. Theorem 6.47.Multiplicativity: The Hesselholt–Madsen T -Tate spectral sequence is mul-tiplicative in the sense that a pairing φ : X ∧ Y → Z of orthogonal T -spectragives rise to a pairing of the associated spectral sequences φ : ( ˆ E ∗ ( X ) , ˆ E ∗ ( Y )) → ˆ E ∗ ( Z ) . Explicitly, φ gives rise to homomorphisms φ r : ˆ E r ( X ⋆ ) ⊗ ˆ E r ( Y ⋆ ) −→ ˆ E r ( Z ⋆ ) for all r ≥ , such that: (1) The Leibniz rule d r φ r = φ r ( d r ⊗ 1) + φ r (1 ⊗ d r ) holds as an equality of homomorphisms ˆ E ri ( X ) ⊗ ˆ E rj ( Y ) −→ ˆ E ri + j − r ( Z ) for all i, j ∈ Z and r ≥ . MULTIPLICATIVE TATE SPECTRAL SEQUENCE 113 (2) The diagram ˆ E r +1 ( X ) ⊗ ˆ E r +1 ( Y ) ˆ E r +1 ( Z ) H ( ˆ E r ( X ) ⊗ ˆ E r ( Y )) H ( ˆ E r ( Z )) φ r +1 ∼ = H ( φ r ) commutes for all r ≥ . (3) The pairing φ : ˆ E ( X ⋆ ) ⊗ ˆ E ( Y ⋆ ) −→ ˆ E ( Z ⋆ ) agrees with the cup product ⌣ : d Ext − i, ∗ S [ T ] ∗ ( S ∗ , π ∗ ( X )) ⊗ d Ext − j, ∗ S [ T ] ∗ ( S ∗ , π ∗ ( Y )) −→ d Ext − i − j, ∗ S [ T ] ∗ ( S ∗ , π ∗ ( Z )) . For r ≥ the same statements hold for the Greenlees–May spectral se-quence, with ˇ E r in place of ˆ E r . Multiplicative abutment: We have an induced pairing φ ∗ : π ∗ ( X t T ) ⊗ π ∗ ( Y t T ) −→ π ∗ ( Z t T ) of abutments with the Hesselholt–Madsen filtrations, which is compatiblewith the pairing φ ∞ of E ∞ -pages. Explicitly, the diagram F i π ∗ ( X t T ) F i − π ∗ ( X t T ) ⊗ F j π ∗ ( Y t T ) F j − π ∗ ( Y t T ) ¯ φ ∗ / / β ⊗ β (cid:15) (cid:15) F i + j π ∗ ( Z t T ) F i + j − π ∗ ( Z t T ) (cid:15) (cid:15) β (cid:15) (cid:15) ˆ E ∞ i ( X ) ⊗ ˆ E ∞ j ( Y ) φ ∞ / / ˆ E ∞ i + j ( Z ) commutes, for all i, j ∈ Z .If the Greenlees–May spectral sequence is strongly convergent, then thesame statements hold for the Greenlees–May filtrations and ˇ E ∞∗ , ∗ .Proof. For the Hesselholt–Madsen T -Tate spectral sequence this is Theorem 6.18and Theorem 6.21. The statements about multiplicativity of the E r -pages and d r -differentials can be transported to the Greenlees–May spectral sequence for r ≥ GM ⋆ ( X ) and HM ⋆ ( X ) induce the same filtration on π ∗ ( X t T ), which holdsunder the hypothesis of strong convergence by Theorem 6.43. (cid:3) Recalling the discussion of Remark 2.54, in the context of the circle group, theHopf algebra Tate cohomology can also be described as the homology of the differ-ential graded S [ T ] ∗ -module π ∗ ( X )[ t, t − ] , | t | = − d ( x ) = txs and d ( t ) = t η . Moreover, given a pairing X ∧ Y → Z we have an induced pairing π ∗ ( X ) ⊗ S ∗ π ∗ ( Y ) → π ∗ ( Z ) on homotopy groups, and the cup product on Tate cohomology is preciselythe one induced by the obvious map π ∗ ( X )[ t, t − ] ⊗ π ∗ ( Y )[ t, t − ] −→ π ∗ ( Z )[ t, t − ]on homology. By Theorem 6.21, the multiplicative structure on the second page of(both versions of) the T -Tate spectral sequence corresponds to this cup product. 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