A construction of some objects in many base cases of an Ausoni-Rognes conjecture
aa r X i v : . [ m a t h . A T ] J u l A CONSTRUCTION OF SOME OBJECTS IN MANY BASECASES OF AN AUSONI-ROGNES CONJECTURE
DANIEL G. DAVIS
Abstract.
Let p be a prime, n ≥ K ( n ) the n th Morava K –theory spec-trum, G n the extended Morava stabilizer group, and K ( A ) the algebraic K –theory spectrum of a commutative S –algebra A . For a type n + 1 complex V n ,Ausoni and Rognes conjectured that the unit map i n : L K ( n ) ( S ) → E n fromthe K ( n )–local sphere to the Lubin-Tate spectrum induces a weak equivalence K ( L K ( n ) ( S )) ∧ v − n +1 V n → ( K ( E n )) h G n ∧ v − n +1 V n , where π ∗ ( − ) of the target is the abutment of a homotopy fixed point spectralsequence. Since G n is profinite, ( K ( E n )) h G n denotes a continuous homotopyfixed point spectrum, and for every n and p , there are no known constructionsof it, the above target, or the spectral sequence. For n = 1, p ≥
5, and V = V (1), we give a way to realize the last two objects by proving that i induces a map K ( L K (1) ( S )) ∧ v − V → ( K ( E ) ∧ v − V ) h G =: K , where K is a continuous homotopy fixed point spectrum, with the expectedspectral sequence associated to it. Though we do not construct ( K ( E )) h G ,we prove that K ≃ ( K ( E )) e h G ∧ v − V , where ( K ( E )) e h G is the homotopyfixed points with G regarded as a discrete group. Introduction
An overview of an Ausoni-Rognes conjecture and statements of ourmain theorems.
Let n ≥ p be a prime. Let E n be the Lubin-Tatespectrum with π ∗ ( E n ) = W ( F p n ) J u , ..., u n − K [ u ± ], where W ( F p n ) is the ring ofWitt vectors of the field F p n (with p n elements), the complete power series ringis in degree zero, and | u | = 2, and let G n be the n th extended Morava stabilizergroup. By [21, 23], E n is a commutative S –algebra and the group G n acts on E n bymaps of commutative S –algebras. Given a commutative S –algebra A , the algebraic K –theory spectrum of A , K ( A ), is a commutative S –algebra. Thus, K ( E n ) is acommutative S –algebra, and by the functoriality of K ( − ), G n acts on K ( E n ) bymaps of commutative S –algebras.Let L K ( n ) ( S ) denote the Bousfield localization of the sphere spectrum withrespect to K ( n ), the n th Morava K –theory spectrum. The group G n is profinite,and by [33, 8], the K ( n )–local unit map(1.1) L K ( n ) ( S ) → E n is a consistent profaithful K ( n )–local profinite G n –Galois extension.Now let V n be a finite p –local complex of type n + 1 and let v : Σ d V n → V n be a v n +1 –self-map, where d is some positive integer (see [25, Theorem 9]). The map v nduces a sequence V n → Σ − d V n → Σ − d V n → · · · of maps of spectra, and we let v − n +1 V n = colim j ≥ Σ − jd V n , the colimit of the above sequence, denote the mapping telescope associated to the v n +1 –self-map v . As hinted at by the notation, the mapping telescope v − n +1 V n isindependent of the choice of self-map v .In [5, Conjecture 4.2], [3, page 46; Remark 10.8], and [4, paragraph containing(0.1)], Christian Ausoni and John Rognes conjectured that the G n –Galois extension L K ( n ) ( S ) → E n induces a map(1.2) K ( L K ( n ) ( S )) ∧ v − n +1 V n → ( K ( E n )) h G n ∧ v − n +1 V n that is a weak equivalence, and associated with the target of this weak equivalence,there exists a homotopy fixed point spectral sequence that has the form E s,t = H sc (cid:0) G n ; ( V n ) t ( K ( E n ))[ v − n +1 ] (cid:1) = ⇒ ( V n ) t − s (( K ( E n )) h G n )[ v − n +1 ] , where the E –term is given by continuous cohomology and the conjectural spectrum( K ( E n )) h G n is a continuous homotopy fixed point spectrum. This conjecture isan extension of the Lichtenbaum-Quillen conjectures (for example, see [37, (0.1),Theorem 4.1]), which can be viewed as corresponding to n = 0 versions of theabove (see [5], [3, Section 10]). More generally, the conjecture is related to tryingto understand ´etale descent for the algebraic K –theory of commutative S –algebras;for more details about this, see [4, Introduction] and [34, Section 4]. Remark 1.3.
The above two conjectural statements are just a piece of an impor-tant family of conjectures – which include the chromatic redshift conjecture – madeby Ausoni and Rognes; we only state the part that we focus on in this paper. Formore information about these conjectures, see [3, 4, 5, 6, 34].Notice that for every integer t , there is an isomorphism( V n ) t ( K ( E n ))[ v − n +1 ] ∼ = π t ( K ( E n ) ∧ v − n +1 V n ) . Thus, when the above homotopy fixed point spectral sequence exists, since itsabutment should be π ∗ ( − ) of a homotopy fixed point spectrum, there should alsobe an equivalence(1.4) ( K ( E n )) h G n ∧ v − n +1 V n ≃ ( K ( E n ) ∧ v − n +1 V n ) h G n , where the right-hand side is a continuous homotopy fixed point spectrum. Obtain-ing equivalence (1.4) and a homotopy fixed point spectral sequence E s,t = H sc ( G n ; π t ( K ( E n ) ∧ v − n +1 V n )) = ⇒ π t − s (cid:0) ( K ( E n ) ∧ v − n +1 V n ) h G n (cid:1) immediately implies the existence of the conjectural spectral sequence above.To make progress on the above two-part conjecture, one obstacle that must beovercome is that currently, there are no known constructions of the continuoushomotopy fixed point spectra( K ( E n )) h G n , ( K ( E n ) ∧ v − n +1 V n ) h G n for any n and p . Directly related to this issue is the fact that there are also no knownconstructions of the above two descent spectral sequences (here and elsewhere, we se the term “descent spectral sequence” in place of “homotopy fixed point spectralsequence”).In this paper, we remedy part of the situation just described in certain basecases: for n = 1, p ≥
5, and V = V (1), the type 2 Smith-Toda complex S / ( p, v ),we give an elementary construction of( K ( E ) ∧ v − V ) h G and we obtain the desired descent spectral sequence E s,t = H sc ( G ; π t ( K ( E ) ∧ v − V )) = ⇒ π t − s (cid:0) ( K ( E ) ∧ v − V ) h G (cid:1) . Remark 1.5.
Our work addresses aspects of an Ausoni-Rognes conjecture involv-ing a certain Galois extension, where the relevant group, G n , is infinite and profinite.For K ( n )–local G –Galois extensions A → B , where G is a finite group, Ausoni andRognes have made a conjecture similar to the one encapsulated above in (1.2) [5,Conjecture 4.2], and in these cases, since G is naturally discrete, it is well-knownthat ( K ( B )) hG always exists, and so there is no issue with the statement of theconjecture. For these cases, progress on the conjecture has been made by [15].Given our hypotheses – n = 1, p ≥
5, and V = V (1), we can be a little moreconcrete about some of the main actors in the scenario that we focus on: E = KU p ,p –completed complex K –theory; G = Z × p , the group of units in the p –adic integers Z p ; and v − V (1) = colim j ≥ Σ − jd V (1) . Then our first result is actually an extension of the aforementioned new n = 1constructions to all closed subgroups of Z × p . Theorem 1.6.
Let p ≥ . Given any closed subgroup K of Z × p , there is a stronglyconvergent descent spectral sequence E s,t = H sc ( K ; π t ( K ( KU p ) ∧ V (1))[ v − ]) = ⇒ π t − s (cid:0)(cid:0) K ( KU p ) ∧ v − V (1) (cid:1) hK (cid:1) , with E s,t = 0 , for all s ≥ and any t ∈ Z . Also, there is an equivalence of spectra (cid:0) K ( KU p ) ∧ v − V (1) (cid:1) hK ≃ colim j ≥ ( K ( KU p ) ∧ Σ − jd V (1)) hK . In the above result, the subgroup K is a profinite group and each application of( − ) hK denotes a continuous homotopy fixed point spectrum (as in [8]), formed inthe setting of symmetric spectra of simplicial sets.Our next two results are about (cid:0) K ( KU p ) ∧ v − V (1) (cid:1) h Z × p . Theorem 1.7.
When p ≥ , there is a canonical map K ( L K (1) ( S )) ∧ v − V (1) → (cid:0) K ( KU p ) ∧ v − V (1) (cid:1) h Z × p , induced by the K (1) –local unit map L K (1) ( S ) → KU p , in the category of symmetricspectra. or n = 1, p ≥
5, and V = V (1), if (1.4) were valid, then Theorem 1.7 wouldyield the conjectural map in (1.2), as a map in the stable homotopy category.However, in this paper, we do not show that the morphism in Theorem 1.7 is aweak equivalence. But we hope that the spectral sequence of Theorem 1.6 when K = Z × p is a useful computational tool for this problem.Though this paper does not construct ( K ( KU p )) h Z × p for any p (so that (1.4)remains conjectural in all n = 1 cases), we do have a related result. Before statingthis result, we recall that if G is any profinite group and X is a (naive) G –spectrum,then G can be regarded as a discrete group and one can always form the “discretehomotopy fixed point spectrum” X e hG = Map G ( EG + , X )(the usual notation for X e hG omits the “ e , ” but we use it here to distinguish ( − ) e hG from the continuous ( − ) hG ). Theorem 1.8.
When p ≥ , there is an equivalence of spectra (cid:0) K ( KU p ) ∧ v − V (1) (cid:1) h Z × p ≃ ( K ( KU p )) e h Z × p ∧ v − V (1) . Remark 1.9.
It is worth pointing out that in proving Theorem 1.8, we show that(for p ≥
5) there is a mapcolim j ≥ ( K ( KU p ) ∧ Σ − jd V (1)) e h Z × p ≃ −→ colim j ≥ ( K ( KU p ) ∧ Σ − jd V (1)) h Z × p that is a weak equivalence.In (1.4), when n = 1, if ( K ( E )) h G = ( K ( KU p )) h Z × p is changed to ( K ( KU p )) e h Z × p ,then Theorem 1.8 is an instance of this “modified (1.4).” But we do not take this ob-servation as evidence that ( K ( KU p )) e h Z × p should be the definition of ( K ( KU p )) h Z × p for some p , and thus, as noted earlier, we believe that for all p , the proper con-struction of ( K ( KU p )) h Z × p is still an open problem.The proofs of Theorems 1.6, 1.7, and 1.8 are given in the first part of Section 8,that section’s second part, and Section 9, respectively.1.2. The construction of the continuous homotopy fixed point spectra inTheorem 1.6.
We now explain our work in more detail. Let G be a profinitegroup and let X be a G –spectrum. Then there is X e hG and one can always formthe associated descent spectral sequence E s,t = H s ( G ; π t ( X )) = ⇒ π t − s ( X e hG ) , with E –term given by (non-continuous) group cohomology. However, it is not( K ( E n ) ∧ v − n +1 V n ) e h G n that the conjecture of Ausoni and Rognes is concerned with.Since G n is profinite and the E –term of the conjectured spectral sequence is givenby continuous cohomology, one wants a continuous homotopy fixed point spectrum( K ( E n ) ∧ v − n +1 V n ) h G n that takes the profinite topology of G n into account; that is,we would like to know that K ( E n ) ∧ v − n +1 V n is a continuous G n –spectrum in somesense, and that ( K ( E n ) ∧ v − n +1 V n ) h G n can be formed with respect to the continuousaction.To address this problem in the n = 1 , p ≥ G , wework with discrete G –spectra (as in [8]) within the framework of symmetric spectra f simplicial sets (for more detail, see the end of the introduction). For the moment,let X be a discrete G –spectrum. Then for all k, l ≥
0, the l -simplices of the k thpointed simplicial set of X , X k,l , is a discrete G –set. Also, the homotopy fixedpoint spectrum X hG is defined (in [8], as recalled at the end of the introduction)in a way that respects the profinite topology of G . Throughout this paper, we use( − ) hG for these “continuous” homotopy fixed points.The following convention and terminology (from [12]) will be helpful to us. Definition 1.10.
Let X be a spectrum (that is, a symmetric spectrum). By“ π ∗ ( X ),” we always mean the homotopy groups π t ( X ) := [ S t , X ] , t ∈ Z , of morphisms S t → X in the homotopy category of symmetric spectra, where here, S t denotes a fixed cofibrant and fibrant model for the t -th suspension of the spherespectrum. Definition 1.11 ([12, page 5]) . A spectrum X is an f –spectrum if π t ( X ) is finitefor every integer t .Recall that a profinite group is strongly complete if every subgroup of finiteindex is open. Let p be any prime: since Z p is strongly complete, it follows thatthe profinite group Z p × H , where H is any finite discrete group and Z p × H isequipped with the product topology, is strongly complete. Thus (see Remark 3.2),if M is any ( Z p × H )–module that is finite, then M is a discrete ( Z p × H )–module.Then, as an immediate consequence of Theorem 3.6 – the proof of which uses [20]in a key way – and our central result, Theorem 4.9, we have the following. Theorem 1.12.
Let p be any prime and let H be any finite discrete group. If X isa ( Z p × H ) –spectrum and an f –spectrum, then X is a discrete ( Z p × H ) –spectrum. We state the conclusion of the above result more precisely: under the hypothesesof Theorem 1.12, there is a zigzag(1.13) X ≃ −→ X ′ ≃ ←− X dis N of ( Z p × H )–spectra and ( Z p × H )–equivariant maps that are weak equivalences ofsymmetric spectra, and X dis N is a discrete ( Z p × H )–spectrum. Thus, as in Definition6.2, it is natural to identify X with the discrete ( Z p × H )–spectrum X dis N and todefine X h ( Z p × H ) = ( X dis N ) h ( Z p × H ) . To go further, we need to introduce some notation and make a few comments.Let ΣSp denote the model category of symmetric spectra (as in [26, Theorem 3.4.4]).We use ( − ) f : ΣSp → ΣSp , Z Z f to denote a fibrant replacement functor, so that given the spectrum Z , there isa natural map Z → Z f that is a trivial cofibration, with Z f fibrant. It is usefulto note that if X is a G –spectrum, then X f is also a G –spectrum and the trivialcofibration X → X f is G –equivariant. Similarly, if p : X → Y is a map of G –spectra(thus, p is G –equivariant), then p f : X f → Y f is a map of G –spectra. e want to highlight the fact that in zigzag (1.13), the construction of X dis N iselementary: by Definition 4.4,(1.14) X dis N = colim m ≥ holim [ n ] ∈ ∆ (cid:0) Sets( Z p × H, · · · , Sets( Z p × H, | {z } ( n +1) times X f ) · · · ) | {z } ( n +1)times (cid:1) ( p m Z p ) ×{ e } , where each ( p m Z p ) × { e } is an (open normal) subgroup of Z p × H and p m Z p hasits usual meaning. We would like the reader to see how basic the construction of X dis N is, and thus, in this introduction, we do not think it is necessary to give anyfurther explanation of (1.14). It turns out that for a ( Z p × H )–spectrum X that isan f –spectrum,(1.15) X h ( Z p × H ) ≃ (cid:16) holim [ n ] ∈ ∆ Sets( Z p × H, · · · , Sets( Z p × H, | {z } ( n +1) times X f ) · · · ) | {z } ( n +1)times (cid:17) Z p × H , by Theorem 6.4. We are confident that without any additional explanation, thereader has at least an almost complete understanding of the meaning of the ex-pression in (1.15); later reading about it (and (1.14)) will mostly just confirm thereader’s “native conclusions.”We now explain our application of Theorem 1.12 to the conjecture of Ausoni andRognes. Let p ≥
5. Then Z × p ∼ = Z p × Z / ( p − , and as discussed earlier, K ( KU p ) is a Z × p –spectrum. By giving V (1) the trivial Z × p –action, K ( KU p ) ∧ V (1) is a Z × p –spectrum under the diagonal action.Let ku p be the p –completed connective complex K –theory spectrum, with coef-ficients π ∗ ( ku p ) = Z p [ u ], where | u | = 2, as before. In [9], Andrew Blumberg andMichael Mandell proved a conjecture of Rognes that there is a localization cofibersequence(1.16) K ( Z p ) → K ( ku p ) → K ( KU p ) → Σ K ( Z p ) , and hence, there is a cofiber sequence(1.17) K ( Z p ) ∧ V (1) → K ( ku p ) ∧ V (1) → K ( KU p ) ∧ V (1) → Σ( K ( Z p ) ∧ V (1)) . By [11], it is known that K ( Z p ) ∧ V (1) is an f –spectrum (see also [2, pages 663–664]for a helpful discussion about V (1) ∗ K ( Z p )). Also, Ausoni [2, Theorems 1.1, 8.1]showed that there exists an element b ∈ V (1) p +2 K ( ku p ) such that if F p [ b ] ⊂ V (1) ∗ K ( ku p )denotes the polynomial F p –subalgebra generated by b , then there is a short exactsequence of graded F p [ b ]–modules0 → Σ p − F p → V (1) ∗ K ( ku p ) → F → , where F is a free F p [ b ]–module on 4 p + 4 generators. (Work of Rognes with Ausoniplayed a role in the Ausoni result: for example, see [32, Section 8]. Also, [2,Theorems 1.1, 8.1] were, in some sense, anticipated by [7, discussion of Lemma6.6], as explained in [2, discussion of Proposition 1.4].)It follows from the last result that K ( ku p ) ∧ V (1) is an f –spectrum, and hence,cofiber sequence (1.17) implies that K ( KU p ) ∧ V (1) is an f –spectrum. Therefore,by setting H = Z / ( p −
1) in Theorem 1.12, we obtain that K ( KU p ) ∧ V (1) is (inthe sense of zigzag (1.13)) a discrete Z × p –spectrum. emark 1.18. Given our conclusion that K ( KU p ) ∧ V (1) is an f –spectrum for p ≥
5, it is natural to wonder if, for an arbitrary prime p , K ( E n ) ∧ V n is an f –spectrum for n ≥
2. A starting point for considering this question would be acofiber sequence analogous to the one in (1.16). For n ≥ E ( n ) p , the p –completionof the Johnson-Wilson spectrum E ( n ), and E n are closely related, and in [4, page5], Ausoni and Rognes state that they expect there to be such a cofiber sequenceinvolving K ( E ( n ) p ) (for a precise description of this sequence, see [ibid.]). But by[1], such cofiber sequences do not exist. However, as Blumberg and Mandell discussin [10, Introduction], there is a localization cofiber sequence K ( π ( E n )) → K ( BP n ) → K ( E n ) → Σ K ( π ( E n )) , where BP n is the connective cover of E n , and we see that it has the attractive featurethat K ( E n ) itself appears as a term, instead of K ( E ( n ) p ) (see [10, Introduction]for more detail about this sequence). Thus, this cofiber sequence provides a wayto begin studying the above question (the author has not pursued the argumentsuggested by cofiber sequences (1.16) and (1.17)).We continue with letting p ≥
5. Our next step is to note that there is anequivalence K ( KU p ) ∧ v − V (1) = K ( KU p ) ∧ (cid:0) colim j ≥ Σ − jd V (1) (cid:1) ≃ colim j ≥ (cid:0) K ( KU p ) ∧ Σ − jd V (1) (cid:1) f , where (cid:8)(cid:0) K ( KU p ) ∧ Σ − jd V (1) (cid:1) f (cid:9) j ≥ is a diagram of Z × p –spectra and Z × p –equivariantmaps (as in the case of V (1), each spectrum Σ − jd V (1) is given the trivial Z × p –action). Since K ( KU p ) ∧ V (1) is an f –spectrum, it is immediate that for each j ≥ (cid:0) K ( KU p ) ∧ Σ − jd V (1) (cid:1) f is an f –spectrum, and hence, Theorem 1.12 impliesthat each (cid:0) K ( KU p ) ∧ Σ − jd V (1) (cid:1) f can be regarded as a discrete Z × p –spectrum. Remark 1.19.
To aid the reader in making connections between the theory de-veloped in this paper and the application of it that is discussed in this introduc-tion, we use the terminology that is set up in later sections to express our mainconclusions above (thus, p ≥ N denote the collection of open normal sub-groups of Z × p that corresponds to the family { ( p m Z p ) × { e }} m ≥ of subgroups of Z p × Z / ( p − Z × p has a good filtration (see Definition 3.3), and we haveshown that ( Z × p , K ( KU p ) ∧ V (1) , N ) is a suitably finite triple (see Definition 6.1)and (cid:0) Z × p , (cid:8)(cid:0) K ( KU p ) ∧ Σ − jd V (1) (cid:1) f (cid:9) j ≥ , N (cid:1) is a suitably filtered triple (Definition 7.1).Let N be as defined in Remark 1.19. As explained (in greater generality) in thediscussion centered around (7.2), there is a zigzag of Z × p –equivariant maps C p := colim j ≥ (cid:0) K ( KU p ) ∧ Σ − jd V (1) (cid:1) f ≃ / / colim j ≥ (cid:0)(cid:0) K ( KU p ) ∧ Σ − jd V (1) (cid:1) f (cid:1) ′ C dis p := colim j ≥ (cid:0)(cid:0) K ( KU p ) ∧ Σ − jd V (1) (cid:1) f (cid:1) dis N≃ O O ith each map a weak equivalence of symmetric spectra, and C dis p is a discrete Z × p –spectrum. The above zigzag is obtained by taking a colimit of the zigzags thatare obtained from (1.13) by setting X (in (1.13)) equal to (cid:0) K ( KU p ) ∧ Σ − jd V (1) (cid:1) f ,for each j ≥ Z × p –spectrum C p with the discrete Z × p –spectrum C dis p and wemake the concomitant definition( C p ) h Z × p = ( C dis p ) h Z × p . Similarly, it is natural to identify the Z × p –spectrum K ( KU p ) ∧ v − V (1) with C p ,and hence, with the discrete Z × p –spectrum C dis p (the mapping telescope v − V (1)has the trivial Z × p –action). Thus, we define (cid:0) K ( KU p ) ∧ v − V (1) (cid:1) h Z × p = ( C dis p ) h Z × p . More explicitly, we have (cid:0) K ( KU p ) ∧ v − V (1) (cid:1) h Z × p = (cid:16) colim j ≥ (cid:0)(cid:0) K ( KU p ) ∧ Σ − jd V (1) (cid:1) f (cid:1) dis N (cid:17) h Z × p . Now let K be an arbitrary closed subgroup of Z × p . By the identification above of K ( KU p ) ∧ v − V (1) with C dis p in the world of Z × p –spectra and as in Definition 7.4,it follows that the K –spectrum K ( KU p ) ∧ v − V (1) can be regarded as the discrete K –spectrum C dis p , and hence, it is natural to define (cid:0) K ( KU p ) ∧ v − V (1) (cid:1) hK = (cid:16) colim j ≥ (cid:0)(cid:0) K ( KU p ) ∧ Σ − jd V (1) (cid:1) f (cid:1) dis N (cid:17) hK . Similarly (and easier; see the discussion just above (8.5)), for each j ≥
0, it isnatural to define( K ( KU p ) ∧ Σ − jd V (1)) hK = (cid:0) ( K ( KU p ) ∧ Σ − jd V (1)) dis N (cid:1) hK . This completes the construction of the continuous homotopy fixed point spectrathat appear in Theorem 1.6.1.3.
Concluding introductory remarks: our underlying framework, ter-minology, etc.
In work in preparation, we use the theory developed in this paperto study ( KU p ) h Z × p , and more generally, E hGn , for G a closed subgroup of G n .We work in the framework of symmetric spectra in this paper because it is asymmetric monoidal category and such a category is important for studying the al-gebraic K –theory of commutative S –algebras. For example, in symmetric spectra,the role of commutative S –algebras is played by commutative symmetric ring spec-tra, and their properties are essential in the statement that Z × p acts on K ( KU p )by morphisms of commutative symmetric ring spectra. Furthermore, use of theframework of symmetric spectra makes available for future work the model cate-gory Alg A,G of discrete commutative G – A –algebras, where G is any profinite groupand A is a commutative symmetric ring spectrum (see [8, Section 5.2]). Since the G n –action on K ( E n ) is by maps of commutative symmetric ring spectra, the modelcategory Alg K ( L K ( n ) ( S )) , G n (or Alg S , G n ) might play a role in understanding theconjectural continuous homotopy fixed point spectrum ( K ( E n )) h G n .We conclude this introduction with some preparatory comments for the upcom-ing work. For the rest of the paper, “spectrum” means symmetric spectrum of implicial sets (except for a few instances in which the exception is clearly noted).It is useful to recall that given any collection { X γ } γ ∈ Γ of fibrant spectra, there isan isomorphism π k (cid:0)Q γ ∈ Γ X γ (cid:1) ∼ = Q γ ∈ Γ π k ( X γ ) of abelian groups, where k is anyinteger, for the product of spectra Q γ ∈ Γ X γ . Also, it is helpful to note that if a map f of spectra is, when regarded as a map of Bousfield-Friedlander spectra, a weakequivalence (in the usual stable model structure on Bousfield-Friedlander spectra),then the map f is a weak equivalence of spectra, by [26, Theorem 3.1.11]. We useholim to denote the homotopy limit for ΣSp, as defined in [24, Definition 18.1.8].Let G be any profinite group. A “discrete G –spectrum” is a discrete symmetric G –spectrum, as defined in [8, Section 2.3] (see also [16, Section 3]); these objects,together with the G –equivariant maps (see [8] for the precise definition), constitutethe category ΣSp G of discrete G –spectra. By [8, Theorem 2.3.2], there is a modelcategory structure on ΣSp G in which a morphism f in ΣSp G is a weak equivalence(cofibration) if and only if f is a weak equivalence (cofibration) in ΣSp. Given afibrant replacement functor( − ) fG : ΣSp G → ΣSp G , X X fG (thus, X fG is fibrant in ΣSp G ), such that there is a natural trivial cofibration η : X → X fG in ΣSp G , there is the induced map η G : X G → ( X fG ) G = X hG . By [8, Section 3.1], the target of η G , the homotopy fixed point spectrum X hG , isthe output of the right derived functor of fixed points.Given any profinite group G , a “ G –spectrum” is a naive symmetric G –spectrumand not a genuine equivariant symmetric G –spectrum. Thus, when G is finite, a G –spectrum need not be an equivariant symmetric G –spectrum in the sense of [28](defined by using the spheres S ( G ) = V G S in the bonding maps). Acknowledgements.
I thank John Rognes for helpful discussions. Also, I thankChristian Ausoni, Andrew Blumberg, Paul Goerss, Arturo Magidin, and PeterSymonds for useful comments.2.
Some preliminaries
In this section, we explain some constructions and a result (Lemma 2.1) that willbe useful for our main work later. As in the introduction, we let G be any profinitegroup.Given a set S , let Sets( G, S ) be the G –set of all functions f : G → S , with G –action defined by ( g · f )( g ′ ) = f ( g ′ g ) , g, g ′ ∈ G. Let U be the forgetful functor from the category of G –sets to the category of sets.Then it is easy to see that Sets( G, − ) is the right adjoint of U . By analogy witha standard construction in group cohomology, Sets( G, S ) can be thought of as the“coinduced G -set on S .”The construction Sets( G, S ) prolongs to the category of G –spectra and the for-getful functor U G from the category of G –spectra to ΣSp has a right adjoint that isgiven by the prolongation Sets( G, − ), so that, given a spectrum Z and any k, l ≥ l -simplices of the pointed simplicial set Sets( G, Z ) k is defined bySets( G, Z ) k,l = Sets( G, Z k,l ) . hus, for any Z ∈ ΣSp, there is an isomorphismSets(
G, Z ) ∼ = Q G Z in ΣSp, where the right-hand side of the isomorphism is the product of | G | copies of Z . Since the functors U G and Sets( G, − ) are an adjoint pair, there is the associatedtriple (e.g., see [39, 8.6.2]), and, for any G -spectrum X , we letSets( G • +1 , X )denote the cosimplicial G –spectrum that is given in the usual way by the triple (formore detail, see [39, 8.6.4]).For any m ≥
0, we use G m to denote the Cartesian product of m copies of G ,with G = ∗ , the point. Then it is not hard to see that, for any G –spectrum X andany m ≥
0, the “ G –spectrum of m –cosimplices” of the cosimplicial G –spectrumSets( G • +1 , X ) satisfies the G –equivariant isomorphismSets( G • +1 , X ) m ∼ = Sets( G, Sets( G m , X )) , where, as before, Sets( G m , X ) is the spectrum defined on the level of sets bySets( G m , X ) k,l = Sets( G m , X k,l ) , for every k, l ≥ L is a discrete group, Z an L –spectrum that is fibrantin ΣSp, and P a subgroup of L , then the descent spectral sequence E s,t ⇒ π t − s (cid:0) Map P ( EL + , Z ) (cid:1) ∼ = π t − s (cid:0) Z e hP (cid:1) has an E –term that satisfies E s,t = H s ( P ; π t ( Z )) , the (non-continuous) group cohomology of P with coefficients in the P –module π t ( Z ). Also, the result below is a “discrete version” of [22, page 210 and the proofof Lemma 5.4] and [16, proof of Lemma 7.12]. But, since Lemma 2.1 is a usefultool for our work later, we give a complete proof. Lemma 2.1.
Let G be a profinite group. If X is a G –spectrum and K is a subgroupof G , then, for every s ≥ and any t ∈ Z , there is an isomorphism lim s ∆ π t (cid:0) Sets( G • +1 , X f ) K (cid:1) ∼ = H s ( K ; π t ( X )) . Remark 2.2.
To avoid any confusion, we note that in the statement of Lemma2.1, K is any subgroup of G (thus, for example, K does not have to be a closedsubgroup of G ). Proof of Lemma 2.1. If A is an abelian group and P is a profinite group, letSets( P, A ) be the abelian group of functions P → A : in fact, Sets( P, A ) is a P –module, with its P –action defined by ( p · f )( p ′ ) = f ( p ′ p ). Then there is anisomorphism lim s ∆ π t (cid:0) Sets( G • +1 , X f ) K (cid:1) ∼ = H s h Sets( G ∗ +1 , π t ( X )) K i , here Sets( G ∗ +1 , π t ( X )) K is the cochain complex obtained by applying, for each m ≥
0, the chain of isomorphisms π t (cid:16)(cid:0) Sets( G • +1 , X f ) K (cid:1) m (cid:17) ∼ = π t (cid:0) Sets( G, Sets( G m , X f )) K (cid:1) ∼ = π t (cid:16)Q G/K Q Gm X f (cid:17) ∼ = Q G/K Q Gm π t ( X f ) ∼ = Sets( G, Sets( G m , π t ( X )) K ∼ = Sets( G m +1 , π t ( X )) K . Above, for m ≥
1, Sets( G m , π t ( X )) is the K –module of functions G m → π t ( X )whose K –action is given by( k · p )( g , g , g , ..., g m ) = p ( g k, g , g , ..., g m ) , for k ∈ K, p ∈ Sets( G m , π t ( X )) , and g , g , ..., g m ∈ G. (In the preceding sentence,since m ≥
1, it goes without saying that this sentence also defines the K –action onthe K –module Sets( G m +1 , π t ( X )) that appears in the last expression in the abovechain of isomorphisms.)Notice that there is a G –equivariant monomorphism π t ( X ) η −→ Sets(
G, π t ( X )) , [ f ] (cid:0) g g · [ f ] (cid:1) and a homomorphismev : Sets( G, π t ( X )) → π t ( X ) , p p (1) , such that ev ◦ η = id π t ( X ) . Then, since the cochain complex Sets( G ∗ +1 , π t ( X ))originally comes from a triple, there is an exact sequence(2.3) 0 → π t ( X ) η −→ Sets( G ∗ +1 , π t ( X ))of K –modules (for example, see the dual of [39, Corollary 8.6.9]).There is a chainSets( G, Sets( G m , π t ( X )) ∼ = Q K Q G/K
Sets( G m , π t ( X )) ∼ = Hom Z (cid:0)L K Z , Q G/K
Sets( G m , π t ( X )) (cid:1) of isomorphisms of K –modules, where Hom Z (cid:0)L K Z , Q G/K
Sets( G m , π t ( X )) (cid:1) is acoinduced K –module, and hence, Shapiro’s Lemma implies that H s (cid:0) K ; Sets( G, Sets( G m , π t ( X ))) (cid:1) ∼ = H s (cid:0) K ; Hom Z (cid:0)L K Z , Q G/K
Sets( G m , π t ( X )) (cid:1)(cid:1) = 0 , whenever s > , for all m ≥ K –module π t ( X ) by ( − ) K –acyclic K –modules, and therefore, H s h Sets( G ∗ +1 , π t ( X )) K i ∼ = H s ( K ; π t ( X )) , as desired. (cid:3) . Profinite groups that have a good filtration
As usual, let G be a profinite group. In this section, after explaining the notionof a good filtration for G and making several comments about it, we show that Z p × H , where p is any prime and H is a finite discrete group, has a good filtration. Definition 3.1.
Given a discrete G –module M , let λ sM : H sc ( G ; M ) → H s ( G ; M )be the natural homomorphism between continuous cohomology and non-continuouscohomology that is obtained by regarding each group Map c ( G m , M ) of continuouscochains as a subgroup of the corresponding group Sets( G m , M ) of all cochains.Then, in this paper (see Remark 3.2 below), we say that G is good if λ sM is anisomorphism for all s ≥ G –module M . Remark 3.2.
The above definition is taken from [36, page 13, Exercise 2]: if G isstrongly complete, so that G ∼ = b G , where b G is the profinite completion of G , and λ sM is an isomorphism for all s ≥ G –module M (a finite G –moduleconsists of finite orbits, so that every stabilizer subgroup of G has finite index,and hence, is an open subgroup (since G is strongly complete), so that a finite G –module is automatically a discrete G –module), then, following Serre, G is “bon.” Ingeneral, since G and b G need not be the same, our definition of “good” is differentfrom the usual one (that is, the aforementioned “bon”) in group theory. However,our use of “good” in this paper should cause no confusion, since, throughout thispaper, we only use “good” in the sense of Definition 3.1.We say that G has finite cohomological dimension (“finite c.d.”) if there existssome positive integer r such that the continuous cohomology H sc ( G ; M ) = 0 , for alldiscrete G –modules M , whenever s > r . Definition 3.3.
A profinite group G has a good filtration if(a) there exists a directed poset Λ such that there is an inverse system N = { N α } α ∈ Λ of open normal subgroups of G , with the maps in the diagram given by theinclusions (that is, α ≤ α in Λ if and only if N α is a subgroup of N α );(b) the intersection T α ∈ Λ N α is the trivial group { e } ;(c) each N α is a good profinite group, in the sense of Definition 3.1; and(d) the collection { N α } α ∈ Λ has uniformly bounded finite c.d.; that is, thereexists a fixed natural number r G , such that H sc ( N α ; M ) = 0, for all s > r G ,whenever α ∈ Λ and M is any discrete N α –module. Remark 3.4.
Let G be a profinite group with a good filtration and let N = { N α } α ∈ Λ satisfy (a)–(d) in Definition 3.3. It follows from (a) and (b) that N is acofinal subcollection of the family of all open normal subgroups of G , and hence,the canonical homomorphism G → lim α ∈ Λ G/N α is a homeomorphism. Now chooseany α ∈ Λ, so that N α is good, by (c) above. We give an argument that is suggestedby [36, page 14, Exercise 2, (c)] (for instances of Serre’s argument that are closelyrelated to the version given here, see [20, proof of Theorem 2.10] and [35, proof ofProposition 3.1]). Since λ ∗ M : H ∗ c ( N α ; M ) → H ∗ ( N α ; M ) s an isomorphism in each degree for any finite discrete G –module M , the E –termof the Lyndon-Hochschild-Serre spectral sequence E p,q = H p ( G/N α ; H qc ( N α ; M )) = ⇒ H p + qc ( G ; M )for continuous group cohomology (since G/N α is a finite discrete group, the E –termis given by just group cohomology) is isomorphic to the E –term of the correspond-ing Lyndon-Hochschild-Serre spectral sequence H p ( G/N α ; H q ( N α ; M )) = ⇒ H p + q ( G ; M )for group cohomology, and hence, by comparison of spectral sequences, the map λ sM : H sc ( G ; M ) ∼ = −→ H s ( G ; M )is an isomorphism, for all s ≥ G –module M . Remark 3.5.
Let G be a profinite group that has finite c.d. and let { N α } α ∈ Λ bean inverse system of open normal subgroups of G that satisfies (a)–(c) in Definition3.3. Then the inverse system also satisfies (d), so that G has a good filtration. Thisconclusion follows from the fact that for r as in our definition of finite c.d. above(just before Definition 3.3), Shapiro’s Lemma implies that whenever s > r , givenany α ∈ Λ, H sc ( N α ; M ) ∼ = H sc ( G ; Coind GN α ( M )) = 0 , for all discrete N α –modules M (above, Coind GN α ( M ) is the coinduced module ofcontinuous functions G → M that are N α –equivariant). Theorem 3.6.
Let p be any prime and let G = Z p × H , where H is a finite discretegroup and G is equipped with the product topology. Then G has a good filtration.Proof. Recall that there is a descending chain Z p = U (cid:13) U (cid:13) · · · (cid:13) U m (cid:13) · · · of open normal subgroups of Z p , with U m = p m Z p for each m ≥ T m ≥ U m = { e } . For each m ≥
0, we set N m = U m × { e } , a subgroup of G . We will show that { N m } m ≥ satisfies conditions (a)–(d) in Definition 3.3.It is easy to see that { N m } m ≥ satisfies (a) and (b). By [20, Theorem 2.9], Z p is a good profinite group and, for each m ≥ N m ∼ = Z p , showing that (c) is valid.Finally, since the pro- p -group Z p has cohomological p -dimension equal to one, itfollows that Z p has finite c.d. This fact, coupled with another application of theisomorphisms N m ∼ = Z p for all m ≥
0, shows that (d) holds. (cid:3) An r – Z p –spectrum is a discrete Z p –spectrum In this section, we prove one of the key results of this paper, Theorem 4.9; thetitle above illustrates a special case of this result, and the unfamiliar term in thetitle is defined below.
Definition 4.1.
Let G be a profinite group and X a G –spectrum. If π t ( X ) is afinite discrete G –module for every t ∈ Z , then we say that X is an r – G –spectrum . Remark 4.2.
Since an r – G –spectrum is both a G –spectrum and an f –spectrum,our first thought was to use the term “ f – G –spectrum” for such an object, but thisterm is already used (often) by [18] (see [ibid., Definition 3.1]), and so we removedthe horizontal stroke in the term’s “ f ,” to “obtain” the “ r ” in “ r – G –spectrum” also, “ r estricted” is, roughly speaking, a synonym of “ f inite”). If G is stronglycomplete, then every r – G –spectrum X has an f – G –spectrum associated to it inthe following way: X f is a G –spectrum and since it is a fibrant spectrum, for eachinteger t , there is an isomorphism(4.3) π t ( X f ) ∼ = colim k π t + k ( X k ) = π t ( U ( X f ))of finite abelian groups, where the last expression in (4.3) refers to the t -th (classical)stable homotopy group of the Bousfield-Friedlander spectrum U ( X f ) that underlies X f , and hence, by an application of [31, Theorem 5.15], there is a G –equivariantmap and weak equivalence U ( X f ) ≃ −→ F sG ( U ( X f )) of Bousfield-Friedlander spectra,with F sG ( U ( X f )) an f – G –spectrum.For the remainder of this section (with the exception of Lemma 4.7), G denotesa profinite group that has a good filtration. Thus, we let N = { N α } α ∈ Λ be an inverse system of open normal subgroups of G that satisfies the requirementsof Definition 3.3. Definition 4.4.
Let X be a G –spectrum. We set X dis N = colim α ∈ Λ holim ∆ Sets( G • +1 , X f ) N α , where the colimit is formed in ΣSp.Since each N α is an open normal subgroup of G , with G/N α a finite discretegroup, Sets( G • +1 , X f ) N α is a cosimplicial G/N α –spectrum. Thus, the spectrumholim ∆ Sets( G • +1 , X f ) N α is a G/N α –spectrum, and hence, a discrete G –spectrum(via the canonical projection G → G/N α ). By [8, Section 3.4], colimits in ΣSp G are formed in ΣSp, and hence, we have the following observation. Lemma 4.5. If X is a G –spectrum, where G is a profinite group that has a goodfiltration, then X dis N is a discrete G –spectrum. Remark 4.6.
Let X be a G –spectrum. Since N is cofinal in the collection of allopen normal subgroups of G , there is an isomorphism X dis N ∼ = colim N ⊳ o G holim ∆ Sets( G • +1 , X f ) N of discrete G –spectra, where above, N ⊳ o G means that N is an open normalsubgroup of G . Similarly, if N ′ = { N α ′ } α ′ ∈ Λ ′ is another inverse system of opennormal subgroups of G that satisfies Definition 3.3, there is an isomorphism X dis N ′ = colim α ′ ∈ Λ ′ holim ∆ Sets( G • +1 , X f ) N α ′ ∼ = colim N ⊳ o G holim ∆ Sets( G • +1 , X f ) N in ΣSp G , and hence, there is an isomorphism X dis N ∼ = X dis N ′ in ΣSp G . It followsthat the definition of X dis N is independent of the choice of inverse system N up toisomorphism.Now we are ready to prove the central result of this paper: its conclusion can beabbreviated by saying that if X is an r – G –spectrum (as in Definition 4.1), then X is a discrete G –spectrum. We break up our work for this result into two pieces. Thefirst piece, Lemma 4.7 below, can be regarded as a special case of [19, Proposition6.4], in the setting of G –spectra. emma 4.7. If G is any profinite group and X is any G –spectrum, then there isa G –equivariant map i X : X ≃ −→ holim ∆ Sets( G • +1 , X f ) that is a weak equivalence in ΣSp .Proof.
Given a spectrum Z , let cc • ( Z ) denote the constant cosimplicial spectrumon Z . Then the G –equivariant map i X is defined to be the composition i X : X ≃ −→ X f ∼ = −→ lim ∆ cc • ( X f ) → holim ∆ cc • ( X f ) → holim ∆ Sets( G • +1 , X f ) , where the last (rightmost) map is induced by repeated use of the G –equivariantmonomorphism i : Y → Sets(
G, Y ) of G –spectra, that is defined on the level ofsets, for any G –spectrum Y , by the maps Y k,l → Sets(
G, Y k,l ) , y ( g g · y ) . Notice that for each m ≥
0, the spectrum of m -cosimplices of Sets( G • +1 , X f ), (cid:0) Sets( G • +1 , X f ) (cid:1) m ∼ = Q Gm +1 X f , is fibrant, so that Sets( G • +1 , X f ) is a cosimplicial fibrant spectrum. Thus, there isa homotopy spectral sequence(4.8) I E s,t ∼ = H s (cid:2) π t (Sets( G ∗ +1 , X f )) (cid:3) = ⇒ π t − s (cid:0) holim ∆ Sets( G • +1 , X f ) (cid:1) . By Lemma 2.1, we have I E s,t ∼ = H s ( { e } ; π t ( X )) = ( π t ( X ) , s = 0;0 , s > , and hence, spectral sequence I E ∗ , ∗ r of (4.8) collapses, showing that i X is a weakequivalence. (cid:3) Theorem 4.9.
Let G be a profinite group that has a good filtration and let N be adiagram of subgroups of G that satisfies Definition 3.3. If X is an r – G –spectrum,then there is a zigzag of G –equivariant maps (4.10) X ≃ −→ holim ∆ Sets( G • +1 , X f ) ≃ ←− X dis N that are weak equivalences in ΣSp . Remark 4.11.
As stated just before Lemma 4.7, the above theorem says that(given a suitable profinite group G ) an r – G –spectrum can be regarded as a discrete G –spectrum (in a canonical way): the “ G –equivariant zigzag” of weak equivalencesin (4.10) makes this statement precise. Proof of Theorem 4.9.
By Lemma 4.7, it suffices to construct a G –equivariant map φ X : X dis N = colim α ∈ Λ holim ∆ Sets( G • +1 , X f ) N α → holim ∆ Sets( G • +1 , X f )and then show that it is a weak equivalence of spectra. The G –equivariant map φ X is defined to be the compositionlim −→ holim ∆ Ens( G • +1 , X f ) N α φ X −→ holim ∆ lim −→ Ens( G • +1 , X f ) N α φ X −→ holim ∆ Ens( G • +1 , X f ) f canonical maps, where, here (and below), to conserve space, we (sometimes) usethe notation “ lim −→ ” to denote “ colim α ∈ Λ ”, and “Ens” in place of “Sets.”The definition of the map φ X is given explicitly as follows: the collection ofinclusions Sets( G • +1 , X f ) N α ֒ → Sets( G • +1 , X f ) induces the morphism φ X : colim α ∈ Λ Sets( G • +1 , X f ) N α → Sets( G • +1 , X f )of cosimplicial G –spectra, and φ X = holim ∆ φ X . The morphism φ X also induces amap E ∗ , ∗ r (cid:16) φ X (cid:17) : II E ∗ , ∗ r → I E ∗ , ∗ r , from the homotopy spectral sequence(4.12) II E s,t = H s h π t (cid:0) lim −→ Ens( G ∗ +1 , X f ) N α (cid:1)i ⇒ π t − s (cid:16) holim ∆ lim −→ Ens( G • +1 , X f ) N α (cid:17) to spectral sequence (4.8). We point out that the construction of spectral se-quence (4.12) uses the fact that for each m ≥
0, the spectrum of m –cosimplices oflim −→ Ens( G • +1 , X f ) N α satisfies (cid:0) colim α ∈ Λ Sets( G • +1 , X f ) N α (cid:1) m ∼ = colim α ∈ Λ (cid:0)Q G/Nα Q Gm X f (cid:1) , which is a fibrant spectrum, since products and filtered colimits of fibrant spectraare again fibrant (the second fact is justified, for example, in [14, Section 5]), sothat lim −→ Ens( G • +1 , X f ) N α is a cosimplicial fibrant spectrum.Notice that for spectral sequence II E ∗ , ∗ r , there is the chain of isomorphisms II E s,t ∼ = colim α ∈ Λ H s ( N α ; π t ( X )) ∼ = colim α ∈ Λ H sc ( N α ; π t ( X )) ∼ = H sc (cid:0) T α ∈ Λ N α ; π t ( X ) (cid:1) = H s ( { e } ; π t ( X )) , (4.13)where the first isomorphism uses Lemma 2.1 and the fact that filtered colimits offibrant spectra commute with [ S t , − ]; the second isomorphism applies the assump-tion that each N α is a good profinite group; and the last step (involving the equality)is due to property (b) of Definition 3.3. Therefore, there is an isomorphism II E s,t ∼ = I E s,t , for all s and t , so that the map E ∗ , ∗ r (cid:16) φ X (cid:17) of spectral sequences is an isomorphismfrom the E –terms onward. Hence, the map π ∗ ( φ X ) = [ S ∗ , φ X ] between the abut-ments of these conditionally convergent spectral sequences is an isomorphism, sothat φ X is a weak equivalence.As in (4.13), there are isomorphisms(4.14) H s h π t (cid:0) Sets( G ∗ +1 , X f ) N α (cid:1)i ∼ = H s ( N α ; π t ( X )) ∼ = H sc ( N α ; π t ( X ))for each α , and hence, condition (d) of Definition 3.3 implies that(4.15) H s h π t (cid:0) Sets( G ∗ +1 , X f ) N α (cid:1)i = 0 , for all s > r G , every t ∈ Z , and each α. Therefore, the map φ X is a weak equivalence, by [29, Proposition 3.4]. inally, we can conclude that φ X is a weak equivalence, since φ X and φ X are weakequivalences. (cid:3) An extension of the central result, Theorem 4.9
In this section, we show – in Theorem 5.1 – that the hypotheses of Theorem 4.9can be slightly loosened. We give this result in this later section so that Theorem4.9 (and Section 4) is ready-made for the intended applications. Suppose that X isa G –spectrum with homotopy groups that are always torsion discrete G –modules:as explained in the second part of this section, the homotopy groups of such a G –spectrum are closely related to those of r – G –spectra. However, we explain why ourproof of Theorem 4.9 does not extend to this more general “torsion case.”For the rest of this section, we suppose that G is an arbitrary profinite groupand X is any G –spectrum. Given this context, it is easy to see that Definition 4.4and Lemma 4.5 depend only on condition (a) of Definition 3.3, and hence, underonly the additional assumption of condition (a), the spectrum X dis N is defined andis a discrete G –spectrum. Also, the proof of Theorem 4.9 depends only on(i) condition (a);(ii) the assumption that the G –module π t ( X ) is a discrete G –module, for every t ∈ Z ; and(iii) part (b) of Definition 3.3: T α ∈ Λ N α = { e } , except in three spots: • in the second isomorphisms of (4.13) and (4.14), in addition to (i) and (ii)above, the proof of Theorem 4.9 uses both the assumption that π t ( X ) isfinite for every integer t and part (c) of Definition 3.3; and • in (4.15), besides (i) and (ii) above, the proof uses part (d) of Definition3.3.These observations imply the following result. Theorem 5.1.
Let G be a profinite group, with N = { N α } α ∈ Λ an inverse systemof open normal subgroups of G that satisfies properties ( a ) and ( b ) of Definition 3.3,and let X be a G –spectrum such that condition ( ii ) above holds. Also, suppose thatthe map λ sπ t ( X ) : H sc ( N α ; π t ( X )) → H s ( N α ; π t ( X )) is an isomorphism for all s ≥ , every integer t , and each α ∈ Λ . If • there exists a natural number r , such that for all integers t and every α ∈ Λ , H sc ( N α ; π t ( X )) = 0 , for all s > r ; or • there exists some fixed integer l , such that π t ( X ) = 0 , for all t > l ,then there is a zigzag of G –equivariant maps X ≃ −→ holim ∆ Sets( G • +1 , X f ) ≃ ←− X dis N that are weak equivalences in ΣSp , with X dis N ( defined as in Definition 4.4 ) a discrete G –spectrum.Proof. The only part of the theorem that is not justified by the remarks precedingit is the following. In our proof of Theorem 4.9, in (4.15), we assumed condition (d)of Definition 3.3, but by [29, Proposition 3.4], an alternative to assuming condition d) is to require that there exists some fixed integer l , such that for each m ≥ α ∈ Λ ,π t (cid:0) Sets( G m +1 , X f ) N α (cid:1) ∼ = Q G/Nα × Gm π t ( X ) = 0 , for all t > l, which is equivalent to assuming that π t ( X ) = 0, for all t > l. (cid:3) We conclude this section by explaining why the proof of Theorem 4.9 fails toextend to the case when X is a G –spectrum with each homotopy group a (possiblyinfinite) discrete G –module that is also a torsion abelian group. With G as inTheorem 4.9, our assumptions imply that for each t ∈ Z , π t ( X ) = S β M t,β is the union of its finite G –submodules M t,β , each of which is automatically adiscrete G –module.As discussed at the beginning of this section, in the second isomorphisms in(4.13) and (4.14), we need to know that for each α and every integer t , the naturalmap λ sπ t ( X ) : H sc ( N α ; π t ( X )) → H s ( N α ; π t ( X ))is an isomorphism, for all s ≥
0. Since each N α is a good profinite group, there areisomorphisms H sc ( N α ; π t ( X )) ∼ = colim β H sc ( N α ; M t,β ) ∼ = colim β H s ( N α ; M t,β ) ∼ = H s h colim β Sets( N ∗ α , M t,β ) i , where, here, given an N α –module M , Sets( N ∗ α , M ) denotes the usual cochain com-plex such that H s (cid:2) Sets( N ∗ α , M ) (cid:3) = H s ( N α ; M ), for each s ≥
0, with the abeliangroup of m –cochains equal toSets( N ∗ α , M ) m = Sets( N mα , M ) ∼ = Q Nmα
M , for each m ≥ . It follows that the map λ sπ t ( X ) is an isomorphism if and only if the canonical map h s,t : H s h colim β Sets( N ∗ α , M t,β ) i → H s h Sets( N ∗ α , S β M t,β ) i = H s ( N α ; π t ( X ))is an isomorphism.Since filtered colimits and infinite products do not commute in general, the map h s,t above need not be an isomorphism, so that λ sπ t ( X ) need not be an isomorphism:this situation is the crux of what prevents the proof of Theorem 4.9 from goingthrough in the case when each π t ( X ) is a torsion discrete G –module. Remark 5.2.
Let G be as in Theorem 4.9 and suppose that X is a G –spectrumsuch that π t ( X ) is a discrete G –module and torsion abelian group, for every integer t . Then it is clear from the above discussion that if G , as an abstract group, isof type F P ∞ (for background on this notion, we refer to [13]), then H ∗ ( G ; − ) ∼ =Ext ∗ Z [ G ] ( Z , − ) commutes with direct limits, and hence, the conclusion of Theorem4.9 is still valid. Now we add the desirable condition that G is an infinite group, andwe give an argument that we learned from Peter Symonds. As an abstract group,if G is of type F P ∞ , then it is of type F P , and hence, it is finitely generated(abstractly) and thereby countably infinite, contradicting the fact that G must e uncountable (since it is profinite). Therefore, G cannot be both infinite and,abstractly, of type F P ∞ .6. The spectrum X dis N , fibrancy, and homotopy fixed points In this section, we let G be any profinite group and X any G –spectrum. Definition 6.1. If G , X , and N (an inverse system of open normal subgroupsof G ) satisfy the hypotheses of Theorem 4.9 or Theorem 5.1, then we say thatthe triple ( G, X, N ) is suitably finite . (In the preceding sentence, by satisfying thehypotheses of Theorem 5.1, we mean that G , X , and N satisfy the conditions of thefirst two sentences of Theorem 5.1 and at least one of the two “itemized conditions”(that is, the conditions marked by a “ • ”) listed in the third sentence of Theorem5.1.) Notice that if ( G, X, N ) is a suitably finite triple, then there is a zigzag of G –equivariant maps X ≃ −→ holim ∆ Sets( G • +1 , X f ) ≃ ←− X dis N that are weak equivalences in ΣSp. Definition 6.2.
If (
G, X, N ) is a suitably finite triple, then because of the abovezigzag of equivalences between X and X dis N , it is natural to identify X with thediscrete G –spectrum X dis N , and hence, to define X hG = ( X dis N ) hG . Remark 6.3.
Let (
G, X, N ) be a suitably finite triple, with the inverse system N written as { N α } α ∈ Λ , and suppose that X is a discrete G –spectrum (that is,before the identification of Definition 6.2, X ∈ ΣSp G ). In this case, after followingDefinition 6.2, X hG can mean ( X fG ) G or ( X dis N ) hG . Since X ∈ ΣSp G , the weakequivalence i X : X ≃ −→ holim ∆ Sets( G • +1 , X f ) factors into the map δ : X → X dis N ,which is defined to be the composition X ∼ = −→ colim α ∈ Λ X N α colim α ∈ Λ ( i X ) Nα −−−−−−−−→ colim α ∈ Λ (cid:0) holim ∆ Sets( G • +1 , X f ) (cid:1) N α ∼ = −→ X dis N (the first isomorphism in the composition is due to the fact that, since N satisfies(a) and (b) in Definition 3.3, N is a cofinal subcollection of { N | N ⊳ o G } ), followedby the weak equivalence X dis N ≃ −→ holim ∆ Sets( G • +1 , X f ), and hence, the map δ isa weak equivalence of spectra. It follows that δ is a weak equivalence in ΣSp G ;therefore, δ induces a weak equivalence ( X fG ) G ≃ −→ ( X dis N ) hG , showing that thetwo possible interpretations of X hG are equivalent to each other.Several interesting consequences of Definition 6.2 are stated in Theorem 6.4below. Before giving this result, we need to give some background material for itsproof.Let G -ΣSp be the category of G –spectra (as defined at the end of the introduc-tion): G -ΣSp has a model category structure in which a morphism f is a weakequivalence (cofibration) if and only if f is a weak equivalence (cofibration) whenregarded as a morphism in ΣSp. The existence of this model structure follows,for example, from the fact that G -ΣSp is isomorphic to ΣSp {∗ G } , the category offunctors {∗ G } → ΣSp, where {∗ G } is the one-object groupoid associated to G , andthis diagram category can be equipped with an injective model structure, by [27,Proposition A.2.8.2], since ΣSp is a combinatorial model category. ince the forgetful functor U G : G -ΣSp → ΣSp preserves weak equivalences andcofibrations, the adjoint functors ( U G , Sets( G, − )) are a Quillen pair. Also, it will behelpful to recall the standard fact that if Y is fibrant in G -ΣSp, then Y is fibrant inΣSp (since, for example, an injective fibrant object in ΣSp {∗ G } is projective fibrantin ΣSp {∗ G } (one reference for this is [27, Remark A.2.8.5]; ΣSp {∗ G } has a projectivemodel structure by [ibid., Proposition A.2.8.2])).The left adjoint functor ΣSp → G -ΣSp that sends a spectrum to itself, butnow regarded as a G –spectrum with trivial G –action, preserves weak equivalencesand cofibrations. It follows that the right adjoint, the G –fixed points functor( − ) G : G -ΣSp → ΣSp, is a right Quillen functor, and if Y → Y fib is a trivialcofibration to a fibrant object, in G -ΣSp, then Y e hG = ( Y fib ) G . As in [27, Example 1.1.5.8], the category {∗ G } can be regarded as a simplicialcategory by defining the simplicial set Map {∗ G } ( ∗ G , ∗ G ) to be the constant simplicialset on Hom {∗ G } ( ∗ G , ∗ G ). With S equal to the category of simplicial sets, it is easyto see that the category of S -enriched functors from {∗ G } to the simplicial categoryΣSp, with morphisms the S -enriched natural transformations, is identical to theusual functor category ΣSp {∗ G } . Since ΣSp is a simplicial model category, it followsfrom [27, Proposition A.3.3.2, Remark A.3.3.4] that the injective model structureon ΣSp {∗ G } is simplicial, and hence, the model category G -ΣSp is simplicial.Let holim G denote the homotopy limit for G -ΣSp, as defined in [24, Definition18.1.8] (this definition uses the fact that G -ΣSp is a simplicial model category).Since the forgetful functor U G is a right adjoint (its left adjoint is given by thefunctor ΣSp → G -ΣSp that sends a spectrum Z to the G –spectrum W G Z , where G acts only on the indexing set of the coproduct), limits in G -ΣSp are formedin ΣSp. Also, it is a standard fact that the cotensor Y S • in G -ΣSp, where Y isa G –spectrum and S • is a simplicial set, is equal to the corresponding cotensor Y S • in ΣSp equipped with the natural G –action. Since holim G is defined as theequalizer of maps between products of cotensors, it follows that holim G is formedin ΣSp: if { Y c } c ∈C is a small diagram of G –spectra, then holim G C { Y c } c is equal tothe spectrum holim C { Y c } c equipped with the induced G –action.Now we recall [17, Theorem 4.3], but we rewrite it for symmetric spectra ([loc.cit.] is written in the world of Bousfield-Friedlander spectra, but the argument isthe same when using symmetric spectra). The forgetful functor U : ΣSp G → G -ΣSphas a right adjoint, the discretization functor( − ) d : G -ΣSp → ΣSp G , Y ( Y ) d = colim N ⊳ o G Y N . Since U preserves weak equivalences and cofibrations, the functors ( U, ( − ) d ) are aQuillen pair. Theorem 6.4. If ( G, X, N ) is a suitably finite triple, then X hG ≃ ( X dis N ) G ∼ = (cid:0) holim ∆ Sets( G • +1 , X f ) (cid:1) G ≃ X e hG . Proof.
Since X f is fibrant in ΣSp, Sets( G, X f ) is fibrant in G -ΣSp, and hence, itis fibrant in ΣSp. By iterating this argument, we obtain that Sets( G • +1 , X f ) is acosimplicial fibrant G –spectrum (that is, for each m ≥
0, the m –cosimplices are a fi-brant G –spectrum). It follows that holim G ∆ Sets( G • +1 , X f ) is a fibrant G –spectrum. ince holim G ∆ Sets( G • +1 , X f ) is equal to the G –spectrum holim ∆ Sets( G • +1 , X f ), wewrite the latter instead of the former.Let X → X fib be a trivial cofibration to a fibrant object, in G -ΣSp, and noticethat the equivalence X ≃ −→ holim ∆ Sets( G • +1 , X f ) (in Definition 6.1) is a weakequivalence with fibrant target, in G -ΣSp. Then there exists a weak equivalence X fib ≃ −→ holim ∆ Sets( G • +1 , X f )in G -ΣSp, and since ( − ) G : G -ΣSp → ΣSp is a right Quillen functor, the inducedmap X e hG = ( X fib ) G ≃ −→ (cid:0) holim ∆ Sets( G • +1 , X f ) (cid:1) G is a weak equivalence.Since N satisfies conditions (a) and (b) in Definition 3.3, there is an isomorphism X dis N ∼ = colim N ⊳ o G (cid:0) holim ∆ Sets( G • +1 , X f ) (cid:1) N = (cid:0) holim ∆ Sets( G • +1 , X f ) (cid:1) d of discrete G –spectra, as noted in Remark 4.6, and since the functor ( − ) d is a rightQuillen functor, (cid:0) holim ∆ Sets( G • +1 , X f ) (cid:1) d is a fibrant discrete G –spectrum, andhence, so is X dis N . Thus, applying the right Quillen functor ( − ) G : ΣSp G → ΣSpto the fibrant replacement map X dis N → ( X dis N ) fG , which is a trivial cofibrationbetween fibrant objects in ΣSp G , yields the weak equivalence( X dis N ) G ≃ −→ (cid:0) ( X dis N ) fG (cid:1) G = ( X dis N ) hG = X hG . The final step is to note that( X dis N ) G ∼ = (cid:16) colim N ⊳ o G (cid:0) holim ∆ Sets( G • +1 , X f ) (cid:1) N (cid:17) G ∼ = (cid:0) holim ∆ Sets( G • +1 , X f ) (cid:1) G , as desired. (cid:3) Remark 6.5.
Let (
G, X, N ) be a suitably finite triple. In light of the proof ofTheorem 6.4, we reexamine the G –equivariant zigzag X ≃ −→ holim ∆ Sets( G • +1 , X f ) ≃ ←− X dis N of equivalences: the first map is taking an explicit fibrant replacement of X – callit X ′ – in the model category of G –spectra (here, we do not require the fibrantreplacement map to be a cofibration) and the second map is the inclusion into X ′ from its largest discrete G –subspectrum X dis N ∼ = ( X ′ ) d (this description of theoutput of the functor ( − ) d is used in [38, page 861] and it is meant to be takenliterally: if X ′′ is a discrete G –subspectrum of X ′ , then the isomorphism X ′′ ∼ = colim N ⊳ o G ( X ′′ ) N shows that X ′′ is a G –subspectrum ofcolim N ⊳ o G ( X ′ ) N = ( X ′ ) d ∼ = X dis N ) . Therefore, we can think of the above zigzag as saying that X is equivalent toan explicit model – X dis N – for ( R ( − ) d )( X ), the output of the total right derivedfunctor R ( − ) d of ( − ) d (recall from the proof of Theorem 6.4 that there is a weak quivalence X fib ≃ −→ X ′ between fibrant objects in G -ΣSp, and hence, there is aweak equivalence ( R ( − ) d )( X ) = ( X fib ) d ≃ −→ ( X ′ ) d ∼ = −→ X dis N of discrete G –spectra).We can use Theorem 6.4 to build a descent spectral sequence, as follows. Corollary 6.6. If ( G, X, N ) is a suitably finite triple, then there is a conditionallyconvergent descent spectral sequence that has the form E s,t ∼ = H sc ( G ; π t ( X )) ∼ = H s ( G ; π t ( X )) = ⇒ π t − s ( X e hG ) ∼ = π t − s ( X hG ) . Proof.
At the beginning of the proof of Theorem 6.4, we noted that Sets( G • +1 , X f )is a cosimplicial fibrant G –spectrum, and hence, Sets( G • +1 , X f ) G is a cosimplicialfibrant spectrum. Thus, there is a homotopy spectral sequence E s,t = H s (cid:2) π t (cid:0) Sets( G ∗ +1 , X f ) G (cid:1)(cid:3) = ⇒ π t − s (cid:0) holim ∆ Sets( G • +1 , X f ) G (cid:1) . This spectral sequence is the descent spectral sequence described in the corollary,and the isomorphism that occurs in the abutment of the descent spectral sequencefollows immediately from applying Theorem 6.4 to the abutment of the above ho-motopy spectral sequence.Lemma 2.1 yields the isomorphism E s,t ∼ = H s ( G ; π t ( X )), for all s ≥ t . If the triple ( G, X, N ) satisfies the hypotheses of Theorem 4.9, then thereis an isomorphism H sc ( G ; π t ( X )) ∼ = H s ( G ; π t ( X )), for all s ≥ t ∈ Z , byRemark 3.4. If the triple ( G, X, N ) satisfies the hypotheses of Theorem 5.1, thenthis same isomorphism is obtained by applying the spectral sequence argument ofRemark 3.4 to the case where the “ M ” in the remark is changed to π t ( X ). (cid:3) To illustrate the previous result, we have the following special case for G = Z p . Corollary 6.7.
Let p be any prime. If X is a Z p –spectrum and an f –spectrum,then there is a strongly convergent descent spectral sequence E s,t = H sc ( Z p ; π t ( X )) ⇒ π t − s ( X h Z p ) , with E s,t = 0 , whenever s ≥ and t is any integer.Proof. By Theorem 3.6, Z p has a good filtration, with N = { p m Z p } m ≥ . Anysubgroup of finite index in Z p is open in Z p and π ∗ ( X ) is finite in each degree, sothat π t ( X ) is a discrete Z p –module (see Remark 3.2), for every integer t . It followsthat X is an r – G –spectrum, and hence, ( Z p , X, { p m Z p } m ≥ ) is a suitably finitetriple, X can be identified with the discrete Z p –spectrum X dis N , X h Z p is defined,and Corollary 6.6 gives the conditionally convergent spectral sequence describedabove.Since each π t ( X ) is finite and Z p has cohomological p -dimension one, E s,t = H sc ( Z p ; π t ( X )) = 0, whenever s ≥
2, for all integers t (this fact is well-known; as areference for the argument, see, for example, [20, proof of Theorem 2.9]), and thisvanishing result implies that the spectral sequence is strongly convergent, by [37,Lemma 5.48]. (cid:3) . Filtered diagrams of suitably finite triples and their colimits
In this section, we extend Definitions 6.1 and 6.2 to the case of a filtered diagramof G –spectra. Definition 7.1.
Let G be a profinite group with N a fixed inverse system of opennormal subgroups of G , and let { X µ } µ be a filtered diagram of G –spectra (thus,the morphisms in the diagram are G –equivariant), such that for each µ , ( G, X µ , N )is a suitably finite triple and X µ is a fibrant spectrum. We refer to ( G, { X µ } µ , N )as a suitably filtered triple .Let ( G, { X µ } µ , N ) be a suitably filtered triple. Since the colimit of a filtereddiagram of weak equivalences between fibrant spectra is a weak equivalence, thereis a zigzag of G –equivariant maps(7.2) colim µ X µ ≃ −→ colim µ holim ∆ Sets( G • +1 , ( X µ ) f ) ≃ ←− colim µ ( X µ ) dis N that are weak equivalences in ΣSp (since each Sets( G • +1 , ( X µ ) f ) is a cosimplicialfibrant spectrum, each holim ∆ Sets( G • +1 , ( X µ ) f ) is a fibrant spectrum; also, by theproof of Theorem 6.4, each ( X µ ) dis N is a fibrant discrete G –spectrum, and thus, by[8, Corollary 5.3.3], each ( X µ ) dis N is a fibrant spectrum). The right end of zigzag(7.2) satisfies colim µ ( X µ ) dis N = colim µ colim α ∈ Λ holim ∆ Sets( G • +1 , ( X µ ) f ) N α and colim µ ( X µ ) dis N is a discrete G –spectrum. (In zigzag (7.2), since each X µ is afibrant spectrum, the fibrant replacement in each ( X µ ) f is not necessary. However,we believe that by leaving the ( − ) f in each ( X µ ) f and by continuing to use themaps i X µ as previously defined (in the proof of Lemma 4.7), our presentation isless cumbersome.) Notice that for every integer t , our hypotheses on the triple andzigzag (7.2) imply that the composition(7.3) colim µ π t ( X µ ) ∼ = −→ π t (colim µ X µ ) ∼ = −→ π t (colim µ ( X µ ) dis N ) ∼ = −→ colim µ π t (( X µ ) dis N )consists of three isomorphisms in the category of discrete G –modules (in particular,each of the four abelian groups above is a discrete G –module). Definition 7.4.
Given a suitably filtered triple ( G, { X µ } µ , N ), the weak equiva-lences in zigzag (7.2) imply that the G –spectrum colim µ X µ can be identified withthe discrete G –spectrum colim µ ( X µ ) dis N . Thus, it is natural to define(colim µ X µ ) hG = (cid:0) colim µ ( X µ ) dis N (cid:1) hG . We can extend this definition to an arbitrary closed subgroup K in G : since the K –spectrum colim µ X µ can be regarded as the discrete K –spectrum colim µ ( X µ ) dis N ,we define (colim µ X µ ) hK = (cid:0) colim µ ( X µ ) dis N (cid:1) hK . Remark 7.5.
Let ( G, { X µ } µ , N ) be a suitably filtered triple and let K be a closedsubgroup of G . Suppose that K is finite, so that its topology is both profiniteand discrete. It follows that any K –spectrum can itself be regarded as a dis-crete K –spectrum, whenever desired. Thus, the notation (colim µ X µ ) hK can mean (colim µ X µ ) fK (cid:1) K or it can mean (cid:0) colim µ ( X µ ) dis N (cid:1) hK . In the remainder of this re-mark, to avoid any ambiguity, we take (colim µ X µ ) hK to have the latter meaning, (cid:0) colim µ ( X µ ) dis N (cid:1) hK , and for the former meaning, (cid:0) (colim µ X µ ) fK (cid:1) K , we just writeit out as needed. Since (7.2) can be regarded as a zigzag of weak equivalences inthe category of discrete K –spectra, there is a zigzag of weak equivalences (cid:0) (colim µ X µ ) fK (cid:1) K ≃ −→ (cid:0) colim µ holim ∆ Sets( G • +1 , ( X µ ) f ) (cid:1) hK ≃ ←− (cid:0) colim µ ( X µ ) dis N (cid:1) hK . Also, given an arbitrary K –spectrum Y , let Y → Y fib be a trivial cofibration to afibrant object, in K -ΣSp, the category of K –spectra. Then we have Y e hK = ( Y fib ) K ≃ ( Y fK ) K , where the last equivalence follows from the fact that Y fib is fibrant in ΣSp K (andthis fibrancy assertion is true because the functor ( − ) d : K -ΣSp → ΣSp K preservesfibrant objects and ( Y fib ) d ∼ = Y fib is an isomorphism in ΣSp K ). We conclude thatwhen K is finite, there are equivalences(colim µ X µ ) hK ≃ (cid:0) (colim µ X µ ) fK (cid:1) K ≃ (colim µ X µ ) e hK , as one would expect.We say that a profinite group G has finite virtual cohomological dimension (“fi-nite v.c.d.”) if G contains an open subgroup that has finite c.d. Under the assump-tion that G has this property, the following result gives a descent spectral sequencefor the situation described by Definition 7.1. Theorem 7.6.
Let G be a profinite group with finite v.c.d. If ( G, { X µ } µ , N ) is asuitably filtered triple and K is a closed subgroup of G , then there is a conditionallyconvergent descent spectral sequence E ∗ , ∗ r ( K ) that has the form (7.7) E s,t ( K ) = H sc ( K ; π t (colim µ X µ )) = ⇒ π t − s (cid:0) (colim µ X µ ) hK (cid:1) . Remark 7.8. If G has a good filtration, then condition (d) of Definition 3.3 impliesthat G has finite v.c.d. Thus, if ( G, { X µ } µ , N ) is a suitably filtered triple such thatthere is some µ ∈ { µ } µ for which the triple ( G, X µ , N ) satisfies the hypotheses ofTheorem 4.9, then G has finite v.c.d. and the first sentence of Theorem 7.6 can beomitted. Proof of Theorem 7.6.
Let U be an open subgroup of G that has finite c.d. Then U ∩ K is an open subgroup of K , and since U has finite c.d. and U ∩ K is closedin U , there exists some r such that for any discrete ( U ∩ K )–module M , H sc ( U ∩ K ; M ) ∼ = H sc ( U ; Coind UU ∩ K ( M )) = 0 , whenever s > r, by Shapiro’s Lemma. This shows that K has finite v.c.d. Therefore, [8, proofs ofTheorem 3.2.1, Proposition 3.5.3] and [16, proof of Theorem 7.9] yield the condi-tionally convergent spectral sequence E s,t = H sc ( K ; π t (colim µ ( X µ ) dis N )) = ⇒ π t − s (cid:16)(cid:0) colim µ ( X µ ) dis N (cid:1) hK (cid:17) , and this is the desired spectral sequence, since the middle map in composition (7.3)is an isomorphism of discrete K –modules. e provide some more detail (based on the above two references) because it willbe useful to us later. Since K has finite v.c.d., (cid:0) colim µ ( X µ ) dis N (cid:1) hK ≃ holim ∆ Γ • K colim µ ( X µ ) dis N , and for each m ≥
0, the m -cosimplices of cosimplicial spectrum Γ • K colim µ ( X µ ) dis N satisfy the isomorphism(7.9) (cid:0) Γ • K colim µ ( X µ ) dis N (cid:1) m ∼ = colim V ⊳ o K m Q K m /V colim µ ( X µ ) dis N , where K m is the m -fold Cartesian product of K ( K is the trivial group { e } ,equipped with the discrete topology). (For more detail about this, we refer thereader to [8, Sections 2.4, 3.2].)The above spectral sequence is the homotopy spectral sequence for the spectrumholim ∆ Γ • K colim µ ( X µ ) dis N . Based on [8, proof of Theorem 3.2.1] and [16, proof ofTheorem 7.9], the reader might expect us to instead form the homotopy spectralsequence for holim ∆ Γ • K (cid:0) colim µ ( X µ ) dis N (cid:1) fK . But since each ( X µ ) dis N is a fibrant spec-trum, colim µ ( X µ ) dis N is already a fibrant spectrum, so that we do not need to apply( − ) fK to it (so that we are taking the homotopy limit of a cosimplicial fibrantspectrum). (cid:3) Notice that if ( G, { X µ } µ , N ) is a suitably filtered triple, then for each µ ′ ∈ { µ } µ ,( G, { X µ } µ ∈{ µ ′ } , N ) is a suitably filtered triple, so that Definition 7.4 gives( X µ ′ ) hK = (cid:0) ( X µ ′ ) dis N (cid:1) hK , for any closed subgroup K of G . Theorem 7.10.
Let G be a profinite group with finite v.c.d., let ( G, { X µ } µ , N ) bea suitably filtered triple such that { µ } µ is a directed poset, and let K be a closedsubgroup of G . If there exists a nonnegative integer r such that for all t ∈ Z andeach µ , H sc ( K ; π t ( X µ )) = 0 whenever s > r , then descent spectral sequence E ∗ , ∗ r ( K ) in ( ) is strongly convergent and there is an equivalence of spectra (colim µ X µ ) hK ≃ colim µ ( X µ ) hK . Proof.
For all t ∈ Z , when s > r , we have E s,t ( K ) = H sc ( K ; π t (colim µ X µ )) ∼ = colim µ H sc ( K ; π t ( X µ )) = 0 , so that the spectral sequence is strongly convergent, by [37, Lemma 5.48].If V is an open normal subgroup of K m , where m ≥
0, then K m /V is finite, andhence, isomorphism (7.9) implies that (cid:0) Γ • K colim µ ( X µ ) dis N (cid:1) m ∼ = colim µ colim V ⊳ o K m Q K m /V ( X µ ) dis N ∼ = colim µ (cid:0) Γ • K ( X µ ) dis N (cid:1) m , so that there is an isomorphismΓ • K colim µ ( X µ ) dis N ∼ = colim µ Γ • K ( X µ ) dis N of cosimplicial spectra. Therefore, we have (cid:0) colim µ ( X µ ) dis N (cid:1) hK ≃ holim ∆ Γ • K colim µ ( X µ ) dis N ∼ = holim ∆ colim µ Γ • K ( X µ ) dis N , hich gives(colim µ X µ ) hK ≃ holim ∆ colim µ Γ • K ( X µ ) dis N ←− colim µ holim ∆ Γ • K ( X µ ) dis N ≃ colim µ (cid:0) ( X µ ) dis N (cid:1) hK = colim µ ( X µ ) hK , and the canonical colim/holim exchange map above is a weak equivalence if thereexists a nonnegative integer r such that for every t and all µ , H s (cid:2) π t (cid:0) Γ ∗ K ( X µ ) dis N (cid:1)(cid:3) = 0 , when s > r, by [29, Proposition 3.4]. The proof is completed by noting that there are isomor-phisms H s (cid:2) π t (cid:0) Γ ∗ K ( X µ ) dis N (cid:1)(cid:3) ∼ = H sc ( K ; π t (( X µ ) dis N )) ∼ = H sc ( K ; π t ( X µ )) , for all s ≥ (cid:3) The proofs of Theorems 1.6 and 1.7
After proving Theorem 1.6, a task which ends with (8.6), we prove Theorem 1.7.Let p ≥ K be any closed subgroup of Z × p . As noted in the proof ofTheorem 3.6, Z p has finite c.d., and since it is open in Z × p , Z × p has finite v.c.d.Also, in the introduction (see Remark 1.19), we showed that (cid:0) Z × p , (cid:8)(cid:0) K ( KU p ) ∧ Σ − jd V (1) (cid:1) f (cid:9) j ≥ , N (cid:1) , where N is as defined in Remark 1.19, is a suitably filtered triple. Therefore, byTheorem 7.6, there is a conditionally convergent descent spectral sequence that hasthe form(8.1) E s,t ⇒ π t − s (cid:0)(cid:0) K ( KU p ) ∧ v − V (1) (cid:1) hK (cid:1) , where E s,t = H sc (cid:0) K ; π t (cid:0) colim j ≥ (cid:0) K ( KU p ) ∧ Σ − jd V (1) (cid:1) f (cid:1)(cid:1) ∼ = H sc ( K ; π t ( K ( KU p ) ∧ V (1))[ v − ]) , as desired.Since p ≥ V (1) is a homotopy commutative and homotopy associative ringspectrum [30], so that π ∗ ( K ( KU p ) ∧ V (1)) is a graded right π ∗ ( V (1))–module, andhence, π ∗ ( K ( KU p ) ∧ V (1)) is a unitary F p –module. It follows that for every integer t , the finite abelian group π t ( K ( KU p ) ∧ V (1)) is a p –torsion group (that is, pm = 0,for all m ∈ π t ( K ( KU p ) ∧ V (1))).Given any profinite group G , we use cd p ( G ) to denote its cohomological p –dimension. Since K is closed in Z × p ,cd p ( K ) ≤ cd p ( Z × p ) = cd p ( Z p ) = 1 , where the first equality is due to the fact that Z p is the p –Sylow subgroup of Z × p ,and hence,(8.2) H sc ( K ; M ) = 0 , for all s ≥ , whenever M is a discrete K –module that is also p –torsion. Now choose any j ≥ t ∈ Z and all s ≥
0, there is an isomorphism H sc (cid:0) K ; π t (cid:0)(cid:0) K ( KU p ) ∧ Σ − jd V (1) (cid:1) f (cid:1)(cid:1) ∼ = H sc ( K ; π t + jd ( K ( KU p ) ∧ V (1))) . hen (8.2) implies that for every integer t ,(8.3) H sc (cid:0) K ; π t (cid:0)(cid:0) K ( KU p ) ∧ Σ − jd V (1) (cid:1) f (cid:1)(cid:1) = 0 , for all s ≥ , since the discrete K –module π t + jd ( K ( KU p ) ∧ V (1)) is p –torsion.We have now verified the hypotheses of Theorem 7.10, so that descent spectralsequence (8.1) is strongly convergent, E s,t = 0 for all integers t whenever s ≥ (cid:0) K ( KU p ) ∧ v − V (1) (cid:1) hK ≃ colim j ≥ (cid:0)(cid:0) K ( KU p ) ∧ Σ − jd V (1) (cid:1) f (cid:1) hK . Let G be any profinite group and let X and X be arbitrary G –spectra, suchthat ( G, X , U ) and ( G, X , U ) are suitably finite triples (the inverse system U isthe same in each triple) and there is a weak equivalence w : X → X in G -ΣSp.The equivalence w induces the commutative diagram X ≃ / / ≃ w (cid:15) (cid:15) holim ∆ Sets( G • +1 , ( X ) f ) (cid:15) (cid:15) ( X ) dis U≃ o o w dis U (cid:15) (cid:15) X ≃ / / holim ∆ Sets( G • +1 , ( X ) f ) ( X ) dis U≃ o o in which each “ ≃ ” denotes a weak equivalence in G -ΣSp. From the left commutativesquare, it follows that the middle vertical map in the diagram is a weak equivalencein G -ΣSp, and hence, the right commutative square implies that the G –equivariantmap w dis U is a weak equivalence of spectra, which allows us to conclude that w dis U isa weak equivalence in ΣSp G .As in Definition 7.4, for any suitably finite triple ( G, X, U ) and any closed sub-group P of G , it is natural to define X hP = ( X dis U ) hP (this extends Definition 6.2). For any P , since w dis U is a weak equivalence in thecategory of discrete P –spectra, it follows that the induced map( X ) hP = (( X ) dis U ) hP ≃ −→ (( X ) dis U ) hP = ( X ) hP is a weak equivalence.For each j ≥
0, the triples( Z × p , K ( KU p ) ∧ Σ − jd V (1) , N ) and (cid:0) Z × p , (cid:0) K ( KU p ) ∧ Σ − jd V (1) (cid:1) f , N (cid:1) are suitably finite, the natural fibrant replacement map K ( KU p ) ∧ Σ − jd V (1) ≃ −→ (cid:0) K ( KU p ) ∧ Σ − jd V (1) (cid:1) f is a weak equivalence in the category of Z × p –spectra, and K ( KU p ) ∧ Σ − jd V (1)can be identified with the discrete Z × p –spectrum ( K ( KU p ) ∧ Σ − jd V (1)) dis N , as inDefinition 6.2. It follows from the above discussion that for each j ≥ K , there is the definition( K ( KU p ) ∧ Σ − jd V (1)) hK = (cid:0) ( K ( KU p ) ∧ Σ − jd V (1)) dis N (cid:1) hK and there is a weak equivalence( K ( KU p ) ∧ Σ − jd V (1)) hK ≃ −→ (cid:0)(cid:0) K ( KU p ) ∧ Σ − jd V (1) (cid:1) f (cid:1) hK etween fibrant spectra, giving a weak equivalence(8.5) colim j ≥ ( K ( KU p ) ∧ Σ − jd V (1)) hK ≃ −→ colim j ≥ (cid:0)(cid:0) K ( KU p ) ∧ Σ − jd V (1) (cid:1) f (cid:1) hK . From (8.4) and (8.5), we obtain an equivalence(8.6) (cid:0) K ( KU p ) ∧ v − V (1) (cid:1) hK ≃ colim j ≥ ( K ( KU p ) ∧ Σ − jd V (1)) hK . Proof of Theorem 1.7.
Setting n = 1 in (1.1) gives the K (1)–local profinite Z × p –Galois extension L K (1) ( S ) → KU p , and this map yields a Z × p –equivariant map K ( L K (1) ( S )) → K ( KU p ), with Z × p acting trivially on K ( L K (1) ( S )). Thus, foreach j ≥
0, the induced map K ( L K (1) ( S )) ∧ Σ − jd V (1) → K ( KU p ) ∧ Σ − jd V (1) ≃ −→ (cid:0) K ( KU p ) ∧ Σ − jd V (1) (cid:1) f is Z × p –equivariant, giving the canonical map to the fixed points,(8.7) K ( L K (1) ( S )) ∧ Σ − jd V (1) → (cid:0)(cid:0) K ( KU p ) ∧ Σ − jd V (1) (cid:1) f (cid:1) Z × p . It follows that there is the map(8.8) K ( L K (1) ( S )) ∧ v − V (1) → colim j ≥ (cid:0)(cid:0) K ( KU p ) ∧ Σ − jd V (1) (cid:1) f (cid:1) Z × p , which is defined to be the composition K ( L K (1) ( S )) ∧ v − V (1) ∼ = −→ colim j ≥ ( K ( L K (1) ( S )) ∧ Σ − jd V (1)) → colim j ≥ ( KV j ) Z × p , where here and below, we use the notation KV j := (cid:0) K ( KU p ) ∧ Σ − jd V (1) (cid:1) f , for j ≥ , to keep certain expressions from being too long (and the second map in the com-position is obtained by taking a colimit of the maps given by (8.7)).For the diagram of Z × p –equivariant maps (cid:8) i KVj : KV j ≃ −→ holim ∆ Sets(( Z × p ) • +1 , ( KV j ) f ) (cid:9) j ≥ , taking fixed points and then the colimit gives the canonical map(8.9) colim j ≥ ( KV j ) Z × p → colim j ≥ (cid:0) holim ∆ Sets(( Z × p ) • +1 , ( KV j ) f ) (cid:1) Z × p . Also, for each j ≥ N as defined in Remark 1.19), there are naturalisomorphisms (cid:0) holim ∆ Sets(( Z × p ) • +1 , ( KV j ) f ) (cid:1) Z × p ∼ = (cid:16) colim N ⊳ o Z × p (cid:0) holim ∆ Sets(( Z × p ) • +1 , ( KV j ) f ) (cid:1) N (cid:17) Z × p ∼ = (cid:0) ( KV j ) dis N (cid:1) Z × p , where the last step is due to the isomorphismcolim N ⊳ o Z × p (cid:0) holim ∆ Sets(( Z × p ) • +1 , ( KV j ) f ) (cid:1) N ∼ = ( KV j ) dis N in the category of discrete Z × p –spectra (which itself is valid by Remark 3.4 (see itsfirst two sentences)), and hence, there is the isomorphism(8.10) colim j ≥ (cid:0) holim ∆ Sets(( Z × p ) • +1 , ( KV j ) f ) (cid:1) Z × p ∼ = −→ colim j ≥ (cid:0) ( KV j ) dis N (cid:1) Z × p . inally, there is the composition of canonical maps(8.11) colim j ≥ (cid:0) ( KV j ) dis N (cid:1) Z × p → (cid:0) colim j ≥ ( KV j ) dis N (cid:1) Z × p = ( C dis p ) Z × p → (cid:0) ( C dis p ) f Z × p (cid:1) Z × p , where the first map is due to the universal property of the colimit and the secondmap is obtained by applying fixed points to the fibrant replacement map. Thetarget of map (8.11) is equal to (cid:0) K ( KU p ) ∧ v − V (1) (cid:1) h Z × p , and the composition ofmaps (8.8), (8.9), (8.10), and (8.11) (that is, after omitting the source and targetfrom each map, the composition ( . ) −−−→ ( . ) −−−→ ( . ) −−−−→ ( . ) −−−−→ ) defines the desiredmap K ( L K (1) ( S )) ∧ v − V (1) → (cid:0) K ( KU p ) ∧ v − V (1) (cid:1) h Z × p . (cid:3) The proof of Theorem 1.8
As in the preceding section, we continue with letting p ≥
5. By Theorem 1.6 (inparticular, see (8.6)), there is an equivalence (cid:0) K ( KU p ) ∧ v − V (1) (cid:1) h Z × p ≃ colim j ≥ ( K ( KU p ) ∧ Σ − jd V (1)) h Z × p , and for each j ≥
0, ( Z × p , K ( KU p ) ∧ Σ − jd V (1) , N ) (with N as defined in Remark1.19) is a suitably finite triple. Then by the proof of Theorem 6.4 (the spectrum( K ( KU p ) ∧ Σ − jd V (1)) dis N is a fibrant discrete Z × p –spectrum, for each j ), there areweak equivalencescolim j ≥ ( K ( KU p ) ∧ Σ − jd V (1)) h Z × p = colim j ≥ (cid:16)(cid:0) ( K ( KU p ) ∧ Σ − jd V (1)) dis N (cid:1) f Z × p (cid:17) Z × p ≃ ←− colim j ≥ (cid:0) ( K ( KU p ) ∧ Σ − jd V (1)) dis N (cid:1) Z × p ≃ ←− colim j ≥ ( K ( KU p ) ∧ Σ − jd V (1)) e h Z × p . The last weak equivalence above requires a little more justification. Let J denotethe indexing category { j ≥ } for the above colimits. For any profinite group G ,the model structure on G -ΣSp is combinatorial, by [27, Proposition A.2.8.2], andhence, ( G -ΣSp) J , the category of J –shaped diagrams in G -ΣSp, can be equippedwith a projective model structure (again, by [27, Proposition A.2.8.2]). Thus, weregard ( Z × p -ΣSp) J as having a projective model structure, and we let { K ( KU p ) ∧ Σ − jd V (1) } j ≥ ≃ −→ { ( K ( KU p ) ∧ Σ − jd V (1)) pf } j ≥ be a trivial cofibration to a fibrant object, in ( Z × p -ΣSp) J . Notice that by the proofof Theorem 6.4, the morphism { K ( KU p ) ∧ Σ − jd V (1) } j ≥ ≃ −→ (cid:8) holim ∆ Sets(( Z × p ) • +1 , ( K ( KU p ) ∧ Σ − jd V (1)) f ) (cid:9) j ≥ is a weak equivalence to a fibrant object, in ( Z × p -ΣSp) J , and hence, there is amorphism { ( K ( KU p ) ∧ Σ − jd V (1)) pf } j ≥ ≃ −→ (cid:8) holim ∆ Sets(( Z × p ) • +1 , ( K ( KU p ) ∧ Σ − jd V (1)) f ) (cid:9) j ≥ that is a weak equivalence between fibrant objects, in ( Z × p -ΣSp) J . As in the proofof Theorem 6.4, the application of the right Quillen functor( − ) Z × p : Z × p -ΣSp → ΣSp o the last morphism induces compositions(( K ( KU p ) ∧ Σ − jd V (1)) pf ) Z × p ≃ −→ (cid:0) holim ∆ Sets(( Z × p ) • +1 , ( K ( KU p ) ∧ Σ − jd V (1)) f ) (cid:1) Z × p ∼ = −→ (cid:0) ( K ( KU p ) ∧ Σ − jd V (1)) dis N (cid:1) Z × p for all j ≥
0, with each composition a weak equivalence between fibrant spectra.Taking the colimit over J of these weak equivalences yields the weak equivalence ω : colim j ≥ (( K ( KU p ) ∧ Σ − jd V (1)) pf ) Z × p ≃ −→ colim j ≥ (cid:0) ( K ( KU p ) ∧ Σ − jd V (1)) dis N (cid:1) Z × p . By [27, Remark A.2.8.5], every projective cofibration is an injective cofibration, sothat for each j , the map K ( KU p ) ∧ Σ − jd V (1) ≃ −→ ( K ( KU p ) ∧ Σ − jd V (1)) pf is a trivial cofibration to a fibrant object in Z × p -ΣSp. It follows from this that thesource of weak equivalence ω satisfies the equalitycolim j ≥ (( K ( KU p ) ∧ Σ − jd V (1)) pf ) Z × p = colim j ≥ ( K ( KU p ) ∧ Σ − jd V (1)) e h Z × p , and thus, ω is the weak equivalence that we set out in this paragraph to obtain.Fix j ≥
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