Detecting and describing ramification for structured ring spectra
aa r X i v : . [ m a t h . A T ] J a n DETECTING AND DESCRIBING RAMIFICATION FOR STRUCTUREDRING SPECTRA
EVA H ¨ONING AND BIRGIT RICHTER
Abstract.
Ramification for commutative ring spectra can be detected by relative topologicalHochschild homology and by topological Andr´e-Quillen homology. In the classical algebraiccontext it is important to distinguish between tame and wild ramification. Noether’s theoremcharacterizes tame ramification in terms of a normal basis and tame ramification can also bedetected via the surjectivity of the trace map. We transfer the latter fact to ring spectra anduse the Tate cohomology spectrum to detect wild ramification in the context of commutativering spectra. We study ramification in examples in the context of topological K-theory andtopological modular forms. Introduction
Classically, ramification is studied in the setting of extensions of rings of integers in numberfields. If K ⊂ L is an extension of number fields and if O K → O L is the corresponding extensionof rings of integers, then a prime ideal p ⊂ O K ramifies in L , if p O L = p e · . . . · p e s s in O L and e i > i s . The ramification is tame when the ramification indices e i areall relatively prime to the residue characteristic of p and it is wild otherwise. Auslander andBuchsbaum [AB59] considered ramification in the setting of general noetherian rings. If K ⊂ L is a finite G -Galois extension, then O K → O L is unramified, if and only if O K = O GL → O L isa Galois extension of commutative rings and this in turn says that O L ⊗ O K O L ∼ = Q G O L (see[CHR65, Remark 1.5 (d)], [AB59] or [Rog08a, Example 2.3.3] for more details).Our main interest is to investigate notions of ramified extensions of ring spectra and to studyexamples.Rognes [Rog08a, Definition 4.1.3] introduces G -Galois extensions of ring spectra. A map A → B of commutative ring spectra is a G -Galois extension for a finite group G , if certaincofibrancy conditions are satisfied, if G acts on B from the left through commutative A -algebramaps and if the following two conditions are satisfied:(1) The map from A to the homotopy fixed points of B with respect to the G -action, i : A → B hG is a weak equivalence.(2) The map(1.1) h : B ∧ A B → Y G B is a weak equivalence.Here, h is right adjoint to the composite map B ∧ A B ∧ G + / / B ∧ A B / / B , induced by the G -action B ∧ G + ∼ = G + ∧ B → B on B and the multiplication on B . Date : February 1, 2021.2000
Mathematics Subject Classification.
Key words and phrases.
Tame ramification, wild ramification, topological Andr´e-Quillen homology, topologicalHochschild homology, topological modular forms, Tate vanishing.
Condition (1) is the fixed point condition familiar from ordinary Galois theory. Condition(2) is needed to ensure that the map A → B is unramified . Among other things, it implies forinstance that the A -endomorphisms of B correspond to the group elements in the sense that j : B ∧ G + → F A ( B, B ) , is a weak equivalence, where j is right adjoint to the composite map( B ∧ G + ) ∧ A B → B ∧ A B → B, which is again induced by the G -action and the multiplication on B .If A is the Eilenberg-MacLane spectrum H O K and B = H O L for a G -Galois extension K ⊂ L , then H O K → H O L is a G -Galois extension of ring spectra if and only if O K → O L isa G -Galois extension of commutative rings.For certain Galois extensions Ausoni and Rognes [AR08] conjecture a version of Galois descentfor algebraic K-theory. A descent result that covers many of the conjectured cases is establishedin [CMNN20]. In some cases, descent can be established even in the presence of ramification.Ausoni [Aus05, Theorem 10.2] shows for instance that the canonical map K ( ℓ p ) → K ( ku p ) hC p − is an equivalence after p -completion despite the fact that the inclusion of the p -completedconnective Adams summand, ℓ p , into p -completed topological connective K-theory, ku p , shouldbe viewed as a tamely ramified extension of commutative ring spectra. In other cases that arenot Galois extensions, for instance in cases, that we will identify as wildly ramified, one canconsider a modified version of descent [CMNN20, § A → B is separable [Rog08a, Definition 9.1.1] and this in turn implies that the canonicalmap from B to the relative topological Hochschild homology, THH A ( B ), is an equivalence andthat the spectrum of topological Andr´e-Quillen homology TAQ A ( B ) [Bas99] is trivial. So ifwe know for a map of commutative ring spectra A → B that B → THH A ( B ) is not a weakequivalence or that π ∗ TAQ A ( B ) = 0, then this is an indicator for ramification. We will studyexamples of non-vanishing TAQ in 2.1 and study relative topological Hochschild homology inexamples related to level-2-structures on elliptic curves in 2.2.An interesting class of examples arises as connective covers of G -Galois extensions. AkhilMathew shows in [Mat16a, Theorem 6.17] that connective Galois extensions are algebraically´etale: the induced map on homotopy groups is ´etale in a graded sense. So, in particular, connec-tive covers of Galois extensions are rarely Galois extensions, because several known examplesof Galois extensions such as KO → KU , L p → KU p and examples of Galois extensions in thecontext of topological modular forms are far from behaving nicely on the level of homotopygroups.We use relative topological Hochschild homology and topological Andr´e-Quillen homologyin order to detect ramification in the cases ko → ku , ℓ → ku ( p ) , tmf (3) (2) → tmf (3) (2) , tmf (3) → tmf (2) (3) , Tmf (3) → Tmf (2) (3) and tmf (2) (3) → tmf (2) (3) . We also study a versionof the discriminant map in the context of structured ring spectra and apply it to the examples ℓ → ku ( p ) and ko → ku in 2.3.For certain finite extensions of discrete valuation rings tame ramification is equivalent tobeing log-´etale (see for instance [Rog09, Example 4.32]). It is known by work of Sagave [Sag14],that ℓ → ku ( p ) is log-´etale if one considers the log structures generated by v ∈ π p − ℓ and u ∈ π ( ku ). We show that ko → ku is not log-´etale if one considers the log structures generatedby the Bott elements ω ∈ π ( ko ) and u ∈ π ( ku ).Emmy Noether shows [Noe32, §
2] that tame ramification is equivalent to the existence ofa normal basis. Tame ramification can also be detected by the surjectivity of the trace map[CF67, Theorem 2, Chapter 1, § ETECTING AND DESCRIBING RAMIFICATION 3 G -action X is in the thick subcategory generated by spectra ofthe form G + ∧ W , then X tG ≃ ∗ , so in particular, if B has a normal basis, B ≃ G + ∧ A , then B tG ≃ ∗ .We use the Tate spectrum in order to propose a definition of tame and wild ramification ofmaps of ring spectra and study examples in the context of topological K-theory, topologicalmodular forms and cochains on classifying spaces with coefficients in Morava E-theory akaLubin-Tate spectra.Several of our examples use topological modular forms with level structures. The spectrumof topological modular forms, TMF , arises as the global sections of a structure sheaf of E ∞ -ringspectra on the moduli stack of elliptic curves, M ell . A variant of it, Tmf , lives on a compactifiedversion, M ell . Its connective version is denoted by tmf . There are other variants correspondingto level structures on elliptic curves. Recall that a Γ( n )-structure (or level n -structure for short)carries the datum of a chosen isomorphism between the n -torsion points of an elliptic curve andthe group ( Z /n Z ) . A Γ ( n )-structure corresponds to the choice of a point of exact order n whereas a Γ ( n )-structure comes from the choice of a subgroup of order n of the n -torsion points.See [KM85, Chapter 3] for the precise definitions and for background. These level structuresgive rise to a tower of moduli problems (see [KM85, p. 200])[Γ( n )][Γ ( n )] ∗ ∗ ! [Γ ( n )] ( Z /n Z ) × (Ell) GL ( Z /n Z ) with corresponding spectra TMF ( n ), TMF ( n ) and TMF ( n ) and their compactified versions Tmf ( n ), Tmf ( n ) and Tmf ( n ) [HL16, Theorem 6.1].In [MM15] Mathew and Meier prove that the maps Tmf [ n ] → Tmf ( n ) are not Galois exten-sions but they satisfy Tate vanishing, which might be seen as an indication of tame ramification.In contrast, we will show that tmf ( n ) tGL ( Z /n Z ) is non-trivial if 2 does not divide n or if n is apower of 2. Acknowledgements.
We thank Lennart Meier for sharing [Mei] with us and we thank him, JohnRognes and Mark Behrens for several helpful comments. Mike Hill helped us a lot with patientexplanations about tmf and friends. The first named author was funded by the DFG priorityprogram SPP 1786
Homotopy Theory and Algebraic Geometry . The last named author wouldlike to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitalityduring the programme
Homotopy Harnessing Higher Structures when work on this paper wasundertaken. This work was supported by EPSRC grant number EP/R014604/1.2.
Detecting ramification
Topological Andr´e-Quillen homology.
For a map of connective commutative ring spec-tra i : A → B we use the connectivity of the map to determine the bottom homotopy group of TAQ A ( B ) [Bas99]. The non-triviality of TAQ A ( B ) indicates that the map i is ramified. Algebraic cases. If O K → O L is an extension of number rings with corresponding extension ofnumber fields K ⊂ L , then of course we cannot use a connectivity argument for understanding TAQ , but here, the algebraic module of K¨ahler differentials, Ω O L |O K , is isomorphic to the first EVA H ¨ONING AND BIRGIT RICHTER
Hochschild homology group HH O K ( O L ) which in turn is π TAQ H O K ( H O L ) [BR04, Theorem2.4]. The ramification type of O K → O L can be read off the different , i.e. , the annihilator ofthe module of K¨ahler differentials. The connective Adams summand.
Let ℓ denote the Adams summand of connective p -localizedtopological complex K-theory, ku ( p ) . Here, p is an odd prime.The inclusion i : ℓ → ku ( p ) induces an isomorphism on π and π . Thus, by the Hurewicztheorem for topological Andr´e-Quillen homology [Bas99, Lemma 8.2], [BGR08, Lemma 1.2] weget that π TAQ ℓ ( ku ( p ) ) is the bottom homotopy group and is isomorphic to the second homotopygroup of the cone of i , cone( i ) and this in turn can be determined by the long exact sequence . . . → π ( ℓ ) = 0 → π ( ku ( p ) ) → π (cone( i )) → π ( ℓ ) = 0 → . . . hence π TAQ ℓ ( ku ( p ) ) ∼ = Z ( p ) .We know from [DLR20] that ℓ → ku ( p ) shows features of a tamely ramified extension ofnumber rings and Sagave shows [Sag14, Theorem 6.1] that ℓ → ku ( p ) is log-´etale. Real and complex connective topological K-theory.
The complexification map c : ko → ku in-duces an isomorphism on π and an epimorphism on π , so it is a 1-equivalence. Hence again π cone( c ) ∼ = π ( TAQ ko ( ku )) is the bottom homotopy group, but here we obtain an extension0 → π ku = Z → π cone( c ) → π ( ko ) = Z / Z → . In order to understand π cone( c ) we consider the cofiber sequenceΣ KO η / / KO c / / KU δ / / Σ KO and the commutative diagram on homotopy groups Z / Z = π ko π ( c )0 / / π ( τ ko ) ∼ = (cid:15) (cid:15) π ku / / π ( τ ku ) ∼ = (cid:15) (cid:15) π cone( c ) / / g (cid:15) (cid:15) π ko π ( c ) / / π ( τ ko ) ∼ = (cid:15) (cid:15) π ku π ( τ ku ) ∼ = (cid:15) (cid:15) Z / Z = π KO π ( c )0 / / π KU π ( δ ) / / π Σ KO π (Σ η ) / / π KO π ( c ) / / π KU.
Here τ e : e → E denotes the map from the connective cover e of E to E . The middle verticalmap g is the map induced by the cofiber sequences. By the five lemma, g is an isomorphismhence π cone( c ) ∼ = π Σ KO ∼ = Z . So this group is also torsion free. We will later see that ko → ku is not log-´etale and we willsee some other indicators for wild ramification, but the bottom homotopy group of TAQ ko ( ku )does not detect that. Connective topological modular forms with level structure (case n = 3 ). We consider tmf (3). Itshomotopy groups are π ∗ ( tmf (3)) ∼ = Z [ ][ a , a ] with | a i | = 2 i . See [HL16] for some background.There is a C -action on tmf (3) coming from the permutation of elements of exact order threeand one denotes by tmf (3) the connective cover of the homotopy fixed points, tmf (3) hC .There is a homotopy fixed point spectral sequence that was studied in detail in [MR09] forthe periodic versions. In [HL16, p. 407] it is explained how to adapt this calculation to theconnective variants: The terms in the spectral sequence with s > t − s > C -action on the a i ’s is given by the sign-action, so if τ generates C , then τ ( a ni ) = ( − n a ni .This implies that only H ( C ; π ( tmf (3))) ∼ = Z [ ] survives to π ( tmf (3)). For π we get acontribution from H ( C ; π ( tmf (3))), giving a Z / Z generated by the class of a (this detectsan η ). For π ( tmf (3)) the class of a generates a copy of Z / Z . ETECTING AND DESCRIBING RAMIFICATION 5
Hence the map j : tmf (3) (2) → tmf (3) (2) is 1-connected, so π TAQ tmf (3) (2) ( tmf (3) (2) ) isthe bottom homotopy group and is isomorphic to π (cone( j )) which sits in an extension. Wecan use the commutative diagram of commutative ring spectra from [HL16, Theorem 63] tmf (3) / / (cid:15) (cid:15) tmf (3) (cid:15) (cid:15) ko [ ] / / ku [ ]in order to determine to π (cone( j )). By [LN14, Theorem 1.2] there is a cofiber sequence of tmf (3) (2) -modules Σ tmf (3) (2) v / / tmf (3) (2) / / ku (2) and hence π ( tmf (3) (2) ) ∼ = π ( ku (2) ).The diagram π tmf (3) (2) 0 / / ∼ = (cid:15) (cid:15) π tmf (3) (2) ∼ = (cid:15) (cid:15) / / π cone( j ) (cid:15) (cid:15) / / π tmf (3) (2) ∼ = (cid:15) (cid:15) / / π tmf (3) (2) = 0 ∼ = (cid:15) (cid:15) π ko (2) 0 / / π ku (2) / / π cone( c ) / / π ko (2) 0 / / π ku (2) = 0 . commutes and the 5-lemma implies that π (cone( j )) ∼ = Z (2) . Connective topological modular forms with level structure (case n = 2 , p = 3 ). Forgetting aΓ (2)-structure yields a map f : tmf (3) → tmf (2) (3) such that f is a 3-equivalence. We willrecall more details about these spectra at the beginning of 2.2. Again, we obtain that thebottom non-trivial homotopy group of the spectrum of topological Andr´e-Quillen homology is π ( TAQ tmf (3) ( tmf (2) (3) )) ∼ = π (cone( f )). There is a short exact sequence0 = π tmf (3) → π tmf (2) (3) = Z (3) → π cone( f ) → π tmf (3) ∼ = Z / Z → π cone( f ) could be isomorphic to Z (3) or to Z (3) ⊕ Z / Z .There is an equivalence [Beh06, § tmf (3) ∧ T ≃ tmf (2) (3) where T = S ∪ α e ∪ α e with α denoting the generator of π s at 3. Thus T is part of acofiber sequence S → T → Σ cone( α )and we obtain a cofiber sequence tmf (3) = tmf (3) ∧ S → tmf (3) ∧ T ≃ tmf (2) (3) → tmf (3) ∧ Σ cone( α )and thus π (cone( f )) ∼ = π ( tmf (3) ∧ Σ cone( α )) ∼ = π ( tmf (3) ∧ cone( α )) . But as we have a short exact sequence0 = π (Σ tmf (3) ) → π ( tmf (3) ) ∼ = Z (3) → π ( tmf (3) ∧ cone( α )) → π ( TAQ tmf (3) ( tmf (2) (3) )) ∼ = Z (3) . EVA H ¨ONING AND BIRGIT RICHTER
Relative topological Hochschild homology.
In [DLR20] (see also [DLR] for a cor-rection) we show that the relative topological Hochschild homology spectra
THH ℓ ( ku ( p ) ) and THH ko ( ku ) have highly non-trivial homotopy groups. Here, we extend these results to the rela-tive THH -spectra of tmf (3) → tmf (2) (3) , Tmf (3) → Tmf (2) (3) and tmf (2) (3) → tmf (2) (3) . Forformulas concerning the coefficients of elliptic curves we refer to [Del75].Recall that we have tmf (2) (3) ≃ τ > tmf (2) hC (3) . By [Sto12, §
7] we know that π ∗ tmf (2) (3) ∼ = Z (3) [ λ , λ ] with | λ i | = 4 and with C -action given by λ λ and λ λ [Sto12, Lemma7.3]. Since | C | is invertible in π ∗ tmf (2) (3) the E -page of the homotopy fixed point spectralsequence is given by H ∗ ( C , π ∗ tmf (2) (3) ) = H ( C , π ∗ tmf (2) (3) ) = π ∗ ( tmf (2) (3) ) C . Thus, we have π ∗ tmf (2) (3) = Z (3) [ λ + λ , λ λ ] = Z (3) [ a , a ]with a = − ( λ + λ ) and a = λ λ . Recall the following facts about the homotopy of tmf (3) (see for instance [DFHH14, p. 192]): We have π ∗ tmf (3) = Z (3) { } , ∗ = 0; Z / Z { α } ∗ = 3; Z (3) { c } , ∗ = 8; Z (3) { c } , ∗ = 12;0 , ∗ = 4 , , , α is the image of α ∈ π ( S (3) ) under π ( S (3) ) → π tmf (3) . By [Sto12, Proof of Propo-sition 10.3] we have that the map π ∗ tmf (3) → π ∗ tmf (2) (3) satisfies c a − a and c
7→ − a + 288 a a . (There is a discrepancy between our sign for c and that in [Sto12].)We know from personal communication with Mike Hill that there is a fiber sequence of tmf (2) (3) -modules Tmf (2) (3) tmf (2) (3) [ a − ] × tmf (2) (3) [ a − ] tmf (2) (3) [( a a ) − ] . f See [HM17, Proposition 4.24] for the analogous statement at p = 2. The kernel of π ∗ ( f ) has Z (3) -basis { ( a n a m , − a n a m ) | n, m ∈ N } and the cokernel has Z (3) -basis { a n a m | n > , m > } . We get that in negative degrees π ∗ Tmf (2) (3) is given by M n,m > Z (3) { a n a m } , where a n a m has degree − n − m −
1. The π ∗ tmf (2) (3) -action is given by a · a n a m = ( a n − a m , if n > , otherwise , and analogously for a .By the gap theorem (see for instance [Kon]) we have π ∗ Tmf (3) ∼ = 0 for − < ∗ < Lemma 2.1. π ∗ ( tmf (2) (3) ∧ tmf (3) tmf (2) (3) ) ∼ = Z (3) [ a , a , r ] /r + a r + a r =: Z (3) [ a , a , r ] /I where r has degree and is mapped to zero under the multiplication map. ETECTING AND DESCRIBING RAMIFICATION 7
Proof.
As above we use that we have an equivalence of tmf (3) -modules tmf (3) ∧ T ≃ tmf (2) (3) .Here, T is defined by the cofiber sequences S S cone( α ) S α and S cone( α ) T S , φ where S φ −→ cone( α ) → S is equal to α . We get an equivalence of left tmf (2) (3) -modules tmf (2) (3) ∧ tmf (3) tmf (2) (3) ≃ tmf (2) (3) ∧ tmf (3) ( tmf (3) ∧ T ) ≃ tmf (2) (3) ∧ T. Smashing the above cofiber sequences with tmf (2) (3) gives cofiber sequences of tmf (2) (3) -modulesΣ tmf (2) (3) tmf (2) (3) tmf (2) (3) ∧ cone( α ) Σ tmf (2) (3)¯ α δ and Σ tmf (2) (3) tmf (2) (3) ∧ cone( α ) tmf (2) (3) ∧ T Σ tmf (2) (3) . ¯ φ ∆ The map ¯ α is zero in the derived category of tmf (2) (3) -modules, because π ∗ ( tmf (2) (3) ) isconcentrated in even degrees. We therefore get an equivalence of tmf (2) (3) -modules tmf (2) (3) ∧ cone( α ) ≃ tmf (2) (3) ∨ Σ tmf (2) (3) . This implies that tmf (2) (3) ∧ cone( α ) has non-trivial homotopy groups only in even degreesand therefore that ¯ φ is zero in the derived category of tmf (2) (3) -modules. We get an equivalenceof tmf (2) (3) -modules tmf (2) (3) ∧ T ≃ tmf (2) (3) ∨ Σ tmf (2) (3) ∨ Σ tmf (2) (3) . We can assume that the map tmf (3) → tmf (2) (3) factors in the derived category of tmf (3) -modules as tmf (3) tmf (3) ∧ cone( α ) tmf (3) ∧ T tmf (2) (3) ≃ This implies that the inclusion in the first smash factor η L : tmf (2) (3) → tmf (2) (3) ∧ tmf (3) tmf (2) (3) is given by tmf (2) (3) tmf (2) (3) ∧ cone( α ) tmf (2) (3) ∧ T tmf (2) (3) ∧ tmf (2) tmf (2) (3) . ≃ We obtain that the map tmf (2) (3) ∧ cone( α ) tmf (2) (3) ∧ T ≃ tmf (2) (3) ∧ tmf (3) tmf (2) (3) tmf (2) (3) is a left inverse for tmf (2) (3) → tmf (2) (3) ∧ cone( α ). It is also clear that the inclusion in thesecond smash factor η R : tmf (2) (3) → tmf (2) (3) ∧ tmf (3) tmf (2) (3) is given by tmf (2) (3) tmf (3) ∧ T tmf (2) (3) ∧ T tmf (2) (3) ∧ tmf (3) tmf (2) (3) . ≃ ≃ We claim that η R ( a ) ∈ π ( tmf (2) (3) ∧ tmf (3) tmf (2) (3) ) ∼ = π ( tmf (2) (3) ∧ T ) ∼ = π ( tmf (2) (3) ∧ cone( α ))maps to 3 times a unit under δ : π ( tmf (2) (3) ∧ cone( α )) → π (Σ tmf (2) (3) ) ∼ = Z (3) . EVA H ¨ONING AND BIRGIT RICHTER
By commutativity of the diagram π ( tmf (2) (3) ∧ T ) π ( tmf (2) (3) ∧ cone( α )) π (Σ tmf (2) (3) ) π ( tmf (3) ∧ T ) π ( tmf (3) ∧ cone( α )) π (Σ tmf (3) ) ∼ = δ η R ∼ = it suffices to show that a ∈ π ( tmf (3) ∧ T ) maps to 3 times a unit under the bottom map. Thisfollows by the exact sequence π ( tmf (3) ) π ( tmf (3) ∧ cone( α )) π (Σ tmf (3) ) π (Σ tmf (3) )0 Z (3) Z (3) Z / Z { α } . ∼ = ∼ = ∼ = ∼ = We define r to be the unique element in π ( tmf (2) (3) ∧ cone( α )) that maps to that unit under δ and that is in the kernel of the composition of π ( tmf (2) (3) ∧ cone( α )) ∼ = π ( tmf (2) (3) ∧ T )and the multiplication map π ( tmf (2) (3) ∧ T ) ∼ = π ( tmf (2) (3) ∧ tmf (3) tmf (2) (3) ) → π ( tmf (2) (3) ) . We have that 3 r − η R ( a ) is in the image of π ( tmf (2) (3) ) → π ( tmf (2) (3) ∧ cone( α )) and thuscan be written as 3 r − η R ( a ) = n · a for an n ∈ Z (3) . Applying the map π ( tmf (2) (3) ∧ cone( α )) ∼ = π ( tmf (2) (3) ∧ T ) ∼ = π ( tmf (2) (3) ∧ tmf (3) tmf (2) (3) ) → π ( tmf (2) (3) )gives n = − η R ( a ) ∈ π ( tmf (2) (3) ∧ T ) maps to 3 times a unit under∆ : π ( tmf (2) (3) ∧ T ) → π (Σ tmf (2) (3) ) . As above one sees that it suffices to show that a maps to 3 times a unit under the map π ( tmf (3) ∧ T ) → π (Σ tmf (3) ). For this we consider the exact sequence π ( tmf (3) ∧ cone( α )) π ( tmf (3) ∧ T ) π (Σ tmf (3) ) π (Σ tmf (3) ∧ cone( α )) . Z (3) { a } ⊕ Z (3) { a } Z (3) ∼ = ∼ = Using that π ( tmf (3) ) = 0 = π ( tmf (3) ) one gets that π ( tmf (3) ) ∼ = π ( tmf (3) ∧ cone( α )), andunder this isomorphism the first map in the exact sequence identifies with π ( tmf (3) ) ∼ = Z (3) { c } → π ( tmf (2) (3) ) ∼ = Z (3) { a } ⊕ Z (3) { a } , c a − a . As π ( tmf (3) ) = 0 = π ( tmf (3) ), one gets that π ( tmf (3) ∧ cone( α )) ∼ = π (Σ tmf (3) ), and underthis isomorphism the third map in the exact sequence identifies with π (Σ tmf (3) ) ∼ = Z (3) → π ( tmf (3) ) ∼ = Z / Z { α } , α . One obtains that the second map in the exact sequence maps a to 3 · m and a to 9 · m for aunit m ∈ Z (3) .Since the map π ∗ ( tmf (3) ) → π ∗ ( tmf (2) (3) ) maps c to 16 a − a , we have the equation16 · a − · a = 16 · η R ( a ) − · η R ( a )in π ∗ ( tmf (2) (3) ∧ tmf (3) tmf (2) (3) ). Replacing η R ( a ) by 3 r + a and using torsion-freeness onegets the equation η R ( a ) = a + 3 r + 2 a r. ETECTING AND DESCRIBING RAMIFICATION 9
We apply the map ∆ : π ( tmf (2) (3) ∧ T ) → π (Σ tmf (2) (3) ) to this equation and obtain bytorsion-freeness of π ∗ ( tmf (2) (3) ) ∆ ( r ) = m. We thus have an isomorphism of left π ∗ ( tmf (2) (3) )-modules π ∗ ( tmf (2) (3) ∧ tmf (3) tmf (2) (3) ) ∼ = π ∗ ( tmf (2) (3) ) ⊕ π ∗ ( tmf (2) (3) ) { r } ⊕ π ∗ ( tmf (2) (3) ) { r } . Since the map π ∗ ( tmf (3) ) → π ∗ ( tmf (2) (3) ) maps c to − a + 288 a a , we have − · a + 288 · a · a = − · η R ( a ) + 288 · η R ( a ) · η R ( a )in π ∗ ( tmf (2) (3) ∧ tmf (3) tmf (2) (3) ). Replacing η R ( a ) by 3 r + a and η R ( a ) by a + 3 r + 2 a r and using torsion-freeness one gets r + a r + a r = 0 . This implies the lemma. (cid:3)
Theorem 2.2.
The relative
THH -spectrum
THH tmf (3) ( tmf (2) (3) ) is not trivial. More precisely, THH tmf (3) ∗ ( tmf (2) (3) ) ∼ = Z (3) [ a , a ] ⊕ M i > Σ i +5 Z (3) [ a ] ∼ = π ∗ tmf (2) ⊕ M i > Σ i +5 π ∗ tmf (2) / ( a ) . Proof.
We use the Tor spectral sequence E ∗ , ∗ = Tor π ∗ ( tmf (2) (3) ∧ tmf (3) tmf (2) (3) ) ∗ , ∗ ( π ∗ tmf (2) (3) , π ∗ tmf (2) (3) ) ⇒ π ∗ THH tmf (3) ( tmf (2) (3) )in order to calculate relative topological Hochschild homology. For determining Tor Z (3) [ a ,a ,r ] /I ∗ , ∗ ( Z (3) [ a , a ] , Z (3) [ a , a ])we consider the free resolution of Z (3) [ a , a ] as a Z (3) [ a , a , r ] /I -module . . . / / Σ Z (3) [ a , a , r ] /I r + a r + a / / Σ Z (3) [ a , a , r ] /I r / / Z (3) [ a , a , r ] /I. Applying ( − ) ⊗ Z (3) [ a ,a ,r ] /I Z (3) [ a , a ] yields . . . / / Σ Z (3) [ a , a ] a / / Σ Z (3) [ a , a ] / / Z (3) [ a , a ]and hence we get E n, ∗ = π ∗ ( tmf (2) (3) ) , n = 0;Σ k Z (3) [ a ] , n = 2 k + 1 , k > , otherwise . We note that all non-trivial classes in positive filtration degree have an odd total degree. Sincethe edge morphism π ∗ ( tmf (2) (3) ) → THH tmf (3) ∗ ( tmf (2) (3) ) is the unit, the classes in filtrationdegree zero cannot be hit by a differential and the spectral seqence collapses at the E -page.Since E n,m = E ∞ n,m is a free Z (3) -module for all n, m , there are no additive extensions. (cid:3) As for the connective covers we have an equivalence of
Tmf (3) -modules
Tmf (3) ∧ T ≃ Tmf (2)[Mat16b, § Tmf (3) → Tmf (2) (3) factors in the derived category of Tmf (3) -modules as
Tmf (3) → Tmf (3) ∧ cone( α ) → Tmf (3) ∧ T ≃ Tmf (2) (3) . Using the gap theorem, one can argue analogously to the proof of Lemma 2.1 to show that π ∗ ( Tmf (2) (3) ∧ Tmf (3)
Tmf (2) (3) ) ∼ = π ∗ Tmf (2) (3) [ r ] / ( r + a r + a r ) . Theorem 2.3.
There is an additive isomorphism
THH
Tmf (3) ( Tmf (2) (3) ) ∼ = π ∗ Tmf (2) (3) ⊕ M i > Σ i +5 Z (3) [ a ] ⊕ M i > Σ i Z (3) { a i a } . Proof.
As above we have the following free resolution of π ∗ ( Tmf (2) (3) ) as a module over C ∗ = π ∗ ( Tmf (2) (3) ∧ Tmf (3)
Tmf (2) (3) ) : . . . Σ C ∗ Σ C ∗ C ∗ π ∗ Tmf (2) (3) . r r + a r + a r We get that the E -page of the Tor spectral sequence E ∗ , ∗ = Tor C ∗ ∗ , ∗ ( π ∗ Tmf (2) (3) , π ∗ Tmf (2) (3) ) = ⇒ π ∗ THH
Tmf (3) ( Tmf (2) (3) )is given by E n, ∗ = π ∗ Tmf (2) (3) , n = 0;Σ k π ∗ Tmf (2) (3) /a , n = 2 k + 1 , k > (cid:0) Σ k π ∗ Tmf (2) (3) · a −−→ Σ k − π ∗ Tmf (2) (3) (cid:1) , n = 2 k, k > π ∗ Tmf (2) (3) , n = 0;Σ k Z (3) [ a ] , n = 2 k + 1 , k > k L n > Z (3) { a n a } , n = 2 k, k > . Since all non-trivial classes in positive filtration have an odd total degree, the spectral sequencecollapses at the E -page. There are no additive extensions, because the E ∞ = E -page is a free Z (3) -module. (cid:3) Theorem 2.4.
We have an additive isomorphism π ∗ THH tmf (2) (3) ( tmf (2) (3) ) ∼ = Z (3) [ λ , λ ] ⊕ M i > Σ i +5 Z (3) [ λ ] . Proof.
The map π ∗ tmf (2) (3) → π ∗ tmf (2) (3) is given by Z (3) [ λ + λ , λ λ ] → Z (3) [ λ , λ ]. Oneeasily sees that Z (3) [ λ , λ ] ∼ = Z (3) [ λ + λ , λ λ ] ⊕ Z (3) [ λ + λ , λ λ ] λ , so π ∗ tmf (2) (3) is a free π ∗ tmf (2) (3) -module. We get π ∗ ( tmf (2) (3) ∧ tmf (2) (3) tmf (2) (3) ) = π ∗ tmf (2) (3) ⊗ π ∗ tmf (2) (3) π ∗ tmf (2) (3) = Z (3) [ λ , λ , a ] /a + λ a − λ a, where a = η R ( λ ) − λ . Let C ∗ = Z (3) [ λ , λ , a ] /a + λ a − λ a . We have the following freeresolution of π ∗ tmf (2) (3) as a C ∗ -module: . . . Σ C ∗ Σ C ∗ C ∗ π ∗ tmf (2) (3) · a · ( a + λ − λ ) · a Thus, the E -page of the Tor spectral sequence E ∗ , ∗ = Tor C ∗ ∗ , ∗ ( π ∗ tmf (2) (3) , π ∗ tmf (2) (3) ) = ⇒ π ∗ THH tmf (2) (3) ( tmf (2) (3) )is given by E n, ∗ = π ∗ tmf (2) (3) , n = 0;Σ k +4 π ∗ tmf (2) (3) / ( λ − λ ) , n = 2 k + 1 , k > , otherwise . Since the non-trivial classes in positive filtration have odd total degree, the spectral sequencecollapses at the E -page. There are no additive extensions, because the E = E ∞ -page is a free Z (3) -module. (cid:3) ETECTING AND DESCRIBING RAMIFICATION 11
The discriminant map. If A → B is a G -Galois extension for a finite group G , then thediscriminant map d B | A : B → F A ( B, A ) is a weak equivalence [Rog08a, Proposition 6.4.7]. Themap d B | A is right adjoint to the trace pairing B ∧ A B µ / / B tr / / A where ( A → B ) ◦ tr is homotopic to P g ∈ G g and µ is the multiplication map of B . Rognesproposes that the deviation of d B | A from being a weak equivalence might be used for measuringramification. We show in the examples of ℓ p → ku p and ko → ku that d does detect ramification,but it does not give any information about the type of ramification. Proposition 2.5.
There is a cofiber sequence ku p d kup | ℓp / / F ℓ p ( ku p , ℓ p ) / / W p − i =1 Σ − p +2 i +2 H Z p . Proof.
We know that F ℓ p ( ku p , ℓ p ) can be decomposed as F ℓ p ( ku p , ℓ p ) ≃ F ℓ p ( p − _ i =0 Σ i ℓ p , ℓ p ) ∼ = p − Y i =0 Σ − i ℓ p ≃ p − _ i =0 Σ − i ℓ p and d ku p | ℓ p can be identified with a map p − _ i =0 Σ i ℓ p → p − _ i =0 Σ − i ℓ p . As π ∗ ku p is a free graded π ∗ ℓ p -module, we can calculate the effect of d ku p | ℓ p algebraically viathe trace pairing: The element Σ i ∈ π ∗ Σ i ℓ p maps an element u i to tr ( u i + j ) and this is tr ( u i + j ) = ( , ( p − ∤ i + j, ( p − u i + j , ( p − | i + j. Hence on the level of homotopy groups d ku p | ℓ p maps 1 ∈ π Σ ℓ p to ( p − · ∈ π ∗ Σ ℓ p = π ∗ ℓ p and it maps Σ i ∈ π ∗ Σ i ℓ p via multiplication with ( p − v = ( p − u p − to π ∗ Σ − p +2 i +2 ℓ p .On the summands Σ i ℓ p we get the following maps:Σ p − ℓ p ( p − v % % ❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑ ...Σ ℓ p ( p − v % % ❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑ ℓ p ( p − / / ℓ p Σ − ℓ p ...Σ − p +4 ℓ p As ( p −
1) is a unit in π ( ℓ p ) the cofiber ofΣ i ℓ p ( p − v / / Σ − p +2 i +2 ℓ p is Σ − p +2 i +2 H Z p . (cid:3) Note that ko ≃ τ > ku hC , but as the trace map tr : ku → ku hC has the connective spectrum ku as a source, it factors through τ > ku hC ≃ ko and we obtain a discriminant d ku | ko : ku → F ko ( ku, ko ). We fix notation for π ∗ ko as π ∗ ko = Z [ η, y, ω ] / (2 η, η , ηy, y − ω )with | η | = 1, | y | = 4 and | ω | = 8. Proposition 2.6.
There is a cofiber sequence ku d ku | ko / / F ko ( ku, ko ) / / Σ − H Z .Proof. The cofiber sequence Σ ko η / / ko c / / ku δ / / Σ ko η / / Σ ko induces a cofiber sequence F ko (Σ ko, ko ) η / / F ko (Σ ko, ko ) δ / / F ko ( ku, ko ) c / / F ko ( ko, ko ) η / / F ko (Σ ko, ko )which is equivalent toΣ − ko η / / Σ − ko δ / / F ko ( ku, ko ) c / / ko η / / Σ − ko. This is the 2-fold desuspension of the cofiber sequence of ku and hence F ko ( ku, ko ) ≃ Σ − ku. We consider the composition c ∗ ◦ d ku | ko : ku → F ko ( ku, ko ) → F ko ( ku, ku ). As c ∗ is part ofthe cofiber sequence F ko ( ku, Σ ko ) η ∗ / / F ko ( ku, ko ) c ∗ / / F ko ( ku, ku )and as η is trivial on ku , we know that c ∗ induces a monomorphism on the level of homotopygroups.As d ku | ko is adjoint to the trace pairing, the composite π ∗ ku / / π ∗ F ko ( ku, ko ) / / π ∗ F ko ( ku, ku )can be identified with π ∗ ku / / π ∗ F ko ( ku, ku ) (id+ t ) ∗ / / F ko ( ku, ku )where t denotes the generator of C and the first map is adjoint to the multiplication ku ∧ ko ku → ku .The target of c ∗ is F ko ( ku, ku ) ≃ F ku ( ku ∧ ko ku, ku ), and we know by work of the first author,documented in [DLR, Proof of Lemma 0.1] that π ∗ F ku ( ku ∧ ko ku, ku ) ∼ = Hom ku ∗ ( ku ∗ [ s ] / ( s − su ) , Σ −∗ ku ∗ ) , so we can control the effect of c ∗ ◦ d ku | ko on homotopy groups.Note that t induces a ku -linear map t ∗ : ku → t ∗ ku , where t ∗ ku is the ku -module given byrestriction of scalars along t .As t = id, we therefore obtain F ko ( ku, ku ) t ∗ / / F ko ( ku, t ∗ ku )and a commutative diagram F ko ( ku, ku ) t / / ≃ (cid:15) (cid:15) F ko ( ku, ku ) ≃ (cid:15) (cid:15) F ku ( ku ∧ ko ku, ku ) β / / F ku ( ku ∧ ko ku, ku )Here, β induces the map on π ∗ that sends an f : ( ku ∧ ko ku ) ∗ → Σ − i ku ∗ to( ku ∧ ko ku ) ∗ ( t ∧ id) ∗ / / ( ku ∧ ko ku ) ∗ f / / Σ − i ku ∗ t / / Σ − i ku ∗ . ETECTING AND DESCRIBING RAMIFICATION 13
If we denote the right unit η R : ku → ku ∧ ko ku applied to u by u r , then we have the relation2 s + u r = u . As ( t ∧ id) ∗ ( u ) = − u and ( t ∧ id) ∗ ( u r ) = u r , this implies that( t ∧ id) ∗ (2 s ) = 2 s − u. Torsionfreeness then yields ( t ∧ id) ∗ ( s ) = s − u .The adjoint of the multiplication map π ∗ ku → π ∗ F ko ( ku, ku ) maps u i to the map that sends1 to Σ − i u i and s to zero. Therefore, the composite c ∗ ◦ d ku | ko maps u i to the map with values1 Σ − i ( u i + ( − i u i ) and s ( s − u ) u i
7→ − t ( u i +1 ) = ( − i u i +1 . In order to understand the effect of d ku | ko we consider the diagram π ∗ ku ( d ku | ko ) ∗ / / π ∗ F ko ( ku, ko ) ∼ = π ∗ +2 ( ku ) c ∗ (cid:15) (cid:15) c ∗ / / π ∗ F ko ( ku, ku ) ∼ = π ∗ (Σ − ku ∨ ku ) c ∗ (cid:15) (cid:15) π ∗ F ko ( ko, ko ) ∼ = π ∗ ( ko ) c ∗ / / π ∗ F ko ( ko, ku ) ∼ = π ∗ ( ku )where we can identify c ∗ : π ∗ F ko ( ku, ko ) ∼ = π ∗ +2 ( ku ) → π ∗ ( ko ) with π ∗ Σ − δ .The application of c ∗ gives the restriction to the unit c : ko → ku . Say ( d ku | ko ) ∗ ( u ) = x ∈ π ( ku ). Then π ∗ Σ − δ ( x ) = λy , and as c ∗ ( y ) = 2 u , we obtain that c ∗ ( d ku | ko ) ∗ ( u ) = Σ − y andtherefore ( d ku | ko ) ∗ ( u ) = u . Similarly c ∗ ( d ku | ko ) ∗ ( u ) = Σ − ω and ( d ku | ko ) ∗ ( u ) = u and in general( d ku | ko ) ∗ ( u i ) = u i +1 . Restriction to the unit of the odd powers of u gives zero.All the u i send s to ± u i +1 under c ∗ ◦ ( d ku | ko ) ∗ , so also the odd powers of u have to hit agenerator under ( d ku | ko ) ∗ , so as a map from ku to Σ − ku the map d ku | ko has cofiber Σ − H Z . (cid:3) Describing ramification
Log-´etaleness.
It is shown in [RSS15] and [Sag14] that ℓ → ku ( p ) is log-´etale with respectto the log structures that are generated by v and by u . We will use the class u ∈ π ku (2) inorder to define a pre-log structure for ko (2) → ku (2) and show that ko (2) → ku (2) is not log-´etalewith respect to this pre-log structure. This indicates that the map is not tamely ramified. Weuse the notation from [Sag14].Let ω denote the Bott element ω ∈ π ko (2) . The complexification map sends ω to u .By [Sag14, Lemma 6.2] we have an exact sequence π TAQ C ( ku (2) ) EDBCGF@A / / π (cid:16) ku (2) ∧ γ ( D ( u )) /γ ( D ( w )) (cid:17) / / π TAQ ( ko (2) ,D ( w )) ( ku (2) , D ( u )) / / π TAQ C ( ku (2) ) , where C = ko (2) ∧ S J D ( w ) S J D ( u ) and D ( u ), D ( ω ) are the pre-log structures for the elements u and ω as in [Sag14, Construction 4.2]. We have that γ ( D ( w )) and γ ( D ( u )) have the homotopytype of the sphere and that γ ( D ( w )) → γ ( D ( u )) is multiplication by 4. Therefore we get π (cid:16) ku (2) ∧ γ ( D ( u )) /γ ( D ( w )) (cid:17) = Z / Z . We want to show that π TAQ C ( ku (2) ) = 0 = π TAQ C ( ku (2) ). By [Bas99, Lemma 8.2] it sufficesto show that C → ku (2) is an 1-equivalence. Since π ( ku (2) ) = 0, it is enough to show thatthe map is an isomorphism on π . Since S J D ( w ) and S J D ( u ) are concentrated in nonnegative J -space degrees by [RSS15, Example 6.8], they are connective. Thus, it is enough to showthat S J D ( w ) → S J D ( u ) induces an isomorphism on π . For this, we only have to prove that H ( S J D ( w ) , Z ) → H ( S J D ( u ) , Z ) is an isomorphism. Since this map is a ring map we onlyneed to know that both sides are Z . This follows from [RSS18, Proposition 5.2, Corollary 5.3].Hence we obtain the following result: Theorem 3.1.
The map ( ko (2) , D ( ω )) → ( ku (2) , D ( u )) is not log-´etale. One could try to distinguish between tame and wild ramification by testing for log-´etaleness.In many examples, however, it is less obvious what a suitable log structure would be.3.2.
Wild ramification and Tate cohomology.
In the algebraic context of Galois extensionsof number fields and corresponding extension of number rings tame ramification yields a normalbasis and a surjective trace map. Both facts are actually also sufficient in order to distinguishtame from wild ramification. For structured ring spectra it does not work to impose theseproperties on the level of homotopy groups, because even for finite faithful Galois extensionsthese would not hold. Instead we propose a different criterion that uses the Tate construction.
Remark . Let G be a finite group. Usually one calls a G -module M cohomologically trivial ,if ˆ H i ( H ; M ) = 0, for all i ∈ Z and all H < G . If M is a commutative ring S , however, it sufficesto require ˆ H i ( G ; S ) = 0 for all i ∈ Z : In particular, ˆ H ( G ; S ) = 0, and hence the trace map tr G : S → S G is surjective. Thus 1 S G is in the image of the norm, say N G [ x ] = 1 S G for [ x ] ∈ S G .If H < G then we consider the diagram H ( G ; S ) = S G i ∗ / / H ( H ; S ) = S H H ( G ; S ) = S GN G O O tr GH / / H ( H ; S ) = S HN H O O and therefore we can express can express 1 S H as1 S H = i ∗ (1 S G ) = i ∗ N G [ x ] = N H tr GH [ x ] , so 1 S H is in the image of N H and ˆ H ( H ; S ) = 0. But ˆ H ∗ ( H ; S ) is a graded commutative ringwith unit [1 S H ] = 0, and thus ˆ H ∗ ( H ; S ) = 0.The same argument shows that the surjectivity of the trace map suffices for being cohomo-logically trivial.Even if A → B is a G -Galois extension of ring spectra, it is not true, that this impliesthat B is faithful as an A -module. An example, due to Wieland is the C -Galois extension F (( BC ) + , H F ) → F (( EC ) + , H F ) ≃ H F which is not faithful: The F (( BC ) + , H F )-module spectrum ( H F ) tC is not trivial, but H F ∧ F (( BC ) + ,H F ) ( H F ) tC ∼ ∗ .Note that for a map A → B between connective commutative ring spectra with a finitegroup G acting on B via commutative A -algebra maps it makes sense to replace the usualhomotopy fixed point condition by the condition that A is weakly equivalent to τ > B hG . Inmany examples B hG won’t be connective. The map A → B factors through A → B hG → B ,but as A is connective, we can consider the induced map on connective covers and obtain a mapof commutative ring spectra τ > A = A → τ > B hG → τ > B = B, that turns τ > B hG into a commutative A -algebra spectrum.For any spectrum X we denote by τ < X the cofiber of the map τ > X → X .The following result is probably well-known. ETECTING AND DESCRIBING RAMIFICATION 15
Lemma 3.3.
Let G be a finite group and let e be a naive connective G -spectrum. Then τ > e hG → e hG → τ < e tG is a cofiber sequence and τ < e tG ≃ τ < e hG .Proof. We consider the norm sequence e hG N / / e hG / / e tG . As e hG is a connective spectrum, we have that π − e hG = 0. Hence, applying τ > still gives riseto a cofiber sequence τ > e hG = e hG τ > N / / τ > e hG / / τ > e tG . We combine the norm cofiber sequences with the defining cofiber sequence of τ < and obtain τ > e hGτ > N (cid:15) (cid:15) e hGN (cid:15) (cid:15) / / ∗ (cid:15) (cid:15) τ > e hG / / (cid:15) (cid:15) e hG / / (cid:15) (cid:15) τ < e hG (cid:15) (cid:15) τ > e tG / / e tG / / τ < e tG . Thus τ < e hG ≃ τ < e tG and the cofiber sequence in the second row then yields the claim. (cid:3) Remark . In many cases, if B tG
6≃ ∗ , then π ∗ ( B tG ) is actually periodic. As the canonicalK¨unneth map π ∗ ( B tG ) ⊗ π ∗ ( B hG ) π ∗ ( B ) → π ∗ ( B tG ∧ B hG B )is a map of graded commutative rings and as π ∗ ( B tG ) ∼ = π ∗ ( B hG ) in negative degrees, a peri-odicity generator in a negative degree would map to zero in π ∗ B for connective B and hence π ∗ ( B tG ) ⊗ π ∗ ( B hG ) π ∗ ( B ) is the zero ring. But then also π ∗ ( B tG ∧ B hG B ) ∼ = 0 and B tG ∧ B hG B ≃ ∗ . Therefore B would not be a faithful B hG -module in these cases. This emphasizes the importanceof replacing the condition that A be weakly equivalent to B hG by the requirement that A ≃ τ > ( B hG ).From Lemma 3.3 we also know that in order to show that B tG
6≃ ∗ for connective B it issufficient to show that τ < B hG is not trivial.In [Rog08a, Proposition 6.3.3] Rognes assumes that A ≃ B hG , but that assumption is actuallynot needed for the following: Theorem 3.5.
Assume that G is a finite group, B is a cofibrant commutative A -algebra onwhich G acts via maps of commutative A -algebras. If B is dualizable and faithful as an A -moduleand if h : B ∧ A B ∼ / / F ( G + , B ) , then B tG ≃ ∗ . The proof is given in [Rog08a]. We repeat it in order to convince the reader that one doesnot need to assume that A ≃ B hG . Proof.
We consider the following commutative diagram in which the columns arise from thenorm cofiber sequence. B ∧ A ( B hG ) / / B ∧ A N (cid:15) (cid:15) ( B ∧ A B ) hG h hG / / N (cid:15) (cid:15) F ( G + , B ) hGN (cid:15) (cid:15) B ∧ A ( B hG ) / / (cid:15) (cid:15) ( B ∧ A B ) hG h hG / / (cid:15) (cid:15) F ( G + , B ) hG (cid:15) (cid:15) B ∧ A ( B tG ) / / ( B ∧ A B ) tG h tG / / F ( G + , B ) tG As F ( G + , B ) is free over G , we have F ( G + , B ) tG ≃ ∗ and the norm map is an equivalencebetween F ( G + , B ) hG and F ( G + , B ) hG . The map h is equivariant and hence it induces weakequivalences on homotopy orbits, homotopy fixed points and the Tate construction. This showsthat N : ( B ∧ A B ) hG → ( B ∧ A B ) hG is a weak equivalence.The left horizontal map is a weak equivalence because B is dualizable as an A -module. Themap B ∧ A ( B hG ) → ( B ∧ A B ) hG is always a weak equivalence. Therefore the map B ∧ A N : B ∧ A ( B hG ) → B ∧ A ( B hG )is a weak equivalence and thus B ∧ A ( B tG ) ≃ ∗ . As B is a faithful A -module, this implies that B tG ≃ ∗ . (cid:3) Remark . Therefore, if G is a finite group, B is a cofibrant commutative A -algebra on which G acts via maps of commutative A -algebras. If B is dualizable and faithful as an A -moduleand if B tG
6≃ ∗ , then we know that h : B ∧ A B → F ( G + , B ) cannot be a weak equivalence, andhence A → B is ramified.We propose a definition of tame and wild ramification for commutative ring spectra andjustify our definition by investigating several examples.In the following we denote by τ > X the connective cover of a spectrum X . Definition 3.7.
Assume that A → B is a map of commutative ring spectra such that G actson B via commutative A -algebra maps and B is faithful and dualizable as an A -module.If A ≃ B hG (or A ≃ τ > B hG if A and B are connective), then we call A → B tamely ramified if B tG ≃ ∗ . Otherwise, A → B is wildly ramified . Remark . To compute the homotopy of B tG we use the Tate spectral sequence E n,m = ˆ H − n ( G ; π m ( B )) = ⇒ π n + m B tG which is of standard homological type, multiplicative and conditionally convergent. In particularby [Boa99, Theorem 8.2], it converges strongly if it collapses at a finite stage.We will now investigate our criterion for wild ramification in examples. First, we establishfaithfulness: Lemma 3.9.
The map tmf (2) (3) → tmf (2) (3) identifies tmf (2) (3) as a faithful tmf (2) (3) -module.Proof. For the map tmf (2) (3) → tmf (2) (3) we know that C acts on tmf (2) (3) via commutative tmf (2) (3) -algebra maps and that tmf (2) (3) ≃ τ > ( tmf (2) hC (3) ). The trace map tr : tmf (2) (3) → ETECTING AND DESCRIBING RAMIFICATION 17 tmf (2) hC (3) factor through τ > ( tmf (2) hC (3) ) ≃ tmf (2) (3) , because tmf (2) (3) is connective. As in[Rog08a, Lemma 6.4.3] one can show that the composite tmf (2) (3) ≃ τ > ( tmf (2) hC (3) ) / / tmf (2) (3) tr / / τ > ( tmf (2) hC (3) ) ≃ tmf (2) (3) is homotopic to the map that is the multiplication by | C | = 2. As 2 is invertible in π tmf (2) (3) ,the trace map tr : tmf (2) (3) → tmf (2) (3) is a split surjective map of tmf (2) (3) -modules andhence tmf (2) (3) → tmf (2) (3) is faithful. (cid:3) Lemma 3.10.
The spectrum tmf (2) (3) is faithful as a tmf (3) -module spectrum.Proof. We already mentioned Behrens’ identification [Beh06, Lemma 2] tmf (2) (3) ≃ tmf (3) ∧ T where T = S ∪ α e ∪ α e with α ∈ ( π S ) (3) . Note that α is nilpotent of order 2 because( π S ) (3) = 0.Assume that M is a tmf (3) -module with ∗ ≃ M ∧ tmf (3) tmf (2) (3) ≃ M ∧ T. Then the cofiber sequences S / / T / / Σ cone( α ) and cone( α ) / / T / / S imply that Σ cone( α ) ∧ M ≃ Σ M and Σ M ≃ Σcone( α ) ∧ M and thereforeΣ M ≃ M. The equivalence is induced by a class in π S (3) ∼ = Z / Z { β } . As this is nilpotent, we getthat M ≃ ∗ . (cid:3) Remark . It is known that ko → ku is faithful [Rog08a, Proposition 5.3.1] and dualizableand it is clear that ℓ → ku ( p ) is faithful and dualizable as the inclusion of a summand. As tmf (3) (2) can be identified with tmf (2) ∧ DA (1) as a tmf (2) -module [Mat16b, Theorem 4.12],where DA (1) is a finite cell complex realizing the double of A (1) = h Sq , Sq i , it is dualizable.An argument as in [Rog08a, Proof of Proposition 5.4.5] shows that tmf (2) → tmf (3) (2) isfaithful.At the moment we don’t know whether tmf (3) (2) → tmf (3) (2) is faithful. The diagram tmf (3) (2) / / · & & ▼▼▼▼▼▼▼▼▼▼ tmf (3) (2) tr x x qqqqqqqqqq tmf (3) (2) commutes, so if M is a tmf (3) (2) -module spectrum with M ∧ tmf (3) (2) tmf (3) (2) ≃ ∗ , thenmultiplication by 2 is a trivial self-map on M .Meier shows [Mei] that tmf (3) is not perfect as a tmf (3)-module, hence tmf (3) is not adualizable tmf (3)-module.Meier also proves that tmf [ n ] → tmf ( n ) is dualizable for all n . By combining his result withLemma 3.9 and Lemma 3.10 we obtain that tmf (2) (3) is dualizable and faithful as a tmf (3) -module.We show that the extensions tmf (3) (2) → tmf (3) (2) and tmf (3) → tmf (2) (3) have non-trivialTate spectra. For ku the Tate spectrum with respect to the complex conjugation C -actionsatisfies ku tC ≃ _ i ∈ Z Σ i H Z / Z . This result is due to Rognes (compare [Rog08a, § Theorem 3.12.
For tmf (3) (2) with its C -action we obtain an equivalence of spectra tmf (3) tC (2) ≃ _ i ∈ Z Σ i H Z / Z . Proof.
We use the calculations in [MR09]. They compute the homotopy fixed point spectralsequence E n,m = H − n ( C ; π m TMF (3) (2) ) = ⇒ π n + m TMF (3) (2) , where π ∗ TMF (3) (2) = Z (2) [ a , a ][∆ − ] with ∆ = a ( a − a ). From their computations wededuce the following behaviour of the Tate spectral sequence(3.1) E n,m = ˆ H − n ( C ; π m TMF (3) (2) ) = ⇒ π n + m TMF (3) tC (2) :Let R n,m be the bigraded ring Z / a , a ][∆ − ][ ζ ± ] with | ζ | = ( − , a , a and ζ , then the E -page of the Tate spectral sequence is the even part of R n,m .Alternatively, it is given by E ∗ , ∗ = S ∗ [∆ − ][ x ± ] , where S ∗ is the subalgebra of Z / Z [ a , a ] generated by a , a a , a , and where x = ζa ∈ E − , .Note that a is invertible in this ring with a − = (( a a ) a − a )∆ − . By Mahowald-Rezk’scomputations the first non-trivial differential is d and we have d ( a ) =( x ( a a ) a − ) , d ( a a ) = 0 , d ( a ) = x ( a a ) a − ,d ( x ) =0 , d (∆ − ) = 0 . Using the Leibniz rule we get that the class c n,m,k,l,i = ( a ) n ( a a ) m ( a ) k ∆ − l x i with n, m, k, l ∈ N and i ∈ Z has differential d ( c n,m,k,l,i ) = ( n + k ) x ( a a ) a − c n,m,k,l,i . It follows that ker d is generated as F -vector space by the classes c n,m,k,l,i with n + k = 0 in F . We claim that E ∗ , ∗ ∼ = F [ x ± , ∆ ± ] . To see this, note the following: If n + k = 0 in F and m >
0, then c n,m,k,l,i is zero in E ∗ , ∗ because d ( c n,m − ,k +5 ,l,i − ) = c n,m,k,l,i . If n + k = 0 in F and n, k >
0, then we have c n, ,k,l,i = c n − , ,k − ,l,i . This is in the image of d ,because n − k − F and 2 >
0. If n = 0 in F and n >
0, then c n, , ,l,i = ( a ) n ∆ − l x i = ( a a ) ( a ) n − a − ∆ − l x i = ( a a ) ( a ) n − (( a a ) a + a )∆ − ∆ − l x i = c n, , ,l +1 ,i + c n − , , ,l +1 ,i , and both of these summands are in the image of d . Furthermore, note that in E ∗ , ∗ we have∆ = ( a a ) + a = c , , , , + a = a . This implies that for k = 0 in F we have c , ,k,l,i = ( a ) k ∆ − l x i ≡ ∆ − l + k x i in E ∗ , ∗ . We thus get a surjective map F [ x ± , ∆ ± ] → E ∗ , ∗ , which is injective, because the classes∆ l x i for l, i ∈ Z are not divisible by ( a a ) in S ∗ [∆ − ][ x ± ].From Mahowald-Rezk’s computations we get that the next non-trivial differential is d andthat we have d ( x ) = 0 and d (∆) = x ∆ − . ETECTING AND DESCRIBING RAMIFICATION 19
This gives E ∗ , ∗ = 0.We now want to determine the behaviour of the Tate spectral sequence(3.2) E n,m = ˆ H − n ( C ; π m tmf (3) (2) ) = ⇒ π n + m tmf (3) tC (2) . If we assign again odd weight to a , a and ζ , then the E -page is the even part of Z / Z [ a , a ][ ζ ± ] , and one sees that the map of spectral sequences from (3.2) to (3.1) is injective. We get that d is the first non-trivial differential in (3.2) and that we have d ( a a ) = 0 , d ( a ) = ( a ζ ) ,d ( a ) = a a ζ , d ( a ζ ) = ( a a ) ζ ,d ( a ζ ) = 0 , d ( ζ ) = a ζ . Note that an F -basis of the E -page is given by the classes d n,m,i =( a ) n ( a ) m ( ζ ) i ,e n,m,i =( a ) n ( a a )( a ) m ( ζ ) i ,f n,m,i =( a ) n ( a ) m ( a ζ )( ζ ) i ,g n,m,i =( a ) n ( a ) m ( a ζ )( ζ ) i , for n, m ∈ N and i ∈ Z .The d -differential on these classes is given by d ( d n,m,i ) = ( n + m + i ) · f n,m,i +1 ,d ( e n,m,i ) = ( n + m + i ) · g n +1 ,m,i +1 ,d ( f n,m,i ) = ( n + m + i ) · d n +1 ,m,i +2 ,d ( g n,m,i ) = ( n + m + i + 1) · e n,m,i +2 . We get E ∗ , ∗ = M m ∈ N ,i ∈ Z m + i =0 in F F { d ,m,i } ⊕ M m ∈ N ,i ∈ Z m + i +1=0 in F F { g ,m,i } . The map of spectral sequences from (3.2) to (3.1) satisfies d ,m,i ∆ m − i x i , g ,m,i ∆ m − i − x i +1 . In particular, one sees that it is injective on E -pages. We conclude that the next non-trivialdifferential in spectral sequence (3.2) is d and that we have d ( d ,m,i ) = m − i g ,m,i +3 , d ( g ,m,i ) = m − i − d ,m +1 ,i +4 . We obtain that E ∗ , ∗ = M i ∈ Z F { d , , i } = M i ∈ Z F { ζ i } . Since the E -page is concentrated in the zeroth row, the spectral sequence collapses at thisstage. This gives the answer on the level of homotopy groups. As tmf (3) tC is an E ∞ -ringspectrum [McC96] it is in particular an E -ring spectrum and therefore a result by Hopkins-Mahowald (see [MNN15, Theorem 4.18]) implies that tmf (3) tC receives a map from H F andtherefore is a generalized Eilenberg-MacLane spectrum of the claimed form. (cid:3) Theorem 3.13.
The Σ -action on tmf (2) (3) yields tmf (2) t Σ (3) ≃ _ i ∈ Z Σ i H Z / Z . Proof.
We use the calculation of [Sto12]. She proves that
Tmf (2) t Σ (3) ≃ ∗ via the Tate spectralsequence E n.m = ˆ H − n (cid:0) Σ ; π m ( Tmf (2) (3) ) (cid:1) = ⇒ π n + m ( Tmf (2) t Σ (3) ) . The E -page is given by Z / Z [ α, β ± , ∆ ± ] /α with | α | = ( − , | β | = ( − ,
12) and | ∆ | = (0 , d (∆) = αβ and d ( α ∆ ) = β . Since tmf (2) (3) is the connective cover of
Tmf (2) (3) the E -page of the Tate spectral sequence¯ E n,m = ˆ H − n (cid:0) Σ ; π m ( tmf (2) (3) ) (cid:1) = ⇒ π n + m ( tmf (2) t Σ (3) )is the Z / Z -module M k,l ∈ Z k +2 l > Z / Z { β k ∆ l } ⊕ M k,l ∈ Z k +6 l > Z / Z { αβ k ∆ l } . Using the map of Tate spectral sequences ¯ E ∗∗ , ∗ → E ∗∗ , ∗ one sees that¯ E ∗ , ∗ = M k,l ∈ Z k +6 l > Z / Z { β k (∆ ) l } ⊕ M k,l ∈ Z k +18 l > Z / Z { ( α ∆ ) β k (∆ ) l } . Since E ∗ , ∗ = Z / Z [ α ∆ , β ± , ∆ ± ] / ( α ∆ ) the map ¯ E ∗ , ∗ → E ∗ , ∗ is injective. Thus, ¯ d is deter-mined by d and one gets ¯ E ∗ , ∗ = M k ∈ Z Z / Z { ( β − ∆ ) k } . The class β − ∆ has bidegree (12 , E ∗ , ∗ is concentrated in line zero and the spectralsequence collapses at this stage. (cid:3) So we can view the extensions ko (2) → ku (2) , and tmf (3) → tmf (2) (3) as being wildly ramifiedand tmf (3) (2) → tmf (3) (2) has a non-trivial Tate construction.In contrast, KO → KU is a faithful C -Galois [Rog08a, § TMF (3) → TMF (3) and Tmf (3) → Tmf (3) are both faithful C -Galois extensions [MM15, Theorem 7.12]. In general, TMF [1 /n ] → TMF ( n ) is a faithful GL ( Z /n Z )-Galois extension [MM15, Theorem 7.6] and theTate spectrum Tmf ( n ) tGL ( Z /n Z ) is contractible [MM15, Theorem 7.11].For general n , constructions of tmf ( n ) and tmf ( n ) are tricky: For some large n , π Tmf ( n )is non-trivial. Lennart Meier constructs a connective version of Tmf ( n ) with trivial π as an E ∞ -ring spectrum in [Mei], so that there are E ∞ -models of tmf ( n ) for all n .As π ( tmf ( n )) ∼ = Z [ n , ζ n ] where ζ n is a primitive n th root of unity, the defining cofibersequence of tmf ( n ) tGL ( Z /n Z ) gives . . . / / π tmf ( n ) hGL ( Z /n Z ) N / / π tmf ( n ) hGL ( Z /n Z ) / / π tmf ( n ) tGL ( Z /n Z ) / / tmf ( n ) hGL ( Z /n Z ) is connective.By the homotopy orbit spectral sequence we get that π tmf ( n ) hGL ( Z /n Z ) ∼ = Z [ n , ζ n ] GL ( Z /n Z ) .As τ > tmf ( n ) hGL ( Z /n Z ) ≃ tmf [ n ], we know that Z [ n ] ∼ = π tmf [ n ].For every n > | GL ( Z /n Z ) | = ϕ ( n ) n Q p | n (1 − p ). Here, ϕ denotes the Euler ϕ -function and p runs over all primes dividing n .Meier shows [Mei], that tmf ( n ) is a perfect tmf [1 /n ]-module spectrum and hence dualizable.In general we do not know whether tmf ( n ) is faithful as a tmf [1 /n ]-module. For n = 2, tmf (2) (3) is faithful and dualizable over tmf (3) , as we saw in Remark 3.11. ETECTING AND DESCRIBING RAMIFICATION 21
We cannot determine the homotopy type of the GL ( Z /n Z )-Tate construction of tmf ( n ) forarbitrary n , but we can identify cases where it is non-trivial: Theorem 3.14.
Assume n > , ∤ n or n = 2 k for some k > . Then tmf ( n ) tGL ( Z /n Z )
6≃ ∗ . Proof.
By [KM85, p. 282] we know that GL ( Z /n Z ) acts on ζ n via the determinant map: For A ∈ GL ( Z /n Z ) and ζ n we get A.ζ in = ζ i · det( A ) n . Therefore, the norm map N : Z [ n , ζ n ] GL ( Z /n Z ) → Z [ n ] sends ζ in to X A ∈ GL ( Z /n Z ) ζ i · det( A ) n = | SL ( Z /n Z ) | X r ∈ ( Z /n Z ) × ζ irn , in particular it sends a primitive n -th root of unity ζ to | SL ( Z /n Z ) | µ ( n ) with µ denoting theM¨obius function.If n is square-free, then µ ( n ) = ±
1. Any power of ζ n has order d with d | n , so this d issquarefree as well, so N ( ζ in ) is equal to | SL ( Z /n Z ) | µ ( n ) or a multiple of it.If n contains a square of a prime, then µ ( n ) = 0, so N ( ζ n ) = 0 but of course N (1) = | GL ( Z /n Z ) | . If d | n , and d is squarefree, then the corresponding power of ζ n can give anon-trivial multiple of | SL ( Z /n Z ) | .If 2 ∤ n then | GL ( Z /n Z ) | and | SL ( Z /n Z ) | are not units in Z [ n ]: Let p be an odd primefactor of n . Then in n Q p | n (1 − p ) we have a factor of p − Z [ n ].If n = 2 k for some k >
1, we obtain | SL ( Z /n Z ) | = 2 k (cid:3) Remark . For many n the Tate construction tmf ( n ) tGL ( Z /n Z ) is actually trivial. If n = 2 k ℓ with k, ℓ > GL ( Z /n Z ) is invertible in Z [ n ]. Similarly, if n =2 · · . . . · p m is the product of the first m prime numbers for any m >
2, then the group orderis invertible as well.We close with a periodic example: Let E n be the Lubin-Tate spectrum for the Honda formalgroup law over W ( F p n ). For any finite group G , F ( BG + , E n ) → E n is faithful in the K n -localcategory [BR11, Theorem 4.4]. At the moment we don’t know whether E n is a dualizable F ( BG + , E n )-module for any finite group G .In [BR11, Theorem 5.1] it is shown that F (( BC p r ) + , E n ) → E n is ramified and one can alsoconsider more general groups than C p r , but the type of ramification was not determined. Thefollowing result indicates that F (( BC p r ) + , E n ) → E n is wildly ramified. Theorem 3.16.
For all r > and n > E tC pr n
6≃ ∗ . Proof.
The Tate spectral sequence E s,t = ˆ H − s ( C p r ; π t E n ) ⇒ π s + t ( E tC pr n )has as E -term ˆ H − s ( C p r ; π t E n ) ∼ = ( π t E C pr n /p r = π t E n /p r , for s even , ker( N ) / im( t −
1) = 0 , for s odd . As π ∗ ( E n ) is concentrated in even degrees, the whole E -term is concentrated in bidegrees ( s, t )where s and t are even. Therefore, all differentials have to be trivial and E = E ∞ . Thus π ∗ ( E tC pr n ) is highly non-trivial. (cid:3) Proposition 3.17.
Assume that G is a finite group with a non-trivial cyclic subgroup C p k < G for some prime p . Then E tGn is non-trivial when E n is the Lubin-Tate spectrum at the prime p .Proof. The restriction map induces a map on Tate constructions E tGn → E tC pk n . McClure[McC96] shows that the E ∞ -structure on Tate constructions E tGn = t (( E n ) G ) G is compatiblewith inclusions of subgroups and Greenlees-May show [GM95, Proposition 3.7] that for anysubgroup H < G the H -spectrum t (( E n ) G ) is equivalent to t (( E n ) H ). Therefore the inclusionof fixed points t (( E n ) G ) G → t (( E n ) G ) H is a map of E ∞ -ring spectra. As we know that E tC pk n = t (( E n ) G ) C pk is non-trivial by Theorem 3.16, E tGn cannot be trivial, either. (cid:3) References [AB59] Maurice Auslander and David A. Buchsbaum,
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