The E_2-term of the K(n)-local E_n-Adams spectral sequence

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Topology and its Applications

www.elsevier.com/locate/topol

The E

-term of the K(n)-local E

-Adams spectral sequence

Tobias Barthel^∗, Drew Heard^∗

MaxPlanckInstituteforMathematics,Bonn,Germany

a r t i c l e i n f o a b s t r a c t

Article history:

Received1December2014

Receivedinrevisedform25January 2016

Accepted28March2016 Availableonline12April2016

Keywords:

K(n)-localhomotopytheory Chromatichomotopytheory MoravaE-theory

Spectralsequence

LetE=EnbeMoravaE-theoryofheightn.In[8]DevinatzandHopkinsintroduced the K(n)-local En-Adams spectral sequence and showed that, under certain conditions,theE₂-termofthisspectralsequencecanbeidentifiedwithcontinuous groupcohomology.WeworkwiththecategoryofL-completeE_∗^∨E-comodules,and showthatinanumberofcasestheE2-termoftheabovespectralsequencecanbe computedbyarelativeExt groupinthiscategory.Wegivesuitableconditionsfor whenwecanidentifythisExt groupwithcontinuousgroupcohomology.

© 2016ElsevierB.V.All rights reserved.

0. Introduction

LetEn denotethen-thMoravaE-theory(atafixedprimep),theLandweberexactcohomologytheory with coefficientring

π_∗(En) =W(Fpⁿ)[[u1, . . . , u_n−1]]][u^±1],

where each ui isindegree0,and uhasdegree−2.Here W(Fpⁿ) refers tothe Wittvectors overthe finite fieldFpⁿ (anunramified extensionofZp of degreen). NotethatE₀is acomplete localregularNoetherian ringwith maximalidealm= (p,u1,. . . ,u_n−1).

Unless indicated otherwise, let us fix an integer n ≥ 1 and write E instead of En throughout. The cohomologytheoryE playsaveryimportantroleinthechromaticapproachto stablehomotopytheory,in particular intheunderstandingoftheK(n)-localhomotopycategory (see,forexample,[21]).

TheformalgrouplawassociatedtoE_∗istheuniversaldeformationoftheHondaformalgrouplawΓn of height n overFpⁿ.LetGn = Aut(Γn)Gal(Fpⁿ/Fp) denotethen-th (extended)Morava stabilizergroup.

Lubin–Tate theory impliesthatGn actson thering E_∗,and Brown representability impliesthat Gn acts

* Correspondingauthors.

E-mailaddresses:[email protected](T. Barthel),[email protected](D. Heard).

http://dx.doi.org/10.1016/j.topol.2016.03.024 0166-8641/© 2016ElsevierB.V.All rights reserved.

onE itself inthestable homotopycategory. TheGoerss–Hopkins–Miller theorem[31,12]impliesthatthis actioncanbetakentobe viaE_∞-ringmaps.

In general Gn is a profinite group, and it is not clear how to form the homotopy fixed points with respect to such groups (although progress has been made in this area; see [6,2,29]). Nonetheless, in [8]

DevinatzandHopkinsdefinedE_∞-ringspectraE^hG forG⊂GnaclosedsubgroupoftheMoravastabilizer group,whichbehavelikecontinuoushomotopyfixedpointspectra(andindeedifGisfinitetheyagreewith the usual construction of homotopy fixed points). Remarkably they showed that there is an equivalence E^hGnL_K(n)S⁰,aresultexpectedsincetheworkofMorava[26].Davis[6],Behrens–Davis[2]andQuick[29]

havegivenconstructionsofhomotopyfixed pointspectrawithrespectto thecontinuousactionofGonE, andthese agreewiththeconstructionofDevinatzand Hopkins.

DevinatzandHopkinsadditionallyshowedthatforanyspectrumZthereisastronglyconvergentspectral sequence

E₂^∗,∗=H_c^∗(G, E^∗Z)⇒(E^hG)^∗Z

which isa particular caseof aspectral sequence known as theK(n)-local E-Adams spectral sequence [8, Appendix A]. Here,for aclosed subgroup G⊂Gn the continuouscohomology of Gwith coefficients ina topologicalGn-moduleN isdefinedusingthecochaincomplexHom^c(G^•,N) (seethediscussionbeforethe proofofTheorem 4.3).

Using homologyinstead ofcohomology Devinatzand Hopkins identified conditions[8, Proposition 6.7]

underwhichtheK(n)-localE-Adamsspectral sequencetakestheform E^∗,∗₂ =H_c^∗(Gn, E_∗Z)⇒π_∗L_K(n)Z.

Itwasremarkedthatthiswasprobablynotthemostgeneralresult.InmanycasestheE₂-termofAdams-type spectralsequencescanbecalculatedbyExt groups(forexample[30,Chapter 2]).Thusweaskthefollowing twoquestions:

(a) CantheE₂-termoftheK(n)-localE-Adamsspectral sequencebecalculatedbyasuitableExt group?

(b) InwhatgeneralitycanweidentifytheE₂-termwithcontinuousgroupcohomology?

Inthisdocumentwegivepartialanswerstoboththese questions.Someworkonthesecondproblemhas beendonepreviously,andweprovide acomparison betweensomeknownresultsandourresults.

IntheK(n)-localsettingthenaturalfunctortoconsiderforaspectrumX isnotE_∗X butratherE_∗^∨X :=

π_∗L_K(n)(E∧X).TheuseofthiscompletedversionofE-homologybecomesveryimportantinunderstanding theE2-termofthisspectralsequence.¹ThisisnotjustanE_∗-module,butratheranL-completeE_∗-module, and basedon workof Baker [1] wework inthecategory ofL-complete E_∗^∨E-comodules. This category is notabelian,and soweusethemethodsofrelative homologicalalgebrato definearelativeExt functor for certain classesof objectsinthe category, whichwe denote by Ext^s,t_E∨

∗E(−,−).The following isour answer for (a).

Theorem 3.1. Let X and Y be spectra and suppose that E_∗^∨X is pro-free, and E_∗^∨Y is either a finitely- generated E_∗-module, pro-free, or has bounded m-torsion (i.e., is annihilated by some power of m). Then theE2-termof theK(n)-localE-Adamsspectral sequencewithabutment πt−sF(X,L_K(n)Y)is

1 Thisalsogives one reason whythecase ofcontinuouscohomology withcoefficientsin E^∗Z iseasier thanin E_∗^∨Z forany spectrumZ; since F(Z,Σ^kE) isalready K(n)-local forany k ∈ Z (sinceE is),there is noneed fora ‘completed’versionof E-cohomology.

E₂^s,t=Ext^s,t_E∨

∗E(E^∨_∗X, E_∗^∨Y).

Ouranswertoquestion(b)isthefollowing.

Theorem4.3. SupposethatX isaspectrumsuchthatE_∗^∨X iseitherafinitely-generatedE_∗-module,pro-free, or has boundedm-torsion.Then, for theK(n)-localE-Adamsspectral sequence with abutment π_∗L_K(n)X, there isanisomorphism

E₂^∗,∗=Ext^∗,∗_E∨

∗E(E_∗, E_∗^∨X)H_c^∗(Gn, E_∗^∨X).

Wethencompare thistosomeoftheknownresultsintheliterature.

Wehavetwoapplicationsoftheseresults.Firstly,wecanalmostimmediatelyextendaresultofGoerss–

Henn–Mahowald–Rezk,usedintheirconstructionofaresolutionoftheK(2)-localsphereattheprime 3[13], from finitesubgroupsofGn to arbitraryclosed subgroups.

The second application appears to work at height n = 1 only. Here we construct a spectral sequence withE2-termLiExt^s,t_E∗E(E_∗X,E_∗Y),whereE_∗X isaprojectiveE_∗-module,E_∗Y aflatE_∗-module,andLi

referstothederivedfunctorofcompletiononthecategoryofZ(p)-modules.Weshowthattheabutmentof this spectralsequenceisExt^s−i,t_E∨

∗E(E_∗^∨X,E^∨_∗Y) andcalculatethiswhenX =Y =S⁰ attheprime2.

1. L-completionandL-completecomodules 1.1. L-completion

It is now well understood (see, for example [21]) that in the K(n)-local setting the functor E_∗^∨(−) = π_∗LK(n)(E∧ −), from spectra to E_∗-modules, mentioned in theintroduction is a morenatural covariant analogueofE^∗(−) thanordinaryE_∗-homology,despitethefactthatitisnotahomologytheory.Itisequally well understoodthatthis functoris naturallythought ofas landing inthecategory ModE_∗ ofL-complete E_∗-modules, ratherthanthecategory of E_∗-modules. Wereviewthebasics ofthis category now;for more detailssee[21,3,18,32].

Remark 1.1. Since we always work with E-modules there is some ambiguity to the type of Bousfield localisationweareusing.RecallthatL_K(n)denotesBousfieldlocalisationwithrespecttoMoravaK-theory K(n) on the category of spectra. Let L^E_K(n) denote Bousfield localisation on the category of E-modules.

SupposenowthatMisanE-module.Thenby[4,Lemma 4.3]thereisanequivalenceLK(n)M L^E_K(n)∧EM.

Butby[19,Proposition2.2] thelatterisjustL^E_K(n)M andsoitdoesnotmatterifweuseLK(n)or L^E_K(n). To keep thetheory general,suppose thatR is acomplete local Noetheriangraded ring with aunique maximal homogeneousidealm, generatedbyaregularsequence ofnhomogeneouselements.Our assump- tionsimplythatthe(Krull)dimensionofRisn.LetModRdenotethecategoryofgradedR-modules,where themorphismsarethemorphismsofR-modulesthatpreservethegrading.

Recall thatgivenanR-moduleM,thecompletionof M (atm)is M_m^∧= lim_←−−

k

M/m^kM.

Here wemusttakethelimitinthegradedsense.Thisisfunctorial,butcompletionisnotright(norinfact left) exact;theideaistothenreplacecompletionwithitszerothderivedfunctor.

Definition 1.2.For s ≥ 0 let L_s(−) : Mod_R → Mod_R be the s-th left derivedfunctor of the completion functor (−)^∧_m.

Since completionis notright exactit isnot true thatL₀M M_m^∧. Infactthe naturalmap M →M_m^∧ factorsasthecomposite M−−→^ηM L₀M −−→^M M_m^∧.

Definition1.3. WesaythatM isL-complete ifη_M isanisomorphismofR-modules.

ThemapM issurjectivewithkernel lim←−−1

kTor^R₁(R/m^k, M); (1.1)

ingeneralthese derivedfunctorsfit intoanexactsequence[21, TheoremA.2]

0→lim_←−−¹_kTor^R_s+1(R/m^k, M)→LsM→lim_←−−

k

Tor^R_s(R/m^k, M)→0, andvanish ifs<0 ors> n.

LetModRdenotethesubcategoryofModR consistingofthosegradedR-modulesM forwhichηM isan isomorphism.This categoryis abicomplete fullabeliansubcategory ofthecategory of gradedR-modules, and is closed under extensions and inverse limits formed in ModR. One salient feature of this category is that Ext^s

ModR(M,N) Ext^s_R(M,N) for all s ≥0 whenever M and N are L-complete R-modules [18, Theorem 1.11].Thetensor productof L-complete modulesneednotbe L-complete; we writeMRN :=

L0(M⊗RN).By[21, Proposition A.6]thisgivesModRthestructure ofasymmetricmonoidalcategory.

Remark1.4. WewillusethefollowingpropertiesofL-completionrepeatedly:

(i) IfM isaflatR-module,thenL0M=M_m^∧ isflatasanR-moduleandthusLsM = 0 fors>0 (see[18, Corollary1.3]or [3,Proposition A.15]);

(ii) IfM isafinitely-generatedR-module,then L₀M =M and L_sM = 0 for s>0[21, Proposition A.4, Theorem A.6];and,

(iii) IfM isabounded m-torsionmodule,then L0M=M andLsM= 0 fors>0.

Thelastitemfollowsfrom[21,Theorem A.6]andtheobservationthatforlargeenoughkthereareequiva- lences(by[21,PropositionA.4])

L0ML0(M⊗RR/m^k) L0M⊗RR/m^k M⊗RR/m^k M,

so that M is L-complete. Modules M that have LsM = 0 for s > 0 are known as tame. For example, L-complete modulesarealwaystame.

Example 1.5. Let R = Z(p) and m = (p). Since Z(p) has Krull dimension 1 the only potential non-zero derivedfunctorsareL₀and L₁.By[3,Proposition 5.2],L-completionwith respectto Z(p)naturallylands inthecategory ofZp-modules.

Itisimmediatefrom theremarkabovethatL0Z(p)=Zp andLiZ(p)= 0 fori>0.By[21,Theorem A.2]

foranyZ(p)-moduleM wehave

L₀M= Ext¹_Z(p)(Z/p^∞, M) and L₁M = Hom_Z(p)(Z/p^∞, M)lim_←−−

r

Hom_Z(p)(Z/p^r, M).

IfM isanyinjectiveZ(p)-moduleM,forexampleifM=Q/Z(p),thenitfollowsfrom thisdescriptionthat L0M = 0.OntheotherhandtheinversesystemdefinedabovegivesL1(Q/Z(p))=Zp.

The discussion abovealsoshows thatifM isany bounded p-torsion Z(p)-module thenit isL-complete and hencetame.

Suppose nowthatM isaflat R-moduleso that,by Lazard’s theorem,wecanwrite itcanonically as a filtered colimitover finite freemodules, M = lim_−−→jFj.Since HomR(Fj,L0N) is L-complete for any j ∈J, thesameistrueforlim_←−−jHom_R(F_j,L₀N)= Hom_R(M,L₀N),andhencewegetanaturalfactorization

L0HomR(M, N)

Hom_R(M, N) Hom_R(M, L₀N) forarbitraryN.

Proposition1.6. IfM isprojectiveandN is flat,thenthenatural map

L0HomR(M, N) HomR(M, L0N) is anisomorphismof L-completeR-modules.

Proof. ItisenoughtoshowtheclaimforM=

IR free.SinceRisNoetherian,productsofflatmodules are flat,so weget

L0HomR(M, N) =L0

I

N = lim_←−−

k

((

I

N)⊗R/m^k)

and similarly

Hom_R(M, L₀N) =

I

lim←−−

k

(N⊗R/m^k) = lim_←−−

k

I

(N⊗R/m^k).

Therefore, itsufficestoshowthatthenaturalmap : (

I

N)⊗R/m^k→

I

(N⊗R/m^k)

is anisomorphismforallk.Since R/m^k isfinitely-presented,thisistrueby[23,Proposition 4.44],andthe propositionfollows. 2

Corollary 1.7. ForM projective andN flat,there areisomorphisms

LsHomR(M, N) =

Hom_Mod

R(L0M, L0N) ifs= 0

0 otherwise.

Proof. Thefirst statementisadirectconsequence ofthe previousproposition.For thecase ofs>0 note thatHomR(M,N) isflat,hencetame. 2

Remark1.8.UsingworkofValenzuela[35]itispossible toconstructaspectral sequence E₂^s,t=L_pExt^q_R(M, N)⇒Ext^q_R^−p(M, L_R/mN),

where M and N are arbitraryR-modules andL_R/m is thetotalleft derivedfunctorof L₀.Specialising to M projectiveand N tame givestheabovecorollary.

1.2. Completed E-homology

Wenow specialiseto the casewhere R =E_∗. By[21, Proposition 8.4] thefunctor E_∗^∨(−) always takes valuesinModE_∗.Thisisinfactaspecialcaseofthefollowingtheorem.

Proposition 1.9. ([3, Corollary 3.14]) An E-module M is K(n)-local if and only if π_∗M isan L-complete E_∗-module.

Remark 1.10.The case where M = E∧X, for X an arbitrary spectrum,proved in[21], uses adifferent method.InparticularthereisatowerofgeneralisedMoorespectraM_I suchthatL_K(n)X holim_IL_nX∧ M_I [21, Proposition 7.10]. ThisgivesrisetoaMilnorsequence

0→lim_←−−

I

1E_∗+1(X∧M_I)→E_∗^∨X →lim_←−−

I

E_∗(X∧M_I)→0, (1.2)

whichby[21,Theorem A.6] impliesE^∨_∗X isL-complete.

TheprojectiveobjectsinModE_∗ willbeimportantforus. Thesearecharacterisedin[21,Theorem A.9]

and[3,Proposition A.15].

Definition1.11.AnL-completeE_∗-moduleispro-freeifitisisomorphictothecompletion(or,equivalently, L-completion)ofafreeE_∗-module.Equivalently,thesearetheprojectiveobjectsinModE_∗.

Proposition1.12. If E_∗^∨X iseither finitely-generatedasanE_∗-module,pro-free, orhasboundedm-torsion, thenE_∗^∨X iscomplete inthem-adictopology.

Proof. ThecasewhereE_∗^∨X isfinitely-generatedfollowsfromthefactthatE_∗iscompleteandNoetherian.

SinceE_∗^∨X isalwaysL-completeandL₀-completionisidempotent,whenE^∨_∗X ispro-free(andhenceflat) L₀(E_∗^∨X)E_∗^∨X = (E_∗^∨X)^∧_m, so thatE_∗^∨X is complete.The casewhere E^∨_∗X hasbounded m-torsionis clear. 2

Remark 1.13. The condition that E_∗^∨X is pro-free is not overly restrictive. Let K denote the2-periodic version ofMorava K-theory with coefficientring K_∗ =E_∗/m =Fpⁿ[u^±1]. If K_∗X is concentratedineven degrees,thenE_∗^∨Xispro-free[21,Proposition8.4].Forexample,thisimpliesthatE_∗^∨E_n^hF ispro-freeforany closed subgroupF ⊂Gn.By[21, Theorem 8.6]E_∗^∨X is finitelygenerated ifand onlyifX isK(n)-locally dualisable.

We will need the following version of the universal coefficient theorem (for Y = S this is [18, Corol- lary 4.2]).

Proposition1.14. LetX andY be spectra.If E_∗^∨X is pro-free,then

Hom_E∗(E_∗^∨X, E_∗^∨Y)π_∗F(X, L_K(n)(E∧Y)).

Proof. Let M, N be K(n)-local E-module spectra. Note that π_∗M and π_∗N are always L-complete by Proposition 1.9. Undersuch conditions Hovey [18, Theorem 4.1]has constructed a natural, strongly and conditionallyconvergent,spectral sequenceofE_∗-modules²

E^s,t₂ = Ext^s,t

ModE∗

(π_∗M, π_∗N)Ext^s,t_E∗(π_∗M, π_∗N)⇒πt−sFE(M, N).

SetM =L_K(n)(E∧X) andN =L_K(n)(E∧Y).Notethenthat

FE(LK(n)(E∧X), LK(n)(E∧Y))FE(E∧X, LK(n)(E∧Y))F(X, LK(n)(E∧Y)), where thesecondisomorphismis[10,Corollary III.6.7],givingaspectral sequence

E₂^s,t= Ext^s,t

ModE∗(E_∗^∨X, E_∗^∨Y)Ext^s,t_E∗(E_∗^∨X, E_∗^∨Y)⇒πt−sF(X, L_K(n)(E∧Y)).

Since E_∗^∨X is pro-freeitis projectiveinModE_∗ andso the spectralsequence collapses, givingthedesired isomorphism. 2

Remark1.15.Themapabovecanbedescribed inthefollowing way:given f :X →L_K(n)(E∧Y)

then thehomomorphismtakes

g:S →L_K(n)(E∧X) to theelement

S −→^g L_K(n)(E∧X)−−−→^1∧f L_K(n)(E∧E∧Y)−−−→^μ∧1 L_K(n)(E∧Y).

1.3. L-completeHopfalgebroids

Since E_∗^∨X always lands in the category of L-complete E_∗-modules, one is led to wonder if E_∗^∨X is a comodule over a suitable L-complete Hopf algebroid. The category of L-complete Hopf algebroids has previouslybeen studiedbyBaker[1],andwe nowbrieflyreviewthiswork.

SupposethatRisasinSection1.1and,additionally,Risanalgebraoversomelocalsubring(k0,m0) of (R,m),suchthatm0=k0∩m.

WesayA∈Modk0 isaringobjectifithasanassociativeproductφ:A⊗k0A→A.AnR-unitforφisa k₀-algebrahomomorphismη:R→A.AringobjectAisR-biunitalifithastwounitsη_L,η_R:R→Awhich extendto giveamorphismη_L⊗η_R:R⊗k0R→A.Suchanobjectiscalled L-completeifitisL-complete as bothaleftandright R-module.

Definition 1.16.([1, Definition 2.3]) Suppose thatΓ is an L-complete commutativeR-biunitalring object with leftandrightunitsηL,ηR:R→Γ,alongwiththefollowing maps:

Δ : Γ→ΓRΓ (composition) : Γ→R (identity)

c: Γ→Γ (inverse)

2 Notethatwehaveregradedthespectralsequencein[18]toreflectthefactweusehomologyratherthancohomology.

satisfying the usual identities (as in [30, Appendix A]) for a Hopf algebroid. Then the pair (R,Γ) is an L-complete Hopfalgebroid ifΓ ispro-freeasaleftR-module,andtheidealmisinvariant,i.e., mΓ= Γm.

Lemma1.17. ([1,Proposition 5.3]) Thepair(E_∗,E_∗^∨E) isanL-completeHopf algebroid.

Definition1.18.([1,Definition 2.4]) Let(R,Γ) beanL-completeHopfalgebroid.Aleft(R,Γ)-comoduleM isanL-complete R-moduleM togetherwith aleftR-linearmapψ:M→ΓRM whichiscounitaryand coassociative.

Wewill usually refertoaleft (R,Γ)-comodule asan L-completeΓ-comodule and wewrite Comod _Γ for thecategoryofsuchcomodules.

Remark1.19.Inallcaseswewillconsider,E_∗^∨X willbeacompleteE_∗-module,andsowecouldworkinthe category ofcompleteE_∗^∨E-comodules,as studiedpreviouslybyDevinatz[7].However,whilstthecategory ofL-complete E_∗-modulesisabelian,thesameisnot truefor thecategoryofcomplete E_∗-modules,sowe prefertoworkwithL-complete E_∗^∨E-comodules.

Given anL-complete R-moduleN,letΓRN bethe comodulewithstructure map ψ= ΓRΔ.This is called an extended L-complete Γ-comodule. The following is thestandard adjunctionbetween extended comodulesandordinarymodules.

Lemma1.20. LetN beanL-completeR-moduleandletM beanL-completeΓ-comodule.Thenthereisan isomorphism

Hom_Mod

R(M, N) = Hom_Comod

Γ(M,ΓRN).

SupposethatF isaringspectrum(inthestablehomotopycategory)suchthatF_∗F isaflatF_∗-module.

Inthis case thepair (F_∗,F_∗F) isan (ordinary) Hopfalgebroid. To showthat F_∗(X) is anF_∗F-comodule for any spectrumX requires knowing that F_∗(F ∧X) F_∗F⊗F_∗F_∗X. The same is true here; to show thatE_∗^∨X isanL-completeE_∗^∨E-comoduleweneedtoshowthatE_∗^∨(E∧X)E_∗^∨EE_∗E_∗^∨X.Wedonot knowifitistrueingeneral;ournextgoalwillbetogivetheexamplesofL-completeE_∗^∨E-comodulesthat weneed.Wefirststartwithapreliminarylemma.

Lemma1.21. LetMandNbeE_∗-modulessuchthatMisflatandN iseitherafinitely-generatedE_∗-module, pro-free,or has boundedm-torsion.Then M⊗E_∗N istame.

Proof. FirstassumeN is finitely-generated.SinceE_∗ isNoetherianthere isashort exactsequence 0→K→F →N →0

whereF =⊕IE_∗isfreeandKandFarefinitely-generated.TensoringwiththeflatmoduleMgivesanother short exactsequence,and by[21, Theorem A.2]there isalongexactsequence

· · · →Lk+1(M⊗E_∗N)→Lk(M⊗E_∗K)→Lk(M⊗E_∗F)→Lk(M⊗E_∗N)→ · · ·. (1.3) The functors Lk are additivefor all k ≥ 0,and sinceM is flatwe see thatL0(M⊗E_∗F) = ⊕IM_m^∧ and Lk(M⊗E_∗F)= 0 fork >0.ItfollowsthatLk+1(M⊗E_∗N)Lk(M⊗E_∗K) fork≥1.

SinceK,F andN areallfinitely-generatedE_∗-modulesweuse[21,Theorem A.4]toseethattheendof thelongexactsequence(1.3)takestheform

0→L₁(M⊗E∗N)→L₀(M)⊗E∗K→L₀(M)⊗E∗F →L₀(M)⊗E∗N →0.

Since M isflat,L₀(M) ispro-free,andhenceflat[3,PropositionA.15],so L₀(M)⊗E_∗K→L₀(M)⊗E_∗F is injective,forcing L₁(M⊗E_∗N) = 0. Since N wasan arbitrary finitely-generated E_∗-module and K is finitely generated,wesee thatL1(M⊗E_∗K)= 0,also.It followsthatL2(M⊗E_∗N)L1(M⊗E_∗K)= 0, and arguinginductivelyweseethatLk(M⊗E_∗N)= 0 for k >0,so thatM⊗E_∗N istame.

Now assumethatN ispro-free,and henceflat.Itfollows thatM⊗E_∗N isalsoflat,andhencetame.

Forthefinalcase,whereN hasboundedm-torsion,notethatM⊗E_∗N alsohasboundedm-torsion,and so istame(see Remark 1.4). 2

Wenow identifyconditionsonaspectrumX so thatE_∗^∨X isanL-completeE_∗^∨E-comodule.

Proposition 1.22. LetX be aspectrum.If E_∗^∨E⊗E∗E_∗^∨X istame, then

E_∗^∨(E∧X)E_∗^∨EE∗E_∗^∨X (1.4) andE_∗^∨X isanL-completeE_∗^∨E-comodule.InparticularthisoccurswhenE^∨_∗X iseitherafinitely-generated E_∗-module,pro-free or hasbounded m-torsion.

Proof. Thereisaspectralsequence[10, Theorem IV.4.1]

E_s,t² = Tor^E_s,t^∗(E^∨_∗E, E_∗^∨X)⇒πs+t(LK(n)(E∧E)∧ELK(n)(E∧X)). (1.5) Forany E-moduleM wealsohavethespectral sequenceofE_∗-modules[19,Theorem 2.3]

E_s,t² = (Lsπ_∗M)t⇒πs+tL_K(n)M.

Inparticular thereisaspectralsequencestartingfrom theabutmentof(1.5)thathastheform (Liπ_∗(L_K(n)(E∧E)∧EL_K(n)(E∧X)))s+t⇒πi+s+tL_K(n)(L_K(n)(E∧E)∧EL_K(n)(E∧X)).

ByRemark 1.1wededucethatthere isanequivalence

L_K(n)(L_K(n)(E∧E)∧EL_K(n)(E∧X))L_K(n)(E∧E∧X),

and so thelatter spectralsequence abuts toE_∗^∨(E∧X).Since E_∗^∨E isaflatE_∗-module thefirst spectral sequence alwayscollapses,andthesecondspectral sequencebecomes

(L_i(E_∗^∨E⊗E∗E_∗^∨X))_s+t⇒E^∨_i+s+t(E∧X). (1.6) Thus, ifE_∗^∨E⊗E∗E_∗^∨X istame,this givesanisomorphism

E_∗^∨(E∧X)E_∗^∨EE_∗E_∗^∨X,

and soE_∗^∨X isanL-complete E_∗^∨E-comodule.SinceE_∗^∨E ispro-freeitisflat,and Lemma 1.21applies to show thatE_∗^∨E⊗E_∗E_∗^∨X istame inthegivencases. 2

Remark1.23.Thisraisesthequestion:whatisthemostgeneralclassofL-completecomodulesMsuchthat E^∨_∗E⊗E∗M is tame? Inlightof Baker’s example [1,Appendix B] ofan L-complete – and hencetame – module N suchthatL₁(_∞

i=0N)= 0,this seemsto be asubtleproblem.In particular,we note thatthis exampleimpliesthatthecollectionoftame modulesitselfneednotsatisfytheabovecondition.

Thefollowingcorollaryshowsthattheequivalenceof(1.4)canbe iterated.

Corollary 1.24. LetY be a spectrum such that E_∗^∨Y is either a finitely-generated E_∗-module, pro-free or has boundedm-torsion.Thenforalls≥0there isanisomorphism

E^∨_∗(E^∧s∧Y)(E_∗^∨E)^sE∗E_∗^∨Y.

Proof. Wewillprovethisbyinductionons,thecases= 0 beingtrivial.AssumenowthatE_∗^∨(E^∧(s−1)∧Y) (E_∗^∨E)^(s−1)E_∗E_∗^∨Y; wewillshow thatE_∗^∨E⊗E_∗((E^∨_∗E)^(s−1)E_∗E_∗^∨Y) istame.Weclaimthatthis istrueinthethree casesweconsider.

1. If E_∗^∨Y isflat,thenso is(E_∗^∨E)^(s−1)E_∗E_∗^∨Y,and wecanapplyLemma 1.21to seethatE_∗^∨E⊗E_∗

((E_∗^∨E)^(s−1)E_∗E_∗^∨Y) istame.

2. If E_∗^∨Y is finitely-generatedthen(E_∗^∨E)^(s−1)E_∗E_∗^∨Y (E_∗^∨E)^(s−1)⊗E_∗E_∗^∨Y [21, Theorem A.4].

Since E_∗^∨E⊗E∗(E_∗^∨E)^(s−1) is a flat E_∗-module, once again we can apply Lemma 1.21 to see that E^∨_∗E⊗E_∗((E_∗^∨E)^(s−1)E_∗E_∗^∨Y) istame.

3. If E^∨_∗Y has bounded m-torsion, then thesame is true for E_∗^∨E⊗E_∗((E_∗^∨E)^(s−1)E_∗E_∗^∨Y), and it follows thatitistame,asrequired.

Therefore,Proposition 1.22 appliedtoX =E^∧(s−1)∧Y impliesthat

E_∗^∨(E^∧s∧Y)E^∨_∗EE_∗E_∗^∨(E^∧(s−1)∧Y)(E^∨_∗E)^sE_∗E_∗^∨Y, wherethelast isomorphismusestheinductivehypothesisoncemore. 2

2. Relative homologicalalgebra 2.1. Motivation

Recall[30,Appendix A]thatthecategoryofcomodulesoveraHopfalgebroid(A,Γ) isabelianwhenever Γ isflatoverA,andthatifIisaninjectiveA-modulethenΓ⊗AI isaninjectiveΓ-comodule.Thisimplies thatthecategory ofΓ-comoduleshasenoughinjectives.

Given Γ-comodules M and N we can define Extⁱ_Γ(M,N) in the usual way as the i-th derivedfunctor of HomΓ(M,N), functorial in N. However, the category of L-complete Γ-comodules does not need to be abelian.Inthiscase,inordertodefineL-completeExt-groups,weneedtouserelativehomologicalalgebra, forwhichthefollowingismeant toprovidesomemotivation.

The following two lemmas show that we canform aresolution byrelative injective objects, insteadof absoluteinjectives.

Lemma 2.1. Let(A,Γ) be a Hopf algebroid(over a commutative ring K) suchthat Γ isa flat A-module, andlet

0→N →R⁰→R¹→ · · ·

beasequenceofleftΓ-comoduleswhichisexact (overK)andsuchthat foreach i,Extⁿ_Γ(M,Rⁱ)= 0forall n>0.ThenExt_Γ(M,N)is thecohomologyof thecomplex

Ext⁰_Γ(M, R⁰)→Ext⁰_Γ(M, R¹)→ · · ·.

Proof. See[27, Lemma1.1] or[30,LemmaA1.2.4]. 2

Definition2.2.A Γ-comoduleS isarelative injectiveΓ-comodule ifitisadirectsummand ofanextended comodule, i.e.,oneoftheform Γ⊗AN.

Lemma2.3. LetS be arelativelyinjective comodule.If M isaprojective A-module,then Extⁱ_Γ(M,S)= 0 fori>0.Henceif I^∗ isaresolutionof N byrelativelyinjectivecomodulesthen

Extⁿ_Γ(M, N) =Hⁿ(Hom_Γ(M, I^∗)) (2.1) foralln≥0.

Proof. ThesecondstatementfollowsfromthefirstandLemma 2.1.Forthefirststatementproceedasin[30, A1.2.8(b)]. 2

In the case of L-complete Γ-comodules, we will take the analogue of Equation (2.1) as a definition of Ext_Γ(−,−) (seeDefinition 2.13).

Remark 2.4.Thereader maywonder aboutprojective objects. Ingeneral, comodules overaHopf algebra do nothaveenoughprojectives. Forexample,when (A,Γ)= (Fp,A), where Aisthedual of theSteenrod algebra,itisbelievedthattherearenonon-zero projectiveobjects[28].

2.2. Homologicalalgebra forL-completecomodules

ThecategoryComod ΓofL-completeΓ-comodulesisnotabelian;itisanadditivecategorywithcokernels.

The absenceofkernels isdue to thefailureof tensoringwith Γ tobe flat.Ifθ :M →N isamorphism of L-completecomodules,thenthere isacommutativediagram[1]

0 kerθ M N

ΓRkerθ ΓRM ΓRN,

θ

ψM ψN

idRθ

butthedashedarrowneednotexistor beunique.

Since Comod Γ is not abelian we need to use the methods of relative homological algebra to define a suitable Ext functor, which we briefly review now. For amore thorough exposition see [11] (although in general oneneeds to dualisewhatthey say,since theymainlywork with relative projectiveobjects). Our workisinfactsimilar tothatof Millerand Ravenel[27].

Definition2.5. AninjectiveclassI inacategory Cisapair(D,S) whereDisaclassofobjectsandS isa class ofmorphismssuchthat:

1. I isinDifandonlyifforeachf :A→B inS

f^∗: Hom_C(B, I)→Hom_C(A, I) isanepimorphism.Wecall suchobjectsrelative injectives.

2. Amorphismf :A→B is inS ifandonlyifforeachI∈ D f^∗: Hom_C(B, I)→Hom_C(A, I)

is anepimorphism.Thesearecalled therelativemonomorphisms.

3. Forany objectA∈ C thereexists anobjectQ∈ Dandamorphismf :A→QinS.

Remark2.6.Note thatgiven eitherS or D, theother classis determined bytherequirements above,and thatthethirdconditionensurestheexistenceofenoughrelativeinjectives.

Itisnothardtocheck thatDisclosedunderretractsandthatifthecompositemorphismA−→^f B→C isinS thensoisf :A→B.

Example 2.7 (The split injective class).The split injective class Is = (Ds,Ss) has Ds equalto allobjects ofC andSsallmorphismsthatsatisfyDefinition 2.5,i.e.,Hom_C(f,−) issurjectiveforallobjects.Onecan easilycheck thatthis isequivalenttotherequirement thatf :A→B isasplitmonomorphism.

Example2.8(Theabsolute injectiveclass).LetS betheclassofallmonomorphismsandthenletDbe the objectsasneeded. Thissatisfies(3)ifthereareenoughcategoricalinjectives.

Onewaytoconstructaninjectiveclassisviaamethodknownas reflectionofadjointfunctors.

Proposition2.9. Supposethat C andF are additivecategorieswithcokernels,andthere isapairofadjoint functors

T :CF:U.

Then,if(D,S)isaninjectiveclassinC,wedefineaninjectiveclass(D,S)inF,wheretheclassofobjects isgivenbythesetofallretractsofT(D)andtheclassofmorphismsisgivenbyallmorphismswhoseimage (underU)isin S.

Sketch of proof. ³ Firstnotethat,sincerelative injectivesareclosed underretracts, toshow thatD isas claimed,itsufficestoshowthatT(I) isrelativeinjective,wheneverI∈ D.LetA→B beinS andI∈ D; thenthemap

Hom_F(B, T(I))→Hom_F(A, T(I)) isequivalentto theepimorphism

Hom_C(U(B), I)→Hom_C(U(A), I).

A similar methodshows that therelative monomorphisms are as claimed. Finally weobserve thatfor all A∈ F thereexistsaQ∈ DsuchthatU(A)→Q∈ S.ThentheadjointA→T(Q) satisfiesCondition3.To see this notethatU(A)→Q factorsas U(A)→U(T(Q))→Q; sincerelative monomorphisms areclosed underleftfactorisation(seeabove) U(A)→U(T(Q))∈ S.Then A→T(Q)∈ S as required. 2

Werecallthefollowingdefinition.

Definition 2.10.An extended L-complete E_∗^∨E-comodule is a comodule isomorphic to one of the form E_∗^∨EE_∗M,where M isanL-completeE_∗-module.Here thecomultiplicationisgivenbythemap

E_∗^∨EE_∗M−−−−→^Δid E_∗^∨EE_∗E_∗^∨EE_∗M.

3 Forfulldetailssee[11,p. 15]– hereitisprovedforrelativeprojectives,butitisessentiallyformaltodualisethegivenargument.

Example 2.11. GiveMod_E∗ thesplit injectiveclass.Then theadjunction Hom_Mod

E∗(A, B) = Hom_Comod

E∨∗E(A, E_∗^∨EE_∗B) produces aninjectiveclassinComod E_∗^∨E.Inparticularwehave

1. S is theclassofallcomodule morphismsf :A→B whose underlyingmapofL-completeE_∗-modules isasplit monomorphism.

2. DistheclassofL-completeE^∨_∗E-comoduleswhichareretractsofextendedcompleteE_∗^∨E-comodules.

NotethatforanycompleteE_∗^∨E-comoduleM thecoactionmapM−→^ψ E_∗^∨E⊗E_∗M isarelativemonomor- phism intoarelativeinjective.

Wewill saythatathree termcomplexM−→^f N −→^g P ofcomodulesisrelative shortexactifgf= 0 and f :M → N is arelative monomorphism.A relative injective resolutionof acomodule M isacomplex of theform

0→M →J⁰→J¹→ · · · where eachJⁱ isrelatively injective,andeachthree-termsubsequence

J^s−1→J^s→J^s+1,

where J⁻¹ = M and J^s = 0 for s <−1, is relative short exact. Note that, by definition, relative exact sequencesarepreciselythosethatgiveexactsequencesofabeliangroupsafterapplyingHom_Comod

E∨∗E(−,I), wheneverI isrelative injective.

Wehavetheusual comparisontheoremforrelativeinjectiveresolutions.Theproofisnearly identicalto thestandardinductivehomologicalalgebraproof– inthiscontext see[14,Theorem 2.2].

Proposition 2.12. LetM andM beobjects inan additivecategoryC with relative injectiveresolutionsP^∗ and P ^∗,respectively.Suppose thereis amapf :M →M .Then, there existsachain mapf^∗:P^∗→P ^∗ extending f that isuniqueuptochain homotopy.

Definition2.13.(Cf.[11,p. 7]) LetM andN beL-completeE_∗^∨E-comodules,andletM bepro-free.LetI^∗ be arelative injectiveresolution ofN. Then,foralls≥0,we define

Ext^s_ComodE∨

∗E(M, N) =H^s(Hom_Comod

E∨∗E(M, I^∗)).

Forbrevity wewillwriteExt^s_E∨

∗E(M,N) forthisExtfunctor.

Note that Proposition 2.12 implies that the derived functor is independent of the choice of relative injectiveresolution.

Remark2.14.

1. Thereadershouldcomparethis definitionto Lemma 2.3.

2. ThecategoryofL-completeE_∗-moduleshasnonon-zeroinjectives[3,p. 40];thissuggeststhatthesame istrueofL-completeE_∗^∨E-comodules,whichisyetanotherreasonweneedtouserelativehomological algebra.