Local Gorenstein duality for cochains on spaces

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Journal of Pure and Applied Algebra

www.elsevier.com/locate/jpaa

Local Gorenstein duality for cochains on spaces

Tobias Barthel^a, Natàlia Castellana^b, Drew Heard^c,∗, Gabriel Valenzuela^a

a Max-Planck-InstitutfürMathematik,Vivatsgasse7,53111Bonn,Germany

bDepartamentdeMatemàtiques,UniversitatAutònomadeBarcelona,08193Bellaterra,Spain

cDepartmentofMathematicalSciences,NorwegianUniversityofScienceandTechnology,Trondheim, Norway

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

Articlehistory:

Received15January2020 Receivedinrevisedform24June 2020

Availableonline15July2020 CommunicatedbyS.Iyengar

MSC:

Primary55U30;55R35;secondary 13H10;13D45

Keywords:

Gorensteinduality Localcohomology Structuredringspectra p-compactgroups p-localfinitegroups

WeinvestigatewhenacommutativeringspectrumRsatisfiesahomotopicalversion oflocalGorensteinduality,extendingthenotionpreviouslystudiedbyGreenlees.

Inordertodothis,weproveanascenttheoremforlocalGorensteindualityalong morphisms ofk-algebras. Ourmainexamplesareof theformR=C^∗(X;k),the ring spectrumof cochainson a spaceX for a field k.In particular,weestablish localGorensteindualityincharacteristicpforp-compactgroupsandp-localfinite groupsaswellasfork=QandX asimplyconnectedspacewhichisGorenstein inthesenseofDwyer,Greenlees,andIyengar.

©2020TheAuthor(s).PublishedbyElsevierB.V.Thisisanopenaccessarticle undertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).

Contents

1. Introduction . . . . 2

2. Gorensteinringspectra . . . . 5

3. Gorensteinascent . . . 11

4. Examples . . . 15

5. LocalGorensteindualityforp-localcompactgroups . . . 20

References . . . 23

* Correspondingauthor.

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

https://doi.org/10.1016/j.jpaa.2020.106495

0022-4049/©2020TheAuthor(s). PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).

1. Introduction

Given aNoetheriancommutativelocalring(A,m,k),therearenumerousequivalentconditionsforwhen A isGorenstein.Inparticular,ifAhasKrulldimensionn,thenA isGorensteinifandonlyif

Extⁱ_A(k, A)∼=

k i=n

0 otherwise.

In the derived category D(A), this can be restated in terms of the derived hom as an equivalence RHomA(k,A) Σⁿk. Inspired by this, Dwyer, Greenlees, and Iyengar [19] introduced the notion of a Gorenstein ringspectrum.More specially,ifk isafieldand R acommutativering spectrum,amorphism R →k ofringspectra¹ (alwaysassumed to becommutative) is saidto be Gorensteinof shiftr ifthere is anequivalence Hom_R(k,R)Σ^rk forsomeintegerr.

One is particularly interested in the duality that the Gorenstein condition implies. Assume that R is a k-algebra. IfR → k is Gorenstein then, undersomeadditional orientablehypothesis and coconnective- ness, R automatically satisfies theproperty thatCellk(R) Σ^rHomk(R,k),where Cellk isthe k-cellular approximation to R, see Section2.2.As weshall see, thisis theanalogofthe classicalcharacterizationof Gorenstein rings as those commutativelocal NoetherianringsA of Krull dimensionn forwhich thelocal cohomology withrespecttomsatisfies

H_mⁱ(A)∼=

Im i=n 0 otherwise,

whereIm∼= Homk(A,k) denotestheinjectivehullofk.WhenevertheequivalenceCellk(R)Σ^rHomk(R,k) issatisfiedforamorphismR→kofringspectra,wesaythatR satisfiesGorensteinduality.Notehowever that,incontrasttothealgebraicsituation,R→kbeingGorensteindoesnotimplythatRsatisfiesGoren- steinduality,seeRemark2.11.Thestructuralimplicationsforπ_∗RwhenRsatisfiesGorensteindualityhave been investigatedpreviouslybyGreenleesand Lyubeznik[24].

Our examples will mostly come from ring spectra of the form R = C^∗(X;k), the ring spectrumof k- valued cochains on asuitable space X. If X =BG where Gis a finite group, then C^∗(BG;F_p) → F_p is always Gorenstein of shift 0, and C^∗(BG;Fp) satisfies Gorenstein duality of the same shift, even though the cohomology ring π_−∗C^∗(BG;Fp) ∼= H^∗(BG;Fp) is not Gorenstein in general [19, Section 10.3]. A consequenceofthestructuralimplicationsmentionedearlieristhatH^∗(BG;Fp) isCohen–Macaulayifand onlyifH^∗(BG;Fp) isGorenstein,aresultoriginally shownbyBensonand Carlson[1].

If A is acommutative local Gorenstein ring, then the localizationAp at any prime idealp ∈ Spec(A) is still Gorenstein. One way to see this is to use yet another interpretation of Gorenstein rings as those commutativelocalringswithfinite injectivedimensionasanA-module.Alternatively,weobserve thatifp hasdimensiond,thentheringAp islocalNoetherianofdimensionn−d,andGreenlees–Lyubeznik’s dual localization[24,Section2] canbeused toshowthat

H_p^∗(Ap)∼=Ip[n−d] (1.1)

where Ip is theinjectivehullofA/p,² andhenceAp isstillGorenstein.

Now supposethatR isaringspectrum.Aswewillexplainbelow,foranyp∈Spec^h(π_∗R),wecanform spectralversionsofthetermsintheequation(1.1);thatis,R_pforlocalizationatp,Γ_pRforlocalcohomology

1 Ifkisafield,wewillalsodenotebyktheEilenberg–MacLanespectrumHk.

2 Wenotethatby[32,Proposition3.77],IphasanaturalstructureasanAp-module,andisisomorphictotheinjectivehullof Ap/p.

ofR_p andT_R(I_p) forinjectivehulls,whereI_p istheinjectivehullof(π_∗R)/pandπ_∗(T_R(I_p))∼=I_p.Wethus saythatR→k satisfieslocalGorensteindualityofshiftaifthespectralversionof(1.1) holdsforallp,

ΓpRΣ^a+dTR(Ip).

AnoutcomeofthisequivalenceisthefactthatthehomotopyofthespectrumΓpRisdeterminedbyalgebraic informationinπ_∗(R),seeTheorem2.15andRemark2.16ThisspectralversionwasintroducedbyBarthel, Heardand Valenzuelain[10, Definition4.21] under theterminology ofabsolute Gorensteinduality. Fora formaldefinitionof localGorensteinduality,see Section2.3. TheconsequencesoflocalGorenstein duality fortheringπ_∗RarereviewedinTheorem2.15below.Forexample,itimpliesthattheringπ_∗Risgenerically Gorenstein,i.e.,thelocalizationofπ_∗RatanyminimalprimeidealisGorenstein.Infact,themainresultof [7] isthatC^∗(BG;F_p)→F_psatisfieslocalGorensteindualitywhenGisafinitegroup,ormoregenerallyfor certaincompactLiegroupswithanorientabilityproperty.InthiscontextthemodulesΓp(R) areunderstood as generalizationsoftheidempotent Rickardmodulesinthestable modulecategory StMod_G when Gisa finite group(see [5,Theorem 2]), whichwere firstused to classify thethicksubcategories of thecompact objectsofStMod_G [4],see Remark2.16.

Afundamentaldifferencebetweenthealgebraicandtopologicalsituationsisthatintopologywedonot knowingeneralthatGorensteindualityimplieslocalGorensteinduality.Thefirstobjectiveforthisworkis toidentifyconditionswherelocalGorensteindualityholds.Themaintechniquestodeterminewhetheraring spectrumR→kisGorensteinaretheGorensteinascentanddescenttheoremsofDwyer–Greenlees–Iyengar, see[27, Section19] forasummary,and[10] forascenttechniques inlocal Gorensteinduality.

Inspired by theGorenstein ascentof Dwyer–Greenlees–Iyengar, we provethe following ascent theorem forlocalGorenstein duality.

Theorem (Proposition 3.4).Let S −→^f R be a finite morphismof augmented k-algebras and Q = R⊗S k.

Assumethat thefollowingconditions aresatisfied:

(1) R→k andS→kare orientableGorenstein (Definition 2.4)of shiftr andsrespectively.

(2) Q→kis cosmall,i.e., Qisin thethicksubcategory inMod_Q generatedby k.

(3) R andS are dc-complete(Section 2.2).

Then, ifS satisfieslocal Gorensteinduality of shifts,thenR satisfieslocal Gorenstein dualityof shiftr.

Ourmainexamplescomefrom ringspectraoftheformC^∗(X;k) withemphasis onclassifyingspacesof compactLie groupsand itshomotopicalgeneralizations.Here thetechnicalconditionsof orientabilityand dc-completenessaresatisfiedautomaticallywhenX isasuitablynicespace.Inparticular,theyaresatisfied for spacesof Eilenberg–Moore type(EM-type),see Definition3.5. The previouspropositionspecializes to thefollowingstatement.

Theorem(Theorem3.12).Letg:Y →X beamorphismof spacesofEM-type(p-completeifthecharacter- istic of k is p) with fiberF. Supposethat H^∗(F;k) isfinite-dimensional,and that C^∗(X;k) isGorenstein of shifts.

(1) IfC^∗(Y;k)isGorensteinofshiftr,theng^∗:C^∗(X;k)→C^∗(Y;k)isrelativelyGorensteinofshifts−r.

(2) If,inaddition,C^∗(X;k)satisfieslocalGorensteindualityofshifts,thenC^∗(Y;k)satisfieslocalGoren- steinduality ofshift r.

Let G be a compact Lie group such that (ifp > 2) the adjoint representation of G is orientable, and consider a unitary embedding f:G → U(n). Taking g = Bf_p^∧ in the above theorem, one recovers the

result of Benson and Greenlees thatC^∗(BG;F_p) → F_p satisfies local Gorenstein duality of shift dim(G).

The sameresult(withnoorientabilityhypothesis)holds forthe classifyingspaceof ap-compactgroupG, that is, C^∗(BG,Fp) → Fp satisfies local Gorenstein dualityof shiftdimp(G), where dimp(X) denotes the Fp-cohomologicaldimensionofaspaceX.

It is not true in general that if G is a compact Lie group, then C^∗(BG;F_p) satisfies local Gorenstein duality ofshift dim(G). Asimple exampledescribed in[28] isgiven byG=O(2) atan oddprime,which satisfieslocalGorensteindualityofshift3,whiledim(G)= 1.However,thetriple(Ω(BO(2)^∧_p)),BO(2)^∧_p,id) is ap-compactgroup,even thoughΩ(BO(2)^∧_p)O(2)^∧_p. InfactΩ(BO(2)^∧_p)Ω(B(S³)^∧_p)(S³)^∧_p atodd primes,whichexplainstheshiftobtainedastheF_p-cohomologicaldimension.Moregenerally,wesaythata compact Liegroupisof p-compacttypeifthetriple (Ω(BG^∧_p),BG^∧_p,id) isap-compactgroup(thisistrue if and onlyif Ω(BG^∧_p) isFp-finite). Inthis case, C^∗(BG;Fp)→Fp will satisfyGorenstein dualityof shift dimp(Ω(BG^∧_p)) by theprevious resulton p-compact groups.Taking G=O(2) at an oddprimeas above, we havedim_p(Ω(BO(2)^∧_p))= 3,asexpected.

A common generalization of both p-compact groupsand compact Lie groupsis the notionof ap-local compactgroupofBroto,Levi,andOliver[14].Givensuchap-localcompactgroupG,thereisanassociated classifyingspace BG, andonecanask ifC^∗(BG;Fp)→Fp satisfieslocalGorenstein duality,or evenifitis Gorenstein. Wedo notknow theanswerto this questioninfullgenerality, howeverwe identifyconditions for this to occur in Section5. In thecase ofp-local finite groups[13] we deduce from work ofCantarero, Castellana,andMorales[16],thatC^∗(BG;Fp)→FpsatisfieslocalGorensteindualityofshift0.Insummary, we obtainthefollowingresults.

Theorem (Theorems 4.12 and 4.17 and Corollary 5.12).Let G be a p-local compact group of one of the following types:

(1) associatedtoaLie groupof p-compacttype, (2) Ω(BGp^∧)isap-compact group, or

(3) ap-localfinite group.

Then, C^∗(BG,Fp)→Fp satisfies local Gorenstein dualityof shiftdimp(Ω(BG^∧p))(Cases (1)and(2)) or0 (Case(3)),respectively.

Forexample,thisshowsthatO(2n) atanoddprimesatisfieslocalGorensteindualityofshiftn(2n+ 1), while dim(O(2n))=n(2n−1).

In the rational case, a stronger result holds. Using the fact thatalgebraic Noether normalization can be lifted to the spectrum level, we show that for any simple connected rational space with Noetherian cohomology, GorensteinimplieslocalGorensteindualityofthesameshift.

Theorem (Theorem 4.19).Let X be a simply connected rational space with Noetherian cohomology. If C^∗(X;Q)→QisGorenstein of shiftr,thenC^∗(X;Q)satisfieslocal Gorenstein dualityof shiftr.

Convention. Throughout thisdocument, all ringsand structuredringspectra will be assumed to be com- mutative, whichinthecaseofcommutativeringspectrameanscarryinganE_∞-ringstructure.

Acknowledgments.WewouldliketothanktheCRMforfundingavisitoftheauthorsthroughtheResearch in Pairs program. The first author was partially supportedby the DanishNational Research Foundation (DNRF92) and the European Union’sHorizon 2020 research and innovation programme undertheMarie Sklodowska-CuriegrantagreementNo. 751794,thesecondauthorwaspartiallysupportedbyFEDER-MEC grant MTM2016-80439-Pand acknowledges financial supportfrom theSpanish Ministry ofEconomy and

Competitivenessthroughthe“MaríadeMaeztu”ProgrammeforUnitsofExcellenceinR&D(MDM-2014- 0445), the third author was partially supported by the SFB 1085 “Higher Invariants”, and the first and fourthauthor would like to thank the Max Planck Institute forMathematics for itshospitality. The first andsecond authorwouldfurthermorelike to thanktheIsaacNewtonInstitute forMathematicalSciences, Cambridge, for support and hospitality during the programme Homotopy Harnessing Higher Structures viagrantnumberEP/R014604/1,whereworkonthis paperwasundertaken.Thisworkwassupportedby EPSRCgrantNo.EP/K032208/1.Finally,wethanktherefereefortheirhelpfulcomments.

2. Gorenstein ringspectra

InthissectionwefirstreviewthenotionsofGorensteinringspectrum,relativeGorensteinmorphism,and Gorensteinduality,asstudiedbyDwyer,Greenlees,andIyengar[19].Then,weprovearesultforrecognizing when certainmorphisms of ringspectra are relative Gorenstein, anduse this inthenext sectionto prove anascenttheorem forlocalGorensteinduality.

2.1. The Gorenstein condition

SupposeAisa(discrete)Noetheriancommutativelocalringwithresiduefieldk,thenitisatheorem of SerrethatAisregularifandonlyifkhasaresolutionoffinitelengthbyfreeA-modules.Fortheassociated map of Eilenberg–MacLane spectra HA → k, this implies that k is in the thick subcategory of Mod_HA generated byHAitself. This leadsmoregenerally tothedefinition ofaregularmorphism ofringspectra, wherewesaythatamorphismofringspectraR→kisregularifkisacompactR-module,i.e.,kisinthe thicksubcategoryofMod_R generatedbyR.However,aweakernotionofregularity isoftenuseful.

Definition 2.1.Letk be acommutativering spectrum(unlessotherwise stated,k is assumedto be afield throughout this paper). Amorphism of ringspectra R →k iscalled proxy-regular ifthere exists another R-moduleK called aKoszulcomplex, suchthatK isacompactR-module,K isinthethicksubcategory ofModR generated byk, andk isinthelocalizingsubcategory generatedbyK inModR.If K=R itself, thenwesaythatR→kiscosmall.IfK=k thenwesaythatR→kissmall.

Returning to thecommutative algebraexample,if A isa commutativelocal Noetherianring as above, thenHA→kisalwaysproxy-regular,wherewecantakeKtocorrespondtotheusualKoszulcomplex[19, Section5.1] associatedto asequenceofgeneratorsofthemaximalidealofA.

We recallthat acommutative Noetherianlocal ring A with residuefield k is Gorensteinif and only if Ext^∗_A(k,A) isaone-dimensional k-vector space. This leadsto thefollowing definition, whichis actually a specialcaseofamoregeneraldefinitiondue toDwyer–Greenlees–Iyengar,see[19,Proposition8.4].

Definition2.2. WesaythatamorphismR→kofringspectraisGorensteinofshiftaifitisproxy-regular and there is anequivalence HomR(k,R) Σ^akof k-modules. More generally,we saythat amorphism of ringspectraS→Risrelatively GorensteinofshiftaifHomS(R,S)Σ^aR.

2.2. Gorenstein duality

NotethatifRadmits thestructureofak-algebra,andR→k isGorenstein,we haveanequivalenceof k-modules

Hom_R(k, R)Σ^akΣ^aHom_R(k,Hom_k(R, k)), (2.3)

where the last equivalencefollows byadjunction. Oneimportantobservation ofDwyer–Greenlees–Iyengar is thatsinceHomR(k,R) hasanaction byE = HomR(k,k),if R→k isGorenstein, then Σ^akadmits the structureofarightE-module.Likewise,thesecondequivalenceof(2.3) equipsΣ^akwithanaprioridifferent right E-modulestructure.

Restrictingto thecaseofaugmented k-algebras,thisleadsto thefollowingdefinition.³

Definition 2.4. LetR be anaugmented k-algebra.IfR is Gorenstein, then itis orientableifthe two right E-actionsonΣ^akagree,i.e.,

HomR(k, R)Σ^aHomR(k,Homk(R, k)) is anequivalence ofrightE-modules.

Oneinteresting consequenceoftheGorenstein conditionisthe dualitythatitoftenimplies. Toexplain this, weintroduce somefurtherterminology. IfM is anR-module,we letCell^R_k(M) denote thek-cellular approximationofM;thatis,Cell^R_k(M) isinthelocalizingsubcategory inMod_Rgenerated byk,andthere isamorphismCell^R_k(M)→M thatinducesanequivalenceonHom_R(k,−).IftheringspectrumRisclear, we willusually justwriteCellk(M).Forexample,ifAis alocal Noetherianringwithresiduefieldk, then taking R=HA andM adiscrete A-module,we havethatπ_∗Cellk(HM) isthelocal cohomology H_m^∗(M) of M.

IfR→k isproxy-regular,thenk-cellularizationhasaparticularly simpleformula, namely

Cellk(M)HomR(k, M)⊗E k, (2.5)

where,aspreviously,E= HomR(k,k).Moreover,k-cellularapproximationissmashing,thatis,Cellk(M) (Cell_k(R))⊗RM for any M ∈ Mod_R. Proofs of these claims can be found in [27, Lemma 6.3] and [27, Lemma6.6].Wethen havethefollowing,see [27,Section18.B].

Lemma 2.6.Supposethat Risan augmentedk-algebrasuchthat R→k isorientableGorenstein ofshifta.

Then, there isanequivalenceof R-modules

Cell_k(R)Σ^aCellk(Homk(R, k)).

Proof. UsingtheGorensteinorientableconditionthere areequivalencesofrightE-modules HomR(k,Cellk(R))HomR(k, R)Σ^aHomR(k,Homk(R, k)).

By(2.5),applying−⊗Ek tothelatterequivalencegivesriseto anequivalence ofR-modules Cellk(R)Σ^aCellk(Homk(R, k)).

Definition 2.7.LetR be anaugmented k-algebra.WesaythatRhasGorenstein dualityofshifta ifthere is anequivalence ofR-modules

Cellk(R)Σ^aHomk(R, k).

3 WithouttheassumptionthatRisak-algebra,thedefinitionismorecomplicated,andinvolvesthenotionofMatlisliftofk, see[19,Section6].

Proposition 2.8. If R is acoconnective augmented k-algebra with π₀R ∼=k such that R →k isGorenstein orientable,then itsatisfiesGorenstein duality.

Proof. The R-moduleHomk(R,k) is k-cellularsinceHomk(R,k) is bounded below[19,Remark3.17]. We canthendeducefromLemma2.6thatCellk(R)Σ^aHomk(R,k).

Finally, we point out a useful way of recognizing when R → k is orientable Gorenstein. We need the followingdefinition[19,Section8.11].

Definition 2.9.A k-algebraR is said to satisfy Poincaréduality of dimension a ifthere is an equivalence R→Σ^aHomk(R,k) ofR-modules.

Lemma 2.10.Let R be an augmented k-algebra which is proxy-regular. If R satisfies Poincaré duality of dimensiona,thenitisorientableGorenstein ofshift a.If R isadditionallycosmallandcoconnective,with π₀R∼=k,thenthereverseimplication istrue.

Proof. By[19,Proposition8.12],RisGorensteinofshiftaandwegetorientabilitybyapplyingHomR(k,−) totheequivalenceRΣ^aHomk(R,k).

On the other hand, if R is also cosmall and coconnective with π0R ∼= k, then by Lemma 2.6 and Proposition2.8,wehavethat

RCellk(R)Σ^aCellk(Homk(R, k))Σ^aHomk(R, k), sothatRsatisfies Poincarédualityofdimensiona.

Remark2.11.Unlike inalgebra,if amap of commutativeringspectra R →k is Gorenstein, thenR does not necessarily satisfy Gorenstein duality. For example, let M be a non-oriented compact manifold, e.g., M=RP²,andtakeR=C^∗(M;Z/4) theringspectrumofZ/4-valuedcochainsonM.ThenRisGorenstein, see[26, Example11.2(ii)],butcannot satisfyGorensteinduality.

Indeed,wewillarguethatforaconnectedfiniteCW-complexX andadiscretecommutativeself-injective ringk,theringspectrumR=C^∗(X;k) satisfiesGorensteindualityifandonlyifXsatisfiesPoincaréduality with respect to k. Since M is notorientable, itdoes notsatisfy Poincarédualitywith respect to theself- injectiveringk=Z/4,thusC^∗(M;Z/4) cannotsatisfyGorensteinduality.

Inorder toprovetheclaim, wenote that,ontheonehand,X being afinite CW-compleximpliesthat C^∗(X;k) → k is cosmall [19, Section 5.5(1)], which shows that Cellk(R) R. On the other hand, the spectralsequence

Ext^∗_k(π_∗R, k) =⇒ π_∗Hom_k(R, k)

collapsesto anisomorphismπ_∗Homk(R,k)∼= Ext⁰_k(π_∗R,k). Therefore,there isanequivalence Cellk(R) Σ^aHomk(R,k) ifandonlyif

H^∗(X;k)∼=π_−∗(R)∼=π_−∗Cell_k(R)∼=π_−∗Σ^aHom_k(R, k)∼= Ext⁰_k(H^∗−a(X;k), k), i.e.,ifandonlyifX satisfiesPoincarédualitywithcoefficientsink.

2.3. Local Gorensteinduality

Classically,ifA isadiscrete commutativeGorensteinring,then so isthelocalizationA_p forany prime ideal p ∈ Spec(A). The proof involves the characterization of Gorenstein rings as those rings with finite

injective dimension, and so there is noobvious generalization to the caseof ring spectra. Rather, we will identifyconditionswhere thedualityconditionofDefinition2.7localizes.Forthisweneedtoexplainwhat wemeanbylocalizingaringspectrumRataprimeidealp∈Spec^h(π_∗R).Namely,following[11] or[10] we explain how, given any p∈Spec^h(π_∗R), wecandefine afunctor Γp: ModR →ModR,which isaspectral versionof classicallocal cohomology.

WebrieflydescribeonewaytoconstructΓp.ForanysuchpthereexistsaringspectrumRpwithhomotopy (π_∗R)p,andanaturalmorphismR→Rp,see[22, ChapterV.1] forexample.Byextensionofscalarsthere is a functor Mod_R → Mod_Rp sending M to M_p = M ⊗R R_p. We can then construct a Koszul object R_p//pinsideMod_Rp, see[10,Section3.1].Thelocalizing subcategory generated byR_p//pinsideofMod_Rp isdenotedMod^p_R^−tors_p ,thecategoryofp-localandp-torsionobjects.TheinclusionMod^p_R^−tors_p →ModRp has arightadjointΓ_V(p)(thisisequivalenttothelocalcohomologyfunctorconstructedbyGreenleesandMay in[25,Section3]).Wethen defineΓpM = Γ_V(p)Mp.

UndersomeconditionswecanidentifyCellk withalocal cohomologyfunctor.

Lemma 2.12.Let k be a field, and R a coconnective commutative augmented k-algebra. Assume that π_∗R is a Noetherian local ring and that the augmentation mapinduces an isomorphism π0R ∼= k∼= (π_∗R)/m.

Then, thefunctors Cellk andΓm are equivalent.

Proof. This isaconsequenceof theproof of[19, Proposition9.3].We sketchthedetailsfor thebenefitof thereader.FirstnotethatΓm= Γ_V(m).Itthensufficestoshow thatΓ_V(m)M Cellk(M).

Suppose now thatm= (x1,. . . ,xn). LetK_∞=K_∞(x1)⊗R· · · ⊗RK_∞(xn),where K_∞(xi) is thefiber of the map R → R[1/x_i]. By [25] there is an equivalence Γ_V(m)M = K_∞⊗RM. By the proof of [19, Proposition 9.3] theassumptions ofthe lemmagive riseto anequivalence K_∞⊗RM Cell_k(M) andwe are done.

Thus, forourlocal version ofGorenstein dualitywewill replaceCell_k(R) withΓ_pR. Thenextquestion is whattheanalogofHom_k(R,k) shouldbe ingeneral. Fromnow onwewriteIR= Hom_k(R,k).Suppose thatwe arestill undertheconditionsof Lemma2.12, then π_∗IR ∼=π_∗Homk(R,k)∼= Homk(π_∗R,k)∼=Im, theinjectivehullof(π_∗R)/m∼=k,see[32,Example3.41].ByBrownrepresentability, thereisanR-module spectrum TR(Im) such that π_∗TR(Im) ∼= Im. Then IR TR(Im). More generally, for any injective π_∗R- moduleI,as in[10, Section4] wecanconstructanR-moduleTR(I) suchthatπ_∗TR(I)∼=I.These spectra arecharacterized bythepropertythatforanyM∈Mod_R thereisanisomorphismofgradedπ_∗R-modules

π_∗Hom_R(M, T_R(I))∼= Homπ_∗R(π_∗M, I).

WeletIpdenotetheinjectivehullof(π_∗R)/p.ThespectraTR(Ip) areourlocalsubstitutesforIR.Together, we getthefollowing definition.

Definition 2.13.LetR bearingspectrum.WesaythatR satisfieslocal Gorensteindualitywith shiftaif, foreachp∈Spec^h(π_∗R) ofdimensiond,⁴ thereisanequivalenceΓpRΣ^a+dTR(Ip).

Remark 2.14.As we havediscussed, underthe conditionsof Lemma 2.12, this reduces to the Gorenstein dualitycondition

Cellk(R)Σ^aHomk(R, k) inthecasethatpisthemaximal idealm.

4 Thedimension(alsoknownasthecoheight)ofaprimeidealpistheKrulldimensionofπ_∗R/p.

Finally, we point out some properties of rings satisfying local Gorenstein duality. Here we denote the internalshiftfunctor ingradedmodulesbyΣ aswell.

Theorem 2.15.Let R be aring spectrum satisfying local Gorenstein duality of shift a.Then the following hold.

(1) Thereisanisomorphism ofR-modulesπ_∗ΓpR∼= Σ^a+dIp. (2) Thereisaspectralsequence

E₂^s,t∼=H_p^−s,t(π_∗R)p =⇒ πs+t−a−dTR(Ip)∼= Σ^a+d−s−tIp.

(3) π_∗R isgenericallyGorenstein, i.e.,thelocalization atany minimalprimeidealisGorenstein.

(4) Thereareno nontrivial R-modulephantom mapsintoΓ_pR.

Proof. (1) is an immediate consequence from the definition of local Gorenstein duality and the factthat π_∗TR(Ip) ∼= Ip. (2) follows from (1) and the local cohomology spectral sequence, see for example [10, Proposition3.19]. (3)follows from thespectral sequencein(2). Indeed,ifpis minimal,then thelocalized ring (π_∗R)_p is of dimension 0, and hence H_p^−s,t(π_∗R)_p = 0 whenever s = 0, and the spectral sequence collapses. For (4), it is shownin [9, Lemma3.2] that there are no nontrivial phantom maps into TR(Ip), andhencenophantommapsintoΓpR.

Remark2.16.Thefunctors Γ_p canbe definedmore generallyforany triangulated categorywith acentral actionofa(discrete)commutativeNoetherianringA.Inthecaseofthestablemodulecategory ofafinite group G over a field k, the ring A = H^∗(BG;k) is the group cohomology, and Γpk corresponds to the Rickardidempotent denotedκV in[4] forV anirreduciblesubvarietyof Spec^h(H^∗(BG;k)) corresponding top.Foradetaileddiscussionofthecomparison,seepage30of[11].Asshownin[7,Theorem2.4],theTate cohomology H^∗(G;κ_V)∼= Σ^dI_p, theinjectivehull ofH^∗(BG;k)/pinthe category ofH^∗(BG;k)-modules, shiftedbythedimensionofp.TheresultTheorem2.15(1)aboveisthespectral generalizationofthis,and showshowthehomotopyofthespectrumΓpR iscompletelydeterminedbythehomotopygroupsπ_∗R.

FinallyweintroduceausefulwayofidentifyingspectrasatisfyinglocalGorensteinduality.Thefollowing notionwasintroducedin[10,Definition4.5].

Definition2.17.AringspectrumR withπ_∗(R) localNoetherianofdimensionnisalgebraicallyGorenstein ofshiftaifπ_∗(R) isagradedGorensteinringofthesameshift.

Proposition2.18.[10,Proposition4.7] LetR bearingspectrum. IfRisalgebraicallyGorenstein ofshifta, thenR satisfies localGorenstein dualityof shifta.

2.4. The relative Gorensteincondition

LetR→kandS →k be morphismsofringspectra andS−→^f Rbe amorphismofringspectra overk.

Wehaverestrictionofscalarsf^∗: Mod_R→Mod_S withleftadjointf_∗ andrightadjointf_!.Notethatiff is relatively Gorenstein,thenf!(S)Σ^aR.Thepurposeofthis sectionistoidentifyconditionsguaranteeing thatf:S →Risrelatively Gorenstein.Firstweobservethefollowing:

Lemma2.19.LetR→kandS→kbeproxy-regularmorphismsofring spectra,andf:S→R bearelative Gorenstein ringmorphismoverk.ThenS isGorenstein ifand onlyif R isso.

Proof. Thisfollowsfromthefollowingidentitieswherethethirdoneholdsbecausef isrelativeGorenstein:

Hom_S(k, S)Hom_S(f^∗(k), S)Hom_R(k, f_!(S))Hom_R(k,Σ^aR)Σ^aHom_R(k, R).

At this point, we need to recall the notion of dc-completeness. There is a canonical morphism R → End_E(k)=R inducedfrom theR-moduleactiononk,and wesaythatR isdc-complete ifthismap isan equivalence. Recall that we write IR = Homk(R,k). We want to describe hypothesis on R and S which allowusto identifywhen amorphismisrelative Gorenstein.

Proposition 2.20.Suppose that R is an augmented k-algebra, and that R → k is orientable Gorenstein of shift a,then

Hom_R(Cell_k(R),IR)Σ^−aR.

Proof. By(2.5) andtheorientableGorensteincondition,thereisanequivalence Cell_k(R)Hom_R(k, R)⊗E kΣ^ak⊗E k.

BythedefinitionofIR,

HomR(Cellk(R),IR)Homk(Cellk(R), k).

SubstitutingforCellk(R),weseethat

Hom_R(Cell_k(R),IR)Hom_k(Σ^ak⊗E k, k)Σ^−aHom_E(k, k) = Σ^−aR, as required.

Werequiretwomorelemmas.Thefirstfollowsbyasimpleadjunctionargument.

Lemma 2.21.Let R → k and S → k be ring homomorphisms and S −→^f R be a morphism of ring spectra over k.Thereisan equivalenceofS-modules HomS(R,IS) IR.

Forthesecondlemma,weobservethatkisbothnaturallyanS-moduleandanR-module.Thefollowing comparesthek-cellularizationfunctorinthetwocategories.

Lemma 2.22.Let S −→^f R be amorphism of ring spectra over k and Q =R⊗S k. Assume that S → k is proxy-regular withKoszulobject K(S).Considerthefollowingconditions:

(1) Q→k iscosmall.

(2) R→k isproxy-regularwith KoszulobjectR⊗SK(S).

(3) Thereisan equivalenceofS-modules Cell^S_k(R)Cell^R_k(R).

Then (1) =⇒ (2) =⇒ (3).

Proof. That(1) =⇒ (2) is shownintheproof of[19,Proposition4.18(1)].

Assuming(2)then,wehavetheequivalencesCell^R_k(R)Cell^R_K(R)(R)Cell^R_R⊗SK(S)(R) andCell^S_k(R) Cell^S_K(S)(R). To show (2) =⇒ (3) it thus suffices to prove that Cell^R_R⊗SK(S)(R) Cell^S_K(S)(R) as S- modules.Accordingto[38,Lemma3.1(2)] thisfollowsifR⊗SK(S) isK(S)-cellularasanS-module.Since

thecategoryofS-modulesisgeneratedbyS,wededucethatR⊗SK(S)∈Loc_ModS(K(S)),i.e.,R⊗SK(S) isK(S)-cellularas anS-module,asneeded.

Wenowobtainourresultfordeducing thatamorphismf:S →Risrelatively Gorenstein.Forthis,we saythatamorphismf:R→S isfiniteifS iscompactas anR-module.

Proposition 2.23.Suppose that wehave a finite morphismof augmentedk-algebrasf:S →R over k, and letQ=R⊗Sk.Assumethat thefollowingconditionsare satisfied:

(1) R→k andS→kare orientableGorenstein of shiftrand srespectively.

(2) Q→kis cosmall.

(3) R andS are dc-complete.

Then, f isrelativelyGorenstein ofshift s−r.

Proof. The prooffollows [7, Theorem 7.3(m)],whichisactually aspecialcaseofthis theorem.Wehavea chainofequivalences

Hom_S(R, S)Σ^sHom_S(R,Hom_S(Cell^S_k(S),IS)) [Proposition 2.20and (3)]

Σ^sHomS((Cell^S_k(S))⊗SR,IS)

Σ^sHomS(Cell^S_k(R),IS) [Lemma2.5]

Σ^sHom_S(Cell^R_k(R),IS) [Lemma2.22]

Σ^sHomR(Cell^R_k(R),HomS(R,IS))

Σ^sHomR(Cell^R_k(R),IR) [Lemma2.21]

Σ^s−rR. [Proposition 2.20and (3)]

ThisisexactlytheclaimthatHomS(R,S) isrelatively Gorensteinofshifts−r.

3. Gorenstein ascent

In this section we describe ascent techniques which allow to construct new examples of ring spectra satisfying(local)Gorensteindualityfromknownexamples.

3.1. Ascent forGorensteinrings

LetR→kandS →k be morphismsofringspectra. Supposewearegiven amorphismf: S→Rover k withf relatively Gorenstein,then S is Gorensteinifand onlyifR is Gorenstein,see Lemma2.19.Now letQ=R⊗Sk.WeconsiderthesituationwheretwooutofR,S andQsatisfyGorensteinduality.Part(1) ofthefollowingwasalreadyshownin[19,Section8.6] or[27,Lemma19.3].

Theorem 3.1.Let S−→^f R be amorphismover k, andlet Q=R⊗S k.Suppose that thenatural morphism ν: HomS(k,S)⊗SR→HomS(k,R)isanequivalenceofS-modules,andthatoneofthefollowingconditions issatisfied:

(i) S→kisproxy-regular andQ→k iscosmall.

(ii) S→kissmall andQ→k isproxy-regular.

Then thefollowinghold.

(1) If Q→k and S → k are Gorenstein of shift q ands respectively,then R→ k is Gorenstein of shift s+q.

(2) If S → k and R →k are Gorenstein of shift s and r respectively, then Q →k is Gorenstein of shift r−s.

Proof. Either of the conditions implies that R → k is proxy-regular, see [19, Proposition 4.18], so that R → k is Gorensteinif and onlyif Hom_R(k,R) Σ^ak. Butby [19, Proposition 8.6] or [27, Lemma19.3]

theassumption onν impliesthatthere isanequivalenceofk-modules HomR(k, R)HomQ(k,HomS(k, S)⊗kQ).

Theresultfollows.

Remark3.2.Thenaturalmapν isanequivalenceif(butnotonlyif)eitherR orkaresmallasS-modules.

3.2. Ascentforlocal Gorensteinduality

In Section3.1we recalledtheGorenstein ascenttheorem ofDwyer–Greenlees–Iyengar, namelythatfor afinitemorphism S−→^f Roverk ifS and Q=R⊗Sk areGorenstein,thenso isR.In[10,Theorem 4.27], which we recallnow, we gaveacriterion forascentof local Gorensteinduality.Note thatinthe following we donotneedtoassumethatRandS are k-algebras.

Proposition 3.3. Let R and S be ring spectra. Suppose that S satisfies local Gorenstein duality of shift s and that f: S →R is a finite morphism and is relativelyGorenstein of shift r−s, then R satisfies local Gorenstein duality ofshift r.

Combined withProposition2.23wededucethefollowing.

Proposition 3.4. Let S −→^f R bea finite morphism of augmented k-algebras andQ=R⊗Sk. Assume that thefollowingconditions are satisfied:

(1) R→k andS→k areorientable Gorensteinof shiftr andsrespectively.

(2) Q→k iscosmall.

(3) R andS aredc-complete.

Then, if S satisfies localGorenstein dualityof shifts,then R satisfieslocal Gorensteinduality ofshift r.

3.3. Cochainalgebras

In this sectionwe specialize theresults ofthe previoussubsectionto ring spectra obtainedas cochains onspaces.WerecallthatwewriteC^∗(X;k) fortheringspectrumofk-valuedcochainsonX definedasthe functionspectrumHomSp(Σ^∞₊X,Hk).Inparticular,thereis anisomorphismπ_∗C^∗(X;k)∼=H^−∗(X;k).

It is important to have in mind that the object we are interested in is R = C^∗(X;k), not X itself, which means that we can have different spaces giving rise to the same ring spectrum R. For example, R=C^∗(BZ/p;F_q)C^∗(∗;F_q) ifpandqare coprime.

If kisafieldand X isaspace,then wedenotetheBousfieldk-completionof X byX_k^∧. IfX isk-good, then C^∗(X;k)C^∗(X_k^∧;k) andinthiscase wecanassumethatX isk-complete. For example,ifπ₁X is

finite,then X is k-good fork =Qand k=F_p forany prime p, and thereforeX_k^∧ is k-complete (see [12, I.5.2,VII.3.2,VII.5.1]).

GivenaspaceX,theringspectrumofcochainswillbewell behavedwhenitsatisfiescertainhypothesis whichwewillassumemostlythroughtherestofthepaper:

Definition 3.5.[19, Section 4.22] A space X is said to be of Eilenberg–Moore type (EM-type) if X is connected,H^∗(X;k) isoffinite type,and

(1) X issimplyconnectedwhen k=Q,or

(2) kisofcharacteristicpandπ1X isafinitep-group.

Remark 3.6.A space of EM-type is k-good and so we can assume always that X is k-complete when considering itsringspectrumofcochainswithcoefficientsink.

Weareinterestedinthesepropertiesforthefollowingreason.Supposewearegivenahomotopypullback squareofspaces

Y ×XZ Z

Y X.

IfX isofEM-type,thenthehomotopypullbackgivesrisetoanequivalence[34,Corollary1.1.10]

C^∗(Y ×XZ;k)C^∗(Y;k)⊗C^∗(X;k)C^∗(Z;k).

Inparticular,weobtain:

Lemma3.7. LetF →Y →X be afibersequenceof spaceswhereX isof EM-type.Then C^∗(F;k)C^∗(Y;k)⊗C^∗(X;k)k.

Undersome hypothesis on X, a morphism C^∗(X;k) →k thatis Gorenstein is also automatically ori- entable.

Lemma3.8. LetX beaconnectedspacesuch thatH^∗(X;k)isof finite type.Supposethat:

(1) X issimplyconnectedwithk=Q,or

(2) kisfield ofcharacteristicp,andπ1X isafinite p-group.

(3) kisafinite fieldof characteristicpandπ1X is apro-pgroup.

Then, if C^∗(X;k)→k isGorenstein,itis orientable.In particular,theconditions of thelemmaholdif X isaspaceof EM-type.

Proof. We first check that E = HomR(k,k) C_∗(ΩX;k). If X is simply connected with k = Q then it follows from the strong convergence of the Eilenberg–Moore spectral sequence. Otherwise, the action of π1X on H^∗(X;k) is nilpotent: if k is a finite fieldof characteristic p, the action factors then through a finitequotientwhichisap-groupsinceH^∗(X;k) isoffinitetype.Thenagainthestrongconvergenceofthe Eilenberg–Moorespectral sequenceshowsthatE = Hom_R(k,k)C_∗(ΩX;k).

WeshowthatEhasauniqueactiononk.Thisactionfactorsthroughπ₀E ∼=π₁XsincekisanEilenberg–

MacLanespectrum.Thecasewherekisofcharacteristicpandπ1X isafinitep-groupis[27,Lemma18.2], and followsbecauseE isak-algebra,and actsthroughπ0(E)∼=H0(ΩX)∼=k[π1X].Becauseπ1X isafinite p-group,andkhascharacteristicp,thisactionmustbe unique.

If, π1X is apro-pgroupand k is finite,then the action map factorsthrough afinite quotient of π1X, which is a finite p-group, and hence the result also follows inthis case. In the rational case, since X is simply connected,thesameargumentas [27,Lemma18.2] showsthatkhasauniqueE-modulestructure, and henceisorientablyGorenstein.

In the previous subsection we identified conditions to ascend local Gorenstein duality along a finite morphism. InlightofRemark3.2,weareinterestedinconditionswhichensurethattheinducedmorphism on cochainsforamapof spacesf: Y →X is finite.To thisend,wehavethefollowing,dueto Greenlees–

Hess–Shamir[23] intherationalcase,andBenson–Greenlees–Shamir[8] inthecharacteristicpcase.

Lemma 3.9. Letf:Y →X be a mapof spaces with homotopyfiber F andH^∗(F;k) finite dimensional.If k=Q,thenf:C^∗(X;k)→C^∗(Y;k) isfinite. If k has characteristicp,then thesame conclusionholds if additionally X andY are p-complete spaceswithfundamental groups finitep-groups.

Proof. Therationalcaseisprovedin[23,Lemma4.7],whilethecharacteristicpcaseis[8,Lemma3.4].

Wenow presentacochainversionof Theorem3.1.

Theorem3.10.Supposethatg:Y →X isamorphismofspacesofEM-type(p-completeifthecharacteristic of k is p) with fiberF, suchthat H^∗(F;k) isfinite-dimensional, and that C^∗(X;k)→k isproxy-regular.

Then thefollowinghold:

(1) If C^∗(F;k)→k andC^∗(X;k)→k are Gorensteinof shift q ands respectively, thenC^∗(Y;k)→k is Gorenstein ofshift s+q.

(2) If C^∗(Y;k)→k and C^∗(X;k)→k are Gorenstein of shift rand s respectively,then C^∗(F;k)→k is Gorenstein ofshift r−s.

Proof. By Lemmas 3.7 and 3.9 there is a finite morphism C^∗(X;k) −→^f C^∗(Y;k) and C^∗(F;k) C^∗(Y;k)⊗C^∗(X;k)k. Moreover, H^∗(F) finite-dimensional implies that C^∗(F;k) → k is cosmall [19, Sec- tion5.5(2)].WearethusinthesituationofTheorem3.1(i).

Remark 3.11.It worth pointing out that C^∗(Y;k) is Gorenstein if, for example, C^∗(F;k) is a Poincaré dualityalgebrabyTheorem3.10andLemma2.10.

We now specialize the results on relative Gorenstein duality and local Gorenstein duality to cochain algebras.

Theorem 3.12.Letg: Y →X beamorphismof spaces ofEM-type (p-completeif thecharacteristicof kis p) withfiberF.Supposethat H^∗(F;k)isfinite-dimensional,andthat C^∗(X;k)isGorenstein ofshift s.

(1) IfC^∗(Y;k)isGorensteinofshiftr,theng^∗:C^∗(X;k)→C^∗(Y;k)isrelativelyGorensteinofshifts−r.

(2) If,inaddition,C^∗(X;k)satisfieslocalGorensteindualityofshifts,thenC^∗(Y;k)satisfieslocalGoren- stein dualityof shiftr.