On support varieties and tensor products for finite dimensional algebras

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Journal of Algebra

www.elsevier.com/locate/jalgebra

On support varieties and tensor products for finite dimensional algebras

Petter Andreas Bergh,Mads Hustad Sandøy^∗, Øyvind Solberg

Instituttformatematiskefag,NTNU,N-7491Trondheim,Norway

a r t i c l e i n f o a bs t r a c t

Article history:

Received18June2019

Availableonline2December2019 CommunicatedbyMarkus Linckelmann

MSC:

16D20 16E40 16S80 16T05 18D10 18E30 81R50

Keywords:

Supportvarieties Tensorproducts

Quantumcompleteintersections

It has been asked whether there is a version of the tensor productpropertyforsupportvarietiesoverfinitedimensional algebrasdefinedintermsofHochschildcohomology.Weshow that in generalno such version can exist. Inparticular, we show thatfor certainquantumcompleteintersections,there aremodulesandbimodulesforwhichthevarietyofthetensor productisnotevencontainedinthevarietyoftheone-sided module.

©2019TheAuthors.PublishedbyElsevierInc.Thisisan openaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).

1. Introduction

In[11,12],Carlsonintroducedcohomologicalsupportvarietiesformodulesovergroup algebrasoffinitegroups,usingthemaximalidealspectrumofthegroupcohomologyring.

* Correspondingauthor.

E-mailaddresses:[email protected](P.A. Bergh),[email protected](M. Hustad Sandøy), [email protected](Ø. Solberg).

https://doi.org/10.1016/j.jalgebra.2019.10.059

0021-8693/©2019TheAuthors.PublishedbyElsevierInc. ThisisanopenaccessarticleundertheCC BYlicense(http://creativecommons.org/licenses/by/4.0/).

These varietiesbehave wellwith respect to thetypical operationssuchas directs sums and syzygies.Moreover, theyencode importanthomologicalinformation.For example, thedimensionof thesupportvarietyofamoduleequals thecomplexityof themodule.

Inparticular,thevarietyofamoduleistrivialifand onlyifthemoduleisprojective.

Shortlyafter thesecohomologicalsupportvarietieswereintroduced,it wasshownin [1] thatthevarietyofatensorproductofmodulesequalstheintersectionofthevarieties of themodules. This property iscommonly referred to as the tensor product property.

Asshownin[14],itholdsalso formodulesoverfinitedimensionalcocommutativeHopf algebras; for such algebras,there is a theory of support varieties generalizingthat for groups.Infact,onecandefinesupportvarietiesoveranyfinitedimensionalHopfalgebra, cocommutativeornot,usingtheHopfalgebracohomologyring.However,itisnotknown ifthiscohomologyringisfinitelygeneratedingeneral.Whatisknownisthatthetensor product property may or may not hold for non-cocommutative Hopf algebras having finitely generated cohomologyrings. Namely, asshown in[6,18,19], thereare examples of suchalgebras where the tensorproduct property holds, and exampleswhere it does not.

Why dowe careaboutthe tensor productproperty? There are several reasons.Not onlydoesitlook good;itindicatesthatthehomologicalbehaviorofatensorproduct is closelyrelated to each ofthe factors. Whentheproperty does nothold,some peculiar thingscanhappen; examplesin[6] show thatthetensorproduct oftwomodulesinone ordercanbe projective, butnon-projective intheotherorder. Anotherreasonwhythe tensor product property is of interest is that in many cases, it is connected with the classificationofthicksubcategories.ItisaningredientinBalmer’sclassificationofthick tensor idealsof tensor triangulated categories (cf. [2]), and anecessary consequence of Benson,Iyengarand Krause’sstratification approachin[4,5],as shownin[4, Theorem 7.3].Ingeneral,oneisofteninasituationwheresometriangulatedtensorcategory(where the tensor product is not necessarily symmetric) acts on a triangulated category, and where thelatter comes witha theoryof supportvarieties relative to somecohomology ring; this isstudied indetailin[10]. Ifthe appropriatetensor product property holds, thenitissometimesthecasethatthethicksubcategoriesare actuallytensorideals.

In [13,20,21], a theory of support varieties for arbitrary finite dimensional algebras was developed,using Hochschild cohomology rings. Forsuch an algebra A, there is in general no naturaltensor product between one-sided modules, as is the case for Hopf algebras.However,onecantensoranyleftA-modulewithabimodule,andobtainanew left A-module.It hastherefore been askedwhethersome versionof thetensor product propertyholds inthis setting.Inotherwords, givenabimoduleB and aleft A-module M,isthere anequality

V(B⊗AM) = V(B)∩V(M)

of supportvarieties?This does notimmediately make sense:how shouldwe define the supportvarietyofabimodule?Ifwejustusethesamedefinitionasforone-sidedmodules,

then thesupportvarietyofanybimodule whichisone-sidedprojectiveistrivial.Inthis case,thevarietyofthetensorproduct A⊗AM wouldbe V(M),whereasV(A)∩V(M) wouldalwaysbetrivial.However,asweexplainattheendofSection2,thereareactually severalpossiblemeaningfulwaysofdefiningasupportvarietytheoryforbimodules,using Hochschild cohomology.On theother hand,we showthatthe tensorproduct property cannever holdingeneral,regardlessofwhichbimoduleversionofsupportvarietytheory weuse.Infact,weshowinTheorem2.2thatwhenAisaquantumcompleteintersection of acertaintype,thenthere existsaleftA-moduleM andabimoduleB forwhich

V(B⊗AM)V(M)

Oneconsequenceofthefailureofsuchaninclusionisthatinthestablemodulecategory and theboundedderivedcategory ofA-modules,thereare thicksubcategories thatare nottensorideals.

2. Supportvarietiesandtensorproducts

Let usfirst recallthebasics onthe theoryof supportvarietiesforfinite dimensional algebras, usingHochschild cohomology.We onlygiveavery briefoverview;for details, we referthereaderto[13,20,21].

Let k be a field and A a finite dimensional k-algebra with radical r. All modules considered will be finitely generated left modules, and we denote thecategory of such A-modules bymodA. A bimodule over A is the samething as aleft moduleover the enveloping algebra A^e = A⊗k A^op, and the Hochschild cohomology ring of A is the graded ring

HH^∗(A) = ∞ n=0

Extⁿ_Ae(A, A)

withtheYonedaproduct.Thisringisgraded-commutative,andsoitsevenpartHH^2∗(A) iscommutativeintheordinarysense.NowletMandN beA-modules,andconsiderthe graded vectorspace

Ext^∗_A(M, N) = ∞ n=0

Extⁿ_A(M, N)

TheYonedaproductmakesthisintoagradedleftmoduleoverExt^∗_A(N,N),andagraded right moduleover Ext^∗_A(M,M).Sincefor everyL∈modA thetensorproduct −⊗AL induces ahomomorphism

ϕL: HH^∗(A)→Ext^∗_A(L, L)

ofgraded rings,we see thatExt^∗_A(M,N) becomesamoduleover HH^∗(A) intwo ways:

viathering homomorphismsϕN andϕM. However,thescalar multiplicationviathese tworinghomomorphismscoincideupto asign.

NowsupposethatH isagradedsubalgebraofHH^2∗(A).Thenforeverypair(M,N) of A-modules, we can define the support variety V_H(M,N) using the maximal ideal spectrumofH:

VH(M, N) ={m∈MaxSpecH|AnnH(Ext^∗_A(M, N))⊆m} Thereareequalities

V_H(M, M) = V_H(M, A/r) = V_H(A/r, M)

and we define this to be the support variety VH(M) of the single module M. These supportvarietiessharemanyofthepropertiesenjoyedbythecohomologicalsupportva- rietiesformodulesovergrouprings,inparticularwhenH isnoetherianandExt^∗_A(M,N) isafinitely generated H-module forallM,N ∈modA.If this isthe case,we saythat thealgebraAsatisfiesFgwithrespecttoH.Notethatby[21,Proposition5.7],the(even partofthe)Hochschildcohomologyringisuniversalwiththisproperty,inthefollowing sense:thealgebraAsatisfiesFgwithrespecttosomeH ⊆HH^∗(A) ifandonlyifHH^∗(A) isnoetherianandExt^∗_A(A/r,A/r) isafinitelygeneratedHH^∗(A)-module.

The finite dimensional algebras we shall study are of a very special form, namely quantumcompleteintersections.Thesearequantumcommutativeanaloguesoftruncated polynomialrings.Letusthereforefix somenotationthatweshallusethroughout.

Setup.(1)Fixanalgebraicallyclosedfieldk,togetherwithtwointegersc≥2 anda≥2.

(2)Defineaninteger¯aby

¯ a=

a if chark= 0

a/gcd(a,chark) if chark >0 andfixaprimitive¯athrootofunityq∈k.

(3)DenotebyA^c_q thequantumcomplete intersection

kx₁, . . . , x_c /(x^a₁, . . . , x^a_c,{x_ix_j−qx_jx_i}i<j)

Thisisalocalselfinjectivealgebraofdimensiona^c,andby[8,Theorem5.5] itsatisfies FgwithrespecttoHH^2∗(A^c_q).In[3],itwasshownthatonecanactuallydefinerankvari- etiesoverthisalgebra,andthatthesevarietiesbehaveverymuchlike therankvarieties for group algebras. It was then shown in [7] that these rank varieties are isomorphic to the support varieties one obtains by using asuitable polynomial subalgebra of the Hochschild cohomology ring. We now point out some factsabout this algebra and its supportvarieties.

Fact 2.1. (1) By[8, Theorem 5.3],the Ext-algebra Ext^∗_Ac

q(k,k) ofthe simplemodule k admits apresentation

kz1, . . . , zc, y1, . . . , yc /a where aistheidealgenerated bytherelations

⎛

⎜⎜

⎜⎜

⎜⎝

z_iz_j−z_jz_i for alli, j z_iy_j−y_jz_i for alli, j y_iy_j+qy_jy_i for alli > j y²_i for alliifa >2 y²_i −zi for alliifa= 2

⎞

⎟⎟

⎟⎟

⎟⎠

Here,thehomologicaldegreeofeachy_iisone,whereasthatofeachz_iistwo.Inparticular, the z_i generate apolynomialsubalgebra k[z₁,. . . ,z_c] over which Ext^∗_Ac

q(k,k) is finitely generated asamodule.

(2)Asexplainedin[7,Section2],itfollowsfrom[17,Corollary3.5] thattheimageof theringhomomorphism

ϕk: HH^2∗(A^c_q)→Ext^∗_Ac q(k, k)

isthewholepolynomialsubalgebrak[z1,. . . ,zc].Consequently,thereexistsapolynomial subalgebra k[η1,. . . ,ηc] of HH^2∗(A^c_q) with the following properties: each ηi is ahomo- geneous element inHH^2∗(A^c_q) of degreetwo with ϕk(ηi)=zi, and A^c_q satisfies Fgwith respect tok[η1,. . . ,ηc].

We now prove our main result. It shows that there exists an A^c_q-module M and a bimoduleBforwhichthesupportvarietyofthetensorproductB⊗A^c_qMisnotcontained inthesupportvarietyofM.

Theorem 2.2.Let k[η1,. . . ,ηc] be a polynomial subalgebra of HH^2∗(A^c_q) as in Fact 2.1.

Then foreverygradedsubalgebraH of HH^∗(A^c_q)with k[η₁, . . . , η_c]⊆H ⊆HH^2∗(A^c_q) thefollowinghold:

(1) thealgebra H is noetherian,and A^c_q satisfiesFgwith respecttoH;

(2) thereexistsan A^c_q-moduleM and abimoduleB withVH(B⊗A^c_qM)VH(M).

Proof. Letus simplifynotation abit and write A for ouralgebraA^c_q. Sinceit satisfies Fgwithrespecttok[η₁,. . . ,η_c],itfollowsfrom[13,Proposition2.4] thattheHochschild cohomology ringHH^∗(A) is finitely generatedas amoduleoverk[η₁,. . . ,η_c].Note that theassumption in[13,Proposition2.4] isthatFgholdswithrespect toagradedsubal- gebra ofHH^∗(A) whose degreezeropartcoincides withHH⁰(A), whichisthecenterof

A.Thisisnotthecaseforthepolynomialsubalgebrak[η1,. . . ,ηc],sincethecenterofA isnotofdimensionone.However,thisassumption isnotneededintheresult.

SinceHH^∗(A) isfinitely generatedasamoduleoverthenoetherianringk[η1,. . . ,ηc], the same is true for H, since this is a k[η1,. . . ,ηc]-submodule of HH^∗(A). Then H is noetherian as a ring, since it contains k[η₁,. . . ,η_c] as a subring. Moreover, since Ext^∗_A(k,k) isfinitely generatedoverk[η₁,. . . ,η_c],itmustalsobefinitelygenerated over thebiggeralgebraH.Thisproves (1).

Toprove(2),we firstshowthatwemaywithoutloss ofgeneralityassume thatH = k[η1,. . . ,ηc]. Todothis,consider theringhomomorphism

ϕk: HH^∗(A)→Ext^∗_A(k, k)

By Fact 2.1, the image of HH^2∗(A) is the polynomial subalgebra k[z1,. . . ,zc] of Ext^∗_A(k,k), and this is also the image of k[η1,. . . ,ηc]; after all, that is how we con- structed k[η1,. . . ,ηc] inthe first place.Therefore, since k[η1,. . . ,ηc] ⊆ H ⊆HH^2∗(A), weseethattheimageofk[η1,. . . ,ηc] isthesameasthatofH,namelyk[z1,. . . ,zc].Now takeany A-module X, and consider itssupport varietyV_H(X), which bydefinition is theset

{m∈MaxSpecH |Ann_H(Ext^∗_A(X, X))⊆m} By[20,Theorem3.2],thereisanequality

VH(X) ={m∈MaxSpecH |AnnH(Ext^∗_A(X, k))⊆m}

and so by [9, Proposition 3.6] the variety VH(X) is isomorphic to the set of maximal idealsofk[z1,. . . ,zc] containingtheannihilatorofExt^∗_A(X,k).HereweviewExt^∗_A(X,k) asaleftmoduleoverExt^∗_A(k,k),andinthiswayitbecomesamoduleoverthesubalgebra k[z₁,. . . ,z_c].Theisomorphismrespectsinclusionsofvarieties,andthisprovestheclaim.

Inlightof theabove,we now takeH =k[η₁,. . . ,η_c]. Sincek is algebraicallyclosed, wemayidentifythemaximalidealspectrumofH withtheaffine spacek^c. Forapoint λ= (λ1,. . . ,λc) ink^c,wedenotethecorrespondingmaximalideal(η1−λ1,. . . ,ηc−λc) inH bymλ,and whenλis nonzerowe denotethecorrespondingline

{(γλ1, . . . , γλc)|γ∈k} ink^c by λ. Moreover, wedenote the element c

i=1λixi inA byuλ, andby F(λ) the point (λ^a₁,. . . ,λ^a_c) in k^c. By [7, Proposition 3.5], the support variety VH(Auλ) of the cyclicA-moduleAuλ equals_F(λ),thatis,thereisanequality

VH(Auλ) =

m_γF(λ)|γ∈k

={(η1−γλ^a₁, . . . , ηc−γλ^a_c)|γ∈k} NotethatF(λ)= 0 ifandonlyifλ= 0.

Now take any point μ = (μ1,. . . ,μc) in k^c with μi = 0 for all i, and consider the automorphism ψμ:A →A given byxi →μixi. Whathappens to thecyclic A-module Auλ when we twist it by this automorphism? In general, for an A-module X and an automorphism ψ ofA, thetwistedmodule ψX is thesameas X as avectorspace, but for w∈A andx∈X thescalarmultiplication isw·x=ψ(w)x.Nowdenote thepoint (μ⁻¹₁ λ₁,. . . ,μ⁻¹_c λ_c) in k^c byμ⁻¹λ,andconsider themap

Au_μ−1λ→ψμ(Auλ) wu_μ−1λ→ψ_μ(w)u_λ

Note thatsince u_μ−1λ = ψ⁻¹_μ (uλ), this map is obtained bysimply applying ψμ to the elements in Au_μ−1λ.It is k-linear, andfor everyelement v ∈A and wu_μ−1λ ∈Au_μ−1λ

there areequalities

ψμ

v·(wu_μ−1λ)

=ψμ

vwu_μ−1λ

=ψ_μ(u)ψ_μ(w)u_λ

=u·(ψ_μ(w)u_λ)

ThusthemapisanA-homomorphism.Similarly,theinverseautomorphismψ_μ⁻¹induces anA-homomorphismintheotherdirection,henceAu_μ−1λ andψμ(Auλ) areisomorphic A-modules.Using[7,Proposition3.5] again,wenow seethatV_H

ψμ(Au_λ)

equals the line _F(μ−1λ).

Twisting anA-module X by an automorphism ψ is the sameas tensoring with the bimoduleψA1,i.e.ψXψA1⊗AX.Therefore,withλandμasabove,thesupportvariety VH

ψμA1⊗AAuλ

istheline_F(μ−1λ).Ontheotherhand,thesupportvarietyVH(Auλ) istheline_F(λ),whichgenericallydiffersfrom_F(μ−1λ).Forexample,withλ= (1,. . . ,1), any μ whose components are not all the same when raised to the ath power will do.

Consequently,forthisλandsuchaμ,weseethatV_H

ψμA₁⊗AAu_λ

V_H(Au_λ). 2 Asaconsequenceofthetheorem,therecannotexistabimoduleversionofthetensor product propertyforsupportvarietiesoverthealgebraA^c_q.

Corollary 2.3. Let H,M and B be as in Theorem 2.2, and suppose that V_H^b is some supportvarietytheory onthecategoryofA^c_q-bimodules, definedintermsofthemaximal ideal spectrumof H.ThenV_H(B⊗A^c_qM)= V^b_H(B)∩V_H(M).

ForafinitedimensionalalgebraA,thereareactuallyseveralpossiblewaysofdefining supportvarietiesforbimodules.Namely,takeanycommutativegradedsubalgebraH of HH^∗(A).ForabimoduleB,wecanviewExt^∗_Ae(B,A) asaleftmoduleoverHH^∗(A),and inthisway itbecomesanH-module.Wecanthendefine

V^b_H(B) ={m∈MaxSpecH|AnnH(Ext^∗_Ae(B, A))⊆m}

Similarly, we can use the fact that Ext^∗_Ae(A,B) is a right module over HH^∗(A) and obtainanothersupportvariety.These typesofone-sidedsupportvarietieswerestudied in [9], where it was shownthat they satisfy many of the properties oneexpects for a meaningfultheoryofsupport.

NowsupposethatwetakeabimoduleB whichisprojectiveasaleftA-module.Then ifwetakeanyexactsequenceη ofbimodules,thesequenceη⊗ABremainsexact.Thus weobtainaringhomomorphism

HH^∗(A)→Ext^∗_Ae(B, B) η →η⊗AB

ofgradedrings,andwecandefine

V^b_H(B) ={m∈MaxSpecH |AnnH(Ext^∗_Ae(B, B))⊆m}

Similarly,ifB is projectiveas arightA-module, weobtainaversionbytensoring with B onthe left. Consequently, for bimodules which are projective as both left and right A-modules,therearetotallyatleastfourwaysofdefiningsupportvarietiesusingH,and thereisingeneralnoreasonto expectthemto beequivalent.

Suppose now that A is a finite dimensional selfinjective algebra satisfying Fg with respect to some subalgebra H of its Hochschild cohomology ring. We then ask: what are the consequences of having a tensor product formula for bimodules acting on left modules?Inordertoinvestigatethis,assumethat

V_H(B⊗AM) = V^b_H(B)∩V_H(M)

forallBinatensorclosedsubcategory_X ofbimodulesandallleftA-modulesM,where VH istheusualsupportvarietytheoryonleftmodulesandV^b_H issomesupportvariety theoryforbimodulesin X (definedintermsofthesamegeometricspaceasVH,namely themaximalidealspectrumofH).Then

V^b_H(B₁⊗AB₂)∩V_H(M) = V_H((B₁⊗AB₂)⊗AM)

= VH(B1⊗A(B2⊗AM))

= V^b_H(B₁)∩V_H(B₂⊗AM)

= V^b_H(B1)∩V^b_H(B2)∩VH(M)

= V^b_H(B₂)∩V^b_H(B₁)∩V_H(M)

= VH(B2⊗A(B1⊗AM))

= V_H((B₂⊗AB₁)⊗AM)

= V^b_H(B2⊗AB1)∩VH(M)

forallB1and B2 inX ogallleftA-modulesM.Thenweclaimthattheequality V^b_H(B1⊗AB2) = V^b_H(B2⊗AB1)

holds for all bimodules B₁ and B₂ in _X. To see this, choose M = A/r, where r is the radical of A. Then VH(M) is thewholedefining maximal idealspectrum ofH, so that V^b_H(B1⊗AB2) = V^b_H(B2⊗AB1). Hence, one consequence is that the bimodule supportvarietyV^b_H must be independentofthe orderof thetermsinatensorproduct of bimodules,and therefore forcing some typeof symmetry on the tensor products of bimodules in_X.

Letη: Ωⁿ_Ae(A)→ArepresentahomogeneouselementinH,whereΩⁿ_Ae(A) isthenth syzygy inaminimal projective resolution of Aover A^e. Taking thepushout alongthis homomorphismand theminimal projectiveresolutionofAover A^e givesrisetoashort exactsequence

0→A→M_η →Ωⁿ⁻¹_Ae (A)→0

asdefinedin[13].ThebimodulesMηforhomogeneouselementsηinHhavethefollowing property

V_H(M_η1⊗A· · · ⊗AM_ηt⊗AM) = V_H(η₁, . . . , η_t )∩V_H(M).

Ifthere isasupportvarietyV^b_H ofbimodules suchthat

V^b_H(M_η1⊗A· · · ⊗AM_ηt) = V(η₁, . . . , η_t ), then V^b_H mustinparticularsatisfy

V^b_H(M_η1⊗AM_η2) = V^b_H(M_η2⊗AM_η1).

Forexample,letV^b_H(B)= VH(B⊗AA/r) forabimoduleB. Thenitfollowsthat V^b_H(Mη1⊗A· · · ⊗AMηt) = VH(η1, . . . , ηt )

for allhomogeneouselements ηi inH,and V^b_H satisfies theabovesymmetrycondition.

Since

Ext^∗_A(B⊗AA/r, A/r)Ext^∗_Ae(B,Hom_A(A/r, A/r)) Ext^∗_Ae(B, A/r⊗kA/r) Ext^∗_Ae(B, A^e/radA^e)

asH-modules,andA/r⊗kA/rA^e/radA^ewhenA/risseparableoverthefieldk,then applying similarargumentsas in[20] weobtainthat

V^b_H(B) = V(AnnHExt^∗_Ae(B, A^e/radA^e))

= V(Ann_HExt^∗_Ae(B, B))

= V(AnnHExt^∗_Ae(A^e/radA^e, B)).

Inotherwords,adaptingthenotionfrom [20],

V^b_H(B) = V^b_H(B, A^e/radA^e) = V^b_H(B, B) = V^b_H(A^e/radA^e, B).

Then itis naturalto askhow we can/shouldchoose_X. Ifwe arethinkinginterms of subcategories of the stable category of bimodules,can we choose X to be the tensor closed subcategory generated bythe bimodules Mη for allhomogeneous elements η in H?IfallM_η’sarein X,wedonotknowhowM_η1⊗AM_η2 andM_η2⊗AM_η1 arerelated asbimodules ingeneral.

LetusnowreturntoourquantumcompleteintersectionA^c_q.Corollary2.3,whichisa directconsequenceofTheorem 2.2,showsthatthetensor productpropertyforsupport varietiesover this algebracannothold ingeneral,now matter how onedefinessupport varieties for bimodules. Another consequence of Theorem 2.2 is thatnot all the thick subcategories of thederivedcategory andthe stable module category ofA^c_q are tensor ideals. Inorder to explain this, let us first briefly describe ageneral framework where one typically is interested in such questions; for details, we refer to [10]. Let C be a triangulatedtensorcategory,thatis,atriangulatedcategorywhichisatthesametimea (possiblynon-symmetric)tensorcategory,andwherethetwostructuresarecompatible.

Furthermore,supposethatC actsonatriangulatedcategoryD.Thismeansthatthere existsanadditivebifunctor

C ×D→D (C, D)→C∗D

which is compatible in a natural way with the structures of both C and D. Finally, suppose thatH isacommutative gradedsubalgebra ofthe graded endomorphismring End^∗_C(I) of the unit object I in_C, or,more generally, thatthere exists a ring homo- morphismH →End^∗_C(I).Thenfor allobjectsD1,D2 ∈D,thegraded homomorphism groupHom^∗_D(D1,D2) becomesaleftandarightH-module,andleftandrightscalarmul- tiplicationcoincide upto asign. One canthen define the supportvariety V_H(D₁,D₂) as usual, in terms of the variety of the annihilator ideal Ann_H(Hom^∗_D(D₁, D₂)). For asingle object D ∈ D, onedefines the supportvariety by VH(D) = VH(D,D). If H is Noetherian and the graded H-modules Hom^∗_D(D1,D2) are finitely generated for all objectsD1 andD2 in D,thenoneobtainsameaningfultheoryofsupportvarieties.

Given any triangulated category, it is of great interest to classify its thicksubcate- gories. Thefirst example of such aclassification wasthe celebrated result of Hopkins- Neeman,forthe categoryof perfectcomplexes overacommutativenoetherianring(cf.

[15,16]).Thatparticular classificationresult showedforfreethatall thethicksubcate- goriesareactuallythicktensorideals.Nowgiven C and D as above,onemayaskfora similarclassificationofthicksubcategoriesof D,andwhetherthesearealltensorideals.

Here,thenotionoftensoridealsin_Dreferstotheactionof_C on_D:athicksubcategory A ⊆D isatensor idealifC∗A∈A forallC∈C andA∈A.

Suppose thatV isaclosedhomogeneous subvarietyofMaxSpecH,and defineafull subcategory AV of D by

AV ={D∈D|V_H(D)⊆V}

Thisisathicksubcategoryof D,andthereareseveralclassesofexamplesoftriangulated categorieswhereallthethicksubcategoriesareofthisform.Forexample,thisisthecase for the category of perfectcomplexes over acommutative noetherianring. Thecrucial pointnowisthatwheneverVH(C∗D)⊆VH(D) forallobjectsC∈C andD∈D,then AV isautomaticallyathicktensoridealforallV. Thisindicatestheimportanceof the inclusionproperty

V_H(C∗D)⊆V_H(D)

for supportvarieties inthesetting of atriangulated tensor category actingon atrian- gulated category.

NowconsiderourquantumcompleteintersectionA=A^c_qagain.Thisisaselfinjective algebra,andsothestablemodulecategorymodAistriangulated.Theenvelopingalgebra A^eisalsoselfinjective,anditsstablemodulecategorymodA^e,thatis,thestablemodule category ofA-bimodules,is atriangulated tensorcategory. It acts onmodA bytensor products over A, and so we are in a setting where all of the above applies. However, let H,M and B be as in Theorem 2.2. Since VH(B ⊗AM) VH(M), not all thick subcategories of modA can be tensor ideals. Namely, take V = V_H(M) and define AV as above. This is a thick subcategory of modA, butit is not atensor ideal since M ∈ AV butB⊗AM /∈AV. Finally,note thatthebimodule B we used in theproof of Theorem 2.2 is actually projective as a left and as aright A-module. Thebounded derivedcategoryofsuchbimodulesisalsoatriangulatedtensorcategory,anditactson theboundedderivedcategory D^b(modA) ofA-modules.ThusalsoinD^b(modA) there are thicksubcategoriesthatarenottensorideals.

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