A TOUR OF SUPPORT THEORY FOR TRIANGULATED CATEGORIES THROUGH TENSOR TRIANGULAR GEOMETRY 6 1 GREGSTEVENSON 0 2 Abstract. Thesenotesattempttogiveashortsurveyoftheapproachtosup- n porttheoryandthestudyoflatticesoftriangulatedsubcategoriesthroughthe a machineryoftensortriangulargeometry. Onemainaimistointroducethema- J terialnecessarytostateandprovethelocal-to-globalprinciple. Inparticular, 4 wediscussBalmer’sconstructionofthespectrum,generalisedRickardidempo- 1 tentsandsupportforcompactlygeneratedtriangulatedcategories,andactions of tensor triangulated categories. Several examples are also given along the ] T way. ThesenotesarebasedonaseriesoflecturesgivenduringtheSpring2015 programon‘InteractionsbetweenRepresentationTheory,AlgebraicTopology C andCommutativeAlgebra’(IRTATCA)attheCRMinBarcelona. . h t a m [ Contents 1 v Introduction 2 5 1. The Balmer spectrum 2 9 5 1.1. Rigid tensor triangulated categories 3 3 1.2. Ideals and the spectrum 4 0 1.3. Supports and the Zariski topology 5 . 1 1.4. The classification theorem 8 0 1.5. An explicit example 11 6 2. Generalised Rickard idempotents and supports 12 1 2.1. Rigidly-compactly generated tensor triangulated categories 12 : v 2.2. Localising sequences and smashing localisations 13 i X 2.3. Generalised Rickard idempotents and supports 16 3. Tensor actions and the local-to-globalprinciple 20 r a 3.1. Actions and submodules 20 3.2. Supports 23 3.3. The local-to-globalprinciple and parametrising submodules 23 3.4. An example: commutative noetherian rings 28 4. Further applications and examples 30 4.1. Singularity categories of hypersurfaces 30 4.2. Representations of categories over commutative noetherian rings 34 References 36 I am grateful to the CRM for both the very pleasant and productive environment provided and the opportunity to give the course which led to these notes during the IRTATCA program. Thanksarealsoduetotheparticipantsinthecourseforhelpfulfeedbackandpointingoutseveral typographical misfortunesintheoriginalversionofthesenotes. 1 2 GREGSTEVENSON Introduction The aim of these notes is to give both an introduction to, and an overview of, certain aspects of the developing field of tensor triangular geometry and abstract supportvarieties. Weseektounderstandthecoarsestructure,i.e.latticesofsuitable subcategories, of triangulated categories. Put another way, given a triangulated categoryT and objects X,Y ∈T, we study the questionof when one canobtain Y from X by taking cones, suspensions, and (possibly infinite) coproducts. The notes are structured as follows. In the first section we review work of Paul Balmer in the case of essentially small (rigid) tensor triangulated categories. Here one can introduce a certaintopologicalspace, the spectrum, which solves the classificationproblemforthicktensorideals. Thespectrumistheuniversalexample of a so-called“supportvariety” and provides a conceptual frameworkwhich unites earlier results in various examples such as perfect complexes over schemes, stable categories of modular representations, and the finite stable homotopy category. In thissectionwefirstmeetthenotionofsupportsandlaythefoundationsfordefining supports in more general settings. Inthesecondsectionweturntotheinfinitecase,i.e.compactlygeneratedtensor triangulated categories. Following work of Balmer and Favi we use the compact objects and the Balmer spectrum to define a notion of support for objects of such categories. Along the way we discuss smashing localisations and the associated generalisedRickardidempotents which are the key to the definition of the support given in [BF11]. The third section serves as an introduction to actions of tensor triangulated categories. This allows us to define a relative version of the supports introduced in the second section; in this way we can, at least somewhat, escape the tyranny of monoidal structures. After introducing actions, the associated support theory, andoutliningsomeofthefundamentallemmasconcerningsupportsandactions,we come to the main abstractresult of this course - the local-to-globalprinciple. This theorem, which already provides new insight in the situation of Section 2, reduces the study oflattices of localisingsubcategoriesto the computationofthese lattices in smaller (and hopefully simpler) subcategories. Finally, the fourth section focusses on illustrating some applications of the ab- stract machinery from the preceding sections; there will be, of course, examples along the way, but in this section we concentrate on giving more details on some more recent examples where the machinery of actions has been successfully ap- plied. In particular, we discuss singularity categories of affine hypersurfaces and the corresponding classification problem for localising subcategories as well as ap- plicationstostudyingderivedcategoriesofrepresentationsofquiversoverarbitrary commutative noetherian rings. 1. The Balmer spectrum The main reference for this section is the paper [Bal05] by Paul Balmer. We follow his exposition fairly closely, albeit with two major differences: firstly, in order to simplify the discussion, and since we will not require it later on, we do not work in full generality, and secondly we omit many of the technical details. The interested reader should consult [Bal05], [Bal10] and the references within for further details. SUPPORT THEORY FOR TRIANGULATED CATEGORIES 3 This section mainly consists of definitions, but we provide several examples on the wayandultimately gettoBalmer’sclassificationofthick tensoridealsinterms of the spectrum. 1.1. Rigid tensor triangulated categories. Throughout this section K will de- note an essentially small triangulated category. We use lowercase letters k,l,m for objects of K and denote its suspension functor by Σ. We begin by introducing the main player in this section. Definition1.1. Anessentiallysmalltensortriangulatedcategory isatriple(K,⊗,1), where K is an essentially small triangulated category and (⊗,1) is a symmetric monoidal structure on K such that ⊗ is an exact functor in each variable. Slightly more explicitly, −⊗−: K×K−→K is a symmetric monoidal structure on K with unit 1 and with the property that, for all k ∈K, the endofunctors k⊗− and −⊗k are exact. Remark 1.2. Throughout we shall not generally make explicit the associativity, symmetry, and unit constraints for the symmetric monoidal structure on a tensor triangulated category. By standard coherence results for monoidal structures this will not get us into any trouble. Definition1.3. LetKbeanessentiallysmalltensortriangulatedcategory. Assume that K is closed symmetric monoidal, i.e. for each k ∈ K the functor k⊗− has a right adjoint which we denote hom(k,−). These functors can be assembled into a bifunctor hom(−,−) which we call the internal hom of K. By definition one has, for all k,l,m∈K, the tensor-hom adjunction K(k⊗l,m)∼=K(l,hom(k,m)), with corresponding units and counits η : l −→hom(k,k⊗l) and ǫ : hom(k,l)⊗k −→l. k,l k,l The dual of k ∈K is the object k∨ =hom(k,1). Given k,l∈K there is a natural evaluation map k∨⊗l −→hom(k,l), which is defined by following the identity map on l through the composite K(l,l) ∼ // K(l⊗1,l) K(l⊗ǫk,1,l) //K(k⊗k∨⊗l,l) ∼ // K(k∨⊗l,hom(k,l)). We say that K is rigid if for all k,l ∈ K this natural evaluation map is an isomor- phism k∨⊗l −∼→hom(k,l). Remark 1.4. If K is rigid then, given k ∈K, there is a natural isomorphism (k∨)∨ ∼=k, andthefunctork∨⊗−isbothaleftandarightadjointtok⊗−,thefunctorgiven by tensoring with k. Example 1.5. Let us provide some standard examples of essentially small rigid tensor triangulated categories: 4 GREGSTEVENSON (1) GivenacommutativeringRthecategoryDperf(R)ofperfect complexes i.e., those complexes in the derived category which are quasi-isomorphic to a bounded complex of finitely generated projectives, is symmetric monoidal L via the left derived tensor product ⊗ (which we will usually denote just R as ⊗ or ⊗ if the ring is clear) with unit the stalk complex R sitting in R degree 0. The category Dperf(R) is easily checked to be rigid. (2) LetGbeafinitegroupandkbeafieldwhosecharacteristicdividestheorder of G (this is not necessary but rules out trivial cases). We write modkG for the category of finite dimensional kG-modules. This is a Frobenius category,so its stable categorymodkG, whichis obtainedby factoringout mapsfactoringthroughprojectives,istriangulated(see[Hap88]forinstance for more details). Moreover, it is a rigid tensor triangulated category via the usual tensor product ⊗ with the diagonal action and unit object the k trivial representation k. (3) ThefinitestablehomotopycategorySHfin togetherwiththesmashproduct ofspectraisarigidtensortriangulatedcategorywithunitobjectthesphere spectrum S0. 1.2. Ideals and the spectrum. Fromthis pointonwardK denotes anessentially smallrigidtensortriangulatedcategorywithtensorproduct⊗andunit1. Formuch of what follows the rigidity assumption is overkill, but it simplifies the discussion in several places and will be a necessary hypotheses in the sections to come. We begin by recalling the subcategories of K with which we will be concerned. All subcategoriesof K are assumedto be full andreplete (i.e. closedunder isomor- phisms). The simple,but beautiful, idea is toview K asaverystrangesortofring. One takes thick subcategories to be the analogue of additive subgroups and then defines ideals and prime ideals in the naive way. Definition 1.6. A triangulated subcategory I of K is thick if it is closed under taking direct summands, i.e. if k⊕k′ ∈ I then both k and k′ lie in I. To be very explicit: a full subcategory I of K is thick if it is closed under suspensions, cones, and direct summands. A thick subcategory I of K is a (thick) tensor-ideal if given any k ∈ K and l ∈I the tensor product k⊗l lies in I. Put another way we require the functor K×I−⊗→K to factor through I. Finally, a proper thick tensor-ideal P of K is prime if given k,l ∈ K such that k ⊗ l ∈ P then at least one of k or l lies in P. We will often just call prime tensor-ideals in K prime ideals. Remark 1.7. Since we haveassumedK is essentiallysmallthe collectionsof thick subcategories,thick tensor-ideals, and prime tensor-ideals each form a set. Remark1.8. RigidityofKprovidesthefollowingsimplificationwhendealingwith tensor-ideals. Givenk ∈K the adjunctionsbetweenk⊗− andk∨⊗− imply thatk is a summand of k⊗k⊗k∨ and that k∨ is a summand of k⊗k∨⊗k∨. The former implies that every tensor-ideal is radical, i.e. if I is a tensor-ideal and k⊗n ∈I then k ∈I. The latter implies that every tensor-ideal is closed under taking duals. An important special case of the above discussion is the following: if k ∈ K is nilpotent, i.e. k⊗n ∼=0, then k∼=0. SUPPORT THEORY FOR TRIANGULATED CATEGORIES 5 Given a collection of objects S ⊆ K we denote by thick(S) (resp. thick⊗(S)) the smallest thick subcategory (resp. thick tensor-ideal) containing S. We use Thick(K)andThick⊗(K)respectivelyto denotethesets ofthicksubcategoriesand thick tensor-ideals of K. Both of these sets of subcategories are naturally ordered by inclusion and form complete lattices whose meet is given by intersection. Lemma 1.9. Suppose that K is generated by the tensor unit, i.e. thick(1) = K. Then every thick subcategory is a tensor-ideal: Thick(K)=Thick⊗(K). Proof. Exercise. (cid:3) Example1.10. ThelemmaappliestobothDperf(R)andSHfinwhicharegenerated bytheirrespectivetensorunitsRandS0. ItisnotnecessarilythecasethatmodkG isgeneratedbyitstensorunitk. However,ifGisap-groupthenthetrivialmodule k is the unique simple kG-module and thus generates modkG. The prime ideals, somewhat unsurprisingly, receive special attention (and nota- tion). Definition 1.11. The spectrum of K is the set SpcK={P⊆K|Pis prime} of prime ideals of K. We nextrecordsomeelementarybutcrucialfacts aboutprime idealsandSpcK. Proposition 1.12 ([Bal05, Proposition 2.3]). Let K be as above. (a) Let S be a set of objects of K containing 1 and such that if k,l ∈ S then k⊗l ∈ S. If S does not contain 0 then there exists a P ∈ SpcK such that P∩S =∅. (b) For any proper thick tensor-ideal I(K there exists a maximal proper thick tensor-ideal M with I⊆M. (c) Maximal proper thick tensor-ideals are prime. (d) The spectrum is not empty: SpcK6=∅. Remark 1.13. It is worth noting that, as in the common proof of the analogous statement in commutative algebra, the proof of this proposition appeals to Zorn’s lemma. 1.3. Supports and the Zariski topology. Wenowdefine,foreachobjectk ∈K, a subset of prime ideals at which k is “non-zero”. This is the central construction of the section, allowing us to both put a topology on SpcK and to understand Thick⊗(K) in terms of the resulting topological space. Definition 1.14. Let k be an object of K. The support of k is the subset suppk ={P∈SpcK|k ∈/ P}. We denote by U(k)={P∈SpcK|k∈P} the complement of suppk. 6 GREGSTEVENSON Let us give some initial intuition for the way in which one can think of k as being supported at P ∈ suppk. We can form the Verdier quotient K/P of K by P, whichcomes with a canonicalprojectionπ: K−→K/P. Since P is thick the kernel of π is precisely P and so, as k ∈/ P, the object π(k) is non-zero in the quotient; this is the sense in which k is supported at P. The reason this may, at first, look at odds with the analogous definition in commutative algebra is that one inverts morphisms in a triangulated category by taking such quotients. One can, at least in many cases, make a precise connection between the support defined above and the usual support of modules over a (graded) commutative ring. We now list the main properties of the support. Lemma 1.15 ([Bal05, Lemma 2.6]). The assignment k 7→ suppk given by the support satisfies the following properties: (a) supp1=SpcK and supp0=∅; (b) supp(k⊕l)=supp(k)∪supp(l); (c) supp(Σk)=suppk; (d) for any distinguished triangle k −→l −→m−→Σk in K there is a containment suppl⊆(suppk∪suppm); (e) supp(k⊗l)=supp(k)∩supp(l). Proof. We refer to Balmer’s paper for the details, where the corresponding results are proved for the subsets U(k). These properties, as stated above, appear in [Bal05, Definition 3.1]. However, we suggest proving these statements as doing so provides an instructive exercise. (cid:3) Another veryimportantpropertyofthe supportisthat itcandetect whetheror not an object is zero. Lemma 1.16. Given k ∈K we have suppk =∅ if and only if k ∼=0. Proof. By[Bal05,Corollary2.4]thesupportofk isemptyifandonlyifk istensor- nilpotenti.e., k⊗n ∼=0. As K is rigidthereareno non-zerotensor-nilpotentobjects by Remark 1.8. (cid:3) A fairly immediate consequence of Lemma 1.15 is that the family of subsets {suppk |k ∈K} form a basis of closed subsets for a topology on SpcK. Definition 1.17. The Zariski topology on SpcK is the topology defined by the basis of closed subsets {suppk |k∈K} The closed subsets of SpcK are of the form Z(S)={P∈SpcK|S∩P=∅} = supp(k) k\∈S where S ⊆K is an arbitrary family of objects. Before continuing to the universal property of SpcK and the classificationtheo- rem we discuss some topological properties of SpcK. SUPPORT THEORY FOR TRIANGULATED CATEGORIES 7 Proposition 1.18 ([Bal05, Proposition 2.9]). Let P∈SpcK. The closure of P is {P}={Q∈SpcK|Q⊆P}. In particular, SpcK is T i.e., for P ,P ∈SpcK 0 1 2 {P }={P } ⇒ P =P . 1 2 1 2 Proof. LetS =K\Pdenotethe complementofP. Itisimmediate thatP∈Z(S ) 0 0 and one easily checks that if P∈Z(S) then S ⊆S . Thus for any such S we have 0 Z(S )⊆Z(S), i.e. Z(S ) is the smallest closed subset containing P. This shows 0 0 {P}=Z(S )={Q∈SpcK|Q⊆P} 0 as claimed. The assertion that SpcK is T follows immediately. (cid:3) 0 Remark 1.19. This proposition is the first indication of the mental gymnastics that occur when dealing with SpcK versus SpecR for a commutative ring R. A wealth of further information on the relationship between prime ideals in R and prime tensor-ideals in SpcDperf(R) can be found in [Bal10]. In fact SpcK is much more than just a T space. 0 Definition1.20. AtopologicalspaceX isaspectralspace ifitverifiesthefollowing properties: (1) X is T ; 0 (2) X is quasi-compact; (3) the quasi-compact open subsets of X are closed under finite intersections and form an open basis of X; (4) every non-empty irreducible closed subset of X has a generic point. GivenspectralspacesX andY,aspectralmapf: X −→Y isacontinuousmapsuch that for any quasi-compact open U ⊆Y the preimage f−1(U) is quasi-compact. Typical examples of spectral spaces are given by SpecR where R is a commu- tative ring. In fact Hochster has shown in [Hoc69] that any spectral topological spaceisofthisform. Furtherexamplesincludethetopologicalspaceunderlyingany quasi-compact and quasi-separated scheme. Another class of examples is provided by the following result. Theorem 1.21 ([BKS07]). Let K be as above. The space SpcK is spectral. Thus the spaces we produce by taking spectra of tensor triangulated categories are particularly nice and enjoy many desirable properties. Aside 1.22. Thereis a deepconnectionbetweenspectralspacesandthe theory of particularly nice lattices (coherent frames to be precise). One consequence of this isthatthe topologyofaspectralspaceis determinedbythe specialisationordering onpoints. Thisorderingis givenbydefining, foraspectralspaceX,apointy ∈X to be a specialisation of x∈X if y is in the closure of x. Viewed through the lens oflattice theory,the keyfactwhichmakesBalmer’stheoryworksowellis thatthe lattice of thick tensor-ideals is distributive. More details on this point of view can be found in [KP13]. AsacomplementtoTheorem1.21letusrecordafewrelatedfactswhichappear in Balmer’s work. 8 GREGSTEVENSON Proposition 1.23. For any rigid tensor triangulated category K the following as- sertions hold. (a) Any non-empty closed subset of SpcK contains at least one closed point. (b) For any k ∈K the open subset U(k)=SpcK\supp(k) is quasi-compact. (c) Anyquasi-compactopensubsetofSpcKisoftheformU(k)forsomek ∈K. Proof. Both results can be found in [Bal05]: the first is Corollary 2.12, and the second two are the content of Proposition 2.14. (cid:3) In particular, it follows from the above proposition and Lemma 1.15 that the U(k) for k ∈K give a basis of quasi-compact open subsets for SpcK that is closed under finite intersections as is required in the definition of a spectral space. Remark 1.24. Recall, or prove as an exercise, that a space is noetherian (i.e. satisfiesthe descendingchainconditionforclosedsubsets)ifandonlyifeveryopen subset is quasi-compact. Combining this with (c) above we see that if SpcK is noetherian then every open subset is of the form U(k) for some k ∈ K and hence every closed subset is the support of some object. 1.4. The classification theorem. We now come to the first main result of these notes,namelytheabstractclassificationofthicktensor-idealsofKintermsofSpcK. The starting point to this story is an abstract axiomatisation of the properties of the support, based on Lemma 1.15. Definition1.25. Asupportdata on(K,⊗,1)isapair(X,σ)whereX isatopolog- ical space and σ is an assignment associating to each object k of K a closed subset σ(k) of X such that: (a) σ(1)=X and σ(0)=∅; (b) σ(k⊕l)=σ(k)∪σ(l); (c) σ(Σk)=σ(k); (d) for any distinguished triangle k −→l −→m−→Σk in K there is a containment σ(l)⊆(σ(k)∪σ(m)); (e) σ(k⊗l)=σ(k)∩σ(l). Amorphismofsupportdata f: (X,σ)−→(Y,τ)onKisacontinuousmapf: X −→ Y such that for every k ∈K the equality σ(k)=f−1(τ(k)) issatisfied. Anisomorphismofsupportdataisamorphismf ofsupportdatawhere f is a homeomorphism. It should, given the definition of the support, come as no surprise that the pair (SpcK,supp) is a universal support data for K. Theorem1.26([Bal05,Theorem3.2]). LetKbeasthroughout. Thepair(SpcK,supp) is the terminal support data on K. Explicitly, given any support data (X,σ) on K there is a unique morphism of support data f: (X,σ) −→ (SpcK,supp), i.e. a unique continuous map f such that σ(k)=f−1supp(k) SUPPORT THEORY FOR TRIANGULATED CATEGORIES 9 for all k ∈K. This unique morphism is given by sending x∈X to f(x)={k ∈K|x∈/ σ(k)}. Sketch of proof. Wegiveabriefsketchoftheargument. WehaveseeninLemma1.15 that the pair (SpcK,supp) is in fact a support data for K. By the same lemma it is clear that for x∈ X the full subcategory f(x), defined as in the statement, is a properthicktensor-ideal: itisproperby(a),closedundersummandsby(b),closed under Σ by (c), closed under extensions by (d), and a tensor-ideal by (e). To see thatitis prime suppose k⊗l lies inf(x). Byaxiom(e) fora supportdatawe have x∈/ σ(k⊗l)=σ(k)∩σ(l). and hence x must fail to lie in at least one of σ(k),σ(l) implying that one of k or l lies in f(x). The map f is a map of support data as f(x) ∈suppk if and only if k ∈/ f(x) if and only if x∈σ(k), i.e. f−1supp(k)={x∈X |f(x)∈suppk}={x∈X |x∈σ(k)}=σ(k). This also provescontinuity of f by definition of the Zariskitopology on SpcK. We leavethe unicity of f to the reader(or it canbe found as[Bal05, Lemma 3.3]). (cid:3) Before stating the promised classification theorem we need one more definition (we also provide two bonus definitions which will be useful both here and later). Definition 1.27. Let X be a spectralspace andlet W ⊆X be a subset ofX. We sayW isspecialisation closed ifforanyw∈W andw′ ∈{w}wehavew′ ∈W. That is, W is specialisationclosedif it is the union ofthe closuresofits elements. Given w,w′ as above we call w′ a specialisation of w. Dually, we say W is generisation closed if given any w′ ∈W and a w ∈X such that w′ ∈{w} then we have w ∈W. In this situation we call w a generisation of w′. A Thomason subset of X is a subset of the form V λ λ[∈Λ where each V is a closed subset of X with quasi-compact complement. Note λ that any Thomason subset is specialisation closed. We denote by Thom(X) the collection of Thomason subsets of X. It is a poset with respect to inclusion and in factalsocarriesthestructureofacompletelatticewherethejoinisgivenbytaking unions. Remark 1.28. As observed in Remark 1.24 if X is noetherian then every open subset is quasi-compact. Thus the Thomason subsets of X are precisely the spe- cialisation closed subsets. Remark1.29. ByProposition1.23foranyk ∈Kthesubsetsupp(k)isaThomason subset of SpcK, as is any union of supports of objects. Part (c) of the same proposition implies the converse, namely that any Thomason subset V can be written as a union of supports of objects of k. This observation explains the role of Thomason subsets in the following theorem. Theorem 1.30 ([Bal05, Theorem 4.10]). The assignments σ: Thick⊗(K)−→Thom(SpcK), σ(I)= supp(k) k[∈I 10 GREGSTEVENSON and τ: Thom(SpcK)−→Thick⊗(K), τ(V)={k∈K| supp(k)⊆V}, for I∈Thick⊗(K) and V ∈Thom(K) give an isomorphism of lattices Thick⊗(K)∼=Thom(K). Sketch of proof. By the previous remark and the properties of the support given in Lemma 1.15 both assignments are well defined, i.e. σ(I) is a Thomason subset and τ(V) is a thick tensor-ideal. It is clear that both morphisms are inclusion preserving, so we just need to check that they are inverse to one another. Consider the thick tensor-ideal τσ(I). It is clear that I ⊆ τσ(I). The equality is proved by showing that I= P=τσ(I). P∈\SpcK I⊆P We refer to [Bal05] for the details. On the other hand consider the Thomasonsubset στ(V). In this case it is clear that στ(V)⊆V. By the remark preceding the theorem we know V can be written as a union of supports of objects and so this containment is in fact an equality. (cid:3) Aside 1.31. ByStone duality the lattice Thom(SpcK)actually determinesSpcK. So by the theorem if we know Thick⊗(K) we can recover SpcK. In fact this gives another way of describing the universality of SpcK. To conclude we briefly explain what the spectrum is in each of our current running examples and then give a slightly more detailed treatment of a particular case. Example 1.32. Let R be a commutative ring. Then SpcDperf(R)∼=SpecR. This computation is due to Neeman [Nee92] (and Hopkins) in the case that R is noetherian and Thomason [Tho97] in general. Applying the above theorem, in combination with Lemma 1.9, tells us that thick subcategories of Dperf(R) are in bijection with Thomason subsets of SpecR. Example1.33. AspreviouslyletSHfin denotethefinitestablehomotopycategory. ThedescriptionofthethicksubcategoriesofSHfinbyDevinatz,Hopkins,andSmith [DHS88]allowsonetocomputeSpcSHfin. Anicepictureofthisspacecanbefound in [Bal10, Corollary 9.5]. Example 1.34. Let G be a finite group and k a field whose characteristic divides the order of G. By a result of Benson, Carlson, and Rickard [BCR97] we have SpcmodkG∼=ProjH•(G;k) i.e.thespectrumofthestablecategoryisthespaceunderlyingtheprojectivescheme associated to the group cohomology ring.