THE STABLE DERIVED CATEGORY OF A NOETHERIAN SCHEME HENNING KRAUSE Abstract. For a noetherian scheme, we introduce its unbounded stable deri* *ved cate- gory. This leads to a recollement which reflects the passage from the bou* *nded derived category of coherent sheaves to the quotient modulo the subcategory of pe* *rfect com- plexes. Some applications are included, for instance an analogue of maxim* *al Cohen- Macaulay approximations, a construction of Tate cohomology, and an extens* *ion of the classical Grothendieck duality. 1.Introduction Let X be a separated noetherian scheme and denote by Qcoh X the category of q* *uasi- coherent sheaves on X. We consider the derived category D(Qcoh X) and two full * *sub- categories Dperf(coh X) Db(coh X) D(Qcoh X) which are of particular interest. Here, Db(coh X) denotes the bounded derived c* *ategory of coherent sheaves, and Dperf(coh X) denotes the subcategory of perfect comple* *xes. Now let InjX be the full subcategory of injective objects in Qcoh X, and deno* *te by K(InjX) its homotopy category. The composite Q: K(InjX) _inc//_K(Qcoh X)can//_D(Qcoh X) gives rise to a localization sequence Q S(Qcoh X) __I_//_K(InjX)___//_D(Qcoh X) where S(Qcoh X) denotes the full subcategory of all acyclic complexes in K(InjX* *). Thus Q induces an equivalence K(InjX)=S(Qcoh X) -~! D(Qcoh X). Next we`recall that an object X in some category with coproducts is compact i* *f every map X ! iYi into an arbitrary coproduct factors through a finite coproduct. * *For instance, an object in D(Qcoh X) is compact if and only if it is isomorphic to * *a perfect complex. It is well known that the derived category D(Qcoh X) is compactly gene* *rated, that is, there is a set of compact objects which generate D(Qcoh X) [26]. To fo* *rmulate our main result, let us denote by Kc(InjX) and Sc(Qcoh X) the full subcategorie* *s of compact objects in K(InjX) and S(Qcoh X) respectively. Theorem 1.1. Let X be a separated noetherian scheme. ____________ 2000 Mathematics Subject Classification. Primary: 14F05, 18E30; Secondary: 16* *E45, 16G50, 55U35. Version from March 2004. 1 2 HENNING KRAUSE (1)The functors I, Q have left adjoints I~, Q~ and right adjoints Ij, Qj res* *pectively. We have therefore a recollement S(Qcoh X) ____//_K(InjX)___//_oo_oo_D(QcohoX).o_oo_ (2)The triangulated category K(InjX) is compactly generated, and Q induces an equivalence Kc(InjX) ! Db(coh X). (3)The sequence Q~ I~ D(Qcoh X) _____//K(InjX)____//_S(Qcoh X) is a localization sequence. Therefore S(Qcoh X) is compactly generated, * *and I~ OQj induces (up to direct factors) an equivalence Db(coh X)=Dperf(coh X) -~! Sc(Qcoh X). Note that this theorem is a special case of a general result about Grothendie* *ck cat- egories. All we need is a locally noetherian Grothendieck category A, for inst* *ance A = Qcoh X, such that D(A) is compactly generated. There is a surprising conseq* *uence which seems worth mentioning. Corollary 1.2. Let X be a separated noetherian scheme. Then a product of acycl* *ic complexes of injective objects in Qcoh X is acyclic. We call the category S(Qcoh X) the stable derived category of Qcoh X. A first* * system- atic study of the bounded stable derived category Db(coh X)=Dperf(coh X) can be found in work of Buchweitz [12]. Unfortunately this beautiful paper has * *never been published; see however [11]. He identifies for a Gorenstein ring the bo* *unded derived category of finitely generated -modules modulo perfect complexes Db(mod )=Dperf(mod ) with the stable category of maximal Cohen-Macaulay -modules and with the categ* *ory of acyclic complexes of finitely generated projective -modules. The same ident* *ification appears in [30] for selfinjective algebras and plays an important role in modul* *ar repre- sentation theory of finite groups; see also [23]. The approach in the present p* *aper differs from that of Buchweitz substantially because we work in the unbounded setting a* *nd we use injective objects instead of projectives. This has some advantages. For ins* *tance, we have in any Grothendieck category enough injectives but often not enough projec* *tives. On the other hand, we obtain a recollement in the unbounded setting which does * *not ex- ists in the bounded setting. In fact, the celebrated theory of maximal Cohen-Ma* *caulay approximations [2] is described as `decomposition' [2] or `glueing' [12], but f* *inds a nat- ural interpretation as `recollement' in the sense of [6] if one passes to the u* *nbounded setting. To be precise, the recollement oIæo_ oQæo_ S(Mod )____//_K(Injo)___//_o_D(Modoo_) I~ Q~ induces for any Gorenstein ring the Gorenstein injective approximation functor I~OQæ Z0 T :Mod can--!D(Mod ) ----! S(Mod ) --! Mod__ THE STABLE DERIVED CATEGORY OF A NOETHERIAN SCHEME 3 where Mod__ denotes the stable category modulo injective objects. For any -mo* *dule A, the Gorenstein injective approximation A ! T A is the `dual' of the maximal * *Cohen- Macaulay approximation which is based on projective resolutions. Let us stress * *again that this approach generalizes to any locally noetherian Grothendieck category * *A pro- vided that D(A) is compactly generated. Next we explain the connection between Gorenstein injective approximations an* *d Tate cohomology. We fix a locally noetherian Grothendieck category A and pass from * *the stable derived category S(A) to the full subcategory T(A) of totally acyclic co* *mplexes. An object in A is by definition Gorenstein injective if it is of the form Ker(X* *0 ! X1) for some X in T(A). The inclusion J :T(A) ! K(InjA) has a left adjoint J~. Give* *n an object A in A with injective resolution iA, we may think of J~iA as a complete * *injective resolution of A. This leads to the following definition of Tate cohomology grou* *ps dExtnA(A, B) = Hn Hom K(A)(A, n(J~iB)) for any A, B in A and n 2 Z. This cohomology theory is symmetric in the sense t* *hat * * 0 for any A in A, we have dExtA(A, -) = 0 iff dExtA(-, A) = 0 iff dExtA(A, A) = 0* *. Let X * denote the class of all objects A such that dExtA(A, -) vanishes, and let Y be * *the class of Gorenstein injective objects in A. Theorem 1.3. Let A be a locally noetherian Grothendieck category and suppose th* *at D(A) is compactly generated. (1)X = {A 2 A | Ext1A(A, B) = 0 for all B 2 Y}. (2)Y = {B 2 A | Ext1A(A, B) = 0 for all A 2 X }. (3)Every object A in A fits into exact sequences 0 ! YA ! XA ! A ! 0 and 0 ! A ! Y A ! XA ! 0 in A with XA , XA in X and YA , Y A in Y. (4)X \ Y = InjA. After explaining some historical backround, let us mention more recent work o* *n stable derived categories. For instance, Beligiannis develops a general theory of `sta* *bilization' in the framework of relative homological algebra [4], and Jørgensen studies the* * category of `spectra' for a module category [19]. Also, Orlov discusses the category Db(coh X)=Dperf(coh X) under the name `triangulated category of singularities' and points out some con* *nection with the Homological Mirror Symmetry Conjecture [28]. In any case, our notation S(Qcoh X) reflects this terminology. Our main results suggests that the homotopy category K(InjX) deserves some mo* *re attention. We may think of this category as the `compactly generated completion* *' of the category Db(coh X). In fact, the category cohX of coherent sheaves carries a na* *tural DG structure and its derived category Ddg(coh X) is equivalent to K(InjX). This f* *ollows from Keller's work [22] and complements a recent result of Bondal and van den B* *ergh [9] which says that D(Qcoh X) is equivalent to Ddg(A) for some DG algebra A. As another application of our main result, let us mention that the adjoint pa* *ir of functors Rf* and f! which establish the Grothendieck duality for a morphism f :* *X ! Y between schemes [16, 26], can be extended to a pair of adjoint functors between* * K(InjX) and K(InjY). 4 HENNING KRAUSE Theorem 1.4. Let f :X ! Y be a morphism between separated noetherian schemes. Denote by Rf*: D(Qcoh X) ! D(Qcoh Y) the right derived direct image functor and* * by f! its right adjoint. Then there is an adjoint pair of functors ^Rf* and bf!ma* *king the following diagram commutative. Rf* f! D(Qcoh X) __________//_D(QcohOY)O D(Qcoh Y) __________//_D(QcohOX)O |Q~| |Q| Qæ|| |Q| fflffl| R^f* | fflffl| fb! | K(InjX) ____________//_K(InjY) K(InjY) ____________//_K(InjX) Again, this theorem is really a lot more general. It is irrelevant that the * *functor f* comes from a morphism f :X ! Y. All we need is that f* and its right derived functor Rf* preserve coproducts. On the other hand, there is a strengthened ver* *sion of Theorem 1.4 which uses the special properties of f*. I am grateful to Amnon Nee* *man for pointing out the following. Theorem 1.5 (Neeman). Let f :X ! Y be a morphism between separated noetherian schemes. Then ^Rf* sends acyclic complexes to acyclic complexes. Thus we have a* *n ad- joint pair of functors between S(Qcoh X) and S(Qcoh Y), making the following di* *agrams commutative. Q S(Qcoh X) _____I____//_K(InjX)_________//_D(Qcoh X) |Sf*||| R^f*|| Rf*|| fflffl|fflffl|I fflffl| Q fflffl| S(Qcoh Y) __________//_K(InjY)_________//_D(Qcoh Y) Iæ Qæ S(Qcoh Y) oo_________K(InjY) oo_________D(Qcoh Y) || ! |b |! |Sf| f!| f| fflffl|fflffl|Iæ fflffl| Qæ fflffl| S(Qcoh X) oo_________K(InjX) oo_________D(Qcoh X) It seems an interesting project to study the functor Sf*, for instance to fin* *d out when it is an equivalence. The following result demonstrates the geometric content * *of this question; it generalizes a result of Orlov for the bounded stable derived categ* *ory [28]. Theorem 1.6. Let Y be a seperated noetherian scheme of finite Krull dimension. * * If f :X ! Y denotes the inclusion of an open subscheme which contains all singular* * points of Y, then Sf*: S(Qcoh X) ! S(Qcoh Y) is an equivalence. Having stated some of the main results, let us sketch the outline of this pap* *er. The paper deals with locally noetherian Grothendieck categories and covers therefor* *e various applications, for instance in algebraic geometry or representation theory. Thus* * we fix a locally noetherian Grothendieck category A and study the recollement (1.1) S(A) _____//K(InjA)____//_oo_oo_D(A).oo_oo_ More specifically, we begin in Section 2 with the basic properties of the homot* *opy category K(InjA). The recollement (1.1) is established in Sections 3 and 4. In * *Section 5, we discuss the essential properties of the stable derived category S(A). Then w* *e extend THE STABLE DERIVED CATEGORY OF A NOETHERIAN SCHEME 5 derived functors in Section 6, and Section 7 is devoted to studying Gorenstein * *injective approximations and Tate cohomology. An appendix provides additional material ab* *out DG categories. 2. The homotopy category of injectives We fix a locally noetherian Grothendieck category A. Thus A is an abelian Gro* *then- dieck category and has a set A0 of noetherian objects which generate A, that is* *, every object in A is a quotient of a coproduct of objects in A0. We denote by noethA * *the full subcategory formed by the noetherian objects in A, and InjA denotes the full su* *bcate- gory of injective objects. Note that InjA is closed under taking coproducts. We write K(A) for the homotopy category and D(A) for the derived category of unbounded complexes in A; for their definitions and basic properties, we refer * *to [32]. We do not distinguish between an object in A and the corresponding complex concent* *rated in degree zero in the homotopy category K(A). The inclusion noethA ! A induces a fully faithful functor Db(noeth A) -! D(A) which identifies Db(noeth A) with the full subcategory of objects X in D(A) suc* *h that HnX is noetherian for all n and HnX = 0 for almost all n 2 Z. In this section we study the basic properties of the homotopy category K(InjA* *). We shall see that this category solves a completion problem for the triangulated c* *ategory Db(noeth A). Let us begin with some elementary observations. Lemma 2.1. Let A be an object in A and denote by iA an injective resolution. Th* *en the natural map (2.1) Hom K(A)(iA, X) -! Hom K(A)(A, X) is an isomorphism for all X in K(InjA). Therefore iA is a compact object in K(I* *njA) if A is noetherian. Proof.Denote for any n 2 Z by oe>n X the truncation satisfying ( Xp ifp > n, (oe>n X)p = 0 ifp < n. We complete the map A ! iA to an exact triangle aA -! A -! iA -! (aA) and obtain 0 = Hom K(A)(aA, oe>-1 X) ~=Hom K(A)(aA, X) since aA is acyclic and concentrated in non-negative degrees. Thus Hom K(A)(iA,* * X) ~= Hom K(A)(A, X). Now assume that A is noetherian. Clearly, A is a compact object in A and ther* *efore a compact object in K(A). The isomorphism (2.1) shows that iA is a compact obje* *ct in K(InjA). Lemma 2.2. Let X be a non-zero object in K(InjA). Then there exists a noetherian object A in A such that Hom K(A)(A, nX) 6= 0 for some n 2 Z. 6 HENNING KRAUSE Proof.Suppose first HnX 6= 0 for some n. Choose a noetherian object A and a map A ! ZnX inducing a non-zero map A ! HnX. We obtain a chain map A ! nX which induces a non-zero element in Hom K(A)(A, nX). Now suppose HnX = 0 for all n. We can choose n such that ZnX is non-injective. Using Baer's criterion, there exists a noetherian object A in A such that Ext1A* *(A, ZnX) is non-zero. Now observe that Hom K(A)(A, n+pX) ~=ExtpA(A, ZnX) for all p > 1. Thus Hom K(A)(A, n+1X) 6= 0. This completes the proof. Let T be a triangulated category with arbitrary coproducts. Recall that an ob* *ject X in T is compact if Hom T (X, -) preserves all coproducts. The triangulated cat* *egory is compactly generated if there is a set T0 of compact objects such that Hom T(X, * * nY ) = 0 for all X 2 T0 and n 2 Z implies Y = 0 for every object Y in T . Proposition 2.3. Let A be a locally noetherian Grothendieck category, and let K* *c(InjA) denote the full subcategory of compact objects in K(InjA). (1)The triangulated category K(InjA) is compactly generated. (2)The canonical functor K(A) ! D(A) induces an equivalence Kc(InjA) -~! Db(noeth A). Proof.It follows from Lemma 2.1 and Lemma 2.2 that K(InjA) is compactly generat* *ed. A standard argument shows that Kc(InjA) equals the thick subcategory of K(InjA) which is generated by the injective resolutions of the noetherian objects in A;* * see [25, 2.2]. The equivalence K+ (InjA) ! D+ (A) restricts to an equivalence K+,b(InjA)* * ! Db(A) and identifies Kc(InjA) with Db(noeth A). Note that we obtain a functor Db(noeth A) ! K(InjA) which identifies Db(noeth* * A) with the full subcategory of compact objects. Therefore the formation of the ca* *tegory K(InjA) solves a completion problem which we explain by an analogy. The categor* *y A is a completion of noethA in the following sense. o A is an additive category with filtered colimts. o The inclusion noethA ! A identifies noethA with the full subcategory of f* *initely presented objects. o A coincides with the smallest subcategory which contains all finitely pre* *sented objects and is closed under forming filtered colimits. Recall that an object X in A is finitely presented if the functor Hom A(X, -) p* *reserves filtered colimits. Similarly, we have the following for T = K(InjA). o T is a triangulated category with coproducts. o The functor Db(noeth A) ! T identifies Db(noeth A) with the full subcateg* *ory of compact objects. o T coincides with the smallest subcategory which contains all compact obj* *ects and is closed under forming triangles and coproducts. The category A is, up to an equivalence, uniquely determined by noethA. It woul* *d be interesting to know to what extent K(InjA) is uniquely determined by Db(noeth A* *). Example 2.4. Suppose there is a noetherian object A in A such that Db(noeth A) * *is generated by A, that is, there is no proper thick subcategory containing A. Ta* *ke an THE STABLE DERIVED CATEGORY OF A NOETHERIAN SCHEME 7 injective resolution iA and denote by EndA (A) the endomorphism DG algebra of i* *A. Then Hom A (iA, -) inducs an equivalence between K(InjA) and the derived catego* *ry Ddg(End A(A)) of DG EndA (A)-modules; see [22]. If one replaces a single genera* *tor by a set of generating objects, then one obtains an analogue which involves a DG c* *ategory instead of a DG algebra. In particular, noethA carries the structure of a DG ca* *tegory such that K(InjA) and Ddg(noeth A) are equivalent. We refer to Appendix A for d* *etails. Example 2.5. Let G be a finite p-group and k be a field of characteristc p > 0. We consider the category A = Mod kG of modules over the group algebra kG. Take an injective resolution ik of the trivial representation k, and denote by EndkG* * (k) the endomorphism DG algebra of ik. Then its derived category Ddg(End kG(k)) is equi* *valent to K(InjA). The tensor product k on A restricts to a product on InjA and induc* *es therefore a (total) tensor product on K(InjA). On the other hand, the E1 -struc* *ture of EndkG(k) induces a product on Ddg(End kG(k)). We conjecture that these products* * are naturally isomorphic. Example 2.6. Let be a finite dimensional algebra over a field k. Then E = Hom k( op, k) is an injective cogenerator for A = Mod , and Hom (E, -) indu* *ces an equivalence InjA ! ProjA since Hom (E, E) ~= . Thus the homotopy category K(ProjA) is compactly generated. For more on K(ProjA), see [19, 20]. 3. A localization sequence Let A be a locally noetherian Grothendieck category and let Kac(InjA) = K(InjA) \ Kac(A). In this section we prove that the canonical functors I :Kac(InjA) inc--!K(InjA) and Q: K(InjA) inc--!K(A) can--!D(A) form a localization sequence Q (3.1) Kac(InjA) __I_//_K(InjA)____//D(A). Let us start with some preparations. In particular, we need to give the definit* *ion of a localization sequence. Definition 3.1. We say that a sequence _F__//___G__//00 T 0 T T of exact functors between triangulated categories is a localization sequence if* * the following holds. (L1)The functor F has a right adjoint Fj: T ! T 0satisfying FjO F ~=IdT 0. (L2)The functor G has a right adjoint Gj: T 00! T satisfying G OGj ~=IdT 00. (L3)Let X be an object in T . Then GX = 0 if and only if X ~= F X0 for some X02 T 0. The sequence (F, G) of functors is called colocalization sequence if the sequen* *ce (F op, Gop) of opposite functors is a localization sequence. The basic properties of a localization sequence are the following [32, II.2]. (1)The functors F and Gj are fully faithful. 8 HENNING KRAUSE (2)Identify T 0= Im F and T 00= Im Gj. Given objects X, Y 2 T , then X 2 T 0 () Hom T (X, T 00) = 0, Y 2 T 00 () Hom T (T 0, Y ) = 0. (3)Identify T 0= Im F . Then the functor G induces an equivalence T =T 0! T * *00. (4)Let X be an object in T . Then there is an exact triangle (F OFj)X -! X -! (GjO G)X -! ((F OFj)X) which is functorial in X. (5)The sequence Gæ Fæ T 00____//T____//_T 0 is a colocalization sequence. The next lemma is well known; it provides useful criteria for a sequence to b* *e a localization sequence. Recall that a full subcategory of a triangulated categor* *y is thick if it is a triangulated subcategory which is closed under taking direct factors. Lemma 3.2. Let T be a triangulated category and S be a thick subcategory. Then* * the following are equivalent. (1)The sequence S inc--!T -can-!T =S is a localization sequence. (2)The inclusion functor S ! T has a right adjoint. (3)The quotient functor T ! T =S has a right adjoint. Proof.Condition (1) implies (2) and (3). Also, (2) and (3) together imply (1). * *Thus we need to show that (2) and (3) are equivalent. Let us write F :S ! T and G: T ! * *T =S for the functors which are involved. (2) ) (3): Let T 00denote the full subcategory of objects Y in T such that Ho* *m T(S, Y ) = 0. We obtain a left adjoint L: T ! T 00for the inclusion by completing for each* * X in T the natural map FjX ! X to an exact triangle FjX -! X -! LX -! (FjX). The functor L annihilates S and factors therefore through G via an exact functor ~L:T =S ! T 00. The composite of ~L with the inclusion T 00! T is a right adj* *oint for G. (3) ) (2): We obtain a right adjoint Fj: T ! S for the inclusion F by complet* *ing for each X in T the natural map X ! (GjO G)X to an exact triangle FjX -! X -! (GjO G)X -! (FjX). Note that FjX belongs to S since G(FjX) = 0. We need to construct left and right adjoints for functors starting in a compa* *ctly generated triangulated category. Our basic tool for this is the following resul* *t which is due to Neeman. Proposition 3.3. Let F :S ! T be an exact functor between triangulated categor* *ies, and suppose S is compactly generated. (1)There is a right adjoint T ! S if and only if F preserves all coproducts. (2)There is a left adjoint T ! S if and only if F preserves all products. THE STABLE DERIVED CATEGORY OF A NOETHERIAN SCHEME 9 Proof.For (1), see [26, Theorem 4.1]. The proof of (2) is analogous and uses co* *variant Brown representability [27, Theorem 8.6.1]; see also [24]. We record a similar result for later use. Proposition 3.4. Let T be a compactly generated triangulated category and S0 b* *e a set of objects in T . Denote by U the full subcategory of objects Y in T suc* *h that Hom T( nX, Y ) = 0 for all X 2 S0 and n 2 Z. Then the inclusion U ! T has a le* *ft adjoint. Proof.The localizing subcategory S generated by S0 is well generated and the in* *clusion S ! T has therefore a right adjoint; see [27]. We obtain a localization sequenc* *e S inc--! T can--!T =S by Lemma 3.2, and the right adjoint of the canonical functor T !* * T =S identifies T =S with U. There is a useful criterion when a left adjoint preserves compactness. Lemma 3.5. Let F :S ! T be an exact functor between compactly generated triang* *u- lated categories which has a right adjoint G. Then F preserves compactness if a* *nd only if G preserves coproducts. Proof.See [26, Theorem 5.1]. The following result establishes the localization sequence for the homotopy c* *ategory of injective objects. Proposition 3.6. Let A be a locally noetherian Grothendieck category. Then the * *canon- ical functors Kac(InjA) ! K(InjA) and K(InjA) ! D(A) form a localization sequen* *ce Q Kac(InjA) __I_//_K(InjA)____//D(A). Proof.We know from Proposition 2.3 that K(InjA) is compactly generated. In addi* *tion, we use Lemma 3.2 and Proposition 3.3. The inclusion J :K(InjA) ! K(A) preserves products and has therefore a left adjoint J~ satisfying J~ OJ ~=IdK(InjA). We o* *btain a localization sequence _inc_// _J~_//_ K K(A) K(InjA) where K denotes the kernel of J~. Thus Hom K(A)(X, Y ) = 0 for allX 2 K and Y 2 K(InjA). This implies K Kac(A) and gives the following commutative diagram of exact fu* *nctors. _____inc____//_ _____J~___//_ K K(A) K(InjA) |inc| |||| |F| fflffl| inc || can fflffl| Kac(A) __________//_K(A)___________//D(A) The functor F is induced by the canonical functor K(A) ! D(A), and we have F ~=Q since J~ OJ ~= IdK(InjA). Moreover, F preserves coproducts and has therefore a* * right adjoint Fj. The composite J OFj is a right adjoint for the canonical functor K(* *A) ! D(A). This implies F OFj ~=IdD(A). On the other hand, Kac(InjA) is the kernel o* *f F . Thus we conclude that the sequence (3.1) is a localization sequence. 10 HENNING KRAUSE We add some useful remarks which are immediate consequences. Remark 3.7. Let J~: K(A) ! K(InjA) be the left adjoint of the inclusion K(InjA)* * ! K(A). Then the composite Q OJ~ is naturally isomorphic to the canonical functor K(A) ! D(A). Remark 3.8. The right adjoint Qj of Q induces an equivalence Db(noeth A) -~! Kc(InjA) which is a quasi-inverse for the equivalence Kc(InjA) ! Db(noeth A) induced by * *Q. Let us denote by Kinj(A) the full subcategory of complexes Y in K(InjA) such * *that Hom K(A)(X, Y ) = 0 for all acyclic complexes X in K(A). Following Spaltenstei* *n's terminology [31], the objects in Kinj(A) are precisely the K-injective complexe* *s having injective components. There are various results about K-injective resolutions * *in the literature; see for instance [31, 10]. The following is certainly not the most * *general one; however it is sufficient in our context. Corollary 3.9. The inclusion Kinj(A) ! K(A) has a left adjoint i: K(A) ! Kinj(A) which has the following properties. (1)Every object X in K(A) fits into an exact triangle aX -! X -! iX -! (aX) such that aX is an acyclic complex. (2)The functor i: K(A) ! Kinj(A) induces an equivalence D(A) = K(A)=Kac(A) -~! Kinj(A). (3)We have for all X, Y in K(A) Hom D(A)(X, Y ) ~=Hom K(A)(X, iY ). Proof.Put iX = QjX for each X in K(A), where Qj denotes the right adjoint of Q: K(InjA) ! D(A). The functor R: D(A) = K(A)=Kac(A) -~! Kinj(A) inc-!K(A) provides a right adjoint for the canonical functor K(A) ! D(A). Let us mention * *as an application that the right derived functor of any additive functor F :A ! B is * *obtained as composite K(F) can RF :D(A) -R! K(A) -! K(B) -! D(B). Example 3.10. Suppose every object in A has finite injective dimension. Then t* *he functor K(InjA) ! D(A) is an equivalence since Kac(InjA) = 0. In particular, t* *he compact objects in D(A) are precisely those from Db(noeth A). Example 3.11. Suppose products in A are exact. For instance, let A be a module category. Then one can show that Kinj(A) is the smallest triangulated subcatego* *ry of K(A) which is closed under taking products and contains the injective objects o* *f A (viewed as complexes concentrated in degree zero). THE STABLE DERIVED CATEGORY OF A NOETHERIAN SCHEME 11 4.A recollement In this section we provide a criterion for A such that the sequence Q Kac(InjA) __I__//K(InjA)____//_D(A) induces a recollement Kac(InjA) _____//K(InjA)____//_oo_oo_D(A)oo_oo_ in the sense of [6]. It is important to note that one cannot expect a recolleme* *nt (4.1) Kac(A) _____//K(A)____//_oo_oo_D(A)oo_oo_ without severe restrictions on A; see Example 4.9. In fact, a recollement (4.1)* * implies that a product of exact sequences in A remains exact. We begin with a lemma. Lemma 4.1. Let A be a locally noetherian Grothendieck category. Then a compact object in D(A) belongs to Db(noeth A). Proof.Suppose X is compact in D(A). We need to show that HnX is noetherian for all n, and that HnX vanishes for almost all n in Z. We have for any injective o* *bject E in A an isomorphism Hom D(A)(X, E) ~=Hom A(H0X, E). Therefore Hom A (H0X, -) preserves coproducts in InjA. This implies that each H* *nX is noetherian; see [29]. Now fix for each n an injective envelope HnX ! E(HnX) * *and consider the induced map Y ff: X -! -n E(HnX) n2Z in D(A). The canonical map a Y -n E(HnX) -! -n E(HnX) n2Z n2Z is an isomorphism in D(A), and therefore ff factors though a finite number of f* *actors in Y -n E(HnX). n2Z Thus HnX vanishes for almost all n in Z, and the proof is complete. We denote by Dc(A) the full subcategory of D(A) which is formed by all compact objects. Theorem 4.2. Let A be a locally noetherian Grothendieck category and suppose D(* *A) is compactly generated. Then the canonical functor Q: K(InjA) ! D(A) has a left adjoint and therefore the sequence Q Kac(InjA) __I__//K(InjA)____//_D(A) is a colocalization sequence. 12 HENNING KRAUSE Proof.Let K be the localizing subcategory of K(InjA) which is generated by all * *compact objects X in K(InjA) such that QX is compact in D(A). We claim that Q|K :K ! D(A) is an equivalence. First note that K and D(A) are both compactly generated* *. We have seen in Lemma 4.1 that Dc(A) Db(noeth A), and Q induces an equivalence Kc(InjA) -~! Db(noeth A), by Proposition 2.3. Thus Q induces an equivalence between the subcategories of * *compact objects in K and D(A). Then a standard argument shows that Q|K is an equivalence since Q preserves all coproducts. Now fix a left adjoint L: D(A) ! K. We claim * *that the composite D(A) -L! K inc-!K(InjA) is a left adjoint for Q. To see this, consider for objects X in D(A) and Y in K* *(InjA) the natural map ffX,Y :Hom K(InjA)(LX, Y ) -! Hom D(A)(QLX, QY ) -~! Hom D(A)(X, QY ) which is induced by Q. If X and Y are compact, then ffX,Y is bijective by Propo* *sition 2.3. We use a standard argument to show that ffX,Y is bijective for arbitrary X and * *Y . Fix a compact object X. Then the objects Y such that ffX,Y is bijective form a tria* *ngulated subcategory which is closed under taking coproducts and contains all compact ob* *jects. Thus ffX,Y is bijective for all Y because K(InjA) is compactly generated. Now f* *ix any object Y . The same argument shows that ffX,Y is bijective for all X because D(* *A) is compactly generated. We conclude that Q has a left adjoint. Moreover, Lemma 3* *.2 implies that I and Q form a colocalization sequence. Following Beilinson, Bernstein, and Deligne [6], we say that a sequence (4.2) T 0____//_T____//T 00 of exact functors between triangulated categories induces a recollement oo___ oo___ 00 T 0____//_T____//oo_Too_ if the sequence (4.2) is a localization sequence and a colocalization sequence * *in the sense of Definition 3.1. Corollary 4.3. Let A be a locally noetherian Grothendieck category and suppose * *D(A) is compactly generated. Then the sequence Q Kac(InjA) __I__//K(InjA)____//_D(A) induces a recollement Kac(InjA) ____//_K(InjA)____//oo_oo_D(A).oo_oo_ Corollary 4.4. Let A be a locally noetherian Grothendieck category and suppose * *D(A) is compactly generated. Then a product of acyclic complexes of injective object* *s in A is acyclic. Let us give a criterion for A such that the derived category D(A) is compactl* *y gen- erated. THE STABLE DERIVED CATEGORY OF A NOETHERIAN SCHEME 13 Lemma 4.5. Let A be a locally noetherian Grothendieck category. Suppose there i* *s a set A0 of objects in A which are compact when viewed as objects in D(A). If A0 gene* *rates A, then D(A) is compactly generated by A0. The lemma is an immediate consequence of the following statement. Lemma 4.6. Let A be a locally noetherian Grothendieck category and fix a set A0* * of generating objects. Let X be a complex in A such that H0X 6= 0. Then there ex* *ists some object A in A0 such that Hom K(A)(A, X) 6= 0 and Hom D(A)(A, X) 6= 0. Proof.Choose A in A0 and a map A ! Z0X such that the composite with Z0X ! H0X is non-zero. This induces a non-zero element in H0(Hom A (A, X)) ~=Hom K(A)(A, X). The second assertion follows from the first since for any object A in A we have Hom D(A)(A, X) ~=Hom K(A)(A, iX) and H0(iX) ~=H0X. We give examples of Grothendieck categories such that objects in A become com* *pact objects in D(A). Example 4.7. Let be an associative ring. Denote by A = Mod the category of (right) -modules and by proj the full subcategory of finitely generated proje* *ctive -modules. Then every object in proj is compact when viewed as object in D(A). Thus the inclusion Db(proj ) ! D(A) identifies Db(proj ) with the full subcateg* *ory of compact objects in D(A). Suppose now that is right noetherian. Then the fu* *lly faithful functor Q~: D(A) ! K(InjA) identifies D(A) with the localizing subcate* *gory of K(InjA) which is generated by the injective resolution i of . Let us return to the completion problem for triangulated categories which has* * been adressed in Section 2. Keeping the analogy between the completion with respect* * to filtered colimits and the completion with respect to triangles und coproducts, * *we ob- tain the following diagram for a right noetherian ring . The vertical arrows * *denote completions and the horizontal ones the appropriate inclusions. proj _____//mod Db(proj ) ____//_Db(mod ) | | | | | | | | | fflffl| fflffl| fflffl|Q~ fflffl| Flat _____//Mod D(Mod ) _____//_K(Inj ) Here, Flat denotes the full subcategory of flat -modules, which is the closur* *e of proj under forming filtered colimits. Example 4.8. Let X be a quasi-compact and separated scheme, and let L be a loca* *lly free sheaf of finite rank. Then Hom D(QcohX)(L, -) ~=Hom D(QcohX)(OX, L_ OX -) ~=H0(L_ OX -), where L_ = Hom OX(L, OX). Thus L is a compact object in D(Qcoh X); see [26]. If* * X has an ample family of line bundles, then the locally free sheaves of finite ra* *nk generate QcohX. 14 HENNING KRAUSE Our final example shows that products in Qcoh X need not to be exact. We incl* *ude this as an illustration for Corollary 4.4 Example 4.9. Let k be a field and X = P1kthe projective line with homogeneous coordiante ring S = k[x0, x1]. For each n 2 Z, we have an exact sequence S(n - 1) q S(n - 1) -! S(n) -! k(n) -! 0 of S-modules which gives rise to an epimorphism O(n-1)qO(n-1) ! O(n) in Qcoh X. We claim that the product Y Y ß : (O(n - 1) q O(n - 1)) -! O(n) n2Z n2Z is not an epimorphism.QIn fact, the cokernel of ß is isomorphic to the sheaf co* *rresponding to`the module n2Zk(n). But this module is not torsion. For instance, it con* *tains n>0 k(n) as a finitely generated submodule which is not of finite length. We c* *onclude that ß has a non-trivial cokernel. 5.The stable derived category Let A be a locally noetherian Grothendieck category. We suppose that D(A) is compactly generated. Definition 5.1. The stable derived category S(A) of A is by definition the full* * subcat- egory of K(A) which is formed by all acyclic complexes of injective objects in * *A. The full subcategory of compact objects is denoted by Sc(A). In this section, we show that the stable derived category is compactly genera* *ted, and the description of the category of compact objects justifies our terminology. O* *ur basic tool is the (co)localization sequence Q S(A) __I__//K(InjA)____//_D(A). Thus we use the fact that I and Q have left adjoints I~, Q~ and right adjoints * *Ij, Qj. The stabilization functor is by definition the composite I~OQæ S :D(A) ----! S(A). We begin with the following lemma. Lemma 5.2. Let A be a locally noetherian Grothendieck category and suppose D(A) is compactly generated. The functors Q~, Qj: D(A) ! K(InjA) admit a natural tra* *ns- formation j :Q~ ! Qj, and j is an isomorphism when restricted to the subcategor* *y of compact objects in D(A). Proof.Each object in D(A) is isomorphic to one in the image of Q, and each map QX ! QY is a fraction of maps in the image of Q. Thus the natural transformation Q~ OQ -! IdK(InjA)-! QjO Q induces a natural transformation j :Q~ ! Qj. Now observe that Q(j) induces an isomorphism Q OQ~ -~! Q OQj. THE STABLE DERIVED CATEGORY OF A NOETHERIAN SCHEME 15 We know from Proposition 2.3 that Q induces an equivalence Kc(InjA) -~! Db(noeth A). On the other hand, Q~(Dc(A)) Kc(InjA) since a left adjoint preserves compactness if the right adjoint preserves copro* *ducts; see Lemma 3.5. Also, Qj(Dc(A)) Kc(InjA), since Dc(A) Db(noeth A) by Lemma 4.1, and Qj(Db(noeth A)) = Kc(InjA) by Remark 3.8. We conclude that j|Dc(A) is an isomorphism. Proposition 5.3. Let A be a locally noetherian Grothendieck category, and suppo* *se D(A) is compactly generated. Then we have a localization sequence Q~ I~ (5.1) D(A) _____//K(InjA)____//_S(A) which induces the following commutative diagram. Dc(A) ____inc___//_Db(noeth A)__can___//_Db(noeth A)=Dc(A) || Q|| | || o|æ Db(noethA) |F || fflffl| |fflffl Dc(A) ___________//_Kc(InjA)________________//_Sc(A) |inc| inc|| |inc| fflffl| Q~ fflffl| I~ |fflffl D(A) ____________//_K(InjA)__________________//S(A) Proof.It follows from Theorem 4.2 that the sequence (5.1) is a localization seq* *uence. Let us explain the commutativity of the diagram. First observe that a left adjoint * *preserves compactness if the right adjoint preserves coproducts; see Lemma 3.5. Therefor* *e I~ and Q~ preserve compactness, and this explains the commutativity of the lower s* *quares. Now observe that Dc(A) Db(noeth A), by Lemma 4.1, and that Qj|Db(noethA)is a quasi-inverse for Q|Kc(InjA). It follo* *ws from Lemma 5.2 that the upper left hand square commutes. The functor F is by definit* *ion the unique functor making the upper right hand square commutative. It exists be* *cause I~ OQ~ = 0. We have seen that the stable derived category S(A) is a localization of the h* *omotopy category K(InjA). This has some interesting consequences. Corollary 5.4. The stable derived category S(A) is compactly generated, and the* * functor I~ OQj: D(A) ! S(A) induces (up to direct factors) an equivalence F :Db(noeth A)=Dc(A) -~! Sc(A). 16 HENNING KRAUSE Proof.We know from Proposition 2.3 that K(InjA) is compactly generated. This pr* *op- erty carries over to S(A) since I~ sends a set of compact generators of K(InjA)* * to a set of compact generators of S(A). The functor Q~ identifies D(A) with the loca* *lizing subcategory of K(InjA) which is generated by all compact objects in the image o* *f Q~. Now apply the localization theorem of Neeman-Ravenel-Thomason-Trobaugh-Yao [25]. This result describes the category of compact objects of the quotient S(A), up * *to di- rect factors, as the quotient of the compact objects in K(InjA) modulo those fr* *om the localizing subcategory. To be precise, F is fully faithful and every object in * *Sc(A) is a direct factor of some object in the image of F . Corollary 5.5. The composite I~OQæ A can--!D(A) ----! S(A) preserves all coproducts and annihilates the objects in A \ Dc(A). Proof.The diagram in Proposition 5.3 shows that I~ OQj annihilates A \ Dc(A). * *To show that I~ OQj preserves all coproducts, observe that Qj sends an object in A* * to an injective resolution. A coproduct of injective resolutions is again an injectiv* *e resolution, and the left adjoint I~ preserves all coproducts. This finishes the proof. Using the stabilization functor S :D(A) ! S(A), we define for objects X, Y in* * D(A) and n 2 Z the stable cohomology group Ext_nA(X, Y ) = Hom K(A)(SX, n(SY )). Note that in both arguments each exact sequence in A induces a long exact seque* *nce in stable cohomology. We do not go into details but refer to our discussion of* * Tate cohomology in Section 7. In fact, both cohomology theories coincide in case A s* *atisfies some appropriate Gorenstein property, and we shall see explicit formulae for th* *e Tate cohomology groups. Example 5.6. Suppose A is a module category. Then the stabilization functor an- nihilates all modules of finite projective dimension. Similarly, if A is a cat* *egory of quasi-coherent sheaves, then the stabilization functor annihilates all sheaves * *having a finite resolution with locally free sheaves. Given a noetherian scheme X, the stable derived category S(Qcoh X) vanishes i* *f X is regular. Nonetheless, a classical result of Bernstein-Gelfand-Gelfand [8] s* *hows that stable derived categories are relevant when one studies regular schemes. This i* *s sketched in the following example. Example 5.7. Let be a Koszul algebra and !its Koszul dual. Then we have under appropriate assumptions an equivalence K(Inj ) ~!K(Inj !) which induces an equi* *v- alence Db(mod ) ~! Db(mod !) when restricted to the full subcategories of com* *pact objects [7, 22]. Note that we consider the categories of graded modules over * *and ! respectively. The classical example is the symmetric algebraV = SV of a d-dime* *nsional space V over a field k, where !is the exterior algebra V *of the dual space * *V *. The equivalence K(Inj ) ~! K(Inj !) takes an injective resolution ik of 0 = k to * *! and identifies the localizing subcategory K generated by ik with the localizing sub* *category generated by !, which is D(Mod !). Note that the quotient K(Inj )=K identif* *ies with the derived category of the quotient Mod =(Mod )0, where (Mod )0 den* *otes THE STABLE DERIVED CATEGORY OF A NOETHERIAN SCHEME 17 the subcategory of torsion modules. This quotient is equivalent to Qcoh Pd-1kby* * Serre's Theorem. Thus we obtain an equivalence V D(Qcoh Pd-1k) -~! S(Mod kd). V V Note that S(Mod kd) is equivalent to the stable module category Mod__ kd bec* *ause the exterior algebra is self-injective; see Example 7.15. Passing to the subca* *tegory of compact objects, one obtains the equivalence V Db(coh Pd-1k) -~! mod_ kd V of Bernstein-Gelfand-GelfandV[8], where mod_ kd denotes the stable category of* * all finite dimensional kd-modules. This example generalizes to non-commutative algebras,* * for instance, to Artin-Schelter regular algebras [21]. 6. Extending derived functors An additive functor F :A ! B between locally noetherian Grothendieck categori* *es admits a right derived functor RF :D(A) ! D(B). In this section we extend this * *to a functor ^RF :K(InjA) ! K(InjB) and investigate its right and left adjoints. A* *s an application, we consider for F the direct image functor f*: Qcoh X ! Qcoh Y cor* *re- sponding to a morphism f :X ! Y between noetherian schemes. We use the following functors I :K(InjA) inc-!K(A) and Q: K(InjA) inc-!K(A) can-!D(A) simultanously for A and B. Moreover, we use the fact that both functors have le* *ft and right adjoints. Theorem 6.1. Let F :A ! B be an additive functor between locally noetherian Gro* *then- dieck categories. Suppose D(A) and D(B) are compactly generated. Then the compo* *site ^RF :K(InjA) -I! K(A) K(F)-!K(B) -I~!K(InjB) makes the following diagram commutative. D(A) _____RF_____//_D(B)OO |Qæ| |Q| |fflffl ^RF | K(InjA) __________//_K(InjB) (1)Suppose F preserves coproducts. Then ^RF preserves coproducts and has the* *re- fore a right adjoint (R^F )j. (2)Suppose F and RF preserve coproducts. Then RF has a right adjoint (RF )j making the following diagram commutative. (RF)æ D(A) ______RF_____//D(B)OO D(B) _____________//D(A)OO |Q~| |Q| |Qæ| |Q| fflffl| ^RF | fflffl| (R^F)æ | K(InjA) __________//_K(InjB) K(InjB) __________//_K(InjA) 18 HENNING KRAUSE Proof.The composite K(A) -I~!K(InjA) -Q! D(A) is naturally isomorphic to the canonical functor K(A) ! D(A); see Remark 3.7. C* *learly, I OQj is its right adjoint. We denote by RF the right derived functor of F and * *have RF = Q OI~ OK(F ) OI OQj. Using the definition ^RF = I~ OK(F ) OI, we obtain RF = Q O^RF OQj. (1) Suppose F preserves coproducts. Then K(F ) preserves coproducts. It follo* *ws that ^RF preserves coproducts since I and I~ preserve coproducts. Now apply Proposit* *ion 3.3 to obtain a right adjoint for ^RF . (2) Suppose F and RF preserve coproducts. Then RF has a right adjoint by Prop* *o- sition 3.3. Next we show that Q O^RF OQ~ ~=Q O^RF OQj. We have a natural transformation Q~ ! Qj which is induced from the natural tran* *s- formation Q~ OQ ! QjO Q. Now apply Q O^RF to get a natural transformation ~: Q O^RF OQ~ -! Q O^RF OQj. It is shown in Lemma 5.2 that Q~ ! Qj is an isomorphism when restricted to com- pact objects in D(A). On the other hand, Q O^RF OQ~ and Q O^RF OQj both preserve coproducts by our assumption on RF . It follows that ~ is an isomorphism since * *D(A) is compactly generated. Clearly, Q O(R^F )jO Qj is a right adjoint for Q O^RF O* *Q~. This completes the proof. The extended derived functor and its right adjoint admit some alternative des* *cription. I am indebted to Bernhard Keller for providing this remark. Remark 6.2. It is possible to express ^RF as the tensor functor and its right a* *djoint (R^F )j as the Hom functor with respect to a bimodule of DG categories; see [22* *, 6.4]. This depends on the appropriate choice of DG categories A0 and B0 such that K(I* *njA) ~= Ddg(A0) and K(InjB) ~=Ddg(B0) respectively. Next we consider as an example a morphism f :X ! Y between separated noetheri* *an schemes. Let f*: Qcoh X ! Qcoh Y denote the direct image functor. Note that the* * right derived functor Rf*: D(Qcoh X) ! D(Qcoh Y) preserves coproducts [26, Lemma 1.4]. Thus Rf* and its right adjoint Grothendieck duality functor f! extend to functo* *rs be- tween K(InjX) and K(InjY), by Theorem 6.1. This is the statement of Theorem 1.4 from the introduction. In fact, the situation is in this case much nicer. I am * *grateful to Amnon Neeman for pointing out that the functor ^Rf* and its right adjoint (R* *^f*)j make the following diagram commutative. R^f* (R^f*)æ (6.1) K(InjX) ____________//_K(InjY) K(InjY)O____________//_K(InjX)OOO |Q| |Q| Qæ|| Qæ|| fflffl| Rf* fflffl| | f! | D(Qcoh X) __________//_D(Qcoh Y) D(Qcoh Y) __________//_D(Qcoh X) This is essentially the statement of Theorem 1.5 from the introduction. The pro* *of which is due to Neeman uses the following lemma. THE STABLE DERIVED CATEGORY OF A NOETHERIAN SCHEME 19 Lemma 6.3. Keep the assumptions from Theorem 6.1 and consider the following dia- gram. ^RF K(InjA) __________//_K(InjB) |Q| |Q| |fflffl RF fflffl| D(A) ____________//_D(B) There is a natural transformation Q O^RF ! RF OQ which is an isomorphism if and only if F sends every acyclic complex of injective objects to an acyclic comple* *x. Proof.We apply the the localization sequence Q Kac(InjA) __I__//K(InjA)____//_D(A) from Proposition 3.6. Let X be an object in K(InjA) and consider the triangle (I OIj)X -! X -! (QjO Q)X -! (I OIj)X in K(InjA). Now apply Q O^RF which gives a map (Q O^RF )X -! (RF OQ)X since Q O^RF OQj ~=RF , by Theorem 6.1. Clearly, this map is an isomorphism if * *and only if Q O^RF annihilates (I OIj)X. We include a simple example which illustrates the preceding lemma. Example 6.4. Let k be a field and = k[t]=(t2). We take the functor F :Mod -! Mod k, X 7! Hom (k, X), and observe that the following diagram does not commute. ^RF K(Inj ) ____________//K(Injk) Q|| o|Q| fflffl| RF fflffl| D(Mod ) __________//_D(Mod k) For instance, we have QX = 0 and (R^F )X 6= 0 if we take for X the acyclic comp* *lex . .-.t! -t! -t! -t! . . . in K(Inj ). Proof of Theorem 1.5.We need to show that both squares in (6.1) commute. Then we use the localization sequence Q S(Qcoh X) __I_//_K(InjX)___//_D(Qcoh X) from Proposition 3.6 and obtain from ^Rf* and (R^f*)j an adjoint pair of functo* *rs between S(Qcoh X) and S(Qcoh Y). In order to show the commutativity of (6.1), we apply Lemma 6.3 and need to s* *how that f* sends an acyclic complex X of injective objects to an acyclic complex. * * The question is local in Y and we may assume Y affine. Cover X by a finite number * *of 20 HENNING KRAUSE affines. Then f* can be computed using the ~Cech cohomology of the cover. If th* *ere are n open sets in the cover, then for any quasi-coherent sheaf A we have Rn+1f*A = 0. Now take our acyclic complex X of injective sheaves on X. Then the sequence 0 -! X0 -! X1 -! X2 -! . . . is an injective resolution of the kernel A of the map X0 ! X1. Applying f*, the* * sequence computes for us Rif*A, which vanishes if i > n + 1. Thus f*X is acyclic above d* *egree n, but by shifting we conclude that it is acyclic everywhere. Having shown the commutativity of the left hand square, the commutativity of * *the right hand square follows, because it is obtained by taking right adjoints. Th* *us the proof is complete. Next we investigate for an exact functor F :A ! B an extension ^LF of the der* *ived functor LF :D(A) ! D(B). For this we need some assumptions, and it is convenient to introduce the following notation. As before, A and B denote locally noether* *ian Grothendieck categories. Let f :noeth A ! noethB be an additive functor. Then t* *here is, up to isomorphism, a unique functor f* :A ! B which extends f and preserves filtered colimits. This has a right adjoint f*: B ! A if and only if f is right* * exact. Note that f is exact iff f* is exact iff f* sends injective objects to injective obj* *ects. Here is an example. Example 6.5. Let f :X ! Y be a morphism between noetherian schemes. Then the inverse image functor f* :Qcoh Y ! Qcoh X sends coherent sheaves to coherent sh* *eaves and preserves filtered colimits. Moreover, the direct image functor f* is a rig* *ht adjoint of f*. Our notation is therefore consistent if we identify the morphism f :X ! * *Y with the functor cohY ! cohX. Theorem 6.6. Let A and B locally noetherian Grothendieck categories such that D* *(A) and D(B) are compactly generated. Let f :noeth A ! noeth B be an exact functor. Then R^f* has a left adjoint ^Lf* which induces a functor Sf* making the follow* *ing diagram commutative. (6.2) Kc(InjA)xx______~_______//Db(noeth A) qqq | qqqxxq| qqqq | qqqq | I xxqqqq || Q xxqq || S(A) __________//_K(InjA)_________________//_D(A) Db(f)| | | | | | | | | | | |Lf* | || || fflffl| || fflffl| Sf*| |^Lf* Kc(InjB) _______|___~___//_Db(noeth B) | | qxxq | qqxx | | qqq | qqq | | qqqq | qqq fflffl|I fflffl|xxqq Q fflffl|xxqq S(B) __________//_K(InjB)_________________//_D(B) THE STABLE DERIVED CATEGORY OF A NOETHERIAN SCHEME 21 If Rf* preserves coproducts, then in addition the following diagram commutes. Db(noeth A)=Dc(A) oo________________Kc(InjA)xxoo____________ Dc(A)zz FA nnnnnn | qqqqqq | vvvv | nnn | qqq | vvvv || vvnnnn || I~ xxqq || Q~ zzv | S(A) oo__________________________ K(InjA) oo_________________D(A) | | |_____ | | | | | |Db(f)| |^ * || | * || || fflffl| |Lf| fflffl| Lf|| fflf* *fl| |Sf* Db(noeth B)=Dc(B) oo_______|_________Kc(InjB) oo______|_______Dc(B) | nn | qxxq | vzzv | FBnnnn | qqq | vv | nnnn | qqqq | vvv fflffl|vvnnn I~ fflffl|xxqq Q~ fflffl|zzvv S(B) oo___________________________K(InjB) oo_________________D(B) Note that FA and FB induce, up to direct factors, equivalences_onto_the full * *subcat- egories of compact objects in S(A) and S(B) respectively. Thus Db(f) determines* * the functor Sf*. Proof.The exactness of f implies the exactness of f*. Thus f* sends injective o* *bjects to injective objects and we have the following commutative diagram. (6.3) K(InjB) //___J____//_K(B) R^f*|| |K(f*)| fflffl| J fflffl| K(InjB) //________//_K(B) The right adjoint f* preserves products and we have therefore a left adjoint fo* *r ^Rf*, by Proposition 3.3, which we denote by ^Lf*. We obtain the following diagram J~ Q K(A) __________//_K(InjA)_________//_D(A) K(f*)|| |^Lf*| |Lf*| fflffl|J~ fflffl| Q fflffl| K(B) __________//_K(InjB)_________//_D(B) and claim it is commutative. The left hand square commutes because it is obtai* *ned from (6.3) by taking left adjoints. The outer square commutes because the compo* *site Q OJ~ is naturally isomorphic to the canonical functor K(A) ! D(A); see Remark * *3.7. We conclude the commutativity of the right hand square, using that J~ OJ ~=IdK(* *InjA). Clearly, ^Lf* sends acyclic complexes to acyclic complexes and we obtain the fu* *nctor Sf* making the diagram (6.2) commutative. Now assume in addition that Rf* preserves coproducts. We use the fact that a * *left adjoint preserves compactness if the right adjoint preserves coproducts; see Le* *mma 3.5. The functor ^Rf* preserves coproducts since f* preserves coproducts. Thus Lf* a* *nd ^Lf* preserve compactness. Note that Q~ identifies D(A) with the localizing subcateg* *ory of K(InjA) which is generated by Dc(A) Db(noeth A) ~=Kc(InjA) K(InjA). 22 HENNING KRAUSE Of course, the same applies for B. We obtain the following diagram I~ Q~ S(A) oo________K(InjA)_ oo________D(A)_ S|| |^Lf*| Lf*|| fflffl|I~ fflffl| Q~ fflffl| S(B) oo________K(InjB)_ oo________D(B)_ where the right hand square commutes. The horizontal sequences are localizatio* *n se- quences by Theorem 4.2, and ^Lf* induces a functor S :S(A) ! S(B) making the le* *ft hand square commutative. Moreover, we have S = S OI~ OI = I~ O^Lf* OI = I~ OI OSf* = Sf*. The functors FA and FB are both induced by I~, and the commutativity ______ Sf* OFA = FB ODb(f) is easily checked; see Corollary 5.4. This completes the proof. Remark 6.7. Let f :X ! Y be a morphism between noetherian schemes and suppose f* is exact. Then Sf* is the left adjoint of Sf* which appears in Theorem 1.5. Next we investigate the inclusion f :X ! Y of an open subscheme. In this case* *, the adjoint pair of functors f* and f* between Qcoh X and Qcoh Y restricts to an ad* *joint pair of functors between InjX and InjY; see [15, VI]. Moreover, f* Of* ~=IdQcoh* *X. Thus we can identify ^Rf* = f* and ^Lf* = f*. Note that both functors send acyclic c* *omplexes with injective components to acyclic complexes. This is clear for f* because it* * is exact, and follows for f* from Theorem 1.5. We denote for each sheaf A in Qcoh Y by Su* *pp A the support of A and observe that f* annihilates A if and only Supp A is contai* *ned in Y \ X. In fact, the natural map A ! (f*O f*)A induces a split exact sequence 0 -! A0- ! A -! (f*O f*)A -! 0 if A is injective. In particular, the support of A0is contained in Y \ X. Now fix a complex X in K(InjY). The support of X is by definition [ Supp X = Supp Xn. n2Z We write XX = (f*O f*)X, and the natural map X ! XX induces an exact triangle (6.4) XY\X -! X -! XX -! (XY\X) in K(InjY) where the support of XY\X is contained in Y \ X. Lemma 6.8. Let Y be a seperated noetherian scheme and f :X ! Y be the inclusion of an open subscheme. If X is a complex in K(InjY), then f*X = 0 if and only if* * X is isomorphic to a complex with support contained in Y \ X. Proof.We have f*X = 0 if and only if the first map in the triangle (6.4) is an * *isomor- phism. It is well known that f* induces an equivalence D(Qcoh Y)=DY\X(Qcoh Y) -~! D(Qcoh X), THE STABLE DERIVED CATEGORY OF A NOETHERIAN SCHEME 23 where DY\X(Qcoh Y) denotes the full subcategory of all complexes in D(Qcoh Y) s* *uch that the support of the cohomology is contained in Y \ X. We obtain an analogue* * for K(InjY) and S(Qcoh Y) if we define KY\X(InjY) = {X 2 K(InjY) | Supp X Y \ X}, SY\X(Qcoh Y) = {X 2 S(Qcoh Y) | Supp X Y \ X}. Proposition 6.9. Let Y be a seperated noetherian scheme and f :X ! Y be the inc* *lu- sion of an open subscheme. Then f* induces equivalences K(InjY)=KY\X(InjY) -~! K(InjX), S(Qcoh Y)=SY\X(Qcoh Y) -~! S(Qcoh X). Proof.We have f* Of* ~=IdQcohX, and this carries over to complexes of injective* *s. On the other hand, we have for X in K(InjY) a natural map X ! (f*O f*)X which indu* *ces an isomorphism in K(InjY)=KY\X(InjY), by Lemma 6.8. Let us give a more elaborate formulation of Proposition 6.9. The functor R^f* ** = f*: K(InjX) ! K(InjY) admits a left and a right adjoint. Therefore ^Rf* induce* *s a recollement K(InjX) _____//K(InjY)____//_oo_oo_KY\X(InjY).oo_oo_ This recollement is compatible with the recollement S(Qcoh Y) ____//_K(InjY)___//_oo_oo_D(QcohoY),o_oo_ and we obtain the following diagram. S(QcohOX)O_______//_K(InjX)______//_oo_oo_D(QcohoX)o_oo_OOOOOOO* *OOO ||| ||| ||| ||| ||| ||| |fflffl||oo___ |fflffl||oo__ |fflffl|| S(QcohOY)O_______//_K(InjY)______//_oo_D(QcohoY)o_OOOOOOOOOO ||| ||| ||| ||| ||| ||| |fflffl|| |fflffl|| |fflffl|| SY\X(Qcoh Y)_____//KY\X(InjY)____//_oo_oo_DY\X(QcohoY)o_oo_ In this diagram, each row and each column is a recollement. Moreover, the diagr* *am is commutative if one restricts to arrows in south and east direction. All other c* *ommuta- tivity relations follow by taking left adjoints or right adjoints. Proposition 6.9 tells us precisely when the inclusion of a subscheme induces * *an equiv- alence for the stable derived category. In [28], Orlov observed that the bounde* *d stable derived category of a noetherian scheme depends only on the singular points. We* * extend this result to the unbounded stable derived category, using a completely differ* *ent proof. Corollary 6.10. Let Y be a seperated noetherian scheme of finite Krull dimensio* *n. If f :X ! Y denotes the inclusion of an open subscheme which contains all singular* * points of Y, then Sf* :S(Qcoh Y) ! S(Qcoh X) is an equivalence. Proof.We apply Proposition 6.9 and need to show that SY\X(Qcoh Y) = 0. But this* * is clear from our assumptions on X and Y. Orlov's result [28, Proposition 1.14] is an immediate consequence if one rest* *ricts the equivalence Sf* to compact objects; see Theorem 6.6. 24 HENNING KRAUSE Corollary 6.11. Let Y be a seperated noetherian scheme of finite Krull dimensio* *n. If f :X ! Y denotes the inclusion of an open subscheme which contains all singular* * points of Y, then f* induces (up to direct factors) an equivalence Db(coh Y)=Dperf(coh Y) -! Db(coh X)=Dperf(coh X). 7. Gorenstein injective approximations and Tate cohomology Let A be a locally noetherian Grothendieck category and suppose that the deri* *ved category D(A) is compactly generated. In this section we study the category of * *complete injective resolutions. We assign functorially to each complex of injectives a * *complete resolution. This yields Gorenstein injective approximations and Tate cohomology* * groups for objects in A. The classical definition of Tate cohomology is based on comp* *lete projective resolutions. Our approach is essentially the same, using however res* *olutions with injective instead of projective components. Another aspect in this sectio* *n is the interplay between the stable derived category S(A) and the stable category A_mo* *dulo injective objects, which is obtained from A by identifying two maps if their di* *fference factors through some injective object. Given objects A, B in A, we write Hom__A(A, B) = Hom A_(A, B). The functor K(InjA) -! A_, X 7! Z0X = Ker(X0 ! X1) provides a link between the stable categories S(A) and A_. In particular, we ob* *tain an explicit description of the stabilization functor I~OQæ S :A can--!D(A) ----! S(A) provided that A has some appropriate Gorenstein property. Most of the concepts in this section are classical, but seem to be new in thi* *s setting and this generality. We refer to the end of this section for historical remarks and* * references to the literature. Let us start with the relevant definitions. A complex X in InjA is called to* *tally acyclic if Hom A(A, X) and Hom A(X, A) are acyclic complexes of abelian groups * *for all A in InjA. We denote by Ktac(A) the full subcategory of all totally acyclic com* *plexes in K(InjA). Following [13], we call an object A in A Gorenstein injective if it* * is of the form Z0X for some X in Ktac(A). We write GInjA for the full subcategory formed * *by all Gorenstein injective objects. Lemma 7.1. Let A be an abelian category and let X, Y be objects in K(InjA). Sup* *pose HnX = 0 for all n > 0 and Y is totally acyclic. Then the canonical map oeX,Y :Hom K(InjA)(X, Y ) -! Hom__A(Z0X, Z0Y ) is bijective. Proof.Fix a map ff: Z0X ! Z0Y in A. We need to extend ff to a chain map ~ff:X !* * Y such that Z0~ff= ff. We use the assumption on X to extend ff in non-negative de* *grees, and the assumption on Y allows to extend ff in negative degrees. Thus oeX,Y is * *surjective. To show that oeX,Y is injective, let OE: X ! Y be a chain map such that Z0OE f* *actors through some injective object. A similar argument as before yields a chain homo* *topy X ! Y which shows that OE is null-homotopic. Thus the proof is complete. THE STABLE DERIVED CATEGORY OF A NOETHERIAN SCHEME 25 Let us denote by GInjA_ the full subcategory of A_formed by the objects in GI* *njA. Observe that GInjA is a Frobenius category with respect to the class of exact s* *equences from A. With respect to this exact structure, an object A in GInjA is projecti* *ve iff A is injective iff A belongs to InjA. Thus the category GInjA_ carries a trian* *gulated structure. The shift takes an object A to the cokernel A of a monomorphism A !* * E into an injective object E. The exact triangles are induced from short exact se* *quences in A. Proposition 7.2. Let A be an abelian category. Then the functor Ktac(InjA) -! GInjA_, X 7! Z0X, is an equivalence of triangulated categories. Proof.We need to show that the functor is fully faithful and surjective on isom* *orphism classes of objects. The last property is clear from the definition of GInjA. Th* *e functor is fully faithful by Lemma 7.1. Finally, observe that an exact triangle of com* *plexes comes, up to isomorphism, from a sequence of complexes which is split exact in * *each degree. Thus we obtain an exact sequence in A and an exact triangle in A_if we * *apply Z0. The following lemma is crucial because it provides the existence of complete * *injective resolutions. Lemma 7.3. Let A be a locally noetherian Grothendieck category and suppose that D(A) is compactly generated. Then the inclusion J :Ktac(InjA) ! K(InjA) has a l* *eft adjoint J~: K(InjA) -! Ktac(InjA). Proof.The inclusion I :Kac(InjA) ! K(InjA) has a left adjoint I~ by Theorem 4.2. Thus it is sufficient to show that the inclusion Ktac(InjA) ! Kac(InjA) has a l* *eft adjoint. Let us fix an injective cogenerator E in A. By definition, Ktac(InjA) * *consists of all objects X in Kac(InjA) such that Hom K(InjA)( nE, X) ~=Hom Kac(InjA)(I~( nE), X) vanishes for all n 2 Z. The category Kac(InjA) is compactly generated by Coroll* *ary 5.4, and we can apply Proposition 3.4 to obtain a left adjoint for the inclusion Kta* *c(InjA) ! Kac(InjA). Given an object A in A with injective resolution iA, we call the natural map iA -! J~iA a complete injective resolution of A. If we apply the functor Z0 to this map, w* *e obtain a Gorenstein injective approximation of A. Theorem 7.4. Let A be a locally noetherian Grothendieck category and suppose th* *at D(A) is compactly generated. Then the inclusion GInjA_ ! A_has a left adjoint T :A_-! GInjA_. Thus we have for each object A in A a natural map A ! T A which induces a bijec* *tion Hom__A(T A, B) -~! Hom__A(A, B) for all B 2 GInjA. 26 HENNING KRAUSE Proof.Fix an object A in A and choose an injective resolution iA. We put T A = Z0(J~iA), and this induces a functor T :A_! GInjA_. Let B in GInjA and fix a totally acyc* *lic complex tB such that Z0tB = B. The natural map iA ! J~iA induces a map A ! T A in A_which makes the following square commutative. Hom K(InjA)(J~iA, tB)~__//_HomK(InjA)(iA, tB) |Z0| |Z0| fflffl| fflffl| Hom__A(T A, B)__________//Hom_A(A, B) The vertical maps are bijective by Lemma 7.1, and we conclude that T is a left * *adjoint for the inclusion GInjA_! A_. Next we use complete injective resolutions to define Tate cohomology groups f* *or objects in A. Definition 7.5. Given objects A, B in A and n 2 Z, the Tate cohomology group is dExtnA(A, B) = Hn Hom A(A, J~iB) Remark 7.6. The correct term for this cohomology theory would be `injective Tate cohomology' in order to distinguish it from the usual `projective Tate cohomolo* *gy' which is defined via complete projective resolutions. For simplicity, we drop the ext* *ra adjective `injective'. Note that confusion is not possible because we do not consider pr* *ojective Tate cohomology in this paper. Tate cohomology is natural in both arguments because the formation of complete injective resolutions is functorial. In addition, we have a comparison map n Ext nA(A, B) -! dExtA(A, B), which is induced by the map iB ! J~iB. There is an alternative description of T* *ate cohomology which is based on the left adjoint T :A_! GInjA_. Proposition 7.7. Given objects A, B in A and n 2 Z, there is a natural isomorph* *ism dExtnA(A, B) ~=Hom__A(A, n(T B)). Proof.Using Lemmas 2.1 and 7.1, we have the following sequence of isomorphisms Hn Hom A(A, J~iB) ~= Hom K(A)(A, n(J~iB)) ~= Hom K(A)(iA, n(J~iB)) ~= Hom__A(A, n(T B)). We have a more conceptual definition of Tate cohomology for objects in D(A) w* *hich uses the composite J~OQæ U :D(A) ----! Ktac(InjA). Thus we define for objects X, Y in D(A) and n 2 Z dExtnA(X, Y ) = Hom K(A)(UX, n(UY )). THE STABLE DERIVED CATEGORY OF A NOETHERIAN SCHEME 27 Note that this definition is consistent with the original definition of Tate co* *homology if we take objects in A and view them as complexes concentrated in degree 0. T* *his follows from the fact that QjX is nothing but an injective resolution of X0 whe* *n X is concentrated in degree 0. From now on, we will use one of the alternative descr* *iptions of Tate cohomology whenever this is convenient. Next we show that each exact sequence in A induces a long exact sequence in T* *ate cohomology. This is based on the following simple lemma. Lemma 7.8. The left adjoint T :A_! GInjA_ has the following properties. (1)An exact sequence 0 ! A0! A ! A00! 0 in A induces an exact triangle T A0- ! T A -! T A00-! (T A0) in GInj A_. (2)Let A, B be in A and n 2 Z. The natural map A ! T A induces an isomorphism EdxtnA(T A, B) ~=dExtnA(A, B). Proof.(1) We have an exact triangle A0! A ! A00! (A0) in D(A). Now use that the exact functor Z0 OJ~ OQj computes T . (2) The adjointness property of T implies Hom__A(T A, T B) ~=Hom__A(A, T B). Proposition 7.9. Let 0 ! B0 ! B ! B00! 0 be an exact sequence in A. Then we have for A and C in A the following long exact sequences. n 0 n n 00 . .-.! dExtA(A, B ) -! dExtA(A, B) -! dExtA(A, B ) -! n+1 0 n+1 n+1 00 -! dExtA (A, B ) -! dExtA (A, B) -! dExtA (A, B ) -! . . . n 00 n n 0 . .-.! dExtA(B , C) -! dExtA(B, C) -! dExtA(B , C) -! n+1 00 n+1 n+1 0 -! dExtA (B , C) -! dExtA (B, C) -! dExtA (B , C) -! . . . Proof.We apply Lemma 7.8 and use the fact that Hom__A(T A, -) and Hom__A(-, T C* *) are cohomological functors. We compute Tate cohomology for Gorenstein injective objects. Proposition 7.10. Let A, B be objects in A and suppose B is Gorenstein injectiv* *e. Then the comparison map n ExtnA(A, B) -! dExtA(A, B) is an isomorphism for n > 0 and induces an isomorphism 0 Hom__A(A, B) -~! dExtA(A, B) for n = 0. Proof.Our assumption implies T B = B. The case n = 0 is clear. For n = 1, choos* *e an exact sequence 0 ! B ! E ! B ! 0 with E injective and apply Hom A (A, -). The cokernel of Hom A(A, E) -! Hom A(A, B) is isomorphic Ext1A(A, B); it is isomorphic to Hom__A(A, B) since B is Gorenst* *ein in- jective. For n > 1, use dimension shift. 28 HENNING KRAUSE * Next we describe those objects A in A such that dExtA(A, -) vanishes. For ins* *tance, Tate cohomology vanishes for all objects having finite projective or finite inj* *ective di- mension. Proposition 7.11. For an object A in A, the following are equivalent. * (1)dExtA(A, -) = 0. 0 (2)dExtA(A, A) = 0. * (3)dExtA(-, A) = 0. (4)Ext1A(A, B) = 0 for all B 2 GInjA. (5)Hom__A(A, B) = 0 for all B 2 GInjA. * * 1 Proof.Use the isomorphism dExtA(-, -) ~=Hom__A(T -, T -) and the fact that dExt* *A(-, B) can be computed via Ext1A(-, T B). The following result formulates our analogue of the maximal Cohen-Macaulay ap* *prox- imation in the sense of Auslander and Buchweitz [2]. Note that a Gorenstein inj* *ective object is the `dual' of a maximal Cohen-Macaulay object which one defines in a * *category with enough projectives. Let X = {A 2 A | Ext1A(A, B) = 0 for allB 2 GInjA} and Y = GInjA. Theorem 7.12. Let A be a locally noetherian Grothendieck category and suppose t* *hat D(A) is compactly generated. (1)Every object A in A fits into exact sequences 0 ! YA ! XA ! A ! 0 and 0 ! A ! Y A ! XA ! 0 in A with XA , XA in X and YA , Y A in Y. (2)The map A 7! XA induces a right adjoint for the inclusion X_! A_. (3)The map A 7! Y A induces a left adjoint for the inclusion Y_! A_. (4)X \ Y = InjA. Note that this is essentially the statement of Theorem 1.3 from the introduct* *ion, since X is precisely the subcategory of objects A in A such that the Tate cohomology * *functor dExt*A(A, -) vanishes. Proof.We use the basic properties of Tate cohomology. (1) Fix an object A in A and a complete injective resolution iA -! J~iA = yA. We complete this map to an exact triangle (7.1) iA -! yA -! xA -! (iA). in K(InjA) and have therefore a sequence 0 ! iA ! yA ! xA ! 0 of complexes which is split exact in each degree. Applying Z0: K(InjA) ! A_produces an exact seque* *nce 0 -! A -ff!Y A -fi!XA -! 0 * in A. Clearly, Y A belongs to Y. On the other hand, dExtA(ff, -) is an isomor* *phism. * A A Thus dExtA(X , -) vanishes and X belongs to X . The second sequence ending in* * A is obtained by rotating the triangle (7.1). THE STABLE DERIVED CATEGORY OF A NOETHERIAN SCHEME 29 (2) We consider the exact sequence 0 -! YA -~! XA -! A -! 0 and need to show that Hom__A(X, ) is bijective for all X in X . To see this, l* *et OE: X ! A be a map with X in X . The map OE factors through since Ext1A(X, YA ) = 0. Th* *erefore Hom__A(X, ) is surjective. To show that Hom__A(X, ) is injective, let _ :X !* * XA be 0 ffi00 a map such that O_ has a factorization X ffi!E ! A with injective E. We obta* *in a factorization OE00= OØ, since E belongs to X . We have O(_ -Ø OOE0) = 0, an* *d _ -Ø OOE0 needs to factor through ~. Therefore _ - Ø OOE0 factors through some injective * *object, since Hom__A(X, YA ) = 0. We conclude that _ factors through an injective objec* *t. Thus the map Hom__A(X, ) is bijective. (3) We consider the exact sequence 0 -! A -ff!Y A -fi!XA -! 0 and need to show that Hom__A(ff, Y ) is bijective for all Y in Y. But this is * *clear from 0 the long exact sequence for Tate cohomology, since Hom__A(-, Y ) ~= dExtA(-, Y * *) and dExt*A(XA , -) vanishes. (4) Clearly, InjA is contained in X \ Y. Now let A be in X \ Y. Thus 0 Hom__A(A, A) ~=dExtA(A, A) = 0. Therefore the identity map A ! A factors through an injective object. We concl* *ude that A is injective. Let us comment on the interplay between the stable category A_and the stable * *derived category S(A). We have already seen that the definition of Tate cohomology is p* *ossible in both settings. It is more elementary in A_, but more conceptual using the ca* *tegory of complete injective resolutions Ktac(InjA) which is a subcategory of S(A). Th* *e same phenomenon appears when one studies Gorenstein injective approximations. The pr* *oof of Theorem 7.12 we have given uses the category of complexes K(InjA). There is * *an alternative proof which avoids complexes and uses instead the left adjoint T :A* *_ ! GInjA_. Gorenstein rings and schemes play an important role in applications and have a number of interesting homological properties. It is therefore important to form* *ulate a Gorenstein property for a locally noetherian Grothendieck category A. Let us de* *note by 1 A the full subcategory of objects A in A which fit into an exact sequence . .-.! E2 -! E1 -! E0 -! A -! 0 with En injective for all n. We say that A has the injective Gorenstein propert* *y if the equivalent conditions in the following proposition are satisfied. This property* * has been studied by Beligiannis in [4]. Proposition 7.13. Let A be a locally noetherian Grothendieck category and suppo* *se that D(A) is compactly generated. Then the following are equivalent. (1)Ext1A(A, B) = 0 for all A 2 InjA and B 2 1 A. (2)Every acyclic complex in InjA is totally acyclic. (3)GInj A = 1 A. I~OQæ (4)S :A can--!D(A) ----! S(A) annihilates all injective objects. 30 HENNING KRAUSE (5)S induces an equivalence GInjA_ ! S(A). Proof.The conditions (1) - (3) are pairwise equivalent. This follows from the f* *ormula ExtnA(A, Z0Y ) ~=Hom K(A)(A, nY ) ~=Hn Hom A(A, Y ) where A is any object in A and Y is an acyclic complex in InjA. The first isomo* *rphism is valid for all n > 1, and the second for all n 2 Z. Now observe that I~ annihilates precisely those objects X in K(InjA) such that Hom K(InjA)(X, Y ) = 0 for every acyclic complex Y in InjA. On the other hand, * *A ! D(A) and Qj are faithful. Thus (1) - (3) are equivalent to (4). Also, (5) impli* *es (4). So, it remains to show that (1) - (4) imply (5). Suppose (2) and (4) hold. We have already seen in Proposition 7.2 that S(A) -! GInjA_, X 7! Z0X, is an equivalence, since every acyclic complex is totally acyclic. On the other* * hand, S annihilates all injective objects and induces therefore a functor GInjA_ ! S(A)* *. It is easily checked that both functors are quasi-inverse to each other. We are now in the position that we can describe the stabilization functor S :* *A ! S(A), provided that A has the injective Gorenstein property. We use the left a* *djoint T :A_! GInjA_ of the inclusion GInjA_ ! A_. For A in A, choose any acyclic comp* *lex X of injective objects such that Z0X ~=T A. Then SA ~=X. Example 7.14. Let be a ring and suppose is Gorenstein, that is, is two-si* *ded noetherian and has finite injective dimension as left and right -module. In* * this case, the category Mod has the injective Gorenstein property. This follows fr* *om the fact that every injective -module has finite projective*dimension if is Gore* *nstein; see Example 5.6. Given a -module A, Tate cohomology dExt(A, -) vanishes iff A has * *finite injective dimension iff A has finite projective dimension. Note that for Gorens* *tein rings, the classical Tate cohomology defined via complete projective resolutions coinc* *ides with our Tate cohomology, which is defined via complete injective resolutions; see [* *5]. Example 7.15. Let be a ring and suppose that projective and injective -modul* *es coincide. Then every -module is Gorenstein injective. In particular, S(Mod * * ) is equivalent to the stable category Mod__ . Given a -module A, there is an exact* * triangle pA -! iA -! tA -! (pA) in K(InjA) where pA denotes a projective, iA an injective, and tA a Tate resolu* *tion of A. This triangle is isomorphic to the canonical triangle (Q~ OQ)A~- ! ~A-! (I OI~)A~- ! (Q~ OQ)A~ where A~ = QjA. For details in case is a finite dimensional cocommutative Ho* *pf algebra, see [18]. Example 7.16. Let X be a noetherian scheme and suppose that every injective obj* *ect E in Qcoh X admits a finite resolution 0 -! Lr -! . .-.! L2 -! L1 -! L0 -! E -! 0 with Ln locally free for each n. Then the category Qcoh X has the injective Gor* *enstein property. THE STABLE DERIVED CATEGORY OF A NOETHERIAN SCHEME 31 Historical remarks. Gorenstein injective approximations and Tate cohomology hav* *e a long history. Auslander and Bridger [1] introduce the stable module category an* *d assign to each module a G-dimension. Over Gorenstein rings, the modules of G-dimension* * 0 are precisely the maximal Cohen-Macaulay or Gorenstein projective modules. Ausl* *an- der and Buchweitz establish maximal Cohen-Macaulay approximations in [2], and t* *here is an alternative unpublished approach by Buchweitz [12] which involves the der* *ived cat- egory. Enochs and his collaborators drop finiteness conditions on modules and p* *rove the existence of Gorenstein projective and Gorenstein injective approximations for * *arbitrary modules, for instance over Gorenstein rings [14]. Further generalizations can b* *e found in work of Beligiannis [4]. Jørgensen [19] constructs Gorenstein projective approx* *imations for artin algebras via Bousfield localization, using the category of complete p* *rojective resolutions. Papers of Hovey [17] and Beligiannis and Reiten [5] employ the for* *malism of model category structures and cotorsion pairs. The exposition of Buchweitz [12] discusses the close connection between maxim* *al Cohen-Macaulay approximations and Tate cohomology over Gorenstein rings. For mo* *re general settings, we refer to the work of Beligiannis and Reiten [5]. Another e* *xposition of Tate cohomology over noetherian rings can be found in a paper of Avramov and Martsinkovsky [3]. Appendix A. The DG category of noetherian objects Let A be a locally noetherian Grothendieck category. We give an alternative d* *escrip- tion of the homotopy category K(InjA) as the derived category of some DG catego* *ry. Here, we follow closely Keller's exposition in [22]. Let C be a small DG category. We recall the definition of the derived catego* *ry Ddg(C) of C. The category Cdg(C) of cochain complexes has by definition as obj* *ects all DG C-modules. A map in Cdg(C) is a map of DG C-modules which is homogeneous of degree zero and commutes with the differential. The homotopy category Kdg(C* *) is obtained from Cdg(C) by identifying homotopy equivalent maps, where f, g :X ! Y are homotopy equivalent if there exists a map s: X ! Y of graded modules which * *is homogeneous of degree -1 and satisfies (f - g)n = sn+1 Od + d Osn for all n 2 Z. Finally, the derived category of C is obtained from Kdg(C) as the localization Ddg(C) = Kdg(C)[Q-1] with respect to the class Q of all maps f which induce an isomorphism H*f. Given two cochain complexes X and Y in A, we define the cochain complex Hom A* *(X, Y ). The nth component is Y Hom A(Xp, Y n+p) p2Z and the differential is given by d(fp) = d Ofp - (-1)nfp+1 Od. Now fix a class C of objects in A. We obtain a DG category ~Cby taking as obj* *ects for each A in C an injective resolution ~A, and as maps Hom C~(A~, ~B) = Hom A(A~, ~B). 32 HENNING KRAUSE Proposition A.1. Let A be a locally noetherian Grothendieck category, and let C* * be a class of noetherian objects which generate Db(noeth A), that is, there is no pr* *oper thick subcategory containing C. Then the functor K(InjA) -! Ddg(C~), X 7! Hom A(-, X)|C~, is an equivalence of triangulated categories. Proof.The functor is exact. To see that it preserves coproducts, fix an object * *A in C and a family of objects Xi in K(InjA). Then we have for every n 2 Z a a a Hn Hom A(A~, Xi) ~= Hn Hom A(A~, Xi) ~= Hom K(InjA)( -n ~A, Xi) i i a i a ~= Hom K(InjA)( -n ~A, Xi) ~=Hn Hom A(A~, Xi) i i since ~Ais compact in K(InjA) by Lemma 2.1. Thus the canonical map a a Hom A(-, Xi)|C~-! Hom A(-, Xi)|C~ i i is an isomorphism. Furthermore, the functor induces for objects A and B in C bi* *jections Hom K(InjA)(A~, nB~) ~=Hn Hom A(A~, ~B) ~=Hn Hom ~C(A~, ~B) ~=Hom Ddg(C~)(A~^,* * nB~^), where A~^denotes the free module Hom ~C(-, ~A). Using infinite d'evissage, we c* *onclude that the functor is fully faithful since C generates K(InjA). The functor is, * *up to isomorphism, surjective on objects since the image contains the free ~C-modules* * which generate Ddg(C~). Corollary A.2. Viewing noethA as DG category, we have an equivalence K(InjA) -~! Ddg(noeth A). We remark that the proof of Proposition A.1 works for any homotopy category. * *To be precise, let X be an additive category with arbitrary coproducts and let C b* *e a set of objects in K(X ) which are compact (when viewed as objects in the locali* *zing subcategory generated by C). Define ~Cas before by Hom ~C(A, B) = Hom X (A, B) for A and B in C. Then the functor K(X ) -! Ddg(C~), X 7! Hom X (-, X)|C~, induces an equivalence between the localizing subcategory which is generated by* * C, and Ddg(C~). Acknowledgements. 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