THE GORENSTEIN DEFECT CATEGORY

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1 THE GORENSTEIN DEFECT CATEGORY PETTER ANDREAS BERGH, DAVID A. JORGENSEN & STEFFEN OPPERMANN Dedicated to Ranar-Ola Buchweitz on the occasion o his sixtieth birthday Abstract. We consider the homotopy cateory o complexes o projective modules over a Noetherian rin. Truncation at deree zero induces a ully aithul trianle unctor rom the totally acyclic complexes to the stable derived cateory. We show that i the rin is either Artin or commutative Noetherian local, then the unctor is dense i and only i the rin is Gorenstein. Motivated by this, we deine the Gorenstein deect cateory o the rin, a cateory which in some sense measures how ar the rin is rom bein Gorenstein. 1. Introduction Given a rin, one can associate to it certain trianulated cateories: derived cateories, homotopy cateories, and various trianulated subcateories o these, such as bounded derived cateories or homotopy cateories o acyclic complexes. When the rin is Gorenstein, classical results by Buchweitz (c. [Buc]) show that some o these trianulated cateories are equivalent: the stable cateory o maximal Cohen- Macaulay modules, the stable derived cateory o initely enerated modules, and the homotopy cateory o totally acyclic complexes o initely enerated projective modules. Thus, or Gorenstein rins, these trianulated cateories (toether with their respective cohomoloy theories) virtually coincide. In this paper, we provide a cateorical characterization o Gorenstein rins. Let A be a let Noetherian rin, and proj A the cateory o initely enerated projective let A-modules. Brutal truncation at deree zero induces a map rom the homotopy cateory K tac (proj A) o totally acyclic complexes to the homotopy cateory K,b (proj A) o riht bounded eventually acyclic complexes. However, this map is not a unctor. We thereore consider instead the stable derived cateory de = K,b (proj A)/K b (proj A) o A, where K b (proj A) is the homotopy cateory o bounded complexes. Brutal truncation now induces a trianle unctor, and we show that this unctor is always ull and aithul. The main result, Theorem 2.7, shows that the properties o this unctor actually characterize Gorenstein rins. Namely, i A is either an Artin rin or a commutative Noetherian local rin, then the unctor β proj A is dense i and only i A is Gorenstein. The i part here is classical: it is part o [Buc, Theorem 4.4.1]. Motivated by this, we deine the Gorenstein deect cateory D b G (A) o A as the Verdier quotient D b G(A) de = / Im β proj A, where Im β proj A is the isomorphism closure o the imae o β proj A in : this is a thick subcateory. Then Theorem 2.7 translates to the ollowin: i A is either 2010 Mathematics Subject Classiication. 13H10, 16E65, 18E30. Key words and phrases. Trianulated cateories, Gorenstein rins. 1

2 2 PETTER ANDREAS BERGH, DAVID A. JORGENSEN & STEFFEN OPPERMANN an Artin rin or a commutative Noetherian local rin, then D b G (A) = 0 i and only i A is Gorenstein. The dimension o the Gorenstein deect cateory is thereore in some sense a measure o how ar the rin is rom bein Gorenstein. 2. The results Let P be an additive cateory, and KP the homotopy cateory o complexes in P. This is a trianulated cateory, with suspension Σ: KP KP iven by shitin a complex one deree to the let, and chanin the sin o its dierential. That is, or a complex C KP with dierential d, the complex Σ C has C n 1 in deree n, and d as dierential. The (distinuished) trianles in KP are the sequences o objects and maps that are isomorphic to sequences o the orm C 1 C2 C() Σ C 1 or some map and its mappin cone C(). We say that a complex C in P is acyclic i the complex Hom P (P, C) o abelian roups is acyclic or all objects P P. I in addition Hom P (C, P ) is acyclic or all P P, then C is totally acyclic. Moreover, C is eventually acyclic i or all objects P P, the complex Hom P (P, C) is eventually acyclic, i.e. H n (Hom P (P, C)) = 0 or n 0. Note that we cannot deine acyclicity directly or complexes in P, since the cateory is only assumed to be additive. We shall be workin with the ollowin ull subcateories o KP (the deinitions are up to isomorphism in KP): K tac P = {C KP C is totally acyclic} K,b P = {C KP C n = 0 or n 0 and C is eventually acyclic} K b P = {C KP C n = 0 or n 0}. These are all trianulated subcateories o KP. For example, when P is the cateory o initely enerated let projective modules over a let Noetherian rin A, then K b P is by deinition the cateory o perect complexes. Moreover, in this settin it is well known that the trianulated cateories K,b P and D b (mod A) are equivalent, where mod A is the cateory o initely enerated let A-modules. The cateory K b P is a thick subcateory o K,b P, that is, a trianulated subcateory closed under direct summands. The Verdier quotient K,b P/K b P is thereore a well deined trianulated cateory. Recall that the objects in this quotient are the same as the objects in K,b P. A morphism C 1 C 2 in the quotient is an equivalence class o diarams o the orm D C 1 C 2 where and are morphisms in K,b P, and has the property that its mappin cone C() belons to K b P. Two such morphisms (, D, ) and (, D, ) are equivalent i there exists a third such morphism (, D, ), and two morphisms h: D D and h : D D in K,b P, such that the diaram D h D C 1 C 2 h D

3 THE GORENSTEIN DEFECT CATEGORY 3 is commutative. For urther details, we reer to [Nee, Chapter 2]. The natural trianle unctor K,b P K,b P/K b P maps an object to itsel, and a morphism : C 1 C 2 to the equivalence class o the diaram C 1 1 C 1 C 2 In Theorem 2.2, we establish a ully aithul trianle unctor rom K tac P to the quotient K,b P/K b P. To avoid too many technicalities in the proo, we irst prove the ollowin lemma. It allows us to complete certain morphisms and homotopies o truncated complexes. Given an inteer n and a complex C : dn+3 d n+2 d n+1 C n+2 Cn+1 Cn in P, we denote its brutal truncation C : d n Cn 1 d n 1 Cn 2 dn+3 d n+2 d n+1 C n+2 Cn+1 Cn 0 0 d n 2 at deree n by β n (C). Note that when C K tac P, then β n (C) K,b P. Lemma 2.1. Let P be an a additive cateory, and C, D two complexes in P with C totally acyclic. Furthermore, let n be an inteer, and : β n (C) β n (D) a chain map. (a) The map can be extended to a chain map : C (b) Let π : C D (with β n ( ) = ). β n (C) be the natural chain map, and consider the composite chain map π : C β n (D). I π is nullhomotopic throuh a homotopy h, then h can be extended to a homotopy ĥ makin nullhomotopic. Proo. (a) The chain map is iven by the solid part o the commutative diaram d C n+3 d C n+2 d C n+1 C n+2 C n+1 C n C n 1 C n 2 d C n d C n 1 d C n 2 d D n+3 n+2 n+1 n n 1 d D n+2 d D n+1 D n+2 D n+1 D n D n 1 D n 2 and it suices to ind a map n 1 as indicated, makin the square to its let commutative. The composition d D n n d C n+1 is zero, and by assumption the sequence d D n d D n 1 d D n 2 Hom P (C n 1, D n 1 ) (dc n ) Hom P (C n, D n 1 ) (dc n+1 ) Hom P (C n+1, D n 1 ) is exact. Thereore, there exists a map n 1 : C n 1 D n 1 such that n 1 d C n = d D n n. (b) The homotopy h is iven by maps h n 1, h n, h n+1,... as in the diaram d C n+3 d C n+2 d C n+1 C n+2 C n+1 C n C n 1 C n 2 d C n d C n 1 d C n 2 h n+2 n+2 h n+1 n+1 h n n h n 1 n 1 h n 2 n 2 h n 3 d D n+3 d D n+2 d D n+1 D n+2 D n+1 D n D n 1 D n 2 with i = d D i+1 h n + h i 1 d C i or all i n. We must ind maps h n 2, h n 3,... completin the homotopy. d D n d D n 1 d D n 2

4 4 PETTER ANDREAS BERGH, DAVID A. JORGENSEN & STEFFEN OPPERMANN To ind the map h n 2, consider the map n 1 d D n h n 1 in Hom P (C n 1, D n 1 ), or which we obtain Since the sequence ( n 1 d D n h n 1 ) d C n = d D n n d D n ( n d D n+1 h n ) = 0. Hom P (C n 2, D n 1 ) (dc n 1 ) Hom P (C n 1, D n 1 ) (dc n ) Hom P (C n, D n 1 ) is exact, there exists a map h n 2 : C n 2 D n 1 such that n 1 d D n h n 1 = h n 2 d C n 1. Iteratin this procedure, we obtain the maps h n 3, h n 4,... ivin ĥ. We are now ready to prove Theorem 2.2. It establishes a ully aithul trianle unctor rom K tac P to the quotient K,b P/K b P, mappin a complex C to its brutal truncation β 0 (C) at deree zero. Note that brutal truncation does not deine a unctor between homotopy cateories. Theorem 2.2. For an additive cateory P, brutal truncation at deree zero induces a ully aithul trianle unctor β P : K tac P K,b P/K b P. Proo. To simpliy notation, we denote β P by just β. The irst issue to address is well-deinedness. Let : C D be a map o complexes in K tac P, as indicated in the ollowin diaram: d C 3 d C 2 d C 1 C 2 C 1 C 0 C 1 d C 0 d C 1 d D 3 h 2 2 d D 2 h 1 1 d D 1 h 0 D 2 D 1 D 0 D 1 It suices to show that i vanishes in K tac P, that is, i there is a homotopy h as indicated by the dashed arrows above, then β 0 () vanishes in K,b P/K b P. Usin this homotopy h, we see that the map d C 3 d C 2 d C 1 C 2 C 1 C d D 0 h 1 1 d D 1 h h 1 d C 0 d D 3 d D 2 d D 1 D 2 D 1 D 0 0 is nullhomotopic in K,b P. Consequently, the map β 0 () is homotopic to the map d C 3 d C 2 d C 1 C 2 C 1 C h 1 d C 0 d D 3 d D 2 d D 1 D 2 D 1 D 0 0 in K,b P. Clearly, this map actors throuh the stalk complex with C 1 in deree zero. Thereore β(), which equals the imae o β 0 () in the quotient K,b P/K b P, vanishes. This shows that the unctor β is well-deined.

5 THE GORENSTEIN DEFECT CATEGORY 5 Σ It is easy to check that β is a trianle unctor. The natural isomorphism β Σ β is iven by de 2 de 1 de 0 de 1 β(σc): C 1 C 0 C 1 0 Σ(β(C)): C 1 C This is indeed an isomorphism in K,b P/K b P, since its mappin cone in K,b P is isomorphic to the stalk complex with C 1 in deree one, which belons to K b P. Usin a similar isomorphism, one checks that β commutes with mappin cones. Next, we prove that β is aithul. Let : C D be a morphism in K tac P such that β() = 0. We may think o as a morphism o complexes. Then the condition β() = 0 means that, up to homotopy, the brutal truncation β 0 () actors throuh a bounded complex C K b P. Choose a positive inteer n such that C i = 0 or i n. By truncatin at deree n, we see that the induced map β n () π : C β n (D) o complexes is nullhomotopic. It then ollows rom Lemma 2.1(b) that itsel is nullhomotopic, and this shows that the unctor β is aithul. It remains to show that β is ull. Let ψ : β(c) β(d) be a morphism in K,b P/K b P between two complexes in the imae o β. Then ψ is represented by a diaram β 0 (C) C β 0 (D) o complexes and maps in K,b P, where the mappin cone C() o the map belons to K b P. Up to homotopy, in suiciently hih derees, the complex C then coincides with β 0 (C), and then also with C. Thereore, or some positive inteer n, there is an equality β n (C) = β n (C ), and the truncation β n () is the identity. Furthermore, the truncation β n () is a morphism β n (): β n (C) β n (D). By Lemma 2.1(a), it admits and extension : C shall prove that ψ = β( ). Consider the solid part o the diaram D o complexes in K tac P: we C θ β 0 (C) π β 1 (C) β 0 ( ) π β 0 (D) π 1 β 0 ( ) β 0 (C) o complexes and maps in K,b P, where π is the natural projection. The lower two trianles obviously commute. Furthermore, by Lemma 2.1(a), the identity chain map 1: β n (C) β n (C ) admits an extension θ : β 1 (C) C, and by

6 6 PETTER ANDREAS BERGH, DAVID A. JORGENSEN & STEFFEN OPPERMANN construction the equalities β n (π) = β n ( θ) β n (β 0 ( ) ) π = β n ( θ) hold. The chain map π θ can be viewed as a chain map C β 0 (C), and its truncation β n (π θ) is trivially nullhomotopic. Thus, by Lemma 2.1(b), the map π θ itsel is nullhomotopic, and this shows that the top let trianle commutes. Similarly, the top riht trianle commutes, hence the diaram is commutative. Consequently, the map ψ equals β( ), and so the unctor β is ull. We shall apply Theorem 2.2 to the case when the additive cateory P is the cateory proj A o initely enerated projective modules over a let Noetherian rin A. The ollowin result shows that, in this situation, i the unctor is dense (i.e. an equivalence in view o Theorem 2.2), then all hiher extensions between any module and the rin vanish. Recall irst that the Verdier quotient K,b (proj A)/K b (proj A) is the classical stable derived cateory o A. Proposition 2.3. Let A be a let Noetherian rin and proj A the cateory o initely enerated projective let A-modules. I the unctor is dense, then Ext n A(M, A) = 0 or all n 0 and every initely enerated let A-module M. Proo. Let M be a initely enerated let A-module, and P : d4 d P 3 d 3 2 d P2 1 P1 P0 its projective resolution: this is a complex in K,b (proj A). Since the unctor β proj A is dense, the complex P is isomorphic in to β proj A (T ) or some totally acyclic complex T K tac (proj A). For some n 0, the two complexes β n (P ) and β n (T ) coincide up to homotopy, ivin Ext i A(M, A) H i (Hom A (P, A)) H i (Hom A (T, A)) or all i n + 1. Since the complex T is totally acyclic, the roup H j (Hom A (T, A)) vanishes or all j Z, and this proves the result. Specializin to the case when the rin is either let Artin or commutative Noetherian local, we obtain the ollowin two corollaries. Recall that a commutative local rin is Gorenstein i its injective dimension (as a module over itsel) is inite. Corollary 2.4. Let A be a let Artin rin, and proj A the cateory o initely enerated projective let A-modules. I the unctor is dense, then the injective dimension o A as a let module is inite. Proo. There are initely many simple let A-modules S 1,..., S t, and by Proposition 2.3 there exists an inteer n such that Ext i A( S j, A) = 0 or i n + 1. The injective dimension o A is thereore at most n. Corollary 2.5. Let A be a commutative Noetherian local rin, and proj A the cateory o initely enerated projective (i.e. ree) A-modules. I the unctor is dense, then A is Gorenstein.

7 THE GORENSTEIN DEFECT CATEGORY 7 Proo. Let k be the residue ield o A. By Proposition 2.3 there exists an inteer n such that Ext i A(k, A) = 0 or i n + 1. The injective dimension o A is thereore at most n. The ollowin result shows that, in the situation o Proposition 2.3, every injective module has inite projective dimension. Proposition 2.6. Let A be a let Noetherian rin and proj A the cateory o initely enerated projective let A-modules. I the unctor is dense, then the projective dimension o every initely enerated injective let A- module is inite. Proo. Let I be a initely enerated injective let A-module, and P I K,b (proj A) a projective resolution o I. For every n 1, denote by Ω n A (I) the imae o the nth dierential in P I. It suices to show that the identity on P I actors throuh an object in K b (proj A), or this would imply that Ω n A (I) is projective or hih n. Let T be a totally acyclic complex in K tac (proj A), and M the imae o its zeroth dierential. Then there is a monomorphism : M P or some P proj A (take or example P = T 1 ). Since I is injective, every map M I actors throuh. Now or every n 1, denote by Ω n A (M) the imae o the nth dierential in T. We claim that every map : Ω n A (M) Ωn A (I) actors throuh a projective module. To see this, note that every such map lits to a chain map β n (T ) β n (P I ), and by Lemma 2.1(a) this chain map can be extended to a chain map T P I. This ives a map : M I, which actors throuh a projective module by the above. Since = Ω n A ( ), the map also actors throuh a projective module, as claimed. We show next that Hom D b st (A)(β proj A (T ), P I ) = 0. Any morphism ψ in this roup is (represented by) a diaram C β 0 (T ) P I in K,b (proj A), with the cone o belonin to K b (proj A). As in the proo o Theorem 2.2, we can assume that (up to homotopy) the two complexes β 0 (T ) and C aree in hih derees. From the above, it then ollows that or hih n, every map B n (C) B n (P I ) actors throuh a projective module, where B n (D) denotes the imae o the nth dierential in a complex D. This shows that ψ = 0. We can now show that the identity on P I actors throuh an object in K b (proj A). Namely, since the unctor β proj A is dense, the complex P I is isomorphic in to β proj A (T ) or some acyclic complex T K tac (proj A). From what we showed above, we obtain and the result ollows. Hom D b st (A)(P I, P I ) Hom D b st (A)(β proj A (T ), P I ) = 0, Recall that a Noetherian rin (i.e. a rin that is both let and riht Noetherian) is Gorenstein i its injective dimensions both as a let and as a riht module are inite. By a classical result o Zaks (c. [Zak, Lemma A]), the two injective dimensions then coincide. However, it is an open question whether a Noetherian rin o inite selinjective dimension on one side is o inite selinjective dimension on both sides, and thereore Gorenstein.

8 8 PETTER ANDREAS BERGH, DAVID A. JORGENSEN & STEFFEN OPPERMANN We have now come to the main result. It deals with Artin rins (i.e. rins that are both let and riht Artin) and commutative Noetherian local rins. Namely, or such a rin A, the unctor β proj A is dense i and only i A is Gorenstein. As mentioned in the introduction, the i part o this result is classical: it is part o [Buc, Theorem 4.4.1]. We include a proo or the convenience o the reader. Theorem 2.7. Let A be either an Artin rin or a commutative Noetherian local rin, and proj A the cateory o initely enerated projective let A-modules. Then the unctor is dense i and only i A is Gorenstein. Proo. Suppose the unctor β proj A is dense. I A is local, then it is Gorenstein by Corollary 2.5. I A is Artin, then the injective dimension o A as a let module is inite by Corollary 2.4. Moreover, by Proposition 2.6, every initely enerated injective let A-module has inite projective dimension. The duality between initely enerated let and riht modules then implies that every initely enerated projective riht A-module has inite injective dimension. Thereore A is Gorenstein. Conversely, suppose that A is Gorenstein, and let C be a complex in K,b (proj A). Usin the same notation as in the previous proo, there is an inteer n such that the A-module B n (C) is maximal Cohen-Macaulay, and such that β n (C) is a projective resolution o B n (C). Since B n (C) is maximal Cohen-Macaulay, it admits a projective co-resolution C, and splicin β n (C) and C at B n (C) ives a totally acyclic complex T K tac (proj A). Since β n (C) = β n (T ), the complexes C and β proj A (T ) are isomorphic in, hence the unctor β proj A is dense. Motivated by Theorem 2.2 and Theorem 2.7, we now introduce a new trianulated cateory or any let Noetherian rin A. Since the trianle unctor is ully aithul by Theorem 2.2, the cateory K tac (proj A) embeds in as the imae o β proj A. The isomorphism closure Im β proj A is a thick subcateory o, hence we may orm the correspondin Verdier quotient. Deinition. The Gorenstein deect cateory o a let Noetherian rin A is the Verdier quotient D b G(A) de = / Im β proj A, where proj A is the cateory o initely enerated projective let A-modules. In terms o the Gorenstein deect cateory, Theorem 2.7 takes the ollowin orm. Theorem 2.8. I A is either an Artin rin or a commutative Noetherian local rin, then D b G (A) = 0 i and only i A is Gorenstein. Theorem 2.8 suests that the size o the Gorenstein deect cateory (o an Artin rin or a commutative Noetherian local rin) measures in some sense how ar the rin is rom bein Gorenstein. It would thereore be interestin to ind criteria which characterize the rins whose Gorenstein deect cateories are n-dimensional, in the sense o Rouquier (c. [Rou]). An answer to the ollowin question would be a natural start. Question. What characterizes Artin rins and commutative Noetherian local rins with zero-dimensional Gorenstein deect cateories?

9 THE GORENSTEIN DEFECT CATEGORY 9 Reerences [Buc] R.-O. Buchweitz, Maximal Cohen-Macaulay modules and Tate-cohomoloy over Gorenstein rins, 1987, 155 pp; see [Nee] A. Neeman, Trianulated cateories, Annals o Mathematics Studies, 148, Princeton University Press, Princeton, NJ, 2001, viii+449 pp. [Rou] R. Rouquier, Dimensions o trianulated cateories, J. K-theory 1 (2008), no. 2, [Zak] A. Zaks, Injective dimension o semi-primary rins, J. Alebra 13 (1969), Petter Andreas Berh, Institutt or matematiske a, NTNU, N-7491 Trondheim, Norway address: berh@math.ntnu.no David A. Jorensen, Department o mathematics, University o Texas at Arlinton, Arlinton, TX 76019, USA address: djorens@uta.edu Steen Oppermann, Institutt or matematiske a, NTNU, N-7491 Trondheim, Norway address: steen.oppermann@math.ntnu.no