AUSLANDER REITEN TRIANGLES AND A THEOREM OF ZIMMERMANN

Size: px
Start display at page:

Download "AUSLANDER REITEN TRIANGLES AND A THEOREM OF ZIMMERMANN"

Transcription

1 Bull. London Math. Soc. 37 (2005) C 2005 London Mathematical Society doi: /s AUSLANDER REITEN TRIANGLES AND A THEOREM OF ZIMMERMANN HENNING KRAUSE Abstract A classical theorem of Zimmermann describes the relation between almost split sequences in the category of finitely presented modules and those in the category of all modules over some fixed ring. An analogue of Auslander Reiten triangles in triangulated categories is proved in this paper. This is used to explain the relation between different existence results for Auslander Reiten triangles, which are based either on Brown s representability theorem, or on the existence of Serre functors. 1. Introduction This paper is a continuation of [14], where Brown representability was introduced as a foundation for a general Auslander Reiten theory in triangulated categories. More recently, Auslander Reiten triangles have played an important role in work of Reiten and van den Bergh [18] and Jørgensen [11]. In [18], the existence of Auslander Reiten triangles is translated into the existence of a Serre functor in the sense of Bondal and Kapranov [4]. This is used for a classification of the noetherian hereditary abelian categories satisfying Serre duality. In [11], the singular cochain differential graded algebra of a simply connected space is studied. The existence of Auslander Reiten triangles over this algebra characterizes Poincaré duality spaces. Our aim in this paper is to explain a relation between the approach adopted in [14], which is based on Brown representability, and the existence theorems for Auslander Reiten triangles that are based on variants of Serre duality. It was Neeman [16] who noticed that Brown representability can be applied to prove Grothendieck s duality theorem, which contains Serre duality as a special case. The extra ingredient that is relevant for the existence of Auslander Reiten triangles is the analogue of a theorem of Zimmermann [21] about the existence of almost split sequences in module categories. We present Zimmermann s result in a form that is different from his own account. In particular, some unnecessary assumptions are removed; this might be of independent interest. An alternative discussion of Zimmermann s result can be found in work of Herzog [9], who uses some ideas from model theory, in particular those of Ziegler [19]. More recently, Beligiannis [3] has analysed the Ziegler spectrum of a triangulated category, and has established a connection with its Auslander Reiten theory. 2. A theorem of Zimmermann Let A be an associative ring with identity. We consider the category Mod A of (right) A-modules, and we denote by mod A the full subcategory that is formed Received 17 April 2003; revised 12 February Mathematics Subject Classification 18E30, 16G70 (primary), 14F05, 55U35 (secondary).

2 362 henning krause by the finitely presented A-modules. In this section we present a theorem of Zimmerman on the existence of almost split sequences. For the definition and the basic properties of almost split maps and sequences, we refer to [1, 2]. Recall that a map α: X Y of A-modules is a pure monomorphism if it induces an exact sequence 0 Hom A (C, X) Hom A (C, Y ) Hom A (C, Coker α) 0 for every finitely presented A-module C. For basic facts about purity in module categories, we refer to [10]. Theorem (Zimmermann). Let Z be a finitely presented and non-projective module over some associative ring A. Suppose that Z has a local endomorphism ring, and let 0 X Y Z 0 be an almost split sequence in the category of A-modules (which always exists). Then there exists an almost split sequence 0 X Y Z 0 in the category of finitely presented A-modules if and only if X has a finitely presented non-zero pure submodule. This result appears as [21, Theorem 1], where the ring A is assumed to be semiperfect and the pure submodule of X is assumed to have a local endomorphism ring. The existence of an almost split sequence in the category of all A-modules is due to Auslander [1, Proposition II.5.1]. We deduce Zimmermann s theorem from the following result, which generalizes an earlier result of Zimmermann; see [20, Proposition 3]. Proposition 2.1. Suppose that we have the following map between short exact sequences in the category of A-modules. δ : 0 X Y Z 0 (2.1) δ : 0 X Y Z 0 Let the modules in the first row be finitely presented, and let the map X X be a pure-injective envelope. Then the following statements are equivalent. (1) The sequence δ is an almost split sequence in the category of finitely presented A-modules. (2) The sequence δ is an almost split sequence in the category of all A-modules. Proof. It follows from classical Auslander Reiten theory that an exact sequence 0 L M N 0 is almost split if and only if the map L M is left almost split and the endomorphism ring of N is local; see [1, Proposition II.4.4]. Recall that α: L M is left almost split if α is not a section and if every map L M that is not a section factors through α. (1) = (2). We assume that X Y is left almost split, and we need to show that X Y is left almost split. If X Y were a section, then X X

3 auslander reiten triangles and a theorem of zimmermann 363 would factor through X Y.ThusX Y would be a pure monomorphism, contradicting the fact that X Y is not a section. Thus X Y is not a section. Now let X M be a map in Mod A that is not a section. We consider the composite α: X M with X X. Note that α is not a pure monomorphism, since X X is a pure-injective envelope. The maps α i : X M i in mod A such that α factors through α i form a filtered system, and its colimit is α. There is an index j such that α j is not a section, because a filtered colimit of sections is a pure monomorphism. Thus α j factors through X Y. We conclude that X M factors through X Y because the diagram (2.1) is a push-out. Thus X Y is left almost split. (2) = (1). We assume that X Y is left almost split, and we need to show that X Y is left almost split. We shall work in the abelian category F = (mod A op, Ab) of additive functors from finitely presented A op -modules to abelian groups. The full subcategory formed by the finitely presented functors in F is denoted by F. The diagram (2.1) induces in F the following commutative diagram with exact rows. 0 F φ X A α A Y A φ α 0 F A X A Y A In [5, Theorem 2.3] it is shown that α is left almost split in Mod A if and only if φ is the injective envelope of a simple object in F. On the other hand, in [15, Proposition 3.4] it is shown that α is left almost split in mod A if and only if φ is the injective envelope of a simple object in F. Thus our assumption on α implies that φ is the injective envelope of a simple object in F. Now observe that X A X A is a monomorphism in F, since X X is a pure monomorphism. Moreover, F 0, because otherwise δ would split, contradicting the fact that δ does not split. This implies that F F is an isomorphism, and therefore φ is an injective envelope of a simple object in F. It follows that X Y is left almost split, and this completes the proof. The proof of Proposition 2.1 has the following corollary. Corollary 2.2. Let X Y be a map between finitely presented A-modules, and let the following diagram be the push-out with a pure-injective envelope X X. X α Y X α Y Then α is left almost split in Mod A if and only if α is left almost split in mod A. We are now in a position to prove our version of Zimmermann s theorem. Proof of Zimmermann s theorem. Let X 0 be a finitely presented pure submodule of X. The projection X X/X factors through X Y because

4 364 henning krause X Y is a left almost split map. Thus we obtain the following commutative diagram, with exact rows and columns. δ : 0 X Y Z 0 δ : 0 X Y Z 0 X/X X/X 0 0 We conclude from Proposition 2.1 that the sequence δ is almost split. To prove the converse, suppose there is an almost split sequence 0 X Y Z 0 in the category of finitely presented A-modules. Let X X be a pure-injective envelope that induces the following commutative diagram with exact rows. δ : 0 X Y Z 0 δ : 0 X Y Z 0 It follows from Proposition 2.1 that the sequence δ is almost split. Thus X = X, and we therefore have a finitely presented pure submodule of X. This completes the proof. 3. Auslander Reiten triangles for compact objects In this section we prove the analogue of Zimmermann s theorem for triangulated categories. We fix a compactly generated triangulated category T. ThusT has arbitrary coproducts and is generated by a set of compact objects. Recall that an object X in T is compact if for every family (Y i ) i I in T, the canonical map ( Hom(X, Y i ) Hom X, ) Y i i i is an isomorphism. We denote by T c the full subcategory of compact objects in T, and we observe that T c is a triangulated subcategory of T. For basic properties of compactly generated triangulated categories, we refer to [16]. The analogue of an almost split sequence for a triangulated category was introduced by Happel [6]. Definition 3.1. A triangle X α Y β Z γ X[1] is called an Auslander Reiten triangle if the following conditions hold. (1) Every map X Y that is not a section factors through α. (2) Every map Y Z that is not a retraction factors through β. (3) γ 0.

5 auslander reiten triangles and a theorem of zimmermann 365 Note that the end terms X and Z of an Auslander Reiten triangle are indecomposable objects with local endomorphism rings. Moreover, each end term determines an Auslander Reiten triangle up to isomorphism. Now fix a compact object Z. Suppose that the endomorphism ring Γ = End(Z) is local, and denote by E a minimal injective cogenerator in the category of Γ- modules. Brown s representability theorem [16] provides a representing object TZ in T such that Hom Γ (Hom(Z, ),E) = Hom(,TZ). (3.1) This yields an Auslander Reiten triangle (TZ)[ 1] Y Z TZ in T, where the map Z TZ corresponds under (3.1) to a non-zero map Hom(Z, Z) E annihilating the radical of Γ; see [14, Theorem 2.2]. Next, we recall from [13] the notion of purity in T. A map X Y in T is a pure monomorphism if the induced map Hom(C, X) Hom(C, Y ) is a monomorphism for every compact object C. An object X is pure-injective if every pure monomorphism X Y is a split monomorphism. Finally, a map α: X Y is a pure-injective envelope if Y is pure-injective and if the composite β α with a map β : Y Z is a pure monomorphism if and only if β is a pure monomorphism. We have the following analogue of Proposition 2.1. Proposition 3.2. Suppose that we have the following map between triangles in a compactly generated triangulated category. δ : X Y Z X [1] δ : X Y Z X[1] (3.2) Let the objects in the first triangle be compact, and let the map X X be a pure-injective envelope. Then the following statements are equivalent. (1) δ is an Auslander Reiten triangle in the category of compact objects. (2) δ is an Auslander Reiten triangle in the category of all objects. Recall that a map α: X Y is left almost split if it is not a section and if every map X X that is not a section factors through α. We need the following simple reformulation of this definition. Lemma 3.3. For a triangle X α Y β Z γ X[1], the following statements are equivalent. (1) Every map X X that is not a section factors through α. (2) The map γ[ 1] factors through every non-zero map X X. Proof. This is clear, since a map X X is not a section if and only if its fibre X X is non-zero. Proof of Proposition 3.2. We use the fact that a triangle X Y Z X[1] is an Auslander Reiten triangle if and only if End(Z) is local and the map X Y is left almost split; see [14, Lemma 2.6].

6 366 henning krause Rotating the triangles δ and δ, we obtain from (3.2) the following commutative diagram. Z[ 1] γ [ 1] X Z[ 1] γ[ 1] α β X Y φ α Y Note that our assumption on φ implies that α is not a section if and only if α is not a section. (1) = (2). We assume that α is left almost split, and we need to show that α is left almost split. Take as given, a map ρ: X M that is not a section. Thus ρ is not a pure monomorphism because X is pure-injective. This implies that ρ φ is not a pure monomorphism, since φ is a pure-injective envelope. So there exists a non-zero map σ : C X from a compact object C such that the composite with ρ φ is zero. The map γ [ 1] factors through σ by Lemma 3.3, since α is left almost split by our assumption. This implies that ρ γ[ 1] = 0, since ρ φ σ =0. Thus ρ factors through α, and we conclude that α is left almost split. (2) = (1). We assume that α is left almost split, and we need to show that α is left almost split. We want to apply Lemma 3.3. Take as given, a non-zero map σ : C X from a compact object C. The composite φ σ is non-zero, since φ is a pure monomorphism. Thus γ[ 1] factors through φ σ, since α is left almost split. We obtain γ[ 1] = φ σ ρ for some map ρ: Z[ 1] C. This implies that γ [ 1] = σ ρ, since γ[ 1] = φ γ [ 1] and φ is a pure monomorphism. We conclude that α is left almost split. This completes the proof. We leave it to the reader to formulate the analogue of Corollary 2.2. The following consequence is the analogue of Zimmermann s existence result for almost split sequences in the category of finitely presented modules. Corollary 3.4. Let Z be a compact object with local endomorphism ring, and let X Y Z X[1] be an Auslander Reiten triangle in the category of all objects (which always exists). Then there exists an Auslander Reiten triangle X Y Z X [1] in the category of compact objects if and only if X has a compact non-zero pure subobject. Proof. Suppose first there is a pure monomorphism X X such that X is a non-zero compact object. Complete the map to a triangle X X X X [1], and observe that X X factors through the map X Y, since X Y is left almost split. Using the octahedral axiom, we obtain a commutative diagram of the form (3.2). We conclude, from Proposition 3.2, the existence of an Auslander Reiten triangle X Y Z X [1] in the category of compact objects. To prove the converse, suppose that we have an Auslander Reiten triangle X Y Z X [1] in the category of compact objects. Choose a pure-injective envelope X X, which exists by [13, Corollary 1.13]. Compose this map with Z[ 1] X,and ψ β Z Z

7 auslander reiten triangles and a theorem of zimmermann 367 complete the composite to a triangle Z[ 1] X Y Z. This gives rise to the following map between triangles. δ : X Y Z X [1] δ : X Y Z X [1] It follows from Proposition 3.2 that δ is an Auslander Reiten triangle. Thus X = X, and we therefore have a compact pure subobject of X. This completes the proof. Examples of Auslander Reiten triangles in the category of compact objects that are not Auslander Reiten triangles in the category of all objects can be obtained from Zimmerman s work [21]. He constructs almost split sequences in the category of finitely presented modules over some hereditary Artinian ring A that are not almost split in the category of all modules. The canonical embedding Mod A D(Mod A) identifies such a sequence with an Auslander Reiten triangle in the bounded derived category D b (mod A) that is not an Auslander Reiten triangle when viewed as a triangle in the unbounded derived category D(Mod A). This follows, for example, from the fact that an A-module is pure-injective if and only if it is pure-injective when viewed as a complex in D(Mod A) that is concentrated in a single degree. Note that the inclusion D b (mod A) D(Mod A) identifies D b (mod A) with the compact objects in D(Mod A), since A has finite global dimension. 4. Brown representability and Serre duality Brown s representability theorem has been used to construct an Auslander Reiten triangle for each compact object having a local endomorphism ring. In this section, we make this construction functorial; we obtain in this way the connection with the Serre functors arising in the work of Bondal and Kapranov [4], and also in recent work of Reiten and Van den Bergh [18]. Throughout this section, we fix a commutative noetherian ring k that is complete local; that is, k = lim k/m i, where m denotes the unique maximal ideal of k. We assume that T is a compactly generated k-linear category such that Hom(X, Y )is a finitely generated k-module for every pair of compact objects X, Y. We denote by D =Hom k (,E): Modk Mod k the duality that is induced by the injective envelope E = E(k/m). Lemma 4.1. There exists a fully faithful functor T : T c T, together with a natural isomorphism D Hom(X, Y ) = Hom(Y,TX) (4.1) for every compact object X and every object Y in T.

8 368 henning krause Proof. Given a compact object X in T, we apply Brown s representability theorem [16], and we obtain for D Hom(X, ) a representing object TX in T such that D Hom(X, ) = Hom(,TX). The assignment X TX induces a fully faithful functor T : T c T, since we have a natural map Hom(X, Y ) D 2 Hom(X, Y ) D Hom(Y,TX) Hom(TX,TY), where the first map is an isomorphism by our assumption that Hom(X, Y ) is finitely generated over k. The functor T provides the Auslander Reiten translation in T. Proposition 4.2. Let Z be a compact object that is indecomposable. Then there exists an Auslander Reiten triangle (TZ)[ 1] Y Z TZ. Proof. First observe that Γ = End(Z) is local because it is a noetherian k- algebra. Now apply [14, Theorem 2.2]. Note that D =Hom k (,E) is isomorphic to the functor Hom Γ (,I)thatisusedin[14]. The k-linear structure of T implies the pure-injectivity of the compact objects. Lemma 4.3. Every compact object in T is pure-injective. Proof. Fix a compact object X in T. We use the category F =(Tc op, Mod k) of additive functors from compact objects to modules over k. The duality D : Modk Mod k induces a natural map F D 2 F for each F in F, whichis an isomorphism for H X =Hom(,X) Tc.ThusH X is a pure-injective object in F; see [1, Proposition I.3.8]. It follows from [13, Theorem 1.8] that X is a pure-injective object in T. We say that a triangulated category has Auslander Reiten triangles if, for every indecomposable object X, there are Auslander Reiten triangles of the form X Y Z X[1] and V W X V [1]. Theorem 4.4. The following statements are equivalent for T. (1) The category T c of compact objects in T has Auslander Reiten triangles. (2) An object in T is compact if and only if it is isomorphic to some object in the image of T. (3) The functor T : T c T defined by (4.1) induces an equivalence T c T c. Proof. (1) = (2). We need to show that an object X in T is compact if and only if X = TX for some compact object X. Suppose first that X is compact. Let X Y Z X[1] be the corresponding Auslander Reiten triangle. Then X is a pure subobject of (TZ)[ 1], by Corollary 3.4 and Proposition 4.2. However, X is pure-injective, by Lemma 4.3, and therefore isomorphic to (TZ)[ 1] = T (Z[ 1]). The same argument shows that every object in the image of T is compact.

9 auslander reiten triangles and a theorem of zimmermann 369 (2) = (1). Apply Proposition 4.2. (2) (3). This is clear, since T is fully faithful. We comment on how to apply this result. Suppose that we are given a triangulated category C, and we ask whether C has Auslander Reiten triangles. In many cases, there is a natural choice for a compactly generated triangulated category T such T c is equivalent to C; see the examples given below. In fact, many interesting examples are of the form C = D b (A) for some exact category A in the sense of Quillen [17], where D b (A) denotes the bounded derived category of A. One can show that A carries a DG structure such that C is equivalent to the category of compact objects in the unbounded derived category of DG modules over A. Thus Theorem 4.4 can be applied to characterize the existence of Auslander Reiten triangles in C. We end this paper by giving some examples that illustrate our approach towards the existence of Auslander Reiten triangles. Example 1. Let A be a k-algebra, and denote by D(Mod A) the unbounded derived category of Mod A. Recall that a complex of A-modules is perfect if it is quasi-isomorphic to a bounded complex of finitely generated projectives. The perfect complexes are precisely the compact objects in D(Mod A) (see[12, 5.3]), and we identify them with the homotopy category K b (proj A) of finitely generated projective modules. Now fix complexes X and Y in D(A) =D(Mod A), and suppose that X is perfect. Note that we have isomorphisms RHom A (X, Y ) = Y L A X and X = X, where X = RHom A (X, A). Using the adjointness of RHom and L, and viewing the injective k-module E as a complex that is concentrated in degree zero, we obtain the following isomorphism: D Hom D(A) (X, Y ) = Hom k (H 0 (RHom A (X, Y )),E) = Hom D(k) (RHom A (X, Y ),E) = Hom D(k) (Y L A X,E) = Hom D(A) (Y,RHom k (X,E)) = Hom D(A) (Y,RHom k (A L A X,E)) = Hom D(A) (Y,RHom A (X, RHom k (A, E))) = Hom D(A) (Y,X L A Hom k (A, E)), (4.2) which is natural in X and Y. We might view this as an analogue of Serre duality, where the dualizing complex is the A op -module DA =Hom k (A, E), concentrated in degree zero. Now suppose that the algebra A is noetherian over k. Then the functor T defined via (4.1) identifies with the restriction of L A DA: D(Mod A) D(Mod A) to K b (proj A). This functor preserves compactness if and only if the right adjoint RHom A (DA, ) preserves coproducts; see [16, Theorem 5.1]. By definition, this happens if and only if DA is a compact object in D(Mod A). Note that an A-module, when viewed as a complex that is concentrated in a single degree, is compact if and only if it is finitely generated and has finite projective dimension.

10 370 henning krause Also, observe that DA is Artinian over k. It follows that L ADA induces a functor K b (proj A) K b (proj A) if and only if A as an A op -module has finite composition length and is of finite injective dimension. Of course, we have the same result for the opposite algebra A op, and combining both we conclude that T induces an equivalence K b (proj A) K b (proj A) if and only A is an Artin k-algebra which is Gorenstein; thatis,ahas finite injective dimension when viewed as an A-module and also as an A op -module. In particular, we obtain Happel s criterion [6] for the existence of Auslander Reiten triangles in K b (proj A) if we apply Theorem 4.4. This result carries over to differential graded algebras; see [11]. Example 2. Let A be an Artin k-algebra, and denote by K(Inj A) the homotopy category of complexes of injective A-modules. One can show that the injective resolution of each finitely presented A-module is a compact object in K(Inj A). In fact, the category is compactly generated, and the bounded derived category D b (mod A) ofmoda is equivalent to the category of compact objects in K(Inj A) via the following sequence of canonical functors: The functor D b (mod A) D b (Mod A) K +,b (Inj A) K(Inj A). T : D b (mod A) K(Inj A) induces an equivalence D b (mod A) D b (mod A) if and only if A has finite global dimension; see [7]. Example 3. Let k be a field, and let X be a projective scheme over k; thatis, we have a projective morphism f : X Y =Speck. The derived category D(Qcoh X) of quasi-coherent sheaves on X is compactly generated, and the subcategory of compact objects is equivalent to the bounded derived category D b (Vect X) of vector bundles on X; see [16, Example 1.10]. Note that the inclusion Vect X Coh X into the category of coherent sheaves induces an equivalence D b (Vect X) = D b (Coh X), provided that X is smooth. In [16], Neeman shows that Grothendieck s duality theorem is a formal consequence of Brown s representability theorem; that is, the functor Rf : D(Qcoh X) D(Qcoh Y) has a right adjoint f! : D(Qcoh Y) D(Qcoh X). This yields Serre duality for X. Fix complexes X and Y in D(X) =D(Qcoh X), and suppose that X is perfect. Note that we have isomorphisms RHom X (X, Y ) = Y L O X X and X = X, where X = RHom X (X, O X ). Using Grothendieck duality, we obtain the following

11 auslander reiten triangles and a theorem of zimmermann 371 isomorphism: D Hom D(X) (X, Y ) = Hom D(Y) (Rf RHom X (X, Y ),k) = Hom D(X) (RHom X (X, Y ),f! k) = Hom D(X) (Y L O X X,f! k) = Hom D(X) (Y,RHom X (X,f! k)) = Hom D(X) (Y,X L O X f! k), (4.3) which is natural in X and Y. Notice the analogy between (4.2) and (4.3). There is a notion for a scheme to be Gorenstein (see [8, p. 144]), which means that the dualizing complex f! k is isomorphic to an invertible sheaf and therefore inducing an equivalence L O X f! k : D(Qcoh X) D(Qcoh X). For instance, if X is smooth of dimension n, then this equivalence takes a familiar form, since f! k =Ω n X/k [n]. In particular, the category Db (Coh X) has Auslander Reiten triangles. Acknowledgements. It is a pleasure to thank Apostolos Beligiannis for several stimulating discussions about Auslander Reiten triangles. For his help with the algebraic geometry in this paper, I would like to thank Amnon Neeman. In addition, I would like to thank Karsten Schmidt and the referee for several suggestions that have improved the exposition of this paper. References 1. M. Auslander, Functors determined by objects, Representation theory of algebras, Proc. Conf. Philadelphia 1976 (ed. R. Gordon, Dekker, New York, 1978) M. Auslander, I. Reiten and S. O. Smalø, Representation theory of Artin algebras (Cambridge Univ. Press, 1995). 3. A. Beligiannis, Auslander Reiten triangles, Ziegler spectra and Gorenstein rings, K-Theory 32 (2004) A. I. Bondal and M. M. Kapranov, Representable functors, Serre functors, and mutations, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989) (in Russian); Math. USSR Izv. 35 (1990) (English translation). 5. W. W. Crawley-Boevey, Modules of finite length over their endomorphism ring, Representations of algebras and related topics, London Math. Soc. Lecture Note Ser. 168 (ed. S. Brenner and H. Tachikawa, Cambridge Univ. Press, 1992) D. Happel, Triangulated categories in the representation theory of finite dimensional algebras, London Math. Soc. Lecture Note Ser. 119 (Cambridge Univ. Press, 1988). 7. D. Happel, Auslander Reiten triangles in derived categories of finite-dimensional algebras, Proc. Amer. Math. Soc. 112 (1991) R. Hartshorne, Residues and duality, Lecture Notes in Math. 20 (Springer, New York, 1966). 9. I. Herzog, The Auslander Reiten translate, Abelian groups and noncommutative rings, Contemp. Math. 130 (Amer. Math. Soc., Providence, RI, 1992) C. U. Jensen and H. Lenzing, Model theoretic algebra (Gordon and Breach, New York, 1989). 11. P. Jørgensen, Auslander Reiten theory over topological spaces, Comment. Math. Helv. 79 (2004) B. Keller, Deriving DG categories, Ann. Sci. École. Norm. Sup. 27 (1994) H. Krause, Smashing subcategories and the telescope conjecture an algebraic approach, Invent. Math. 139 (2000) H. Krause, Auslander Reiten theory via Brown representability, K-Theory 20 (2000) H. Krause, The spectrum of a module category, Mem. Amer. Math. Soc. 707 (2001).

12 372 auslander reiten triangles and a theorem of zimmermann 16. A. Neeman, The Grothendieck duality theorem via Bousfield s techniques and Brown representability, J. Amer. Math. Soc. 9 (1996) D. Quillen, Higher algebraic K-theory, I, Algebraic K-theory, Lecture Notes in Math. 341 (Springer, New York, 1973) I. Reiten and M. van den Bergh, Noetherian hereditary abelian categories satisfying Serre duality, J. Amer. Math. Soc. 15 (2002) M. Ziegler, Model theory of modules, Ann. Pure Appl. Logic 26 (1984) W. Zimmermann, Existenz von Auslander Reiten Folgen, Arch. Math. 40 (1983) W. Zimmermann, Auslander Reiten sequences over artinian rings, J. Algebra 119 (1988) Henning Krause Institut für Mathematik Universität Paderborn Paderborn Germany hkrause@math.uni-paderborn.de

AUSLANDER-REITEN THEORY VIA BROWN REPRESENTABILITY

AUSLANDER-REITEN THEORY VIA BROWN REPRESENTABILITY AUSLANDER-REITEN THEORY VIA BROWN REPRESENTABILITY HENNING KRAUSE Abstract. We develop an Auslander-Reiten theory for triangulated categories which is based on Brown s representability theorem. In a fundamental

More information

An Axiomatic Description of a Duality for Modules

An Axiomatic Description of a Duality for Modules advances in mathematics 130, 280286 (1997) article no. AI971660 An Axiomatic Description of a Duality for Modules Henning Krause* Fakulta t fu r Mathematik, Universita t Bielefeld, 33501 Bielefeld, Germany

More information

ON MINIMAL APPROXIMATIONS OF MODULES

ON MINIMAL APPROXIMATIONS OF MODULES ON MINIMAL APPROXIMATIONS OF MODULES HENNING KRAUSE AND MANUEL SAORÍN Let R be a ring and consider the category ModR of (right) R-modules. Given a class C of R-modules, a morphism M N in Mod R is called

More information

2 HENNING KRAUSE AND MANUEL SAOR IN is closely related is that of an injective envelope. Recall that a monomorphism : M! N in any abelian category is

2 HENNING KRAUSE AND MANUEL SAOR IN is closely related is that of an injective envelope. Recall that a monomorphism : M! N in any abelian category is ON MINIMAL APPROXIMATIONS OF MODULES HENNING KRAUSE AND MANUEL SAOR IN Let R be a ring and consider the category Mod R of (right) R-modules. Given a class C of R-modules, a morphism M! N in Mod R is called

More information

THE TELESCOPE CONJECTURE FOR HEREDITARY RINGS VIA EXT-ORTHOGONAL PAIRS

THE TELESCOPE CONJECTURE FOR HEREDITARY RINGS VIA EXT-ORTHOGONAL PAIRS THE TELESCOPE CONJECTURE FOR HEREDITARY RINGS VIA EXT-ORTHOGONAL PAIRS HENNING KRAUSE AND JAN ŠŤOVÍČEK Abstract. For the module category of a hereditary ring, the Ext-orthogonal pairs of subcategories

More information

HOMOLOGICAL DIMENSIONS AND REGULAR RINGS

HOMOLOGICAL DIMENSIONS AND REGULAR RINGS HOMOLOGICAL DIMENSIONS AND REGULAR RINGS ALINA IACOB AND SRIKANTH B. IYENGAR Abstract. A question of Avramov and Foxby concerning injective dimension of complexes is settled in the affirmative for the

More information

Extensions of covariantly finite subcategories

Extensions of covariantly finite subcategories Arch. Math. 93 (2009), 29 35 c 2009 Birkhäuser Verlag Basel/Switzerland 0003-889X/09/010029-7 published online June 26, 2009 DOI 10.1007/s00013-009-0013-8 Archiv der Mathematik Extensions of covariantly

More information

Derived Canonical Algebras as One-Point Extensions

Derived Canonical Algebras as One-Point Extensions Contemporary Mathematics Derived Canonical Algebras as One-Point Extensions Michael Barot and Helmut Lenzing Abstract. Canonical algebras have been intensively studied, see for example [12], [3] and [11]

More information

Dedicated to Helmut Lenzing for his 60th birthday

Dedicated to Helmut Lenzing for his 60th birthday C O L L O Q U I U M M A T H E M A T I C U M VOL. 8 999 NO. FULL EMBEDDINGS OF ALMOST SPLIT SEQUENCES OVER SPLIT-BY-NILPOTENT EXTENSIONS BY IBRAHIM A S S E M (SHERBROOKE, QUE.) AND DAN Z A C H A R I A (SYRACUSE,

More information

TRIVIAL MAXIMAL 1-ORTHOGONAL SUBCATEGORIES FOR AUSLANDER 1-GORENSTEIN ALGEBRAS

TRIVIAL MAXIMAL 1-ORTHOGONAL SUBCATEGORIES FOR AUSLANDER 1-GORENSTEIN ALGEBRAS J. Aust. Math. Soc. 94 (2013), 133 144 doi:10.1017/s1446788712000420 TRIVIAL MAXIMAL 1-ORTHOGONAL SUBCATEGORIES FOR AUSLANDER 1-GORENSTEIN ALGEBRAS ZHAOYONG HUANG and XIAOJIN ZHANG (Received 25 February

More information

One-point extensions and derived equivalence

One-point extensions and derived equivalence Journal of Algebra 264 (2003) 1 5 www.elsevier.com/locate/jalgebra One-point extensions and derived equivalence Michael Barot a, and Helmut Lenzing b a Instituto de Matemáticas, UNAM, Mexico 04510 D.F.,

More information

The Diamond Category of a Locally Discrete Ordered Set.

The Diamond Category of a Locally Discrete Ordered Set. The Diamond Category of a Locally Discrete Ordered Set Claus Michael Ringel Let k be a field Let I be a ordered set (what we call an ordered set is sometimes also said to be a totally ordered set or a

More information

Triangulated categories and the Ziegler spectrum. Garkusha, Grigory and Prest, Mike. MIMS EPrint:

Triangulated categories and the Ziegler spectrum. Garkusha, Grigory and Prest, Mike. MIMS EPrint: Triangulated categories and the Ziegler spectrum Garkusha, Grigory and Prest, Mike 2005 MIMS EPrint: 2006.122 Manchester Institute for Mathematical Sciences School of Mathematics The University of Manchester

More information

arxiv: v1 [math.ag] 18 Feb 2010

arxiv: v1 [math.ag] 18 Feb 2010 UNIFYING TWO RESULTS OF D. ORLOV XIAO-WU CHEN arxiv:1002.3467v1 [math.ag] 18 Feb 2010 Abstract. Let X be a noetherian separated scheme X of finite Krull dimension which has enough locally free sheaves

More information

A BRIEF INTRODUCTION TO GORENSTEIN PROJECTIVE MODULES. Department of Mathematics, Shanghai Jiao Tong University Shanghai , P. R.

A BRIEF INTRODUCTION TO GORENSTEIN PROJECTIVE MODULES. Department of Mathematics, Shanghai Jiao Tong University Shanghai , P. R. A BRIEF INTRODUCTION TO GORENSTEIN PROJECTIVE MODULES PU ZHANG Department of Mathematics, Shanghai Jiao Tong University Shanghai 200240, P. R. China Since Eilenberg and Moore [EM], the relative homological

More information

arxiv: v1 [math.kt] 27 Jan 2015

arxiv: v1 [math.kt] 27 Jan 2015 INTRODUCTION TO DERIVED CATEGORIES AMNON YEKUTIELI arxiv:1501.06731v1 [math.kt] 27 Jan 2015 Abstract. Derived categories were invented by Grothendieck and Verdier around 1960, not very long after the old

More information

FILTRATIONS IN ABELIAN CATEGORIES WITH A TILTING OBJECT OF HOMOLOGICAL DIMENSION TWO

FILTRATIONS IN ABELIAN CATEGORIES WITH A TILTING OBJECT OF HOMOLOGICAL DIMENSION TWO FILTRATIONS IN ABELIAN CATEGORIES WITH A TILTING OBJECT OF HOMOLOGICAL DIMENSION TWO BERNT TORE JENSEN, DAG MADSEN AND XIUPING SU Abstract. We consider filtrations of objects in an abelian category A induced

More information

AN AXIOMATIC CHARACTERIZATION OF THE GABRIEL-ROITER MEASURE

AN AXIOMATIC CHARACTERIZATION OF THE GABRIEL-ROITER MEASURE AN AXIOMATIC CHARACTERIZATION OF THE GABRIEL-ROITER MEASURE HENNING KRAUSE Abstract. Given an abelian length category A, the Gabriel-Roiter measure with respect to a length function l is characterized

More information

Gorenstein Homological Algebra of Artin Algebras. Xiao-Wu Chen

Gorenstein Homological Algebra of Artin Algebras. Xiao-Wu Chen Gorenstein Homological Algebra of Artin Algebras Xiao-Wu Chen Department of Mathematics University of Science and Technology of China Hefei, 230026, People s Republic of China March 2010 Acknowledgements

More information

Ideals in mod-r and the -radical. Prest, Mike. MIMS EPrint: Manchester Institute for Mathematical Sciences School of Mathematics

Ideals in mod-r and the -radical. Prest, Mike. MIMS EPrint: Manchester Institute for Mathematical Sciences School of Mathematics Ideals in mod-r and the -radical Prest, Mike 2005 MIMS EPrint: 2006.114 Manchester Institute for Mathematical Sciences School of Mathematics The University of Manchester Reports available from: And by

More information

What is an ind-coherent sheaf?

What is an ind-coherent sheaf? What is an ind-coherent sheaf? Harrison Chen March 8, 2018 0.1 Introduction All algebras in this note will be considered over a field k of characteristic zero (an assumption made in [Ga:IC]), so that we

More information

RECOLLEMENTS GENERATED BY IDEMPOTENTS AND APPLICATION TO SINGULARITY CATEGORIES

RECOLLEMENTS GENERATED BY IDEMPOTENTS AND APPLICATION TO SINGULARITY CATEGORIES RECOLLEMENTS GENERATED BY IDEMPOTENTS AND APPLICATION TO SINGULARITY CATEGORIES DONG YANG Abstract. In this note I report on an ongoing work joint with Martin Kalck, which generalises and improves a construction

More information

DERIVED EQUIVALENCES AND GORENSTEIN PROJECTIVE DIMENSION

DERIVED EQUIVALENCES AND GORENSTEIN PROJECTIVE DIMENSION DERIVED EQUIVALENCES AND GORENSTEIN PROJECTIVE DIMENSION HIROTAKA KOGA Abstract. In this note, we introduce the notion of complexes of finite Gorenstein projective dimension and show that a derived equivalence

More information

SMASHING SUBCATEGORIES AND THE TELESCOPE CONJECTURE AN ALGEBRAIC APPROACH

SMASHING SUBCATEGORIES AND THE TELESCOPE CONJECTURE AN ALGEBRAIC APPROACH SMASHING SUBCATEGORIES AND THE TELESCOPE CONJECTURE AN ALGEBRAIC APPROACH HENNING KRAUSE Abstract. We prove a modified version of Ravenel s telescope conjecture. It is shown that every smashing subcategory

More information

arxiv:math/ v1 [math.rt] 26 Jun 2006

arxiv:math/ v1 [math.rt] 26 Jun 2006 arxiv:math/0606647v1 [math.rt] 26 Jun 2006 AUSLANDER-REITEN TRIANGLES IN SUBCATEGORIES PETER JØRGENSEN Abstract. This paper introduces Auslander-Reiten triangles in subcategories of triangulated categories.

More information

ALGEBRAIC STRATIFICATIONS OF DERIVED MODULE CATEGORIES AND DERIVED SIMPLE ALGEBRAS

ALGEBRAIC STRATIFICATIONS OF DERIVED MODULE CATEGORIES AND DERIVED SIMPLE ALGEBRAS ALGEBRAIC STRATIFICATIONS OF DERIVED MODULE CATEGORIES AND DERIVED SIMPLE ALGEBRAS DONG YANG Abstract. In this note I will survey on some recent progress in the study of recollements of derived module

More information

LOCAL DUALITY FOR REPRESENTATIONS OF FINITE GROUP SCHEMES

LOCAL DUALITY FOR REPRESENTATIONS OF FINITE GROUP SCHEMES LOCAL DUALITY FOR REPRESENTATIONS OF FINITE GROUP SCHEMES DAVE BENSON, SRIKANTH B. IYENGAR, HENNING KRAUSE AND JULIA PEVTSOVA Abstract. A duality theorem for the stable module category of representations

More information

An extension of Dwyer s and Palmieri s proof of Ohkawa s theorem on Bousfield classes

An extension of Dwyer s and Palmieri s proof of Ohkawa s theorem on Bousfield classes An extension of Dwyer s and Palmieri s proof of Ohkawa s theorem on Bousfield classes Greg Stevenson Abstract We give a proof that in any compactly generated triangulated category with a biexact coproduct

More information

A BRIEF INTRODUCTION TO GORENSTEIN PROJECTIVE MODULES. Shanghai , P. R. China

A BRIEF INTRODUCTION TO GORENSTEIN PROJECTIVE MODULES. Shanghai , P. R. China A BRIEF INTRODUCTION TO GORENSTEIN PROJECTIVE MODULES PU ZHANG Department of Mathematics, Shanghai 200240, P. R. China Shanghai Jiao Tong University Since Eilenberg and Moore [EM], the relative homological

More information

DERIVED CATEGORIES IN REPRESENTATION THEORY. We survey recent methods of derived categories in the representation theory of algebras.

DERIVED CATEGORIES IN REPRESENTATION THEORY. We survey recent methods of derived categories in the representation theory of algebras. DERIVED CATEGORIES IN REPRESENTATION THEORY JUN-ICHI MIYACHI We survey recent methods of derived categories in the representation theory of algebras. 1. Triangulated Categories and Brown Representability

More information

On the Existence of Gorenstein Projective Precovers

On the Existence of Gorenstein Projective Precovers Rend. Sem. Mat. Univ. Padova 1xx (201x) Rendiconti del Seminario Matematico della Università di Padova c European Mathematical Society On the Existence of Gorenstein Projective Precovers Javad Asadollahi

More information

Introduction and preliminaries Wouter Zomervrucht, Februari 26, 2014

Introduction and preliminaries Wouter Zomervrucht, Februari 26, 2014 Introduction and preliminaries Wouter Zomervrucht, Februari 26, 204. Introduction Theorem. Serre duality). Let k be a field, X a smooth projective scheme over k of relative dimension n, and F a locally

More information

RELATIVE HOMOLOGY. M. Auslander Ø. Solberg

RELATIVE HOMOLOGY. M. Auslander Ø. Solberg RELATIVE HOMOLOGY M. Auslander Ø. Solberg Department of Mathematics Institutt for matematikk og statistikk Brandeis University Universitetet i Trondheim, AVH Waltham, Mass. 02254 9110 N 7055 Dragvoll USA

More information

STRATIFYING TRIANGULATED CATEGORIES

STRATIFYING TRIANGULATED CATEGORIES STRATIFYING TRIANGULATED CATEGORIES DAVE BENSON, SRIKANTH B. IYENGAR, AND HENNING KRAUSE Abstract. A notion of stratification is introduced for any compactly generated triangulated category T endowed with

More information

On the Hochschild Cohomology and Homology of Endomorphism Algebras of Exceptional Sequences over Hereditary Algebras

On the Hochschild Cohomology and Homology of Endomorphism Algebras of Exceptional Sequences over Hereditary Algebras Journal of Mathematical Research & Exposition Feb., 2008, Vol. 28, No. 1, pp. 49 56 DOI:10.3770/j.issn:1000-341X.2008.01.008 Http://jmre.dlut.edu.cn On the Hochschild Cohomology and Homology of Endomorphism

More information

ALGEBRAS OF DERIVED DIMENSION ZERO

ALGEBRAS OF DERIVED DIMENSION ZERO Communications in Algebra, 36: 1 10, 2008 Copyright Taylor & Francis Group, LLC ISSN: 0092-7872 print/1532-4125 online DOI: 10.1080/00927870701649184 Key Words: algebra. ALGEBRAS OF DERIVED DIMENSION ZERO

More information

HIGHER DIMENSIONAL AUSLANDER-REITEN THEORY ON MAXIMAL ORTHOGONAL SUBCATEGORIES 1. Osamu Iyama

HIGHER DIMENSIONAL AUSLANDER-REITEN THEORY ON MAXIMAL ORTHOGONAL SUBCATEGORIES 1. Osamu Iyama HIGHER DIMENSIONAL AUSLANDER-REITEN THEORY ON MAXIMAL ORTHOGONAL SUBCATEGORIES 1 Osamu Iyama Abstract. Auslander-Reiten theory, especially the concept of almost split sequences and their existence theorem,

More information

Relations for the Grothendieck groups of triangulated categories

Relations for the Grothendieck groups of triangulated categories Journal of Algebra 257 (2002) 37 50 www.academicpress.com Relations for the Grothendieck groups of triangulated categories Jie Xiao and Bin Zhu Department of Mathematical Sciences, Tsinghua University,

More information

Applications of exact structures in abelian categories

Applications of exact structures in abelian categories Publ. Math. Debrecen 88/3-4 (216), 269 286 DOI: 1.5486/PMD.216.722 Applications of exact structures in abelian categories By JUNFU WANG (Nanjing), HUANHUAN LI (Xi an) and ZHAOYONG HUANG (Nanjing) Abstract.

More information

REFLECTING RECOLLEMENTS

REFLECTING RECOLLEMENTS Jørgensen, P. Osaka J. Math. 47 (2010), 209 213 REFLECTING RECOLLEMENTS PETER JØRGENSEN (Received October 17, 2008) Abstract A recollement describes one triangulated category T as glued together from two

More information

SUBCATEGORIES OF EXTENSION MODULES BY SERRE SUBCATEGORIES

SUBCATEGORIES OF EXTENSION MODULES BY SERRE SUBCATEGORIES PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 140, Number 7, July 2012, Pages 2293 2305 S 0002-9939(2011)11108-0 Article electronically published on November 23, 2011 SUBCATEGORIES OF EXTENSION

More information

REPRESENTATION DIMENSION OF ARTIN ALGEBRAS

REPRESENTATION DIMENSION OF ARTIN ALGEBRAS REPRESENTATION DIMENSION OF ARTIN ALGEBRAS STEFFEN OPPERMANN In 1971, Auslander [1] has introduced the notion of representation dimension of an artin algebra. His definition is as follows (see Section

More information

arxiv:math/ v1 [math.rt] 9 Apr 2006

arxiv:math/ v1 [math.rt] 9 Apr 2006 AN AXIOMATIC CHARACTERIZATION OF THE GABRIEL-ROITER MEASURE HENNING KRAUSE arxiv:math/0604202v1 [math.rt] 9 Apr 2006 Abstract. Given an abelian length category A, the Gabriel-Roiter measure with respect

More information

Stable equivalence functors and syzygy functors

Stable equivalence functors and syzygy functors Stable equivalence functors and syzygy functors Yosuke OHNUKI 29 November, 2002 Tokyo University of Agriculture and Technology, 2-24-16 Nakacho, Koganei, Tokyo 184-8588, Japan E-mail: ohnuki@cc.tuat.ac.jp

More information

THE DERIVED CATEGORY OF A GRADED GORENSTEIN RING

THE DERIVED CATEGORY OF A GRADED GORENSTEIN RING THE DERIVED CATEGORY OF A GRADED GORENSTEIN RING JESSE BURKE AND GREG STEVENSON Abstract. We give an exposition and generalization of Orlov s theorem on graded Gorenstein rings. We show the theorem holds

More information

Correct classes of modules

Correct classes of modules Algebra and Discrete Mathematics Number?. (????). pp. 1 13 c Journal Algebra and Discrete Mathematics RESEARCH ARTICLE Correct classes of modules Robert Wisbauer Abstract. For a ring R, call a class C

More information

Derived categories, perverse sheaves and intermediate extension functor

Derived categories, perverse sheaves and intermediate extension functor Derived categories, perverse sheaves and intermediate extension functor Riccardo Grandi July 26, 2013 Contents 1 Derived categories 1 2 The category of sheaves 5 3 t-structures 7 4 Perverse sheaves 8 1

More information

ON SPLIT-BY-NILPOTENT EXTENSIONS

ON SPLIT-BY-NILPOTENT EXTENSIONS C O L L O Q U I U M M A T H E M A T I C U M VOL. 98 2003 NO. 2 ON SPLIT-BY-NILPOTENT EXTENSIONS BY IBRAHIM ASSEM (Sherbrooke) and DAN ZACHARIA (Syracuse, NY) Dedicated to Raymundo Bautista and Roberto

More information

IndCoh Seminar: Ind-coherent sheaves I

IndCoh Seminar: Ind-coherent sheaves I IndCoh Seminar: Ind-coherent sheaves I Justin Campbell March 11, 2016 1 Finiteness conditions 1.1 Fix a cocomplete category C (as usual category means -category ). This section contains a discussion of

More information

ON THE DERIVED DIMENSION OF ABELIAN CATEGORIES

ON THE DERIVED DIMENSION OF ABELIAN CATEGORIES ON THE DERIVED DIMENSION OF ABELIAN CATEGORIES JAVAD ASADOLLAHI AND RASOOL HAFEZI Abstract. We give an upper bound on the dimension of the bounded derived category of an abelian category. We show that

More information

Pure-Injectivity in the Category of Gorenstein Projective Modules

Pure-Injectivity in the Category of Gorenstein Projective Modules Pure-Injectivity in the Category of Gorenstein Projective Modules Peng Yu and Zhaoyong Huang Department of Mathematics, Nanjing University, Nanjing 2193, Jiangsu Province, China Abstract In this paper,

More information

DERIVED CATEGORIES OF STACKS. Contents 1. Introduction 1 2. Conventions, notation, and abuse of language The lisse-étale and the flat-fppf sites

DERIVED CATEGORIES OF STACKS. Contents 1. Introduction 1 2. Conventions, notation, and abuse of language The lisse-étale and the flat-fppf sites DERIVED CATEGORIES OF STACKS Contents 1. Introduction 1 2. Conventions, notation, and abuse of language 1 3. The lisse-étale and the flat-fppf sites 1 4. Derived categories of quasi-coherent modules 5

More information

INJECTIVE CLASSES OF MODULES. Introduction

INJECTIVE CLASSES OF MODULES. Introduction INJECTIVE CLASSES OF MODULES WOJCIECH CHACHÓLSKI, WOLFGANG PITSCH, AND JÉRÔME SCHERER Abstract. We study classes of modules over a commutative ring which allow to do homological algebra relative to such

More information

arxiv: v2 [math.rt] 27 Nov 2012

arxiv: v2 [math.rt] 27 Nov 2012 TRIANGULATED SUBCATEGORIES OF EXTENSIONS AND TRIANGLES OF RECOLLEMENTS PETER JØRGENSEN AND KIRIKO KATO arxiv:1201.2499v2 [math.rt] 27 Nov 2012 Abstract. Let T be a triangulated category with triangulated

More information

The homotopy categories of injective modules of derived discrete algebras

The homotopy categories of injective modules of derived discrete algebras Dissertation zur Erlangung des Doktorgrades der Mathematik (Dr.Math.) der Universität Bielefeld The homotopy categories of injective modules of derived discrete algebras Zhe Han April 2013 ii Gedruckt

More information

STABLE MODULE THEORY WITH KERNELS

STABLE MODULE THEORY WITH KERNELS Math. J. Okayama Univ. 43(21), 31 41 STABLE MODULE THEORY WITH KERNELS Kiriko KATO 1. Introduction Auslander and Bridger introduced the notion of projective stabilization mod R of a category of finite

More information

TRIANGULATED SUBCATEGORIES OF EXTENSIONS, STABLE T-STRUCTURES, AND TRIANGLES OF RECOLLEMENTS. 0. Introduction

TRIANGULATED SUBCATEGORIES OF EXTENSIONS, STABLE T-STRUCTURES, AND TRIANGLES OF RECOLLEMENTS. 0. Introduction TRIANGULATED SUBCATEGORIES OF EXTENSIONS, STABLE T-STRUCTURES, AND TRIANGLES OF RECOLLEMENTS PETER JØRGENSEN AND KIRIKO KATO Abstract. In a triangulated category T with a pair of triangulated subcategories

More information

Relative singularity categories and Gorenstein-projective modules

Relative singularity categories and Gorenstein-projective modules Math. Nachr. 284, No. 2 3, 199 212 (2011) / DOI 10.1002/mana.200810017 Relative singularity categories and Gorenstein-projective modules Xiao-Wu Chen Department of Mathematics, University of Science and

More information

REFLECTING RECOLLEMENTS. A recollement of triangulated categories S, T, U is a diagram of triangulated

REFLECTING RECOLLEMENTS. A recollement of triangulated categories S, T, U is a diagram of triangulated REFLECTING RECOLLEMENTS PETER JØRGENSEN Abstract. A recollement describes one triangulated category T as glued together from two others, S and. The definition is not symmetrical in S and, but this note

More information

Model-theoretic imaginaries and localisation for additive categories

Model-theoretic imaginaries and localisation for additive categories Model-theoretic imaginaries and localisation for additive categories Mike Prest Department of Mathematics Alan Turing Building University of Manchester Manchester M13 9PL UK mprest@manchester.ac.uk December

More information

NONCOMMUTATIVE GRADED GORENSTEIN ISOLATED SINGULARITIES

NONCOMMUTATIVE GRADED GORENSTEIN ISOLATED SINGULARITIES NONCOMMUTATIVE GRADED GORENSTEIN ISOLATED SINGULARITIES KENTA UEYAMA Abstract. Gorenstein isolated singularities play an essential role in representation theory of Cohen-Macaulay modules. In this article,

More information

Aisles in derived categories

Aisles in derived categories Aisles in derived categories B. Keller D. Vossieck Bull. Soc. Math. Belg. 40 (1988), 239-253. Summary The aim of the present paper is to demonstrate the usefulness of aisles for studying the tilting theory

More information

CONTRAVARIANTLY FINITE RESOLVING SUBCATEGORIES OVER A GORENSTEIN LOCAL RING

CONTRAVARIANTLY FINITE RESOLVING SUBCATEGORIES OVER A GORENSTEIN LOCAL RING CONTRAVARIANTLY FINITE RESOLVING SUBCATEGORIES OVER A GORENSTEIN LOCAL RING RYO TAKAHASHI Introduction The notion of a contravariantly finite subcategory (of the category of finitely generated modules)

More information

The Mock Homotopy Category of Projectives and Grothendieck Duality

The Mock Homotopy Category of Projectives and Grothendieck Duality The Mock Homotopy Category of Projectives and Grothendieck Duality Daniel Murfet September 2007 A thesis submitted for the degree of Doctor of Philosophy of the Australian National University Declaration

More information

An introduction to derived and triangulated categories. Jon Woolf

An introduction to derived and triangulated categories. Jon Woolf An introduction to derived and triangulated categories Jon Woolf PSSL, Glasgow, 6 7th May 2006 Abelian categories and complexes Derived categories and functors arise because 1. we want to work with complexes

More information

A note on the singularity category of an endomorphism ring

A note on the singularity category of an endomorphism ring Ark. Mat., 53 (2015), 237 248 DOI: 10.1007/s11512-014-0200-0 c 2014 by Institut Mittag-Leffler. All rights reserved A note on the singularity category of an endomorphism ring Xiao-Wu Chen Abstract. We

More information

arxiv:math/ v1 [math.ra] 20 Feb 2007

arxiv:math/ v1 [math.ra] 20 Feb 2007 ON SERRE DUALITY FOR COMPACT HOMOLOGICALLY SMOOTH DG ALGEBRAS D.SHKLYAROV arxiv:math/0702590v1 [math.ra] 20 Feb 2007 To Leonid L vovich Vaksman on his 55th birthday, with gratitude 1. Introduction Let

More information

1. THE CONSTRUCTIBLE DERIVED CATEGORY

1. THE CONSTRUCTIBLE DERIVED CATEGORY 1. THE ONSTRUTIBLE DERIVED ATEGORY DONU ARAPURA Given a family of varieties, we want to be able to describe the cohomology in a suitably flexible way. We describe with the basic homological framework.

More information

Lecture 9: Sheaves. February 11, 2018

Lecture 9: Sheaves. February 11, 2018 Lecture 9: Sheaves February 11, 2018 Recall that a category X is a topos if there exists an equivalence X Shv(C), where C is a small category (which can be assumed to admit finite limits) equipped with

More information

Duality, Residues, Fundamental class

Duality, Residues, Fundamental class Duality, Residues, Fundamental class Joseph Lipman Purdue University Department of Mathematics lipman@math.purdue.edu May 22, 2011 Joseph Lipman (Purdue University) Duality, Residues, Fundamental class

More information

Singularity Categories, Schur Functors and Triangular Matrix Rings

Singularity Categories, Schur Functors and Triangular Matrix Rings Algebr Represent Theor (29 12:181 191 DOI 1.17/s1468-9-9149-2 Singularity Categories, Schur Functors and Triangular Matrix Rings Xiao-Wu Chen Received: 14 June 27 / Accepted: 12 April 28 / Published online:

More information

SERRE FINITENESS AND SERRE VANISHING FOR NON-COMMUTATIVE P 1 -BUNDLES ADAM NYMAN

SERRE FINITENESS AND SERRE VANISHING FOR NON-COMMUTATIVE P 1 -BUNDLES ADAM NYMAN SERRE FINITENESS AND SERRE VANISHING FOR NON-COMMUTATIVE P 1 -BUNDLES ADAM NYMAN Abstract. Suppose X is a smooth projective scheme of finite type over a field K, E is a locally free O X -bimodule of rank

More information

Weak Proregularity, Weak Stability, and the Noncommutative MGM Equivalence

Weak Proregularity, Weak Stability, and the Noncommutative MGM Equivalence Weak Proregularity, Weak Stability, and the Noncommutative MGM Equivalence Amnon Yekutieli Department of Mathematics Ben Gurion University email: amyekut@math.bgu.ac.il Notes available at http://www.math.bgu.ac.il/~amyekut/lectures

More information

1 Replete topoi. X = Shv proét (X) X is locally weakly contractible (next lecture) X is replete. D(X ) is left complete. K D(X ) we have R lim

1 Replete topoi. X = Shv proét (X) X is locally weakly contractible (next lecture) X is replete. D(X ) is left complete. K D(X ) we have R lim Reference: [BS] Bhatt, Scholze, The pro-étale topology for schemes In this lecture we consider replete topoi This is a nice class of topoi that include the pro-étale topos, and whose derived categories

More information

RELATIVE EXT GROUPS, RESOLUTIONS, AND SCHANUEL CLASSES

RELATIVE EXT GROUPS, RESOLUTIONS, AND SCHANUEL CLASSES RELATIVE EXT GROUPS, RESOLUTIONS, AND SCHANUEL CLASSES HENRIK HOLM Abstract. Given a precovering (also called contravariantly finite) class there are three natural approaches to a homological dimension

More information

Model Theory and Modules, Granada

Model Theory and Modules, Granada Model Theory and Modules, Granada Mike Prest November 2015 Contents 1 Model Theory 1 2 Model Theory of Modules 4 2.1 Duality......................................... 7 3 Ziegler Spectra and Definable Categories

More information

Basic results on Grothendieck Duality

Basic results on Grothendieck Duality Basic results on Grothendieck Duality Joseph Lipman 1 Purdue University Department of Mathematics lipman@math.purdue.edu http://www.math.purdue.edu/ lipman November 2007 1 Supported in part by NSA Grant

More information

Recollement of Grothendieck categories. Applications to schemes

Recollement of Grothendieck categories. Applications to schemes ull. Math. Soc. Sci. Math. Roumanie Tome 56(104) No. 1, 2013, 109 116 Recollement of Grothendieck categories. Applications to schemes by D. Joiţa, C. Năstăsescu and L. Năstăsescu Dedicated to the memory

More information

ATOM SPECTRA OF GROTHENDIECK CATEGORIES

ATOM SPECTRA OF GROTHENDIECK CATEGORIES ATOM SPECTRA OF GROTHENDIECK CATEGORIES RYO KANDA Abstract. This paper explains recent progress on the study of Grothendieck categories using the atom spectrum, which is a generalization of the prime spectrum

More information

ABSOLUTELY PURE REPRESENTATIONS OF QUIVERS

ABSOLUTELY PURE REPRESENTATIONS OF QUIVERS J. Korean Math. Soc. 51 (2014), No. 6, pp. 1177 1187 http://dx.doi.org/10.4134/jkms.2014.51.6.1177 ABSOLUTELY PURE REPRESENTATIONS OF QUIVERS Mansour Aghasi and Hamidreza Nemati Abstract. In the current

More information

Higher dimensional homological algebra

Higher dimensional homological algebra Higher dimensional homological algebra Peter Jørgensen Contents 1 Preface 3 2 Notation and Terminology 5 3 d-cluster tilting subcategories 6 4 Higher Auslander Reiten translations 10 5 d-abelian categories

More information

Cohomological quotients and smashing localizations

Cohomological quotients and smashing localizations Cohomological quotients and smashing localizations Krause, Henning, 1962- American Journal of Mathematics, Volume 127, Number 6, December 2005, pp. 1191-1246 (Article) Published by The Johns Hopkins University

More information

A note on standard equivalences

A note on standard equivalences Bull. London Math. Soc. 48 (2016) 797 801 C 2016 London Mathematical Society doi:10.1112/blms/bdw038 A note on standard equivalences Xiao-Wu Chen Abstract We prove that any derived equivalence between

More information

Cohomology operations and the Steenrod algebra

Cohomology operations and the Steenrod algebra Cohomology operations and the Steenrod algebra John H. Palmieri Department of Mathematics University of Washington WCATSS, 27 August 2011 Cohomology operations cohomology operations = NatTransf(H n ( ;

More information

Dualizing complexes and perverse sheaves on noncommutative ringed schemes

Dualizing complexes and perverse sheaves on noncommutative ringed schemes Sel. math., New ser. Online First c 2006 Birkhäuser Verlag Basel/Switzerland DOI 10.1007/s00029-006-0022-4 Selecta Mathematica New Series Dualizing complexes and perverse sheaves on noncommutative ringed

More information

This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and

This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution

More information

arxiv:math/ v2 [math.ac] 25 Sep 2006

arxiv:math/ v2 [math.ac] 25 Sep 2006 arxiv:math/0607315v2 [math.ac] 25 Sep 2006 ON THE NUMBER OF INDECOMPOSABLE TOTALLY REFLEXIVE MODULES RYO TAKAHASHI Abstract. In this note, it is proved that over a commutative noetherian henselian non-gorenstein

More information

UNIVERSAL DERIVED EQUIVALENCES OF POSETS

UNIVERSAL DERIVED EQUIVALENCES OF POSETS UNIVERSAL DERIVED EQUIVALENCES OF POSETS SEFI LADKANI Abstract. By using only combinatorial data on two posets X and Y, we construct a set of so-called formulas. A formula produces simultaneously, for

More information

arxiv: v2 [math.ct] 27 Dec 2014

arxiv: v2 [math.ct] 27 Dec 2014 ON DIRECT SUMMANDS OF HOMOLOGICAL FUNCTORS ON LENGTH CATEGORIES arxiv:1305.1914v2 [math.ct] 27 Dec 2014 ALEX MARTSINKOVSKY Abstract. We show that direct summands of certain additive functors arising as

More information

The preprojective algebra revisited

The preprojective algebra revisited The preprojective algebra revisited Helmut Lenzing Universität Paderborn Auslander Conference Woodshole 2015 H. Lenzing Preprojective algebra 1 / 1 Aim of the talk Aim of the talk My talk is going to review

More information

WIDE SUBCATEGORIES OF d-cluster TILTING SUBCATEGORIES

WIDE SUBCATEGORIES OF d-cluster TILTING SUBCATEGORIES WIDE SUBCATEGORIES OF d-cluster TILTING SUBCATEGORIES MARTIN HERSCHEND, PETER JØRGENSEN, AND LAERTIS VASO Abstract. A subcategory of an abelian category is wide if it is closed under sums, summands, kernels,

More information

Author's personal copy. Journal of Algebra 324 (2010) Contents lists available at ScienceDirect. Journal of Algebra

Author's personal copy. Journal of Algebra 324 (2010) Contents lists available at ScienceDirect. Journal of Algebra Journal of Algebra 324 2) 278 273 Contents lists available at ScienceDirect Journal of Algebra www.elsevier.com/locate/jalgebra Homotopy equivalences induced by balanced pairs iao-wu Chen Department of

More information

MODULE CATEGORIES WITH INFINITE RADICAL SQUARE ZERO ARE OF FINITE TYPE

MODULE CATEGORIES WITH INFINITE RADICAL SQUARE ZERO ARE OF FINITE TYPE MODULE CATEGORIES WITH INFINITE RADICAL SQUARE ZERO ARE OF FINITE TYPE Flávio U. Coelho, Eduardo N. Marcos, Héctor A. Merklen Institute of Mathematics and Statistics, University of São Paulo C. P. 20570

More information

The Cohomology of Modules over a Complete Intersection Ring

The Cohomology of Modules over a Complete Intersection Ring University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Dissertations, Theses, and Student Research Papers in Mathematics Mathematics, Department of 12-2010 The Cohomology of Modules

More information

Non characteristic finiteness theorems in crystalline cohomology

Non characteristic finiteness theorems in crystalline cohomology Non characteristic finiteness theorems in crystalline cohomology 1 Non characteristic finiteness theorems in crystalline cohomology Pierre Berthelot Université de Rennes 1 I.H.É.S., September 23, 2015

More information

Journal of Algebra 328 (2011) Contents lists available at ScienceDirect. Journal of Algebra.

Journal of Algebra 328 (2011) Contents lists available at ScienceDirect. Journal of Algebra. Journal of Algebra 328 2011 268 286 Contents lists available at ScienceDirect Journal of Algebra www.elsevier.com/locate/jalgebra Generalized Serre duality Xiao-Wu Chen Department of Mathematics, University

More information

MODULE CATEGORIES WITHOUT SHORT CYCLES ARE OF FINITE TYPE

MODULE CATEGORIES WITHOUT SHORT CYCLES ARE OF FINITE TYPE proceedings of the american mathematical society Volume 120, Number 2, February 1994 MODULE CATEGORIES WITHOUT SHORT CYCLES ARE OF FINITE TYPE DIETER HAPPEL AND SHIPING LIU (Communicated by Maurice Auslander)

More information

arxiv: v2 [math.ac] 21 Jan 2013

arxiv: v2 [math.ac] 21 Jan 2013 COMPLETION BY DERIVED DOUBLE CENTRALIZER arxiv:1207.0612v2 [math.ac] 21 Jan 2013 MARCO PORTA, LIRAN SHAUL AND AMNON YEKUTIELI Abstract. Let A be a commutative ring, and let a be a weakly proregular ideal

More information

BROWN REPRESENTABILITY FOLLOWS FROM ROSICKÝ

BROWN REPRESENTABILITY FOLLOWS FROM ROSICKÝ BROWN REPRESENTABILITY FOLLOWS FROM ROSICKÝ Abstract. We prove that the dual of a well generated triangulated category satisfies Brown representability, as long as there is a combinatorial model. This

More information

Relative Left Derived Functors of Tensor Product Functors. Junfu Wang and Zhaoyong Huang

Relative Left Derived Functors of Tensor Product Functors. Junfu Wang and Zhaoyong Huang Relative Left Derived Functors of Tensor Product Functors Junfu Wang and Zhaoyong Huang Department of Mathematics, Nanjing University, Nanjing 210093, Jiangsu Province, China Abstract We introduce and

More information