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 In this note, K is a fixed field. An algebra means a finite dimensional selfinjective K-algebra. An module means a finite dimensional left module. For an algebra Λ, we denote by modλ the category of Λ-modules. The stable category of an algebra Λ is denoted by modλ. The triangularity of stable categories was stedied by Happel, Rickard and many mathematcian [2], [3], [5], [9], [10]. Our aim in this note is to show that for an equivalence functor F : modλ modλ, F is a triangle functor if and only if F commutes with the syzygy functors i.e., F Ω Ω F, where Ω and Ω, respectively, are the syzygy functors of modλ and modλ, respectively. Note that a stable equivalence functor commutes with the syzygy functors if and only if it commutes with the Nakayama functors [1], and therefore if Λ and Λ are stable equivalence symmetric algebras, then modλ and modλ are triangle equivalent. The detail in this note is referred to [6]. 1 A stable category Let Λ be an algebra. The stable (module) category modλ is the factor category of modλ by morphisms factoring through projective Λ-modules. Objects of modλ are Λ-modules. For Λ-modules X, Y, a morphism from X to Y in modλ is given by its residue class in Hom Λ (X, Y )/P(X, Y ), where P(X, Y ) is the subset of Hom Λ (X, Y ) consisting of morphisms which factor through projective Λ-modules. For each morphism u : X Y in modλ, the residue class of u is denoted by u. 1
For a Λ-module X, we denote by µ X : X I X the injective hull in modλ, and let Ω 1 X be the cokernel of µ X. Then for a morphism u : X Y in modλ, Ω 1 u is given by the following commutative diagram with exact rows 0 X u 0 Y µ X IX Ω 1 X 0 Ω 1 u µ Y IY Ω 1 Y 0. Note that Ω 1 u is uniquely determined by u in modλ i.e., u factors through projective Λ-modules if and only if so does Ω 1 u. Then Ω 1 u is denoted by Ω 1 u, and Ω 1 : modλ modλ is the equivalence functor [4]. The quasi-inverse of Ω 1 is called the syzygy (or the Heller s loop-space) functor. For a morphism u : X Y in modλ, we have the following commutative diagram with exact rows in modλ 0 X u 0 Y µ X π Ω 1 X IX Ω 1 X 0 x v w C(u) Ω 1 X 0. (1) C(u) is called a mapping cone of u. Note that a mapping cone is uniquely determined by a morphism in modλ, and C(u) is also called a mapping cone of a morphism u in modλ. The diagram (1) induces the sequence X u v w Y C(u) Ω 1 X which is called a standard triangle in modλ. Let T be a collection of sextuples which are isomorphic to standard triangles in modλ. The stable category modλ is regarded as a triangulated category [2], [3] whose translation is the equivalence functor Ω 1 : modλ modλ, and T is the collection of triangles. A short exact sequence 0 X 1 f X2 g X3 0 is called quasiindecomposable provided that if there exist short exact sequences 0 Y 1 f 1 g 1 f 2 g 2 Y 2 Y3 ( 0 and) 0 Z( 1 Z2 ) Z3 0 such that X i Y i Z i (i = 1, 2, 3) f1 0 g1 0 and f =, g =, then either Y 0 f 2 0 g i are projective for any i and 2 Z j is nonzero nonprojective for some j, or Z i are projective for any i and Y j is nonzero nonprojective for some j. Then, 0 X v 1 X w 2 X 3 0 is quasi-indecomposable if and only if the induced triangle (X 1, X 2, X 3, v, w, u) is indecomposable in modλ. 2
2 A stable equivalence functor which commutes with the syzygy functors Let Λ and Λ be algebras and F : modλ modλ an equivalence functor. We denote by Ω and Ω, respectively, are the syzygy functors of Λ and Λ, respectively. For a morphism u in modλ, the mapping cone of u denotes by C(u), and for a morphism u in modλ, the mapping cone of u denotes by C (u ). We denote by ι X : X Q X the injective hull of a Λ -module X. Lemma 2.1 ([6]). Assume that F : modλ modλ is an equivalence functor with Ω F F Ω. For a morphism u : X Y in modλ, we set u := F (u). Let 0 @ u 1 A µ X v x 0 X Y I X C(u) 0 be a quasi-indecomposable short exact sequence in modλ, and 0 1 @ u A ι F X v 1 x 1 0 F X F Y Q F X C (u ) 0 a short exact sequence in modλ. Then there exists an isomorphism z : C(u) F 1 C (u ) in modλ such that F (zv) = v 1. An additive functor F : modλ modλ is called a triangle (or exact) functor provided that Ω 1 F F Ω 1 and sextuple (F X, F Y, F Z, F u, F v, F w) is a triangle in modλ for any triangle (X, Y, Z, u, v, w) in modλ. If F is an equivalence and a triangle functor, then modλ and modλ are equivalent as triangulated categories. Theorem 2.2 ([6]). The following conditions are equivalent for an equivalence functor F : modλ modλ 1. F is a triangle functor. 2. F commutes with the syzygy functors i.e., Ω F F Ω. The outline of proof. Assume that Ω F F Ω in order to prove 2 = 1. Note that modλ is a Krull-Schmidt category, because so is modλ. Any triangle in modλ decomposes finite direct sum of indecomposable triangles. Let u be a morphism in modλ with an indecomposable standard triangle (X, Y, C(u), u, v, w) in modλ, and we use the notation of Lemma 2.1. In particular, let (F X, F Y, C (u ), u, v 1, w 1 ) be a standard triangle in modλ. 3
We have the following commutative diagram with exact rows in modλ 0 Y v 0 C(u) µ Y π Ω 1 Y IY Ω 1 Y 0 0 0 @ 0 1 @ w 1 A A s 1 Ω 1 u π Ω 1 Ω X I 1 Y Y Ω 1 Y 0. Note that the bottom rows is quasi-indecomposable. We also have the following commutative diagram with exact rows 0 F X u ι F X ρ Ω 1 F X QF X Ω 1 F X 0 v 1 x 1 0 F Y 0 1 C (u ) Ω 1 F X 0 @ v A ι F Y t 0 F Y F C(u) Q F Y Ω 1 F X Q 0 v 1 0 F Y C (u w 1 ) Ω 1 F X 0 ( ) w 0 where t F = F w and Q is some projective Λ s 1 -module, by Lemma 2.1. Since the top and bottom rows induce a standard triangle, (F X, F Y, F C(u), F u, F v, F w) is also a triangle in modλ. F is a triangle functor. In [1, Chapter X], we obtain the following corollary. Corollary 2.3. Let F : modλ modλ be an equivalence functor. If it holds N F F N, then F is a triangle functor. In particular, if Λ and Λ are symmetric algebras, then any equivalence functor F : modλ modλ is a triangle functor. In [7], we constructed the trivial extension algebra and the non-symmetric Hochschild extension algebra given by 2-cocycle. They are stably equivalent by [8, Example 4.4]. Therefore, we obtain the non-triangle stable equivalence functor. Howevere, we do not know the example which is a non-triangle stable equivalence functor if base field is algebraically closed. w 1 References [1] M. Auslander, I. Reiten and S. O. Smalø, Representation theory of Artin algebras, Cambridge Studies in Adv. Math. 36, Cambridge 1995. 4
[2] D. Happel, On the derived category of a finite-dimensional algebra, Comment. Math. Helv. 62 (1987) 339 389. [3] D. Happel, Triangulated categories in the representation theory of finitedimensional algebras, London Math. Soc. Lecture Notes 119, University Press, Cambridge (1988). [4] A. Heller, The loop-space functor in homological algebra, Trans. Amer. Math. Soc. 96 (1960), 382 394. [5] B. Keller and D. Vossieck, Sous les catégories dérivées, C. R. Acad. Soc. Paris, 305 (1987), 225 228. [6] Y. Ohnuki, The triangularity of stable equivalence functors, preprint. [7] Y. Ohnuki, K. Takeda and K. Yamagata Symmetric Hochschild extension algebras, Colloq. Math. 80 (1999), 155 174. [8] Y. Ohnuki, K. Takeda and K. Yamagata Automorphisms of repetitive algebras, J. Alg. 232 (2000), 708 724. [9] J. Rickard, Derived categories and stable equivalence, J. Pure Appl. Alg. 61 (1989), 303 317. [10] J. Rickard, Derived equivalences as derived functors, J. London Math. Soc. 43 (1991), 37 48. 5