Auslander-Reiten Triangles in Derived Categories of Finite-Dimensional Algebras
Proceedings of the American Mathematical Society
Let A be a finite-dimensional algebra. The category mod A of finitely generated left /1-modules canonically embeds into the derived category D (A) of bounded complexes over triodo and the stable category mod T(A) of Z-graded modules over the trivial extension algebra of A by the minimal injective cogenerator. This embedding can be extended to a full and faithful functor from D (A) to mod T(A). Using the concept of Auslander-Reiten triangles it is shown that both categories are equivalent only
... A has finite global dimension. Let k be a field and A a finite-dimensional k-algebra. By mod A we denote the category of finitely generated left ¿(-modules. Let T(A) be the trivial extension algebra by the bimodule Homk(A, k). Then T(A) is a Z-graded algebra and the category mod T(A) of finitely generated Z-graded r(^)-modules is a Frobenius category in the sense of [HI]. The stable category of modz T(A) is denoted by mod2 T(A). In [HI] (see also [H2, KV]) we show that the derived category D (A) = D (mod ,4) and mod T(A) are triangle-equivalent, if the global dimension of A is finite. The purpose of this note is to prove the converse of this result. We point out that examples of this fact have been obtained in [TW], Moreover we will give a surprisingly simple construction of a full and faithful exact functor F from D (A) to mod T(A). The proof is based on the fact that mod T(A) has Auslander-Reiten triangles in the sense of [HI], whereas we will show in section one that D (A) has Auslander-Reiten triangles only if A is of finite global dimension. Thus obtaining the converse of a result established in [HI]. In §2 we will give the construction of F and the proof of the aforementioned theorem. The composition of two morphisms /:I-»V and g : Y -* Z in a given category is denoted by fg.