Tilting Theory and Functor Categories I. Classical Tilting

Abstract

Tilting theory has been a very important tool in the classification of finite dimensional algebras of finite and tame representation type, as well as, in many other branches of mathematics. Happel [Ha] proved that generalized tilting induces derived equivalences between module categories, and tilting complexes were used by Rickard [Ri] to develop a general Morita theory of derived categories.In the other hand, functor categories were introduced in representation theory by M. Auslander [A], [AQM] and used in his proof of the first Brauer-Thrall conjecture [A2] and later on, used systematically in his joint work with I. Reiten on stable equivalence [AR], [AR2] and many other applications.Recently, functor categories were used in [MVS3] to study the Auslander-Reiten components of finite dimensional algebras.The aim of the paper is to extend tilting theory to arbitrary functor categories, having in mind applications to the functor category Mod(mod Λ ), with Λ a finite dimensional algebra.

Representation theory of strongly locally finite quivers

“…Note that Reiten and Van den Bergh have done so (with no proof) for each type of infinite Dynkin quivers with the alternating orientation; see [39,(III.3)]. More partial results in this subject can be found in [15,36,37].…”

Tilting Theory and Functor Categories II. Generalized Tilting

“…In the first paper [17] we generalized classical tilting to the category of contravariant functors from a preadditive skeletally small category C, to the category of abelian groups and generalized Bongartz's proof [10] of Brenner-Butler's theorem [11]. We then applied the theory so far developed, to the study of locally finite infinite quivers with no relations, and computed the Auslander-Reiten components of infinite Dynkin diagrams.…”

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In this paper we continue the project of generalizing tilting theory to the category of contravariant functors Mod(C), from a skeletally small preadditive category C to the category of abelian groups, initiated in [17]. In [18] we introduced the notion of a a generalized tilting category T , and extended Happel's theorem to Mod(C).

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The Auslander–Reiten Components Seen as Quasi-Hereditary Categories

“…We will assume in this section that C is a dualizing Krull-Schmidt K-variety. In this way, finitely presented functors have projective covers (see Theorem 2 in [MVO1]), and the category of finitely presented functors mod(C) has enough projective and injective objects. In the rest of this work, all the C-modules we are considering are finitely presented.…”

“…On the other hand, functor categories were introduced in representation theory by Auslander [A] and used in his proof of the first Brauer-Thrall conjecture [A2] and later used systematically in his joint work with I. Reiten on stable equivalence and many other applications [AR,AR2]. Recently, functor categories were employed by Martínez-Villa and Solberg to study the Auslander-Reiten components of finitedimensional algebras [MVS3] and to develop tilting theory in arbitrary functor categories [MVO1,MVO2].…”