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.
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). We proved that there is an equivalence of triangulated categories D b (Mod(C)) ∼ = D b (Mod(T )). In the case of dualizing varieties, we proved a version of Happel's theorem for the categories of finitely presented functors. We also proved in this paper, that there exists a relation between covariantly finite coresolving categories, and generalized tilting categories. Extending theorems for artin algebras proved in [4], [5]. In this article we consider the category of maps, and relate tilting categories in the category of functors, with relative tilting in the category of maps. Of special interest is the category mod(modΛ) with Λ an artin algebra.
Abstract. Quasi-hereditary were introduced by L. Scott [Scott,CPS1,CPS2] in order to deal highest weight categories as they arise in the representation theory of semi-simple complex Lie algebras and algebraic groups, and they have been a very important tool in the study of finite-dimensional algebras. On the other hand, functor categories were introduced in representation theory by M. to study the Auslander-Reiten components of finite-dimensional algebras. The aim of the paper is to introduce the concept of quasi-hereditary category, and we can think of the components of the Auslander-Reiten components as quasi-hereditary categories. In this way, we have applications to the functor category Mod(C), with C a component of the Auslander-Reiten quiver.
we introduce the notion of Gabriel filter for a preadditive category C and we show that there is a bijective correspondence between Gabriel filters of C and hereditary torsion theories in the category of additive functors (C, Ab), obtaining a generelization of the theorem given by Gabriel [Ga] and Maranda [Ma] which establishes a bijective correspondence between Gabriel filters for a ring and hereditary torsion theories in the corresponding category of modules.
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