Tilting theory and functor categories I. Classical tilting

TL;DR

This paper extends classical tilting theory to functor categories, aiming to enhance the understanding of derived equivalences and module categories in representation theory.

Contribution

It introduces a framework for tilting theory within arbitrary functor categories, broadening its applicability to categories like Mod(mod{ abla}) for finite dimensional algebras.

Findings

Extended tilting theory to functor categories.

Established connections between tilting and derived equivalences.

Potential applications to the classification of algebra representations.

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 and used in his proof of the first Brauer- Thrall conjecture and later on, used systematically in his joint work with I. Reiten on stable equivalence and many other applications. Recently, functor categories were used to study the Auslander- Reiten components of finite dimensional algebras. The aim of the paper is to extend tilting theory to arbitrary functor cate- gories, having in mind applications to the functor category Mod(mod{\Lambda}), with {\Lambda} a finite dimensional algebra.