Tilting Theory and Functor Categories III. The Maps Category

TL;DR

This paper advances tilting theory within functor categories, establishing equivalences of derived categories and exploring relations between tilting and coresolving categories, especially in the context of artin algebras.

Contribution

It generalizes tilting theory to the category of maps and relates tilting categories in functor categories to relative tilting in the category of maps.

Findings

Established an equivalence of derived categories for functor categories.

Extended Happel's theorem to finitely presented functors in dualizing varieties.

Linked covariantly finite coresolving categories with generalized tilting categories.

Abstract

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. 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 $D^b (Mod(C))\cong Db (Mod(T))$. 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. 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 $mod(mod \Lambda)$ with $\Lambda$ an artin algebra.