Relative tilting theory in abelian categories II: $n$-$\mathcal{X}$-tilting theory

TL;DR

This paper develops a unified relative tilting theory in abelian categories, connecting various existing notions and providing tools for constructing tilting classes and cotorsion pairs, with applications to quiver representations.

Contribution

It introduces a comprehensive relative tilting framework in abelian categories, unifying previous theories and enabling new constructions in related categories.

Findings

Unified framework for relative tilting in abelian categories

Coincidence of tilting theory in exact and extriangulated categories

Construction of tilting classes and cotorsion pairs in quiver representation categories

Abstract

We introduce a relative tilting theory in abelian categories and show that this work offers a unified framework of different previous notions of tilting, ranging from Auslander-Solberg relative tilting modules on Artin algebras to infinitely generated tilting modules on arbitrary rings. Furthermore, we see that it presents a tool for developing new tilting theories in categories that can be embedded nicely in an abelian category. In particular, we will show how the tilting theory in exact categories built this way, coincides with tilting objects in extriangulated categories introduced recently. We will review Bazzoni\textquoteright s tilting characterization, the relative homological dimensions on the induced tilting classes and parametrise certain cotorsion-like pairs by using $n$-$\mathcal{X}$-tilting classes. As an application, we show how to construct relative tilting classes and cotorsion pairs in $\operatorname{Rep}(Q,\mathcal{C})$ (the category of representations of a quiver $Q$ in an abelian category $\mathcal{C}$) from tilting classes in $\mathcal{C},$ where $Q$ is finite-cone-shape.