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

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.