Extending ($\tau$-)tilting subcategories and (co)silting modules

TL;DR

This paper explores how tilting, $ au$-tilting, silting, and cosilting modules and subcategories behave under restriction and extension functors between a finite-dimensional algebra and its one-point extension, extending classical results.

Contribution

It extends the study of tilting and support $ au$-tilting subcategories and related modules from small to large module categories, revealing new behaviors under functorial operations.

Findings

Behavior of tilting and support $ au$-tilting subcategories under functors

Extension of classical results from modules to subcategories

Analysis of finendo quasi-tilting, silting, and cosilting modules

Abstract

Let $B$ be a finite dimensional algebra and $A=B[P_0]$ be the one-point extension algebra of $B$ with respect to the finitely generated projective $B$-module $P_0$. The categories of $B$-modules and $A$-modules are related by two adjoint functors $\mathcal{R}$ and $\mathcal{E}$, called the restriction and the extension functors, respectively. Based on the nice homological properties of these two functors, restriction and extension of some notions such as tilting and $\tau$-tilting modules have been studied in the category of finitely presented modules, i.e. small mod. In this paper, we investigate the behaviour of tilting and support $\tau$-tilting subcategories with respect to these two functors. Moreover, we investigate the restriction and the extension of special related modules such as finendo quasi-tilting modules, silting modules, and cosilting modules. Our studies will be done in the category of all modules, which will be called large Mod. Based on such study, in addition to the new results, classical results are extended, not only from modules to subcategories but also from small mod to large Mod.