TTF classes generated by silting modules

TL;DR

This paper investigates when TTF classes in module categories over rings are silting, establishing conditions involving idempotent ideals and projective modules, with results applicable to broad classes of rings.

Contribution

It provides necessary and sufficient conditions for $R/I$ to be silting, linking silting TTF classes to traces of projective modules, especially in semiperfect rings.

Findings

$R/I$ is silting if $I$ is the trace of a projective module.

The converse holds for semiperfect rings.

Characterizes when TTF classes are silting based on idempotent ideals.

Abstract

We study the conditions under which a TTF class in a module category over a ring is silting. Using the correspondence between idempotent ideals over a ring and TTF classes in the module category, we focus on finding the necessary and sufficient conditions for $R/I$ to be a silting $R$-module, and hence for the TTF class $\mathbf{Gen}(R/I)$ to be silting, where $I$ is an idempotent two-sided ideal of $R$. In our main result, we show that $R/I$ is a silting module whenever $I$ is the trace of a projective $R$-module. Furthermore, we demonstrate that the converse holds for a broad class of rings, including semiperfect rings.