Minimal silting modules and ring extensions

TL;DR

This paper investigates minimal silting and cosilting modules over rings, especially tame hereditary and commutative noetherian rings, analyzing their properties and behavior under ring extensions.

Contribution

It characterizes minimal tilting and cotilting modules over tame hereditary algebras and explores how minimality is preserved under ring extensions.

Findings

A large cotilting module is minimal iff it contains an adic module as a summand.

Minimal cosilting modules extend along flat ring epimorphisms in commutative noetherian rings.

Results apply to rings of small homological dimension.

Abstract

Ring epimorphisms often induce silting modules and cosilting modules, termed minimal silting or minimal cosilting. The aim of this paper is twofold. Firstly, we determine the minimal tilting and minimal cotilting modules over a tame hereditary algebra. In particular, we show that a large cotilting module is minimal if and only if it has an adic module as a direct summand. Secondly, we discuss the behaviour of minimality under ring extensions. We show that minimal cosilting modules over a commutative noetherian ring extend to minimal cosilting modules along any flat ring epimorphism. Similar results are obtained for commutative rings of small homological dimension.