Silting modules and ring epimorphisms

TL;DR

This paper explores the relationship between silting modules and ring epimorphisms, providing explicit constructions and parametrizations, especially for hereditary rings, and revealing a lattice structure among homological ring epimorphisms.

Contribution

It introduces the concept of silting modules as a broader framework linking ring epimorphisms and tilting modules, with explicit descriptions and parametrizations for hereditary rings.

Findings

Partial silting modules correspond to explicit ring epimorphisms.

Homological ring epimorphisms form a lattice structure.

Parametrization of epimorphisms by silting modules for hereditary rings.

Abstract

There are well-known constructions relating ring epimorphisms and tilting modules. The new notion of silting module provides a wider framework for studying this interplay. To every partial silting module we associate a ring epimorphism which we describe explicitly as an idempotent quotient of the endomorphism ring of the Bongartz completion. For hereditary rings, this assignment is used to parametrise homological ring epimorphisms by silting modules. We further show that homological ring epimorphisms of a hereditary ring form a lattice which completes the poset of noncrossing partitions in the case of finite dimensional algebras.