Silting modules and ring epimorphisms

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

Silting modules

We introduce the new concept of silting modules. These modules generalise tilting modules over an arbitrary ring, as well as support \u3c4-tilting modules over a finite dimensional algebra recently introduced by Adachi, Iyama and Reiten. We show that silting modules generate torsion classes that provide left approximations, and that every partial silting module admits an analogue of the Bongartz complement. Furthermore, we prove that silting modules are in bijection with 2-term silting complexes and with certain t-structures and co-t-structures in the derived module category. We also see how some of these bijections hold for silting complexes of arbitrary finite length

A characterisation of τ-tilting finite algebras

We prove that a finite dimensional algebra is τ-tilting finite if and only if it does not admit large silting modules. Moreover, we show that for a τtilting finite algebra A there is a bijection between isomorphism classes of basic support τ-tilting (that is, finite dimensional silting) modules and equivalence classes of ring epimorphisms A → B with TorA1 (B, B) = 0. It follows that a finite dimensional algebra is τ-tilting finite if and only if there are only finitely many equivalence classes of such ring epimorphisms