On cotilting cotorsion and torsion pairs

TL;DR

This paper explores the relationship between cotilting modules and induced cotorsion and torsion pairs, establishing conditions under which these pairs are of finite type and their implications for ring properties.

Contribution

It demonstrates the equivalence of finite type and $ ext{Sigma}$-pure injectivity for certain cotilting modules and characterizes cotilting torsion pairs over noetherian rings.

Findings

Finite type cotorsion pairs correspond to $ ext{Sigma}$-pure injective cotilting modules.

Torsion pairs cogenerated by such modules have locally noetherian hearts.

Rings with $ ext{Sigma}$-pure injective cotilting modules of low injective dimension are coherent.

Abstract

In this paper we study cotorsion and torsion pairs induced by cotilting modules. We prove the existence of a strong relationship between the $\Sigma$-pure injectivity of the cotilting module and the property of the induced cotorsion pair to be of finite type. In particular for cotilting modules of injective dimension at most 1, or for noetherian rings, the two notions are equivalent. On the other hand we prove that a torsion pair is cogenerated by a $\Sigma$-pure injective cotilting module if and only if its heart is a locally noetherian Grothendieck category. Moreover we prove that any ring admitting a $\Sigma$-pure injective cotilting module of injective dimension at most 1 is necessarily coherent. Finally, for noetherian rings, we characterize cotilting torsion pairs induced by $\Sigma$-pure injective cotilting modules.