Cotilting modules over commutative noetherian rings

TL;DR

This paper constructs minimal n-cotilting modules over commutative noetherian rings using injective precovers, explores their localization properties, and provides examples distinguishing different cotilting classes.

Contribution

It introduces a method to construct minimal n-cotilting modules and analyzes their localization behavior, advancing the classification of cotilting modules.

Findings

Constructed minimal n-cotilting modules via injective precovers.

Proved existence of ample cotilting modules for 1-cotilting classes.

Provided counterexample for 2-cotilting classes lacking ample modules.

Abstract

Recently, tilting and cotilting classes over commutative noetherian rings have been classified in arXiv:1203.0907. We proceed and, for each n-cotilting class C, construct an n-cotilting module inducing C by an iteration of injective precovers. A further refinement of the construction yields the unique minimal n-cotilting module inducing C. Finally, we consider localization: a cotilting module is called ample, if all of its localizations are cotilting. We prove that for each 1-cotilting class, there exists an ample cotilting module inducing it, but give an example of a 2-cotilting class which fails this property.