Change of rings and singularity categories

TL;DR

This paper explores how singularity and Gorenstein stable categories change under ring morphisms, identifying conditions for functorial relationships and introducing new categorical concepts.

Contribution

It provides new conditions for functors between singularity categories induced by ring morphisms and introduces the notion of 0-cocompact objects in triangulated categories.

Findings

Established conditions for functors between singularity categories.

Introduced the concept of 0-cocompact objects in triangulated categories.

Constructed explicit right adjoint functors in homotopy categories.

Abstract

We investigate the behavior of singularity categories and stable categories of Gorenstein projective modules along a morphism of rings. The natural context to approach the problem is via change of rings, that is, the classical adjoint triple between the module categories. In particular, we identify conditions on the change of rings to induce functors between the two singularity categories or the two stable categories of Gorenstein projective modules. Moreover, we study this problem at the level of `big singularity categories' in the sense of Krause. Along the way we establish an explicit construction of a right adjoint functor between certain homotopy categories. This is achieved by introducing the notion of 0-cocompact objects in triangulated categories and proving a dual version of Bousfield's localization lemma. We provide applications and examples illustrating our main results.