Cleft extensions of rings and singularity categories

TL;DR

This paper systematically studies Gorenstein homological properties and singularity categories in cleft extensions of rings, unifying various known results and comparing different singularity categories.

Contribution

It provides a unified framework for Gorenstein homological aspects in cleft extensions, including singularity categories and equivalences.

Findings

Established singular equivalences between algebras in cleft extensions.

Unified treatment of Gorenstein projective modules in various ring extensions.

Compared big singularity categories of cleft extensions.

Abstract

This paper provides a systematic treatment of Gorenstein homological aspects for cleft extensions of rings. In particular, we investigate Goresnteinness, Gorenstein projective modules and singularity categories in the context of cleft extensions of rings. This setting includes triangular matrix rings, trivial extension rings and tensor rings, among others. Under certain conditions, we prove singular equivalences between the algebras in a cleft extension, unifying an abundance of known results. Moreover, we compare the big singularity categories of cleft extensions of rings in the sense of Krause.