Homological Aspects of Separable Extensions of Triangulated Categories

TL;DR

This paper studies how homological properties of triangulated categories behave under separable extensions, showing invariance of key invariants and relating singularity categories.

Contribution

It demonstrates that homological invariants are preserved under separable extensions and connects singularity categories in this context, extending classical algebraic results.

Findings

Homological invariants like global dimension are preserved under separable extensions.

The singularity category of an extension relates to that of the original category via separable extension.

Provides new examples involving rings, schemes, and dg algebras with separable extensions.

Abstract

We investigate the homological behaviour of compactly generated triangulated categories under separable extensions. We show that homological invariants (finiteness of global dimension, gorensteinness and regularity) are preserved under such extensions. We also establish a relation between singularity categories in this setting, proving that the singularity category of a separable extension is equivalent, up to retracts, to a separable extension of the singularity category. Our results unify and extend classical phenomena from commutative and equivariant algebra, and provide new examples involving separable extensions of rings, quotient schemes, and skew group dg algebras.