On Gorenstein Projective, Injective and Flat Dimensions - A Functorial Description with Applications

TL;DR

This paper explores Gorenstein homological dimensions, providing new criteria for their finiteness over a broader class of rings, including those in algebraic geometry and non-commutative algebra.

Contribution

It extends the class of rings known to have practical criteria for Gorenstein dimension finiteness, including complex algebraic and non-commutative rings.

Findings

Established criteria for Gorenstein dimensions over new classes of rings

Included rings from algebraic geometry and non-commutative algebra

Enhanced understanding of homological properties in advanced ring contexts

Abstract

Gorenstein homological dimensions are refinements of the classical homological dimensions, and finiteness singles out modules with amenable properties reflecting those of modules over Gorenstein rings. As opposed to their classical counterparts, these dimensions do not immediately come with practical and robust criteria for finiteness, not even over commutative noetherian local rings. In this paper we enlarge the class of rings known to admit good criteria for finiteness of Gorenstein dimensions: It now includes, for instance, the rings encountered in commutative algebraic geometry and, in the non-commutative realm, $k$--algebras with a dualizing complex.