Some homological criteria for regular, complete intersection and Gorenstein rings

TL;DR

This paper introduces a broad class of local homomorphisms that unify existing homological criteria for characterizing regular, complete intersection, and Gorenstein rings, extending the applicability of these characterizations.

Contribution

It generalizes known homological characterizations of ring properties to a larger class of local homomorphisms, providing a unified framework.

Findings

Characterizations hold for the new class of homomorphisms

Extends the use of homological conditions beyond Frobenius endomorphisms

Provides a unified approach to ring property characterizations

Abstract

Regularity, complete intersection and Gorenstein properties of a local ring can be characterized by homological conditions on the canonical homomorphism into its residue field (Serre, Avramov, Auslander). It is also known that in positive characteristic, the Frobenius endomorphism can also be used for these characterizations (Kunz, ...), and more generally any contracting endomorphism. We introduce here a class of local homomorphisms, in some sense larger than all above, for which these characterizations still hold, providing an unified treatment for this class of homomorphisms.