Homological invariants of modules over contracting endomorphisms

TL;DR

This paper generalizes Kunz's theorem by showing that certain homological invariants over contracting endomorphisms characterize regularity in local rings, extending results to arbitrary characteristic.

Contribution

It proves that finite flat or injective dimension over a Frobenius endomorphism implies regularity, generalizing Kunz's criterion to broader contracting endomorphisms.

Findings

Betti and Bass numbers grow exponentially at the same rate over contracting endomorphisms.

Finite homological dimensions over Frobenius imply regularity in positive characteristic.

Growth rates of invariants over contracting endomorphisms mirror those of the residue field.

Abstract

It is proved that when R is a local ring of positive characteristic, $\phi$ is its Frobenius endomorphism, and some non-zero finite R-module has finite flat dimension or finite injective dimension for the R-module structure induced through $\phi$, then R is regular. This broad generalization of Kunz's characterization of regularity in positive characteristic is deduced from a theorem concerning a local ring R with residue field of k of arbitrary characteristic: If $\phi$ is a contracting endomorphism of R, then the Betti numbers and the Bass numbers over $\phi$ of any non-zero finitely generated R-module grow at the same rate, on an exponential scale, as the Betti numbers of k over R.