Homological invariants of modules over contracting endomorphisms

It is proved that when R is a local ring of positive characteristic, $ϕ$ 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 $ϕ$, 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 $ϕ$ is a contracting endomorphism of R, then the Betti numbers and the Bass numbers over $ϕ$ 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.

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