Homological invariants of modules over contracting endomorphisms

It is proved that when R is a local ring of positive characteristic, \({\phi\colon R{\to} R}\) 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.