Homological invariants of modules over contracting endomorphisms
arXiv:1010.3029
Abstract
It is proved that when R is a local ring of positive characteristic, 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 , 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 is a contracting endomorphism of R, then the Betti numbers and the Bass numbers over 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.
14 pages. This has been accepted for publication in the Math. Ann