| Affiliation: | 1.Department of Mathematics,University of Nebraska,Lincoln,USA;2.Department of Mathematics,University of Michigan,Ann Arbor,USA;3.Department of Mathematics and Statistics,Georgia State University,Atlanta,USA |
| Abstract: | It is proved that when R is a local ring of positive characteristic, ({phicolon 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. |
