2010/10/14 by Avramov, Luchezar L., Hochster, Melvin, Iyengar, Srikanth B. +1
#13D02 #13D05 #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.1010.3029
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.