Observations on the F-signature of local rings of characteristic p

Let (R, m, k) be a d-dimensional Noetherian reduced local ring of prime characteristic p such that R-1/pe are finite over R for all e is an element of N (i.e. R is F-finite). Consider the sequence [a(e)/q(alpha(R)+d)}(e=0)(infinity) in which alpha(R) = log(p)[k: k(p)], q = p(e) and a, is the maximal rank of free R-modules appearing as direct summands of R-module R-1/q. Denote by s(-) (R) and s(+) (R) the liminf and limsup, respectively, of the above sequence as e -> infinity. If s- (R) = s(+) (R), then the limit, denoted by s (R), is called the F-signature of R. It turns out that the F-signature can be defined in a way that is independent of the module finite property of R-1/q over R. We show that: (1) If s(+) (R) >= 1 - 1/(d!p(d)), then R is regular; (2) If R is excellent such that R-P is Gorenstein for every P is an element of Spec(R)\{m}, then s(R) exists; (3) If (R, m) -> (S, n) is a local flat ring homomorphism, then s(+/-) (R) >= s +/- (S) and, if furthermore S/mS is Gorenstein, s(+/-)(S) >= s(+/-)(R)s(S/mS). (c) 2005 Elsevier Inc. All rights reserved.