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 R1/pe are finite over R for all e∈N (i.e. R is F-finite). Consider the sequence , in which α(R)=logp[k:kp], q=pe, and ae is the maximal rank of free R-modules appearing as direct summands of R-module R1/q. Denote by s−(R) and s+(R) the liminf and limsup, respectively, of the above sequence as e→∞. 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 R1/q over R. We show that: (1) If , then R is regular; (2) If R is excellent such that RP is Gorenstein for every P∈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).