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/p e are finite over R for all e ∈ N (i.e. R is F -finite). Consider the sequence { ae qα(R)+d }e=0, 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 s+(R) ≥ 1 − 1 d!pd , 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).