The F-signature and strong F-regularity

We show that the F-signature of a local ring of characteristic p, defined by Huneke and Leuschke, is positive if and only if the ring is strongly F-regular. In [7], Huneke and Leuschke define the F-signature of an F-finite local ring of prime characteristic with perfect residue field. The F-signature, denoted s(R), is an asymptotic measure of the proportion of R-free direct summands in a direct-sum decomposition of R1/pe, the ring of peth roots of R. This proportion seems to give subtle information on the nature of the singularity defining R. For example, the F-signature of any of the two-dimensional quotient singularities (An), (Dn), (E6), (E7), (E8) is the reciprocal of the order of the group G defining the singularity [7, Example 18]. The main theorem of [7] on F-signatures is as follows. Theorem 0.1. [7, Theorem 11] Let (R, m) be a reduced complete F-finite Cohen–Macaulay local ring containing a field of prime characteristic p. Assume that R/m is perfect. Then 1. If s(R)> 0, then R is weakly F-regular. 2. If in addition R is Gorenstein, then s(R) exists, and is positive if and only if R is weakly F-regular.