[Preliminary Version] THE F-RATIONAL SIGNATURE AND DROPS IN THE HILBERT-KUNZ MULTIPLICITY

Abstract. Let (R,m) be a Noetherian local ring of prime characteristic p. We define the F-rational signature of R, denoted by r(R), as the infimum of drops in Hilbert-Kunz multiplicities when one enlarges the ideals generated by systems of parameters. In particular, if R is excellent, then R is F-rational if and only if r(R)> 0, the proof of which depends on the following result: Given an m-primary ideal I in R, there exists a positive δI ∈ R+ such that, for any ideal J ⊇ I, eHK(I,R)−eHK(J,R) is either 0 or ≥ δI. Then we study how F-rational signature behaves under deformations, local (and flat) ring extensions, and localizations. Throughout this paper we assume that (R,m, k) is a Noetherian local ring of prime characteristic p, where m is the maximal ideal and k = R/m is the residue field of R. Then there is the Frobenius homomorphism F: R → R defined by r 7 → rp for any r ∈ R. Therefore, for any e ∈ N, we have the iterated Frobenius homomorphis