[Preliminary Version] SECOND COEFFICIENTS OF HILBERT-KUNZ FUNCTIONS FOR DOMAINS

Abstract. Let (R,m, k) be an excellent (e.g., F-finite) equidimensional local Noe-therian ring of prime characteristic p with dim(R) = d, I an ideal of R such that λ(R/I) < ∞ and M a finitely generated R-module. We study the existence of β(M) ∈ R such that λ(M/I [q]M) = eHK(I,M)qd + β(M)qd−1 + O(qd−2). We re-fer to β(M) as the second coefficient of the Hilbert-Kunz function. In particular, we show the existence of such β(M) when the defining ideal of the singular locus of R has height at least 2. Throughout this paper R is a Noetherian commutative ring of prime characteristic p with dim(R) = d and I is an arbitrarily given ideal of R such that λR(R/I) < ∞. We write q = pn where n is a varying non-negative integer. For any q, we denote by I [q] the ideal generated by {rq | r ∈ I}