Speaker: Chin-Yi Jean Chan

Title: Hilbert-Kunz functions of modules over a Noetherian ring regular in codimension one

Abstract: Let R be an excellent local ring of characteristic p, dim R = d and assume R has a perfect residue field. For a finitely generated module M of dimension d, Monsky (1983) proved the existence of the Hilbert-Kunz multiplicity, denoted α, and that the Hilbert-Kunz function is in the form of

λ_n(M) = αq^{d} + O(q^{{d - 1}})
where q = pⁿ.

Huneke, McDermott and Monsky (Math. Res. Lett. 11 (2004) 539-546) further proved that if in addition R is normal, then there exists a constant β such that

λ_n(M) = αq^{d} + βq^{{d - 1}} + O(q^{{d - 2}})
for n >> 0 in Z.

In this talk, I will weaken the normality condition on the ring and extend Huneke-McDermott-Monsky's proof to rings that are regular in codimension one. Each finitely generated module defines a unique cycle class in the Chow group (replacing the divisor class group in the normal case). I will demonstrate a useful property on these cycle classes and apply it to achieve the desired proof.