Hilbert-Kunz Functions for Normal Rings

Let $(R,\m,k)$ be an excellent, local, normal ring of characteristic $p$ with a perfect residue field and $\dim R=d$. Let $M$ be a finitely generated $R$-module. We show that there exists $\beta(M) \in$ {\normalsize$\mathbb R$} such that $\lambda(M/I^{[q]}M) = e_{HK}(M) q^d + \beta(M) q^{d-1} + O(q^{d-2})$.

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Published 1 January 2004