The Hilbert-Kunz function of rings of finite Cohen-Macaulay type

Abstract.

Let $(R,{\frak m},k)$ be a Noetherian local ring of prime characteristic p and d its Krull dimension. It is known that for an $\frak m$-primary ideal I of R and a finitely generated R-module N the limit $\lim \limits_{n\to \infty } l_R(N/I^{[n]}N)/p^{dn}$ exists where I ^[n] denotes the ideal of R generated by $x^{p^n}$, $x \in I$, and l _R (M) the length of an R-module M.¶ We will show that the ordinary generating function¶¶$\sum _{n = 0}^\infty l_R (N / I^{[n]} N) t^n \in {\Bbb Q} [[t]]$¶¶of the Hilbert-Kunz function ${\Bbb N} \to {\Bbb N}, n \mapsto l_R (N/I^{[n]}N)$ is rational, i.e., an element of ${\Bbb Q} (t)$, if R ₍₁₎ is a finite R-module, N a maximal Cohen-Macaulay module and R is of finite Cohen-Macaulay type, i.e., the number of isomorphism classes of finite, indecomposable maximal Cohen-Macaulay modules over R is finite. From this result, we deduce that $\lim _ {n\to \infty } l_R (N / I^{[n]} N) / p^{dn} \in {\Bbb Q}$. Here R ₍₁₎ denotes R considered as an R-algebra via the Frobenius map $R \to R, x \mapsto x^{p}$. Actually we will consider a somewhat more general situation using the Frobenius functor.