THE HILBERT-KUNZ FUNCTION OF RINGS OF FINITE COHEN-MACAULAY TYPE

Let (R, m, k) be a Noetherian local ring of prime characteristic p and d its Krull dimension. It is known that for an m-primary ideal l of R and a finitely generated R-module N the limit lim(n=infinity l?R(N/Ga mma?(\n\N)/p(dn) exists where I-[n] denotes the ideal of R generated b y x(Pn), x is an element of I, and I-R(M) the length of an R-module M. We will show that the ordinary generating function (n=0)Sigma(infinit y)l(R)(N/(IN)-N-\n\)t(n) is an element of Q[[t]] of the Hilbert-Kunz f unction N --> N, n --> I-R(N/Gamma(\n\)N) is rational, i.e., an elemen t of Q(t), if R-(1) is a finite R-module, N a maximal Cohen-Macaulay m odule and R is of finite Cohen-Macaulay type, i.e., the number of isom orphism classes of finite, indecomposable maximal Cohen-Macaulay modul es over R is finite. From this result, we deduce that lim l(R)(N/I-(N) N)/p(dn) is an element of Q. Here R-(1) denotes R considered as an R-a lgebra via the Frobenius map R --> R, x --> x(P). Actually we will con sider a somewhat more general situation using the Frobenius functor.