The Hilbert-Kunz function of rings of finite Cohen-Macaulay type
Let \((R,m)\) be a \(d\)-dimensional local ring of prime characteristic \(p\). For an ideal \(I\) of \(R\), \(I^{[n]}\) denotes the ideal of \(R\) generated by \(x^{p^n}, x\in I\). For an \(R\)-module \(M\), the \(R\)-module \(M_{(n)}\) is \(M\) as an additive group with scalar multiplication given by \(r\cdot m=r^{p^n}m\) for \(r\in R ,\;m\in M\). It is known that for an \(m\)-primary ideal \(I\) and a finitely generated \(R\)-module \(M\), \(\lim_{n\rightarrow \infty} l_R(M/I^{[n]}M)/p^{dn}\) exists (\(l_R\) denotes the length). The purpose of the paper is to show that \(\sum_{n=0}^\infty l_R(M/I^{[n]}M)t^n\) is a rational function if \(R_{(1)}\) is a finitely generated \(R\)-module, \(M\) is a maximal Cohen-Macaulay \(R\)-module and \(R\) is of finite Cohen-Macaulay type, i.e., the number of the isomorphism classes of indecomposable maximal Cohen-Macaulay \(R\)-modules is finite. From this result, it follows that \(\lim_{n\rightarrow \infty} l_R(M/I^{[n]}M)/p^{dn}\) is a rational number (section 4). Actually the author gives a theorem in a general setting in section 3. For this purpose, some lemmas on analysis and linear algebra are given in section 2.
scientific article; zbMATH DE number 1145425