The cone spanned by maximal Cohen-Macaulay modules and an application

Abstract:

The aim of this paper is to define the notion of the Cohen- Macaulay cone of a Noetherian local domain $R$ and to present its applications to the theory of Hilbert-Kunz functions. It has been shown by the second author that with a mild condition on $R$, the Grothendieck group $\overline {G_0(R)}$ of finitely generated $R$-modules modulo numerical equivalence is a finitely generated torsion-free abelian group. The Cohen-Macaulay cone of $R$ is the cone in $\overline {G_0(R)}_{\mathbb R}$ spanned by cycles represented by maximal Cohen-Macaulay modules. We study basic properties on the Cohen-Macaulay cone in this paper. As an application, various examples of Hilbert-Kunz functions in the polynomial type will be produced. Precisely, for any given integers $\epsilon _i = 0, \pm 1$ ($d/2 < i < d$), we shall construct a $d$-dimensional Cohen-Macaulay local ring $R$ (of characteristic $p$) and a maximal primary ideal $I$ of $R$ such that the function $\ell _R(R/I^{[p^n]})$ is a polynomial in $p^n$ of degree $d$ whose coefficient of $(p^n)^i$ is the product of $\epsilon _i$ and a positive rational number for $d/2 < i < d$. The existence of such ring is proved by using Segre products to construct a Cohen-Macaulay ring such that the Chow group of the ring is of certain simplicity and that test modules exist for it.