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

TL;DR

This paper introduces the Cohen-Macaulay cone of a Noetherian local domain and demonstrates its application in constructing Cohen-Macaulay rings with specific polynomial Hilbert-Kunz functions.

Contribution

It defines the Cohen-Macaulay cone in the numerical Grothendieck group and applies it to produce examples of rings with tailored Hilbert-Kunz functions.

Findings

Cohen-Macaulay cone is a finitely generated cone in the Grothendieck group.

Constructed Cohen-Macaulay rings with polynomial Hilbert-Kunz functions of prescribed coefficients.

Provided explicit examples using Segre products and Chow group analysis.

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 application to the theory of Hilbert-Kunz functions. It has been shown in Kurano's paper "Numerical equivalence defined on Chow groups of Noetherian local rings", Invent. Math. (2004), that, with a mild condition on R, the numerical Grothendieck group is a finitely generated torsion-free abelian group. The Cohen-Macaulay cone of R is a cone in the numerical Grothendieck group 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, -1 or 1 for d/2<i<d, where d = dim R, 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 Hilbert-Kunz function of R 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 exists for it.