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

The aim of this paper is to define the notion of the Cohen- Macaulay cone of a Noetherian local domain RR 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 RR, the Grothendieck group G0(R)¯\overline {G_0(R)} of finitely generated RR-modules modulo numerical equivalence is a finitely generated torsion-free abelian group. The Cohen-Macaulay cone of RR is the cone in G0(R)¯R\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 ϵi=0,±1\epsilon _i = 0, \pm 1 (d/2>i>dd/2 > i > d), we shall construct a dd-dimensional Cohen-Macaulay local ring RR (of characteristic pp) and a maximal primary ideal II of RR such that the function ℓR(R/I[pn])\ell _R(R/I^{[p^n]}) is a polynomial in pnp^n of degree dd whose coefficient of (pn)i(p^n)^i is the product of ϵi\epsilon _i and a positive rational number for d/2>i>dd/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.