The cone spanned by maximal Cohen-Macaulay modules and an application (Q2796511)

Let \(R\) be a Noetherian local domain of dimension \(d\). The authors introduce two notions: the \textit{Cohen-Macaulay cone} and \textit{test modules} of \(R\). They recall the Grothendieck group \(\overline{G_0(R)}\) of finitely generated \(R\)-modules modulo numerical equivalence, with rank \(\rho(R)\). Inside \(\mathbb R^{\rho(R)} = \overline{G_0(R)} \otimes_{\mathbb Z} \mathbb R\) they define the Cohen-Macaulay cone of \(R\) to be the cone consisting of all non-negative linear combinations of maximal Cohen-Macaulay modules. A module \(M\) is a test module if \(M\) is a maximal Cohen-Macaulay module such that its Todd class consists of only the top term. As an application, they produce various examples of Hilbert-Kunz functions in the polynomial type: for any given integers \(\epsilon_i = 0, \pm 1 \;(d/2 < i < d)\) they 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 rationl number for \(d/2 < i < d\). The existence of such a ring is proved by using Segre products. Test modules are not known to always exist, but they are shown to exist in this case.