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

C. Y. Jean Chan, Kazuhiko Kurano

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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 (formula presented) 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 (formula presented) 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 (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 ℓR(R/I[pn]) is a polynomial in pn of degree d whose coefficient of (pn)i is the product of ϵi 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.

Original languageEnglish
Pages (from-to)939-964
Number of pages26
JournalTransactions of the American Mathematical Society
Issue number2
StatePublished - Feb 2016


  • Cohen-Macaulay cone
  • Hilbert-Kunz function
  • Numerical rational equivalence
  • Segre product
  • Test module


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