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


Authors: C-Y. Jean Chan and Kazuhiko Kurano
Journal: Trans. Amer. Math. Soc. 368 (2016), 939-964
MSC (2010): Primary 13C14, 13D15, 13D40, 14C17, 14C40
DOI: https://doi.org/10.1090/tran/6457
Published electronically: May 29, 2015
MathSciNet review: 3430354
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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.


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Additional Information

C-Y. Jean Chan
Affiliation: Department of Mathematics, Central Michigan University, Mt. Pleasant, Michigan 48858
Email: chan1cj@cmich.edu

Kazuhiko Kurano
Affiliation: Department of Mathematics, School of Science and Technology, Meiji University, Higashimita 1-1-1, Tama-ku, Kawasaki 214-8571, Japan
Email: kurano@isc.meiji.ac.jp

DOI: https://doi.org/10.1090/tran/6457
Keywords: Numerical rational equivalence, Cohen-Macaulay cone, test module, Hilbert-Kunz function, Segre product
Received by editor(s): November 15, 2012
Received by editor(s) in revised form: November 16, 2012, and December 5, 2013
Published electronically: May 29, 2015
Additional Notes: The first author was partially supported by Early Career Investigator’s Grant #C61368 of Central Michigan University
The second author was partially supported by KAKENHI (24540054)
Article copyright: © Copyright 2015 American Mathematical Society