The cone spanned by maximal Cohen-Macaulay modules and an application
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- by C-Y. Jean Chan and Kazuhiko Kurano PDF
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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.References
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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
- 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) - © Copyright 2015 American Mathematical Society
- 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
- MathSciNet review: 3430354