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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Open loci of graded modules
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by Christel Rotthaus and Liana M. Şega PDF
Trans. Amer. Math. Soc. 358 (2006), 4959-4980 Request permission


Let $A=\bigoplus _{i\in \mathbb {N}}A_i$ be an excellent homogeneous Noetherian graded ring and let $M=\bigoplus _{n\in \mathbb {Z}}M_n$ be a finitely generated graded $A$-module. We consider $M$ as a module over $A_0$ and show that the $(S_k)$-loci of $M$ are open in $\operatorname {Spec}(A_0)$. In particular, the Cohen-Macaulay locus $U^0_{CM}=\{\mathfrak {p}\in \operatorname {Spec}(A_0) \mid M_\mathfrak {p} \; \mbox {is Cohen-Macaulay}\}$ is an open subset of $\operatorname {Spec}(A_0)$. We also show that the $(S_k)$-loci on the homogeneous parts $M_n$ of $M$ are eventually stable. As an application we obtain that for a finitely generated Cohen-Macaulay module $M$ over an excellent ring $A$ and for an ideal $I\subseteq A$ which is not contained in any minimal prime of $M$, the $(S_k)$-loci for the modules $M/I^nM$ are eventually stable.
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Additional Information
  • Christel Rotthaus
  • Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
  • Email:
  • Liana M. Şega
  • Affiliation: Department of Mathematics and Statistics, University of Missouri, Kansas City, Missouri 64110-2499
  • Email:
  • Received by editor(s): March 23, 2004
  • Received by editor(s) in revised form: September 28, 2004
  • Published electronically: April 11, 2006
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 358 (2006), 4959-4980
  • MSC (2000): Primary 13A02, 13C15, 13F40; Secondary 13A30, 13C14
  • DOI:
  • MathSciNet review: 2231880