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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Open loci of graded modules


Authors: Christel Rotthaus and Liana M. Sega
Journal: Trans. Amer. Math. Soc. 358 (2006), 4959-4980
MSC (2000): Primary 13A02, 13C15, 13F40; Secondary 13A30, 13C14
Published electronically: April 11, 2006
MathSciNet review: 2231880
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Abstract: 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} \;$   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: rotthaus@math.msu.edu

Liana M. Sega
Affiliation: Department of Mathematics and Statistics, University of Missouri, Kansas City, Missouri 64110-2499
Email: segal@umkc.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-06-03876-1
PII: S 0002-9947(06)03876-1
Received by editor(s): March 23, 2004
Received by editor(s) in revised form: September 28, 2004
Published electronically: April 11, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.