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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On endomorphism rings and dimensions of local cohomology modules
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by Peter Schenzel PDF
Proc. Amer. Math. Soc. 137 (2009), 1315-1322 Request permission

Abstract:

Let $(R,\mathfrak m)$ denote an $n$-dimensional complete local Gorenstein ring. For an ideal $I$ of $R$ let $H^i_I(R), i \in \mathbb Z,$ denote the local cohomology modules of $R$ with respect to $I.$ If $H^i_I(R) = 0$ for all $i \not = c = \operatorname {height} I,$ then the endomorphism ring of $H^c_I(R)$ is isomorphic to $R$. Here we prove that this is true if and only if $H^i_I(R) = 0$ for $i = n, n-1$, provided $c \geq 2$ and $R/I$ has an isolated singularity, resp. if $I$ is set-theoretically a complete intersection in codimension at most one. Moreover, there is a vanishing result of $H^i_I(R)$ for all $i > m, m$ a given integer, and an estimate of the dimension of $H^i_I(R).$
References
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Additional Information
  • Peter Schenzel
  • Affiliation: Institut für Informatik, Martin-Luther-Universität Halle-Wittenberg, 06099 Halle (Saale), Germany
  • MR Author ID: 155825
  • ORCID: 0000-0003-1569-5100
  • Email: peter.schenzel@informatik.uni-halle.de
  • Received by editor(s): April 21, 2008
  • Received by editor(s) in revised form: June 17, 2008
  • Published electronically: November 12, 2008
  • Communicated by: Bernd Ulrich
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 137 (2009), 1315-1322
  • MSC (2000): Primary 13D45; Secondary 13H10, 14M10
  • DOI: https://doi.org/10.1090/S0002-9939-08-09676-7
  • MathSciNet review: 2465654