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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On endomorphism rings and dimensions of local cohomology modules


Author: Peter Schenzel
Journal: Proc. Amer. Math. Soc. 137 (2009), 1315-1322
MSC (2000): Primary 13D45; Secondary 13H10, 14M10
Published electronically: November 12, 2008
MathSciNet review: 2465654
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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).$


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Additional Information

Peter Schenzel
Affiliation: Institut für Informatik, Martin-Luther-Universität Halle-Wittenberg, 06099 Halle (Saale), Germany
Email: peter.schenzel@informatik.uni-halle.de

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09676-7
PII: S 0002-9939(08)09676-7
Keywords: Local cohomology, vanishing, cohomological dimension
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
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.