Proceedings of the American Mathematical Society

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



A note on the injective dimension of local cohomology modules

Author: M. Hellus
Journal: Proc. Amer. Math. Soc. 136 (2008), 2313-2321
MSC (2000): Primary 13D45; Secondary 13C05
Published electronically: February 28, 2008
MathSciNet review: 2390497
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Abstract: For a Noetherian ring $ R$ we call an $ R$-module $ M$ cofinite if there exists an ideal $ I$ of $ R$ such that $ M$ is $ I$-cofinite; we show that every cofinite module $ M$ satisfies $ \dim _R(M)\leq \operatorname{injdim}_R(M)$. As an application we study the question which local cohomology modules $ H^i_I(R)$ satisfy $ \operatorname{injdim}_R(H^i_I(R))=\dim_R(H^i_I(R))$. There are two situations where the answer is positive. On the other hand, we present two counterexamples, the failure in these two examples coming from different reasons.

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M. Hellus
Affiliation: Fakultät für Mathematik und Informatik, Universität Leipzig, PF 10 09 20, D-04009 Leipzig, Germany

Keywords: Local cohomology, injective dimension, Bass formula
Received by editor(s): October 26, 2006
Received by editor(s) in revised form: February 28, 2007, and April 2, 2007
Published electronically: February 28, 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.