Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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A NOTE ON ENDOMORPHISMS OF LOCAL COHOMOLOGY MODULES

  • Mahmood, Waqas (Department of Mathematics Quaid-I-Azam University) ;
  • Zahid, Zohaib (Department of Mathematics University of Management and Technology(UMT))
  • Received : 2016.02.10
  • Published : 2017.01.31
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Abstract

Let I denote an ideal of a Noetherian local ring (R, m). Let M denote a finitely generated R-module. We study the endomorphism ring of the local cohomology module $H^c_I(M)$, c = grade(I, M). In particular there is a natural homomorphism $$Hom_{\hat{R}^I}({\hat{M}}^I,\;{\hat{M}}^I){\rightarrow}Hom_R(H^c_I(M),\;H^c_I(M))$$, $where{\hat{\cdot}}^I$ denotes the I-adic completion functor. We provide sufficient conditions such that it becomes an isomorphism. Moreover, we study a homomorphism of two such endomorphism rings of local cohomology modules for two ideals $J{\subset}I$ with the property grade(I, M) = grade(J, M). Our results extends constructions known in the case of M = R (see e.g. [8], [17], [18]).

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Acknowledgement

Supported by : Higher Education Commission

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