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http://dx.doi.org/10.4134/BKMS.b160111

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))
Publication Information
Bulletin of the Korean Mathematical Society / v.54, no.1, 2017 , pp. 319-329 More about this Journal
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]).
Keywords
local cohomology; endomorphism ring; completion functor; cohomologically complete intersection;
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1 L. Alonso Tarrio, A. Jeremias Lopez, and J. Lipman, Local homology and cohomology on schemes, Ann. Sci. Ecole. Norm. Sup. 30 (1997), no. 1, 1-39.   DOI
2 M. Brodmann and Y. Sharp, Local Cohomology, An algebraic introduction with geometric applications. Cambridge Studies in Advanced Mathematics No. 60. Cambridge University Press, 1998.
3 W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Univ. Press, 1993.
4 M. Eghbali and P. Schenzel, On an endomorphism ring of local cohomology, Commu. Algebra 40 (2012), no. 11, 4295-4305.   DOI
5 A. Grothendieck, Local cohomology, Notes by R. Hartshorne, Lecture Notes in Math. Vol. 41, Springer, 1967.
6 A. R. Hartshorne, Residues and duality, Lecture Notes in Mathematics, Vol. 20, Springer, 1966.
7 M. Hellus and P. Schenzel, On cohomologically complete intersections, J. Algebra 320 (2008), no. 10, 3733-3748.   DOI
8 M. Hellus and J. Stuckrad, On endomorphism rings of local cohomology modules, Proc. Amer. Math. Soc. 136 (2008), no. 7, 2333-2341.   DOI
9 C. Huneke, Lectures on local cohomology, Appendix 1 by Amelia Taylor.Contemp. Math., 436, Interactions between homotopy theory and algebra, 51-99, Amer. Math. Soc., Providence, RI, 2007.
10 C. Huneke and M. Hochster, Indecomposable canonical modules and connectedness, In: Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992), 197-208, Contemp. Math., 159, Amer. Math. Soc., Providence, RI, 1994.
11 E. Matlis, Injective modules over Noetherian rings, Pacific J. Math. 8 (1958), 511-528.   DOI
12 H. Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics 8, Cambr. Uni. Press, Cambridge, 1986.
13 P. Schenzel, On endomorphism rings and dimensions of local cohomology modules, Proc. Amer. Math. Soc. 137 (2009), no. 4, 1315-1322.   DOI
14 J. Rotman, An Introduction to Homological Algebra, Springer, United Kingdom Edition published by Academic press limited, Facultat de Matemiltiques i EstaciHs1, 1993.
15 P. Schenzel, Proregular sequences, local cohomology and completion, Math. Scand. 92 (2003), no. 2, 161-180.   DOI
16 P. Schenzel, On birational Macaulayfications and Cohen-Macaulay canonical modules, J. Algebra 275 (2004), no. 2, 751-770.   DOI
17 P. Schenzel, Matlis dual of local cohomology modules and their endomorphism rings, Arch. Math. 95 (2010), no. 2, 115-123.   DOI
18 P. Schenzel, On the structure of the endomorphism ring of a certain local cohomology module, J. Algebra 344 (2011), 229-245.   DOI
19 C. Weibel, An Introduction to Homological Algebra, Cambridge Univ. Press, 1994.
20 M. Porta, L. Shaul, and A. Yekutieli, On the homology of completion and torsion, Algebr. Represent. Theory 17 (2014), no. 1, 31-67.   DOI