On natural homomorphisms of local cohomology modules

Document Type: Research Paper


Quaid-i-Azam University, Islamabad-Pakistan.


‎Let $M$ be a non-zero finitely generated module over a commutative Noetherian local ring $(R,\mathfrak{m})$ with $\dim_R(M)=t$‎. ‎Let $I$ be an ideal of $R$ with $grade(I,M)=c$‎. ‎In this article we will investigate several natural homomorphisms of local cohomology modules‎. ‎The main purpose of this article is to investigate when the natural homomorphisms $\gamma‎: ‎Tor^{R}_c(k,H^c_I(M))\to k\otimes_R M$ and $\eta‎: ‎Ext^{d}_R(k,H^c_I(M))\to Ext^{t}_R(k‎, ‎M)$ are non-zero where $d:=t-c$‎. ‎In fact for a Cohen-Macaulay module $M$ we will show that the homomorphism $\eta$ is injective (resp‎. ‎surjective) if and only if the homomorphism $H^{d}_{\mathfrak{m}}(H^c_{I}(M))\to H^t_{\mathfrak{m}}(M)$ is injective (resp‎. ‎surjective) under the additional assumption of vanishing of Ext modules‎. ‎The similar results are obtained for the homomorphism $\gamma$‎. ‎Moreover we will construct the natural homomorphism $Tor^{R}_c(k‎, ‎H^c_I(M))\to Tor^{R}_c(k‎, ‎H^c_J(M))$ for the ideals $J\subseteq I$ with $c = grade(I,M)= grade(J,M)$‎. ‎There are several sufficient conditions on $I$ and $J$ to provide this homomorphism is an isomorphism.


Main Subjects

L. L. Avramov and H. B. Foxby, Homological dimensions of unbounded complexes, J. Pure Appl. Algebra 71 (1991), no. 2-3, 129--155.

M. Brodmann and R. Y. Sharp, Local Cohomology. An Algebraic Introduction with Geometric Applications, Cambridge Stud. Adv. Math. 60, Cambridge Univ. Press, 1998.

W. Bruns and J. Herzog, Cohen-Macaulay Rings, 39, Cambridge Univ. Press, 1998.

K. Divaani-Aazar, R. Naghipour and M. Tousi, Cohomological dimension of certain algebraic varieties, Proc. Amer. Math. Soc. 130 (2002), no. 12, 3537--3544.

H. B. Foxby, Isomorphisms between complexes with applications to the homological theory of modules, Math. Scand. 40 (1977), no. 1, 5--19.

A. Grothendieck, Local Cohomology, Notes by R. Hartshorne, Lecture Notes in Math. 41, Springer, 1967.

R. Hartshorne, Residues and Duality, Lecture Notes in Math. 20, Springer, 1966.

M. Hellus, Local Cohomology and Matlis Duality, Habilitationsschrift, Leipzig, 2006.

M. Hellus and J. Stuckrad, On endomorphism rings of local cohomology modules, Proc. Amer. Math. Soc. 136 (2008), no. 7, 2333--2341.

M. Hellus and P. Schenzel, On cohomologically complete intersections, J. Algebra 320 (2008), no. 10, 3733--3748.

M. Hellus and P. Schenzel, Notes on local cohomology and duality, J. Algebra 401 (2014) 48--61.

M. Hochster, Canonical elements in local cohomology and the direct summand conjecture, J. Algebra 84 (1983), no. 2, 503--553.

W. Mahmood, On endomorphism rings of local cohomology modules, ArXiv:1308. 2584v1 [math.AC] (2013).

W. Mahmood, On cohomologically complete intersections in Cohen-Macaulay rings, Math. Rep. (Bucur.) 18(68) (2016), no. 1, 21- 40.

W. Mahmood, Cohomologically complete intersections with vanishing of Betti numbers, Math. Rep., to appear.

W. Mahmood and P. Schenzel, On invariants and endomorphism rings of certain local cohomology modules, J. Algebra 372 (2012) 56--67.

W. Mahmood and Z. Zahid, A note on endomorphisms of local cohomology modules, Bull. Korean Math. Soc., to appear.

P. Schenzel, Proregular sequences, local cohomology, and completion, Math. Scand. 92 (2003), no. 2, 161--180.

P. Schenzel, On birational Macaulayfications and Cohen-Macaulay canonical modules, J. Algebra 275 (2004), no. 2, 751--770.

P. Schenzel, On endomorphism rings and dimensions of local cohomology modules, Proc. Amer. Math. Soc. 137 (2009), no. 4, 1315--1322.

P. Schenzel, Matlis duals of local cohomology modules and their endomorphism rings, Arch. Math. 95 (2010), no. 2, 115--123.

P. Schenzel, On the structure of the endomorphism ring of a certain local cohomology module, J. Algebra 344 (2011) 229--245.

C. Weibel, An Introduction to Homological Algebra, Cambridge Univ. Press, 1994.

M. R. Zargar, On a duality of local cohomology modules of relative Cohen-Macaulay rings, ArXiv:1308.3071 [math.AC] (2013).

Volume 42, Issue 6
November and December 2016
Pages 1343-1361
  • Receive Date: 05 January 2015
  • Revise Date: 26 August 2015
  • Accept Date: 03 September 2015