Local cohomology modules and their properties

  • J. Azami Univ. Mohaghegh Ardabili, Ardabil, Iran
  • M. Hasanzad Univ. Mohaghegh Ardabili, Ardabil, Iran


UDC 512.5

Let $(R, \mathfrak{m})$ be a complete Noetherian local ring and let $M$ be a generalized Cohen-Macaulay $R$-module of dimension $d \geq 2.$
We show that
D \left(H_{\mathfrak{m}}^d\Big(D \big(H_{\mathfrak{m}}^d
(D_{\mathfrak{m}}(M))\big)\Big)\right) \approx D_{\mathfrak{m}} (M),
where $D = {\rm Hom} (-, E)$ and $D_{\mathfrak{m}} (-)$ is the ideal transform functor.
Also, assuming that $I$ is a proper ideal of a local ring $R$, we obtain some results on the finiteness of Bass numbers, cofinitness, and cominimaxness of local cohomology modules with respect to $I.$


K. Bahmanpour, R. Naghipour, M. Sedghi, On the finiteness of Bass numbers of local cohomology modules and cominimaxness>, Houston J. Math., 40, № 2, 319 – 337 (2014), https://doi.org/10.1007/s10468-014-9498-3 DOI: https://doi.org/10.1007/s10468-014-9498-3

M. P. Brodmann, R. Y. Sharp, Local cohomology; an algebraic introduction with geometric applications, Cambridge Univ. Press, Cambridge (1998), https://doi.org/10.1017/CBO9780511629204 DOI: https://doi.org/10.1017/CBO9780511629204

W. Bruns, J. Herzog, Cohen Macualay rings, Cambridge Stud. Adv. Math. (1997). DOI: https://doi.org/10.1017/CBO9780511608681

A. Grothendieck, Local cohomology, Notes by R. Hartshorne, Lect. Notes Math., 862, Springer, New York (1966).

Y. Irani, K. Bahmanpour, Cominimaxness of local cohomology modules for ideals of dimension one, preprint.

H. Matsumura, Commutative ring theory, Cambridge Univ. Press, Cambridge, UK (1986).

T. Marley, The associated primes of local cohomology modules over rings of small dimension, Manuscripta Math., 104, 519 – 525 (2001), https://doi.org/10.1007/s002290170024 DOI: https://doi.org/10.1007/s002290170024

L. Melkersson, Modules cofinite with respect to an ideal, J. Algebra, 285, 649 – 668 (2005), https://doi.org/10.1016/j.jalgebra.2004.08.037 DOI: https://doi.org/10.1016/j.jalgebra.2004.08.037

R. Naghipour, K. Bahmanpour, I. Khalili Gorji, Cofiniteness of torsion functors of cofinite modules, Colloq. Math., 136, 221 – 230 (2014), https://doi.org/10.4064/cm136-2-4 DOI: https://doi.org/10.4064/cm136-2-4

P. Schenzel, Proregular sequences, local cohomology, and completion, Math. Scand., 92, 161 – 180 (2003), https://doi.org/10.7146/math.scand.a-14399 DOI: https://doi.org/10.7146/math.scand.a-14399

P. Schenzel, Einige Anwendungen der lokalen Dualitat und verallgemeinerte Cohen – Macaulay – Moduln, Math. Nachr., 69, 227 – 242 (1975), https://doi.org/10.1002/mana.19750690121 DOI: https://doi.org/10.1002/mana.19750690121

How to Cite
Azami, J., and M. Hasanzad. “ Local Cohomology Modules and Their Properties”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 2, Feb. 2021, pp. 268 -74, doi:10.37863/umzh.v73i2.127.
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