General local cohomology modules in view of low points and high points

  • M. Y. Sadeghi Department of Mathematics, Payame Noor University, Tehran, Iran
  • Kh. Ahmadi Amoli Department of Mathematics, Payame Noor University, Tehran, Iran
  • М. Chaghamirza Department of Mathematics, Payame Noor University, Tehran, Iran
Keywords: general local cohomology modules; local cohomology modules; Serre subcategory; associated primes; Faltings' Theorem; Local- global Principle.

Abstract

UDC 512.5

Let $R$ be a commutative Noetherian ring, let $\Phi $ be a system of ideals of $R,$ let $M$ be a finitely generated $R$-module, and let $t$ be a nonnegative integer.  We first show that a general local cohomology module $H_{\Phi}^{i}(M)$ is a finitely generated $R$-module for all $i<t$ if and only if $\mathrm{Ass}_{R} (H_{\Phi}^{i}(M))$ is a finite set and $H_{\Phi_{\mathfrak{p}}}^{i}(M_{\mathfrak{p}})$ is a finitely generated $R_{\mathfrak{p}}$-module for all $i<t$ and all $\mathfrak{p}\in \mathrm{Spec}(R)$. Then, as a consequence, we prove that if $(R,\mathfrak{m})$ is a complete local ring, $\Phi$ is countable, and $n\in \mathbb{N}_{0}$ is such that $\big(\mathrm{Ass}_{R} \big(H_{\Phi}^{h_{\Phi}^{n}(M)}(M)\big)\big)_{\geq n}$ is a finite set, then $f_{\Phi}^{n}(M)=h_{\Phi}^{n}(M)$. In addition, we show that the properties of vanishing and finiteness of general local cohomology modules are equivalent on high points over an arbitrary Noetherian (not necessary local) ring. For each covariant $R$-linear functor $T$ from $\mathrm{Mod}(R)$ into itself, which has  the global vanishing property on $\mathrm{Mod}(R)$ and for an arbitrary Serre subcategory $\mathcal{S}$ and $t\in \mathbb{N},$ we prove that  $\mathcal{R}^{i}T(R)\in \mathcal{S}$ for all $i\geq t$ if and only if $\mathcal{R}^{i}T(M)\in \mathcal{S}$ for any finitely generated  $R$-module $M$ and all $i\geq t$.  Then we obtain some results on general local cohomology modules.

References

M. Aghapournahr, L. Melkersson, A natural map in local cohomology, Ark. Mat., 48, 243–251 (2010). DOI: https://doi.org/10.1007/s11512-009-0115-3

M. Aghapournahr, L. Melkersson, Finiteness properties of minimax and coatomic local cohomology modules, Arch. Math., 94, 519–528 (2010). DOI: https://doi.org/10.1007/s00013-010-0127-z

D. Asadollahi, R. Naghipour, Faltings' local-global principle for the finiteness of local cohomology modules, Comm. Algebra., 43, 953–958 (2015). DOI: https://doi.org/10.1080/00927872.2013.849261

M. Asgharzadeh, M. Tousi, A unified approach to local cohomology modules using Serre classes, Canad. Math. Bull., 53, 577–586 (2010). DOI: https://doi.org/10.4153/CMB-2010-064-0

K. Bahmanpour, R. Naghipour, M. Sedghi, Minimaxness and cofiniteness properties of local cohomology modules, Comm. Algebra., 41, 2799–2814 (2013). DOI: https://doi.org/10.1080/00927872.2012.662709

M. H. Bijan-Zadeh, Torsion theories and local cohomology over commutative Noetherian ring, J. London Math. Soc., 19, 402–410 (1979). DOI: https://doi.org/10.1112/jlms/s2-19.3.402

M. H. Bijan-Zadeh, A common generalization of local cohomology theories, Glasgow Math. J., 21, 173–181 (1980). DOI: https://doi.org/10.1017/S0017089500004158

M. P. Brodmann, A. Lashgari Faghani, A finiteness result for associated primes of local cohomology modules, Proc. Amer. Math. Soc., 128, 2851–2853 (2000). DOI: https://doi.org/10.1090/S0002-9939-00-05328-4

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

K. Divaani-Aazar, M. A. Esmkhani, Artinianness of local cohomology modules of ZD-modules, Comm. Algebra, 33, 2857–2863 (2005). DOI: https://doi.org/10.1081/AGB-200063983

E. G. Evans, Zero divisors in Noetherian-like rings, Trans. Amer. Math. Soc., 155, 505–512 (1971). DOI: https://doi.org/10.1090/S0002-9947-1971-0272773-9

G. Faltings, Der Endlichkeitssatz in der lokalen Kohomologie, Math. Ann., 255, 45–56 (1981). DOI: https://doi.org/10.1007/BF01450555

A. Hajikarimi, Cofiniteness with respect to a Serre subcategory, Math. Notes, 89, 121–130 (2011). DOI: https://doi.org/10.1134/S0001434611010135

T. Marley, J. C. Vassilev, Cofiniteness and associated primes of local cohomology modules, J. Algebra, 256, 180–193 (2002). DOI: https://doi.org/10.1016/S0021-8693(02)00151-5

P. H. Quy, On the finiteness of associated primes of local cohomology modules, Proc. Amer. Math. Soc., 138, 1965–1968 (2010). DOI: https://doi.org/10.1090/S0002-9939-10-10235-4

J. J. Rotman, An introduction to homological algebra, Second ed., Springer Sci. (2009). DOI: https://doi.org/10.1007/b98977

K. I. Yoshida, Cofiniteness of local cohomology modules for ideals of dimension one, Nagoya Math. J., 147, 179–191 (1997). DOI: https://doi.org/10.1017/S0027763000006371

T. Zink, Endlichkeitsbedingungen für Moduln über einem Noetherschen Ring, Math. Nachr., 64, 239–252 (1974). DOI: https://doi.org/10.1002/mana.19740640114

H. Zöschinger, Minimax Moduln, J. Algebra, 102, № 1, 1–32 (1986). DOI: https://doi.org/10.1016/0021-8693(86)90125-0

H. Zöschinger, über die Maximalbedingung für radikalvolle Untermoduln, Hokkaido Math. J., 17, 101–116 (1988). DOI: https://doi.org/10.14492/hokmj/1381517790

Published
24.05.2023
How to Cite
Sadeghi, M. Y., K. Ahmadi Amoli, and ChaghamirzaМ. “General Local Cohomology Modules in View of Low Points and High Points”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 75, no. 5, May 2023, pp. 698 -11, doi:10.37863/umzh.v75i5.7008.
Section
Research articles