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

Authors

  • 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

DOI:

https://doi.org/10.37863/umzh.v75i5.7008

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 Φ 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 HiΦ(M) is a finitely generated R-module for all i<t if and only if AssR(HiΦ(M)) is a finite set and HiΦp(Mp) is a finitely generated Rp-module for all i<t and all pSpec(R). Then, as a consequence, we prove that if (R,m) is a complete local ring, Φ is countable, and nN0 is such that (AssR(HhnΦ(M)Φ(M)))n is a finite set, then fnΦ(M)=hnΦ(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 Mod(R) into itself, which has  the global vanishing property on Mod(R) and for an arbitrary Serre subcategory S and tN, we prove that  RiT(R)S for all it if and only if RiT(M)S for any finitely generated  R-module M and all it.  Then we obtain some results on general local cohomology modules.

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Published

24.05.2023

Issue

Section

Research articles

How to Cite

Sadeghi, M. Y., et al. “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, https://doi.org/10.37863/umzh.v75i5.7008.