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.


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.


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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.
Research articles