About a variation of local cohomology

Abstract

Let $\mathfrak{q}$ denote an ideal of a local ring $(A,\mathfrak{m})$. For a system of elements $\underline{a}=a_1,\dots ,a_t$ such that $a_i\in {\mathfrak{q}}^{c_i},i=1,\dots ,t,$ and $n\in \mathbb{Z}$ we investigate a subcomplex resp. a factor complex of the Čech complex ${\check{C}}_{\underline{a}}{\otimes }_{A}M$ for a finitely generated $A$-module $M$. We start with the inspection of these cohomology modules that approximate in a certain sense the local cohomology modules $H_{\underline{a}}^i(M)$ for all $i\in \mathbb{N}$. In the case of an $\mathfrak{m}$-primary ideal $\underline{a}A$ we prove the Artinianness of these cohomology modules and characterize the last non-vanishing among them.

arXiv commentTitle is changed; publiahed in J. of Commutative Algebra 12 (2020), 353-370. arXiv admin note: text overlap with arXiv:1712.10144