About a variation of local cohomology

Abstract

Let $\mathfrak{𝔮}$ denote an ideal of a local ring $\left(A,\mathfrak{𝔪}\right)$. For a system of elements $\underset{¯}{a}=a_1,\dots ,a_t$ such that $a_i\in {\mathfrak{𝔮}}^{c_i},i=1,\dots ,t$, and $n\in \mathbb{Z}$ we investigate a subcomplex and a factor complex of the Čech complex ${Č}_{\underset{¯}{a}}{\otimes }_{A}M$ 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 $H_{\underset{¯}{a}}^i\left(M\right)$ for all $i\in \mathbb{N}$. In the case of an $\mathfrak{𝔪}$-primary ideal $\underset{¯}{a}A$ we prove the Artinianness of these cohomology modules and characterize the last nonvanishing among them.