About a variation of local cohomology

Let $\mathfrak{q}$ denote an ideal of a local ring $(A,\mathfrak{m})$. For a system of elements $\underset{‾}{a}=a_1,\dots ,a_t$ such that $a_i \in \mathfrak{q}^{c_i}, i = 1, \ldots,t,$and$n \in $\mathbb{Z}$$ we investigate a subcomplex resp. a factor complex of the \v{C}ech complex ${\check{C}}_{\underset{‾}{a}}{\otimes }_{A}M$ 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 $H_{\underset{‾}{a}}^i(M)$ 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 non-vanishing among them.