A few results about variations of local cohomology

Let $\mathfrak{q}$ denote an ideal of a local ring $(A,\mathfrak{m})$. For a system of elements $\underline{a} = a_1,\ldots,a_t$ 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 $\check{C}_{\underline{a} \otimes_A M$ 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 $H^i_{\underline{a}(M)$ 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.