About a variation of local cohomology

Let 𝔮 denote an ideal of a local ring (A,𝔪). For a system of elements a¯=a1,…,at such that ai∈𝔮ci,i=1,…,t, and n∈ℤ we investigate a subcomplex and a factor complex of the Čech complex Ča¯⊗AM 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 Ha¯i(M) for all i∈ℕ. In the case of an 𝔪-primary ideal a¯A we prove the Artinianness of these cohomology modules and characterize the last nonvanishing among them.