Let A be a regular ring containing field of characteristic zero and let $$R = A[X_1,\ldots , X_m]$$ . Consider R as standard graded with $$\deg 0$$ X_i 1$$ for all i. In this paper we present comprehensive study components local cohomology s $$H^i_I(R)$$ where I is an arbitrary homogeneous ideal in R. Our seems to the first regard.

We define two (n + 1) graded Lie brackets on spaces of multilinear mappings. The first one is able to recognize n-graded as-sociative algebras and their modules and gives immediately the correct differential for Hochschild cohomology. The second one recognizes n-graded Lie algebra structures and their modules and gives rise to the notion of Chevalley cohomology.

Let G and K be groupoids. We present the notion of a (Gα,Kβ)-set we prove duality theorem in this context, which extends for graded algebras by groups. For A unital G-graded algebra X finite split G-set, show that there is an isomorphism between category X-graded left A-modules A#αGX-modules. As application isomorphism, construct Morita context.