A cohomological point of view on gradings on algebras with multiplicative basis

In this paper we generalize the concepts of good and elementary gradings for an associative algebra A with a fixed multiplicative basis B. When the group G considered in the grading is abelian, we equip the set of good G-gradings of A with a structure of abelian group, which is denoted by G(B,G). Moreover, when A admits elementary G-gradings, we show the set E(B,G) of all elementary G-gradings of A is a subgroup of G(B,G). In this case, we introduce a cohomology for the pair (A,B) and we show that G(B,G)/E(B,G) is isomorphic to the first cohomology group of (A,B) with coefficients in G.