The g-graded identities of the grassmann algebra

Let G be a finite abelian group with identity element 1G and (Formula present) be an infinite dimensional G-homogeneous vector space over a field of characteristic 0. Let E = E(L) be the Grassmann algebra generated by L. It follows that E is a G-graded algebra. Let |G| be odd, then we prove that in order to describe any ideal of G-graded identities of E it is sufficient to deal with G′-grading, where |G′| ≤ |G|, dimF L1G′ = ∞ and dimF Lg′ < ∞ if g′ = 1G′. In the same spirit of the case |G| odd, if |G| is even it is sufficient to study only those G-gradings such that dimF Lg = ∞, where o(g) = 2, and all the other components are finite dimensional. We also compute graded cocharacters and codimensions of E in the case dim (Formula present) and dim (Formula present) if g ≠ 1G.