Representability and Specht problem for G-graded algebras

Abstract

Let W be an associative PI-algebra over a field F of characteristic zero, graded by a finite group G. Let id G ( W ) denote the T-ideal of G-graded identities of W. We prove: 1. [ G-graded PI-equivalence] There exists a field extension K of F and a finite-dimensional Z / 2 Z × G -graded algebra A over K such that id G ( W ) = id G ( A ∗ ) where A ∗ is the Grassmann envelope of A. 2. [ G-graded Specht problem] The T-ideal id G ( W ) is finitely generated as a T-ideal. 3. [ G-graded PI-equivalence for affine algebras] Let W be a G-graded affine algebra over F. Then there exists a field extension K of F and a finite-dimensional algebra A over K such that id G ( W ) = id G ( A ) .