Exponential codimension growth of PI algebras: An exact estimate

Let A be an associative PI-algebra over a field F of characteristic zero. B y studying the exponential behavior of the sequence of codimensions {c(n)(A )} of A, we prove that Inv(A)=lim(n-->infinity )(n)root c(n)(A) always exis ts and is an integer. We also give an explicit way for computing such integ er: let B be a finite dimensional Z(2)-graded algebra whose Grassmann envel ope G(B) satisfies the same identities of A; then Inv(A)= Inv(G(B))= dim C- (0)+ dim C-(1) where C-(0)+ C-(1) is a suitable Z(2)-graded semisimple suba lgebra of B. (C) 1999 Academic Press.