The polynomial part of the codimension growth of affine PI algebras

Abstract

Let F be a field of characteristic zero and W an associative affine F-algebra satisfying a polynomial identity (PI). The codimension sequence {cn(W)} associated to W is known to be of the form Θ(ntdn), where d is the well known PI-exponent of W. In this paper we establish an algebraic interpretation of the polynomial part (the constant t) by means of Kemer's theory. In particular, we show that in case W is a basic algebra (hence finite dimensional), t=q−d2+s, where q is the number of simple component in W/J(W) and s+1 is the nilpotency degree of J(W) (the Jacobson radical of W). Thus proving a conjecture of Giambruno.