Kemer's theorem for affine PI algebras over a field of characteristic zero

Abstract

We present a proof of Kemer's representability theorem for affine PI algebras over a field of characteristic zero.

The polynomial part of the codimension growth of affine PI algebras

“…In other words, there exist nonidentities in an arbitrary number of alternating small sets, but only a finite number of big sets (actually at most s big sets). For a general finite dimensional algebra one has κ A = (d, s) ≤ Par(A) in the lexicographic order [1]. Kemer showed that equality characterizes basic algebras.…”

“…Theorem 2.3. (Kemer, see [10] or ( [1], Prop. 7.13)) A finite dimensional algebra A is basic if and only if κ A = Par(A).…”

“…In order to get their result for arbitrary affine algebras (namely lim n→∞ n c n (W ) exists and is an integer) one needs to invoke the fundamental representability theorem of Kemer [10,1]. Theorem 1.1.…”

Kemer's theory for H-module algebras with application to the PI exponent

“…For this we follow, for the most part, the exposition of Kemer's proof given in [4]. However, there are two major differences.…”

“…As far as the author knows, this step in all other frameworks (e.g. group graded algebras, algebras with involutions) relies heavily on precise knowledge of all the simple, finite dimensional objects of the category in question (see [4,3]). However, in such general framework as H-module algebras it seems that one must consider more "subtle" approach.…”