One proof of the original Kemer's theorems (concerning the text of C. Procesi "What happened to PI-theory", arxiv.org/abs/1403.5673)

TL;DR

This paper provides a proof of Kemer's theorems on polynomial identities, showing that the identities of certain associative algebras are finitely generated and correspond to finite dimensional superalgebras.

Contribution

It offers a version of Kemer's theorems proof, establishing finite generation of T-ideals for PI-algebras over characteristic zero fields.

Findings

The ideal of graded identities of finitely generated PI-superalgebras matches that of finite dimensional superalgebras.

Polynomial identities of any PI-algebra are finitely based and correspond to Grassmann envelopes.

The results confirm the Specht problem solution for associative algebras in characteristic zero.

Abstract

We consider associative algebras over a field of characteristic zero. We give a version of the proof of the Kemer's theorems concerning the Specht problem solution. It is proved that the ideal of graded identities of a finitely generated PI-superalgebra coincides with the ideal of graded identities of some finite dimensional superalgebra. This implies that the ideal of polynomial identities of any (not necessary finitely generated) PI-algebra coincides with the ideal of identities of the Grassmann envelope of a finite dimensional superalgebra, and is finitely generated as a T-ideal.