The Braun--Kemer--Razmyslov theorem for affine $PI$-algebras

Alexei Kanel Belov ; Louis Rowen

Čebyševskij sbornik, Tome 21 (2020) no. 3, pp. 89-128

Résumé

A self-contained, combinatoric exposition is given for the Braun–Kemer–Razmyslov Theorem over an arbitrary commutative Noetherian ring.At one time, the community did not believe in the validity of this result, and contrary to public opinion, the corresponding question was posed by V.N. Latyshev in his doctoral dissertation. One of the major theorems in the theory of PI algebras is the Braun-Kemer-Razmyslov Theorem. We preface its statement with some basic definitions. 1. An algebra $A$ is affine over a commutative ring $C$ if $A$ is generated as an algebra over $C$ by a finite number of elements $a_1, \dots, a_\ell;$ in this case we write $A = C \{ a_1, \dots, a_\ell \}.$ We say the algebra $A$ is finite if $A$ is spanned as a $C$-module by finitely many elements. 2. Algebras over a field are called PI algebras if they satisfy (nontrivial) polynomial identities. 3. The Capelli polynomial $\mathrm{Cap}_k$ of degree $2k$ is defined as $$\mathrm{Cap}_k(x_1, \dots, x_k; y_1, \dots, y_k)= \sum_{\pi \in S_k}\mathrm{sgn} (\pi) x_{\pi (1)}y_1 \cdots x_{\pi (k)} y_k.$$ 4. $\operatorname{Jac}(A)$ denotes the Jacobson radical of the algebra $A$ which, for PI-algebras is the intersection of the maximal ideals of $A$, in view of Kaplansky's theorem. The aim of this article is to present a readable combinatoric proof of the theorem: The Braun-Kemer-Razmyslov Theorem The Jacobson radical $\operatorname{Jac}(A)$ of any affine PI algebra $A$ over a field is nilpotent.

Détail

Keywords: algebras with polynomial identity, varieties of algebras, representable algebras, relatively free algebras, Hilbert series, Specht problem.