Representability of affine algebras over an arbitrary field

TL;DR

This paper provides a comprehensive proof that relatively free affine PI-algebras over any field are representable, extending previous results to arbitrary fields using quiver techniques.

Contribution

It offers a full exposition of Belov's theorem on the representability of affine PI-algebras over arbitrary fields, building on prior work with quivers and PI-varieties.

Findings

Relatively free affine PI-algebras over any field are representable.

Extended Belov's theorem to arbitrary fields.

Utilized quiver techniques in the proof.

Abstract

In a series of papers, we used full quivers as tools in describing PI-varieties of algebras and providing a complete proof of Belov's solution of Specht's problem for affine algebras over an arbitrary Noetherian ring. In this paper, utilizing ideas from that work, we give a full exposition of Belov's theorem that relatively free affine PI-algebras over an arbitrary field are representable. (Kemer proved the theorem over an infinite field.)