Specht's problem for associative affine algebras over commutative Noetherian rings

TL;DR

This paper proves Belov's solution to Specht's problem for affine algebras over Noetherian rings using full quivers and pseudo-quivers, extending previous methods to a broader class of rings.

Contribution

It provides a complete proof of Specht's problem for affine algebras over Noetherian rings by applying quiver techniques and a local-global reduction approach.

Findings

Established existence of characteristic coefficient-absorbing polynomials in T-ideals

Extended Specht's problem solution from fields to Noetherian rings

Utilized localizations and reduction to field case for general base rings

Abstract

In a series of papers \cite{BRV1}, \cite{BRV2}, \cite{BRV3} we introduced full quivers and pseudo-quivers of representations of algebras, and used them as tools in describing PI-varieties of algebras. In this paper we apply them to obtain a complete proof of Belov's solution of Specht's problem for affine algebras over an arbitrary Noetherian ring. The inductive step relies on a theorem that enables one to find a "$\bar q$-characteristic coefficient-absorbing polynomial in each T-ideal $\Gamma$," i.e., a non-identity of the representable algebra $A$ arising from $\Gamma$, whose ideal of evaluations in $A$ is closed under multiplication by $\bar q$-powers of the characteristic coefficients of matrices corresponding to the generators of $A$, where $\bar q$ is a suitably large power of the order of the base field. The passage to an arbitrary Noetherian base ring $C$ involves localizing at finitely many elements a kind of $C$, and reducing to the field case by a local-global principle.