Stably Noetherian Algebras of Polynomial Growth

TL;DR

This paper introduces an inductive method to identify when certain finite Gelfand-Kirillov dimension algebras are stably noetherian, with applications to PI algebras and Artin-Schelter regular algebras, considering various field extension types.

Contribution

It develops a new inductive approach to determine stable noetherianity of algebras, especially those with polynomial growth, using critical composition series.

Findings

Finite Gelfand-Kirillov dimension algebras can be shown to be stably noetherian.

Artin-Schelter regular algebras are stably noetherian.

The method applies to various graded rings and specific field extension types.

Abstract

Let $A$ be a right noetherian algebra over a field $k$. If the base field extension $A \otimes_k K$ remains right noetherian for all extension fields $K$ of $k$, then $A$ is called stably right noetherian over $k$. We develop an inductive method to show that certain algebras of finite Gelfand-Kirillov dimension are stably noetherian, using critical composition series. We use this to characterize which algebras satisfying a polynomial identity are stably noetherian. The method also applies to many $\mathbb{N}$-graded rings of finite global dimension; in particular, we see that a noetherian Artin-Schelter regular algebra must be stably noetherian. In addition, we study more general variations of the stably noetherian property where the field extensions are restricted to those of a certain type, for instance purely transcendental extensions.