Affine Noetherian algebras, filtrations and presentations

TL;DR

This paper investigates the structure of affine Noetherian algebras, demonstrating the absence of certain filtrations in specific examples and extending results to a broad class over countable fields.

Contribution

It proves that Resco and Small's affine Noetherian algebra has no finite-dimensional filtrations with Noetherian associated graded algebras and generalizes this to many such algebras over countable fields.

Findings

Resco and Small's algebra lacks finite-dimensional filtrations with Noetherian associated graded algebras

Most affine Noetherian algebras over countable fields also lack such filtrations

Answers a question posed by Irving and Small

Abstract

Resco and Small gave the first example of an affine Noetherian algebra which is not finitely presented. It is shown that their algebra has no finite-dimensional filtrations whose associated graded algebras are Noetherian, affirming their prediction. A modification of their example yields countable fields over which `almost all' (that is, a co-countable continuum of) affine Noetherian algebras lack such a filtration, and an answer to a question suggested by Irving and Small is derived.