Affine Noetherian algebras, filtrations and presentations

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.