The finite presentation problem for Noetherian algebras

Abstract

We prove that there exists an affine Noetherian algebra over a field of characteristic zero which is not finitely presented. Specifically, we prove that the Medvedev–Passman–Resco–Small algebra, recently shown to form a counterexample to the stability problem for Noetherian algebras, is not finitely presented. This answers a question due to Bergman and Irving in characteristic zero, a case explicitly left open by Resco and Small in 1993, who gave a counterexample to this question in positive characteristic.