Locally nilpotent module derivations and the fourteenth problem of Hilbert

TL;DR

This paper investigates the finite generation of modules invariant under locally nilpotent derivations on affine algebras, connecting to Hilbert's fourteenth problem, and explores conditions ensuring finite generation.

Contribution

It extends the study of locally nilpotent derivations to modules, analyzing when invariant modules are finitely generated and relating findings to Hilbert's fourteenth problem.

Findings

Counterexamples to finite generation are related to Hilbert's fourteenth problem.

Sufficient conditions for finite generation are identified.

The work links module derivations to classical invariant theory.

Abstract

Given a locally nilpotent derivation on an affine algebra $B$ over a field $k$ of characteristic zero, we consider a finitely generated $B$-module $M$ which admits a locally nilpotent module derivation $\delta_M$ (see Definition 1.1 below). Let $A=\Ker \delta$ and $M_0=\Ker \delta_M$. We ask if $M_0$ is a finitely generated $A$-module. In general, there exist counterexamples which are closely related to the fourteenth problem of Hilbert. We also look for some sufficient conditions for finite generation.