Two extensions of Hilbert's finiteness theorem

TL;DR

This paper extends Hilbert's finiteness theorem to graded algebras over commutative rings with Lie algebroid actions, proving invariants are noetherian under semi-simplicity and constructing noetherian subalgebras for solvable Lie algebroids.

Contribution

It generalizes Hilbert's theorem to broader algebraic structures involving Lie algebroids and constructs new noetherian subalgebras in this context.

Findings

Invariants under semi-simple Lie algebroid actions are noetherian.

Constructs noetherian subalgebras from characters of solvable Lie algebroids.

Provides results for noetherian modules over the algebra-Lie algebroid pair.

Abstract

Let $S^{\cdot}$ be a noetherian graded algebra over a commutative $k$-algebra $A$, where $k$ is a commutative ring, and assume it is a module over a Lie algebroid ${\mathfrak g}_{A/k}$. If $S^\cdot$ is semi-simple over ${\mathfrak g}_{A/k}$ we prove that its ring of invariants $\bar S^{\cdot}$ is notherian. When ${\mathfrak g}_{A/k}$ is a solvable Lie algebra over $A$ we construct noetherian subalgebras of $S^\cdot$ from subsets of characters of ${\mathfrak g}_{A/k}$. We give similar results for noetherian modules over the pair $(S^{\cdot}, {\mathfrak g}_{A/k})$.