Two extensions of Hilbert's finiteness theorem

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 $(S^{\cdot },{\mathfrak{g}}_{A/k})$.

4 pages