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 $(S^{ \cdot}, { \mathfrak g}_{ A/k})$.
Categories: math.RA math.AC math.RT