Hilbert series of modules over Lie algebroids

TL;DR

This paper develops a framework for defining and studying Hilbert series of modules over Lie algebroids, including derivations preserving ideals and modules over Stanley-Reisner rings, extending classical concepts to new algebraic structures.

Contribution

It introduces a natural definition of Hilbert series for modules over Lie algebroids, encompassing derivations preserving ideals and modules over Stanley-Reisner rings, broadening the scope of Hilbert series applications.

Findings

Defined Hilbert series for modules over Lie algebroids.

Analyzed Hilbert series for modules over derivation Lie algebroids.

Extended Hilbert series concepts to Stanley-Reisner rings.

Abstract

We consider modules $M$ over Lie algebroids ${\mathfrak g}_A$ which are of finite type over a local noetherian ring $A$. Using ideals $J\subset A$ such that ${\mathfrak g}_A \cdot J\subset J $ and the length $\ell_{{\mathfrak g}_A}(M/JM)< \infty$ we can define in a natural way the Hilbert series of $M$ with respect to the defining ideal $J$. This notion is in particular studied for modules over the Lie algebroid of $k$-linear derivations ${\mathfrak g}_A=T_{A/k}(I)$ that preserve an ideal $I\subset A$, for example when $A={\mathcal O}_n$, the ring of convergent power series. Hilbert series over Stanley-Reisner rings are also considered.