Hilbert series of modules over Lie algebroids

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.

42 pages. This is a substantial revision of the previous version

References in corpus (1)

• Initial logarithmic Lie algebras of hypersurface singularities

Cited by in corpus (1)

• D-modules with finite support are semi-simple