Hilbert series of modules over Lie algebroids

We consider modules M   over Lie algebroids gAgA which are of finite type over a local noetherian ring A  . Using ideals J⊂AJ⊂A such that gA⋅J⊂JgA⋅J⊂J and the length ℓgA(M/JM)<∞ℓgA(M/JM)<∞ 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 gA=TA(I)gA=TA(I) that preserve an ideal I⊂AI⊂A, for example when A=OnA=On, the ring of convergent power series. Hilbert series over Stanley–Reisner rings are also considered.