Abstract. We consider modules M over Lie algebroids g A which are of finite type over a local noetherian ring A. Using ideals J ⊂ A such that g A · J ⊂ J and the length ℓg A (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 g A = T A (I) that preserve an ideal I ⊂ A, for example when A = On, the ring of convergent power series. Hilbert series over Stanley-Reisner rings are also considered.

Abstract. Let I be a monomial ideal in a polynomial ring A = k[x 1 , . . . , x n ] over a field k of characteristic 0, T A/k (I) be the module of I-preserving kderivations on A and G be the n-dimensional algebraic torus on k. We compute the weight spaces of T A/k (I) considered as a representation of G. Using this, we show that T A/k (I) preserves the integral closure of I and the multiplier ideals of I.

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