Tangential Derivations, Hilbert Series and Modules over Lie Algebroids

Abstract: Let A/k be a local commutative algebra over a field k of characteristic 0, and T_{A/k} be the module of k-linear derivations on A. We study, in two papers, the set of k-linear derivations on A which are tangential to an ideal I of A (preserves I), defining an A-submodule T_{A/k}(I) of T_{A/k}, which moreover is a k-Lie subalgebra. More generally we consider Lie algebroids g_A over A and modules over g_A.Paper I: Using the action of an algebraic torus on a monomial ideal in a polynomial ring A=k[x_1,..., x_n] we:give a new proof of a description of the set of tangential derivations T_{A/k}(I) along a monomial ideal I, first proven by Brumatti and Simis.give a new and direct proof to the fact that the integral closure of a monomial ideal is monomial. We also prove that a derivation which is tangential to a monomial ideal will remain tangential to its integral closure.prove that a derivation which is tangential to a monomial ideal is also tangential to any of its associated multiplier ideals.Paper II: We consider modules M over a Lie algebroid g_A which are of finite type over A. In particular, we study the Hilbert series of the associated graded module of such a module with respect to an ideal of definition.Our main results are:Hilbert's finiteness theorem in invariant theory is shown to hold also for a noetherian graded g_A-algebra S and a noetherian (S, g_A)-graded module which are semisimple over g_A.We define a class of local system g_A-modules and prove that the Hilbert series of such a graded module is rational.  We also define an ideal of definition for a g_A-module M and prove rationality of the Hilbert series of M with respect to such an ideal.We introduce the notion of toral Lie algebroids over a regular noetherian local algebra R and give some properties of modules over such Lie algebroids. In particular, we compute the Hilbert series of submodules of R over a Lie algebroid containig a toral Lie algebroid.