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Hilbert series of modules over Lie algebroids
University of Gävle, Department of Mathematics. (Algebra, Geometry, Representation Theory)
Stockholm University, Faculty of Science, Department of Mathematics. (Algebra, Geometry, Representation Theory)
(English)Article in journal (Refereed) Submitted
Abstract [en]

We consider modules M over Lie algebroids g_A which are of  finite type over a local noetherian ring A.  Using ideals  J\subseteq A such that g_A . J \subseteq J  and the length  l_{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 g_A=T_A(I) that preserve an ideal  I\subseteq A, for example when A is the ring of convergent  power series.

Keyword [en]
Hilbert series, tangential vector field, Lie algebroid, complete intersection
National Category
Algebra and Logic Geometry
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-62614OAI: oai:DiVA.org:su-62614DiVA: diva2:443519
Available from: 2011-09-26 Created: 2011-09-26 Last updated: 2011-09-29Bibliographically approved
In thesis
1. Tangential Derivations, Hilbert Series and Modules over Lie Algebroids
Open this publication in new window or tab >>Tangential Derivations, Hilbert Series and Modules over Lie Algebroids
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

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.
Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University, 2011. 66 p.
Keyword
Tangential Derivations, Monomials, Multiplier Ideals, Lie Algebroids, Hilbert series
National Category
Algebra and Logic Geometry
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-62642 (URN)978-91-7447-372-8 (ISBN)
Public defence
2011-10-28, lecture room 14, house 5, Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (English)
Opponent
Supervisors
Note
At the time of the doctoral defense, the following paper was unpublished and had a status as follows: Paper 1: Submitted. Available from: 2011-10-06 Created: 2011-09-26 Last updated: 2011-09-29Bibliographically approved

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