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Tangential Derivations, Hilbert Series and Modules over Lie Algebroids
Stockholm University, Faculty of Science, Department of Mathematics. (Algebra, Geometry, D-modules)
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 [en]
Tangential Derivations, Monomials, Multiplier Ideals, Lie Algebroids, Hilbert series
National Category
Algebra and Logic Geometry
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-62642ISBN: 978-91-7447-372-8 (print)OAI: oai:DiVA.org:su-62642DiVA: diva2:443693
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
List of papers
1. Hilbert series of modules over Lie algebroids
Open this publication in new window or tab >>Hilbert series of modules over Lie algebroids
(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
Hilbert series, tangential vector field, Lie algebroid, complete intersection
National Category
Algebra and Logic Geometry
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-62614 (URN)
Available from: 2011-09-26 Created: 2011-09-26 Last updated: 2011-09-29Bibliographically approved
2. Derivations Preserving a Monomial Ideal
Open this publication in new window or tab >>Derivations Preserving a Monomial Ideal
2009 (English)In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 137, no 9, 2935-2942 p.Article in journal (Refereed) Published
Abstract [en]

Let I be a monomial ideal in a polynomial ring A = k[x1, . . . ,xn] over a field k of characteristic 0, TA/k(I) be the module of I-preserving k-derivations on A and G be the n-dimensional algebraic torus on k. We computethe weight spaces of TA/k(I) considered as a representation of G. Using this, we show that TA/k(I) preserves the integral closure of I and the multiplierideals of I.

Keyword
Derivations, monomial ideals, multiplier ideals
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-35969 (URN)10.1090/S0002-9939-09-09922-5 (DOI)000269307400015 ()
Available from: 2010-01-21 Created: 2010-01-21 Last updated: 2017-12-12Bibliographically approved

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