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Tangential Derivations, Hilbert Series and Modules over Lie AlgebroidsPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2011 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### 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

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##### 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At the time of the doctoral defense, the following paper was unpublished and had a status as follows: Paper 1: Submitted.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

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.

1. Hilbert series of modules over Lie algebroids$(function(){PrimeFaces.cw("OverlayPanel","overlay443519",{id:"formSmash:j_idt1077:0:j_idt1083",widgetVar:"overlay443519",target:"formSmash:j_idt1077:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. Derivations Preserving a Monomial Ideal$(function(){PrimeFaces.cw("OverlayPanel","overlay288423",{id:"formSmash:j_idt1077:1:j_idt1083",widgetVar:"overlay288423",target:"formSmash:j_idt1077:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

isbn
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