CiteExport$(function(){PrimeFaces.cw("TieredMenu","widget_formSmash_upper_j_idt184",{id:"formSmash:upper:j_idt184",widgetVar:"widget_formSmash_upper_j_idt184",autoDisplay:true,overlay:true,my:"left top",at:"left bottom",trigger:"formSmash:upper:exportLink",triggerEvent:"click"});}); $(function(){PrimeFaces.cw("OverlayPanel","widget_formSmash_upper_j_idt192_j_idt200",{id:"formSmash:upper:j_idt192:j_idt200",widgetVar:"widget_formSmash_upper_j_idt192_j_idt200",target:"formSmash:upper:j_idt192:permLink",showEffect:"blind",hideEffect:"fade",my:"right top",at:"right bottom",showCloseIcon:true});});

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});
function selectAll()
{
var panelSome = $(PrimeFaces.escapeClientId("formSmash:some"));
var panelAll = $(PrimeFaces.escapeClientId("formSmash:all"));
panelAll.toggle();
toggleList(panelSome.get(0).childNodes, panelAll);
toggleList(panelAll.get(0).childNodes, panelAll);
}
/*Toggling the list of authorPanel nodes according to the toggling of the closeable second panel */
function toggleList(childList, panel)
{
var panelWasOpen = (panel.get(0).style.display == 'none');
// console.log('panel was open ' + panelWasOpen);
for (var c = 0; c < childList.length; c++) {
if (childList[c].classList.contains('authorPanel')) {
clickNode(panelWasOpen, childList[c]);
}
}
}
/*nodes have styleClass ui-corner-top if they are expanded and ui-corner-all if they are collapsed */
function clickNode(collapse, child)
{
if (collapse && child.classList.contains('ui-corner-top')) {
// console.log('collapse');
child.click();
}
if (!collapse && child.classList.contains('ui-corner-all')) {
// console.log('expand');
child.click();
}
}
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. , p. 66
##### 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, id: diva2:443693
##### Public defence

2011-10-28, lecture room 14, house 5, Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (English)
##### Opponent

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt987",{id:"formSmash:j_idt987",widgetVar:"widget_formSmash_j_idt987",multiple:true});
##### Supervisors

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt1004",{id:"formSmash:j_idt1004",widgetVar:"widget_formSmash_j_idt1004",multiple:true});
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt1012",{id:"formSmash:j_idt1012",widgetVar:"widget_formSmash_j_idt1012",multiple:true});
##### 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_idt1073:0:j_idt1098",widgetVar:"overlay443519",target:"formSmash:j_idt1073:0: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_idt1073:1:j_idt1098",widgetVar:"overlay288423",target:"formSmash:j_idt1073:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

isbn
urn-nbn$(function(){PrimeFaces.cw("Tooltip","widget_formSmash_j_idt2096",{id:"formSmash:j_idt2096",widgetVar:"widget_formSmash_j_idt2096",showEffect:"fade",hideEffect:"fade",showDelay:500,hideDelay:300,target:"formSmash:altmetricDiv"});});

CiteExport$(function(){PrimeFaces.cw("TieredMenu","widget_formSmash_lower_j_idt2149",{id:"formSmash:lower:j_idt2149",widgetVar:"widget_formSmash_lower_j_idt2149",autoDisplay:true,overlay:true,my:"left top",at:"left bottom",trigger:"formSmash:lower:exportLink",triggerEvent:"click"});}); $(function(){PrimeFaces.cw("OverlayPanel","widget_formSmash_lower_j_idt2150_j_idt2152",{id:"formSmash:lower:j_idt2150:j_idt2152",widgetVar:"widget_formSmash_lower_j_idt2150_j_idt2152",target:"formSmash:lower:j_idt2150:permLink",showEffect:"blind",hideEffect:"fade",my:"right top",at:"right bottom",showCloseIcon:true});});