Högskolan i Skövde

his.sePublications
Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • apa-cv
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Tangential Derivations, Hilbert Series and Modules over Lie Algebroids
Stockholms universitet, Matematiska institutionen.
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. , p. 66
Keywords [en]
Tangential Derivations, Monomials, Multiplier Ideals, Lie Algebroids, Hilbert series
National Category
Algebra and Logic Geometry
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:his:diva-22287ISBN: 978-91-7447-372-8 (print)OAI: oai:DiVA.org:his-22287DiVA, id: diva2:1737733
Public defence
2011-10-28, lecture room 14, house 5, Kräftriket, 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: 2023-02-17 Last updated: 2023-02-17
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
2015 (English)In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 432, p. 129-184Article in journal (Refereed) Published
Abstract [en]

We consider modules M over Lie algebroids gA which are of finite type over a local noetherian ring A. Using ideals J ⊂ A such that gA ·J ⊂ J and the length ℓgA (M/JM) < ∞ 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 gA = TA(I) that preserve an ideal I ⊂ A, for example when A = On, the ring of convergent power series. Hilbert series over Stanley-Reisner rings are also considered.

Place, publisher, year, edition, pages
Elsevier, 2015
Keywords
Lie algebroids, Local systems, Representations of Lie algebras, Hilbert series, Stanley–Reisner rings, Complex analytic singularities
National Category
Geometry Algebra and Logic
Identifiers
urn:nbn:se:his:diva-22285 (URN)10.1016/j.jalgebra.2015.02.020 (DOI)000354001500007 ()2-s2.0-84925434959 (Scopus ID)
Available from: 2023-02-17 Created: 2023-02-17 Last updated: 2023-02-17Bibliographically 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, p. 2935-2942Article 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.

Place, publisher, year, edition, pages
American Mathematical Society, 2009
Keywords
Derivations, monomial ideals, multiplier ideals
National Category
Algebra and Logic
Research subject
Mathematics
Identifiers
urn:nbn:se:his:diva-22284 (URN)10.1090/S0002-9939-09-09922-5 (DOI)000269307400015 ()2-s2.0-77951052339 (Scopus ID)
Note

The author wishes to express his warmest gratitude to his advisor, Rolf Källström,for introducing him to this subject and for his continuing support. This work is financially supported by International Science Programme, Uppsala University.

Available from: 2010-01-21 Created: 2023-02-17 Last updated: 2023-02-17Bibliographically approved

Open Access in DiVA

No full text in DiVA

Authority records

Tadesse, Yohannes

Search in DiVA

By author/editor
Tadesse, Yohannes
Algebra and LogicGeometry

Search outside of DiVA

GoogleGoogle Scholar

isbn
urn-nbn

Altmetric score

isbn
urn-nbn
Total: 55 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • apa-cv
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf