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Rees algebras of modules and coherent functors
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0001-8893-5211
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We show that several properties of the theory of Rees algebras of modules become more transparent using the category of coherent functors rather than working directly with modules. In particular, we show that the Rees algebra is induced by a canonical map of coherent functors.

National Category
Algebra and Logic
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-156534OAI: oai:DiVA.org:kth-156534DiVA: diva2:767503
Note

QCR 20161110

Available from: 2014-12-01 Created: 2014-11-29 Last updated: 2016-11-10Bibliographically approved
In thesis
1. Rees algebras of modules and Quot schemes of points
Open this publication in new window or tab >>Rees algebras of modules and Quot schemes of points
2014 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of three articles. The first two concern a generalization of Rees algebras of ideals to modules. Paper A shows that the definition of the Rees algebra due to Eisenbud, Huneke and Ulrich has an equivalent, intrinsic, definition in terms of divided powers. In Paper B, we use coherent functors to describe properties of the Rees algebra. In particular, we show that the Rees algebra is induced by a canonical map of coherent functors.

In Paper C, we prove a generalization of Gotzmann's persistence theorem to finite modules. As a consequence, we show that the embedding of the Quot scheme of points into a Grassmannian is given by a single Fitting ideal.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2014. vii, 19 p.
Series
TRITA-MAT-A, 2014:17
National Category
Algebra and Logic Geometry
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-156636 (URN)978-91-7595-400-4 (ISBN)
Presentation
2015-01-23, 3418, Lindstedtsvägen 25, Stockholm, 10:00 (English)
Opponent
Supervisors
Note

QC 20141218

Available from: 2014-12-18 Created: 2014-12-01 Last updated: 2014-12-18Bibliographically approved
2. Hilbert schemes and Rees algebras
Open this publication in new window or tab >>Hilbert schemes and Rees algebras
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The topic of this thesis is algebraic geometry, which is the mathematical subject that connects polynomial equations with geometric objects. Modern algebraic geometry has extended this framework by replacing polynomials with elements from a general commutative ring, and studies the geometry of abstract algebra. The thesis consists of six papers relating to some different topics of this field.

The first three papers concern the Rees algebra. Given an ideal of a commutative ring, the corresponding Rees algebra is the coordinate ring of a blow-up in the subscheme defined by the ideal. We study a generalization of this concept where we replace the ideal with a module. In Paper A we give an intrinsic definition of the Rees algebra of a module in terms of divided powers. In Paper B we show that features of the Rees algebra can be explained by the theory of coherent functors. In Paper C we consider the geometry of the Rees algebra of a module, and characterize it by a universal property.

The other three papers concern various moduli spaces. In Paper D we prove a partial generalization of Gotzmann’s persistence theorem to modules, and give explicit equations for the embedding of a Quot scheme inside a Grassmannian. In Paper E we expand on a result of Paper D, concerning the structure of certain Fitting ideals, to describe projective embeddings of open affine subschemes of a Hilbert scheme. Finally, in Paper F we introduce the good Hilbert functor parametrizing closed substacks with proper good moduli spaces of an algebraic stack, and we show that this functor is algebraic under certain conditions on the stack. 

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2016. vii, 49 p.
Series
TRITA-MAT-A, 2016:11
National Category
Algebra and Logic Geometry
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-195717 (URN)978-91-7729-171-8 (ISBN)
Public defence
2016-12-08, F3, Lindstedtsvägen 26, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

QC 20161110

Available from: 2016-11-10 Created: 2016-11-08 Last updated: 2016-11-10Bibliographically approved

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Sædén Ståhl, Gustav

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