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Equivariant vector bundles and rigid cohomology on Drinfeld's upper half space over a finite field / vorgelegt von Mark Kuschkowitz aus Halle/Saale. Wuppertal, März 2016
Content
Introduction
Content of this Thesis
Acknowledgements
Preliminaries
General Notation
Representations
Induced Representations
Composition Series and Semisimplifications
(Generalized) Steinberg Representations
The Simple (Algebraic) Representations of Gk and G, and Weyl Modules
Drinfeld's Upper Half Space over a Finite Field
Sheaves and Cohomology
Cohomology of Equivariant Vector Bundles on Drinfeld's Upper Half Space over a Finite Field
Orlik's Complex for the Cohomology of Equivariant Vector Bundles
Construction of the Complex
An Equivariant Filtration on the Cohomology of Drinfeld's Upper Half Space
Local Cohomology I: First Descriptions
Bundles arising from Representations of a Levi Subgroup
Using the Canonical Projection onto a Projective Subvariety
Local Cohomology II: Failure of ``Classical'' Lie Algebraic Methods
Generalized Fractions
The Action of the Universal Enveloping Algebra
The Action of the Distribution Algebra
Enriched Crystalline Enveloping Algebras: Adding more Divided Powers to the Distribution Algebra
First Examples, Row Algebras and Column Algebras
Some Subalgebras and Duality
Local Cohomology III: Description via Enriched Crystalline Enveloping Algebra
Employing the Enriched Crystalline Enveloping Algebra
Functorial Reinterpretation: U+-Algebraic Induction and Extension of Duality
Examples
The Regular Representation of G
Some Equivariant Vector Bundles on Pn admitting Actions of the Enriched Crystalline Enveloping Algebra
The (Twisted) Structure Sheaf
Rigid Cohomology of Drinfeld's Upper Half Space over a Finite Field
Construction of Rigid Cohomology and some Properties
Berthelot's Definition of Rigid Cohomology (with and without Supports)
Some Properties of Rigid Cohomology
Rigid Cohomology computed as Hypercohomology
Adaption of Orlik's Complex
Construction of a Spectral Sequence
Evaluation of the Spectral Sequence
Computation of the Rigid Cohomology Modules
Rigid Cohomology computed from the Associated De Rham Complex
Adaption of Orlik's Complex
Construction of a Spectral Sequence
Evaluation of the Spectral Sequence
Computation of the Rigid Cohomology Modules
Bibliography