This volume contains the proceedings of the 2015 Clifford Lectures on Algebraic Groups: Structures and Actions, held from March 2–5, 2015, at Tulane University, New Orleans, Louisiana. This volume consists of six articles on algebraic groups, including an enhanced exposition of the classical results of Chevalley and Rosenlicht on the structure of algebraic groups; an enhanced survey of the recently developed theory of pseudo-reductive groups; and an exposition of the recently developed operational -theory for singular varieties. In addition, there are three research articles containing previously unpublished foundational results on birational automorphism groups of algebraic varieties; solution of Hermite-Joubert problem over -closed fields; and cohomological invariants and applications to classifying spaces. The old and new results presented in these articles will hopefully become cornerstones for the future development of the theory of algebraic groups and applications. Graduate students and researchers working in the fields of algebraic geometry, number theory, and representation theory will benefit from this unique and broad compilation of fundamental results on algebraic group theory.
Author(s): Mahir Bilen Can
Series: Proceedings of Symposia in Pure Mathematics 94
Publisher: AMS
Year: 2017
Language: English
Commentary: decrypted from A2332D75936F0870ED17DC7E265BD774 source file
Pages: 294
Cover
Title page
Contents
Preface
Computing torus-equivariant ?-theory of singular varieties
1. Introduction
2. Some background
3. Bivariant theories
4. Kimura’s exact sequences and the Kan extension property
5. Riemann-Roch theorems
6. Localization theorems
7. Other directions
References
Algebraic structures of groups of birational transformations
1. Introduction
2. Structures given by families of transformations
3. Flat families and scheme structure
References
The Hermite-Joubert problem over ?-closed fields
1. Introduction
2. Geometry of the hypersurfaces ?_{?,?} and ?_{?,?}
3. Proof of Theorem 1.3: (1) ⟹ (2)
4. Proof of Theorem 1.3: (2) ⟹ (3)
5. Proof of Theorem 1.3: (3) ⟹ (1)
6. Proof of Theorem 1.4
7. Density of rational points on hypersurfaces
8. Proof of Assertions (∗) and (∗∗)
9. Remarks on Theorems 1.3 and 1.4
10. The Hermite-Joubert problem for ?=2
11. The Hermite-Joubert problem for ?=3
12. When are there solutions to (1.1) and (1.2)?
13. Proof of Theorem 1.5
14. Beyond Theorem 1.5
Acknowledgements
References
Some structure theorems for algebraic groups
1. Introduction
2. Basic notions and results
2.1. Group schemes
2.2. Actions of group schemes
2.3. Linear representations
2.4. The neutral component
2.5. Reduced subschemes
2.6. Torsors
2.7. Homogeneous spaces and quotients
2.8. Exact sequences, isomorphism theorems
2.9. The relative Frobenius morphism
3. Proof of Theorem 1
3.1. Affine algebraic groups
3.2. The affinization theorem
3.3. Anti-affine algebraic groups
4. Proof of Theorem 2
4.1. The Albanese morphism
4.2. Abelian torsors
4.3. Completion of the proof of Theorem 2
5. Some further developments
5.1. The Rosenlicht decomposition
5.2. Equivariant compactification of homogeneous spaces
5.3. Commutative algebraic groups
5.4. Semi-abelian varieties
5.5. Structure of anti-affine groups
5.6. Commutative algebraic groups (continued)
6. The Picard scheme
6.1. Definitions and basic properties
6.2. Structure of Picard varieties
7. The automorphism group scheme
7.1. Basic results and examples
7.2. Blanchard’s lemma
7.3. Varieties with prescribed connected automorphism group
References
Structure and classification of pseudo-reductive groups
1. Introduction
1.1. Motivation
1.2. Initial definitions and examples
1.3. Terminology and notation
1.4. Simplifications and corrections
2. Standard groups and dynamic methods
2.1. Basic properties of pseudo-reductive groups
2.2. The standard construction
2.3. Dynamic techniques and pseudo-parabolic subgroups
3. Roots, root groups, and root systems
3.1. Root groups
3.2. Pseudo-simplicity and root systems
3.3. Open cell
4. Structure theory
4.1. Bruhat decomposition
4.2. Pseudo-completeness
4.3. Properties of pseudo-parabolic subgroups
5. Refined structure theory
5.1. Further rational conjugacy
5.2. General Bruhat decomposition
5.3. Relative roots
5.4. Applications of refined structure
6. Central extensions and standardness
6.1. Central quotients
6.2. Central extensions
7. Non-standard constructions
7.1. Groups of minimal type
7.2. Rank-1 groups and applications
7.3. A non-standard construction
7.4. Root fields and standardness
7.5. Basic exotic constructions
8. Groups with a non-reduced root system
8.1. Preparations for birational constructions
8.2. Construction via birational group laws
8.3. Properties of birational construction
9. Classification of forms
9.1. Automorphisms and Galois-twisting
9.2. Tits-style classification
10. Structural classification
10.1. Exceptional constructions
10.2. Generalized standard groups
Acknowledgements
References
Index
Invariants of algebraic groups and retract rationality of classifying spaces
1. Introduction
2. Galois cohomology
3. Retract rational varieties
4. Retract rational classifying spaces
5. Cohomology of classifying spaces
6. Invariants of algebraic groups
7. Degree 1 invariants with coefficients in Galois module
8. Brauer invariants
9. Invariants of degree 3 with coefficients in \Q/\Z(2)
References
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