**Proceedings of Symposia in Pure Mathematics**

Volume: 94;
2017;
294 pp;
Hardcover

MSC: Primary 12; 14; 20;

**Print ISBN: 978-1-4704-2601-9
Product Code: PSPUM/94**

List Price: $126.00

AMS Member Price: $100.80

MAA Member Price: $113.40

**Electronic ISBN: 978-1-4704-3751-0
Product Code: PSPUM/94.E**

List Price: $126.00

AMS Member Price: $100.80

MAA Member Price: $113.40

#### You may also like

# Algebraic Groups: Structure and Actions

Share this page *Edited by *
*Mahir Bilen Can*

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 \(K\)-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containing previously unpublished foundational results on birational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 \(p\)-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.

#### Readership

Graduate students and researchers working in algebraic group theory as well as algebraic geometry and number theory.

# Table of Contents

## Algebraic Groups: Structure and Actions

- Cover Cover11
- Title page iii4
- Contents v6
- Preface vii8
- Computing torus-equivariant ๐พ-theory of singular varieties 110
- Algebraic structures of groups of birational transformations 1726
- The Hermite-Joubert problem over ๐-closed fields 3140
- 1. Introduction 3140
- 2. Geometry of the hypersurfaces ๐_{๐,๐} and ๐_{๐,๐} 3443
- 3. Proof of Theorem 1.3: (1) โน (2) 3645
- 4. Proof of Theorem 1.3: (2) โน (3) 3746
- 5. Proof of Theorem 1.3: (3) โน (1) 3948
- 6. Proof of Theorem 1.4 4049
- 7. Density of rational points on hypersurfaces 4150
- 8. Proof of Assertions (โ) and (โโ) 4251
- 9. Remarks on Theorems 1.3 and 1.4 4352
- 10. The Hermite-Joubert problem for ๐=2 4453
- 11. The Hermite-Joubert problem for ๐=3 4554
- 12. When are there solutions to (1.1) and (1.2)? 4756
- 13. Proof of Theorem 1.5 4857
- 14. Beyond Theorem 1.5 4958
- Acknowledgements 5059
- References 5160

- Some structure theorems for algebraic groups 5362
- 1. Introduction 5463
- 2. Basic notions and results 5665
- 2.1. Group schemes 5665
- 2.2. Actions of group schemes 5968
- 2.3. Linear representations 6271
- 2.4. The neutral component 6574
- 2.5. Reduced subschemes 6776
- 2.6. Torsors 6978
- 2.7. Homogeneous spaces and quotients 7180
- 2.8. Exact sequences, isomorphism theorems 7382
- 2.9. The relative Frobenius morphism 7685

- 3. Proof of Theorem 1 7988
- 4. Proof of Theorem 2 8594
- 5. Some further developments 94103
- 6. The Picard scheme 110119
- 7. The automorphism group scheme 115124
- References 123132

- Structure and classification of pseudo-reductive groups 127136
- 1. Introduction 128137
- 2. Standard groups and dynamic methods 137146
- 3. Roots, root groups, and root systems 150159
- 4. Structure theory 161170
- 5. Refined structure theory 173182
- 6. Central extensions and standardness 189198
- 7. Non-standard constructions 198207
- 8. Groups with a non-reduced root system 224233
- 9. Classification of forms 241250
- 10. Structural classification 255264
- Acknowledgements 270279
- References 271280
- Index 273282

- Invariants of algebraic groups and retract rationality of classifying spaces 277286
- 1. Introduction 277286
- 2. Galois cohomology 278287
- 3. Retract rational varieties 280289
- 4. Retract rational classifying spaces 282291
- 5. Cohomology of classifying spaces 285294
- 6. Invariants of algebraic groups 287296
- 7. Degree 1 invariants with coefficients in Galois module 289298
- 8. Brauer invariants 291300
- 9. Invariants of degree 3 with coefficients in \Q/\Z(2) 292301
- References 293302

- Back Cover Back Cover1306