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Algebraic $\bar {\mathbb {Q}}$-groups as abstract groups
About this Title
Olivier Frécon, Laboratoire de Mathématiques et Applications, UMR 7348 du CNRS, Université de Poitiers, Téléport 2 - BP 30179, Bd Marie et Pierre Curie, 86962 Futuroscope Chasseneuil Cedex, France
Publication: Memoirs of the American Mathematical Society
Publication Year:
2018; Volume 255, Number 1219
ISBNs: 978-1-4704-2923-2 (print); 978-1-4704-4815-8 (online)
DOI: https://doi.org/10.1090/memo/1219
Published electronically: June 21, 2018
Keywords: Algebraic groups,
groups of finite Morley rank,
abstract isomorphisms,
elementary equivalence,
Burdges’ unipotence.
MSC: Primary 20F11; Secondary 03C60, 14L17, 20E36, 20G15
Table of Contents
Chapters
- 1. Introduction
- 2. Background material
- 3. Expanded pure groups
- 4. Unipotent groups over $\bar {\mathbb {Q}}$ and definable linearity
- 5. Definably affine groups
- 6. Tori in expanded pure groups
- 7. The definably linear quotients of an $ACF$-group
- 8. The group $D_G$ and the Main Theorem for $K=\bar {\mathbb {Q}}$
- 9. The Main Theorem for $K\neq \bar {\mathbb {Q}}$
- 10. Bi-interpretability and standard isomorphisms
- Acknowledgements
- Index of notations
Abstract
We analyze the abstract structure of algebraic groups over an algebraically closed field $K$.
For $K$ of characteristic zero and $G$ a given connected affine algebraic $\bar {\mathbb {Q}}$-group, the main theorem describes all the affine algebraic $\bar {\mathbb {Q}}$-groups $H$ such that the groups $H(K)$ and $G(K)$ are isomorphic as abstract groups. In the same time, it is shown that for any two connected algebraic $\bar {\mathbb {Q}}$-groups $G$ and $H$, the elementary equivalence of the pure groups $G(K)$ and $H(K)$ implies that they are abstractly isomorphic.
In the final chapter, we apply our results to characterize the connected algebraic groups all of whose abstract automorphisms are standard, when $K$ is either $\bar {\mathbb {Q}}$ or of positive characteristic. In characteristic zero, a fairly general criterion is exhibited.
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