Algebraic $\overline {\mathbb {Q}}$-Groups as Abstract Groups (Memoirs of the American Mathematical Society)

Frecon, Olivier

American Mathematical Society（2018/10発売）

• 製本 Paperback:紙装版/ペーパーバック版／ページ数 99 p.
• 言語 ENG
• 商品コード 9781470429232
• DDC分類 512

Full Description

The author analyzes the abstract structure of algebraic groups over an algebraically closed field $K$.

For $K$ of characteristic zero and $G$ a given connected affine algebraic $\overline{\mathbb Q}$-group, the main theorem describes all the affine algebraic $\overline{\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 $\overline{\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 section, the author applies his results to characterize the connected algebraic groups, all of whose abstract automorphisms are standard, when $K$ is either $\overline {\mathbb Q}$ or of positive characteristic. In characteristic zero, a fairly general criterion is exhibited.

Contents

Introduction
Background material
Expanded pure groups
Unipotent groups over $\overline{\mathbb Q} $ and definable linearity
Definably affine groups
Tori in expanded pure groups
The definably linear quotients of an $ACF$-group
The group $D_G$ and the Main Theorem for $K=\overline{\mathbb Q} $
The Main Theorem for $K\neq \overline{\mathbb Q}$
Bi-interpretability and standard isomorphisms
Acknowledgements
Bibliography
Index of notations
Index