Pseudo-abelian varieties
Pseudo-abelian varieties
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伪阿贝尔变种
DOI：
10.24033/asens.2199
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发表时间：
2011
期刊：
影响因子：
--
通讯作者：
B. Totaro
中科院分区：
文献类型：
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作者：
B. Totaro
Chevalley定理指出，完美域上的每个光滑连通代数群都是阿贝尔簇通过光滑连通仿射群的扩展。当基本场不完美时，它就失败了。定义任意域k上的伪阿贝尔簇为光滑连通k-群，其中每个光滑连通仿射正规k-子群是平凡的。这给代数群的分类提供了一个新的观点：域上的每个光滑连通群都是伪阿贝尔簇通过光滑连通仿射群的唯一扩张。
我们研究了伪阿贝尔簇的许多结构。这些群与特徴为p的幂幺群和Tits和Conrad-Gabber-Prasad研究的伪约化群密切相关。交换簇的许多性质，如Mordell-Weil定理，都可以推广到伪交换簇。最后，我们在Milnor猜想的基础上，利用生成元和关系猜想了任意域上Ext^2（G_a，G_m）的一个刻画。
Chevalley's theorem states that every smooth connected algebraic group over a perfect field is an extension of an abelian variety by a smooth connected affine group. That fails when the base field is not perfect. We define a pseudo-abelian variety over an arbitrary field k to be a smooth connected k-group in which every smooth connected affine normal k-subgroup is trivial. This gives a new point of view on the classification of algebraic groups: every smooth connected group over a field is an extension of a pseudo-abelian variety by a smooth connected affine group, in a unique way.
We work out much of the structure of pseudo-abelian varieties. These groups are closely related to unipotent groups in characteristic p and to pseudo-reductive groups as studied by Tits and Conrad-Gabber-Prasad. Many properties of abelian varieties such as the Mordell-Weil theorem extend to pseudo-abelian varieties. Finally, we conjecture a description of Ext^2(G_a,G_m) over any field by generators and relations, in the spirit of the Milnor conjecture.