Duality of Albanese and Picard 1‐motives
Duality of Albanese and Picard 1‐motives
复制标题
Albanese 和 Picard 1 动机的二元性
DOI：
10.1023/a:1011100812900
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发表时间：
1998
期刊：
影响因子：
--
通讯作者：
N. Ramachandran
中科院分区：
文献类型：
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作者：
N. Ramachandran
我们定义了一个完美域上光滑（简单）方案的Albanese动机和Picard动机。对于光滑的适当方案，这些是经典的Albanese和Picard变种。对于一条曲线，它们是Lichtenbaum的同源动机和Deligne的动机H^1$。本文证明了Deligne关于通过1-动机给出复代数变种的第一同调群和上同调群的代数描述的猜想。（L. Barbieri-Viale和V. Srinivas也独立证明了这一点。）它还包含Lichtenbaum猜想的纯代数证明，即（简单）方案的Albanese动机和Picard 1动机是对偶的。这给出了利希滕鲍姆一个未发表的定理的一个新的证明，即曲线的德利涅的1动机与利希滕鲍姆的1动机是对偶的。
We define Albanese and Picard 1-motives of smooth (simplicial) schemes over a perfect field. For smooth proper schemes, these are the classical Albanese and Picard varieties. For a curve, these are t he homological 1-motive of Lichtenbaum and the motivic $H^1$ of Deligne. This paper proves a conjecture of Deligne about providing an algebraic description, via 1-motives, of the first homology and cohomology groups of a complex algebraic variety. (L. Barbieri-Viale and V. Srinivas have also proved this independently.) It also contains a purely algebraic proof of Lichtenbaum's conjecture that the Albanese and the Picard 1-motives of a (simplicial) scheme are dual. This gives a new proof of an unpublished theorem of Lichtenbaum that Deligne's 1-motive of a curve is dual to Lichtenbaum's 1-motive.