Extended Picard complexes and linear algebraic groups
Extended Picard complexes and linear algebraic groups
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扩展皮卡德复形和线性代数群
DOI：
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发表时间：
2006
期刊：
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通讯作者：
J. van Hamel
中科院分区：
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作者：
M. Borovoi;J. van Hamel
对于特征为0的域k上的光滑几何积分簇X，我们引入并研究了推广的Picard复形UPic（X）。它是长度为2的伽罗瓦模的某种复形，其第零上同调为，第一上同调为Pic（），其中是k的固定代数闭包，通过标量的扩张从X得到。当X是连通线性k-群G的k-扭子时，我们用代数基本群π 1（G）计算UPic（X）= UPic（G）（在导出范畴中）.作为应用，我们计算了X的初等阻塞.
Abstract For a smooth geometrically integral variety X over a field k of characteristic 0, we introduce and investigate the extended Picard complex UPic(X). It is a certain complex of Galois modules of length 2, whose zeroth cohomology is and whose first cohomology is Pic(), where is a fixed algebraic closure of k and is obtained from X by extension of scalars to . When X is a k-torsor of a connected linear k-group G, we compute UPic(X) = UPic(G) (in the derived category) in terms of the algebraic fundamental group π1(G). As an application we compute the elementary obstruction for such X.