Extended Picard complexes and linear algebraic groups

Mikhail Borovoi*, Joost Van Hamel

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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 k̄[X]×/k̄×, where is a fixed algebraic closure of k and is obtained from X by extension of scalars to k̄. 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 π 1(G). As an application we compute the elementary obstruction for such X.

Original languageEnglish
Pages (from-to)53-82
Number of pages30
JournalJournal fur die Reine und Angewandte Mathematik
Issue number627
StatePublished - Feb 2009


FundersFunder number
Hermann Minkowski Center for Geometry


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