TY - JOUR

T1 - Extended Picard complexes and linear algebraic groups

AU - Borovoi, Mikhail

AU - Van Hamel, Joost

N1 - Funding Information:
Borovoi was partially supported by the Hermann Minkowski Center for Geometry. Joost van Hamel died on January 8, 2008.

PY - 2009/2

Y1 - 2009/2

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=58449124718&partnerID=8YFLogxK

U2 - 10.1515/CRELLE.2009.011

DO - 10.1515/CRELLE.2009.011

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AN - SCOPUS:58449124718

SN - 0075-4102

SP - 53

EP - 82

JO - Journal fur die Reine und Angewandte Mathematik

JF - Journal fur die Reine und Angewandte Mathematik

IS - 627

ER -