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Picard groups of punctured spectra of dimension three local hypersurfaces are torsion-free

Part of: Commutative algebra: Homological methods Cycles and subschemes

Published online by Cambridge University Press:  09 November 2011

Hailong Dao
Hailong Dao*
Affiliation:
Department of Mathematics, University of Kansas, Lawrence, KS 66045-7523, USA (email: hdao@math.ku.edu)
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Abstract

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Let (R,m) be a Noetherian local ring and UR=Spec(R)−{m} be the punctured spectrum of R. Gabber conjectured that if R is a complete intersection of dimension three, then the abelian group Pic(UR) is torsion-free. In this note we prove Gabber’s statement for the hypersurface case. We also point out certain connections between Gabber’s conjecture, Van den Bergh’s notion of non-commutative crepant resolutions and some well-studied questions in homological algebra over local rings.

Keywords

Picard groupshypersurfaceslocal ringsnon-commutative crepant resolutions

MSC classification

Secondary: 14C22: Picard groups 13D07: Homological functors on modules (Tor, Ext, etc.)
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 1 , January 2012 , pp. 145 - 152
Copyright
Copyright © Foundation Compositio Mathematica 2011

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