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

摘要

Let (R;m) be a Noetherian local ring and U_R = 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(U_R) 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.