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

TL;DR

This paper proves Gabber's conjecture that the Picard group of the punctured spectrum of a 3-dimensional hypersurface local ring is torsion-free, connecting it to non-commutative resolutions and homological algebra.

Contribution

It establishes the torsion-freeness of Pic(U_R) for 3-dimensional hypersurface rings, confirming Gabber's conjecture in this case and exploring related algebraic connections.

Findings

Pic(U_R) is torsion-free for 3-dimensional hypersurfaces

Connections between Gabber's conjecture and non-commutative resolutions

Insights into homological algebra over local rings

Abstract

Let (R,m) be a 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 3, 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.