The units-Picard complex of a reductive group scheme

TL;DR

This paper computes the units-Picard complex of a reductive group scheme over a regular scheme using the dual algebraic fundamental complex, and establishes an exact sequence relating units, Picard, and Brauer groups for torsors.

Contribution

It introduces a method to compute the units-Picard complex of reductive group schemes via the dual algebraic fundamental complex and establishes a new exact sequence for torsors.

Findings

Computed the units-Picard complex in terms of the dual algebraic fundamental complex.

Established a units-Picard-Brauer exact sequence for torsors under smooth group schemes.

Provided explicit descriptions linking algebraic fundamental complexes to classical invariants.

Abstract

Let S be a locally noetherian regular scheme. We compute the units-Picard complex of a reductive S-group scheme G in terms of the dual algebraic fundamental complex of G. To do so, we establish a units-Picard-Brauer exact sequence for a torsor under a smooth S-group scheme.