On spectral sequences for semiabelian varieties over non-closed fields

TL;DR

This paper provides new proofs and explicit formulas for the differentials in the Hochschild--Serre spectral sequence for semiabelian varieties over non-closed fields, with applications to Jacobians, tori, and curves.

Contribution

It introduces a simplified proof for the spectral sequence differential and derives explicit formulas for various cases, including Jacobians and tori.

Findings

Differential is non-zero for Jacobians with non-zero torsor of theta-characteristics.

Spectral sequence degenerates at the second page when the Albanese torsor is trivial.

Explicit formula for the Brauer group of a torus via the spectral sequence.

Abstract

We give a new, short proof of the formula for the first potentially non-zero differential of the Hochschild--Serre spectral sequence for semiabelian varieties over non-closed fields. We show that this differential is non-zero for the Jacobian of a curve when the image of the torsor of theta-characteristics under the Bockstein map is non-zero. An explicit example is a curve of genus 2 whose Albanese torsor is not divisible by 2. When the Albanese torsor is trivial, we show that the Hochschild--Serre spectral sequence for the Jacobian degenerates at the second page. We give a formula for the differential of the Hochschild--Serre spectral sequence for a torus which computes its Brauer group. Finally, we describe the differentials of the Hochschild--Serre spectral sequence for a smooth projective curve, generalising a lemma of Suslin.