On spectral sequences for semiabelian varieties over non-closed fields

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.