ON SPECTRAL SEQUENCES FOR SEMIABELIAN VARIETIES OVER NON-CLOSED FIELDS

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.