Higher-dimensional analogue of cyclicity in degree three Galois cohomology.

par Saurabh Gossavi (Tel Aviv)

jeudi 9 juin 2022, 11:15 → 12:15 Europe/Paris

112 (ICJ (Braconnier))

 Recall that central simple algebras over global fields are cyclic. In Galois cohomological terms, this may be re-expressed as  every element in the m-torsion part of the Brauer group H^2(F, \mu_m) of a global field F (of characteristic coprime to m) can be written as a cup product of a character and a class in H^1(F, \mu_m). In this talk, we will show a higher-dimensional analogue of this fact for function fields of curves over non-archimedean local fields.  More precisely, let F be the function field of a curve over a non-archimedean local field and m \geq 2 be an integer coprime to the characteristic of the residue field. Using the Harbater-Hartmann-Krashen field patching technique, we will show that every    element in H^{3}(F, \mu_{m}^{\otimes 2}) can be written as a cup product of character with two classes in H^1(F, \mu_m).  This extends a result of Parimala and Suresh where they show this when m is prime and under the assumption that F contains a primitive m^{th} root of unity.