Galois cohomology of function fields of curves over non-archimedean local fields

Gosavi, Saurabh

doi:10.1090/proc/16074

Galois cohomology of function fields of curves over non-archimedean local fields
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by Saurabh Gosavi PDF

Proc. Amer. Math. Soc. 150 (2022), 5179-5191 Request permission

Abstract:

Let $F$ be the function field of a curve over a non-archimedean local field. Let $m \geq 2$ be an integer coprime to the characteristic of the residue field of the local field. In this article, we show that every element in $H^{3}(F, \mu _{m}^{\otimes 2})$ is of the form $\chi \cup (f) \cup (g)$, where $\chi$ is in $H^{1}(F, \mathbb {Z}/m\mathbb {Z})$ and $(f)$, $(g)$ in $H^{1}(F, \mu _{m})$. This extends a result of Parimala and Suresh [Ann. of Math. (2) 172 (2010), pp. 1391–1405], where they show this when $m$ is prime and when $F$ contains $\mu _{m}$.

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