Vanishing theorems and Brauer–Hasse–Noether exact sequences for the cohomology of higher-dimensional fields

Abstract:

Let $k$ be a finite field, a $p$-adic field, or a number field. Let $K$ be a finite extension of the Laurent series field in $m$ variables $k((x_1,\ldots ,x_m))$. When $r$ is an integer and $\ell$ is a prime number, we consider the Galois module $\mathbb {Q}_{\ell }/\mathbb {Z}_{\ell }(r)$ over $K$, and we prove several vanishing theorems for its cohomology. In the particular case in which $K$ is a finite extension of the Laurent series field in two variables $k((x_1,x_2))$, we also prove exact sequences that play the role of the Brauer–Hasse–Noether exact sequence for the field $K$ and that involve some of the cohomology groups of $\mathbb {Q}_{\ell }/\mathbb {Z}_{\ell }(r)$ which do not vanish.