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

Abstract

Let k k be a finite field, a p p -adic field, or a number field. Let K K be a finite extension of the Laurent series field in m m variables k ( ( x 1 , … , x m ) ) k((x_1,\ldots ,x_m)) . When r r is an integer and ℓ \ell is a prime number, we consider the Galois module Q ℓ / Z ℓ ( r ) \mathbb {Q}_{\ell }/\mathbb {Z}_{\ell }(r) over K K , and we prove several vanishing theorems for its cohomology. In the particular case in which K K is a finite extension of the Laurent series field in two variables k ( ( x 1 , x 2 ) ) 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 K and that involve some of the cohomology groups of Q ℓ / Z ℓ ( r ) \mathbb {Q}_{\ell }/\mathbb {Z}_{\ell }(r) which do not vanish.