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Transactions of the American Mathematical Society

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


Author: Diego Izquierdo
Journal: Trans. Amer. Math. Soc. 372 (2019), 8621-8662
MSC (2010): Primary 11R34, 11G25, 11G35, 11S25, 12G05, 14G15, 14G20, 14G25, 14G27, 14J20, 14B05, 14J17
DOI: https://doi.org/10.1090/tran/7861
Published electronically: August 8, 2019
MathSciNet review: 4029707
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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.


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Additional Information

Diego Izquierdo
Affiliation: Département de Mathématiques et Applications, École Normale Supérieure, CNRS, PSL Research University, 45 Rue d’Ulm, 75005 Paris, France
Email: diego.izquierdo@ens.fr

DOI: https://doi.org/10.1090/tran/7861
Keywords: Galois cohomology, Bloch--Kato conjecture, Laurent series fields in two or more variables, function fields in two or more variables, singularities, finite base fields, $p$-adic base fields, global base fields, Hasse principle, Brauer group, Brauer--Hasse--Noether exact sequence.
Received by editor(s): December 20, 2018
Received by editor(s) in revised form: March 27, 2019
Published electronically: August 8, 2019
Article copyright: © Copyright 2019 American Mathematical Society