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

Diego Izquierdo

doi:10.1090/tran/7861

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

Diego Izquierdo

Research output: Contribution to journal › Article › peer-review

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((x1, . . ., x_m)). When r is an integer and φ is a prime number, we consider the Galois module Qφ/Zφ(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((x1, x2)), 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 Qφ/Zφ(r) which do not vanish.

Original language	English
Pages (from-to)	8621-8662
Number of pages	42
Journal	Transactions of the American Mathematical Society
Volume	372
Issue number	12
DOIs	https://doi.org/10.1090/tran/7861
Publication status	Published - 15 Dec 2019

Keywords

Bloch-Kato conjecture
Brauer group
Brauer-Hasse-Noether exact sequence
Finite base fields
Function fields in two or more variables
Galois cohomology
Global base fields
Hasse principle
Laurent series fields in two or more variables
P-adic base fields
Singularities

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10.1090/tran/7861

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title = "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((x1, . . ., xm)). When r is an integer and φ is a prime number, we consider the Galois module Qφ/Zφ(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((x1, x2)), 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 Qφ/Zφ(r) which do not vanish.",

keywords = "Bloch-Kato conjecture, Brauer group, Brauer-Hasse-Noether exact sequence, Finite base fields, Function fields in two or more variables, Galois cohomology, Global base fields, Hasse principle, Laurent series fields in two or more variables, P-adic base fields, Singularities",

author = "Diego Izquierdo",

note = "Publisher Copyright: {\textcopyright} 2019 American Mathematical Society",

year = "2019",

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day = "15",

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T1 - Vanishing theorems and Brauer-Hasse-noether exact sequences for the cohomology of higher-dimensional fields

AU - Izquierdo, Diego

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N2 - 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((x1, . . ., xm)). When r is an integer and φ is a prime number, we consider the Galois module Qφ/Zφ(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((x1, x2)), 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 Qφ/Zφ(r) which do not vanish.

AB - 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((x1, . . ., xm)). When r is an integer and φ is a prime number, we consider the Galois module Qφ/Zφ(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((x1, x2)), 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 Qφ/Zφ(r) which do not vanish.

KW - Bloch-Kato conjecture

KW - Brauer group

KW - Brauer-Hasse-Noether exact sequence

KW - Finite base fields

KW - Function fields in two or more variables

KW - Galois cohomology

KW - Global base fields

KW - Hasse principle

KW - Laurent series fields in two or more variables

KW - P-adic base fields

KW - Singularities

U2 - 10.1090/tran/7861

DO - 10.1090/tran/7861

M3 - Article

AN - SCOPUS:85075127924

SN - 0002-9947

VL - 372

SP - 8621

EP - 8662

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 12

ER -

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

Abstract

Keywords

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