Partial Galois cohomology and related homomorphisms

2018 ◽  
Vol 70 (2) ◽  
pp. 737-766 ◽  
Author(s):  
M Dokuchaev ◽  
A Paques ◽  
H Pinedo
Keyword(s):  
Galois Extension ◽  
Galois Cohomology ◽  
Commutative Rings ◽  
Partial Galois Extension

Abstract For a partial Galois extension of commutative rings, we give a seven terms sequence, which is an analog of the Chase–Harrison–Rosenberg sequence.

scholarly journals Note on Galois Cohomology

1953 ◽  
Vol 5 ◽  
pp. 97-104
Author(s):  
Tadasi Nakayama
Keyword(s):  
Galois Extension ◽  
Class Group ◽  
Present Note ◽  
Galois Cohomology ◽  
Group Cohomology ◽  
Ground Field ◽  
General Group ◽  
3 Dimensional ◽  
The Relationship ◽  
American Authors

Let K/k be a Galois extension. Formerly the writer studied, in [3], [4], a certain correlation of factor sets in K/k with the norm class group of K/k, and extended it, in [5.], to 3-dimensional cocycles. The present note is to study the same relationship for general n-cocycles. As a matter of fact, the constructions which underlie the relationship have become common places in cohomology theory, through the works of French and American authors, and indeed the construction to bring certain (non-Galois) cocycles into the ground field k has been discussed by Baer [1, Theorem C] for general dimensions n under the setting of general group cohomology.

The structure of a partial Galois extension II

2016 ◽  
Vol 15 (04) ◽  
pp. 1650061 ◽  
Author(s):  
Jung-Miao Kuo ◽  
George Szeto
Keyword(s):  
Finite Group ◽  
Galois Extension ◽  
Structure Theorem ◽  
Galois Extensions ◽  
Partial Action ◽  
A Finite Group ◽  
Partial Galois Extension

Let [Formula: see text] be a partial Galois extension where [Formula: see text] is a partial action of a finite group on a ring [Formula: see text] such that the associated ideals are generated by central idempotents. We determine the set of all Galois extensions in [Formula: see text], and give an orthogonality criterion for nonzero elements in the Boolean semigroup generated by those central idempotents. These results lead to a structure theorem for [Formula: see text].

Galois corings and groupoids acting partially on algebras

2020 ◽  
pp. 2140003
Author(s):  
S. Caenepeel ◽  
T. Fieremans
Keyword(s):  
Galois Extension ◽  
Galois Theory ◽  
Type Theorem ◽  
Partial Action ◽  
Morita Theory ◽  
Maschke Type Theorem ◽  
Galois Corings ◽  
Partial Galois Extension ◽  
Special Case

Bagio and Paques [Partial groupoid actions: globalization, Morita theory and Galois theory, Comm. Algebra 40 (2012) 3658–3678] developed a Galois theory for unital partial actions by finite groupoids. The aim of this note is to show that this is actually a special case of the Galois theory for corings, as introduced by Brzeziński [The structure of corings, Induction functors, Maschke-type theorem, and Frobenius and Galois properties, Algebr. Represent. Theory 5 (2002) 389–410]. To this end, we associate a coring to a unital partial action of a finite groupoid on an algebra [Formula: see text], and show that this coring is Galois if and only if [Formula: see text] is an [Formula: see text]-partial Galois extension of its coinvariants.

On three special types of partial Galois extensions

2019 ◽  
Vol 19 (12) ◽  
pp. 2150004
Author(s):  
Xiaolong Jiang ◽  
Jung-Miao Kuo ◽  
George Szeto
Keyword(s):  
Galois Extension ◽  
Galois Extensions ◽  
Partial Galois Extension

In this paper, we study the following three special types of partial Galois extensions: DeMeyer–Kanzaki partial Galois extension, partial Galois Azumaya extension and commutator partial Galois extension.

Hilbert 90 for KnM

2019 ◽  
pp. 42-53
Author(s):  
Christian Haesemeyer ◽  
Charles A. Weibel
Keyword(s):  
Exact Sequence ◽  
Galois Extension ◽  
Galois Cohomology ◽  
Inductive Step ◽  
Etale Cohomology ◽  
Étale Cohomology ◽  
Cyclic Galois Extension

This chapter formulates a norm-trace relation for the Milnor 𝐾-theory and étale cohomology of a cyclic Galois extension, herein called Hilbert 90 for 𝐾𝑀 𝑛. To begin, the chapter uses condition BL(n) to establish a related exact sequence in Galois cohomology. It then establishes that condition BL(n − 1) implies the particular case of condition H90(n) for 𝓁-special fields 𝑘 such that 𝐾𝑀 𝑛(𝑘) is 𝓁-divisible. This case constitutes the first part of the inductive step in the proof of Theorem A. The remainder of this chapter explains how to reduce the general case to this particular one. The chapter concludes with some background on the Hilbert 90 for 𝐾𝑀 𝑛.

scholarly journals A Note on Galois Cohomology Groups of Algebraic Tori

1969 ◽  
Vol 34 ◽  
pp. 121-127 ◽  
Author(s):  
Kazuo Amano
Keyword(s):  
Galois Group ◽  
Galois Extension ◽  
Valuation Ring ◽  
Galois Cohomology ◽  
Splitting Field ◽  
Cohomology Groups ◽  
Algebraic Tori ◽  
Usual Manner ◽  
Algebraic Torus ◽  
Complete Field

Let k be a complete field of characteristic 0 whose topology is defined by a discrete valuation and let T be an algebraic torus of dimension d defined over k. As is well known, T has a splitting field K which is a finite Galois extension of k with Galois group . For a ring R, denote by TR the subgroup of R-rational points of T. Then TK and T0K, DK being a valuation ring of K, become -modules in the usual manner.

The structure of a partial Galois extension

2013 ◽  
Vol 175 (4) ◽  
pp. 565-576 ◽  
Author(s):  
Jung-Miao Kuo ◽  
George Szeto
Keyword(s):  
Galois Extension ◽  
Partial Galois Extension
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