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MORE ON GALOIS COHOMOLOGY, DEFINABILITY, AND DIFFERENTIAL ALGEBRAIC GROUPS

Part of: Model theory Connections with logic

Published online by Cambridge University Press:  11 April 2024

OMAR LEÓN SÁNCHEZ ,
DAVID MERETZKY  and
ANAND PILLAY
OMAR LEÓN SÁNCHEZ*
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF MANCHESTER OXFORD ROAD, MANCHESTER, M13 9PL, UK
DAVID MERETZKY
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF NOTRE DAME 255 HURLEY, NOTRE DAME, IN 46556, USA E-mail: dmeretzk@nd.edu
ANAND PILLAY
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF NOTRE DAME 255 HURLEY, NOTRE DAME, IN 46556, USA E-mail: apillay@nd.edu
*
E-mail: omar.sanchez@manchester.ac.uk
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Abstract

As a continuation of the work of the third author in [5], we make further observations on the features of Galois cohomology in the general model theoretic context. We make explicit the connection between forms of definable groups and first cohomology sets with coefficients in a suitable automorphism group. We then use a method of twisting cohomology (inspired by Serre’s algebraic twisting) to describe arbitrary fibres in cohomology sequences—yielding a useful “finiteness” result on cohomology sets.

Applied to the special case of differential fields and Kolchin’s constrained cohomology, we complete results from [3] by proving that the first constrained cohomology set of a differential algebraic group over a bounded, differentially large, field is countable.

Keywords

model theoryatomic extensionsGalois cohomology

MSC classification

Primary: 03C60: Model-theoretic algebra 03C98: Applications of model theory 12L12: Model theory
Type
Article
Information
The Journal of Symbolic Logic , First View , pp. 1 - 20
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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