An Introduction to Galois Cohomology and its Applications (London Mathematical Society Lecture Note Series Book 377) 1st Edition by Grégory Berhuy (PDF)

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This is the first elementary introduction to Galois cohomology and its applications. The first part is self-contained and provides the basic results of the theory, including a detailed construction of the Galois cohomology functor, as well as an exposition of the general theory of Galois descent. The author illustrates the theory using the example of the descent problem of conjugacy classes of matrices. The second part of the book gives an insight into how Galois cohomology may be used to solve algebraic problems in several active research topics, such as inverse Galois theory, rationality questions or the essential dimension of algebraic groups. Assuming only a minimal background in algebra, the main purpose of this book is to prepare graduate students and researchers for more advanced study.

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⭐Given a field k, how does one classify mathematical structures defined over k which become isomorphic over a finite Galois extension or over an arbitrary Galois extension? This question is answered with great clarity in this book, and the explanations and motivations given in it have to rank its didactic quality as being one of the best in the mathematical literature. And most importantly, it does so without sacrificing mathematical rigor, which proves that the latter and intuitive understanding are not inversely related. Readers will walk away with an appreciation of Galois cohomology that might be difficult to attain by the study of other books or research papers.Readers who intend to study this book will know that they must have a thorough grounding in abstract algebra, and be familiar with field extensions and Galois theory. The author motivates the subject of Galois cohomology by examining the simple case of the descent problem for matrices, namely the problem of determining whether two matrices are conjugate over a (finite) Galois extension of a field k remains so over the k itself. The author shows how this problem involves finding an obstruction to descent, and this obstruction is essentially a map that measures how far a matrix is from being conjugate to another by an element of the matrices over k. Infinite Galois extensions are dealt with by using pro finiteness and then the challenge is to patch the obstructions together. Finding such obstructions in contexts more general than matrices is the subject of Galois cohomology.A context in modern mathematical terms is of course a category and to expose the generality of Galois cohomology the author gives a short review of category theory in the book. One category of particular interest in the book is the category whose objects are field extensions of a field and whose morphisms are morphisms of extensions of this field extension. Also of interest is a covariant functor from this category into the category of sets. For a field extension O, the elements of the Galois group GalO of O gives rise eventually to a continuous action of GalO on the category of sets and a representable functor from the category whose objects are associative unital commutative k-algebras and whose morphisms are k-algebra morphisms to the category whose objects are groups and whose morphisms are group morphisms.This representable functor is known as a group scheme and it is in this context that the author formulates and solves the descent problem using Galois cohomology. As the author shows, group schemes allow one to understand the action of a Galois group on a group, and this allows the definition of cohomology sets of the (pro finite) Galois group GalO. This depends on finding groups on which GalO acts by group automorphism, and this can be accomplished by considering O-points of group-valued functors. Such a strategy allows the definition of the nth Galois cohomology set and the author shows how to obtain the Galois cohomology of GalO to the Galois cohomology of its finite Galois sub-extensions.Towards the goal of formulating a general Galois descent problem, it is advantageous to define an equivalence relation on the category of sets for every field extension K of k. This equivalence relation identifies elements that are in the same G(K)-orbit, where G is the group-valued functor acting on the functor F from field extensions to sets. The natural question here is whether two elements that are equivalent in O are also equivalent in k. An answer to this question involves the notion of a twisted element of F, which is an element of F that is equivalent to a fixed element over O. An element a’ is defined to be a twisted K-form of a if a is equivalent to a’ over O.For the element a, the author then defines a collection of equivalence classes [a’]. This collection, denoted Fa, formulates the Galois descent problem in terms of twisted forms, namely that of showing that Fa is in fact equal to {[a]}. The author then goes on to describe Fa in terms of the Galois cohomology of a group scheme associated to a. The Galois descent condition comes down to showing that every element of F(O) on which GalO acts trivially comes from an element of the value of F on K. Examples of Galois descent for vector spaces and central simple k-algebras are given.

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