%% boi-7.tex (short version)
%% Chapter 7 of the Book of Involutions

\chapter{Galois Cohomology}  
\label{7.chap.nonab.coh}

In the preceding chapters, we have met groupoids $\cat{M}=\cat M(F)$
of ``algebraic objects'' over a base field~$F$, for example finite
dimensional $F$-algebras or algebraic groups of a certain type. If
over a separable closure~$\Fs$ of~$F$ the groupoid $\cat M(\Fs)$ is
connected, i.e., all objects over~$\Fs $ are isomorphic, then in many
cases the objects of~$\cat{M}$ are classified up to isomorphism by a
cohomology set $H^1\bigl(\Gal(\Fs /F), A\bigr)$, where~$A$ is the
automorphism group of a fixed object of~$\cat{M}(\Fs)$. The aim of
this chapter is to develop the general theory of such cohomology sets,
to reinterpret some earlier results in this setting and to give
techniques (like twisting) which will be used in later parts of this
book.

There are four sections. The basic techniques are explained
in~\S\ref{CoProGro.sec}, and \S\ref{GaCohAlGrp.sec} gives an explicit
description of the cohomology sets of various algebraic groups in
terms of algebras with involution. In~\S\ref{GaCoRoUn.sec} we focus on
the cohomology groups of $\gmu_{n}$, which are used
in~\S\ref{CohInv.sec} to reinterpret various invariants of algebras
with involution or of algebraic groups, and to define higher
cohomological invariants.


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