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\begin{document}

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\section*{Galois cohomology}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory}

[[!include group theory - contents]]

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_tale_cohomology}{Relation to \'e{}tale cohomology}\dotfill \pageref*{relation_to_tale_cohomology} \linebreak
\noindent\hyperlink{InCohesiveHomotopyTypeTheory}{In terms of cohesive homotopy type theory}\dotfill \pageref*{InCohesiveHomotopyTypeTheory} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\emph{Galois cohomology} is the [[group cohomology]] of [[Galois groups]] $G$. Specifically, for $G$ the Galois group of a [[field extension]] $L/K$, Galois cohomology refers to the group cohomology of $G$ with [[coefficients]] in a $G$-[[module]] naturally associated to $L$.

Galois cohomology is studied notably in the context of \emph{[[algebraic number theory]]}.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{relation_to_tale_cohomology}{}\subsubsection*{{Relation to \'e{}tale cohomology}}\label{relation_to_tale_cohomology}

Galois cohomology of a [[field]] $k$ is essentially the [[étale cohomology]] of the [[spectrum of a commutative ring|spectrum]] $Spec(k)$.

See also at \emph{[[comparison theorem (étale cohomology)]]}.

\hypertarget{InCohesiveHomotopyTypeTheory}{}\subsubsection*{{In terms of cohesive homotopy type theory}}\label{InCohesiveHomotopyTypeTheory}

We make some comments on the formulation of Galois cohomology in [[cohesive homotopy type theory]].

As discussed at \emph{[[Galois theory]]}, the [[absolute Galois group]] $G_{Galois}$ of a [[field]] $K$ is the [[fundamental group]] of the [[spectrum of a commutative ring|spectrum]] $X \coloneqq Spec(K)$. Hence its [[delooping]] $\mathbf{B}G_{Galois}$ is the [[fundamental groupoid]]

\begin{displaymath}
\Pi_1(X)  \simeq \mathbf{B}G_{Galois}
  \,.
\end{displaymath}
In [[cohesive homotopy type theory]] there exists the [[fundamental ∞-groupoid]]-construction -- the [[shape modality]] $\Pi$ --

\begin{displaymath}
X \colon Type \;\vdash \; \Pi(X) \colon Type
  \,.
\end{displaymath}
Moreover, by the discussion at \emph{[[group cohomology]]} in the section \emph{\href{http://ncatlab.org/nlab/show/group%20cohomology#InHomotopyTypeTheory}{group cohomology - In terms of homotopy type theory}}

\begin{enumerate}%
\item a $G_{Galois}$-[[module]] $A$ is a $\mathbf{B}G_{Galois}$-[[dependent type]];


\item the [[group cohomology]] is the [[dependent product]] over the [[function type]]

\begin{displaymath}
\vdash \; \left( \prod_{x \colon \mathbf{B}G_{Galois}} \left( * \to A \right)\right) \colon Type
  \,.
\end{displaymath}


\end{enumerate}
Hence, generally in [[cohesive homotopy type theory]], for $X$ a [[type]] and

\begin{displaymath}
x \colon \Pi(X) \;\vdash \; A(x) \colon Type
\end{displaymath}
a $\Pi(X)$-[[dependent type]], we can say that the corresponding \emph{$\infty$-Galois cohomology} is

\begin{displaymath}
\vdash \; \left( \prod_{x \colon \Pi(X)} \left( * \to A\right) \right)
  \colon Type
  \,.
\end{displaymath}
\textbf{Warning.} Currently there is not written down yet a [[model]] for [[cohesive homotopy type theory]] given by a [[cohesive (∞,1)-topos]] over a [[site]] like the [[étale site]].

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item [[Hilbert's theorem 90]]

\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[algebraic number theory]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[John Tate]], \emph{Galois cohomology} (\href{http://modular.math.washington.edu/edu/2010/582e/refs/tate-galois_cohomology.pdf}{pdf})


\item [[Jean-Pierre Serre]], \emph{Galois cohomology}, Springer Monographs in Mathematics (1997) doi:\href{https://doi.org/10.1007/978-3-642-59141-9}{10.1007/978-3-642-59141-9}

\emph{Cohomologie galoisienne}, Lecture Notes in Mathematics, 5 (Fifth ed. 1994), Springer-Verlag, MR1324577 doi:\href{https://doi.org/10.1007/BFb0108758}{10.1007/BFb0108758)}


\item Gr\'e{}gory Berhuy, \emph{An introduction to Galois cohomology and its applications} (\href{http://www-fourier.ujf-grenoble.fr/~berhuy/fichiers/NTUcourse.pdf}{pdf})


\item M. Kneser, \emph{Lectures on Galois cohomology of classical groups} (\href{http://www.math.tifr.res.in/~publ/ln/tifr47.pdf}{pdf})


\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Galois_cohomology}{Galois cohomology}}



\end{itemize}
A generalization in the setup of [[coring]]s:

\begin{itemize}%
\item [[Tomasz Brzezi?ski]], \emph{Descent cohomology and corings}, Comm Algebra 36:1894-1900, 2008, \href{http://arxiv.org/abs/math/0601491}{math.RA/0601491}

\end{itemize}


\end{document}