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\begin{document}

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\section*{cohomological descent}

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

\hypertarget{locality_and_descent}{}\paragraph*{{Locality and descent}}\label{locality_and_descent}

[[!include descent and locality - contents]]

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

[[!include cohomology - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \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}

The theory of \textbf{cohomological descent} deals with the question if the [[derived geometry|derived]] analogue of the (co)monadic comparison functor is fully faithful (or more rarely an equivalence of categories) when formulated at the level of total derived functors and derived categories, and usually taken with respect to [[hypercovers]].

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

\begin{itemize}%
\item [[descent]]

\begin{itemize}%
\item [[cover]]


\item \textbf{cohomological descent}


\item [[descent morphism]]


\item [[monadic descent]],

\begin{itemize}%
\item [[Sweedler coring]]


\item [[higher monadic descent]]


\item [[descent in noncommutative algebraic geometry]]



\end{itemize}


\end{itemize}

\item [[sheaf]], [[(2,1)-sheaf]], [[2-sheaf]] [[(∞,1)-sheaf]]



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

The notion has been introduced in

\begin{itemize}%
\item [[Pierre Deligne]], \emph{Th\'e{}orie de Hodge. III}, Inst. Hautes \'E{}tudes Sci. Publ. Math. \textbf{44} (1974), 5--77.

\end{itemize}
A summary is also in

\begin{itemize}%
\item [[Donu Arapura]], \emph{Building mixed Hodge structures}, in: The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), 13--32, CRM Proc. Lecture Notes, 24, Amer. Math. Soc., Providence, RI, 2000.

\end{itemize}
For a readable introduction see

\begin{itemize}%
\item [[Brian Conrad]], \emph{Cohomological descent} (\href{http://math.stanford.edu/~conrad/papers/hypercover.pdf}{pdf})

\end{itemize}
Closely related is the [[monadic descent]] in triangulated context in the sense of page 36-37 in

\begin{itemize}%
\item [[Alexander Rosenberg|A. L. Rosenberg]], \emph{Topics in noncommutative algebraic geometry, homological algebra and K-theory}, preprint MPIM Bonn 2008-57 \href{http://www.mpim-bonn.mpg.de/preprints/send?bid=3589}{pdf}

\end{itemize}


\end{document}