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\section*{rigid analytic geometry}

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

\hypertarget{geometry}{}\paragraph*{{Geometry}}\label{geometry}

[[!include higher geometry - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{cohomology}{Cohomology}\dotfill \pageref*{cohomology} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \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{Rigid analytic geometry} (often just ``rigid geometry'' for short) is a form of [[analytic geometry]] over a [[nonarchimedean field]] $K$ which considers [[spaces]] glued from [[polydiscs]], hence from [[maximal spectra]] of [[Tate algebras]] (quotients of a $K$-algebra of [[convergence|converging]] [[power series]]). This is in contrast to some modern approaches to [[non-Archimedean analytic geometry]] such as [[Berkovich spaces]] which are glued from Berkovich's [[analytic spectra]] and more recent Huber's [[adic spaces]].

The issue is that while the [[p-adic numbers]] are [[complete topological space|complete]] in the [[p-adic norm]], that [[topology]] is exotic: $\mathbb{Q}_p$ is a [[Stone space]], hence in particular a [[totally disconnected topological space]].

For that reason the naive idea of formulating [[p-adic geometry|p-adic]] analytic geometry in analogy to [[complex analytic geometry]] as modeled on domains in $\mathbb{Q}_p^n$, regarded with their [[subspace topology]], fails, as also all these domains are totally disconnected.

Instead there is (\hyperlink{Tate71}{Tate 71}) a suitable [[Grothendieck topology]] on such [[affinoid domains]] -- the \emph{[[G-topology]]} -- with respect to which there is a good theory of [[non-archimedean analytic geometry]] (``[[rigid analytic geometry]]'') and hence in particular of [[p-adic geometry]]. Moreover, one may sensibly assign to a $p$-adic domain a [[topological space]] which \emph{is} well behaved (in particular locally connected and even locally contractible), this is the \emph{[[analytic spectrum]]} construction. The resulting topological spaces equipped with covers by [[affinoid domain]] under the [[analytic spectrum]] are called [[Berkovich spaces]].

According to \hyperlink{Kedlaya}{Kedlaya, p. 18}, the terminology ``rigid'' is because\ldots{}

\begin{quote}%
\ldots{} one develops everything ``rigidly'' by imitating the theory of [[schemes]] in [[algebraic geometry]], but using rings of [[convergence|convergent]] [[power series]] instead of [[polynomials]].


\end{quote}
See also \emph{[[global analytic geometry]]}.

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

The related type of cohomology is called [[rigid cohomology]].

\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

\begin{itemize}%
\item The solution by Raynaud and Harbater of Abyhankar's conjecture concerning fundamental groups of curves in positive characteristic uses the rigid analytic GAGA theorems (whose proofs are very similar to Serre's proofs in the complex-analytic case).


\item Work of Kisin on modularity of Galois representations makes creative use of rigid-analytic spaces associated to Galois deformation rings.



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

\begin{itemize}%
\item [[complex analytic space]]


\item [[analytification]]


\item [[global analytic geometry]]


\item [[overconvergent global analytic geometry]]



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

An original article is

\begin{itemize}%
\item [[John Tate]], \emph{Rigid analytic spaces}, Invent. Math. \textbf{12}:257--289, 1971.

\end{itemize}
and for the construction of the generic fiber of formal schemes over the ring of integers of $K$

\begin{itemize}%
\item Michel Raynaud, \emph{G\'e{}om\'e{}trie analytique rigide d'apr\`e{}s Tate, Kiehl,}, Table Ronde d'Analyse non archim\'e{}dienne (Paris, 1972), pp. 319--327. Bull. Soc. Math. France, Mem. No. 39--40, Soc. Math. France, Paris, 1974, \href{http://www.ams.org/mathscinet-getitem?mr=470254}{MR470254}

\end{itemize}
Introductions are in

\begin{itemize}%
\item Johannes Nicaise, \emph{Formal and rigid geometry: an intuitive introduction, and some applications} (\href{http://gc83.perso.sfr.fr/GTIM/PDF%20GROUPE%20DE%20TRAVAIL/Nicaise/formal%20rigid%20Nicaise.pdf}{pdf})


\item [[Brian Conrad]], \emph{Several approaches to non-Archimedean geometry}, \href{http://math.stanford.edu/~conrad/papers/aws.pdf}{pdf}


\item Peter Schneider, \emph{Basic notions of rigid analytic geometry}, in: Galois representations in arithmetic algebraic geometry (Durham, 1996), 369--378, London Math. Soc. Lecture Note Ser. \textbf{254}, Cambridge Univ. Press 1998, \href{http://dx.doi.org/10.1017/CBO9780511662010.010}{doi}



\end{itemize}
A comprehensive textbook account is in

\begin{itemize}%
\item S. Bosch, U. G\"u{}ntzer, [[Reinhold Remmert]], \emph{[[Non-Archimedean Analysis]] -- A systematic approach to rigid analytic geometry}, Grundlehren der Mathem. Wissen. \textbf{261}, Springer 1984 (\href{http://www.ams.org/mathscinet-getitem?mr=0746961}{MR0746961}, \href{http://math.arizona.edu/~cais/scans/BGR-Non_Archimedean_Analysis.pdf}{pdf})

\end{itemize}
Comparison of various spectra and topologies is in

\begin{itemize}%
\item M. van der Put, P. Schneider, \emph{Points and topologies in rigid geometry}, Math. Ann. 302 (1995), no. 1, 81--103, \href{http://www.ams.org/mathscinet-getitem?mr=1329448}{MR96k:32070}, \href{http://dx.doi.org/10.1007/BF01444488}{doi}

\end{itemize}
Other accounts include

\begin{itemize}%
\item Ahmed Abbes, \emph{\'E{}l\'e{}ments de G\'e{}om\'e{}trie Rigide}, vol. I. \emph{Construction et \'e{}tude g\'e{}om\'e{}trique des espaces rigides}, Progress in Mathematics \textbf{286}, Birkh\"a{}user 2011, 496 p.\href{http://perso.univ-rennes1.fr/ahmed.abbes/egr.html}{book page}


\item Siegfried Bosch, \emph{Lectures on formal and rigid geometry}, Preprints of SFB Geom. Struk. Math. Heft 378, \href{http://wwwmath.uni-muenster.de/sfb/about/publ/heft378.pdf}{pdf} (revised 2008)


\item J. Fresnel, M. van der Put, \emph{Rigid geometry and applications}, Birkh\"a{}user (2004) \href{http://www.ams.org/mathscinet-getitem?mr=2014891}{MR2014891}


\item F. Denef, L. van den Dries, \emph{$p$-adic and real subanalytic sets}, Ann. of Math. \textbf{128} (1988) no. 1, 79--138 \href{http://www.ams.org/mathscinet-getitem?mr=951508}{MR951508}, \href{http://dx.doi.org/10.2307/1971463}{doi}


\item [[Yan Soibelman]], \emph{On non-commutative analytic spaces over non-archimedean fields}, preprint IHES, \href{http://www.math.ksu.edu/~soibel/lapkin-spaces-2.ps}{pdf}


\item Hans Grauert, Reinhold Remmert, \emph{Coherent analytic sheaves}, Springer 1984


\item [[R. Cluckers]], L. Lipshitz, \emph{Fields with analytic structure}, J. Eur. Math. Soc. \textbf{13}, 1147--1223, \href{http://wis.kuleuven.be/algebra/Raf/prints/JEMS.pdf}{pdf} and several articles (in various formalisms) in collection


\item R. Cluckers, J. Nicaise, J. Sebag (Editors), \emph{Motivic Integration and its Interactions with Model Theory and Non-Archimedean Geometry}, 2 vols. London Mathematical Society Lecture Note Series \textbf{383}, \textbf{384}


\item Peter Schneider, \emph{Points of rigid analytic varieties}, J. Reine Angew. Math. 434 (1993), 127--157, \href{http://www.ams.org/mathscinet-getitem?mr=1195693}{MR94b:14017}, \href{http://dx.doi.org/10.1515/crll.1993.434.127}{doi}



\end{itemize}
See also

\begin{itemize}%
\item [[Kiran Kedlaya]], \emph{$p$-Adic differential equations} (\href{http://www-math.mit.edu/~kedlaya/18.787/compiled.pdf}{pdf})

\end{itemize}
[[!redirects rigid analytic geometries]]

[[!redirects rigid geometry]] [[!redirects rigid geometries]]



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