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\section*{Berkovich space}

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

\hypertarget{analytic_geometry}{}\paragraph*{{Analytic geometry}}\label{analytic_geometry}

[[!include analytic geometry -- contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition_of_berkovich_analytic_spaces}{Definition of Berkovich analytic spaces}\dotfill \pageref*{definition_of_berkovich_analytic_spaces} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{Cohomology}{Cohomology}\dotfill \pageref*{Cohomology} \linebreak
\noindent\hyperlink{LocalContractibility}{Local contractibility}\dotfill \pageref*{LocalContractibility} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \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
\noindent\hyperlink{introductions_and_reviews}{Introductions and reviews}\dotfill \pageref*{introductions_and_reviews} \linebreak
\noindent\hyperlink{original_articles}{Original articles}\dotfill \pageref*{original_articles} \linebreak
\noindent\hyperlink{relation_to_other_topics}{Relation to other topics}\dotfill \pageref*{relation_to_other_topics} \linebreak


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

\textbf{Berkovich analytic spaces} are a version of [[analytic space]]s over [[nonarchimedean fields]]. Unlike the \emph{rigid analytic spaces} (see [[rigid analytic geometry]]) of Tate, which are locally defined via [[maximal spectra]] of Tate algebras glued via the [[Grothendieck topology|Grothendieck]] [[G-topology]], the Berkovich analytic spaces are actual [[topological space]] equipped with a cover by [[affinoid domains]] via the [[analytic spectrum]] construction, due to [[Vladimir Berkovich]]. This spectrum can be viewed as consisting of the data of prime ideal plus the extension of the norm to the residue field; thus the Berkovich spectrum has far more points (though fewer than, say, [[Huber's adic spaces]] which may also contain valuations of higher order).

For more background see \emph{[[analytic geometry]]}.

\hypertarget{definition_of_berkovich_analytic_spaces}{}\subsection*{{Definition of Berkovich analytic spaces}}\label{definition_of_berkovich_analytic_spaces}

Let $k$ be a [[non-archimedean field]].

\begin{defn}
\label{}\hypertarget{}{}
Given $n \in \mathbb{N}$ and positive elements $\{r_1, \cdots, r_n \in k\}$, consider the sub-[[power series]] [[associative algebra|algebra]] over $k$ of those series which [[convergence|converge]] inside the radii $k_i$, i.e. the algebra defined by

\begin{displaymath}
\{\frac{1}{r_1} T_1 , \cdots, \frac{1}{r_n}T_n\}
  :=
  \left\{
    \sum_\nu a_\nu T^\nu | \lim_{{\vert \nu\vert} \to \infty} {\vert a_\nu \vert} r^\nu = 0
  \right\}
  \,.
\end{displaymath}
This is a commutative [[Banach algebra]] over $k$ with [[norm]] ${\Vert f \Vert} = max {\vert a_\nu\vert} r^\nu$.

A \textbf{$k$-[[affinoid algebra]]} is a commutative Banach $k$-algebra $A$ for which there exists $n$ and $\{r_i\}$ as above and an [[epimorphism]]

\begin{displaymath}
\{\frac{1}{r_1} T_1 , \cdots, \frac{1}{r_n}T_n\}
  \to 
  A
\end{displaymath}
such that the [[norm]] on $A$ is the [[quotient norm]].

If one can choose here $r_i = 1$ for all $i$ then $A$ is called \textbf{strictly $k$-affinoid}.

The [[category]] of \textbf{$k$-[[affinoid spaces]]} is the [[opposite category]] of the category of $k$-[[affinoid algebras]] and bounded [[homomorphisms]] between them.

\end{defn}
Via the [[analytic spectrum]] $Spec_{an}$ there is a [[topological space]] associated with any $k$-affinoid space. Often this underlying topological space is referred to as \emph{the analytic space}.

\begin{defn}
\label{}\hypertarget{}{}
An \textbf{[[affinoid domain]]} in an [[affinoid space]] $X = Spec_{an} A$ is a [[closed subset]] $V \subset X$ such that there is a [[homomorphism]] of $k$-affinoid spaces

\begin{displaymath}
\phi : Spec_{an} A_V \to X
\end{displaymath}
for some $A_V$, whose [[image]] is $V$, and such that every other morphism of $k$-affinoid spaces into $X$ whose image is contained in $V$ uniquely factors through this morphism.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
A \textbf{$k$-analytic space} is a [[locally Hausdorff topological space|locally Hausforf]] [[topological space]] $X$ equipped with an [[atlas]] by $k$-[[affinoid domains]] and [[affinoid domain embeddings]], such that their underlying [[analytic spectra]] [[topological spaces]] form a [[quasinet|net]] of [[compact subsets]] on $X$.

\end{defn}
(\hyperlink{Berkovich09}{Berkovich 09, def. 3.1.4})

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

\hypertarget{Cohomology}{}\subsubsection*{{Cohomology}}\label{Cohomology}

Under some mild conditions, the algebraic and the analytic [[étale cohomology]] of Berkovich spaces agree. (\hyperlink{Berkovich95}{Berkovich 95})

The underlying [[topological space]] $X^{an}$ given by the [[Berkovich analytic spectrum]] has as [[singular cohomology]] the [[weight filtration|weight 0]]-cohomology of $X$ (\hyperlink{Berkovich09}{Berkovich 09}).

See also MO discussion \href{http://mathoverflow.net/a/171653/381}{here}.

\hypertarget{LocalContractibility}{}\subsubsection*{{Local contractibility}}\label{LocalContractibility}

A [[complex analytic manifold]] and a \emph{smooth} [[complex analytic space]] is locally isomorphic to a [[polydisk]] and hence is trivially a [[locally contractible space]]. But over a [[non-archimedean field]] analytic spaces no longer need to be locally isomorphic to polydisks (but $p$-adic polydisks are still contractible (\hyperlink{Berkovich90}{Berkovich 90})). The following result establishes, under mild conditions, that general analytic spaces are nevertheless locally contractible.

Assume that the [[valuation]] on the ground field $k$ is nontrivial.

\begin{defn}
\label{LocallyEmbeddableInASmoothSpace}\hypertarget{LocallyEmbeddableInASmoothSpace}{}
A $k$-analytic space $X$ is called \emph{locally embeddable in a smooth space} if each point of $X$ has an [[open neighbourhood]] [[isomorphism|isomorphic]] to a strictly $k$-analytic domain in smooth $k$-analytic space.

\end{defn}
\begin{theorem}
\label{}\hypertarget{}{}
Every $k$-analytic space which is locally embeddable in a smooth space, def. \ref{LocallyEmbeddableInASmoothSpace}, is a [[locally contractible space]].

More precisely, every point of a locally smooth $k$-analytic space has an open neighbourhood $U$ which is contractible, and which is a [[union]] $U = \cup_{i = 1}^\infty U_i$ of analytic domains.

\end{theorem}
The local contractibility is \hyperlink{BerkovichContractible}{Berkovich (1999), theorem 9.1}. The refined statment in terms of inductive systems of analytic domains is in \hyperlink{BerkovichContractibleII}{Berkovich (2004)}.

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

\begin{itemize}%
\item [[analytic affine line]]

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

\begin{itemize}%
\item The proof of the [[local Langlands conjecture]] for $GL_n$ by Harris--Taylor uses [[étale cohomology]] on non-archimedean analytic spaces (in the sense of Berkovich) to construct the required Galois representations over local fields.

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

\begin{itemize}%
\item [[non-commutative analytic space]]


\item [[p-adic geometry]]


\item [[G-topology]]


\item [[Huber space]], [[perfectoid space]]


\item [[global analytic geometry]]



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

\hypertarget{introductions_and_reviews}{}\subsubsection*{{Introductions and reviews}}\label{introductions_and_reviews}

A nice survey is in

\begin{itemize}%
\item [[Bernard Le Stum]], \emph{One century of $p$-adic geometry -- From Hensel to Berkovich and beyond}, talk notes, June 2012 (\href{http://www-irma.u-strasbg.fr/IMG/pdf/NotesCoursLeStum.pdf}{pdf})

\end{itemize}
A good introduction to the general idea is at the beginning of

\begin{itemize}%
\item [[Sam Payne]], \emph{Topology of nonarchimedean analytic spaces and relations to complex algebraic geometry} (\href{http://arxiv.org/abs/1309.4403}{arXiv:1309.4403})

\end{itemize}
Basic notions are listed in

\begin{itemize}%
\item M. Temkin, \emph{Non-archimedean analytic spaces} (\href{http://www.math.huji.ac.il/~temkin/lectures/Non-Archimedean_Analytic_Spaces.pdf}{pdf slides})

\end{itemize}
A review of basic definitions and facts about affinoid and rigid $k$-analytic spaces can be found in

\begin{itemize}%
\item Ga\"e{}tan Chenevier, \emph{lecture 5} (\href{http://www.math.polytechnique.fr/~chenevier/coursIHP/chenevier_lecture5.pdf}{pdf})

\end{itemize}
See also the references at [[rigid analytic geometry]].

A review of definitions and results on $k$-analytic spaces is in

\begin{itemize}%
\item [[Vladimir Berkovich]], \emph{$p$-Adic analytic spaces}, in Proceedings of the International Congress of Mathematicians, Berlin, August 1998, Doc. Math. J. DMV, Extra Volume ICM II (1998), 141-151 (\href{http://www.wisdom.weizmann.ac.il/~vova/ICM98_1998.pdf}{pdf})

\end{itemize}
A more detailed set of lecture notes along these lines is

\begin{itemize}%
\item [[Vladimir Berkovich]], \emph{Non-archimedean analytic spaces}, lectures at the \emph{Advanced School on $p$-adic Analysis and Applications}, ICTP, Trieste, 31 August - 11 September 2009 (\href{http://www.wisdom.weizmann.ac.il/~vova/Trieste_2009.pdf}{pdf})

\end{itemize}
Introductory exposition of the Berkovich [[analytic spectrum]] is

\begin{itemize}%
\item [[Jérôme Poineau]], \emph{Global analytic geometry}, pages 20-23 in EMS newsletter September 2007 (\href{http://www.ems-ph.org/journals/newsletter/pdf/2007-09-65.pdf}{pdf})


\item [[Frédéric Paugam]], section 2.1.4 of\_Global analytic geometry and the functional equation\_ (2010) (\href{http://www.math.jussieu.fr/~fpaugam/documents/enseignement/master-global-analytic-geometry.pdf}{pdf})



\end{itemize}
A exposition of examples of Berkovich spectra is in

\begin{itemize}%
\item [[Scott Carnahan]], \emph{Berkovich spaces I} (\href{http://sbseminar.wordpress.com/2007/09/18/berkovich-spaces-i/}{web})

\end{itemize}
\hypertarget{original_articles}{}\subsubsection*{{Original articles}}\label{original_articles}

\begin{itemize}%
\item [[Vladimir Berkovich]], \emph{Spectral theory and analytic geometry over non-Archimedean fields}, Mathematical Surveys and Monographs, vol. 33, American Mathematical Society, Providence, RI, (1990) 169 pp.


\item [[Vladimir Berkovich]], \emph{\'E{}tale cohomology for non-Archimedean analytic spaces}, Publ. Math. IHES 78 (1993), 5-161.



\end{itemize}
Discussion of Berkovich09cohomology of Berkovich analytic spaces includes

\begin{itemize}%
\item [[Vladimir Berkovich]], \emph{On the comparison theorem for \'e{}tale cohomology of non-Archimedean analytic spaces.} Israel Journal of Mathematics 92.1-3 (1995): 45-59.


\item [[Vladimir Berkovich]], \emph{A non-Archimedean interpretation of the weight zero subspaces of limit mixed Hodge structures}, Algebra, Arithmetic, and Geometry. Birkh\"a{}user Boston, 2009. 49-67.



\end{itemize}
Discussion of local contractibility of smooth $k$-analytic spaces is in

\begin{itemize}%
\item [[Vladimir Berkovich]], \emph{Smooth $p$-adic analytic spaces are locally contractible}, Invent. Math. 137 1-84 (1999) (\href{http://www.wisdom.weizmann.ac.il/~vova/Inven_1999_137_contra.pdf}{pdf})


\item [[Vladimir Berkovich]], \emph{Smooth p-adic analytic spaces are locally contractible. II}, in Geometric Aspects of Dwork Theory, Walter de Gruyter \& Co., Berlin, (2004), 293-370. (\href{http://www.wisdom.weizmann.ac.il/~vova/Dworkvol_2004_contraII.pdf}{pdf})



\end{itemize}
and more generally in

\begin{itemize}%
\item [[Ehud Hrushovski]], [[François Loeser]], \emph{Non-archimedean tame topology and stably dominated types} (\href{http://arxiv.org/abs/1009.0252}{arXiv:1009.0252})


\item [[Ehud Hrushovski]], [[François Loeser]], [[Bjorn Poonen]], \emph{Berkovich spaces embed in Euclidean spaces} (\href{http://arxiv.org/abs/1210.6485}{arXiv:1210.6485})



\end{itemize}
\hypertarget{relation_to_other_topics}{}\subsubsection*{{Relation to other topics}}\label{relation_to_other_topics}

On the relation to [[buildings]]:

\begin{itemize}%
\item Annette Werner, \emph{Buildings and Berkovich Spaces} (\href{http://www.uni-frankfurt.de/fb/fb12/mathematik/ag/personen/werner/talks/dmvmuench10.pdf}{pdf})

\end{itemize}
Relation to [[integration]] theory

\begin{itemize}%
\item [[Vladimir Berkovich]], \emph{Integration of 1-forms on $p$-adic analytic spaces}, Princeton University Press,

\end{itemize}
Aspects of the [[homotopy theory]]/[[étale homotopy]] of analytic spaces are discussed in

\begin{itemize}%
\item [[Aise Johan de Jong]], \emph{\'E{}tale fundamental groups of non-archimedean analytic spaces}, Mathematica, 97 no. 1-2 (1995), p. 89-118 (\href{http://www.numdam.org/item?id=CM_1995__97_1-2_89_0}{numdam})

\end{itemize}
Relation to [[formal schemes]]:

\begin{itemize}%
\item J\'e{}r\^o{}me Poineau, \href{http://mathoverflow.net/a/138577/381}{MO comment}

\end{itemize}
Discussion of Berkovich analytic geometry as [[algebraic geometry]] in the general sense of [[Bertrand Toën]] and [[Gabriele Vezzosi]] is in

\begin{itemize}%
\item [[Oren Ben-Bassat]], [[Kobi Kremnizer]], \emph{Non-Archimedean analytic geometry as relative algebraic geometry} (\href{http://arxiv.org/abs/1312.0338}{arXiv:1312.0338})

\end{itemize}
[[!redirects Berkovich analytic space]] [[!redirects Berkovich analytic spaces]] [[!redirects Berkovich spaces]] [[!redirects p-adic analytic space]] [[!redirects p-adic analytic spaces]]



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