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\section*{p-adic Hodge theory}

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

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

[[!include cohomology - contents]]

\hypertarget{nonarchimedean_geometry}{}\paragraph*{{non-archimedean geometry}}\label{nonarchimedean_geometry}

[[!include non-archimedean geometry - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{overview}{Overview}\dotfill \pageref*{overview} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{beilinsonbhatt_approach}{Beilinson-Bhatt approach}\dotfill \pageref*{beilinsonbhatt_approach} \linebreak
\noindent\hyperlink{adic_absolute_hodge_cohomology}{$p$-adic absolute Hodge cohomology}\dotfill \pageref*{adic_absolute_hodge_cohomology} \linebreak


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

A surprising parallel between classical [[Hodge theory]] and the theory of [[Galois representations]] was noted in the beginning of the 60's; shortly [[John Tate|Tate]] and [[Alexander Grothendieck|Grothendieck]] attempted to understand the topology of p-adic [[algebraic variety|varieties]] in light of this analogy. A series of precise conjectures about p-adic Hodge theory was formulated by [[Jean-Marc Fontaine|Fontaine]] at the beginning of the 80's; three different proofs were proposed by [[Gerd Faltings|Faltings]], [[Wieslawa Niziol|Niziol]] and [[Takeshi Tsuji|Tsuji]]. A significantly simpler approach was developed recently in works of [[Alexander Beilinson|Beilinson]] and [[Bhargav Bhatt|Bhatt]].

\emph{This paragraph is translated from the abstract of (\hyperlink{BeilinsonLectures2014}{Beilinson's Yaroslavl' lectures}).}

\hypertarget{overview}{}\subsection*{{Overview}}\label{overview}

\emph{p-adic Hodge theory} is the study of properties of p-adic (\'e{}tale, de Rham, logarithmic cristalline) [[cohomology]] (and [[motives]]) of non-archimedean [[analytic spaces]]. The $p$-adic Hodge structure of a (proper or semi-stably compactified) p-adic analytic variety is essentially given by a relation between three important invariants of the given variety:

\begin{itemize}%
\item p-adic de Rham cohomology equipped with the Hodge filtration,


\item log-cristalline cohomology (of a potentially semi-stable model) equipped with its Frobenius and monodromy map,


\item [[pro-étale cohomology]] of the extension of the given variety to a {\colorbox[rgb]{1.00,0.93,1.00}{\tt chosen}} algebraic closure of the base field, together with its natural action of the Galois group.



\end{itemize}
There is an ``evident'' isomorphism between p-adic de Rham cohomology and log-cristalline cohomology (both are given by a kind of differential calculus).

The main theorem of p-adic Hodge theory is that the datum of these two cohomologies and their relation on the one side, and of pro-\'e{}tale cohomology on the other side, mutually determine each other (once equipped with their natural additional structures).

This result has been conjectured first by Grothendieck in the case of p-divisible groups. He called it the {\colorbox[rgb]{1.00,0.93,1.00}{\tt mysterious\char32functor}}. A conjecture generalizing this statement to other kinds of ---algebraic--- p-adic varieties was formulated by Jannsen in a particular case, and then grounded on the setting of [[p-adic periods]] by Fontaine. After these important Breakthrough, the theory was fully developed by a long list of contributors (see {\colorbox[rgb]{1.00,0.93,1.00}{\tt periodes\char32p\char45adiques}}, Ast\'e{}risque for a first list). The comparison theorem was proved by many authors in various particular cases, and then in full generality (for algebraic varieties) by Faltings using [[almost mathematics]].

During the last few years, there were interesting new developments.

If one works with rational coefficients (no torsion), one may get the isomorphism between both structures in a geometric way (in an algebraic setting), using Beilinson's approach, through (derived de Rham cohomology)[[de Rham complex]].

It is also possible to use Scholze's [[perfectoid spaces]] and Faltings' [[almost mathematics]] to give a very elegant proof of this theorem for proper (or more generally, semi-stably compactified) p-adic (strict, i.e., rigid) analytic varieties. The fact that the classical result extends to the analytic setting is essentially due (up to difficulties related to the resolution in finite characteristic, that are dealt with using Faltings' methods) to the fact that (proper) rigid analytic varieties always have an integral model, given by a formal scheme over the ring of integers of the given non-archimedean field. The same proof may also extend to the non-strict situation by using a convenient base extension, but this work has not yet been done.

The very important case of torsion coefficients has also been addressed by Scholze in his setting, and by Bhatt, who gave a useful refinement of Beilinson's construction. Remark that almost mathematical tools seem to be quite well adapted to the study of completed integral cohomology.

Scholze's original method involves the use of Witt vectors (already used in Fontaine's definition of period rings), that were also studied by Kedlaya and Liu in this Hodge theoretic context. This Witt vector approach are interesting, but they seem to have the drawback of being quite hard to globalize (including the archimedean norm in a context of [[global Hodge theory]]).

It is quite clear to the experts that a nice way to overcome the difficulties that appear when one uses Witt vectors would be to combine directly the (analytic) {\colorbox[rgb]{1.00,0.93,1.00}{\tt derived\char32de\char32Rham}} approach of Beilinson and Bhatt to the {\colorbox[rgb]{1.00,0.93,1.00}{\tt perfectoid\char32and\char32almost\char32mathematics}} approach of Faltings and Scholze. Both approaches may be extended to the analytic setting using overconvergent derived analytic spaces (see [[global analytic geometry]]).

One may say that almost geometrical derived methods could be useful to study integral completed cohomology, while usual derived geometric methods are quite well adapted to the study of torsion phenomena in finite characteristic.

Indeed, as explained by Bhatt at the end of his paper, one may use derived de Rham cohomology over $\Z$ to get a (non-archimedean) period ring isomorphism for an [[arithmetic variety]].

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

\begin{itemize}%
\item [[Fargues-Fontaine curve]]

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

Niziol's [[algebraic K-theory|K-theoretic]] proof:

\begin{itemize}%
\item [[Wiesława Nizioł]], \emph{Crystalline conjecture via $K$-theory}, Annales scientifiques de l'\'E{}cole Normale Sup\'e{}rieure 31.5 (1998): 659-681. \href{http://eudml.org/doc/82474}{web}.

\end{itemize}
\hypertarget{beilinsonbhatt_approach}{}\subsubsection*{{Beilinson-Bhatt approach}}\label{beilinsonbhatt_approach}

\begin{itemize}%
\item [[Alexander Beilinson]], \emph{p-adic periods and derived de Rham cohomology}, \href{http://arxiv.org/abs/1102.1294}{arXiv:1102.1294}.


\item [[Alexander Beilinson]], \emph{On the crystalline period map}, \href{http://arxiv.org/abs/1111.3316}{arXiv:1111.3316}.


\item [[Bhargav Bhatt]], \emph{p-adic derived de Rham cohomology}, \href{http://arxiv.org/abs/1204.6560}{arXiv:1204.6560}.


\item [[Luc Illusie]], \emph{Around the Poincar\'e{} lemma, after Beilinson} (Preliminary notes), 2013, \href{http://www.math.u-psud.fr/~illusie/derived-deRham3.pdf}{pdf}.


\item [[Alexander Beilinson]], \emph{p-adic Hodge theory}, lectures from Yaroslavl' summer school 2014, \href{http://bogomolov-lab.ru/SHKOLA2014/talks/beilinson.html}{videos}.



\end{itemize}
\hypertarget{adic_absolute_hodge_cohomology}{}\subsubsection*{{$p$-adic absolute Hodge cohomology}}\label{adic_absolute_hodge_cohomology}

\begin{itemize}%
\item [[Frédéric Déglise]], [[Wiesława Nizioł]], \emph{On $p$-adic absolute Hodge cohomology and syntomic coefficients, I}, \href{http://arxiv.org/abs/1508.02567}{arXiv:1508.02567}.

\end{itemize}


\end{document}