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\begin{document}

\begin{slide}
	Cohomology: powerful tool in AG.  ``Converts" AG world (schemes, morphisms... {\bf HARD!}) into linear algebra world
	(vector spaces, linear maps... {\bf EASY!}).
	
	Characteristic $p$-methods: try to make a characteristic 0 problem easier by deforming it to characteristic $p$.
	Follows general idea of studying problems in ``families."
	$$\C\xrightarrow{\text{\tiny descend}} \Q \xrightarrow{\text{\tiny put in ``family"}}\Z\xrightarrow{\text{\tiny localize}} \Z_p$$
	Any Scheme over $\Z_p$ has {\em two} fibers; the $\Q_p$ (generic) fiber and the $\F_p$ (special) fiber.
	For cohomological techniques, need a theory for special fiber that relates to usual theories (dR, top, Zariski...) for generic fiber.
	
	We want our cohomologies to have characteristic 0 coefficients (otherwise, can only get information ``mod $p$").
	
	$p$-adic cohomology: cohomology theory for schemes over fields of characteristic $p$, with characteristic 0 coefficients
	(usually an extension of $\Q_p$).
	Makes available powerful tools from characteristic $p$, e.g. Frobenius!
	
	
\end{slide}


\begin{slide}
{\bf Example:} $X_{\C}\rightarrow \Spec\C$ proper and finite type.  There exists $A\subset \C$, a f.g. $\Z$-algebra and a model $X_A\rightarrow\Spec A$
such that:
%\begin{equation*}
%	\xymatrixrowsep{4pc}\xymatrix{
%		{X_{\C}}\ar[d]^-{\parbox{2cm}{\tiny proper, f.~type}} \\
%		{\C}
%	}
%\end{equation*}
$$H^i_{\text{\'{e}t}}(X_{\overline{s}},\Q_{\ell})\simeq H^i_{\text{\'{e}t}}(X_{\overline{\eta}},\Q_{\ell})%\simeq H^i_{\text{\'{e}t}}(X_{\C},\Q_{\ell})
\simeq H^i_{\rm top}(X_{\C},\Q_{\ell}).$$
Choosing $s$ with $p^r=q=\#k(s)$, get Frobenius on $H^i_{\rm top}(X_{\C},\Q_{\ell})$, with eigenvalues $\alpha_j$,
\begin{align*}
&|\alpha_j|=q^{w_j/2} &
&w_j\le i.	
\end{align*}
This gives the {\bf weight} filtration of $H^i_{\rm top}(X_{\C},\Q_{\ell})$ by subspaces where $F$ acts with eigenvalues of
weight at most $w_j$; it is {\em independent} of any choices.


{\bf Basic Problem}: Given two different cohomology theories known to be isomorphic,  how
to compare different constructions in each theory?

{\bf Example:} 
$X/\C$ a smooth projective variety, $\dim X=n$.
\begin{equation*}
\xymatrixcolsep{2pc}\xymatrixrowsep{6pc}\xymatrix{
		{H^n(X,\Omega^n_{X/\C})} \ar[d]_-{\tr} \ar[r]^-{\sim} & 
		{H^n(X^{\rm an},\Omega^n_{X^{\rm an}})} \ar[r]^-{\sim} &
		{H^{2n}_{\rm top}(X^{\rm an},\C)}\ar[lld]^-{\frac{1}{(2\pi i)^n}\int }\\
		{\C}
}
\end{equation*}



\end{slide}



\begin{slide}
\centerline{$p$-adic Cohomology}
%$W(\F_q)$---complete dvr with uniformizer $p$ and residue field $\F_q$  (example: $W(\F_p)=\Z_p$).

Crystalline cohomology:
$$
H^i_{\rm cris}: \left\{\parbox{6.6cm}{\small smooth, proper $\F_p$-sch\\ morphisms of $\F_p$-sch}\right\}\rightarrow 
\left\{\parbox{5.5cm}{\small finite $\Z_p$-modules\\ $\Z_p$-module homs}\right\}
$$
\begin{itemize}
	\item Motivated by observation: ``de~Rham cohomology of a lift is independent of the lift"
	\item Replacement for $p$-adic {\'e}tale cohomology, which is {\em very bad} for varieties over $\F_p$ (e.g. not finitely generated, wrong Betti nos...)
	\item Derived functor cohomology (like Zariski cohomology)
	\item Disadvantages: difficult to calculate explicitly; requires properness
%	\item Requires properness
\end{itemize}
\end{slide}

\begin{slide}

{\bf Theorem.}
	Let $\mathscr{X}/\Z_p$ be smooth and proper.  There is a canonical isomorphism
	$$H^i_{\rm cris}(\mathscr{X}_{\F_p})\simeq H^i_{\rm dR}(\mathscr{X}).$$


Get $\Z_p$-lattice with Frobenius action inside $$H^i_{\rm dR}(\mathscr{X}_{\Q_p})\simeq H^i_{\rm dR}(\mathscr{X})\otimes_{\Z_p}\Q_p.$$

{\bf Remark}:
Very important in $p$-adic Hodgy theory: how does $F$ interact with the Hodge filtration on $H^i_{\rm dR}(\mathscr{X}_{\Q_p})$?


{\bf Question}:
	If $\mathscr{X}/\Z_p$ has smooth $\Q_p$-fiber but only generically smooth $\F_p$-fiber, is there still a functorial
	$\Z_p$-lattice equipped with Frobenius inside $H^i_{\rm dR}(\mathscr{X}_{\Q_p})$?

{\bf Answer}: If $\mathscr{X}_{\F_p}$ is ncd, then yes, but uses log-crystalline cohomology: Frobenius is {\em very} hard to describe
explicitly, and this structure is hard to relate to other theories. 
\end{slide}

\begin{slide}
	Monsky-Washnitzer Cohomology:
		$$
H^i_{\rm MW}: \left\{\parbox{6.5cm}{\small smooth affine $\F_p$-sch\\ morphisms of $\F_p$-sch}\right\}\rightarrow 
\left\{\parbox{4.5cm}{\small finite $\Q_p$-v.s.\\ $\Q_p$-linear maps}\right\}
$$

Motivated by work of Dwork, in the spirit of de~Rham cohomology.  

{\bf Sketch}:
Let $X=\Spec \F_p[X_i]/(f_j)$.  

{\em First idea}: Pick lifts $\widetilde{f}_j$ of the $f_j$ to $\Z_p[X_i]$ and set $A:=\Z_p[X_i]/(\widetilde{f}_j)$.
Use the cohomology of the de~Rham complex of $A\otimes \Q_p$.  {\bf Bad}: Not functorial in $X$, moreover cohomology depends on choice of lift.

{\em Second idea}: To lift maps from characteristic $p$, pass to $p$-adic completions.  
Use the cohomology of the de~Rham complex of 
$$\Q_p\otimes \varprojlim A/p^nA=\left\{\parbox{10cm}{\tiny power series in $X_i$ with radius of convergence $|X_i|_p\le 1$, modulo relations}\right\}$$  
{\bf Better}: Functorial in $X$, cohomology
independent of lift, BUT almost never finite dimensional, roughly because:

Cannot integrate things like
$$1+px^{p-1}+p^2x^{p^2-1}+p^3x^{p^3-1}+\cdots$$
\end{slide}

\begin{slide}
{\em Third idea}: The $p$-adic completion is ``too big," so consider the smaller ``weak completion"
$$A^{\dagger}:=\left\{\parbox{10cm}{\tiny power series in $X_i$ with radius of convergence $|X_i|_p\le 1+\epsilon$, some $\epsilon>0$,\\ modulo relations}\right\}$$
of overconvergent power series; use the cohomology of the de~Rham complex of $\Q_p\otimes A^{\dagger}.$
{\bf Best}: can still lift maps, so functorial in $X$, and independent of choice of lift.  Even finite dimensional! (hard theorem).

We define $H^i_{\rm MW}(X)$ to be the $i$th cohomology group of the de~Rham complex of $\Q_p\otimes A^{\dagger}$.
\begin{itemize}
	\item Many nice properties: K\"unneth formula, cup product, Lefschetz fixed-point formula...
	\item Used to prove first two-thirds of Weil conjectures
	\item {\em Very} explicit, easy to calculate: Kedlaya's algorithm
	\item Requires $X$ to be affine and smooth
\end{itemize}
\end{slide}


\begin{slide}
{\bf Question}: How do Monsky-Washnitzer and crystalline cohomology relate to eachother?

	Rigid Cohomology:
	$$
H^i_{\rm rig}: \left\{\parbox{6.5cm}{\small q-projective $\F_p$-sch\\ morphisms of $\F_p$-sch}\right\}\rightarrow 
\left\{\parbox{4.5cm}{\small finite $\Q_p$-v.s.\\ $\Q_p$-linear maps}\right\}
$$
\begin{itemize}
	\item ``de~Rham type" cohomology, based on the idea of lifting to characteristic 0.  Handles singular varieties roughly as in Hartshorne's paper ``On the de~Rham Cohomology..." by locally embedding in a smooth variety and ``completing" in some sense
%	\item Allows singularities, non-properness
	\item ``Interpolation" of MW and crystalline cohomology
	\item Can often be made very explicit
%	\item Requires properness
\end{itemize}

\end{slide}

\begin{slide}
	\begin{thm}
		If $X/\F_p$ is proper and smooth, there is a natural isomorphism
		$$ H^i_{\rm cris}(X)\otimes_{\Z_p}\Q_p\simeq H^i_{\rm rig}(X).$$
	\end{thm}
	
	\begin{thm}
		If $X/\F_p$ is affine and smooth, there is a natural isomorphism
		$$H^i_{\rm MW}(X)\simeq H^i_{\rm rig}(X).$$
	\end{thm}
	
	{\bf Question}: Suppose $X/\Q_p$ and $Y/{\Q_p}$ are smooth and proper of finite type, and that $Y$ has a smooth and proper
	model $\mathscr{Y}/\Z_p$ but $X$ has only a non-smooth model $\mathscr{X}/\Z_p$.  If $U$ is a subscheme of $\mathscr{X}^{\rm sm}_{\F_p}$
	and $X\rightarrow Y$ is a map, there are maps:
	$$H^i_{\rm cris}(\mathscr{Y}_{\F_p})\otimes \Q_p\simeq H^i_{\rm dR}(Y)\rightarrow H^i_{\rm dR}(X)\rightarrow H^i_{\rm rig}(U).$$
	Is the composite map $F$-equivariant?	
	
\end{slide}



\end{document}