\section{Top Witt vector and $p$-adic cohomology}

In this part of the paper we present a few results concerning top Witt vector cohomology of varieties.  
We begin with the following simple technical result which allows us to describe the top Witt vector cohomology in terms of coherent cohomology under arithmetic assumption of ordinarity.  This result is most likely standard but we could not find a precise reference.

\begin{proposition}\label{lem:wo_o_cohomology_ordinary}
Let $X$ be a proper variety of dimension $d = \dim X$ defined over a perfect field $k$ of characteristic $p>0$.  Assume that  $X$ is weakly ordinary, that is, the natural Frobenius action on $H^d(X,\sO_X)$ is bijective, and that $H^{d-1}(X,\sO_X) = 0$.  Then $H^d(X,W\sO_{X,\QQ})$ is a $K$-vector space of dimension $g = \dim_k H^d(X,\sO_X)$, and 
\[
H^d_{\et}(X,\QQ_p) \otimes_{\QQ_p} K \isomto H^d(X,W\sO_{X,\QQ}),
\]
where $K:= \Frac W(k)$.
\end{proposition}


\begin{proof}
First, we observe that by a straightforward inductive argument $H^{d-1}(X,W_n\sO_X) = 0$, for every $n \geq 1$.  Then, we claim that the map $F \colon H^d(X,W_n\sO_X) \to H^d(X,W_n\sO_X)$ induced by the Witt vector Frobenius is an isomorphism.  This follows directly by induction on $n$ from the five lemma applied to following diagram, the rows of which are exact by the vanishing of finite length Witt vector cohomology in degree $d-1$. 
\[
\xymatrix{
0 \ar[r] & H^d(X,\sO_X) \ar[r]^(0.45){V} \ar[d]^F & H^d(X,W_{n+1}\sO_X) \ar[r]\ar[d]^F & H^d(X,W_n\sO_X) \ar[r]\ar[d]^F & 0 \\
0 \ar[r] & H^d(X,\sO_X) \ar[r]^(0.45){V} & H^d(X,W_{n+1}\sO_X) \ar[r] & H^d(X,W_n\sO_X) \ar[r] &  0,
}
\]
 Taking limit with respect to $n$ this implies that the Frobenius endomorphism of $H^d(X,W\sO_X)$ is also an isomorphism.  Using the vanishing of $H^{d-1}(X,\sO_X)$ again, we see that connecting homomorphism $H^{d-1}(X,\sO_X) \to H^d(X,W\sO_X)$ in the long exact sequence associated with
\[
0 \ra W\sO_X \xrightarrow{V} W\sO_X \ra \sO_X \ra 0.
\]
is zero.  Therefore the map $V \colon H^d(X,W\sO_X) \to H^d(X,W\sO_X)$ is injective and hence $p = FV$ is injective as well.  By \autoref{prop:witt_vectors_and_lim}, the natural map $H^d(X,W\sO_X) \to \varprojlim_{n} H^d(X,W_n\sO_X)$ is an isomorphism, which implies that $\bigcap_n V^n H^d(X,W\sO_X) = \{0\}$ and therefore $H^d(X,W\sO_X)$ is $V$-adically and hence $p$-adically separated.  Consequently, using Nakayama's lemma for the complete ring $W(k)$ (see \cite[Theorem 8.4]{Matsumura_Commuatative_ring_theory}), we see that the torsion-free and $p$-adically separated module $H^d(X,W\sO_X)$ is in fact free of rank:
\[
\dim_k H^d(X,W\sO_X)/pH^d(X,W\sO_X) = \dim_k H^d(X,W\sO_X)/VH^d(X,W\sO_X) = \dim_k H^d(X,\sO_X),
\]
as desired.  After localization, this yields the result concerning Witt vector cohomology.  

For the claim related to $p$-adic cohomology, we take a limit of the cohomology exact sequences associated to \eqref{eq:witt_artin_schreier} in order to obtain a short exact sequence (note that we use the vanishing in degree $d-1$):
\[
0 \ra H^d_{\et}(X,\ZZ_p) \ra H^d(X,W\sO_X) \xrightarrow{1 - F} H^d(X,W\sO_X) \ra 0,
\]
which means that $H^d_{\et}(X,\ZZ_p)$ are in fact Frobenius stable elements in the $F$-crystal $H^d(X,W\sO_X)$.
Since, as proven above, the morphism $F$ acting on $H^d(X,W\sO_X)$ is a $p$-linear automorphism, we may now use the standard result on $F$-crystals (see \cite[2.1.2]{Katz_Travaux_de_Dwork}) which implies that $H^d(X,W\sO_X)$ is generated by Frobenius stable elements.  This yields the claim after localization.
\end{proof}

We now proceed to the proof that both Witt vector and $p$-adic cohomology vanishes for uniruled W$\sO$-rational varieties.  We precede the actual statement with a few lemmata.  Before stating them we recall that: if $X$ and $Y$ are integral, and $Y$ is normal, then $u \colon X \to Y$ is a universal homeomorphism if and only if it is finite, surjective and purely inseparable (see \cite[Prop. (3.5.8)]{EGAI}).


\begin{lemma}[{\cite[Lemma 4.1.6 \& Lemma 4.2.4]{Rulling_Chatzistamatiou_Hodge_Witt_and_Witt_rational}}]
\label{lemma:witt_cohomology_universal_homeo}
Let $f \colon Y \to X$ be a morphism of normal varieties over $k$.  Then the following statements hold true for every $i>0$:
\begin{itemize}
    \item if $f$ is finite then $f^* \colon H^i(Y,W\sO_{Y,\QQ}) \to H^i(X,W\sO_{X,\QQ})$ is injective,
    \item if $f$ is a universal homeomorphism  then $f^* \colon H^i(Y,W\sO_{Y,\QQ}) \to H^i(X,W\sO_{X,\QQ})$ is an isomorphism.
\end{itemize}
\end{lemma}

\begin{lemma}\label{lemma:support_witt_cohomology}
Let $f \colon X \to Y$ be a birational morphism of varieties of dimension $d$.  Then for every $i>0$ we have 
\[
\dim {\rm Supp}(R^if_*W\sO_{X,\QQ}) < d-i.
\]
\end{lemma}

\begin{proof}
We first recall the standard proof that $\dim {\rm Supp}(R^if_*\sO_X) < d-i$.  Let $\eta \in Y$ be the generic point of ${\rm Supp}(R^if_*\sO_X)$.  Localizing and using the formal function theorem, we see that 
\[
(R^if_*\sO_X)^{\wedge}_{\eta} = \lim_{n \to 
 \infty} H^i(X_n,\sO_{X_n}),
\]
for $X_n = Y \times_X \Spec(\sO_{X,\eta}/\mathfrak{m}^{n+1}_{\eta})$ where $\mathfrak{m}_\eta \subset \sO_{X,\eta}$ is the maximal.  This implies that $i \leq \dim X_n = \dim X_0$.  Moreover, since $\eta$ is contained in the image of the exceptional set of $f$ we see that $\dim {\rm Supp}(R^if_*\sO_X) + \dim X_0 < d$.  Combining those inequalities we obtain the claim.

We now proceed to the proof concerning $(R^if_*W\sO_{X,\QQ})$.  Set $S_i = {\rm Supp} (R^if_*\sO_X)$  We begin by observing that a simple inductive argument with respect to the parameter $n$ coming from the exact sequence:
\[
\cdots \ra R^if_*\sO_X \xrightarrow{V^n} R^if_*W_{n+1}\sO_X \ra R^if_*W_n\sO_X \ra \cdots,
\]
implies that ${\rm Supp} (R^if_*W_n\sO_X) \subset S_i$.  Moreover, from the exact sequence
\[
\cdots \ra R^{i-1}f_*W_{n+1}\sO_X \xrightarrow{V} R^{i-1}f_*W_n\sO_X \ra R^if_*\sO_X \ra \cdots,
\]
we see that the maps $R^{i-1}f_*W_{n+1}\sO_X \ra R^{i-1}f_*W_n\sO_X$ are surjective on $X \setminus S_i$.  Consequently, the associated projective system satisfies Mittag-Leffler condition along $X \setminus S_i$ and hence 
\[
{\rm Supp}(R^1\lim R^{i-1}f_*W_n\sO_X) \subset S_i.
\]  
Combination of the above two observations with \autoref{prop:witt_vectors_and_lim} finishes the proof.
\end{proof}

\begin{definition}
We say that a morphism of normal varieties $X \to Y$ is \emph{quasi-birational} if it is surjective purely inseparable and generically finite.  Alternatively, taking Stein factorization, it is a composition of a universal homeomorphism and a birational morphism.
\end{definition}

\begin{lemma}
\label{proposition:quasi_birational}
Let $f \colon X' \to X$ be a quasi-birational morphism of normal varieties of dimension $d$.  Then the map $f^* \colon H^d(X,W\sO_{X,\QQ}) \to H^d(X',W\sO_{X',\QQ})$ is surjective.
\end{lemma}

\begin{proof}
First, taking Stein factorization and using \autoref{lemma:witt_cohomology_universal_homeo}, we see that we may assume that $f$ is birational.  In this situation, we conclude by Leray spectral sequence and \autoref{lemma:support_witt_cohomology}.  More precisely, we observe that the top diagonal of the $E_2$ page of the spectral sequence consists of a single entry $H^d(X,W\sO_{X,\QQ})$, which clearly yields the result.
\end{proof}

\begin{proposition}
\label{lem:uniruled_witt_vector}
Let $X$ be a W$\sO$-rational normal uniruled variety of dimension $d = \dim X$ defined over an algebraically closed field of characteristic $p>0$.  Then $H^d(X,W\sO_{X,\QQ}) = 0$ and $H^d_{\et}(X,\QQ_p) = 0$
\end{proposition}


\begin{proof}
Since $X$ is uniruled there exists a generically finite rational map $g \colon Z \times \PP^1 \dashrightarrow X$.  Taking the quasi-resolution of the closure of the graph $\Gamma_g \subset Z \times \PP^1 \times X$ we obtain a proper generically finite map $Y \to X$ with $H^d(Y,W\sO_{Y,\QQ}) = 0$ by \autoref{proposition:quasi_birational}. 

\[
\xymatrix{
  Z \times \PP^1 \ar@{-->}[d] & Y \ar[l]\ar[d] & Y' \ar[l]\ar[d] \\
  X \ar@{=}[r] & X & \ar[l] X'
}
\]

\noindent Taking the flattening of the morphism $Y \to X$, performing base change along a quasi-resolution of the base and potentially taking the normalization of the dominant component, we obtain a proper finite morphism $Y' \to X'$ with $Y'$ normal and quasi-birational to $Y$ and $X'$ a quasi-resolution of $X$.  By applying \autoref{proposition:quasi_birational} again, we see that $H^d(Y',W\sO_{Y',\QQ}) = 0$, which implies that $H^d(X',W\sO_{X',\QQ}) = 0$ using \autoref{lemma:witt_cohomology_universal_homeo}.  Since $X$ is W$\sO$-rational, this means that $H^d(X,W\sO_{X,\QQ}) = H^d(X',W\sO_{X',\QQ}) = 0$, and hence the proof concerning W$\sO$-cohomology is finished.  The claim about $p$-adic cohomology is now a direct consequence of \autoref{cor:p_adic_slope_zero_of_witt_vector}.
\end{proof}
{"config": "arxiv", "file": "2006.04692/top_Witt.tex"}