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\title{A note on Azumaya algebras and one-forms}
\alttitle{Note sur les algèbres d'Azumaya et les $1$-formes}

\author{\firstname{Siqing} \lastname{Zhang}\CDRorcid{0000-0001-5056-454X}}
\address{Department of Mathematics, Yale University, New Haven, CT, 06511, USA}
\email{siqing.zhang.math@gmail.com}

\begin{abstract}
The crystalline differential operators on a smooth variety $X$ give rise to a non-split Azumaya algebra over the cotangent bundle of the Frobenius twist $X'$. In some cases, this Azumaya algebra splits when restricted to finite covers of $X'$. In this short note, we show that, whenever $X$ has a non-closed global one-form, there is a degree-$1$ cover of $X'$ on which the Azumaya algebra does not split, answering a question of Sasha Petrov.
\end{abstract}

\begin{altabstract}
Les opérateurs différentiels cristallins sur une variété lisse $X$ donnent naissance à une algèbre d'Azumaya non décomposable sur le fibré cotangent de la torsion de Frobenius $X'$. Dans certains cas, cette algèbre d'Azumaya se décompose lorsqu'elle est restreinte aux recouvrements finis de $X'$. Dans cette brève note, nous montrons que, chaque fois que $X$ possède une $1$-forme globale non fermée, il existe un recouvrement de degré~$1$ de $X'$ sur lequel l'algèbre d'Azumaya ne se décompose pas, répondant ainsi à une question posée par Sasha Petrov.
\end{altabstract}

\subjclass{14G17}

\keywords{\kwd{Azumaya algebra} \kwd{positive characteristics}}
\altkeywords{\kwd{Algèbre d'Azumaya} \kwd{caractéristique positive}}

\thanks{This work is supported by an AMS-Simons travel grant}

\COI{The author does not work for, advise, own shares in, or receive funds from any organization that could benefit from this article, and has declared no affiliations other than their research organizations.}

\begin{document}
%\input{CR-pagedemetas}
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\maketitle

\section{The question}

Throughout, $k$ is an algebraically closed field of characteristic $p>0$. Let $X$ be a smooth proper variety over $k$, and let $X'$ be its Frobenius twist. By Bezrukavnikov--Mirkovic--Rumynin~\cite[Theorem~2.2.3]{BMR}, the sheaf of crystalline differential operators on $X$ defines an Azumaya algebra~$\mathcal A_X$ on the cotangent bundle $T^*X'$.

In many useful cases, especially in the characteristic-$p$ geometric Langlands and non-abelian Hodge Theory for curves~\cite{BB,Groechenig,deCataldoZhang}, and for abelian varieties~\cite{Hao}, this Azumaya algebra splits when restricted to spectral varieties; compare also the $p$-adic analogue~\cite{Heuer}. Sasha Petrov asked the following question.

\begin{enonce}{Question}[{\cite[Question 7.3(1)]{Petrov}}]\label{petrov-question}
Does there exist a smooth proper variety $X$ together with a smooth subvariety $Z\subset T^*X'$, finite over $X'$, such that $\restr{\mathcal A_X}{Z}$ does not split?
\end{enonce}

In this note, we answer this question affirmatively in every positive characteristic.

\section{The answer}

Let $X'  \coloneqq  X\times_{k,F_k} k$ be the Frobenius twist. Write $F_X = F_{X/k} \colon X \to X'$ for the relative Frobenius, and $\pi_X \colon X' \to X$ for the projection. Milne's exact sequence, in the form used by Ogus--Vologodsky~\cite[(4.1.1)]{OV}, is the following exact sequence of \'{e}tale sheaves of abelian groups on $X'$:
\begin{equation}\label{eq:milne}
0
\xrightarrow[\hphantom{\pi_X^*-C_X}]{} \mathcal O_{X'}^*
\xrightarrow[\hphantom{\pi_X^*-C_X}]{} F_{X,*}\mathcal O_X^*
\xrightarrow[\hphantom{\pi_X^*-C_X}]{\dif \log} F_{X,*}Z^1_{X/k}
\xrightarrow{\pi_X^*-C_X} \Omega^1_{X'/k}
\xrightarrow[\hphantom{\pi_X^*-C_X}]{} 0.
\end{equation}
Here $Z^1_{X/k}\subset \Omega^1_{X/k}$ is the sheaf of closed one-forms and
$C_X$ is the Cartier operator. The Yoneda $\operatorname{Ext}^2$ class presented by~\eqref{eq:milne} defines the obstruction homomorphism
\begin{equation}\label{eq:obstruction}
\operatorname{ob}_{X'}\colon H^0\parens[\big]{X',\Omega^1_{X'/k}}
\to H^2_{\mathrm{\acute et}}\parens[\big]{X',\mathcal O_{X'}^*}.
\end{equation}

Let $\DD_{X/k}$ be the sheaf of crystalline differential operators on $X$. The center of $F_{X,*}\DD_{X/k}$ is identified with $\operatorname{Sym}_{\mathcal O_{X'}}T_{X'/k}$, and $F_{X,*} \DD_{X/k}$ has the structure of an Azumaya algebra on $T^*X'$. This is the algebra denoted $\mathcal A_X$ above.

\begin{lemm}\label{lem:graph}
Let $X/k$ be smooth, and let $\omega\in H^0\parens[\big]{X',\Omega^1_{X'/k}}$. Let $i_\omega \colon X' \to T^*X'$ be the section corresponding to $\omega$. Let\/ $\Gamma_\omega \coloneqq \operatorname{im}(i_\omega)\subset T^*X'$ be the graph. Then,
\[
[i_\omega^*\mathcal A_X] =\operatorname{ob}_{X'}(\omega)
\quad \text{in $H^2_{\mathrm{\acute et}}\parens[\big]{X',\mathcal O_{X'}^*}$}.
\]
In particular, $\restr{\mathcal A_X}{\Gamma_\omega}$ splits if and only if $\operatorname{ob}_{X'}(\omega)= 0$.
\end{lemm}

\begin{proof}
Ogus--Vologodsky identify $\operatorname{ob}_{X'}(\omega)$ with the class of the $\Gm$-gerbe of line bundles on $X$ with integrable connection and $p$-curvature $\omega$ \cite[Proposition~4.2]{OV}. The same gerbe is the gerbe of splittings of the restriction of $\mathcal A_X$ to the section $i_\omega$ by~\cite[Remark~4.3]{OV}.
\end{proof}

The following proposition reduces Question~\ref{petrov-question} to finding a variety with non-closed global one-forms.

\begin{prop}\label{prop:dimension}
Let $X/k$ be a smooth proper connected variety. If not every global one-form on $X$ is closed, i.e.\ $H^0\parens[\big]{X,Z^1_{X/k}}\subsetneq H^0\parens[\big]{X,\Omega^1_{X/k}}$, then the obstruction map $\operatorname{ob}_{X'}$ as in~\eqref{eq:obstruction} is nonzero.
\end{prop}

\begin{proof}
Let $G \coloneqq \Pic^{\nabla,\mathrm{rig}}_{X/k}$ be the rigidified Picard scheme of line bundles with integrable connection. Let $V \coloneqq H^0\parens[\big]{X',\Omega^1_{X'/k}}$, viewed as a vector group. By~\cite[Proposition~4.11 and the paragraph following it]{OV}, taking $p$-curvature defines a morphism of group schemes
\[
\psi\colon G \longrightarrow V.
\]
By~\cite[Proposition~4.2]{OV}, $\operatorname{ob}_{X'}(\omega)=0$ if and only if $\omega\in\psi\parens[\big]{G(k)}$. Therefore, it is enough to show that $\psi \parens[\big]{G(k)} \ne V(k)$.

By Cartier descent,
\[
\Pic_{X'/k} \longrightarrow G,
\quad M \longmapsto \parens[\big]{F_X^*M,\nabla_{\mathrm{can}}},
\]
identifies $\Pic_{X'/k}$ with $\ker(\psi)$; see~\cite[Theorem~5.1]{Katz}. Forgetting the connection fits into the exact sequence of fppf sheaves
\[
0
\to H^0\parens[\big]{X,Z^1_{X/k}}
\to G
\overset{b}{\to} \Pic_{X/k}
\to H^1\parens[\big]{X,Z^1_{X/k}},
\]
as in~\cite[Proposition~4.11]{OV}. In particular, if $G^0$ is the neutral component of $G$, then
\[
\dim G^0\leq h^0\parens[\big]{X,Z^1_{X/k}}+\dim \Pic^0_{X/k}.
\]
Let $\operatorname{Im}\parens[\big]{\restr{\psi}{(G^0)_{\mathrm{red}}}}$ be the algebraic subgroup image. The quotient morphism
\[
(G^0)_{\mathrm{red}} \to \operatorname{Im}\parens[\big]{\restr{\psi}{(G^0)_{\mathrm{red}}}}
\]
is faithfully flat. Therefore
\[
\dim \operatorname{Im}\parens[\big]{\restr{\psi}{(G^0)_{\mathrm{red}}}}
= \dim (G^0)_{\mathrm{red}}-\dim\ker\parens[\big]{\restr{\psi}{(G^0)_{\mathrm{red}}}}
\leq h^0\parens[\big]{X,Z^1_{X/k}}<\dim V.
\]

Finally, passing from $G^0$ to $G$ only gives finitely many translates: let $G_1=b^{-1}\parens[\big]{\Pic^0_{X/k}}$. Since $G_1$ is of finite type, $G_1/G^0$ is finite. Moreover, $G(k) \big/ G_1(k)$ injects into $\NS(X)$, so its image under the map to $V/\psi\parens[\big]{G_1(k)}$ induced by $\psi$ is finite: $\NS(X)$ is finitely generated, whereas $V(k)$ is killed by~$p$.
\end{proof}

\begin{theo}
Let $X/k$ be a smooth proper connected variety. If not every global one-form on $X$ is closed, then there is a smooth subvariety $Z\subset T^*X'$, finite over $X'$, such that $\restr{\mathcal{A}_{X}}{Z}$ does not split. Moreover, the finite morphism $Z \to X'$ is an isomorphism.
\end{theo}

\begin{proof}
Combine Lemma~\ref{lem:graph} and Proposition~\ref{prop:dimension}, and take $Z$ to be $\Gamma_{\omega}$ for some $\omega$ such that \mbox{$\operatorname{ob}_{X'}(\omega)\neq 0$}.
\end{proof}

\begin{remas}[Relation with liftability]\label{rem:liftability}
\;
\begin{enumerate}[leftmargin=*]\alphenumi
\item \label{rem:liftability_5a} If the Hodge-to-de Rham spectral sequence degenerates at $E_1$, then we must have that $H^0\parens[\big]{X,Z^1_{X/k}}= H^0\parens[\big]{X,\Omega^1_{X/k}}$. Therefore, by Deligne--Illusie~\cite[Theorem~2.1 and Corollary~2.5]{DI}, if a smooth proper $X$ has a non-closed global one-form, then it does not lift to the ring of second Witt vectors $W_2(k)$. In particular, all examples in Section~\ref{sec: examples} do not lift to $W_2(k)$.
\item \label{rem:liftability_5b} Conversely, \cite[Lemma~7.1]{Petrov} shows that, if a smooth proper $X$ lifts to $W_2(k)$, then $\mathcal{A}_X$ splits on any closed subvariety $Z\subset T^*X'$ that is finite \'etale over $X'$. Such a $Z$ is also $W_2(k)$-liftable.
\item \label{rem:liftability_5c} Furthermore, any smooth proper subvariety $Z$ of $T^*X'$ on which $\mathcal{A}_X$ does not split must also have a non-closed global one-form, thus not $W_2(k)$-liftable. Indeed, this follows from the proof of~\cite[Lemma~7.1]{Petrov}.
\end{enumerate}
\end{remas}

\subsection{The examples}\label{sec: examples}

\subsubsection{Surface examples}

The existence of smooth projective surfaces with non-closed global one-forms goes back at least to Mumford~\cite[Corollary on p.~341]{MumfordPathologies}. The author would like to thank Sasha Petrov for pointing out this example.

In~\cite[Corollary~3.4 and the paragraph following it]{Takeda}, Takeda constructs some generalized Raynaud surfaces that also have non-closed global one-forms. \cite{Takeda} is written under the assumption that $p>2$, but one can check that these examples still hold when $p=2$, see e.g.~\cite[Section~5]{Tziolas} for more details.

\subsubsection{Higher-dimensional examples}

For higher-dimensional examples, we can take $X$ to be any of the surfaces above and take $X \times \mathbf{P}^n_k$. Indeed, if $\eta\in H^0\parens[\big]{X,\Omega_{X/k}^1}$ is non-closed, then so is the pullback of $\eta$ to $X \times \mathbf{P}^n_k$.

\section*{Acknowledgments}

I would like to thank Sasha Petrov for very helpful feedback and for pointing out Mumford's example. I would like to thank Gleb Terentiuk for illuminating discussions related to Remark~\ref{rem:liftability}\eqref{rem:liftability_5b}. I would also like to thank an anonymous referee for many helpful suggestions and pointing out the fact in Remark~\ref{rem:liftability}\eqref{rem:liftability_5c}.

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