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\begin{document}

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\section*{weakly étale morphism of schemes}

[[!redirects weakly étale morphism]]

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

\hypertarget{geometry}{}\paragraph*{{Geometry}}\label{geometry}

[[!include higher geometry - contents]]

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

[[!include cohomology - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A variant of [[étale morphism of schemes]] where the finiteness conditions on [[étale morphisms]] are relaxed.

Used in the definition of \emph{[[pro-étale site]]} and \emph{[[pro-étale cohomology]]}.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{WeaklyEtale}\hypertarget{WeaklyEtale}{}
A [[morphism]] $f \colon X \longrightarrow Y$ of [[schemes]] is called \emph{weakly \'e{}tale} if

\begin{enumerate}%
\item $f$ is a [[flat morphism of schemes]];


\item its [[diagonal]] $X \longrightarrow X \times_Y X$ is also flat.



\end{enumerate}
\end{defn}
(\hyperlink{BhattScholze13}{Bhatt-Scholze 13, def. 4.1.1})

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

\begin{prop}
\label{}\hypertarget{}{}
Every [[weakly étale morphism]] is a [[formally étale morphism]].

\end{prop}
(\hyperlink{GabberRamero03}{Gabber-Ramero 03, theorem 2.5.36, prop. 3.2.16} \hyperlink{BhattScholze13}{Bhatt-Scholze 13, prop. 2.3.3. (2)})

\begin{remark}
\label{}\hypertarget{}{}
As discussed there, an [[étale morphism]] is a [[formally étale morphism]] which is [[locally of finite presentation]].

\end{remark}
\begin{cor}
\label{}\hypertarget{}{}
A [[weakly étale morphism]] which is [[locally of finite presentation]] is an [[étale morphism of schemes|étale morphism]].

[[étale morphism of schemes|étale morphism]] $\Rightarrow$ [[weakly étale morphism of schemes|weakly étale morphism]] $\Rightarrow$ [[formally étale morphism of schemes|formally étale morphism]]

\end{cor}
In fact a weakly \'e{}tale morphism is equivalently a [[formally étale morphism]] which is ``locally [[pro-object|pro-finitely]] presentable'' (dually locally of [[ind-object|ind]]-finite rank) in the following sense

\begin{defn}
\label{IndEtale}\hypertarget{IndEtale}{}
For $A \to B$ a [[homomorphism]] of [[rings]], say that it is an \textbf{[[ind-étale morphism]]} if that $A$-[[associative algebra|algebra]] $B$ is a [[filtered colimit]] of $A$-[[étale algebras]].

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
Let $f \;\colon\; A \longrightarrow B$ be a [[homomorphism]] of [[rings]].

\begin{itemize}%
\item If $f$ is ind-\'e{}tale, def. \ref{IndEtale}, then it is weakly \'e{}tale, def. \ref{WeaklyEtale}.

\end{itemize}
Almost conversely

\begin{itemize}%
\item If $f$ is weakly \'e{}tale, then there is a [[faithfully flat morphism]] $g \colon B \to C$ which is ind-\'e{}tale such that the [[composition|composite]] $g\circ f$ is ind-\'e{}tale.

\end{itemize}
\end{prop}
(\hyperlink{BhattScholze13}{Bhatt-Scholze 13, theorem 1.3})

\begin{cor}
\label{}\hypertarget{}{}
The [[sheaf toposes]] over the [[sites]] of weak \'e{}tale morphisms and of [[pro-étale morphisms of schemes]] into some base [[scheme]] are [[equivalence of categories|equivalent]], both define the \emph{[[pro-étale topos]]} over the \emph{[[pro-étale site]]}.

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

[[étale morphism of schemes|étale morphism]] $\Rightarrow$ [[pro-étale morphism of schemes|pro-étale morphism]] $\Rightarrow$ [[weakly étale morphism of schemes|weakly étale morphism]] $\Rightarrow$ [[formally étale morphism of schemes|formally étale morphism]]

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

\begin{itemize}%
\item [[Ofer Gabber]] and Lorenzo Ramero, \emph{Almost ring theory}, volume 1800 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2003. (\href{http://arxiv.org/abs/math/0201175}{arXiv:math/0201175})


\item [[Bhargav Bhatt]], [[Peter Scholze]], \emph{The pro-\'e{}tale topology for schemes} (\href{http://arxiv.org/abs/1309.1198}{arXiv:1309.1198})



\end{itemize}
[[!redirects weakly étale morphisms]]

[[!redirects weakly etale morphism]] [[!redirects weakly etale morphisms]]



\end{document}