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\section*{etale morphism of schemes}

\begin{quote}%
This entry is about \'e{}tale morphisms between [[schemes]]. The term \emph{[[étale map]]} is preferred in the context of [[topology]] and [[differential geometry]], see [[étalé space]] for the topological version.


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

\hypertarget{tale_morphisms}{}\paragraph*{{\'E{}tale morphisms}}\label{tale_morphisms}

[[!include etale morphisms - contents]]

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

[[!include higher geometry - 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{closure_properties}{Closure properties}\dotfill \pageref*{closure_properties} \linebreak
\noindent\hyperlink{classes_of_examples}{Classes of examples}\dotfill \pageref*{classes_of_examples} \linebreak
\noindent\hyperlink{AsLocallyConstantSheaves}{As locally constant sheaves}\dotfill \pageref*{AsLocallyConstantSheaves} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \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}

The notion of \emph{\'e{}tale morphism of schemes} is the realization of the general notion of [[étale morphism]] for maps between [[schemes]], hence it captures roughly the idea of a map of schemes which is a [[local homeomorphism]]/[[local diffeomorphism]].

A central use of \'e{}tale morphisms of schemes is that they appear as [[coverings]] in the [[Grothendieck topology]] of the [[étale site]]. The [[abelian sheaf cohomology]] with respect to these \'e{}tale covers of schemes is accordingly called \emph{[[étale cohomology]]}.

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

\begin{defn}
\label{EquivalentConditionsForEtale}\hypertarget{EquivalentConditionsForEtale}{}
A [[morphism]] of [[schemes]] is an \textbf{\'e{}tale morphism} if the following equivalent conditions hold:

\begin{enumerate}%
\item it is

\begin{enumerate}%
\item [[smooth morphism of schemes|smooth]]


\item [[unramified morphism|unramified]]



\end{enumerate}

\item it is

\begin{enumerate}%
\item [[smooth morphism of schemes|smooth]]


\item of [[relative dimension]] $0$.



\end{enumerate}

\item it is

\begin{enumerate}%
\item [[flat morphism|flat]]


\item [[unramified]];



\end{enumerate}

\item it is

\begin{enumerate}%
\item [[formally étale morphism of schemes|formally étale]];


\item [[locally of finite presentation]]



\end{enumerate}


\end{enumerate}
\end{defn}
(A number of other equivalent definitions are listed at \href{http://en.wikipedia.org/wiki/Etale_morphism}{wikipedia}.)

\begin{remark}
\label{}\hypertarget{}{}
For morphisms $f \colon X \longrightarrow Y$ between [[algebraic varieties]] over an [[algebraically closed field]] this means that for all points $p \in X$ the induced morphism on [[tangent cones]]

\begin{displaymath}
T_p X \longrightarrow T_{f(p)} Y
\end{displaymath}
is an [[isomorphism]]. This is analogous to the corresponding characterization of [[local diffeomorphisms]] of [[smooth manifolds]].

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
Relaxing the finiteness condition in item 4 of \ref{EquivalentConditionsForEtale} yields the notion of \emph{[[weakly étale morphism]]}.

[[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]]

\end{remark}
\begin{defn}
\label{}\hypertarget{}{}
A jointly surjective collection of \'e{}tale morphisms $\{U_i \to X\}$ is called an [[étale cover]].

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
Most of the pairs of conditions in def. \ref{EquivalentConditionsForEtale} can be read as constraining the fiber of the morphism to be first suitably surjective/[[bundle]]-like ([[smooth morphism of schemes|smooth]], [[flat morphism|flat]]) and second suitably locally injective ([[unramified]]).

Specifically the first condition has an [[infinitesimal object|infinitesimal anlog]]: a [[formally étale morphism]] is a [[formally smooth morphism|formally smooth]] and [[formally unramified morphism]]. These notions also have an interpretation in [[synthetic differential geometry]] and there they correspond to the statement that a [[local diffeomorphism]] is a [[submersion]] which is also an [[immersion of smooth manifolds]].

\end{remark}
\begin{defn}
\label{}\hypertarget{}{}
A morphism is \textbf{[[formally étale morphism]]} if it is

\begin{itemize}%
\item [[formally smooth]] (satisfying an infinitesimal lifting property)


\item and [[formally unramified]].



\end{itemize}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
These are sheaf-like properties, which can be formalized in the language of [[Q-categories]] ([[monopresheaf]] and [[epipresheaf]] properties on the $Q$-category of nilpotent thickenings).

See at \emph{[[differential cohesion]]} and at \emph{[[infinitesimal shape modality]]}.

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

\hypertarget{closure_properties}{}\subsubsection*{{Closure properties}}\label{closure_properties}

\begin{prop}
\label{}\hypertarget{}{}
\begin{itemize}%
\item A [[composite]] of two \'e{}tale morphism is itself \'e{}tale.


\item The [[pullback]] of an \'e{}tale morphism is \'e{}tale.


\item If $f_1 \circ f_2$ is \'e{}tale and $f_1$ is, then so is $f_2$.



\end{itemize}
\end{prop}
(e.g. \hyperlink{Milne}{Milne, prop. 2.11})

\begin{proof}
Use that an \'e{}tale morphism is a [[formally étale morphism]] with finite fibers, and that $f \colon X \to Y$ is formally \'e{}tale precisely if the [[infinitesimal shape modality]] [[unit of a monad|unit]] [[natural transformation|naturality square]]

\begin{displaymath}
\itexarray{
    X &\longrightarrow& \Pi_{inf}(X)
    \\
    \downarrow && \downarrow
    \\
    Y &\longrightarrow& \Pi_{inf}(Y)
  }
\end{displaymath}
is a [[pullback]] square. Then the three properties to be shown are equivalently the [[pasting law]] for pullback diagrams.

\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
A [[smooth morphism of schemes]] is \'e{}tale iff there is an [[étale cover]] of the base scheme by [[open subschemes]] such that the pullback of the projection to each of them is an open local isomorphism of [[locally ringed spaces]] (and in particular, the pullback of the projection morphism is an [[étalé space|étale map]] of the corresponding underlying topological spaces).

\end{prop}
\begin{remark}
\label{}\hypertarget{}{}
This disjointness picture of \'e{}tale covers make them suitable for having nontrivial [[cohomology]] in situations where Zariski covers give vanishing cohomology.

\end{remark}
\hypertarget{classes_of_examples}{}\subsubsection*{{Classes of examples}}\label{classes_of_examples}

\begin{prop}
\label{}\hypertarget{}{}
Let $k$ be a [[field]]. A morphism of [[scheme]]s $Y \to Spec k$ is \'e{}tale precisely if $Y$ is a [[coproduct]] $Y \simeq \coprod_i Spec k_i$ for each $k_i$ a finite and separable [[field extension]] of $k$.

\end{prop}
This appears for instance as \hyperlink{deJong}{de Jong, prop. 3.1 i)}.

\begin{remark}
\label{}\hypertarget{}{}
Such \'e{}tale morphisms are classified by the classical [[Galois theory]] of field extensions.

\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
A [[ring]] homomorphism of [[affine varieties]] $Spec(A) \to Spec(B)$ for $Spec(B)$ non-singular and for $A \simeq B[x_1, \cdots, x_n]/(f_1, \cdots, f_n)$ with [[polynomials]] $f_i$ is \'e{}tale precisel if the [[Jacobian]] $det(\frac{\partial f_i}{\partial x_j})$ is invertible.

\end{prop}
This appears for instance as (\hyperlink{Milne}{Milne, prop. 2.1}).

\hypertarget{AsLocallyConstantSheaves}{}\subsubsection*{{As locally constant sheaves}}\label{AsLocallyConstantSheaves}

\begin{prop}
\label{}\hypertarget{}{}
A [[sheaf]] $F$ on a scheme $X$ corresponds to an \'e{}tale morphism $Y \to X$ precisely if there is an [[étale cover]] $\{U_i \to X\}$ such that each restriction

\begin{displaymath}
F|_{U_i} \simeq LConst K_i
\end{displaymath}
is [[isomorphic]] to a [[constant sheaf]] on a [[set]] $K_i$.

\end{prop}
A proof is in (\hyperlink{Deligne}{Deligne}).

\hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples}

\begin{example}
\label{}\hypertarget{}{}
A finite separable [[field extension]] $K \hookrightarrow L$ corresponds dually to an \'e{}tale morphism $Spec L \to Spec K$. These are the morphisms classified by classical [[Galois theory]].

\end{example}
\begin{example}
\label{OpenImmersionIsEtale}\hypertarget{OpenImmersionIsEtale}{}
Every [[open immersion of schemes]] is an \'e{}tale morphism of schemes. In particular a standard open inclusion (a [[cover]] in the [[Zariski topology]]) induced by the [[localization of a commutative ring]]

\begin{displaymath}
Spec(R[S^{-1}]) \longrightarrow Spec(R)
\end{displaymath}
is \'e{}tale.

\end{example}
(e.g. [[The Stacks Project|Stacks Project, lemma 28.37.9]])

\begin{proof}
By one of the equivalent characterizations of [[étale morphism]] it is sufficient to check that the map $Spec(R[S^{-1}]) \longrightarrow Spec(R)$ is a [[formally étale morphism]] and [[locally of finite presentation]].

The latter is clear, since the very definition of

\begin{displaymath}
R[S^{-1}] = R[s_1^{-1}, \cdots, s_n^{-1}](s_1 s_1^{-1} - 1, \cdots , s_n s_n^{-1} - 1)
\end{displaymath}
exhibits a [[finitely presented algebra]] over $R$.

To see that it is formally \'e{}tale we need to check that for every [[commutative ring]] $T$ with [[nilpotent ideal]] $J$ we have a [[pullback]] diagram

\begin{displaymath}
\itexarray{
    Hom(R[S^{-1}], T) &\longrightarrow& Hom(R[S^{-1}],T/J)
    \\
    \downarrow && \downarrow
    \\
    Hom(R, T) &\longrightarrow& Hom(R, T/J)
  }
  \,.
\end{displaymath}
Now by the [[universal property]] of the [[localization of a commutative ring|localization]], a homomorphism $R[S^{-1}] \longrightarrow T$ is a homomorphism $R \longrightarrow T$ which sends all elements in $S \hookrightarrow R$ to invertible elements in $T$. But no element in a [[nilpotent ideal]] can be invertible, Therefore the fiber product of the bottom and right map is the set of maps from $R$ to $T$ such that $S$ is taken to invertibles, which is indeed the top left set.

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

\begin{itemize}%
\item [[étale morphism]]


\item \textbf{\'e{}tale morphism of schemes}

\begin{itemize}%
\item [[étale site]], [[étale cohomology]]


\item [[étale (∞,1)-site]]


\item [[étale morphism of E-∞ rings]]



\end{itemize}


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

The classical references are

\begin{itemize}%
\item [[Pierre Deligne]] et al., \emph{Cohomologie \'e{}tale} , Lecture Notes in Mathematics, no. 569, Springer-Verlag, 1977.

\end{itemize}
\begin{itemize}%
\item [[James Milne]], \emph{[[Étale Cohomology]]}, Princeton Mathematical Series \textbf{33}, 1980. xiii+323 pp.

\end{itemize}
Lecture notes include

\begin{itemize}%
\item [[James Milne]], section 2 of \emph{[[Lectures on Étale Cohomology]]}

\end{itemize}
\begin{itemize}%
\item [[Aise Johan de Jong]], \emph{\'E{}tale cohomology} (\href{http://math.columbia.edu/~pugin/Teaching/Etale_files/EtaleCohomology.pdf}{pdf})  --- link broken, couldn't find another copy online

\end{itemize}
The local structure theorems are discussed in

\begin{itemize}%
\item Leovigildo Alonso, Ana Jeremias, Marta Perez, \emph{Local structure theorems for smooth maps of formal schemes} (\href{http://arxiv.org/abs/math/0605115}{arXiv:math/0605115})

\end{itemize}
Discussion of etale morphisms between [[E-infinity rings]]/[[spectral schemes]] is in

\begin{itemize}%
\item [[Jacob Lurie]], section 8.5 of \emph{[[Higher Algebra]]}

\end{itemize}
and generally in [[E-∞ geometry]] in

\begin{itemize}%
\item [[Jacob Lurie]], section 1.2 of \emph{[[Quasi-Coherent Sheaves and Tannaka Duality Theorems]]}

\end{itemize}
[[!redirects étale morphism of schemes]]

[[!redirects etale morphisms of schemes]] [[!redirects étale morphisms of schemes]]



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