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\section*{étale cohomology}

[[!redirects etale cohomology]]

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

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

[[!include cohomology - contents]]

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

[[!include etale morphisms - 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{BasicProperties}{Basic properties}\dotfill \pageref*{BasicProperties} \linebreak
\noindent\hyperlink{RelationZariskiEtaleCohomology}{Relation to Zariski cohomology}\dotfill \pageref*{RelationZariskiEtaleCohomology} \linebreak
\noindent\hyperlink{WithCoefficientsInCoherentModules}{With coefficients in coherent modules}\dotfill \pageref*{WithCoefficientsInCoherentModules} \linebreak
\noindent\hyperlink{WithCoefficientsInACyclicGroup}{With coefficients in a cyclic group}\dotfill \pageref*{WithCoefficientsInACyclicGroup} \linebreak
\noindent\hyperlink{with_coefficients_in_the_multiplicative_group}{With coefficients in the multiplicative group}\dotfill \pageref*{with_coefficients_in_the_multiplicative_group} \linebreak
\noindent\hyperlink{with_coefficients_in_groups_of_roots_of_unity}{With coefficients in groups of roots of unity}\dotfill \pageref*{with_coefficients_in_groups_of_roots_of_unity} \linebreak
\noindent\hyperlink{MainTheorems}{Main theorems}\dotfill \pageref*{MainTheorems} \linebreak
\noindent\hyperlink{proper_base_change_theorem}{Proper base change theorem}\dotfill \pageref*{proper_base_change_theorem} \linebreak
\noindent\hyperlink{comparison_theorem_relation_to_singular_cohomology}{Comparison theorem: Relation to singular cohomology}\dotfill \pageref*{comparison_theorem_relation_to_singular_cohomology} \linebreak
\noindent\hyperlink{K&#252;nnethFormula}{K\"u{}nneth formula}\dotfill \pageref*{K&#252;nnethFormula} \linebreak
\noindent\hyperlink{CycleMap}{Cycle map}\dotfill \pageref*{CycleMap} \linebreak
\noindent\hyperlink{PoincareDuality}{Poincar\'e{} duality}\dotfill \pageref*{PoincareDuality} \linebreak
\noindent\hyperlink{lefschetz_fixedpoint_formula}{Lefschetz fixed-point formula}\dotfill \pageref*{lefschetz_fixedpoint_formula} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{history_motivation_and_original_accounts}{History, motivation and original accounts}\dotfill \pageref*{history_motivation_and_original_accounts} \linebreak
\noindent\hyperlink{reviews_and_modern_accounts}{Reviews and modern accounts}\dotfill \pageref*{reviews_and_modern_accounts} \linebreak


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

Traditional \emph{\'e{}tale cohomology} (e.g. \hyperlink{Deligne77}{Deligne 77}) is the [[abelian sheaf cohomology]] for [[sheaf|sheaves]] on the [[étale site]] of a [[scheme]] -- which is an analog of the [[category of open subsets]] of a [[topological space]] $X$ , or rather the analog of the category of [[étale spaces]] over $X$, with [[finite set|finite]] [[fibers]].

A certain [[inverse limit]] over \'e{}tale cohomology groups for different [[coefficients]] yields [[ℓ-adic cohomology]], which is a [[Weil cohomology theory]].

More generally, there is \'e{}tale [[generalized cohomology theory]] with [[coefficients]] in [[sheaves of spectra]] on the [[étale site]] (\hyperlink{Jardine97}{Jardine 97}). Still more generally, there is \'e{}tale generalized cohomology on the [[étale (∞,1)-site]] (\hyperlink{AntieauGepner12}{Antieau-Gepner 12}, \hyperlink{Lurie}{Lurie}).

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

\begin{prop}
\label{}\hypertarget{}{}
Given a [[scheme]] $X$ of finite type, the small [[étale site]] $X_{et}$ is the [[category]] whose [[objects]] are [[étale morphisms]] $Spec R \to X$ and whose morphisms $(f:Spec R\to X)\to (f':Spec R'\to X)$ are morphisms $\alpha: Spec(R)\to Spec(R')$ of schemes completing triangles: $f'\circ\alpha=f$ (notice that the morphisms between \'e{}tale morphisms are automatically \'e{}tale). This category naturally carries a [[Grothendieck topology]] that makes it a [[site]], the [[étale site]].

For $A \in Sh(X_{et}, Ab)$ an [[abelian sheaf]] on $X$, the \textbf{\'e{}tale cohomology} $H_{et}^\bullet(X,A)$ of $X$ with [[coefficients]] in $A$ is the [[abelian sheaf cohomology]] with respect to this site.

\end{prop}
\hypertarget{BasicProperties}{}\subsection*{{Basic properties}}\label{BasicProperties}

The following are some basic properties of \'e{}tale [[cohomology groups]] for various standard choices of [[coefficients]].

\hypertarget{RelationZariskiEtaleCohomology}{}\subsubsection*{{Relation to Zariski cohomology}}\label{RelationZariskiEtaleCohomology}

\begin{remark}
\label{}\hypertarget{}{}
A [[cover]] in the [[Zariski topology]] on [[schemes]] is an [[open immersion of schemes]] and hence is in particular an [[étale morphism of schemes]]. Hence the [[étale site]] is finer than the [[Zariski site]] and so every \'e{}tale [[sheaf]] is a Zariski sheaf, but not necessarily conversely.

\end{remark}
\begin{remark}
\label{LerayForInclusionOfZariskiIntoEtale}\hypertarget{LerayForInclusionOfZariskiIntoEtale}{}
For $X$ a [[scheme]], the inclusion

\begin{displaymath}
\epsilon \;\colon\; X_{Zar} \longrightarrow X_{et}
\end{displaymath}
of the [[Zariski site]] into the [[étale site]] is indeed a [[morphism of sites]]. Hence there is a [[Leray spectral sequence]] which computes \'e{}tale cohomology in terms of Zariski cohomology

\begin{displaymath}
E^{p,q}_2 = H^p(X_{Zar}, R^q \epsilon^\ast \mathcal{F})
  \Rightarrow
  E^{p+q} = H^{p+q}(X_{et}, \mathcal{F})
  \,.
\end{displaymath}
\end{remark}
This is originally due to ([[Grothendieck]], [[SGA]] 4 (Chapter VII, p355)). Reviews include (\hyperlink{Tamme}{Tamme, II 1.3}).

\hypertarget{WithCoefficientsInCoherentModules}{}\subsubsection*{{With coefficients in coherent modules}}\label{WithCoefficientsInCoherentModules}

\begin{prop}
\label{CohomologyWithCoeffsInCoherentModules}\hypertarget{CohomologyWithCoeffsInCoherentModules}{}
For $N$ a [[quasi-coherent sheaf]] of $\mathcal{O}_X$-[[modules]] and $N_{et}$ the induced \'e{}tale sheaf (by the discussion at \href{etale+topos#QuasiCoherentModules}{\'e{}tale topos -- Quasicohetent sheaves}), then the [[edge morphism]]

\begin{displaymath}
H^p_{Zar}(X, N)
  \longrightarrow
  H^p_{et}(X,N_{et})
\end{displaymath}
of the [[Leray spectral sequence]] of remark \ref{LerayForInclusionOfZariskiIntoEtale} is an [[isomorphism]] for all $p$, itentifying the [[abelian sheaf cohomology]] on the [[Zariski site]] with [[coefficients]] in $N$ with the \'e{}tale cohomology with coefficients in $N_{et}$.

Moreover, for $X$ affine we have

\begin{displaymath}
H^p_{et}(X, N_{et}) \simeq 0
  \,.
\end{displaymath}
\end{prop}
This is due to ([[Grothendieck]], [[FGA]] 1). See also for instance (\hyperlink{Tamme}{Tamme, II (4.1.2)}).

\begin{proof}
By the discussion at \emph{[[edge morphism]]} it suffices to show that

\begin{displaymath}
R^q \epsilon_\ast (N) = 0 \;\,,\;\;\; for \;\; p \gt 0
  \,.
\end{displaymath}
By the discussion at \emph{[[direct image]]} (also at \emph{[[abelian sheaf cohomology]]}), $R^q \epsilon_\ast N$ is the [[sheaf]] on the [[Zariski topology]] which is the [[sheafification]] of the [[presheaf]] given by

\begin{displaymath}
U \mapsto H^q(X_{et}|U, N)
  \,,
\end{displaymath}
hence it is sufficient that this vanishes, or rather, by locality ([[sheafification]]) it suffices to show this vanishes for $X = U = Spec(A)$ an affine [[algebraic variety]].

By the existence of \href{etale+site#CofinalAffineCovers}{cofinal affine \'e{}tale covers} the [[full subcategory]] $X_{et}^{a} \hookrightarrow X_{at}$ with the induced [[coverage]] is a [[dense subsite]] of affines. Therefore it suffices to show the statement there. Moreover, by the finiteness condition on [[étale morphisms]] every cover of $X_{et}^{a}$ may be refined by a finite cover, hence by an affine covering map

\begin{displaymath}
Spec(B) \longrightarrow Spec(A)
  \,.
\end{displaymath}
It follows (by a discussion such as e.g. at [[Sweedler coring]]) that the corresponding [[Cech cohomology]] complex

\begin{displaymath}
N_{et}(Spec(A)) \to C^0(\{Spec(B) \to Spec(A)\}, N_{et}) \to C^1(\{Spec(B) \to Spec(A)\}, N_{et}) \to \cdots
\end{displaymath}
is of the form

\begin{displaymath}
0 \to N \to N \otimes_A B \to N \otimes_{A} B \otimes_A B \to \cdots
  \,.
\end{displaymath}
known as the [[Amitsur complex]].

Since $A \to B$ is a [[faithfully flat morphism]] it follows by the [[descent theorem]] that this is [[exact sequence|exact]], hence that the cohomology indeed vanishes.

\end{proof}
\hypertarget{WithCoefficientsInACyclicGroup}{}\subsubsection*{{With coefficients in a cyclic group}}\label{WithCoefficientsInACyclicGroup}

\begin{prop}
\label{}\hypertarget{}{}
If $X = Spec(A)$ is an affine [[reduced scheme]] of [[characteristic]] a [[prime number]] $p$, then its [[étale cohomology]] with [[coefficients]] in $\mathbb{Z}/p\mathbb{Z}$ is

\begin{displaymath}
H^q(X, (\mathbb{Z}/p\mathbb{Z})_X)
  \simeq
  \left\{
    \itexarray{
      A/(F - id)A & if\; q = 1
      \\
      0 & if \; q \gt 0
    }
  \right.
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
Under the given assumptions, the [[Artin-Schreier sequence]] (see there) induces a [[long exact sequence in cohomology]] of the form

\begin{displaymath}
\begin{aligned}
    0 & \to 
    H^0(X_{et}, \mathbb{Z}/p\mathbb{Z}) \to H^0(X_{et}, \mathcal{O}_X) \stackrel{F-id}{\to} H^0(X_{et}, \mathcal{O}_X)
   \\
    & \to
     H^1(X_{et}, \mathbb{Z}/p\mathbb{Z}) \to H^1(X_{et}, \mathcal{O}_X) \stackrel{F-id}{\to} H^1(X_{et}, \mathcal{O}_X)
   \\
    & \to
     H^2(X_{et}, \mathbb{Z}/p\mathbb{Z}) \to H^2(X_{et}, \mathcal{O}_X) \stackrel{F-id}{\to} H^2(X_{et}, \mathcal{O}_X) \to \cdots
  \end{aligned}
  \,,
\end{displaymath}
where $F(-) = (-)^p$ is the [[Frobenius endomorphism]]. By prop. \ref{CohomologyWithCoeffsInCoherentModules} the terms of the form $H^{p \geq 1}(X, \mathcal{O}_X)$ vanish, and so from [[exact sequence|exactness]] we find an [[isomorphism]]

\begin{displaymath}
H^0(X_{et}, \mathcal{O}_X)/(F-id)(H^0(X_{et}, \mathcal{O}_X))
    \stackrel{\simeq}{\to}
  H^1(X_{et}, \mathbb{Z}/p\mathbb{Z})
  \,,
\end{displaymath}
hence the claimed isomorphism

\begin{displaymath}
A/(F-id)(A)
    \stackrel{\simeq}{\to}
  H^1(X_{et}, \mathbb{Z}/p\mathbb{Z})
  \,.
\end{displaymath}
By the same argument all the higher cohomology groups vanish, as claimed.

\end{proof}
\hypertarget{with_coefficients_in_the_multiplicative_group}{}\subsubsection*{{With coefficients in the multiplicative group}}\label{with_coefficients_in_the_multiplicative_group}

the \'e{}tale cohomology groups with [[coefficients]] in the [[multiplicative group]] $\mathbb{G}_m$ in the first few degrees go by special names:

\begin{itemize}%
\item $H^0_{et}(-, \mathbb{G}_m)$: [[group of units]];


\item $H^1_{et}(-, \mathbb{G}_m)$: [[Picard group]] ([[Hilbert's theorem 90]], \hyperlink{Tamme}{Tamme, II 4.3.1});


\item $H^2_{et}(-, \mathbb{G}_m)$: [[Brauer group]];



\end{itemize}
\hypertarget{with_coefficients_in_groups_of_roots_of_unity}{}\subsubsection*{{With coefficients in groups of roots of unity}}\label{with_coefficients_in_groups_of_roots_of_unity}

\begin{itemize}%
\item [[Kummer sequence]]

\end{itemize}
(\hyperlink{Tamme}{Tamme, II, 4.4})

(\ldots{})

\hypertarget{MainTheorems}{}\subsection*{{Main theorems}}\label{MainTheorems}

The following are the main theorems characterizing properties of \'e{}tale cohomology. Together these theorems imply that \'e{}tale cohomology, in its variant as [[l-adic cohomology]], is a [[Weil cohomology theory]].

\hypertarget{proper_base_change_theorem}{}\subsubsection*{{Proper base change theorem}}\label{proper_base_change_theorem}

\begin{itemize}%
\item [[proper base change theorem]]

\end{itemize}
(\hyperlink{Milne}{Milne, section 17})

\hypertarget{comparison_theorem_relation_to_singular_cohomology}{}\subsubsection*{{Comparison theorem: Relation to singular cohomology}}\label{comparison_theorem_relation_to_singular_cohomology}

\begin{itemize}%
\item [[comparison theorem (étale cohomology)]]

\end{itemize}
(\hyperlink{Milne}{Milne, section 21})

\hypertarget{K&#252;nnethFormula}{}\subsubsection*{{K\"u{}nneth formula}}\label{K&#252;nnethFormula}

(\hyperlink{Milne}{Milne, section 22})

\hypertarget{CycleMap}{}\subsubsection*{{Cycle map}}\label{CycleMap}

(\hyperlink{Milne}{Milne, section 23})

\hypertarget{PoincareDuality}{}\subsubsection*{{Poincar\'e{} duality}}\label{PoincareDuality}

(\hyperlink{Milne}{Milne, section 24})

\hypertarget{lefschetz_fixedpoint_formula}{}\subsubsection*{{Lefschetz fixed-point formula}}\label{lefschetz_fixedpoint_formula}

\hyperlink{K&#252;nnethFormula}{K\"u{}nneth formula} + \hyperlink{CycleMap}{cycle map} + \hyperlink{PoincareDuality}{Poincar\'e{} duality} $\Rightarrow$ [[Lefschetz fixed-point formula]]

(\hyperlink{Milne}{Milne, section 25})

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

\begin{itemize}%
\item [[basics of étale cohomology]]


\item [[étale morphism]], [[étale site]], \'e{}tale cohomology


\item [[étale (∞,1)-site]], [[étale topos]]


\item [[étale homotopy]]


\item [[Weil conjecture]]


\item [[continuous étale cohomology]]



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

\hypertarget{history_motivation_and_original_accounts}{}\subsubsection*{{History, motivation and original accounts}}\label{history_motivation_and_original_accounts}

\'E{}tale cohomology was conceived by [[Artin]], [[Deligne]], [[Alexander Grothendieck|Grothendieck]] and [[Verdier]] in 1963. It was used by Deligne to prove the [[Weil conjectures]]. Some useful (and also funny) remarks on this are in the beginning of

\begin{itemize}%
\item [[Spencer Bloch]], Review of [[Milne]]`s \emph{[[Étale Cohomology]]} (\href{http://www.ams.org/bull/1981-04-02/S0273-0979-1981-14894-1/S0273-0979-1981-14894-1.pdf}{pdf}; publisher's \href{http://www.worldscibooks.com/mathematics/7773.html}{book page})

\end{itemize}
See also

\begin{itemize}%
\item MathOverflow \emph{\href{http://mathoverflow.net/questions/6070/etale-cohomology-why-study-it}{\'E{}tale cohomology --- Why study it?}}

\end{itemize}
The classical references include [[SGA]], esp.

\begin{itemize}%
\item [[Pierre Deligne]] et al., \emph{Cohomologie \'e{}tale} , Lecture Notes in Mathematics \textbf{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}
See also

\begin{itemize}%
\item [[Barry Mazur]], \emph{Notes on \'e{}tale cohomology of number fields}, Annales scientifiques de l'\'E{}cole Normale Sup\'e{}rieure, S\'e{}r. 4, 6 no. 4 (1973), p. 521-552 (\href{http://www.numdam.org/item?id=ASENS_1973_4_6_4_521_0}{Numdam}, \href{http://modular.math.washington.edu/edu/2010/582e/refs/mazur-notes_on_etale_cohomology_of_number_fields_original.pdf}{pdf})

\end{itemize}
\hypertarget{reviews_and_modern_accounts}{}\subsubsection*{{Reviews and modern accounts}}\label{reviews_and_modern_accounts}

\begin{itemize}%
\item [[Günter Tamme]], \emph{[[Introduction to Étale Cohomology]]}, 1994

\end{itemize}
A modern textbook, though largely based on the material in SGA is

\begin{itemize}%
\item Lei Fu, \emph{\'E{}tale cohomology theory}, Nankai Tracts in Math. \textbf{13}, World Sci. 2011; (\href{http://www.worldscibooks.com/etextbook/7773/7773_toc.pdf}{toc pdf}; Preface \href{http://www.worldscibooks.com/etextbook/7773/7773_preface.pdf}{pdf}; chap. 1 Descent theory \href{http://www.worldscibooks.com/etextbook/7773/7773_chap01.pdf}{pdf})

\end{itemize}
Lecture notes include

\begin{itemize}%
\item [[James Milne]], \emph{[[Lectures on Étale Cohomology]]} (\href{http://www.jmilne.org/math/CourseNotes/lec.html}{html}, \href{http://www.jmilne.org/math/CourseNotes/LEC.pdf}{pdf})

\end{itemize}
\begin{itemize}%
\item [[Aise Johan de Jong]], \emph{\'E{}tale cohomology} 2009, in \emph{[[The Stacks Project]]} (\href{http://math.columbia.edu/~pugin/Teaching/Etale_files/EtaleCohomology.pdf}{pdf})

\end{itemize}
\begin{itemize}%
\item [[Evan Jenkins]], \emph{\'E{}tale cohomology seminar} (\href{http://math.uchicago.edu/~ejenkins/etale/}{web})


\item Donu Arapura, \emph{An introduction to \'E{}tale cohomology} (\href{http://www.math.purdue.edu/~dvb/preprints/etale.pdf}{pdf})


\item Antoine Ducros, \emph{\'E{}tale cohomology of schemes and analytic spaces}, (\href{http://www.math.jussieu.fr/~ducros/Cohetale.pdf}{pdf})



\end{itemize}
See also

\begin{itemize}%
\item Edgar Costa, \emph{\'E{}tale cohomology} (\href{https://dspace.ist.utl.pt/bitstream/2295/686086/1/tese.pdf}{pdf})


\item Thomas H. Geisser, \emph{Weil-etale motivic cohomology}, \href{http://www.math.illinois.edu/K-theory/0565}{K-th archive}



\end{itemize}
Discussion of [[generalized cohomology theory]] on the [[étale site]] but with [[coefficients]] in [[sheaves of spectra]] is in

\begin{itemize}%
\item [[Rick Jardine]], \emph{Generalized \'E{}tale cohomology theories}, 1997 Progress in mathematics volume 146

\end{itemize}
Discussion of generalized \'e{}tale cohomology over the [[étale (∞,1)-site]] (hence in [[higher topos theory]]/[[higher algebra]]) is in

\begin{itemize}%
\item [[Benjamin Antieau]], [[David Gepner]], \emph{Brauer groups and \'e{}tale cohomology in derived algebraic geometry} (\href{http://arxiv.org/abs/1210.0290}{arXiv:1210.0290})

\end{itemize}
\begin{itemize}%
\item [[Jacob Lurie]], \emph{Descent theorems} (\href{http://www.math.harvard.edu/~lurie/papers/DAG-XI.pdf}{pdf})

\end{itemize}
[[!redirects étale cohomology]]

[[!redirects generalized étale cohomology]] [[!redirects generalized étale cohomologies]]

[[!redirects generalized etale cohomology]] [[!redirects generalized etale cohomologies]]



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