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\section*{Introduction to Étale 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]]

This page collects links related to the textbook

\begin{itemize}%
\item [[Günter Tamme]],

\emph{Introduction to \'E{}tale Cohomology}

1994



\end{itemize}
on [[étale cohomology]].

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

\noindent\hyperlink{chapter_0_preliminaries}{Chapter 0 Preliminaries}\dotfill \pageref*{chapter_0_preliminaries} \linebreak
\noindent\hyperlink{1_abelian_categories}{1. Abelian categories}\dotfill \pageref*{1_abelian_categories} \linebreak
\noindent\hyperlink{2_homological_algebra_in_abelian_categories}{2. Homological algebra in abelian categories}\dotfill \pageref*{2_homological_algebra_in_abelian_categories} \linebreak
\noindent\hyperlink{21_functors}{2.1 $\delta$-Functors}\dotfill \pageref*{21_functors} \linebreak
\noindent\hyperlink{22_derived_functors}{2.2 Derived functors}\dotfill \pageref*{22_derived_functors} \linebreak
\noindent\hyperlink{23_spectral_sequences}{2.3 Spectral sequences}\dotfill \pageref*{23_spectral_sequences} \linebreak
\noindent\hyperlink{3_inductive_limits}{3. Inductive limits}\dotfill \pageref*{3_inductive_limits} \linebreak
\noindent\hyperlink{chapter_i_topologies_and_sheaves}{Chapter I Topologies and sheaves}\dotfill \pageref*{chapter_i_topologies_and_sheaves} \linebreak
\noindent\hyperlink{1_topologies}{1. Topologies}\dotfill \pageref*{1_topologies} \linebreak
\noindent\hyperlink{2_abelian_presheaves_on_topologies}{2. Abelian presheaves on topologies}\dotfill \pageref*{2_abelian_presheaves_on_topologies} \linebreak
\noindent\hyperlink{21_the_category_of_abelian_presheaves}{2.1 The category of abelian presheaves}\dotfill \pageref*{21_the_category_of_abelian_presheaves} \linebreak
\noindent\hyperlink{22_cech_cohomology}{2.2 Cech cohomology}\dotfill \pageref*{22_cech_cohomology} \linebreak
\noindent\hyperlink{23_the_functors__and_}{2.3 The functors $f^p$ and $f_p$}\dotfill \pageref*{23_the_functors__and_} \linebreak
\noindent\hyperlink{3_abelian_sheaves_on_topologies}{3. Abelian sheaves on topologies}\dotfill \pageref*{3_abelian_sheaves_on_topologies} \linebreak
\noindent\hyperlink{31_the_associated_sheaf_of_a_presheaf}{3.1 The associated sheaf of a presheaf}\dotfill \pageref*{31_the_associated_sheaf_of_a_presheaf} \linebreak
\noindent\hyperlink{32_the_category_of_abelian_sheaves}{3.2 The category of abelian sheaves}\dotfill \pageref*{32_the_category_of_abelian_sheaves} \linebreak
\noindent\hyperlink{33_cohomology_of_abelian_sheaves}{3.3 Cohomology of abelian sheaves}\dotfill \pageref*{33_cohomology_of_abelian_sheaves} \linebreak
\noindent\hyperlink{34_the_spectral_sequences_for_cech_cohomology}{3.4 The spectral sequences for Cech cohomology}\dotfill \pageref*{34_the_spectral_sequences_for_cech_cohomology} \linebreak
\noindent\hyperlink{35_flabby_sheaves}{3.5 Flabby sheaves}\dotfill \pageref*{35_flabby_sheaves} \linebreak
\noindent\hyperlink{36_the_functors__and_}{3.6 The functors $f^s$ and $f_s$}\dotfill \pageref*{36_the_functors__and_} \linebreak
\noindent\hyperlink{37_the_leray_spectral_sequence}{3.7 The Leray spectral sequence}\dotfill \pageref*{37_the_leray_spectral_sequence} \linebreak
\noindent\hyperlink{38_localization}{3.8 Localization}\dotfill \pageref*{38_localization} \linebreak
\noindent\hyperlink{39_the_comparison_lemma}{3.9 The comparison lemma}\dotfill \pageref*{39_the_comparison_lemma} \linebreak
\noindent\hyperlink{310_noetherian_topologies}{3.10 Noetherian topologies}\dotfill \pageref*{310_noetherian_topologies} \linebreak
\noindent\hyperlink{311_commutation_of_the_functors__with_pseudofiltered_inductive_limits}{3.11 Commutation of the functors $H^q(U,-)$ with pseudofiltered inductive limits}\dotfill \pageref*{311_commutation_of_the_functors__with_pseudofiltered_inductive_limits} \linebreak
\noindent\hyperlink{chapter_ii_tale_cohomology}{Chapter II \'E{}tale Cohomology}\dotfill \pageref*{chapter_ii_tale_cohomology} \linebreak
\noindent\hyperlink{1_the_tale_site_of_a_scheme}{1. The \'E{}tale site of a scheme}\dotfill \pageref*{1_the_tale_site_of_a_scheme} \linebreak
\noindent\hyperlink{11_tale_morphisms}{1.1 \'E{}tale morphisms}\dotfill \pageref*{11_tale_morphisms} \linebreak
\noindent\hyperlink{12_tale_site}{1.2 \'E{}tale site}\dotfill \pageref*{12_tale_site} \linebreak
\noindent\hyperlink{13_relation_between_tale_and_zariski_cohomology}{1.3 Relation between \'E{}tale and Zariski cohomology}\dotfill \pageref*{13_relation_between_tale_and_zariski_cohomology} \linebreak
\noindent\hyperlink{14_the_functors__and_}{1.4 The functors $f_\ast$ and $f^\ast$}\dotfill \pageref*{14_the_functors__and_} \linebreak
\noindent\hyperlink{15_the_restricted_tale_site}{1.5 The restricted \'E{}tale site}\dotfill \pageref*{15_the_restricted_tale_site} \linebreak
\noindent\hyperlink{2_the_case_}{2. The case $X = Spec(k)$}\dotfill \pageref*{2_the_case_} \linebreak
\noindent\hyperlink{3_examples_of_tale_sheaves}{3. Examples of \'e{}tale sheaves}\dotfill \pageref*{3_examples_of_tale_sheaves} \linebreak
\noindent\hyperlink{31_representable_sheaves}{3.1 Representable sheaves}\dotfill \pageref*{31_representable_sheaves} \linebreak
\noindent\hyperlink{32_tale_sheaves_of_modules}{3.2 \'E{}tale sheaves of $\mathcal{O}_X$-modules}\dotfill \pageref*{32_tale_sheaves_of_modules} \linebreak
\noindent\hyperlink{33_big_tale_site}{3.3 Big \'e{}tale site}\dotfill \pageref*{33_big_tale_site} \linebreak
\noindent\hyperlink{4_the_theories_of_artinschreier_and_of_kummer}{4. The theories of Artin-Schreier and of Kummer}\dotfill \pageref*{4_the_theories_of_artinschreier_and_of_kummer} \linebreak
\noindent\hyperlink{41_the_groups_}{4.1 The groups $H^q(X, (\mathbb{G}_a)_X)$}\dotfill \pageref*{41_the_groups_} \linebreak
\noindent\hyperlink{42_the_artinschreier_sequence}{4.2 The Artin-Schreier sequence}\dotfill \pageref*{42_the_artinschreier_sequence} \linebreak
\noindent\hyperlink{43_the_groups_}{4.3 The groups $H^q(X, (\mathbb{G}_m)_X)$}\dotfill \pageref*{43_the_groups_} \linebreak
\noindent\hyperlink{44_the_kummer_sequence}{4.4 The Kummer sequence}\dotfill \pageref*{44_the_kummer_sequence} \linebreak
\noindent\hyperlink{45_the_sheaf_of_divisors_on_}{4.5 The sheaf of divisors on $X_{et}$}\dotfill \pageref*{45_the_sheaf_of_divisors_on_} \linebreak
\noindent\hyperlink{5_stalks_of_tale_sheaves}{5. Stalks of \'e{}tale sheaves}\dotfill \pageref*{5_stalks_of_tale_sheaves} \linebreak
\noindent\hyperlink{6_strict_localizations}{6. Strict localizations}\dotfill \pageref*{6_strict_localizations} \linebreak
\noindent\hyperlink{61_henselian_rings_and_strictly_local_rings}{6.1 Henselian rings and strictly local rings}\dotfill \pageref*{61_henselian_rings_and_strictly_local_rings} \linebreak
\noindent\hyperlink{62_strict_localization_of_a_scheme}{6.2 Strict localization of a scheme}\dotfill \pageref*{62_strict_localization_of_a_scheme} \linebreak
\noindent\hyperlink{63_tale_cohomology_on_projective_limits_of_schemes}{6.3 \'E{}tale cohomology on projective limits of schemes}\dotfill \pageref*{63_tale_cohomology_on_projective_limits_of_schemes} \linebreak
\noindent\hyperlink{7_the_artin_spectral_sequence}{7. The Artin spectral sequence}\dotfill \pageref*{7_the_artin_spectral_sequence} \linebreak
\noindent\hyperlink{8_the_decomposition_theorem_relative_cohomology}{8. The decomposition theorem. Relative cohomology}\dotfill \pageref*{8_the_decomposition_theorem_relative_cohomology} \linebreak
\noindent\hyperlink{9_torsion_sheaves_locally_constant_sheaves_constructible_sheaves}{9. Torsion sheaves, locally constant sheaves, constructible sheaves}\dotfill \pageref*{9_torsion_sheaves_locally_constant_sheaves_constructible_sheaves} \linebreak
\noindent\hyperlink{10_tale_cohomology_of_curves}{10. \'E{}tale cohomology of curves}\dotfill \pageref*{10_tale_cohomology_of_curves} \linebreak
\noindent\hyperlink{11_general_theorems_in_tale_cohomology_theory}{11. General theorems in \'e{}tale cohomology theory}\dotfill \pageref*{11_general_theorems_in_tale_cohomology_theory} \linebreak


\hypertarget{chapter_0_preliminaries}{}\subsection*{{Chapter 0 Preliminaries}}\label{chapter_0_preliminaries}

\hypertarget{1_abelian_categories}{}\subsubsection*{{1. Abelian categories}}\label{1_abelian_categories}

\begin{itemize}%
\item [[abelian category]]

\end{itemize}
\hypertarget{2_homological_algebra_in_abelian_categories}{}\subsubsection*{{2. Homological algebra in abelian categories}}\label{2_homological_algebra_in_abelian_categories}

\begin{itemize}%
\item [[homological algebra]]

\end{itemize}
\hypertarget{21_functors}{}\paragraph*{{2.1 $\delta$-Functors}}\label{21_functors}

\begin{itemize}%
\item [[delta-functor]]

\end{itemize}
\hypertarget{22_derived_functors}{}\paragraph*{{2.2 Derived functors}}\label{22_derived_functors}

\begin{itemize}%
\item [[derived functor]]

\end{itemize}
\hypertarget{23_spectral_sequences}{}\paragraph*{{2.3 Spectral sequences}}\label{23_spectral_sequences}

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


\item [[edge morphism]]


\item (2.3.5) [[Grothendieck spectral sequence]]



\end{itemize}
\hypertarget{3_inductive_limits}{}\subsubsection*{{3. Inductive limits}}\label{3_inductive_limits}

\hypertarget{chapter_i_topologies_and_sheaves}{}\subsection*{{Chapter I Topologies and sheaves}}\label{chapter_i_topologies_and_sheaves}

\hypertarget{1_topologies}{}\subsubsection*{{1. Topologies}}\label{1_topologies}

\begin{itemize}%
\item [[Grothendieck topology]]

\end{itemize}
\hypertarget{2_abelian_presheaves_on_topologies}{}\subsubsection*{{2. Abelian presheaves on topologies}}\label{2_abelian_presheaves_on_topologies}

\hypertarget{21_the_category_of_abelian_presheaves}{}\paragraph*{{2.1 The category of abelian presheaves}}\label{21_the_category_of_abelian_presheaves}

\begin{itemize}%
\item [[group object]]

\end{itemize}
\hypertarget{22_cech_cohomology}{}\paragraph*{{2.2 Cech cohomology}}\label{22_cech_cohomology}

\hypertarget{23_the_functors__and_}{}\paragraph*{{2.3 The functors $f^p$ and $f_p$}}\label{23_the_functors__and_}

\begin{itemize}%
\item [[morphism of sites]]


\item [[direct image]], [[inverse image]] (here: on [[presheaves]])



\end{itemize}
\hypertarget{3_abelian_sheaves_on_topologies}{}\subsubsection*{{3. Abelian sheaves on topologies}}\label{3_abelian_sheaves_on_topologies}

\hypertarget{31_the_associated_sheaf_of_a_presheaf}{}\paragraph*{{3.1 The associated sheaf of a presheaf}}\label{31_the_associated_sheaf_of_a_presheaf}

\begin{itemize}%
\item [[sheafification]]

\end{itemize}
\hypertarget{32_the_category_of_abelian_sheaves}{}\paragraph*{{3.2 The category of abelian sheaves}}\label{32_the_category_of_abelian_sheaves}

\begin{itemize}%
\item [[sheaf of abelian groups]]

\end{itemize}
\hypertarget{33_cohomology_of_abelian_sheaves}{}\paragraph*{{3.3 Cohomology of abelian sheaves}}\label{33_cohomology_of_abelian_sheaves}

\begin{itemize}%
\item [[abelian sheaf cohomology]]

\end{itemize}
\hypertarget{34_the_spectral_sequences_for_cech_cohomology}{}\paragraph*{{3.4 The spectral sequences for Cech cohomology}}\label{34_the_spectral_sequences_for_cech_cohomology}

\begin{itemize}%
\item [[Cech cohomology]]

\end{itemize}
\hypertarget{35_flabby_sheaves}{}\paragraph*{{3.5 Flabby sheaves}}\label{35_flabby_sheaves}

\begin{itemize}%
\item [[flabby sheaf]]

\end{itemize}
\hypertarget{36_the_functors__and_}{}\paragraph*{{3.6 The functors $f^s$ and $f_s$}}\label{36_the_functors__and_}

\begin{itemize}%
\item [[morphism of sites]]


\item [[direct image]], [[inverse image]]



\end{itemize}
\hypertarget{37_the_leray_spectral_sequence}{}\paragraph*{{3.7 The Leray spectral sequence}}\label{37_the_leray_spectral_sequence}

\begin{itemize}%
\item [[Leray spectral sequence]]

\end{itemize}
\hypertarget{38_localization}{}\paragraph*{{3.8 Localization}}\label{38_localization}

\begin{itemize}%
\item [[localization]]

\end{itemize}
\hypertarget{39_the_comparison_lemma}{}\paragraph*{{3.9 The comparison lemma}}\label{39_the_comparison_lemma}

\hypertarget{310_noetherian_topologies}{}\paragraph*{{3.10 Noetherian topologies}}\label{310_noetherian_topologies}

\hypertarget{311_commutation_of_the_functors__with_pseudofiltered_inductive_limits}{}\paragraph*{{3.11 Commutation of the functors $H^q(U,-)$ with pseudofiltered inductive limits}}\label{311_commutation_of_the_functors__with_pseudofiltered_inductive_limits}

\hypertarget{chapter_ii_tale_cohomology}{}\subsection*{{Chapter II \'E{}tale Cohomology}}\label{chapter_ii_tale_cohomology}

\hypertarget{1_the_tale_site_of_a_scheme}{}\subsubsection*{{1. The \'E{}tale site of a scheme}}\label{1_the_tale_site_of_a_scheme}

\hypertarget{11_tale_morphisms}{}\paragraph*{{1.1 \'E{}tale morphisms}}\label{11_tale_morphisms}

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

\end{itemize}
\hypertarget{12_tale_site}{}\paragraph*{{1.2 \'E{}tale site}}\label{12_tale_site}

\begin{itemize}%
\item [[étale site]]

\end{itemize}
\hypertarget{13_relation_between_tale_and_zariski_cohomology}{}\paragraph*{{1.3 Relation between \'E{}tale and Zariski cohomology}}\label{13_relation_between_tale_and_zariski_cohomology}

\begin{itemize}%
\item \href{etale+cohomology#RelationZariskiEtaleCohomology}{relation to Zariski cohomology}

\end{itemize}
\hypertarget{14_the_functors__and_}{}\paragraph*{{1.4 The functors $f_\ast$ and $f^\ast$}}\label{14_the_functors__and_}

\begin{itemize}%
\item [[direct image]], [[inverse image]]


\item [[base change]]



\end{itemize}
\hypertarget{15_the_restricted_tale_site}{}\paragraph*{{1.5 The restricted \'E{}tale site}}\label{15_the_restricted_tale_site}

\hypertarget{2_the_case_}{}\subsubsection*{{2. The case $X = Spec(k)$}}\label{2_the_case_}

\begin{itemize}%
\item [[Galois group]]

\end{itemize}
\hypertarget{3_examples_of_tale_sheaves}{}\subsubsection*{{3. Examples of \'e{}tale sheaves}}\label{3_examples_of_tale_sheaves}

\hypertarget{31_representable_sheaves}{}\paragraph*{{3.1 Representable sheaves}}\label{31_representable_sheaves}

\begin{itemize}%
\item [[sheaf]], [[descent]]


\item [[étale topos]]


\item [[representable presheaf]]


\item [[canonical topology]]


\item [[additive group]], [[multiplicative group]], [[roots of unity]]



\end{itemize}
\hypertarget{32_tale_sheaves_of_modules}{}\paragraph*{{3.2 \'E{}tale sheaves of $\mathcal{O}_X$-modules}}\label{32_tale_sheaves_of_modules}

\begin{itemize}%
\item [[sheaf of coherent modules]]


\item [[constructible sheaf]]



\end{itemize}
\hypertarget{33_big_tale_site}{}\paragraph*{{3.3 Big \'e{}tale site}}\label{33_big_tale_site}

\begin{itemize}%
\item [[big site]]

\end{itemize}
\hypertarget{4_the_theories_of_artinschreier_and_of_kummer}{}\subsubsection*{{4. The theories of Artin-Schreier and of Kummer}}\label{4_the_theories_of_artinschreier_and_of_kummer}

\hypertarget{41_the_groups_}{}\paragraph*{{4.1 The groups $H^q(X, (\mathbb{G}_a)_X)$}}\label{41_the_groups_}

\hypertarget{42_the_artinschreier_sequence}{}\paragraph*{{4.2 The Artin-Schreier sequence}}\label{42_the_artinschreier_sequence}

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


\item [[Artin-Schreier sequence]]



\end{itemize}
\hypertarget{43_the_groups_}{}\paragraph*{{4.3 The groups $H^q(X, (\mathbb{G}_m)_X)$}}\label{43_the_groups_}

\begin{itemize}%
\item [[Hilbert's theorem 90]]


\item [[Azumaya algebra]], [[Brauer group]]



\end{itemize}
\hypertarget{44_the_kummer_sequence}{}\paragraph*{{4.4 The Kummer sequence}}\label{44_the_kummer_sequence}

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

\end{itemize}
\hypertarget{45_the_sheaf_of_divisors_on_}{}\paragraph*{{4.5 The sheaf of divisors on $X_{et}$}}\label{45_the_sheaf_of_divisors_on_}

\hypertarget{5_stalks_of_tale_sheaves}{}\subsubsection*{{5. Stalks of \'e{}tale sheaves}}\label{5_stalks_of_tale_sheaves}

\begin{itemize}%
\item [[stalk]]

\end{itemize}
\hypertarget{6_strict_localizations}{}\subsubsection*{{6. Strict localizations}}\label{6_strict_localizations}

\hypertarget{61_henselian_rings_and_strictly_local_rings}{}\paragraph*{{6.1 Henselian rings and strictly local rings}}\label{61_henselian_rings_and_strictly_local_rings}

\begin{itemize}%
\item [[Henselian ring]]


\item [[strictly local ring]]



\end{itemize}
\hypertarget{62_strict_localization_of_a_scheme}{}\subsubsection*{{6.2 Strict localization of a scheme}}\label{62_strict_localization_of_a_scheme}

\hypertarget{63_tale_cohomology_on_projective_limits_of_schemes}{}\subsubsection*{{6.3 \'E{}tale cohomology on projective limits of schemes}}\label{63_tale_cohomology_on_projective_limits_of_schemes}

\hypertarget{7_the_artin_spectral_sequence}{}\subsubsection*{{7. The Artin spectral sequence}}\label{7_the_artin_spectral_sequence}

\begin{itemize}%
\item [[Artin spectral sequence]]

\end{itemize}
\hypertarget{8_the_decomposition_theorem_relative_cohomology}{}\subsubsection*{{8. The decomposition theorem. Relative cohomology}}\label{8_the_decomposition_theorem_relative_cohomology}

\hypertarget{9_torsion_sheaves_locally_constant_sheaves_constructible_sheaves}{}\subsubsection*{{9. Torsion sheaves, locally constant sheaves, constructible sheaves}}\label{9_torsion_sheaves_locally_constant_sheaves_constructible_sheaves}

\begin{itemize}%
\item [[torsion sheaf]]


\item [[constructible sheaf]]



\end{itemize}
\hypertarget{10_tale_cohomology_of_curves}{}\subsubsection*{{10. \'E{}tale cohomology of curves}}\label{10_tale_cohomology_of_curves}

\hypertarget{11_general_theorems_in_tale_cohomology_theory}{}\subsubsection*{{11. General theorems in \'e{}tale cohomology theory}}\label{11_general_theorems_in_tale_cohomology_theory}

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

\end{itemize}
category: reference



\end{document}