\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{Lectures on 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

\begin{itemize}%
\item [[James Milne]],

\emph{Lectures on \'E{}tale Cohomology}

(\href{http://www.jmilne.org/math/CourseNotes/lec.html}{html}, \href{http://www.jmilne.org/math/CourseNotes/LEC.pdf}{pdf})



\end{itemize}
based on the textbook

\begin{itemize}%
\item \emph{\'E{}tale Cohomology},

Princeton Mathematical Series 33, 1980. xiii+323 pp.



\end{itemize}
on [[étale cohomology]] and the [[proof]] of the [[Weil conjectures]].

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

\noindent\hyperlink{i_basic_theory}{\textbf{I} Basic theory}\dotfill \pageref*{i_basic_theory} \linebreak
\noindent\hyperlink{1_introduction}{1. Introduction}\dotfill \pageref*{1_introduction} \linebreak
\noindent\hyperlink{2_tale_morphisms}{2. \'E{}tale morphisms}\dotfill \pageref*{2_tale_morphisms} \linebreak
\noindent\hyperlink{3_the_tale_fundamental_group}{3. The \'e{}tale fundamental group}\dotfill \pageref*{3_the_tale_fundamental_group} \linebreak
\noindent\hyperlink{4_the_local_ring_for_the_tale_topology}{4. The local ring for the \'e{}tale topology}\dotfill \pageref*{4_the_local_ring_for_the_tale_topology} \linebreak
\noindent\hyperlink{5_sites}{5. Sites}\dotfill \pageref*{5_sites} \linebreak
\noindent\hyperlink{6_sheaves_for_the_tale_topology}{6. Sheaves for the \'e{}tale topology}\dotfill \pageref*{6_sheaves_for_the_tale_topology} \linebreak
\noindent\hyperlink{7_the_category_of_sheaves_on_}{7. The category of sheaves on $X_{et}$}\dotfill \pageref*{7_the_category_of_sheaves_on_} \linebreak
\noindent\hyperlink{8_direct_and_inverse_image_sheaves}{8. Direct and inverse image sheaves}\dotfill \pageref*{8_direct_and_inverse_image_sheaves} \linebreak
\noindent\hyperlink{9_cohomology_definition_and_basic_properties}{9. Cohomology: Definition and basic properties}\dotfill \pageref*{9_cohomology_definition_and_basic_properties} \linebreak
\noindent\hyperlink{10_cech_cohomology}{10. Cech cohomology}\dotfill \pageref*{10_cech_cohomology} \linebreak
\noindent\hyperlink{11_principal_homogeneous_spaces_and_}{11. Principal homogeneous spaces and $H^i$}\dotfill \pageref*{11_principal_homogeneous_spaces_and_} \linebreak
\noindent\hyperlink{12_higher_direct_images_the_leray_spectral_sequence}{12. Higher direct images; the Leray spectral sequence}\dotfill \pageref*{12_higher_direct_images_the_leray_spectral_sequence} \linebreak
\noindent\hyperlink{13_the_weildivisor_exact_sequence_and_the_cohomology_of_}{13. The Weil-divisor exact sequence and the cohomology of $\mathbb{G}_m$}\dotfill \pageref*{13_the_weildivisor_exact_sequence_and_the_cohomology_of_} \linebreak
\noindent\hyperlink{14_the_cohomology_of_curves}{14. The cohomology of curves}\dotfill \pageref*{14_the_cohomology_of_curves} \linebreak
\noindent\hyperlink{15_cohomological_dimension}{15. Cohomological dimension}\dotfill \pageref*{15_cohomological_dimension} \linebreak
\noindent\hyperlink{16_purity_the_gysin_sequence}{16. Purity; the Gysin sequence}\dotfill \pageref*{16_purity_the_gysin_sequence} \linebreak
\noindent\hyperlink{17_the_proper_base_change_theorem}{17. The proper base change theorem}\dotfill \pageref*{17_the_proper_base_change_theorem} \linebreak
\noindent\hyperlink{18_cohomology_groups_with_compact_support}{18. Cohomology groups with compact support}\dotfill \pageref*{18_cohomology_groups_with_compact_support} \linebreak
\noindent\hyperlink{19_finiteness_theorem_sheaves_of_modules}{19. Finiteness theorem; Sheaves of $\mathbb{Z}_l$-modules}\dotfill \pageref*{19_finiteness_theorem_sheaves_of_modules} \linebreak
\noindent\hyperlink{20_the_smooth_base_change_theorem}{20. The smooth base change theorem}\dotfill \pageref*{20_the_smooth_base_change_theorem} \linebreak
\noindent\hyperlink{21_the_comparison_theorem}{21. The comparison theorem}\dotfill \pageref*{21_the_comparison_theorem} \linebreak
\noindent\hyperlink{22_the_knneth_formula}{22. The K\"u{}nneth formula}\dotfill \pageref*{22_the_knneth_formula} \linebreak
\noindent\hyperlink{23_the_cycle_map_chern_classes}{23. The cycle map; Chern classes}\dotfill \pageref*{23_the_cycle_map_chern_classes} \linebreak
\noindent\hyperlink{24_poincar_duality}{24. Poincar\'e{} duality}\dotfill \pageref*{24_poincar_duality} \linebreak
\noindent\hyperlink{25_lefschetz_fixedpoint_formula}{25. Lefschetz fixed-point formula}\dotfill \pageref*{25_lefschetz_fixedpoint_formula} \linebreak
\noindent\hyperlink{ii_proof_of_the_weil_conjectures}{\textbf{II} Proof of the Weil conjectures}\dotfill \pageref*{ii_proof_of_the_weil_conjectures} \linebreak
\noindent\hyperlink{26_the_weil_conjecture}{26. The Weil conjecture}\dotfill \pageref*{26_the_weil_conjecture} \linebreak
\noindent\hyperlink{27_proof_of_the_weil_conjectures_except_for_the_riemann_hypothesis}{27. Proof of the Weil conjectures, except for the Riemann hypothesis}\dotfill \pageref*{27_proof_of_the_weil_conjectures_except_for_the_riemann_hypothesis} \linebreak
\noindent\hyperlink{28_preliminary_reductions}{28. Preliminary reductions}\dotfill \pageref*{28_preliminary_reductions} \linebreak
\noindent\hyperlink{29_the_lefschetz_fixed_point_formula_for_nonconstant_sheaves}{29. The Lefschetz fixed point formula for non-constant sheaves}\dotfill \pageref*{29_the_lefschetz_fixed_point_formula_for_nonconstant_sheaves} \linebreak
\noindent\hyperlink{30_the_main_lemma}{30. The main lemma}\dotfill \pageref*{30_the_main_lemma} \linebreak
\noindent\hyperlink{31_the_geometry_of_lefschetz_pencils}{31. The geometry of Lefschetz pencils}\dotfill \pageref*{31_the_geometry_of_lefschetz_pencils} \linebreak
\noindent\hyperlink{32_the_cohomology_of_lefschetz_pencils}{32. The cohomology of Lefschetz pencils}\dotfill \pageref*{32_the_cohomology_of_lefschetz_pencils} \linebreak
\noindent\hyperlink{33_completion_of_the_proof_of_the_weil_conjecture}{33. Completion of the proof of the Weil conjecture}\dotfill \pageref*{33_completion_of_the_proof_of_the_weil_conjecture} \linebreak
\noindent\hyperlink{34_the_geometry_of_estimates}{34. The geometry of estimates}\dotfill \pageref*{34_the_geometry_of_estimates} \linebreak


\hypertarget{i_basic_theory}{}\subsection*{{\textbf{I} Basic theory}}\label{i_basic_theory}

\hypertarget{1_introduction}{}\subsubsection*{{1. Introduction}}\label{1_introduction}

\hypertarget{2_tale_morphisms}{}\subsubsection*{{2. \'E{}tale morphisms}}\label{2_tale_morphisms}

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


\item [[étale morphism of schemes]]


\item [[tangent cone]]


\item [[unramified morphism of schemes]]



\end{itemize}
\hypertarget{3_the_tale_fundamental_group}{}\subsubsection*{{3. The \'e{}tale fundamental group}}\label{3_the_tale_fundamental_group}

\begin{itemize}%
\item [[étale fundamental group]]


\item [[étale homotopy type]]



\end{itemize}
\hypertarget{4_the_local_ring_for_the_tale_topology}{}\subsubsection*{{4. The local ring for the \'e{}tale topology}}\label{4_the_local_ring_for_the_tale_topology}

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


\item [[local ring]]


\item [[locally ringed topological space]]


\item [[Henselian ring]]


\item [[geometric point]]



\end{itemize}
\hypertarget{5_sites}{}\subsubsection*{{5. Sites}}\label{5_sites}

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


\item [[presheaf]]


\item [[sheaf]]



\end{itemize}
\hypertarget{6_sheaves_for_the_tale_topology}{}\subsubsection*{{6. Sheaves for the \'e{}tale topology}}\label{6_sheaves_for_the_tale_topology}

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


\item [[descent]]


\item [[structure sheaf]]


\item [[sheaf of coherent modules]]


\item [[stalk]]


\item [[skyscraper sheaf]]


\item [[locally constant sheaf]]



\end{itemize}
\hypertarget{7_the_category_of_sheaves_on_}{}\subsubsection*{{7. The category of sheaves on $X_{et}$}}\label{7_the_category_of_sheaves_on_}

\begin{itemize}%
\item [[additive and abelian categories]]


\item [[category of presheaves]]


\item [[category of sheaves]]


\item [[sheafification]]


\item [[algebraic space]]


\item [[étale topos]]



\end{itemize}
\hypertarget{8_direct_and_inverse_image_sheaves}{}\subsubsection*{{8. Direct and inverse image sheaves}}\label{8_direct_and_inverse_image_sheaves}

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


\item [[inverse image]]


\item [[injective module]]


\item [[enough injectives]]



\end{itemize}
\hypertarget{9_cohomology_definition_and_basic_properties}{}\subsubsection*{{9. Cohomology: Definition and basic properties}}\label{9_cohomology_definition_and_basic_properties}

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


\item [[abelian sheaf cohomology]]


\item [[left exact functor]]


\item [[global section functor]]


\item [[derived functor]]


\item [[Ext]]


\item [[excision]]



\end{itemize}
\hypertarget{10_cech_cohomology}{}\subsubsection*{{10. Cech cohomology}}\label{10_cech_cohomology}

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


\item [[Grothendieck spectral sequence]]



\end{itemize}
\hypertarget{11_principal_homogeneous_spaces_and_}{}\subsubsection*{{11. Principal homogeneous spaces and $H^i$}}\label{11_principal_homogeneous_spaces_and_}

\begin{itemize}%
\item [[principal homogeneous space]], [[torsor]], [[principal bundle]]


\item [[nonabelian cohomology]]


\item [[gerbe]]



\end{itemize}
\hypertarget{12_higher_direct_images_the_leray_spectral_sequence}{}\subsubsection*{{12. Higher direct images; the Leray spectral sequence}}\label{12_higher_direct_images_the_leray_spectral_sequence}

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


\item [[Leray spectral sequence]]



\end{itemize}
\hypertarget{13_the_weildivisor_exact_sequence_and_the_cohomology_of_}{}\subsubsection*{{13. The Weil-divisor exact sequence and the cohomology of $\mathbb{G}_m$}}\label{13_the_weildivisor_exact_sequence_and_the_cohomology_of_}

\begin{itemize}%
\item [[Weil divisor]]


\item [[function field]]


\item [[Brauer group]]



\end{itemize}
\hypertarget{14_the_cohomology_of_curves}{}\subsubsection*{{14. The cohomology of curves}}\label{14_the_cohomology_of_curves}

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


\item [[Poincaré duality]]


\item [[Hochschild-Serre spectral sequence]]



\end{itemize}
\hypertarget{15_cohomological_dimension}{}\subsubsection*{{15. Cohomological dimension}}\label{15_cohomological_dimension}

\begin{itemize}%
\item [[cohomological dimension]]

\end{itemize}
\hypertarget{16_purity_the_gysin_sequence}{}\subsubsection*{{16. Purity; the Gysin sequence}}\label{16_purity_the_gysin_sequence}

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

\end{itemize}
\hypertarget{17_the_proper_base_change_theorem}{}\subsubsection*{{17. The proper base change theorem}}\label{17_the_proper_base_change_theorem}

\begin{itemize}%
\item [[proper morphism of schemes]]


\item [[constructible sheaf]]


\item [[Beck-Chevalley condition]]


\item [[proper base change theorem]]



\end{itemize}
\hypertarget{18_cohomology_groups_with_compact_support}{}\subsubsection*{{18. Cohomology groups with compact support}}\label{18_cohomology_groups_with_compact_support}

\begin{itemize}%
\item [[cohomology with compact support]]

\end{itemize}
\hypertarget{19_finiteness_theorem_sheaves_of_modules}{}\subsubsection*{{19. Finiteness theorem; Sheaves of $\mathbb{Z}_l$-modules}}\label{19_finiteness_theorem_sheaves_of_modules}

\begin{itemize}%
\item [[ℓ-adic cohomology]]

\end{itemize}
\hypertarget{20_the_smooth_base_change_theorem}{}\subsubsection*{{20. The smooth base change theorem}}\label{20_the_smooth_base_change_theorem}

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

\end{itemize}
\hypertarget{21_the_comparison_theorem}{}\subsubsection*{{21. The comparison theorem}}\label{21_the_comparison_theorem}

\begin{itemize}%
\item [[complex analytic topology]]


\item [[Riemann existence theorem]]


\item [[comparison theorem (étale cohomology)]]



\end{itemize}
\hypertarget{22_the_knneth_formula}{}\subsubsection*{{22. The K\"u{}nneth formula}}\label{22_the_knneth_formula}

\begin{itemize}%
\item [[Künneth theorem]]

\end{itemize}
\hypertarget{23_the_cycle_map_chern_classes}{}\subsubsection*{{23. The cycle map; Chern classes}}\label{23_the_cycle_map_chern_classes}

\begin{itemize}%
\item [[Chern class]]

\end{itemize}
\hypertarget{24_poincar_duality}{}\subsubsection*{{24. Poincar\'e{} duality}}\label{24_poincar_duality}

\begin{itemize}%
\item [[Poincaré duality]]


\item [[Verdier duality]]



\end{itemize}
\hypertarget{25_lefschetz_fixedpoint_formula}{}\subsubsection*{{25. Lefschetz fixed-point formula}}\label{25_lefschetz_fixedpoint_formula}

\begin{itemize}%
\item [[Lefschetz fixed-point formula]]

\end{itemize}
\hypertarget{ii_proof_of_the_weil_conjectures}{}\subsection*{{\textbf{II} Proof of the Weil conjectures}}\label{ii_proof_of_the_weil_conjectures}

\hypertarget{26_the_weil_conjecture}{}\subsubsection*{{26. The Weil conjecture}}\label{26_the_weil_conjecture}

\begin{itemize}%
\item [[Weil conjecture]]

\end{itemize}
\hypertarget{27_proof_of_the_weil_conjectures_except_for_the_riemann_hypothesis}{}\subsubsection*{{27. Proof of the Weil conjectures, except for the Riemann hypothesis}}\label{27_proof_of_the_weil_conjectures_except_for_the_riemann_hypothesis}

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

\end{itemize}
\hypertarget{28_preliminary_reductions}{}\subsubsection*{{28. Preliminary reductions}}\label{28_preliminary_reductions}

\hypertarget{29_the_lefschetz_fixed_point_formula_for_nonconstant_sheaves}{}\subsubsection*{{29. The Lefschetz fixed point formula for non-constant sheaves}}\label{29_the_lefschetz_fixed_point_formula_for_nonconstant_sheaves}

\hypertarget{30_the_main_lemma}{}\subsubsection*{{30. The main lemma}}\label{30_the_main_lemma}

\hypertarget{31_the_geometry_of_lefschetz_pencils}{}\subsubsection*{{31. The geometry of Lefschetz pencils}}\label{31_the_geometry_of_lefschetz_pencils}

\hypertarget{32_the_cohomology_of_lefschetz_pencils}{}\subsubsection*{{32. The cohomology of Lefschetz pencils}}\label{32_the_cohomology_of_lefschetz_pencils}

\hypertarget{33_completion_of_the_proof_of_the_weil_conjecture}{}\subsubsection*{{33. Completion of the proof of the Weil conjecture}}\label{33_completion_of_the_proof_of_the_weil_conjecture}

\hypertarget{34_the_geometry_of_estimates}{}\subsubsection*{{34. The geometry of estimates}}\label{34_the_geometry_of_estimates}

category: reference

[[!redirects Lectures on Etale Cohomology]]

[[!redirects Étale Cohomology]] [[!redirects Etale Cohomology]]



\end{document}