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\input{preamble}
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\begin{document}
\title{\'Etale Cohomology}
\maketitle
\phantomsection
\label{section-phantom}
\tableofcontents
\section{Introduction}
\label{section-introduction}
\noindent
These are the notes of a course on \'etale cohomology taught by Johan de Jong
at Columbia University in the Fall of 2009. The original note takers were
Thibaut Pugin, Zachary Maddock and Min Lee. Over time we will add references
to background material in the rest of the Stacks project and provide rigorous
proofs of all the statements.
\section{Which sections to skip on a first reading?}
\label{section-skip}
\noindent
We want to use the material in this chapter for the development of
theory related to algebraic spaces, Deligne-Mumford stacks, algebraic stacks,
etc. Thus we have added some pretty technical material to the original
exposition of \'etale cohomology for schemes. The reader can recognize this
material by the frequency of the word ``topos'', or by discussions related
to set theory, or by proofs dealing with very general properties of morphisms
of schemes. Some of these discussions can be skipped on a first reading.
\medskip\noindent
In particular, we suggest that the reader skip the following sections:
\begin{enumerate}
\item Comparing big and small topoi,
Section \ref{section-compare}.
\item Recovering morphisms,
Section \ref{section-morphisms}.
\item Push and pull,
Section \ref{section-monomorphisms}.
\item Property (A),
Section \ref{section-A}.
\item Property (B),
Section \ref{section-B}.
\item Property (C),
Section \ref{section-C}.
\item Topological invariance of the small \'etale site,
Section \ref{section-topological-invariance}.
\item Integral universally injective morphisms,
Section \ref{section-integral-universally-injective}.
\item Big sites and pushforward,
Section \ref{section-big}.
\item Exactness of big lower shriek,
Section \ref{section-exactness-lower-shriek}.
\end{enumerate}
Besides these sections there are some sporadic results that may be skipped
that the reader can recognize by the keywords given above.
%9.08.09
\section{Prologue}
\label{section-prologue}
\noindent
These lectures are about another cohomology theory. The first thing to remark
is that the Zariski topology is not entirely satisfactory. One of the main
reasons that it fails to give the results that we would want is that if $X$ is
a complex variety and $\mathcal{F}$ is a constant sheaf then
$$
H^i(X, \mathcal{F}) = 0, \quad \text{ for all } i > 0.
$$
The reason for that is the following. In an irreducible scheme (a variety in
particular), any two nonempty open subsets meet, and so the restriction
mappings of a constant sheaf are surjective. We say that the sheaf is
{\it flasque}. In this case, all higher {\v C}ech cohomology groups vanish, and
so do all higher Zariski cohomology groups. In other words, there are ``not
enough'' open sets in the Zariski topology to detect this higher cohomology.
\medskip\noindent
On the other hand, if $X$ is a smooth projective complex variety, then
$$
H_{Betti}^{2 \dim X}(X (\mathbf{C}), \Lambda) = \Lambda \quad \text{ for }
\Lambda = \mathbf{Z}, \ \mathbf{Z}/n\mathbf{Z},
$$
where $X(\mathbf{C})$ means the set of complex points of $X$. This is a feature
that would be nice to replicate in algebraic geometry. In positive
characteristic in particular.
\section{The \'etale topology}
\label{section-etale-topology}
\noindent
It is very hard to simply ``add'' extra open sets to refine the Zariski
topology. One efficient way to define a topology is to consider not only open
sets, but also some schemes that lie over them. To define the \'etale topology,
one considers all morphisms $\varphi : U \to X$ which are \'etale. If
$X$ is a smooth projective variety over $\mathbf{C}$, then this means
\begin{enumerate}
\item $U$ is a disjoint union of smooth varieties, and
\item $\varphi$ is (analytically) locally an isomorphism.
\end{enumerate}
The word ``analytically'' refers to the usual (transcendental) topology over
$\mathbf{C}$. So the second condition means that the derivative of $\varphi$
has full rank everywhere (and in particular all the components of $U$
have the same dimension as $X$).
\medskip\noindent
A double cover -- loosely defined as a finite degree $2$ map between varieties
-- for example
$$
\Spec(\mathbf{C}[t])
\longrightarrow
\Spec(\mathbf{C}[t]),
\quad t \longmapsto t^2
$$
will not be an \'etale morphism if it has a fibre consisting of a single point.
In the example this happens when $t = 0$. For a finite map between varieties
over $\mathbf{C}$ to be \'etale all the fibers should have the same number of
points. Removing the point $t = 0$ from the source of the map in the example
will make the morphism \'etale. But we can remove other points from the source
of the morphism also, and the morphism will still be \'etale. To consider the
\'etale topology, we have to look at all such morphisms. Unlike the Zariski
topology, these need not be merely be open subsets of $X$, even though their
images always are.
\begin{definition}
\label{definition-etale-covering-initial}
A family of morphisms $\{ \varphi_i : U_i \to X\}_{i \in I}$ is
called an {\it \'etale covering} if each $\varphi_i$ is an \'etale morphism
and their images cover $X$, i.e.,
$X = \bigcup_{i \in I} \varphi_i(U_i)$.
\end{definition}
\noindent
This ``defines'' the \'etale topology. In other words, we can now say what the
sheaves are. An {\it \'etale sheaf} $\mathcal{F}$ of sets
(resp.\ abelian groups, vector spaces, etc) on $X$ is the data:
\begin{enumerate}
\item for each \'etale morphism $\varphi : U \to X$ a set
(resp.\ abelian group, vector space, etc) $\mathcal{F}(U)$,
\item for each pair $U, \ U'$ of \'etale schemes over $X$,
and each morphism $U \to U'$ over $X$ (which is
automatically \'etale) a restriction map
$\rho^{U'}_U : \mathcal{F}(U') \to \mathcal{F}(U)$
\end{enumerate}
These data have to satisfy the condition that $\rho^U_U = \text{id}$
in case of the identity morphism $U \to U$
and that $\rho^{U'}_U \circ \rho^{U''}_{U'} = \rho^{U''}_U$
when we have morphisms $U \to U' \to U''$ of schemes \'etale over $X$
as well as the following {\it sheaf axiom}:
\begin{itemize}
\item[$(*)$] for every \'etale covering $\{ \varphi_i : U_i \to U\}_{i \in
I}$, the diagram
$$
\xymatrix{
\emptyset \ar[r] &
\mathcal{F} (U) \ar[r] &
\Pi_{i \in I} \mathcal{F} (U_i) \ar@<1ex>[r] \ar@<-1ex>[r] &
\Pi_{i, j \in I} \mathcal{F} (U_i \times_U U_j)
}
$$
is exact in the category of sets (resp.\ abelian groups, vector spaces, etc).
\end{itemize}
\begin{remark}
\label{remark-i-is-j}
In the last statement, it is essential not to forget the case where $i = j$
which is in general a highly nontrivial condition (unlike in the Zariski
topology). In fact, frequently important coverings have only one element.
\end{remark}
\noindent
Since the identity is an \'etale morphism, we can compute the global sections
of an \'etale sheaf, and cohomology will simply be the corresponding
right-derived functors. In other words, once more theory has been developed and
statements have been made precise, there will be no obstacle to defining
cohomology.
\section{Feats of the \'etale topology}
\label{section-feats}
\noindent
For a natural number $n \in \mathbf{N} = \{1, 2, 3, 4, \ldots\}$ it is true that
$$
H_\etale^2 (\mathbf{P}^1_\mathbf{C}, \mathbf{Z}/n\mathbf{Z}) =
\mathbf{Z}/n\mathbf{Z}.
$$
More generally, if $X$ is a complex variety, then its \'etale Betti numbers
with coefficients in a finite field agree with the usual Betti numbers of
$X(\mathbf{C})$, i.e.,
$$
\dim_{\mathbf{F}_q} H_\etale^{2i} (X, \mathbf{F}_q) =
\dim_{\mathbf{F}_q} H_{Betti}^{2i} (X(\mathbf{C}), \mathbf{F}_q).
$$
This is extremely satisfactory. However, these equalities only hold for torsion
coefficients, not in general. For integer coefficients, one has
$$
H_\etale^2 (\mathbf{P}^1_\mathbf{C}, \mathbf{Z}) = 0.
$$
By contrast $H_{Betti}^2(\mathbf{P}^1(\mathbf{C}), \mathbf{Z}) = \mathbf{Z}$
as the topological space $\mathbf{P}^1(\mathbf{C})$ is homeomorphic to
a $2$-sphere.
There are ways to get back to nontorsion coefficients from torsion ones by a
limit procedure which we will come to shortly.
\section{A computation}
\label{section-computation}
\noindent
How do we compute the cohomology of $\mathbf{P}^1_\mathbf{C}$ with coefficients
$\Lambda = \mathbf{Z}/n\mathbf{Z}$?
We use {\v C}ech cohomology. A covering of $\mathbf{P}^1_\mathbf{C}$ is given
by the two standard opens $U_0, U_1$, which are both
isomorphic to $\mathbf{A}^1_\mathbf{C}$, and whose intersection is isomorphic
to $\mathbf{A}^1_\mathbf{C} \setminus \{0\} = \mathbf{G}_{m, \mathbf{C}}$.
It turns out that the Mayer-Vietoris sequence holds in \'etale cohomology.
This gives an exact sequence
$$
H_\etale^{i-1}(U_0\cap U_1, \Lambda) \to
H_\etale^i(\mathbf{P}^1_C, \Lambda) \to
H_\etale^i(U_0, \Lambda) \oplus
H_\etale^i(U_1, \Lambda) \to H_\etale^i(U_0\cap U_1,
\Lambda).
$$
To get the answer we expect, we would need to show that the direct sum in the
third term vanishes. In fact, it is true that, as for the usual topology,
$$
H_\etale^q (\mathbf{A}^1_\mathbf{C}, \Lambda) = 0
\quad \text{ for } q \geq 1,
$$
and
$$
H_\etale^q (\mathbf{A}^1_\mathbf{C} \setminus \{0\}, \Lambda) = \left\{
\begin{matrix}
\Lambda & \text{ if }q = 1\text{, and} \\
0 & \text{ for }q \geq 2.
\end{matrix}
\right.
$$
These results are already quite hard (what is an elementary proof?). Let us
explain how we would compute this once the machinery of \'etale cohomology is
at our disposal.
\medskip\noindent
{\bf Higher cohomology.} This is taken care of by the following general
fact: if $X$ is an affine curve over $\mathbf{C}$, then
$$
H_\etale^q (X, \mathbf{Z}/n\mathbf{Z}) = 0 \quad \text{ for } q \geq 2.
$$
This is proved by considering the generic point of the curve and doing some
Galois cohomology. So we only have to worry about the cohomology in degree 1.
\medskip\noindent
{\bf Cohomology in degree 1.} We use the following identifications:
\begin{eqnarray*}
H_\etale^1 (X, \mathbf{Z}/n\mathbf{Z}) = \left\{
\begin{matrix}
\text{sheaves of sets }\mathcal{F}\text{ on the \'etale site }X_\etale
\text{ endowed with an} \\
\text{action }\mathbf{Z}/n\mathbf{Z} \times \mathcal{F} \to \mathcal{F}
\text{ such that }\mathcal{F}\text{ is a }\mathbf{Z}/n\mathbf{Z}\text{-torsor.}
\end{matrix}
\right\}
\Big/ \cong
\\
= \left\{
\begin{matrix}
\text{morphisms }Y \to X\text{ which are finite \'etale together} \\
\text{ with a free }\mathbf{Z}/n\mathbf{Z}\text{ action such that }
X = Y/(\mathbf{Z}/n\mathbf{Z}).
\end{matrix}
\right\}
\Big/ \cong.
\end{eqnarray*}
The first identification is very general (it is true for any cohomology theory
on a site) and has nothing to do with the \'etale topology. The second
identification is a consequence of descent theory. The last set describes a
collection of geometric objects on which we can get our hands.
\medskip\noindent
The curve $\mathbf{A}^1_\mathbf{C}$ has no nontrivial finite \'etale covering
and hence
$H_\etale^1 (\mathbf{A}^1_\mathbf{C}, \mathbf{Z}/n\mathbf{Z}) = 0$.
This can be seen either topologically or by using the argument in the next
paragraph.
\medskip\noindent
Let us describe the finite \'etale coverings
$\varphi : Y \to \mathbf{A}^1_\mathbf{C} \setminus \{0\}$.
It suffices to consider the case where $Y$ is
connected, which we assume. We are going to find out what $Y$ can be
by applying the Riemann-Hurwitz formula (of course this is a bit silly, and
you can go ahead and skip the next section if you like).
Say that this morphism is $n$ to 1, and consider a
projective compactification
$$
\xymatrix{
{Y\ } \ar@{^{(}->}[r] \ar[d]^\varphi &
{\bar Y} \ar[d]^{\bar\varphi} \\
{\mathbf{A}^1_\mathbf{C} \setminus \{0\}} \ar@{^{(}->}[r] &
{\mathbf{P}^1_\mathbf{C}}
}
$$
Even though $\varphi$ is \'etale and does not ramify, $\bar{\varphi}$ may
ramify at 0 and $\infty$. Say that the preimages of 0 are the points $y_1,
\ldots, y_r$ with indices of ramification $e_1, \ldots e_r$, and that the
preimages of $\infty$ are the points $y_1', \ldots, y_s'$ with indices of
ramification $d_1, \ldots d_s$. In particular, $\sum e_i = n = \sum d_j$.
Applying the Riemann-Hurwitz formula, we get
$$
2 g_Y - 2 = -2n + \sum (e_i - 1) + \sum (d_j - 1)
$$
and therefore $g_Y = 0$, $r = s = 1$ and $e_1 = d_1 = n$.
Hence $Y \cong {\mathbf{A}^1_\mathbf{C} \setminus \{0\}}$, and it is easy to
see that $\varphi(z) = \lambda z^n$ for some $\lambda \in \mathbf{C}^*$.
After reparametrizing $Y$ we may assume $\lambda = 1$. Thus our
covering is given by taking the $n$th root of the coordinate on
$\mathbf{A}^1_{\mathbf{C}} \setminus \{0\}$.
\medskip\noindent
Remember that we need to classify the coverings of
${\mathbf{A}^1_\mathbf{C} \setminus \{0\}}$ together with free
$\mathbf{Z}/n\mathbf{Z}$-actions on them.
In our case any such action corresponds
to an automorphism of $Y$ sending $z$ to $\zeta_n z$, where $\zeta_n$ is a
primitive $n$th root of unity. There are $\phi(n)$ such actions
(here $\phi(n)$ means the Euler function). Thus there are exactly
$\phi(n)$ connected finite \'etale coverings with a given free
$\mathbf{Z}/n\mathbf{Z}$-action, each corresponding to a primitive
$n$th root of unity. We leave it to the reader to see that the
disconnected finite \'etale degree $n$ coverings of
$\mathbf{A}^1_{\mathbf{C}} \setminus \{0\}$ with a given free
$\mathbf{Z}/n\mathbf{Z}$-action correspond one-to-one with $n$th
roots of $1$ which are not primitive.
In other words, this computation shows that
$$
H_\etale^1 (\mathbf{A}^1_\mathbf{C} \setminus \{0\},
\mathbf{Z}/n\mathbf{Z}) =
\Hom(\mu_n(\mathbf{C}), \mathbf{Z}/n\mathbf{Z}) \cong \mathbf{Z}/n\mathbf{Z}.
$$
The first identification is canonical, the second isn't, see
Remark \ref{remark-normalize-H1-Gm}.
Since the proof of Riemann-Hurwitz does not use the computation of
cohomology, the above actually constitutes a proof (provided we
fill in the details on vanishing, etc).
\section{Nontorsion coefficients}
\label{section-nontorsion}
\noindent
To study nontorsion coefficients, one makes the following definition:
$$
H_\etale^i (X, \mathbf{Q}_\ell) :=
\left( \lim_n H_\etale^i(X, \mathbf{Z}/\ell^n\mathbf{Z}) \right)
\otimes_{\mathbf{Z}_\ell} \mathbf{Q}_\ell.
$$
The symbol $\lim_n$ denote the {\it limit} of the system of
cohomology groups $H_\etale^i(X, \mathbf{Z}/\ell^n\mathbf{Z})$ indexed
by $n$, see
Categories, Section \ref{categories-section-posets-limits}.
Thus we will need to study systems of sheaves satisfying some compatibility
conditions.
\section{Sheaf theory}
\label{section-sheaf-theory}
%9.10.09
\noindent
At this point we start talking about sites and sheaves in earnest.
There is an amazing amount of useful abstract material that could fit
in the next few sections. Some of this material is worked out in earlier
chapters, such as the chapter on sites, modules on sites, and cohomology
on sites. We try to refrain from adding too much material here, just
enough so the material later in this chapter makes sense.
\section{Presheaves}
\label{section-presheaves}
\noindent
A reference for this section is
Sites, Section \ref{sites-section-presheaves}.
\begin{definition}
\label{definition-presheaf}
Let $\mathcal{C}$ be a category. A {\it presheaf of sets} (respectively, an
{\it abelian presheaf}) on $\mathcal{C}$ is a functor $\mathcal{C}^{opp} \to
\textit{Sets}$ (resp.\ $\textit{Ab}$).
\end{definition}
\noindent
{\bf Terminology.} If $U \in \Ob(\mathcal{C})$, then elements of
$\mathcal{F}(U)$ are called {\it sections} of $\mathcal{F}$ over
$U$. For $\varphi : V \to U$ in $\mathcal{C}$, the
map $\mathcal{F}(\varphi) : \mathcal{F}(U) \to \mathcal{F}(V)$
is called the {\it restriction map} and is often denoted $s \mapsto s|_V$
or sometimes $s \mapsto \varphi^*s$. The notation $s|_V$ is ambiguous
since the restriction map depends on $\varphi$, but it is a standard
abuse of notation. We also use the notation
$\Gamma(U, \mathcal{F}) = \mathcal{F}(U)$.
\medskip\noindent
Saying that $\mathcal{F}$ is a functor means that if
$W \to V \to U$ are morphisms in $\mathcal{C}$ and
$s \in \Gamma(U, \mathcal{F})$ then
$(s|_V)|_W = s |_W$, with the abuse of
notation just seen. Moreover, the restriction mappings corresponding to
the identity morphisms $\text{id}_U : U \to U$ are the identity.
\medskip\noindent
The category of presheaves of sets (respectively of abelian presheaves) on
$\mathcal{C}$ is denoted $\textit{PSh} (\mathcal{C})$ (resp. $\textit{PAb}
(\mathcal{C})$). It is the category of functors from $\mathcal{C}^{opp}$ to
$\textit{Sets}$ (resp. $\textit{Ab}$), which is to say that the morphisms of
presheaves are natural transformations of functors. We only consider the
categories $\textit{PSh}(\mathcal{C})$ and $\textit{PAb}(\mathcal{C})$
when the category $\mathcal{C}$ is small. (Our convention is that a category
is small unless otherwise mentioned, and if it isn't small it should be
listed in Categories, Remark \ref{categories-remark-big-categories}.)
\begin{example}
\label{example-representable-presheaf}
Given an object $X \in \Ob(\mathcal{C})$, we consider the functor
$$
\begin{matrix}
h_X : & \mathcal{C}^{opp} & \longrightarrow & \textit{Sets} \\
& U & \longmapsto & h_X(U) = \Mor_\mathcal{C}(U, X) \\
& V \xrightarrow{\varphi} U & \longmapsto &
\varphi \circ - : h_X(U) \to h_X(V).
\end{matrix}
$$
It is a presheaf, called the {\it representable presheaf associated to $X$.}
It is not true that representable presheaves are sheaves in every topology on
every site.
\end{example}
\begin{lemma}[Yoneda]
\label{lemma-yoneda}
\begin{slogan}
Morphisms between objects are in bijection with natural transformations
between the functors they represent.
\end{slogan}
Let $\mathcal{C}$ be a category, and $X, Y \in
\Ob(\mathcal{C})$. There is a natural bijection
$$
\begin{matrix}
\Mor_\mathcal{C}(X, Y) &
\longrightarrow &
\Mor_{\textit{PSh}(\mathcal{C})} (h_X, h_Y) \\
\psi &
\longmapsto &
h_\psi = \psi \circ - : h_X \to h_Y.
\end{matrix}
$$
\end{lemma}
\begin{proof}
See
Categories, Lemma \ref{categories-lemma-yoneda}.
\end{proof}
\section{Sites}
\label{section-sites}
\begin{definition}
\label{definition-family-morphisms-fixed-target}
Let $\mathcal{C}$ be a category. A {\it family of morphisms with fixed target}
$\mathcal{U} = \{\varphi_i : U_i \to U\}_{i\in I}$ is the data of
\begin{enumerate}
\item an object $U \in \mathcal{C}$,
\item a set $I$ (possibly empty), and
\item for all $i\in I$, a morphism $\varphi_i : U_i \to U$ of $\mathcal{C}$
with target $U$.
\end{enumerate}
\end{definition}
\noindent
There is a notion of a {\it morphism of families of morphisms with fixed
target}. A special case of that is the notion of a {\it refinement}.
A reference for this material is
Sites, Section \ref{sites-section-refinements}.
\begin{definition}
\label{definition-site}
A {\it site}\footnote{What we call a site is a called a category endowed with
a pretopology in \cite[Expos\'e II, D\'efinition 1.3]{SGA4}.
In \cite{ArtinTopologies} it is called a category with a Grothendieck
topology.} consists of a category $\mathcal{C}$ and a set
$\text{Cov}(\mathcal{C})$ consisting of families of morphisms with fixed target
called {\it coverings}, such that
\begin{enumerate}
\item (isomorphism) if $\varphi : V \to U$ is an isomorphism in $\mathcal{C}$,
then $\{\varphi : V \to U\}$ is a covering,
\item (locality) if $\{\varphi_i : U_i \to U\}_{i\in I}$ is a covering and
for all $i \in I$ we are given a covering
$\{\psi_{ij} : U_{ij} \to U_i \}_{j\in I_i}$, then
$$
\{
\varphi_i \circ \psi_{ij} : U_{ij} \to U
\}_{(i, j)\in \prod_{i\in I} \{i\} \times I_i}
$$
is also a covering, and
\item (base change) if $\{U_i \to U\}_{i\in I}$
is a covering and $V \to U$ is a morphism in $\mathcal{C}$, then
\begin{enumerate}
\item for all $i \in I$ the fibre product
$U_i \times_U V$ exists in $\mathcal{C}$, and
\item $\{U_i \times_U V \to V\}_{i\in I}$ is a covering.
\end{enumerate}
\end{enumerate}
\end{definition}
\noindent
For us the category underlying a site is always ``small'', i.e., its
collection of objects form a set, and the collection of coverings of
a site is a set as well (as in the definition above). We will mostly,
in this chapter, leave out the arguments that cut down the collection
of objects and coverings to a set. For further discussion, see
Sites, Remark \ref{sites-remark-no-big-sites}.
\begin{example}
\label{example-site-topological-space}
If $X$ is a topological space, then it has an associated site $X_{Zar}$
defined as follows: the objects of $X_{Zar}$ are the open subsets of $X$,
the morphisms between these are the inclusion mappings, and the coverings are
the usual topological (surjective) coverings. Observe that if
$U, V \subset W \subset X$ are open subsets then $U \times_W V = U \cap V$
exists: this category has fiber products. All the verifications are trivial and
everything works as expected.
\end{example}
\section{Sheaves}
\label{section-sheaves}
\begin{definition}
\label{definition-sheaf}
A presheaf $\mathcal{F}$ of sets (resp. abelian presheaf) on a site
$\mathcal{C}$ is said to be a {\it separated presheaf} if for all coverings
$\{\varphi_i : U_i \to U\}_{i\in I} \in \text{Cov} (\mathcal{C})$
the map
$$
\mathcal{F}(U) \longrightarrow \prod\nolimits_{i\in I} \mathcal{F}(U_i)
$$
is injective. Here the map is $s \mapsto (s|_{U_i})_{i\in I}$.
The presheaf $\mathcal{F}$ is a {\it sheaf} if for all coverings
$\{\varphi_i : U_i \to U\}_{i\in I} \in \text{Cov} (\mathcal{C})$, the
diagram
\begin{equation}
\label{equation-sheaf-axiom}
\xymatrix{
\mathcal{F}(U) \ar[r] &
\prod_{i\in I} \mathcal{F}(U_i) \ar@<1ex>[r] \ar@<-1ex>[r] &
\prod_{i, j \in I} \mathcal{F}(U_i \times_U U_j),
}
\end{equation}
where the first map is $s \mapsto (s|_{U_i})_{i\in I}$ and the two
maps on the right are
$(s_i)_{i\in I} \mapsto (s_i |_{U_i \times_U U_j})$ and
$(s_i)_{i\in I} \mapsto (s_j |_{U_i \times_U U_j})$,
is an equalizer diagram in the category of sets (resp.\ abelian groups).
\end{definition}
\begin{remark}
\label{remark-empty-covering}
For the empty covering (where $I = \emptyset$), this implies that
$\mathcal{F}(\emptyset)$ is an empty product, which is a final object in the
corresponding category (a singleton, for both $\textit{Sets}$ and
$\textit{Ab}$).
\end{remark}
\begin{example}
\label{example-sheaf-site-space}
Working this out for the site $X_{Zar}$ associated to a topological
space, see Example \ref{example-site-topological-space}, gives the usual
notion of sheaves.
\end{example}
\begin{definition}
\label{definition-category-sheaves}
We denote $\Sh(\mathcal{C})$ (resp.\ $\textit{Ab}(\mathcal{C})$)
the full subcategory of $\textit{PSh}(\mathcal{C})$
(resp.\ $\textit{PAb}(\mathcal{C})$) whose objects are sheaves. This is the
{\it category of sheaves of sets} (resp.\ {\it abelian sheaves}) on
$\mathcal{C}$.
\end{definition}
\section{The example of G-sets}
\label{section-G-sets}
\noindent
Let $G$ be a group and define a site $\mathcal{T}_G$ as follows: the underlying
category is the category of $G$-sets, i.e., its objects are sets endowed
with a left $G$-action and the morphisms are equivariant maps; and the
coverings of $\mathcal{T}_G$ are the families
$\{\varphi_i : U_i \to U\}_{i\in I}$ satisfying
$U = \bigcup_{i\in I} \varphi_i(U_i)$.
\medskip\noindent
There is a special object in the site $\mathcal{T}_G$, namely the $G$-set $G$
endowed with its natural action by left translations. We denote it ${}_G G$.
Observe that there is a natural group isomorphism
$$
\begin{matrix}
\rho : & G^{opp} & \longrightarrow & \text{Aut}_{G\textit{-Sets}}({}_G G) \\
& g & \longmapsto & (h \mapsto hg).
\end{matrix}
$$
In particular, for any presheaf $\mathcal{F}$, the set $\mathcal{F}({}_G G)$
inherits a $G$-action via $\rho$. (Note that by contravariance of
$\mathcal{F}$, the set $\mathcal{F}({}_G G)$ is again a left $G$-set.) In fact,
the functor
$$
\begin{matrix}
\Sh(\mathcal{T}_G) & \longrightarrow & G\textit{-Sets} \\
\mathcal{F} & \longmapsto & \mathcal{F}({}_G G)
\end{matrix}
$$
is an equivalence of categories. Its quasi-inverse is the functor $X \mapsto
h_X$. Without giving the complete proof (which can be found in
Sites, Section \ref{sites-section-example-sheaf-G-sets})
let us try to explain why this is true.
\begin{enumerate}
\item
If $S$ is a $G$-set, we can decompose it into orbits $S = \coprod_{i\in I}
O_i$. The sheaf axiom for the covering $\{O_i \to S\}_{i\in I}$ says that
$$
\xymatrix{
\mathcal{F}(S) \ar[r] &
\prod_{i\in I} \mathcal{F}(O_i) \ar@<1ex>[r] \ar@<-1ex>[r] &
\prod_{i, j \in I} \mathcal{F}(O_i \times_S O_j)
}
$$
is an equalizer. Observing that fibered products in $G\textit{-Sets}$ are
induced from fibered products in $\textit{Sets}$, and using the fact that
$\mathcal{F}(\emptyset)$ is a $G$-singleton, we get that
$$
\prod_{i, j \in I} \mathcal{F}(O_i \times_S O_j) = \prod_{i \in I}
\mathcal{F}(O_i)
$$
and the two maps above are in fact the same. Therefore the sheaf axiom merely
says that $\mathcal{F}(S) = \prod_{i\in I} \mathcal{F}(O_i)$.
\item
If $S$ is the $G$-set $S= G/H$ and $\mathcal{F}$ is a sheaf on $\mathcal{T}_G$,
then we claim that
$$
\mathcal{F}(G/H) = \mathcal{F}({}_G G)^H
$$
and in particular $\mathcal{F}(\{*\}) = \mathcal{F}({}_G G)^G$. To see this,
let's use the sheaf axiom for the covering $\{ {}_G G \to G/H \}$ of $S$. We
have
\begin{eqnarray*}
{}_G G \times_{G/H} {}_G G & \cong & G \times H \\
(g_1, g_2) & \longmapsto & (g_1, g_1 g_2^{-1})
\end{eqnarray*}
is a disjoint union of copies of ${}_G G$ (as a $G$-set). Hence the sheaf axiom
reads
$$
\xymatrix{
\mathcal{F} (G/H) \ar[r] &
\mathcal{F}({}_G G) \ar@<1ex>[r] \ar@<-1ex>[r] &
\prod_{h\in H} \mathcal{F}({}_G G)
}
$$
where the two maps on the right are $s \mapsto (s)_{h \in H}$ and $s \mapsto
(hs)_{h \in H}$. Therefore $\mathcal{F}(G/H) = \mathcal{F}({}_G G)^H$ as
claimed.
\end{enumerate}
This doesn't quite prove the claimed equivalence of categories, but it shows at
least that a sheaf $\mathcal{F}$ is entirely determined by its sections over
${}_G G$. Details (and set theoretical remarks) can be found in
Sites, Section \ref{sites-section-example-sheaf-G-sets}.
\section{Sheafification}
\label{section-sheafification}
\begin{definition}
\label{definition-0-cech}
Let $\mathcal{F}$ be a presheaf on the site $\mathcal{C}$ and
$\mathcal{U} = \{U_i \to U\} \in \text{Cov} (\mathcal{C})$.
We define the {\it zeroth {\v C}ech cohomology group} of
$\mathcal{F}$ with respect to $\mathcal{U}$ by
$$
\check H^0 (\mathcal{U}, \mathcal{F}) =
\left\{
(s_i)_{i\in I} \in \prod\nolimits_{i\in I }\mathcal{F}(U_i)
\text{ such that }
s_i|_{U_i \times_U U_j} = s_j |_{U_i \times_U U_j}
\right\}.
$$
\end{definition}
\noindent
There is a canonical map
$\mathcal{F}(U) \to \check H^0 (\mathcal{U}, \mathcal{F})$,
$s \mapsto (s |_{U_i})_{i\in I}$.
We say that a {\it morphism of coverings} from a covering
$\mathcal{V} = \{V_j \to V\}_{j \in J}$ to $\mathcal{U}$ is a triple
$(\chi, \alpha, \chi_j)$, where
$\chi : V \to U$ is a morphism,
$\alpha : J \to I$ is a map of sets, and for all
$j \in J$ the morphism $\chi_j$ fits into a commutative diagram
$$
\xymatrix{
V_j \ar[rr]_{\chi_j} \ar[d] & & U_{\alpha(j)} \ar[d] \\
V \ar[rr]^\chi & & U.
}
$$
Given the data $\chi, \alpha, \{\chi_j\}_{i\in J}$ we define
\begin{eqnarray*}
\check H^0(\mathcal{U}, \mathcal{F}) & \longrightarrow &
\check H^0(\mathcal{V}, \mathcal{F}) \\
(s_i)_{i\in I} & \longmapsto &
\left(\chi_j^*\left(s_{\alpha(j)}\right)\right)_{j\in J}.
\end{eqnarray*}
We then claim that
\begin{enumerate}
\item the map is well-defined, and
\item depends only on $\chi$ and is independent of the choice of
$\alpha, \{\chi_j\}_{i\in J}$.
\end{enumerate}
We omit the proof of the first fact.
To see part (2), consider another triple $(\psi, \beta, \psi_j)$ with
$\chi = \psi$. Then we have the commutative diagram
$$
\xymatrix{
V_j \ar[rrr]_{(\chi_j, \psi_j)} \ar[dd] & & &
U_{\alpha(j)} \times_U U_{\beta(j)} \ar[dl] \ar[dr] \\
& & U_{\alpha(j)} \ar[dr] & &
U_{\beta(j)} \ar[dl] \\
V \ar[rrr]^{\chi = \psi} & & & U.
}
$$
Given a section $s \in \mathcal{F}(\mathcal{U})$, its image in
$\mathcal{F}(V_j)$ under the map given by
$(\chi, \alpha, \{\chi_j\}_{i\in J})$
is $\chi_j^*s_{\alpha(j)}$, and
its image under the map given by $(\psi, \beta, \{\psi_j\}_{i\in J})$
is $\psi_j^*s_{\beta(j)}$. These
two are equal since by assumption $s \in \check H(\mathcal{U}, \mathcal{F})$
and hence both are equal to the pullback of the common value
$$
s_{\alpha(j)}|_{U_{\alpha(j)} \times_U U_{\beta(j)}} =
s_{\beta(j)}|_{U_{\alpha(j)} \times_U U_{\beta(j)}}
$$
pulled back by the map $(\chi_j, \psi_j)$ in the diagram.
\begin{theorem}
\label{theorem-sheafification}
Let $\mathcal{C}$ be a site and $\mathcal{F}$ a presheaf on $\mathcal{C}$.
\begin{enumerate}
\item The rule
$$
U \mapsto \mathcal{F}^+(U) :=
\colim_{\mathcal{U} \text{ covering of }U}
\check H^0(\mathcal{U}, \mathcal{F})
$$
is a presheaf. And the colimit is a directed one.
\item There is a canonical map of presheaves $\mathcal{F} \to \mathcal{F}^+$.
\item If $\mathcal{F}$ is a separated presheaf then $\mathcal{F}^+$ is a sheaf
and the map in (2) is injective.
\item $\mathcal{F}^+$ is a separated presheaf.
\item $\mathcal{F}^\# = (\mathcal{F}^+)^+$ is a sheaf, and the canonical
map induces a functorial isomorphism
$$
\Hom_{\textit{PSh}(\mathcal{C})}(\mathcal{F}, \mathcal{G}) =
\Hom_{\Sh(\mathcal{C})}(\mathcal{F}^\#, \mathcal{G})
$$
for any $\mathcal{G} \in \Sh(\mathcal{C})$.
\end{enumerate}
\end{theorem}
\begin{proof}
See Sites, Theorem \ref{sites-theorem-plus}.
\end{proof}
\noindent
In other words, this means that the natural map
$\mathcal{F} \to \mathcal{F}^\#$ is a left adjoint to the forgetful functor
$\Sh(\mathcal{C}) \to \textit{PSh}(\mathcal{C})$.
\section{Cohomology}
\label{section-cohomology}
\noindent
The following is the basic result that makes it possible to define cohomology
for abelian sheaves on sites.
\begin{theorem}
\label{theorem-enough-injectives}
The category of abelian sheaves on a site is an abelian category
which has enough injectives.
\end{theorem}
\begin{proof}
See
Modules on Sites, Lemma \ref{sites-modules-lemma-abelian-abelian} and
Injectives, Theorem \ref{injectives-theorem-sheaves-injectives}.
\end{proof}
\noindent
So we can define cohomology as the right-derived functors of the
sections functor: if $U \in \Ob(\mathcal{C})$ and
$\mathcal{F} \in \textit{Ab}(\mathcal{C})$,
$$
H^p(U, \mathcal{F}) :=
R^p\Gamma(U, \mathcal{F}) =
H^p(\Gamma(U, \mathcal{I}^\bullet))
$$
where $\mathcal{F} \to \mathcal{I}^\bullet$ is an injective resolution. To do
this, we should check that the functor $\Gamma(U, -)$ is left exact. This is
true and is part of why the category $\textit{Ab}(\mathcal{C})$ is abelian,
see
Modules on Sites, Lemma \ref{sites-modules-lemma-abelian-abelian}.
For more general discussion of cohomology on sites (including the
global sections functor and its right derived functors), see
Cohomology on Sites, Section \ref{sites-cohomology-section-cohomology-sheaves}.
\section{The fpqc topology}
\label{section-fpqc}
%9.15.09
\noindent
Before doing \'etale cohomology we study a bit the fpqc topology, since
it works well for quasi-coherent sheaves.
\begin{definition}
\label{definition-fpqc-covering}
Let $T$ be a scheme. An {\it fpqc covering} of $T$ is a family
$\{ \varphi_i : T_i \to T\}_{i \in I}$ such that
\begin{enumerate}
\item each $\varphi_i$ is a flat morphism and
$\bigcup_{i\in I} \varphi_i(T_i) = T$, and
\item for each affine open $U \subset T$ there exists a finite
set $K$, a map $\mathbf{i} : K \to I$ and affine opens
$U_{\mathbf{i}(k)} \subset T_{\mathbf{i}(k)}$ such that
$U = \bigcup_{k \in K} \varphi_{\mathbf{i}(k)}(U_{\mathbf{i}(k)})$.
\end{enumerate}
\end{definition}
\begin{remark}
\label{remark-fpqc}
The first condition corresponds to fp, which stands for
{\it fid\`element plat}, faithfully flat in french, and
the second to qc, {\it quasi-compact}. The second part of
the first condition is unnecessary when the second condition holds.
\end{remark}
\begin{example}
\label{example-fpqc-coverings}
Examples of fpqc coverings.
\begin{enumerate}
\item Any Zariski open covering of $T$ is an fpqc covering.
\item A family $\{\Spec(B) \to \Spec(A)\}$ is an fpqc
covering if and only if $A \to B$ is a faithfully flat ring map.
\item If $f: X \to Y$ is flat, surjective and quasi-compact, then $\{ f: X\to
Y\}$ is an fpqc covering.
\item The morphism
$\varphi :
\coprod_{x \in \mathbf{A}^1_k} \Spec(\mathcal{O}_{\mathbf{A}^1_k, x})
\to \mathbf{A}^1_k$,
where $k$ is a field, is flat and surjective. It is not quasi-compact, and
in fact the family $\{\varphi\}$ is not an fpqc covering.
\item Write
$\mathbf{A}^2_k = \Spec(k[x, y])$. Denote $i_x : D(x) \to \mathbf{A}^2_k$
and $i_y : D(y) \hookrightarrow \mathbf{A}^2_k$ the standard opens.
Then the families
$\{i_x, i_y, \Spec(k[[x, y]]) \to \mathbf{A}^2_k\}$
and
$\{i_x, i_y, \Spec(\mathcal{O}_{\mathbf{A}^2_k, 0}) \to \mathbf{A}^2_k\}$
are fpqc coverings.
\end{enumerate}
\end{example}
\begin{lemma}
\label{lemma-site-fpqc}
The collection of fpqc coverings on the category of schemes
satisfies the axioms of site.
\end{lemma}
\begin{proof}
See Topologies, Lemma \ref{topologies-lemma-fpqc}.
\end{proof}
\noindent
It seems that this lemma allows us to define the fpqc site of the category
of schemes. However, there is a set theoretical problem that comes up when
considering the fpqc topology, see
Topologies, Section \ref{topologies-section-fpqc}.
It comes from our requirement that sites are ``small'', but that no small
category of schemes can contain a cofinal system of fpqc coverings of a
given nonempty scheme. Although this does not strictly speaking prevent
us from defining ``partial'' fpqc
sites, it does not seem prudent to do so. The work-around is to allow
the notion of a sheaf for the fpqc topology (see below) but to prohibit
considering the category of all fpqc sheaves.
\begin{definition}
\label{definition-sheaf-property-fpqc}
Let $S$ be a scheme. The category of schemes over $S$ is denoted
$\Sch/S$. Consider a functor
$\mathcal{F} : (\Sch/S)^{opp} \to \textit{Sets}$, in other words
a presheaf of sets. We say $\mathcal{F}$
{\it satisfies the sheaf property for the fpqc topology}
if for every fpqc covering $\{U_i \to U\}_{i \in I}$ of schemes over $S$
the diagram (\ref{equation-sheaf-axiom}) is an equalizer diagram.
\end{definition}
\noindent
We similarly say that $\mathcal{F}$
{\it satisfies the sheaf property for the Zariski topology} if for
every open covering $U = \bigcup_{i \in I} U_i$ the diagram
(\ref{equation-sheaf-axiom}) is an equalizer diagram. See
Schemes, Definition \ref{schemes-definition-representable-by-open-immersions}.
Clearly, this is equivalent to saying that for every scheme $T$ over $S$ the
restriction of $\mathcal{F}$ to the opens of $T$ is a (usual) sheaf.
\begin{lemma}
\label{lemma-fpqc-sheaves}
Let $\mathcal{F}$ be a presheaf on $\Sch/S$. Then
$\mathcal{F}$ satisfies the sheaf property for the fpqc topology
if and only if
\begin{enumerate}
\item $\mathcal{F}$ satisfies the sheaf property with respect to the
Zariski topology, and
\item for every faithfully flat morphism $\Spec(B) \to \Spec(A)$
of affine schemes over $S$, the sheaf axiom holds for the covering
$\{\Spec(B) \to \Spec(A)\}$. Namely, this means that
$$
\xymatrix{
\mathcal{F}(\Spec(A)) \ar[r] &
\mathcal{F}(\Spec(B)) \ar@<1ex>[r] \ar@<-1ex>[r] &
\mathcal{F}(\Spec(B \otimes_A B))
}
$$
is an equalizer diagram.
\end{enumerate}
\end{lemma}
\begin{proof}
See Topologies, Lemma \ref{topologies-lemma-sheaf-property-fpqc}.
\end{proof}
\noindent
An alternative way to think of a presheaf $\mathcal{F}$ on
$\Sch/S$ which satisfies the sheaf condition for the
fpqc topology is as the following data:
\begin{enumerate}
\item for each $T/S$, a usual (i.e., Zariski) sheaf $\mathcal{F}_T$ on
$T_{Zar}$,
\item for every map $f : T' \to T$ over $S$, a restriction mapping
$f^{-1}\mathcal{F}_T \to \mathcal{F}_{T'}$
\end{enumerate}
such that
\begin{enumerate}
\item[(a)] the restriction mappings are functorial,
\item[(b)] if $f : T' \to T$ is an open immersion then the restriction
mapping $f^{-1}\mathcal{F}_T \to \mathcal{F}_{T'}$ is an isomorphism, and
\item[(c)] for every faithfully flat morphism
$\Spec(B) \to \Spec(A)$ over $S$, the diagram
$$
\xymatrix{
\mathcal{F}_{\Spec(A)}(\Spec(A)) \ar[r] &
\mathcal{F}_{\Spec(B)}(\Spec(B)) \ar@<1ex>[r] \ar@<-1ex>[r] &
\mathcal{F}_{\Spec(B \otimes_A B)}(\Spec(B \otimes_A B))
}
$$
is an equalizer.
\end{enumerate}
Data (1) and (2) and conditions (a), (b) give the data of a presheaf
on $\Sch/S$ satisfying the sheaf condition for the Zariski topology.
By Lemma \ref{lemma-fpqc-sheaves} condition (c) then suffices to get the
sheaf condition for the fpqc topology.
\begin{example}
\label{example-quasi-coherent}
Consider the presheaf
$$
\begin{matrix}
\mathcal{F} : & (\Sch/S)^{opp} & \longrightarrow & \textit{Ab} \\
& T/S & \longmapsto & \Gamma(T, \Omega_{T/S}).
\end{matrix}
$$
The compatibility of differentials with localization implies that
$\mathcal{F}$ is a sheaf on the Zariski site.
However, it does not satisfy the sheaf condition for the fpqc topology.
Namely, consider the case
$S = \Spec(\mathbf{F}_p)$ and the morphism
$$
\varphi :
V = \Spec(\mathbf{F}_p[v])
\to
U = \Spec(\mathbf{F}_p[u])
$$
given by mapping $u$ to $v^p$. The family $\{\varphi\}$ is an fpqc covering,
yet the restriction mapping
$\mathcal{F}(U) \to \mathcal{F}(V)$
sends the generator $\text{d}u$ to $\text{d}(v^p) = 0$, so
it is the zero map, and the diagram
$$
\xymatrix{
\mathcal{F}(U) \ar[r]^{0} &
\mathcal{F}(V) \ar@<1ex>[r] \ar@<-1ex>[r] &
\mathcal{F}(V \times_U V)
}
$$
is not an equalizer. We will see later that $\mathcal{F}$ does in fact
give rise to a sheaf on the \'etale and smooth sites.
\end{example}
\begin{lemma}
\label{lemma-representable-sheaf-fpqc}
Any representable presheaf on $\Sch/S$ satisfies the
sheaf condition for the fpqc topology.
\end{lemma}
\begin{proof}
See
Descent, Lemma \ref{descent-lemma-fpqc-universal-effective-epimorphisms}.
\end{proof}
\noindent
We will return to this later, since the proof of this fact uses
descent for quasi-coherent sheaves, which we will discuss in the next
section. A fancy way of expressing the lemma is to say that
{\it the fpqc topology is weaker than the canonical topology}, or
that the fpqc topology is {\it subcanonical}. In the setting of sites
this is discussed in
Sites, Section \ref{sites-section-representable-sheaves}.
\begin{remark}
\label{remark-fpqc-finest}
The fpqc is the finest topology that we will see. Hence any presheaf
satisfying the sheaf condition for the fpqc topology will be a
sheaf in the subsequent sites (\'etale, smooth, etc). In particular
representable presheaves will be sheaves on the \'etale site of a scheme
for example.
\end{remark}
\begin{example}
\label{example-additive-group-sheaf}
Let $S$ be a scheme.
Consider the additive group scheme $\mathbf{G}_{a, S} = \mathbf{A}^1_S$
over $S$, see
Groupoids, Example \ref{groupoids-example-additive-group}.
The associated representable presheaf is given by
$$
h_{\mathbf{G}_{a, S}}(T) =
\Mor_S(T, \mathbf{G}_{a, S}) =
\Gamma(T, \mathcal{O}_T).
$$
By the above we now know that this is a presheaf of sets which satisfies the
sheaf condition for the fpqc topology. On the other hand, it is clearly
a presheaf of rings as well. Hence we can think of this as a functor
$$
\begin{matrix}
\mathcal{O} : &
(\Sch/S)^{opp} &
\longrightarrow &
\textit{Rings} \\
&
T/S &
\longmapsto &
\Gamma(T, \mathcal{O}_T)
\end{matrix}
$$
which satisfies the sheaf condition for the fpqc topology.
Correspondingly there is a notion of $\mathcal{O}$-module, and so on and
so forth.
\end{example}
\section{Faithfully flat descent}
\label{section-fpqc-descent}
\begin{definition}
\label{definition-descent-datum}
Let $\mathcal{U} = \{ t_i : T_i \to T\}_{i \in I}$ be a family of
morphisms of schemes with fixed target. A {\it descent datum} for
quasi-coherent sheaves with respect to
$\mathcal{U}$ is a family $(\mathcal{F}_i, \varphi_{ij})_{i, j\in I}$ where
\begin{enumerate}
\item for all $i$, $\mathcal{F}_i$ is a quasi-coherent sheaf on $T_i$, and
\item for all $i, j \in I$ the map
$\varphi_{ij} : \text{pr}_0^* \mathcal{F}_i \cong \text{pr}_1^* \mathcal{F}_j$
is an isomorphism on $T_i \times_T T_j$ such that the diagrams
$$
\xymatrix{
\text{pr}_0^*\mathcal{F}_i \ar[dr]_{\text{pr}_{02}^*\varphi_{ik}}
\ar[rr]^{\text{pr}_{01}^*\varphi_{ij}} & &
\text{pr}_1^*\mathcal{F}_j \ar[dl]^{\text{pr}_{12}^*\varphi_{jk}} \\
& \text{pr}_2^*\mathcal{F}_k
}
$$
commute on $T_i \times_T T_j \times_T T_k$.
\end{enumerate}
This descent datum is called {\it effective} if there exist a quasi-coherent
sheaf $\mathcal{F}$ over $T$ and $\mathcal{O}_{T_i}$-module isomorphisms
$\varphi_i : t_i^* \mathcal{F} \cong \mathcal{F}_i$ satisfying the cocycle
condition, namely
$$
\varphi_{ij} = \text{pr}_1^* (\varphi_j) \circ \text{pr}_0^* (\varphi_i)^{-1}.
$$
\end{definition}
\noindent
In this and the next section we discuss some ingredients of the proof
of the following theorem, as well as some related material.
\begin{theorem}
\label{theorem-descent-quasi-coherent}
If $\mathcal{V} = \{T_i \to T\}_{i\in I}$ is an fpqc covering, then all
descent data for quasi-coherent sheaves with respect to $\mathcal{V}$
are effective.
\end{theorem}
\begin{proof}
See
Descent, Proposition \ref{descent-proposition-fpqc-descent-quasi-coherent}.
\end{proof}
\noindent
In other words, the fibered category of quasi-coherent sheaves is a stack on
the fpqc site.
The proof of the theorem is in two steps. The first one is to realize that for
Zariski coverings this is easy (or well-known) using standard glueing of
sheaves (see
Sheaves, Section \ref{sheaves-section-glueing-sheaves})
and the locality of quasi-coherence. The second step is the case of an
fpqc covering of the form $\{\Spec(B) \to \Spec(A)\}$
where $A \to B$ is a faithfully flat ring map.
This is a lemma in algebra, which we now present.
\medskip\noindent
{\bf Descent of modules.}
If $A \to B$ is a ring map, we consider the complex
$$
(B/A)_\bullet : B \to B \otimes_A B \to B \otimes_A B \otimes_A B \to \ldots
$$
where $B$ is in degree 0, $B \otimes_A B$ in degree 1, etc, and the maps are
given by
\begin{eqnarray*}
b & \mapsto & 1 \otimes b - b \otimes 1, \\
b_0 \otimes b_1 & \mapsto & 1 \otimes b_0 \otimes b_1 - b_0 \otimes 1 \otimes
b_1 + b_0 \otimes b_1 \otimes 1, \\
& \text{etc.}
\end{eqnarray*}
\begin{lemma}
\label{lemma-algebra-descent}
If $A \to B$ is faithfully flat, then the complex $(B/A)_\bullet$ is exact in
positive degrees, and $H^0((B/A)_\bullet) = A$.
\end{lemma}
\begin{proof}
See Descent, Lemma \ref{descent-lemma-ff-exact}.
\end{proof}
\noindent
Grothendieck proves this in three steps. Firstly, he assumes that the map $A
\to B$ has a section, and constructs an explicit homotopy to the complex where
$A$ is the only nonzero term, in degree 0. Secondly, he observes that to prove
the result, it suffices to do so after a faithfully flat base change $A \to
A'$, replacing $B$ with $B' = B \otimes_A A'$. Thirdly, he applies the
faithfully flat base change $A \to A' = B$ and remark that the map
$A' = B \to B' = B \otimes_A B$ has a natural section.
\medskip\noindent
The same strategy proves the following lemma.
\begin{lemma}
\label{lemma-descent-modules}
If $A \to B$ is faithfully flat and $M$ is an $A$-module, then the
complex $(B/A)_\bullet \otimes_A M$ is exact in positive degrees, and
$H^0((B/A)_\bullet \otimes_A M) = M$.
\end{lemma}
\begin{proof}
See Descent, Lemma \ref{descent-lemma-ff-exact}.
\end{proof}
\begin{definition}
\label{definition-descent-datum-modules}
Let $A \to B$ be a ring map and $N$ a $B$-module. A {\it descent datum} for
$N$ with respect to $A \to B$ is an isomorphism
$\varphi : N \otimes_A B \cong B \otimes_A N$ of $B \otimes_A B$-modules such
that the diagram of $B \otimes_A B \otimes_A B$-modules
$$
\xymatrix{
{N \otimes_A B \otimes_A B} \ar[dr]_{\varphi_{02}} \ar[rr]^{\varphi_{01}} & &
{B \otimes_A N \otimes_A B} \ar[dl]^{\varphi_{12}} \\
& {B \otimes_A B \otimes_A N}
}
$$
commutes where $\varphi_{01} = \varphi \otimes \text{id}_B$ and similarly
for $\varphi_{12}$ and $\varphi_{02}$.
\end{definition}
\noindent
If $N' = B \otimes_A M$ for some $A$-module M, then it has a canonical descent
datum given by the map
$$
\begin{matrix}
\varphi_\text{can}: & N' \otimes_A B & \to & B \otimes_A N' \\
& b_0 \otimes m \otimes b_1 & \mapsto & b_0 \otimes b_1 \otimes m.
\end{matrix}
$$
\begin{definition}
\label{definition-effective-modules}
A descent datum $(N, \varphi)$ is called {\it effective} if there exists an
$A$-module $M$ such that $(N, \varphi) \cong (B \otimes_A M,
\varphi_\text{can})$, with the obvious notion of isomorphism of descent data.
\end{definition}
\noindent
Theorem \ref{theorem-descent-quasi-coherent} is a consequence the
following result.
\begin{theorem}
\label{theorem-descent-modules}
If $A \to B$ is faithfully flat then descent data with respect to $A\to B$
are effective.
\end{theorem}
\begin{proof}
See
Descent, Proposition \ref{descent-proposition-descent-module}.
See also
Descent, Remark \ref{descent-remark-homotopy-equivalent-cosimplicial-algebras}
for an alternative view of the proof.
\end{proof}
\begin{remarks}
\label{remarks-theorem-modules-exactness}
The results on descent of modules have several applications:
\begin{enumerate}
\item The exactness of the {\v C}ech complex in positive degrees for
the covering $\{\Spec(B) \to \Spec(A)\}$ where $A \to B$ is
faithfully flat. This will give some vanishing of cohomology.
\item If $(N, \varphi)$ is a descent datum with respect to a faithfully
flat map $A \to B$, then the corresponding $A$-module is given by
$$
M = \Ker \left(
\begin{matrix}
N & \longrightarrow & B \otimes_A N \\
n & \longmapsto & 1 \otimes n - \varphi(n \otimes 1)
\end{matrix}
\right).
$$
See
Descent, Proposition \ref{descent-proposition-descent-module}.
\end{enumerate}
\end{remarks}
%9.17.09
\section{Quasi-coherent sheaves}
\label{section-quasi-coherent}
\noindent
We can apply the descent of modules to study quasi-coherent sheaves.
\begin{proposition}
\label{proposition-quasi-coherent-sheaf-fpqc}
For any quasi-coherent sheaf $\mathcal{F}$ on $S$ the presheaf
$$
\begin{matrix}
\mathcal{F}^a : & \Sch/S & \to & \textit{Ab}\\
& (f: T \to S) & \mapsto & \Gamma(T, f^*\mathcal{F})
\end{matrix}
$$
is an $\mathcal{O}$-module which satisfies the sheaf condition for the
fpqc topology.
\end{proposition}
\begin{proof}
This is proved in
Descent, Lemma \ref{descent-lemma-sheaf-condition-holds}.
We indicate the proof here. As established in
Lemma \ref{lemma-fpqc-sheaves},
it is enough to check the sheaf property
on Zariski coverings and faithfully flat morphisms of affine schemes. The
sheaf property for Zariski coverings is standard scheme theory, since
$\Gamma(U, i^\ast \mathcal{F}) = \mathcal{F}(U)$ when
$i : U \hookrightarrow S$ is an open immersion.
\medskip\noindent
For $\left\{\Spec(B)\to \Spec(A)\right\}$ with $A\to B$ faithfully
flat and
$\mathcal{F}|_{\Spec(A)} = \widetilde{M}$
this corresponds to the fact that
$M = H^0\left((B/A)_\bullet \otimes_A M \right)$, i.e., that
\begin{align*}
0 \to M \to B \otimes_A M \to B \otimes_A B \otimes_A M
\end{align*}
is exact by
Lemma \ref{lemma-descent-modules}.
\end{proof}
\noindent
There is an abstract notion of a quasi-coherent sheaf on a ringed site.
We briefly introduce this here. For more information please consult
Modules on Sites, Section \ref{sites-modules-section-local}.
Let $\mathcal{C}$ be a category, and let $U$ be an object of $\mathcal{C}$.
Then $\mathcal{C}/U$ indicates the category of objects over $U$, see
Categories, Example \ref{categories-example-category-over-X}.
If $\mathcal{C}$ is a site, then $\mathcal{C}/U$ is a site as well, namely
the coverings of $V/U$ are families $\{V_i/U \to V/U\}$ of morphisms
of $\mathcal{C}/U$ with fixed target such that
$\{V_i \to V\}$ is a covering of $\mathcal{C}$. Moreover, given any
sheaf $\mathcal{F}$ on $\mathcal{C}$ the {\it restriction}
$\mathcal{F}|_{\mathcal{C}/U}$ (defined in the obvious manner)
is a sheaf as well. See
Sites, Section \ref{sites-section-localize}
for details.
\begin{definition}
\label{definition-ringed-site}
Let $\mathcal{C}$ be a {\it ringed site}, i.e., a site endowed with a
sheaf of rings $\mathcal{O}$. A sheaf of $\mathcal{O}$-modules $\mathcal{F}$ on
$\mathcal{C}$ is called {\it quasi-coherent} if for all
$U \in \Ob(\mathcal{C})$ there exists a covering
$\{U_i \to U\}_{i\in I}$ of $\mathcal{C}$ such that the restriction
$\mathcal{F}|_{\mathcal{C}/U_i}$ is isomorphic to the cokernel of
an $\mathcal{O}$-linear map of free $\mathcal{O}$-modules
$$
\bigoplus\nolimits_{k \in K} \mathcal{O}|_{\mathcal{C}/U_i}
\longrightarrow
\bigoplus\nolimits_{l \in L} \mathcal{O}|_{\mathcal{C}/U_i}.
$$
The direct sum over $K$ is the sheaf associated to the presheaf
$V \mapsto \bigoplus_{k \in K} \mathcal{O}(V)$ and similarly for the other.
\end{definition}
\noindent
Although it is useful to be able to give a general definition as above
this notion is not well behaved in general.
\begin{remark}
\label{remark-final-object}
In the case where $\mathcal{C}$ has a final object, e.g.\ $S$, it
suffices to check the condition of the definition for
$U = S$ in the above statement. See
Modules on Sites, Lemma \ref{sites-modules-lemma-local-final-object}.
\end{remark}
\begin{theorem}[Meta theorem on quasi-coherent sheaves]
\label{theorem-quasi-coherent}
Let $S$ be a scheme.
Let $\mathcal{C}$ be a site. Assume that
\begin{enumerate}
\item the underlying category $\mathcal{C}$ is a
full subcategory of $\Sch/S$,
\item any Zariski covering of $T \in \Ob(\mathcal{C})$
can be refined by a covering of $\mathcal{C}$,
\item $S/S$ is an object of $\mathcal{C}$,
\item every covering of $\mathcal{C}$ is an fpqc covering of schemes.
\end{enumerate}
Then the presheaf $\mathcal{O}$ is a sheaf on $\mathcal{C}$ and
any quasi-coherent $\mathcal{O}$-module on $(\mathcal{C}, \mathcal{O})$
is of the form $\mathcal{F}^a$ for some quasi-coherent sheaf
$\mathcal{F}$ on $S$.
\end{theorem}
\begin{proof}
After some formal arguments this is exactly Theorem
\ref{theorem-descent-quasi-coherent}. Details omitted. In
Descent, Proposition \ref{descent-proposition-equivalence-quasi-coherent}
we prove a more precise version of the theorem for the
big Zariski, fppf, \'etale, smooth, and syntomic sites of $S$,
as well as the small Zariski and \'etale sites of $S$.
\end{proof}
\noindent
In other words, there is no difference between quasi-coherent
modules on the scheme $S$ and quasi-coherent $\mathcal{O}$-modules
on sites $\mathcal{C}$ as in the theorem. More precise statements
for the big and small sites $(\Sch/S)_{fppf}$, $S_\etale$, etc
can be found in
Descent, Section \ref{descent-section-quasi-coherent-sheaves}.
In this chapter we will sometimes refer to a
``site as in Theorem \ref{theorem-quasi-coherent}''
in order to conveniently state results which hold in any of those
situations.
\section{{\v C}ech cohomology}
\label{section-cech-cohomology}
\noindent
Our next goal is to use descent theory to show that
$H^i(\mathcal{C}, \mathcal{F}^a) = H_{Zar}^i(S, \mathcal{F})$
for all quasi-coherent sheaves $\mathcal{F}$ on $S$, and
any site $\mathcal{C}$ as in Theorem \ref{theorem-quasi-coherent}.
To this end, we introduce {\v C}ech cohomology on sites.
See \cite{ArtinTopologies} and
Cohomology on Sites, Sections \ref{sites-cohomology-section-cech},
\ref{sites-cohomology-section-cech-functor}
and \ref{sites-cohomology-section-cech-cohomology-cohomology}
for more details.
\begin{definition}
\label{definition-cech-complex}
Let $\mathcal{C}$ be a category,
$\mathcal{U} = \{U_i \to U\}_{i \in I}$ a family of morphisms of $\mathcal{C}$
with fixed target, and $\mathcal{F} \in \textit{PAb}(\mathcal{C})$ an abelian
presheaf. We define the {\it {\v C}ech complex}
$\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})$ by
$$
\prod_{i_0\in I} \mathcal{F}(U_{i_0}) \to
\prod_{i_0, i_1\in I} \mathcal{F}(U_{i_0} \times_U U_{i_1}) \to
\prod_{i_0, i_1, i_2 \in I}
\mathcal{F}(U_{i_0} \times_U U_{i_1} \times_U U_{i_2}) \to \ldots
$$
where the first term is in degree 0, and the maps are the usual ones. Again, it
is essential to allow the case $i_0 = i_1$ etc. The
{\it {\v C}ech cohomology groups} are defined by
$$
\check{H}^p(\mathcal{U}, \mathcal{F}) =
H^p(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})).
$$
\end{definition}
\begin{lemma}
\label{lemma-cech-presheaves}
The functor $\check{\mathcal{C}}^\bullet(\mathcal{U}, -)$
is exact on the category $\textit{PAb}(\mathcal{C})$.
\end{lemma}
\noindent
In other words, if $0\to \mathcal{F}_1\to \mathcal{F}_2\to \mathcal{F}_3\to 0$
is a short exact sequence of presheaves of abelian groups, then
$$
0 \to \check{\mathcal{C}}^\bullet\left(\mathcal{U}, \mathcal{F}_1\right)
\to\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}_2) \to
\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}_3)\to 0
$$
is a short exact sequence of complexes.
\begin{proof}
This follows at once from the definition of a short exact sequence of
presheaves. Namely, as the category of abelian presheaves is the category of
functors on some category with values in $\textit{Ab}$, it is automatically an
abelian category: a sequence $\mathcal{F}_1\to \mathcal{F}_2\to \mathcal{F}_3$
is exact in $\textit{PAb}$ if and only if for all
$U \in \Ob(\mathcal{C})$, the sequence
$\mathcal{F}_1(U) \to \mathcal{F}_2(U) \to \mathcal{F}_3(U)$ is exact in
$\textit{Ab}$. So the complex above is merely a product of short exact
sequences in each degree. See also
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cech-exact-presheaves}.
\end{proof}
\noindent
This shows that $\check{H}^\bullet(\mathcal{U}, -)$ is a $\delta$-functor.
We now proceed to show that it is a universal $\delta$-functor. We thus need to
show that it is an {\it effaceable} functor. We start by recalling the Yoneda
lemma.
\begin{lemma}[Yoneda Lemma]
\label{lemma-yoneda-presheaf}
For any presheaf $\mathcal{F}$ on a category $\mathcal{C}$ there is a
functorial isomorphism
$$
\Hom_{\textit{PSh}(\mathcal{C})}(h_U, \mathcal{F}) =
\mathcal{F}(U).
$$
\end{lemma}
\begin{proof}
See Categories, Lemma \ref{categories-lemma-yoneda}.
\end{proof}
\noindent
Given a set $E$ we denote (in this section)
$\mathbf{Z}[E]$ the free abelian group on $E$. In a formula
$\mathbf{Z}[E] = \bigoplus_{e \in E} \mathbf{Z}$, i.e., $\mathbf{Z}[E]$ is
a free $\mathbf{Z}$-module having a basis consisting of the elements of $E$.
Using this notation we introduce the free abelian presheaf on a
presheaf of sets.
\begin{definition}
\label{definition-free-abelian-presheaf}
Let $\mathcal{C}$ be a category.
Given a presheaf of sets $\mathcal{G}$, we define the
{\it free abelian presheaf on $\mathcal{G}$},
denoted $\mathbf{Z}_\mathcal{G}$, by the rule
$$
\mathbf{Z}_\mathcal{G}(U)
=
\mathbf{Z}[\mathcal{G}(U)]
$$
for $U \in \Ob(\mathcal{C})$
with restriction maps induced by the restriction maps of $\mathcal{G}$.
In the special case $\mathcal{G} = h_U$ we write simply
$\mathbf{Z}_U = \mathbf{Z}_{h_U}$.
\end{definition}
\noindent
The functor $\mathcal{G} \mapsto \mathbf{Z}_\mathcal{G}$ is left adjoint to the
forgetful functor $\textit{PAb}(\mathcal{C}) \to \textit{PSh}(\mathcal{C})$.
Thus, for any presheaf $\mathcal{F}$, there is a canonical isomorphism
$$
\Hom_{\textit{PAb}(\mathcal{C})}(\mathbf{Z}_U, \mathcal{F})
=
\Hom_{\textit{PSh}(\mathcal{C})}(h_U, \mathcal{F})
=
\mathcal{F}(U)
$$
the last equality by the Yoneda lemma. In particular, we have the following
result.
\begin{lemma}
\label{lemma-cech-complex-describe}
The {\v C}ech complex $\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})$
can be described explicitly as follows
\begin{eqnarray*}
\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})
& = &
\left(
\prod_{i_0 \in I}
\Hom_{\textit{PAb}(\mathcal{C})}(\mathbf{Z}_{U_{i_0}}, \mathcal{F}) \to
\prod_{i_0, i_1 \in I}
\Hom_{\textit{PAb}(\mathcal{C})}(
\mathbf{Z}_{U_{i_0} \times_U U_{i_1}}, \mathcal{F}) \to \ldots
\right) \\
& = &
\Hom_{\textit{PAb}(\mathcal{C})}\left(
\left(
\bigoplus_{i_0 \in I} \mathbf{Z}_{U_{i_0}} \leftarrow
\bigoplus_{i_0, i_1 \in I} \mathbf{Z}_{U_{i_0} \times_U U_{i_1}} \leftarrow
\ldots
\right), \mathcal{F}\right)
\end{eqnarray*}
\end{lemma}
\begin{proof}
This follows from the formula above. See
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cech-map-into}.
\end{proof}
\noindent
This reduces us to studying only the complex in the first argument of the
last $\Hom$.
\begin{lemma}
\label{lemma-exact}
The complex of abelian presheaves
\begin{align*}
\mathbf{Z}_\mathcal{U}^\bullet \quad : \quad
\bigoplus_{i_0 \in I} \mathbf{Z}_{U_{i_0}} \leftarrow
\bigoplus_{i_0, i_1 \in I} \mathbf{Z}_{U_{i_0} \times_U U_{i_1}} \leftarrow
\bigoplus_{i_0, i_1, i_2 \in I}
\mathbf{Z}_{U_{i_0} \times_U U_{i_1} \times_U U_{i_2}} \leftarrow
\ldots
\end{align*}
is exact in all degrees except $0$ in $\textit{PAb}(\mathcal{C})$.
\end{lemma}
\begin{proof}
For any $V\in \Ob(\mathcal{C})$ the complex of abelian groups
$\mathbf{Z}_\mathcal{U}^\bullet(V)$ is
$$
\begin{matrix}
\mathbf{Z}\left[
\coprod_{i_0\in I} \Mor_\mathcal{C}(V, U_{i_0})\right]
\leftarrow
\mathbf{Z}\left[
\coprod_{i_0, i_1 \in I}
\Mor_\mathcal{C}(V, U_{i_0} \times_U U_{i_1})\right]
\leftarrow \ldots = \\
\bigoplus_{\varphi : V \to U}
\left(
\mathbf{Z}\left[\coprod_{i_0 \in I} \Mor_\varphi(V, U_{i_0})\right]
\leftarrow
\mathbf{Z}\left[\coprod_{i_0, i_1\in I} \Mor_\varphi(V, U_{i_0}) \times
\Mor_\varphi(V, U_{i_1})\right]
\leftarrow
\ldots
\right)
\end{matrix}
$$
where
$$
\Mor_{\varphi}(V, U_i)
=
\{ V \to U_i \text{ such that } V \to U_i \to U \text{ equals } \varphi \}.
$$
Set $S_\varphi = \coprod_{i\in I} \Mor_\varphi(V, U_i)$, so that
$$
\mathbf{Z}_\mathcal{U}^\bullet(V)
=
\bigoplus_{\varphi : V \to U}
\left(
\mathbf{Z}[S_\varphi] \leftarrow
\mathbf{Z}[S_\varphi \times S_\varphi] \leftarrow
\mathbf{Z}[S_\varphi \times S_\varphi \times S_\varphi] \leftarrow
\ldots \right).
$$
Thus it suffices to show that for each $S = S_\varphi$, the complex
\begin{align*}
\mathbf{Z}[S] \leftarrow
\mathbf{Z}[S \times S] \leftarrow
\mathbf{Z}[S \times S \times S] \leftarrow \ldots
\end{align*}
is exact in negative degrees. To see this, we can give an explicit homotopy.
Fix $s\in S$ and define $K: n_{(s_0, \ldots, s_p)} \mapsto n_{(s, s_0,
\ldots, s_p)}.$ One easily checks that $K$ is a nullhomotopy for the operator
$$
\delta :
\eta_{(s_0, \ldots, s_p)}
\mapsto
\sum\nolimits_{i = 0}^p (-1)^p \eta_{(s_0, \ldots, \hat s_i, \ldots, s_p)}.
$$
See
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-homology-complex}
for more details.
\end{proof}
\begin{lemma}
\label{lemma-hom-injective}
Let $\mathcal{C}$ be a category. If $\mathcal{I}$ is an injective object of
$\textit{PAb}(\mathcal{C})$ and $\mathcal{U}$ is a family of morphisms with
fixed target in $\mathcal{C}$, then $\check H^p(\mathcal{U}, \mathcal{I}) = 0$
for all $p>0$.
\end{lemma}
\begin{proof}
The {\v C}ech complex is the result of applying the functor
$\Hom_{\textit{PAb}(\mathcal{C})}(-, \mathcal{I}) $ to the complex $
\mathbf{Z}^\bullet_\mathcal{U} $, i.e.,
$$
\check H^p(\mathcal{U}, \mathcal{I}) = H^p
(\Hom_{\textit{PAb}(\mathcal{C})} (\mathbf{Z}^\bullet_\mathcal{U},
\mathcal{I})).
$$
But we have just seen that $\mathbf{Z}^\bullet_\mathcal{U}$ is exact in
negative degrees, and the functor $\Hom_{\textit{PAb}(\mathcal{C})}(-,
\mathcal{I})$ is exact, hence $\Hom_{\textit{PAb}(\mathcal{C})}
(\mathbf{Z}^\bullet_\mathcal{U}, \mathcal{I})$ is exact in positive degrees.
\end{proof}
\begin{theorem}
\label{theorem-cech-derived}
On $\textit{PAb}(\mathcal{C})$ the functors $\check{H}^p(\mathcal{U}, -)$ are
the right derived functors of $\check{H}^0(\mathcal{U}, -)$.
\end{theorem}
\begin{proof}
By the Lemma \ref{lemma-hom-injective}, the functors
$\check H^p(\mathcal{U}, -)$ are universal
$\delta$-functors since they are effaceable.
So are the right derived functors of $\check H^0(\mathcal{U}, -)$. Since they
agree in degree $0$, they agree by the universal property of universal
$\delta$-functors. For more details see
Cohomology on Sites,
Lemma \ref{sites-cohomology-lemma-cech-cohomology-derived-presheaves}.
\end{proof}
\begin{remark}
\label{remark-presheaves-no-topology}
Observe that all of the preceding statements are about presheaves so we haven't
made use of the topology yet.
\end{remark}
\section{The {\v C}ech-to-cohomology spectral sequence}
\label{section-cech-ss}
\noindent
This spectral sequence is fundamental in proving foundational results on
cohomology of sheaves.
\begin{lemma}
\label{lemma-forget-injectives}
The forgetful functor $\textit{Ab}(\mathcal{C})\to \textit{PAb}(\mathcal{C})$
transforms injectives into injectives.
\end{lemma}
\begin{proof}
This is formal using the fact that the forgetful functor has a left adjoint,
namely sheafification, which is an exact functor. For more details see
Cohomology on Sites,
Lemma \ref{sites-cohomology-lemma-injective-abelian-sheaf-injective-presheaf}.
\end{proof}
\begin{theorem}
\label{theorem-cech-ss}
Let $\mathcal{C}$ be a site. For any covering
$\mathcal{U} = \{U_i \to U\}_{i \in I}$ of $U \in \Ob(\mathcal{C})$
and any abelian sheaf $\mathcal{F}$ on $\mathcal{C}$
there is a spectral sequence
$$
E_2^{p, q}
=
\check H^p(\mathcal{U}, \underline{H}^q(\mathcal{F}))
\Rightarrow
H^{p+q}(U, \mathcal{F}),
$$
where $\underline{H}^q(\mathcal{F})$ is the abelian presheaf
$V \mapsto H^q(V, \mathcal{F})$.
\end{theorem}
\begin{proof}
Choose an injective resolution $\mathcal{F}\to \mathcal{I}^\bullet$ in
$\textit{Ab}(\mathcal{C})$, and consider the double complex
$\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}^\bullet)$
and the maps
$$
\xymatrix{
\Gamma(U, I^\bullet) \ar[r] &
\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}^\bullet) \\
& \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \ar[u]
}
$$
Here the horizontal map is the natural map
$\Gamma(U, I^\bullet) \to
\check{\mathcal{C}}^0(\mathcal{U}, \mathcal{I}^\bullet)$
to the left column, and the vertical map is induced by
$\mathcal{F}\to \mathcal{I}^0$ and lands in the bottom row.
By assumption, $\mathcal{I}^\bullet$ is a complex of injectives in
$\textit{Ab}(\mathcal{C})$, hence by
Lemma \ref{lemma-forget-injectives}, it is a complex of injectives in
$\textit{PAb}(\mathcal{C})$. Thus, the rows of the double complex are
exact in positive degrees (Lemma \ref{lemma-hom-injective}), and
the kernel of $\check{\mathcal{C}}^0(\mathcal{U}, \mathcal{I}^\bullet)
\to \check{\mathcal{C}}^1(\mathcal{U}, \mathcal{I}^\bullet)$
is equal to
$\Gamma(U, \mathcal{I}^\bullet)$, since $\mathcal{I}^\bullet$
is a complex of sheaves. In particular, the cohomology of the total complex
is the standard
cohomology of the global sections functor $H^0(U, \mathcal{F})$.
\medskip\noindent
For the vertical direction, the $q$th cohomology group of the $p$th column is
$$
\prod_{i_0, \ldots, i_p}
H^q(U_{i_0} \times_U \ldots \times_U U_{i_p}, \mathcal{F})
=
\prod_{i_0, \ldots, i_p}
\underline{H}^q(\mathcal{F})(U_{i_0} \times_U \ldots \times_U U_{i_p})
$$
in the entry $E_1^{p, q}$. So this is a standard double complex spectral
sequence, and the $E_2$-page is as prescribed. For more details see
Cohomology on Sites,
Lemma \ref{sites-cohomology-lemma-cech-spectral-sequence}.
\end{proof}
\begin{remark}
\label{remark-grothendieck-ss}
This is a Grothendieck spectral sequence for the composition of functors
$$
\textit{Ab}(\mathcal{C}) \longrightarrow
\textit{PAb}(\mathcal{C}) \xrightarrow{\check H^0} \textit{Ab}.
$$
\end{remark}
\section{Big and small sites of schemes}
\label{section-big-small}
\noindent
Let $S$ be a scheme.
Let $\tau$ be one of the topologies we will be discussing.
Thus $\tau \in \{fppf, syntomic, smooth, \etale, Zariski\}$.
Of course if you are only interested in the \'etale topology, then
you can simply assume $\tau = \etale$ throughout. Moreover, we will
discuss \'etale morphisms, \'etale coverings, and \'etale sites
in more detail starting in Section \ref{section-etale-site}.
In order to proceed with the discussion of cohomology of
quasi-coherent sheaves it is convenient to introduce the
big $\tau$-site and in case $\tau \in \{\etale, Zariski\}$, the
small $\tau$-site of $S$. In order to do this we first introduce
the notion of a $\tau$-covering.
\begin{definition}
\label{definition-tau-covering}
(See
Topologies, Definitions
\ref{topologies-definition-fppf-covering},
\ref{topologies-definition-syntomic-covering},
\ref{topologies-definition-smooth-covering},
\ref{topologies-definition-etale-covering}, and
\ref{topologies-definition-zariski-covering}.)
Let $\tau \in \{fppf, syntomic, smooth, \etale, Zariski\}$.
A family of morphisms of schemes $\{f_i : T_i \to T\}_{i \in I}$ with fixed
target is called a {\it $\tau$-covering} if and only if
each $f_i$ is flat of finite presentation, syntomic, smooth, \'etale,
resp.\ an open immersion, and we have $\bigcup f_i(T_i) = T$.
\end{definition}
\noindent
It turns out that the class of all $\tau$-coverings satisfies the axioms
(1), (2) and (3) of
Definition \ref{definition-site} (our definition of a site), see
Topologies, Lemmas
\ref{topologies-lemma-fppf},
\ref{topologies-lemma-syntomic},
\ref{topologies-lemma-smooth},
\ref{topologies-lemma-etale}, and
\ref{topologies-lemma-zariski}.
In order to be able to compare any of these new topologies to the fpqc topology
and to use the preceding results on descent on modules we single out a special
class of $\tau$-coverings of affine schemes called standard coverings.
\begin{definition}
\label{definition-standard-tau}
(See
Topologies, Definitions
\ref{topologies-definition-standard-fppf},
\ref{topologies-definition-standard-syntomic},
\ref{topologies-definition-standard-smooth},
\ref{topologies-definition-standard-etale}, and
\ref{topologies-definition-standard-Zariski}.)
Let $\tau \in \{fppf, syntomic, smooth, \etale, Zariski\}$.
Let $T$ be an affine scheme.
A {\it standard $\tau$-covering} of $T$ is a family
$\{f_j : U_j \to T\}_{j = 1, \ldots, m}$ with each $U_j$ is affine,
and each $f_j$ flat and of finite presentation,
standard syntomic, standard smooth, \'etale, resp.\ the immersion of a
standard principal open in $T$ and $T = \bigcup f_j(U_j)$.
\end{definition}
\begin{lemma}
\label{lemma-tau-affine}
Let $\tau \in \{fppf, syntomic, smooth, \etale, Zariski\}$.
Any $\tau$-covering of an affine scheme can be refined by a
standard $\tau$-covering.
\end{lemma}
\begin{proof}
See
Topologies, Lemmas
\ref{topologies-lemma-fppf-affine},
\ref{topologies-lemma-syntomic-affine},
\ref{topologies-lemma-smooth-affine},
\ref{topologies-lemma-etale-affine}, and
\ref{topologies-lemma-zariski-affine}.
\end{proof}
\noindent
Finally, we come to our definition of the sites we will be working with.
This is actually somewhat involved since, contrary to what happens in
\cite{SGA4}, we do not want to choose a universe. Instead we pick a ``partial
universe'' (which is a suitably large set as in
Sets, Section \ref{sets-section-sets-hierarchy}), and consider all schemes
contained in this set. Of course we make sure that our favorite base scheme
$S$ is contained in the partial universe. Having picked the underlying category
we pick a suitably large set of $\tau$-coverings which turns this into a site.
The details are in the chapter on topologies on schemes; there is a lot of
freedom in the choices made, but in the end the actual choices made will not
affect the \'etale (or other) cohomology of $S$ (just as in \cite{SGA4} the
actual choice of universe doesn't matter at the end). Moreover, the way the
material is written the reader who is happy using strongly inaccessible
cardinals (i.e., universes) can do so as a substitute.
\begin{definition}
\label{definition-tau-site}
Let $S$ be a scheme.
Let $\tau \in \{fppf, syntomic, smooth, \etale, \linebreak[0] Zariski\}$.
\begin{enumerate}
\item A {\it big $\tau$-site of $S$} is any of the sites
$(\Sch/S)_\tau$ constructed as explained above and in more detail in
Topologies, Definitions
\ref{topologies-definition-big-small-fppf},
\ref{topologies-definition-big-small-syntomic},
\ref{topologies-definition-big-small-smooth},
\ref{topologies-definition-big-small-etale}, and
\ref{topologies-definition-big-small-Zariski}.
\item If $\tau \in \{\etale, Zariski\}$, then the
{\it small $\tau$-site of $S$}
is the full subcategory $S_\tau$ of $(\Sch/S)_\tau$ whose objects
are schemes $T$ over $S$ whose structure morphism $T \to S$ is \'etale,
resp.\ an open immersion. A covering in $S_\tau$ is a covering
$\{U_i \to U\}$ in $(\Sch/S)_\tau$
such that $U$ is an object of $S_\tau$.
\end{enumerate}
\end{definition}
\noindent
The underlying category of the site $(\Sch/S)_\tau$ has reasonable
``closure'' properties, i.e., given a scheme $T$ in it any locally closed
subscheme of $T$ is isomorphic to an object of $(\Sch/S)_\tau$.
Other such closure properties are: closed under fibre products of schemes,
taking countable disjoint unions,
taking finite type schemes over a given scheme, given an affine scheme
$\Spec(R)$ one can complete, localize, or take the quotient of $R$
by an ideal while staying inside the category, etc.
On the other hand, for example arbitrary disjoint unions
of schemes in $(\Sch/S)_\tau$ will take you outside of it.
Also note that, given an object $T$ of $(\Sch/S)_\tau$ there will exist
$\tau$-coverings $\{T_i \to T\}_{i \in I}$ (as in
Definition \ref{definition-tau-covering})
which are not coverings in $(\Sch/S)_\tau$ for example
because the schemes $T_i$ are not objects of the category
$(\Sch/S)_\tau$. But our choice of the sites $(\Sch/S)_\tau$
is such that there always does exist
a covering $\{U_j \to T\}_{j \in J}$ of $(\Sch/S)_\tau$ which refines
the covering $\{T_i \to T\}_{i \in I}$, see
Topologies, Lemmas
\ref{topologies-lemma-fppf-induced},
\ref{topologies-lemma-syntomic-induced},
\ref{topologies-lemma-smooth-induced},
\ref{topologies-lemma-etale-induced}, and
\ref{topologies-lemma-zariski-induced}.
We will mostly ignore these issues in this chapter.
\medskip\noindent
If $\mathcal{F}$ is a sheaf on $(\Sch/S)_\tau$ or $S_\tau$, then
we denote
$$
H^p_\tau(U, \mathcal{F}), \text{ in particular }
H^p_\tau(S, \mathcal{F})
$$
the cohomology groups of $\mathcal{F}$ over the object $U$ of the site, see
Section \ref{section-cohomology}. Thus we have
$H^p_{fppf}(S, \mathcal{F})$,
$H^p_{syntomic}(S, \mathcal{F})$,
$H^p_{smooth}(S, \mathcal{F})$,
$H^p_\etale(S, \mathcal{F})$, and
$H^p_{Zar}(S, \mathcal{F})$. The last two are potentially ambiguous since
they might refer to either the big or small \'etale or Zariski site. However,
this ambiguity is harmless by the following lemma.
\begin{lemma}
\label{lemma-compare-cohomology-big-small}
Let $\tau \in \{\etale, Zariski\}$.
If $\mathcal{F}$ is an abelian sheaf defined on
$(\Sch/S)_\tau$, then
the cohomology groups of $\mathcal{F}$ over $S$ agree with the cohomology
groups of $\mathcal{F}|_{S_\tau}$ over $S$.
\end{lemma}
\begin{proof}
By
Topologies, Lemmas \ref{topologies-lemma-at-the-bottom} and
\ref{topologies-lemma-at-the-bottom-etale}
the functors $S_\tau \to (\Sch/S)_\tau$
satisfy the hypotheses of
Sites, Lemma \ref{sites-lemma-bigger-site}.
Hence our lemma follows from
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cohomology-bigger-site}.
\end{proof}
\noindent
For completeness we state and prove the invariance under choice of partial
universe of the cohomology groups we are considering. We will prove invariance
of the small \'etale topos in
Lemma \ref{lemma-etale-topos-independent-partial-universe} below.
For notation and terminology used in this lemma we refer to
Topologies, Section \ref{topologies-section-change-alpha}.
\begin{lemma}
\label{lemma-cohomology-enlarge-partial-universe}
Let $\tau \in \{fppf, syntomic, smooth, \etale, Zariski\}$.
Let $S$ be a scheme.
Let $(\Sch/S)_\tau$ and $(\Sch'/S)_\tau$ be two
big $\tau$-sites of $S$, and assume that the first is contained in the second.
In this case
\begin{enumerate}
\item for any abelian sheaf $\mathcal{F}'$ defined on $(\Sch'/S)_\tau$ and
any object $U$ of $(\Sch/S)_\tau$ we have
$$
H^p_\tau(U, \mathcal{F}'|_{(\Sch/S)_\tau}) =
H^p_\tau(U, \mathcal{F}')
$$
In words: the cohomology of $\mathcal{F}'$ over $U$ computed in the bigger site
agrees with the cohomology of $\mathcal{F}'$ restricted to the smaller site
over $U$.
\item for any abelian sheaf $\mathcal{F}$ on $(\Sch/S)_\tau$ there is an
abelian sheaf $\mathcal{F}'$ on $(\Sch/S)_\tau'$ whose restriction to
$(\Sch/S)_\tau$ is isomorphic to $\mathcal{F}$.
\end{enumerate}
\end{lemma}
\begin{proof}
By Topologies, Lemma \ref{topologies-lemma-change-alpha} the inclusion functor
$(\Sch/S)_\tau \to (\Sch'/S)_\tau$ satisfies the assumptions of
Sites, Lemma \ref{sites-lemma-bigger-site}. This implies (2) and (1)
follows from
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cohomology-bigger-site}.
\end{proof}
\section{The \'etale topos}
\label{section-etale-topos}
\noindent
A {\it topos} is the category of sheaves of sets on a site, see
Sites, Definition \ref{sites-definition-topos}. Hence it is customary
to refer to the use the phrase ``\'etale topos of a scheme'' to refer to
the category of sheaves on the small \'etale site of a scheme.
Here is the formal definition.
\begin{definition}
\label{definition-etale-topos}
Let $S$ be a scheme.
\begin{enumerate}
\item The {\it \'etale topos}, or the {\it small \'etale topos}
of $S$ is the category $\Sh(S_\etale)$ of sheaves of sets on
the small \'etale site of $S$.
\item The {\it Zariski topos}, or the {\it small Zariski topos}
of $S$ is the category $\Sh(S_{Zar})$ of sheaves of sets on the
small Zariski site of $S$.
\item For $\tau \in \{fppf, syntomic, smooth, \etale, Zariski\}$ a
{\it big $\tau$-topos} is the category of sheaves of set on a
big $\tau$-topos of $S$.
\end{enumerate}
\end{definition}
\noindent
Note that the small Zariski topos of $S$ is simply the category of sheaves
of sets on the underlying topological space of $S$, see
Topologies, Lemma \ref{topologies-lemma-Zariski-usual}.
Whereas the small \'etale topos does not depend on the choices made in the
construction of the small \'etale site, in general the big topoi do depend
on those choices.
\medskip\noindent
Here is a lemma, which is one of many possible lemmas expressing the
fact that it doesn't matter too much which site we choose to define
the small \'etale topos of a scheme.
\begin{lemma}
\label{lemma-alternative}
Let $S$ be a scheme. Let $S_{affine, \etale}$ denote the
full subcategory of $S_\etale$
whose objects are those $U/S \in \Ob(S_\etale)$ with
$U$ affine. A covering of $S_{affine, \etale}$ will be a standard
\'etale covering, see
Topologies, Definition \ref{topologies-definition-standard-etale}.
Then restriction
$$
\mathcal{F} \longmapsto \mathcal{F}|_{S_{affine, \etale}}
$$
defines an equivalence of topoi
$\Sh(S_\etale) \cong \Sh(S_{affine, \etale})$.
\end{lemma}
\begin{proof}
This you can show directly from the definitions, and is a good exercise.
But it also follows immediately from
Sites, Lemma \ref{sites-lemma-equivalence}
by checking that the inclusion functor
$S_{affine, \etale} \to S_\etale$
is a special cocontinuous functor (see
Sites, Definition \ref{sites-definition-special-cocontinuous-functor}).
\end{proof}
\begin{lemma}
\label{lemma-etale-topos-independent-partial-universe}
Let $S$ be a scheme. The \'etale topos of $S$ is independent
(up to canonical equivalence) of the construction of the small
\'etale site in Definition \ref{definition-tau-site}.
\end{lemma}
\begin{proof}
We have to show, given two big \'etale sites
$\Sch_\etale$ and $\Sch_\etale'$ containing
$S$, then $\Sh(S_\etale) \cong \Sh(S_\etale')$
with obvious notation. By Topologies, Lemma \ref{topologies-lemma-contained-in}
we may assume $\Sch_\etale \subset \Sch_\etale'$.
By Sets, Lemma \ref{sets-lemma-what-is-in-it}
any affine scheme \'etale over $S$ is isomorphic to an object
of both $\Sch_\etale$ and $\Sch_\etale'$.
Thus the induced functor
$S_{affine, \etale} \to S_{affine, \etale}'$
is an equivalence. Moreover, it is clear that both this functor
and a quasi-inverse map transform standard \'etale coverings into
standard \'etale coverings.
Hence the result follows from Lemma \ref{lemma-alternative}.
\end{proof}
\section{Cohomology of quasi-coherent sheaves}
\label{section-cohomology-quasi-coherent}
%9.22.09
\noindent
We start with a simple lemma (which holds in greater generality than
stated). It says that the {\v C}ech complex of a standard covering is
equal to the {\v C}ech complex of an fpqc covering of the form
$\{\Spec(B) \to \Spec(A)\}$ with $A \to B$ faithfully flat.
\begin{lemma}
\label{lemma-cech-complex}
Let $\tau \in \{fppf, syntomic, smooth, \etale, Zariski\}$.
Let $S$ be a scheme.
Let $\mathcal{F}$ be an abelian sheaf on $(\Sch/S)_\tau$, or on
$S_\tau$ in case $\tau = \etale$, and let
$\mathcal{U} = \{U_i \to U\}_{i \in I}$
be a standard $\tau$-covering of this site.
Let $V = \coprod_{i \in I} U_i$. Then
\begin{enumerate}
\item $V$ is an affine scheme,
\item $\mathcal{V} = \{V \to U\}$ is a $\tau$-covering and an fpqc covering,
\item the {\v C}ech complexes
$\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F})$ and
$\check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{F})$ agree.
\end{enumerate}
\end{lemma}
\begin{proof}
As the covering is a standard $\tau$-covering each of the schemes
$U_i$ is affine and $I$ is a finite set. Hence $V$ is an affine scheme.
It is clear that $V \to U$ is flat and surjective, hence
$\mathcal{V}$ is an fpqc covering, see
Example \ref{example-fpqc-coverings}.
Note that $\mathcal{U}$ is a refinement of $\mathcal{V}$
and hence there is a map of {\v C}ech complexes
$\check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{F}) \to
\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F})$, see
Cohomology on Sites,
Equation (\ref{sites-cohomology-equation-map-cech-complexes}).
Next, we observe that if $T = \coprod_{j \in J} T_j$ is a
disjoint union of schemes in the site on which $\mathcal{F}$ is defined
then the family of morphisms with fixed target
$\{T_j \to T\}_{j \in J}$ is a Zariski covering, and so
\begin{equation}
\label{equation-sheaf-coprod}
\mathcal{F}(T) =
\mathcal{F}(\coprod\nolimits_{j \in J} T_j) =
\prod\nolimits_{j \in J} \mathcal{F}(T_j)
\end{equation}
by the sheaf condition of $\mathcal{F}$.
This implies the map of {\v C}ech complexes above is an isomorphism
in each degree because
$$
V \times_U \ldots \times_U V
=
\coprod\nolimits_{i_0, \ldots i_p} U_{i_0} \times_U \ldots \times_U U_{i_p}
$$
as schemes.
\end{proof}
\noindent
Note that Equality (\ref{equation-sheaf-coprod})
is false for a general presheaf. Even for sheaves it does not hold on any
site, since coproducts may not lead to coverings, and may not be disjoint.
But it does for all the usual ones (at least all the ones we will study).
\begin{remark}
\label{remark-refinement}
In the statement of Lemma \ref{lemma-cech-complex} the covering $\mathcal{U}$
is a refinement of $\mathcal{V}$ but not the other way around. Coverings
of the form $\{V \to U\}$ do not form an initial subcategory of the
category of all coverings of $U$. Yet it is still true that
we can compute {\v C}ech cohomology $\check H^n(U, \mathcal{F})$ (which
is defined as the colimit over the opposite of the category of
coverings $\mathcal{U}$ of $U$ of the {\v C}ech cohomology groups of
$\mathcal{F}$ with respect to $\mathcal{U}$) in terms of the coverings
$\{V \to U\}$. We will formulate a precise lemma (it only works for sheaves)
and add it here if we ever need it.
\end{remark}
\begin{lemma}[Locality of cohomology]
\label{lemma-locality-cohomology}
Let $\mathcal{C}$ be a site, $\mathcal{F}$ an abelian sheaf on $\mathcal{C}$,
$U$ an object of $\mathcal{C}$, $p >0$ an integer and $\xi \in
H^p(U, \mathcal{F})$. Then there exists a covering
$\mathcal{U} = \{U_i \to U\}_{i \in I}$ of $U$ in $\mathcal{C}$
such that $\xi |_{U_i} = 0$ for all $i \in I$.
\end{lemma}
\begin{proof}
Choose an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet$. Then
$\xi$ is represented by a cocycle $\tilde{\xi} \in \mathcal{I}^p(U)$
with $d^p(\tilde{\xi}) = 0$. By assumption, the sequence
$\mathcal{I}^{p - 1} \to \mathcal{I}^p \to \mathcal{I}^{p + 1}$ in exact in
$\textit{Ab}(\mathcal{C})$, which means that there exists a covering
$\mathcal{U} = \{U_i \to U\}_{i \in I}$ such that
$\tilde{\xi}|_{U_i} = d^{p - 1}(\xi_i)$ for some
$\xi_i \in \mathcal{I}^{p-1}(U_i)$. Since
the cohomology class $\xi|_{U_i}$ is represented by the cocycle
$\tilde{\xi}|_{U_i}$ which is a coboundary, it vanishes.
For more details see
Cohomology on Sites,
Lemma \ref{sites-cohomology-lemma-kill-cohomology-class-on-covering}.
\end{proof}
\begin{theorem}
\label{theorem-zariski-fpqc-quasi-coherent}
Let $S$ be a scheme and $\mathcal{F}$ a quasi-coherent $\mathcal{O}_S$-module.
Let $\mathcal{C}$ be either $(\Sch/S)_\tau$ for
$\tau \in \{fppf, syntomic, smooth, \etale, Zariski\}$ or
$S_\etale$. Then
$$
H^p(S, \mathcal{F}) = H^p_\tau(S, \mathcal{F}^a)
$$
for all $p \geq 0$ where
\begin{enumerate}
\item the left hand side indicates the usual cohomology of the sheaf
$\mathcal{F}$ on the underlying topological space of the scheme $S$, and
\item the right hand side indicates cohomology
of the abelian sheaf $\mathcal{F}^a$ (see
Proposition \ref{proposition-quasi-coherent-sheaf-fpqc})
on the site $\mathcal{C}$.
\end{enumerate}
\end{theorem}
\begin{proof}
We are going to show that
$H^p(U, f^*\mathcal{F}) = H^p_\tau(U, \mathcal{F}^a)$
for any object $f : U \to S$ of the site $\mathcal{C}$.
The result is true for $p = 0$ by the sheaf property.
\medskip\noindent
Assume that $U$ is affine. Then we want to prove that
$H^p_\tau(U, \mathcal{F}^a) = 0$ for all $p > 0$. We use induction on $p$.
\begin{enumerate}
\item[$p = 1$]
Pick $\xi \in H^1_\tau(U, \mathcal{F}^a)$.
By Lemma \ref{lemma-locality-cohomology},
there exists an fpqc covering $\mathcal{U} = \{U_i \to U\}_{i \in I}$
such that $\xi|_{U_i} = 0$ for all $i \in I$. Up to refining
$\mathcal{U}$, we may assume that $\mathcal{U}$ is a standard
$\tau$-covering. Applying the spectral sequence of
Theorem \ref{theorem-cech-ss},
we see that $\xi$ comes from a cohomology class
$\check \xi \in \check H^1(\mathcal{U}, \mathcal{F}^a)$.
Consider the covering $\mathcal{V} = \{\coprod_{i\in I} U_i \to U\}$. By
Lemma \ref{lemma-cech-complex},
$\check H^\bullet(\mathcal{U}, \mathcal{F}^a) =
\check H^\bullet(\mathcal{V}, \mathcal{F}^a)$.
On the other hand, since $\mathcal{V}$ is a covering of the form
$\{\Spec(B) \to \Spec(A)\}$ and $f^*\mathcal{F} = \widetilde{M}$
for some $A$-module $M$, we see the {\v C}ech complex
$\check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{F})$
is none other than the complex $(B/A)_\bullet \otimes_A M$.
Now by Lemma \ref{lemma-descent-modules},
$H^p((B/A)_\bullet \otimes_A M) = 0$ for $p>0$, hence $\check \xi = 0$
and so $\xi = 0$.
\item[$p > 1$]
Pick $\xi \in H^p_\tau(U, \mathcal{F}^a)$. By
Lemma \ref{lemma-locality-cohomology},
there exists an fpqc covering $\mathcal{U} = \{U_i \to U\}_{i \in I}$
such that $\xi|_{U_i} = 0$ for all $i \in I$. Up to refining
$\mathcal{U}$, we may assume that $\mathcal{U}$ is a standard
$\tau$-covering. We apply the spectral sequence of
Theorem \ref{theorem-cech-ss}.
Observe that the intersections $U_{i_0} \times_U \ldots \times_U U_{i_p}$
are affine, so that by induction hypothesis the cohomology groups
$$
E_2^{p, q} = \check H^p(\mathcal{U}, \underline{H}^q(\mathcal{F}^a))
$$
vanish for all $0 < q < p$. We see that $\xi$ must come from a
$\check \xi \in \check H^p(\mathcal{U}, \mathcal{F}^a)$. Replacing
$\mathcal{U}$ with the covering $\mathcal{V}$ containing only one morphism
and using Lemma \ref{lemma-descent-modules} again,
we see that the {\v C}ech cohomology class $\check \xi$ must be zero,
hence $\xi = 0$.
\end{enumerate}
Next, assume that $U$ is separated. Choose an affine open covering
$U = \bigcup_{i \in I} U_i$ of $U$. The family
$\mathcal{U} = \{U_i \to U\}_{i \in I}$ is then an fpqc covering,
and all the intersections
$U_{i_0} \times_U \ldots \times_U U_{i_p}$ are affine
since $U$ is separated. So all rows of the spectral sequence of
Theorem \ref{theorem-cech-ss}
are zero, except the zeroth row. Therefore
$$
H^p_\tau(U, \mathcal{F}^a) =
\check H^p(\mathcal{U}, \mathcal{F}^a) =
\check H^p(\mathcal{U}, \mathcal{F}) = H^p(U, \mathcal{F})
$$
where the last equality results from standard scheme theory, see
Cohomology of Schemes, Lemma
\ref{coherent-lemma-cech-cohomology-quasi-coherent}.
\medskip\noindent
The general case is technical and (to extend the proof as given here)
requires a discussion about maps of spectral sequences, so we won't treat it.
It follows from
Descent, Proposition \ref{descent-proposition-same-cohomology-quasi-coherent}
(whose proof takes a slightly different approach) combined with
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cohomology-of-open}.
\end{proof}
\begin{remark}
\label{remark-right-derived-global-sections}
Comment on Theorem \ref{theorem-zariski-fpqc-quasi-coherent}.
Since $S$ is a final object in the category $\mathcal{C}$, the cohomology
groups on the right-hand side are merely the right derived functors of the
global sections functor. In fact the proof shows that
$H^p(U, f^*\mathcal{F}) = H^p_\tau(U, \mathcal{F}^a)$
for any object $f : U \to S$ of the site $\mathcal{C}$.
\end{remark}
\section{Examples of sheaves}
\label{section-examples-sheaves}
\noindent
Let $S$ and $\tau$ be as in Section \ref{section-big-small}.
We have already seen that any representable presheaf is a sheaf on
$(\Sch/S)_\tau$ or $S_\tau$, see
Lemma \ref{lemma-representable-sheaf-fpqc}
and
Remark \ref{remark-fpqc-finest}.
Here are some special cases.
\begin{definition}
\label{definition-additive-sheaf}
On any of the sites $(\Sch/S)_\tau$ or $S_\tau$ of
Section \ref{section-big-small}.
\begin{enumerate}
\item The sheaf $T \mapsto \Gamma(T, \mathcal{O}_T)$ is denoted
$\mathcal{O}_S$, or $\mathbf{G}_a$, or $\mathbf{G}_{a, S}$ if we
want to indicate the base scheme.
\item Similarly, the sheaf
$T \mapsto \Gamma(T, \mathcal{O}^*_T)$ is denoted $\mathcal{O}_S^*$, or
$\mathbf{G}_m$, or $\mathbf{G}_{m, S}$ if we want
to indicate the base scheme.
\item The {\it constant sheaf} $\underline{\mathbf{Z}/n\mathbf{Z}}$ on any
site is the sheafification of the constant presheaf
$U \mapsto \mathbf{Z}/n\mathbf{Z}$.
\end{enumerate}
\end{definition}
\noindent
The first is a sheaf by
Theorem \ref{theorem-quasi-coherent}
for example. The second is a sub presheaf of the first, which is easily seen
to be a sheaf itself. The third is a sheaf by definition.
Note that each of these sheaves is representable.
The first and second by the schemes $\mathbf{G}_{a, S}$ and
$\mathbf{G}_{m, S}$, see
Groupoids, Section \ref{groupoids-section-group-schemes}.
The third by the finite \'etale group scheme $\mathbf{Z}/n\mathbf{Z}_S$
sometimes denoted $(\mathbf{Z}/n\mathbf{Z})_S$
which is just $n$ copies of $S$ endowed
with the obvious group scheme structure over $S$, see
Groupoids, Example \ref{groupoids-example-constant-group}
and the following remark.
\begin{remark}
\label{remark-constant-locally-constant-maps}
Let $G$ be an abstract group.
On any of the sites $(\Sch/S)_\tau$ or $S_\tau$ of
Section \ref{section-big-small}
the sheafification $\underline{G}$
of the constant presheaf associated to $G$ in the
{\it Zariski topology} of the site already gives
$$
\Gamma(U, \underline{G}) =
\{\text{Zariski locally constant maps }U \to G\}
$$
This Zariski sheaf is representable by the group scheme $G_S$ according to
Groupoids, Example \ref{groupoids-example-constant-group}.
By
Lemma \ref{lemma-representable-sheaf-fpqc}
any representable presheaf satisfies the sheaf condition for the
$\tau$-topology as well, and hence we conclude that the Zariski
sheafification $\underline{G}$ above is also the $\tau$-sheafification.
\end{remark}
\begin{definition}
\label{definition-structure-sheaf}
Let $S$ be a scheme. The {\it structure sheaf} of $S$ is the sheaf of rings
$\mathcal{O}_S$
on any of the sites $S_{Zar}$, $S_\etale$, or $(\Sch/S)_\tau$
discussed above.
\end{definition}
\noindent
If there is some possible confusion as to which site we are working on
then we will indicate this by using indices. For example we may use
$\mathcal{O}_{S_\etale}$ to stress the fact that we are working on the
small \'etale site of $S$.
\begin{remark}
\label{remark-special-case-fpqc-cohomology-quasi-coherent}
In the terminology introduced above a special case of
Theorem \ref{theorem-zariski-fpqc-quasi-coherent}
is
$$
H_{fppf}^p(X, \mathbf{G}_a) =
H_\etale^p(X, \mathbf{G}_a) =
H_{Zar}^p(X, \mathbf{G}_a) =
H^p(X, \mathcal{O}_X)
$$
for all $p \geq 0$. Moreover, we could use the notation
$H^p_{fppf}(X, \mathcal{O}_X)$ to indicate the cohomology of the
structure sheaf on the big fppf site of $X$.
\end{remark}
\section{Picard groups}
\label{section-picard-groups}
\noindent
The following theorem is sometimes called ``Hilbert 90''.
\begin{theorem}
\label{theorem-picard-group}
For any scheme $X$ we have canonical identifications
\begin{align*}
H_{fppf}^1(X, \mathbf{G}_m) & = H^1_{syntomic}(X, \mathbf{G}_m) \\
& = H^1_{smooth}(X, \mathbf{G}_m) \\
& = H_\etale^1(X, \mathbf{G}_m) \\
& = H^1_{Zar}(X, \mathbf{G}_m) \\
& = \text{Pic}(X) \\
& = H^1(X, \mathcal{O}_X^*)
\end{align*}
\end{theorem}
\begin{proof}
Let $\tau$ be one of the topologies considered in
Section \ref{section-big-small}.
By
Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-h1-invertible}
we see that
$H^1_\tau(X, \mathbf{G}_m) =
H^1_\tau(X, \mathcal{O}_\tau^*) =
\text{Pic}(\mathcal{O}_\tau)$
where $\mathcal{O}_\tau$ is the structure sheaf of the site
$(\Sch/X)_\tau$. Now an invertible $\mathcal{O}_\tau$-module
is a quasi-coherent $\mathcal{O}_\tau$-module.
By Theorem \ref{theorem-quasi-coherent} or the more precise
Descent, Proposition \ref{descent-proposition-equivalence-quasi-coherent}
we see that $\text{Pic}(\mathcal{O}_\tau) = \text{Pic}(X)$.
The last equality is proved in the same way.
\end{proof}
\section{The \'etale site}
\label{section-etale-site}
\noindent
At this point we start exploring the \'etale site of a scheme in
more detail. As a first step we discuss a little the notion of an
\'etale morphism.
\section{\'Etale morphisms}
\label{section-etale-morphism}
\noindent
For more details, see
Morphisms, Section \ref{morphisms-section-etale}
for the formal definition and
\'Etale Morphisms, Sections
\ref{etale-section-etale-morphisms},
\ref{etale-section-structure-etale-map},
\ref{etale-section-etale-smooth},
\ref{etale-section-topological-etale},
\ref{etale-section-functorial-etale}, and
\ref{etale-section-properties-permanence}
for a survey of interesting properties of \'etale morphisms.
\medskip\noindent
Recall that an algebra $A$ over an algebraically closed field $k$ is
{\it smooth} if it is of finite type and the module of differentials
$\Omega_{A/k}$ is finite locally free of rank equal to the dimension.
A scheme $X$ over $k$ is {\it smooth} over $k$ if it is locally of finite
type and each affine open is the spectrum of a smooth $k$-algebra.
If $k$ is not algebraically closed then an $A$-algebra is said to be
a smooth $k$-algebra if $A \otimes_k \overline{k}$ is a smooth
$\overline{k}$-algebra. A ring map $A \to B$ is smooth if it is
flat, finitely presented, and for all primes $\mathfrak p \subset A$
the fibre ring $\kappa(\mathfrak p) \otimes_A B$ is smooth over the residue
field $\kappa(\mathfrak p)$. More generally, a morphism of schemes is
{\it smooth} if it is flat, locally of finite presentation, and the
geometric fibers are smooth.
\medskip\noindent
For these facts please see
Morphisms, Section \ref{morphisms-section-smooth}.
Using this we may define an \'etale morphism as follows.
\begin{definition}
\label{definition-etale-morphism}
A morphism of schemes is {\it \'etale} if it is smooth of relative dimension 0.
\end{definition}
\noindent
In particular, a morphism of schemes $X \to S$ is \'etale if it is smooth
and $\Omega_{X/S} = 0$.
\begin{proposition}
\label{proposition-etale-morphisms}
Facts on \'etale morphisms.
\begin{enumerate}
\item Let $k$ be a field. A morphism of schemes $U \to \Spec(k)$ is
\'etale if and only if $U \cong \coprod_{i \in I} \Spec(k_i)$
such that for each $i \in I$
the ring $k_i$ is a field which is a finite separable extension of $k$.
\item Let $\varphi : U \to S$ be a morphism of schemes. The following
conditions are equivalent:
\begin{enumerate}
\item $\varphi$ is \'etale,
\item $\varphi$ is locally finitely presented, flat, and all its fibres are
\'etale,
\item $\varphi$ is flat, unramified and locally of finite presentation.
\end{enumerate}
\item A ring map $A \to B$ is \'etale if and only if
$B \cong A[x_1, \ldots, x_n]/(f_1, \ldots, f_n)$
such that $\Delta = \det \left( \frac{\partial f_i}{\partial x_j} \right)$
is invertible in $B$.
\item The base change of an \'etale morphism is \'etale.
\item Compositions of \'etale morphisms are \'etale.
\item Fibre products and products of \'etale morphisms are \'etale.
\item An \'etale morphism has relative dimension 0.
\item Let $Y \to X$ be an \'etale morphism.
If $X$ is reduced (respectively regular) then so is $Y$.
\item \'Etale morphisms are open.
\item If $X \to S$ and $Y \to S$ are \'etale, then any
$S$-morphism $X \to Y$ is also \'etale.
\end{enumerate}
\end{proposition}
\begin{proof}
We have proved these facts (and more) in the preceding chapters.
Here is a list of references:
(1) Morphisms, Lemma \ref{morphisms-lemma-etale-over-field}.
(2) Morphisms, Lemmas \ref{morphisms-lemma-etale-flat-etale-fibres}
and \ref{morphisms-lemma-flat-unramified-etale}.
(3) Algebra, Lemma \ref{algebra-lemma-etale-standard-smooth}.
(4) Morphisms, Lemma \ref{morphisms-lemma-base-change-etale}.
(5) Morphisms, Lemma \ref{morphisms-lemma-composition-etale}.
(6) Follows formally from (4) and (5).
(7) Morphisms, Lemmas \ref{morphisms-lemma-etale-locally-quasi-finite}
and \ref{morphisms-lemma-locally-quasi-finite-rel-dimension-0}.
(8) See Algebra, Lemmas \ref{algebra-lemma-reduced-goes-up} and
\ref{algebra-lemma-Rk-goes-up}, see also more results of this kind
in \'Etale Morphisms, Section \ref{etale-section-properties-permanence}.
(9) See Morphisms, Lemma \ref{morphisms-lemma-fppf-open} and
\ref{morphisms-lemma-etale-flat}.
(10) See Morphisms, Lemma \ref{morphisms-lemma-etale-permanence}.
\end{proof}
\begin{definition}
\label{definition-standard-etale}
A ring map $A \to B$ is called {\it standard \'etale} if
$B \cong \left(A[t]/(f)\right)_g$ with $f, g \in A[t]$, with $f$ monic,
and $\text{d}f/\text{d}t$ invertible in $B$.
\end{definition}
\noindent
It is true that a standard \'etale ring map is \'etale. Namely, suppose
that $B = \left(A[t]/(f)\right)_g$ with $f, g \in A[t]$, with $f$ monic,
and $\text{d}f/\text{d}t$ invertible in $B$. Then $A[t]/(f)$ is a finite
free $A$-module of rank equal to the degree of the monic polynomial $f$.
Hence $B$, as a localization of this free algebra is finitely presented
and flat over $A$. To finish the proof that $B$ is \'etale it suffices
to show that the fibre rings
$$
\kappa(\mathfrak p) \otimes_A B
\cong
\kappa(\mathfrak p) \otimes_A (A[t]/(f))_g
\cong
\kappa(\mathfrak p)[t, 1/\overline{g}]/(\overline{f})
$$
are finite products of finite separable field extensions.
Here $\overline{f}, \overline{g} \in \kappa(\mathfrak p)[t]$ are
the images of $f$ and $g$. Let
$$
\overline{f} = \overline{f}_1 \ldots \overline{f}_a
\overline{f}_{a + 1}^{e_1} \ldots \overline{f}_{a + b}^{e_b}
$$
be the factorization of $\overline{f}$ into powers of pairwise distinct
irreducible monic factors $\overline{f}_i$ with $e_1, \ldots, e_b > 0$.
By assumption $\text{d}\overline{f}/\text{d}t$ is invertible in
$\kappa(\mathfrak p)[t, 1/\overline{g}]$. Hence we see that
at least all the $\overline{f}_i$, $i > a$ are invertible. We conclude
that
$$
\kappa(\mathfrak p)[t, 1/\overline{g}]/(\overline{f})
\cong
\prod\nolimits_{i \in I} \kappa(\mathfrak p)[t]/(\overline{f}_i)
$$
where $I \subset \{1, \ldots, a\}$ is the subset of indices $i$ such that
$\overline{f}_i$ does not divide $\overline{g}$. Moreover, the image of
$\text{d}\overline{f}/\text{d}t$ in the factor
$\kappa(\mathfrak p)[t]/(\overline{f}_i)$ is clearly equal to a
unit times $\text{d}\overline{f}_i/\text{d}t$. Hence we conclude that
$\kappa_i = \kappa(\mathfrak p)[t]/(\overline{f}_i)$ is a finite field
extension of $\kappa(\mathfrak p)$ generated by one element whose
minimal polynomial is separable, i.e., the field extension
$\kappa(\mathfrak p) \subset \kappa_i$ is finite separable as desired.
\medskip\noindent
It turns out that any \'etale ring map is locally standard \'etale.
To formulate this we introduce the following notation.
A ring map $A \to B$ is {\it \'etale at a prime $\mathfrak q$} of $B$ if there
exists $h \in B$, $h \not \in \mathfrak q$ such that $A \to B_h$ is \'etale.
Here is the result.
\begin{theorem}
\label{theorem-standard-etale}
A ring map $A \to B$ is \'etale at a prime $\mathfrak q$ if and only if there
exists $g \in B$, $g \not \in \mathfrak q$ such that $B_g$ is standard
\'etale over $A$.
\end{theorem}
\begin{proof}
See
Algebra, Proposition \ref{algebra-proposition-etale-locally-standard}.
\end{proof}
\section{\'Etale coverings}
\label{section-etale-covering}
\noindent
We recall the definition.
\begin{definition}
\label{definition-etale-covering}
An {\it \'etale covering} of a scheme $U$ is a family of morphisms
of schemes
$\{\varphi_i : U_i \to U\}_{i \in I}$ such that
\begin{enumerate}
\item each $\varphi_i$ is an \'etale morphism,
\item the $U_i$ cover $U$, i.e., $U = \bigcup_{i\in I}\varphi_i(U_i)$.
\end{enumerate}
\end{definition}
\begin{lemma}
\label{lemma-etale-fpqc}
Any \'etale covering is an fpqc covering.
\end{lemma}
\begin{proof}
(See also
Topologies,
Lemma \ref{topologies-lemma-zariski-etale-smooth-syntomic-fppf-fpqc}.)
Let $\{\varphi_i : U_i \to U\}_{i \in I}$ be an \'etale covering.
Since an \'etale morphism is flat, and the elements of the covering should
cover its target, the property fp (faithfully flat) is satisfied.
To check the property qc (quasi-compact), let $V \subset U$ be an affine
open, and write $\varphi_i^{-1} = \bigcup_{j \in J_i} V_{ij}$
for some affine opens $V_{ij} \subset U_i$. Since $\varphi_i$ is open
(as \'etale morphisms are open), we see that
$V = \bigcup_{i\in I} \bigcup_{j \in J_i} \varphi_i(V_{ij})$
is an open covering of $V$.
Further, since $V$ is quasi-compact, this covering has a finite
refinement.
\end{proof}
\noindent
So any statement which is true for fpqc coverings
remains true {\it a fortiori} for \'etale coverings. For
instance, the \'etale site is subcanonical.
\begin{definition}
\label{definition-big-etale-site}
(For more details see Section \ref{section-big-small}, or
Topologies, Section \ref{topologies-section-etale}.)
Let $S$ be a scheme.
The {\it big \'etale site over $S$} is the site
$(\Sch/S)_\etale$, see
Definition \ref{definition-tau-site}.
The {\it small \'etale site over $S$} is the site $S_\etale$, see
Definition \ref{definition-tau-site}.
We define similarly the {\it big} and {\it small Zariski sites} on $S$,
denoted $(\Sch/S)_{Zar}$ and $S_{Zar}$.
\end{definition}
\noindent
Loosely speaking the big \'etale site of $S$ is made up out of schemes over $S$
and coverings the \'etale coverings. The small \'etale site of $S$ is made up
out of schemes \'etale over $S$ with coverings the \'etale coverings.
Actually any morphism between objects of $S_\etale$ is \'etale, in
virtue of
Proposition \ref{proposition-etale-morphisms},
hence to check that $\{U_i \to U\}_{i \in I}$ in $S_\etale$
is a covering it suffices to check that $\coprod U_i \to U$ is surjective.
\medskip\noindent
The small \'etale site has fewer objects than the big \'etale site, it
contains only the ``opens'' of the \'etale topology on $S$. It is a full
subcategory of the big \'etale site, and its topology is induced from the
topology on the big site. Hence it is true that the restriction functor
from the big \'etale site to the small one is exact and maps injectives to
injectives. This has the following consequence.
\begin{proposition}
\label{proposition-cohomology-restrict-small-site}
Let $S$ be a scheme and $\mathcal{F}$ an abelian sheaf on
$(\Sch/S)_\etale$.
Then $\mathcal{F}|_{S_\etale}$ is a sheaf on $S_\etale$ and
$$
H^p_\etale(S, \mathcal{F}|_{S_\etale}) =
H^p_\etale(S, \mathcal{F})
$$
for all $p \geq 0$.
\end{proposition}
\begin{proof}
This is a special case of Lemma \ref{lemma-compare-cohomology-big-small}.
\end{proof}
\noindent
In accordance with the general notation introduced in
Section \ref{section-big-small}
we write $H_\etale^p(S, \mathcal{F})$ for the above cohomology group.
%9.24.09
\section{Kummer theory}
\label{section-kummer}
\noindent
Let $n \in \mathbf{N}$ and consider the functor $\mu_n$ defined by
$$
\begin{matrix}
\Sch^{opp} & \longrightarrow & \textit{Ab} \\
S & \longmapsto &
\mu_n(S)
=
\{t \in \Gamma(S, \mathcal{O}_S^*) \mid t^n = 1 \}.
\end{matrix}
$$
By
Groupoids, Example \ref{groupoids-example-roots-of-unity}
this is a representable functor, and the scheme representing it
is denoted $\mu_n$ also. By
Lemma \ref{lemma-representable-sheaf-fpqc}
this functor satisfies the sheaf condition for the fpqc topology
(in particular, it is also satisfies the sheaf condition for the
\'etale, Zariski, etc topology).
\begin{lemma}
\label{lemma-kummer-sequence}
If $n\in \mathcal{O}_S^*$ then
$$
0 \to
\mu_{n, S} \to
\mathbf{G}_{m, S} \xrightarrow{(\cdot)^n}
\mathbf{G}_{m, S} \to 0
$$
is a short exact sequence of sheaves on both the small and
big \'etale site of $S$.
\end{lemma}
\begin{proof}
By definition the sheaf $\mu_{n, S}$ is the kernel of the map
$(\cdot)^n$. Hence it suffices to show that the last map is surjective.
Let $U$ be a scheme over $S$. Let
$f \in \mathbf{G}_m(U) = \Gamma(U, \mathcal{O}_U^*)$.
We need to show that we can find an \'etale cover of
$U$ over the members of which the restriction of $f$ is an $n$th power.
Set
$$
U' =
\underline{\Spec}_U(\mathcal{O}_U[T]/(T^n-f))
\xrightarrow{\pi}
U.
$$
(See
Constructions, Section \ref{constructions-section-spec-via-glueing} or
\ref{constructions-section-spec}
for a discussion of the relative spectrum.)
Let $\Spec(A) \subset U$ be an affine open, and say $f|_{\Spec(A)}$ corresponds
to the unit $a \in A^*$. Then $\pi^{-1}(\Spec(A)) = \Spec(B)$ with
$B = A[T]/(T^n - a)$. The ring map $A \to B$ is finite free of rank $n$,
hence it is faithfully flat, and hence we conclude that
$\Spec(B) \to \Spec(A)$ is surjective. Since this holds for every
affine open in $U$ we conclude that $\pi$ is surjective.
In addition, $n$ and $T^{n - 1}$ are invertible in $B$, so
$nT^{n-1} \in B^*$ and the ring map $A \to B$ is standard \'etale,
in particular \'etale. Since this holds for every affine open of $U$
we conclude that $\pi$ is \'etale. Hence
$\mathcal{U} = \{\pi : U' \to U\}$ is an \'etale covering.
Moreover, $f|_{U'} = (f')^n$ where $f'$ is the class of $T$
in $\Gamma(U', \mathcal{O}_{U'}^*)$, so $\mathcal{U}$ has the desired property.
\end{proof}
\begin{remark}
\label{remark-no-kummer-sequence-zariski}
Lemma \ref{lemma-kummer-sequence} is false when ``\'etale'' is replaced
with ``Zariski''.
Since the \'etale topology is coarser than the smooth topology, see
Topologies, Lemma \ref{topologies-lemma-zariski-etale-smooth}
it follows that the sequence is also exact in the smooth topology.
\end{remark}
\noindent
By
Theorem \ref{theorem-picard-group}
and
Lemma \ref{lemma-kummer-sequence}
and general properties of cohomology we obtain
the long exact cohomology sequence
$$
\xymatrix{
0 \ar[r] &
H_\etale^0(S, \mu_{n, S}) \ar[r] &
\Gamma(S, \mathcal{O}_S^*) \ar[r]^{(\cdot)^n} &
\Gamma(S, \mathcal{O}_S^*) \ar@(rd, ul)[rdllllr]
\\
& H_\etale^1(S, \mu_{n, S}) \ar[r] &
\text{Pic}(S) \ar[r]^{(\cdot)^n} &
\text{Pic} (S) \ar@(rd, ul)[rdllllr] \\
& H_\etale^2(S, \mu_{n, S}) \ar[r] &
\ldots
}
$$
at least if $n$ is invertible on $S$. When $n$ is not invertible on $S$
we can apply the following lemma.
\begin{lemma}
\label{lemma-kummer-sequence-syntomic}
For any $n \in \mathbf{N}$ the sequence
$$
0 \to
\mu_{n, S} \to
\mathbf{G}_{m, S} \xrightarrow{(\cdot)^n}
\mathbf{G}_{m, S} \to 0
$$
is a short exact sequence of sheaves on the site
$(\Sch/S)_{fppf}$ and $(\Sch/S)_{syntomic}$.
\end{lemma}
\begin{proof}
By definition the sheaf $\mu_{n, S}$ is the kernel of the map
$(\cdot)^n$. Hence it suffices to show that the last map is surjective.
Since the syntomic topology is weaker than the fppf topology, see
Topologies, Lemma \ref{topologies-lemma-zariski-etale-smooth-syntomic-fppf},
it suffices to prove this for the syntomic topology.
Let $U$ be a scheme over $S$. Let
$f \in \mathbf{G}_m(U) = \Gamma(U, \mathcal{O}_U^*)$.
We need to show that we can find a syntomic cover of
$U$ over the members of which the restriction of $f$ is an $n$th power.
Set
$$
U' =
\underline{\Spec}_U(\mathcal{O}_U[T]/(T^n-f))
\xrightarrow{\pi}
U.
$$
(See
Constructions, Section \ref{constructions-section-spec-via-glueing} or
\ref{constructions-section-spec}
for a discussion of the relative spectrum.)
Let $\Spec(A) \subset U$ be an affine open, and say $f|_{\Spec(A)}$ corresponds
to the unit $a \in A^*$. Then $\pi^{-1}(\Spec(A)) = \Spec(B)$ with
$B = A[T]/(T^n - a)$. The ring map $A \to B$ is finite free of rank $n$,
hence it is faithfully flat, and hence we conclude that
$\Spec(B) \to \Spec(A)$ is surjective. Since this holds for every
affine open in $U$ we conclude that $\pi$ is surjective.
In addition, $B$ is a global relative complete intersection over $A$, so
the ring map $A \to B$ is standard syntomic,
in particular syntomic. Since this holds for every affine open of $U$
we conclude that $\pi$ is syntomic. Hence
$\mathcal{U} = \{\pi : U' \to U\}$ is a syntomic covering.
Moreover, $f|_{U'} = (f')^n$ where $f'$ is the class of $T$
in $\Gamma(U', \mathcal{O}_{U'}^*)$, so $\mathcal{U}$ has the desired property.
\end{proof}
\begin{remark}
\label{remark-no-kummer-sequence-smooth-etale-zariski}
Lemma \ref{lemma-kummer-sequence-syntomic}
is false for the smooth, \'etale, or Zariski topology.
\end{remark}
\noindent
By
Theorem \ref{theorem-picard-group}
and
Lemma \ref{lemma-kummer-sequence-syntomic}
and general properties of cohomology we obtain
the long exact cohomology sequence
$$
\xymatrix{
0 \ar[r] &
H_{fppf}^0(S, \mu_{n, S}) \ar[r] &
\Gamma(S, \mathcal{O}_S^*) \ar[r]^{(\cdot)^n} &
\Gamma(S, \mathcal{O}_S^*) \ar@(rd, ul)[rdllllr]
\\
& H_{fppf}^1(S, \mu_{n, S}) \ar[r] &
\text{Pic}(S) \ar[r]^{(\cdot)^n} &
\text{Pic} (S) \ar@(rd, ul)[rdllllr] \\
& H_{fppf}^2(S, \mu_{n, S}) \ar[r] &
\ldots
}
$$
for any scheme $S$ and any integer $n$. Of course there is a similar sequence
with syntomic cohomology.
\medskip\noindent
Let $n \in \mathbf{N}$ and let $S$ be any scheme.
There is another more direct way to describe the first cohomology group with
values in $\mu_n$. Consider pairs
$(\mathcal{L}, \alpha)$ where $\mathcal{L}$ is an invertible sheaf on $S$
and $\alpha : \mathcal{L}^{\otimes n} \to \mathcal{O}_S$ is a trivialization
of the $n$th tensor power of $\mathcal{L}$.
Let $(\mathcal{L}', \alpha')$ be a second such pair.
An isomorphism $\varphi : (\mathcal{L}, \alpha) \to (\mathcal{L}', \alpha')$
is an isomorphism $\varphi : \mathcal{L} \to \mathcal{L}'$ of invertible
sheaves such that the diagram
$$
\xymatrix{
\mathcal{L}^{\otimes n} \ar[d]_{\varphi^{\otimes n}} \ar[r]_\alpha &
\mathcal{O}_S \ar[d]^1 \\
(\mathcal{L}')^{\otimes n} \ar[r]^{\alpha'} &
\mathcal{O}_S \\
}
$$
commutes. Thus we have
\begin{equation}
\label{equation-isomorphisms-pairs}
\mathit{Isom}_S((\mathcal{L}, \alpha), (\mathcal{L}', \alpha'))
=
\left\{
\begin{matrix}
\emptyset & \text{if} & \text{they are not isomorphic} \\
H^0(S, \mu_{n, S})\cdot \varphi & \text{if} &
\varphi \text{ isomorphism of pairs}
\end{matrix}
\right.
\end{equation}
Moreover, given two pairs $(\mathcal{L}, \alpha)$, $(\mathcal{L}', \alpha')$
the tensor product
$$
(\mathcal{L}, \alpha) \otimes (\mathcal{L}', \alpha')
=
(\mathcal{L} \otimes \mathcal{L}', \alpha \otimes \alpha')
$$
is another pair. The pair $(\mathcal{O}_S, 1)$ is an identity for this
tensor product operation, and an inverse is given by
$$
(\mathcal{L}, \alpha)^{-1} = (\mathcal{L}^{\otimes -1}, \alpha^{\otimes -1}).
$$
Hence the collection of isomorphism classes of pairs forms an abelian group.
Note that
$$
(\mathcal{L}, \alpha)^{\otimes n}
=
(\mathcal{L}^{\otimes n}, \alpha^{\otimes n})
\xrightarrow{\alpha}
(\mathcal{O}_S, 1)
$$
hence every element of this group has order dividing $n$. We warn the reader
that this group is in general {\bf not} the $n$-torsion in $\text{Pic}(S)$.
\begin{lemma}
\label{lemma-describe-h1-mun}
Let $S$ be a scheme. There is a canonical identification
$$
H_\etale^1(S, \mu_n) =
\text{group of pairs }(\mathcal{L}, \alpha)\text{ up to isomorphism as above}
$$
if $n$ is invertible on $S$. In general we have
$$
H_{fppf}^1(S, \mu_n) =
\text{group of pairs }(\mathcal{L}, \alpha)\text{ up to isomorphism as above}.
$$
The same result holds with fppf replaced by syntomic.
\end{lemma}
\begin{proof}
We first prove the second isomorphism.
Let $(\mathcal{L}, \alpha)$ be a pair as above.
Choose an affine open covering $S = \bigcup U_i$ such that
$\mathcal{L}|_{U_i} \cong \mathcal{O}_{U_i}$. Say $s_i \in \mathcal{L}(U_i)$
is a generator. Then $\alpha(s_i^{\otimes n}) = f_i \in \mathcal{O}_S^*(U_i)$.
Writing $U_i = \Spec(A_i)$ we see there exists a global
relative complete intersection $A_i \to B_i = A_i[T]/(T^n - f_i)$
such that $f_i$ maps to an $n$th power in $B_i$. In other words, setting
$V_i = \Spec(B_i)$ we obtain a syntomic covering
$\mathcal{V} = \{V_i \to S\}_{i \in I}$ and trivializations
$\varphi_i : (\mathcal{L}, \alpha)|_{V_i} \to (\mathcal{O}_{V_i}, 1)$.
\medskip\noindent
We will use this result (the existence of the covering $\mathcal{V}$)
to associate to this pair a cohomology class in
$H^1_{syntomic}(S, \mu_{n, S})$. We give two (equivalent) constructions.
\medskip\noindent
First construction: using {\v C}ech cohomology.
Over the double overlaps $V_i \times_S V_j$ we have the isomorphism
$$
(\mathcal{O}_{V_i \times_S V_j}, 1)
\xrightarrow{\text{pr}_0^*\varphi_i^{-1}}
(\mathcal{L}|_{V_i \times_S V_j}, \alpha|_{V_i \times_S V_j})
\xrightarrow{\text{pr}_1^*\varphi_j}
(\mathcal{O}_{V_i \times_S V_j}, 1)
$$
of pairs. By (\ref{equation-isomorphisms-pairs}) this is given by an
element $\zeta_{ij} \in \mu_n(V_i \times_S V_j)$. We omit the verification
that these $\zeta_{ij}$'s give a $1$-cocycle, i.e., give
an element $(\zeta_{i_0i_1}) \in \check C(\mathcal{V}, \mu_n)$
with $d(\zeta_{i_0i_1}) = 0$. Thus its class is an element in
$\check H^1(\mathcal{V}, \mu_n)$ and by
Theorem \ref{theorem-cech-ss}
it maps to a cohomology class in $H^1_{syntomic}(S, \mu_{n, S})$.
\medskip\noindent
Second construction: Using torsors. Consider the presheaf
$$
\mu_n(\mathcal{L}, \alpha) :
U
\longmapsto
\mathit{Isom}_U((\mathcal{O}_U, 1), (\mathcal{L}, \alpha)|_U)
$$
on $(\Sch/S)_{syntomic}$.
We may view this as a subpresheaf of
$\SheafHom_\mathcal{O}(\mathcal{O}, \mathcal{L})$ (internal hom
sheaf, see
Modules on Sites, Section \ref{sites-modules-section-internal-hom}).
Since the conditions defining this subpresheaf are local, we see that it is
a sheaf.
By (\ref{equation-isomorphisms-pairs}) this sheaf has a free action of
the sheaf $\mu_{n, S}$. Hence the only thing we have to check is that
it locally has sections. This is true because of the existence of the
trivializing cover $\mathcal{V}$. Hence $\mu_n(\mathcal{L}, \alpha)$
is a $\mu_{n, S}$-torsor and by
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-torsors-h1}
we obtain a corresponding element of $H^1_{syntomic}(S, \mu_{n, S})$.
\medskip\noindent
Ok, now we have to still show the following
\begin{enumerate}
\item The two constructions give the same cohomology class.
\item Isomorphic pairs give rise to the same cohomology class.
\item The cohomology class of
$(\mathcal{L}, \alpha) \otimes (\mathcal{L}', \alpha')$
is the sum of the cohomology classes of
$(\mathcal{L}, \alpha)$ and $(\mathcal{L}', \alpha')$.
\item If the cohomology class is trivial, then the pair is trivial.
\item Any element of $H^1_{syntomic}(S, \mu_{n, S})$ is the
cohomology class of a pair.
\end{enumerate}
We omit the proof of (1). Part (2) is clear from the second construction,
since isomorphic torsors give the same cohomology classes.
Part (3) is clear from the first construction, since the resulting
{\v C}ech classes add up. Part (4) is clear from the second construction
since a torsor is trivial if and only if it has a global section, see
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-trivial-torsor}.
\medskip\noindent
Part (5) can be seen as follows (although a direct proof would be
preferable). Suppose $\xi \in H^1_{syntomic}(S, \mu_{n, S})$.
Then $\xi$ maps to an element
$\overline{\xi} \in H^1_{syntomic}(S, \mathbf{G}_{m, S})$
with $n \overline{\xi} = 0$. By
Theorem \ref{theorem-picard-group}
we see that $\overline{\xi}$ corresponds to an invertible sheaf $\mathcal{L}$
whose $n$th tensor power is isomorphic to $\mathcal{O}_S$.
Hence there exists a pair $(\mathcal{L}, \alpha')$ whose cohomology
class $\xi'$ has the same image $\overline{\xi'}$ in
$H^1_{syntomic}(S, \mathbf{G}_{m, S})$. Thus it suffices to show
that $\xi - \xi'$ is the class of a pair. By construction, and the
long exact cohomology sequence above, we see that
$\xi - \xi' = \partial(f)$ for some $f \in H^0(S, \mathcal{O}_S^*)$.
Consider the pair $(\mathcal{O}_S, f)$. We omit the verification
that the cohomology class of this pair is $\partial(f)$, which
finishes the proof of the first identification (with fppf replaced
with syntomic).
\medskip\noindent
To see the first, note that if $n$ is invertible on $S$, then the
covering $\mathcal{V}$ constructed in the first part of the proof
is actually an \'etale covering (compare with the proof of
Lemma \ref{lemma-kummer-sequence}). The rest of the proof is independent
of the topology, apart from the very last argument which uses that
the Kummer sequence is exact, i.e., uses Lemma \ref{lemma-kummer-sequence}.
\end{proof}
\section{Neighborhoods, stalks and points}
\label{section-stalks}
\noindent
We can associate to any geometric point of $S$ a stalk functor which is
exact. A map of sheaves on $S_\etale$ is an isomorphism if and only
if it
is an isomorphism on all these stalks. A complex of abelian sheaves is
exact if and only if the complex of stalks is exact at all geometric points.
Altogether this means that the small \'etale site of a scheme $S$
has enough points. It also turns out that any point of the small \'etale topos
of $S$ (an abstract notion) is given by a geometric point.
Thus in some sense the small \'etale topos of $S$ can be understood in
terms of geometric points and neighbourhoods.
\begin{definition}
\label{definition-geometric-point}
Let $S$ be a scheme.
\begin{enumerate}
\item A {\it geometric point} of $S$ is a morphism
$\Spec(k) \to S$ where $k$ is algebraically closed.
Such a point is usually denoted $\overline{s}$, i.e., by an overlined
small case letter. We often use $\overline{s}$ to denote the scheme
$\Spec(k)$ as well as the morphism, and we use $\kappa(\overline{s})$
to denote $k$.
\item We say $\overline{s}$ {\it lies over} $s$
to indicate that $s \in S$ is the image of $\overline{s}$.
\item An {\it \'etale neighborhood} of a geometric point $\overline{s}$
of $S$ is a commutative diagram
$$
\xymatrix{
& U \ar[d]^\varphi \\
{\overline{s}} \ar[r]^{\overline{s}} \ar[ur]^{\bar u} & S
}
$$
where $\varphi$ is an \'etale morphism of schemes.
We write $(U, \overline{u}) \to (S, \overline{s})$.
\item A {\it morphism of \'etale neighborhoods}
$(U, \overline{u}) \to (U', \overline{u}')$
is an $S$-morphism $h: U \to U'$
such that $\overline{u}' = h \circ \overline{u}$.
\end{enumerate}
\end{definition}
\begin{remark}
\label{remark-etale-between-etale}
Since $U$ and $U'$ are \'etale over $S$, any $S$-morphism
between them is also \'etale, see
Proposition \ref{proposition-etale-morphisms}.
In particular all morphisms of \'etale neighborhoods are \'etale.
\end{remark}
\begin{remark}
\label{remark-etale-neighbourhoods}
Let $S$ be a scheme and $s \in S$ a point. In
More on Morphisms,
Definition \ref{more-morphisms-definition-etale-neighbourhood}
we defined the notion of an \'etale neighbourhood $(U, u) \to (S, s)$
of $(S, s)$. If $\overline{s}$ is a geometric point of $S$ lying over
$s$, then any \'etale neighbourhood $(U, \overline{u}) \to (S, \overline{s})$
gives rise to an \'etale neighbourhood $(U, u)$ of $(S, s)$ by taking
$u \in U$ to be the unique point of $U$ such that $\overline{u}$
lies over $u$. Conversely, given an \'etale neighbourhood $(U, u)$
of $(S, s)$ the residue field extension $\kappa(s) \subset \kappa(u)$
is finite separable (see
Proposition \ref{proposition-etale-morphisms})
and hence we can find an embedding $\kappa(u) \subset \kappa(\overline{s})$
over $\kappa(s)$. In other words, we can find a geometric point
$\overline{u}$ of $U$ lying over $u$ such that $(U, \overline{u})$
is an \'etale neighbourhood of $(S, \overline{s})$.
We will use these observations to go between the two types of
\'etale neighbourhoods.
\end{remark}
\begin{lemma}
\label{lemma-cofinal-etale}
Let $S$ be a scheme, and let $\overline{s}$ be a geometric point of $S$.
The category of \'etale neighborhoods is cofiltered. More precisely:
\begin{enumerate}
\item Let $(U_i, \overline{u}_i)_{i = 1, 2}$ be two \'etale neighborhoods of
$\overline{s}$ in $S$. Then there exists a third \'etale neighborhood
$(U, \overline{u})$ and morphisms
$(U, \overline{u}) \to (U_i, \overline{u}_i)$, $i = 1, 2$.
\item Let $h_1, h_2: (U, \overline{u}) \to (U', \overline{u}')$ be two
morphisms between \'etale neighborhoods of $\overline{s}$. Then there exist an
\'etale neighborhood $(U'', \overline{u}'')$ and a morphism
$h : (U'', \overline{u}'') \to (U, \overline{u})$
which equalizes $h_1$ and $h_2$, i.e., such that
$h_1 \circ h = h_2 \circ h$.
\end{enumerate}
\end{lemma}
\begin{proof}
For part (1), consider the fibre product $U = U_1 \times_S U_2$.
It is \'etale over both $U_1$ and $U_2$ because \'etale morphisms are
preserved under base change, see
Proposition \ref{proposition-etale-morphisms}.
The map $\overline{s} \to U$ defined by $(\overline{u}_1, \overline{u}_2)$
gives it the structure of an \'etale neighborhood mapping to both
$U_1$ and $U_2$. For part (2), define $U''$ as the fibre product
$$
\xymatrix{
U'' \ar[r] \ar[d] & U \ar[d]^{(h_1, h_2)} \\
U' \ar[r]^-\Delta & U' \times_S U'.
}
$$
Since $\overline{u}$ and $\overline{u}'$ agree over $S$ with $\overline{s}$,
we see that $\overline{u}'' = (\overline{u}, \overline{u}')$ is a geometric
point of $U''$. In particular $U'' \not = \emptyset$.
Moreover, since $U'$ is \'etale over $S$, so is the fibre product
$U'\times_S U'$ (see
Proposition \ref{proposition-etale-morphisms}).
Hence the vertical arrow $(h_1, h_2)$ is \'etale by
Remark \ref{remark-etale-between-etale} above.
Therefore $U''$ is \'etale over $U'$ by base change, and hence also
\'etale over $S$ (because compositions of \'etale morphisms are \'etale).
Thus $(U'', \overline{u}'')$ is a solution to the problem.
\end{proof}
\begin{lemma}
\label{lemma-geometric-lift-to-cover}
Let $S$ be a scheme.
Let $\overline{s}$ be a geometric point of $S$.
Let $(U, \overline{u})$ an \'etale neighborhood of $\overline{s}$.
Let $\mathcal{U} = \{\varphi_i : U_i \to U \}_{i\in I}$ be an \'etale covering.
Then there exist $i \in I$ and $\overline{u}_i : \overline{s} \to U_i$
such that $\varphi_i : (U_i, \overline{u}_i) \to (U, \overline{u})$
is a morphism of \'etale neighborhoods.
\end{lemma}
\begin{proof}
As $U = \bigcup_{i\in I} \varphi_i(U_i)$, the fibre product
$\overline{s} \times_{\overline{u}, U, \varphi_i} U_i$ is not empty
for some $i$. Then look at the cartesian diagram
$$
\xymatrix{
\overline{s} \times_{\overline{u}, U, \varphi_i} U_i
\ar[d]^{\text{pr}_1} \ar[r]_-{\text{pr}_2} & U_i
\ar[d]^{\varphi_i} \\
\Spec(k) = \overline{s} \ar@/^1pc/[u]^\sigma
\ar[r]^-{\overline{u}} & U
}
$$
The projection $\text{pr}_1$ is the base change of an \'etale morphisms so it
is \'etale, see
Proposition \ref{proposition-etale-morphisms}.
Therefore, $\overline{s} \times_{\overline{u}, U, \varphi_i} U_i$
is a disjoint union of finite separable extensions of $k$, by
Proposition \ref{proposition-etale-morphisms}. Here
$\overline{s} = \Spec(k)$. But $k$ is algebraically closed, so all
these extensions are trivial, and there exists a section $\sigma$ of
$\text{pr}_1$. The composition
$\text{pr}_2 \circ \sigma$ gives a map compatible with $\overline{u}$.
\end{proof}
\begin{definition}
\label{definition-stalk}
Let $S$ be a scheme.
Let $\mathcal{F}$ be a presheaf on $S_\etale$.
Let $\overline{s}$ be a geometric point of $S$.
The {\it stalk} of $\mathcal{F}$ at $\overline{s}$ is
$$
\mathcal{F}_{\overline{s}}
=
\colim_{(U, \overline{u})} \mathcal{F}(U)
$$
where $(U, \overline{u})$ runs over all \'etale
neighborhoods of $\overline{s}$ in $S$.
\end{definition}
\noindent
By Lemma \ref{lemma-cofinal-etale}, this colimit is over a filtered
index category, namely the opposite of the category of \'etale neighbourhoods.
In other words, an element of $\mathcal{F}_{\overline{s}}$ can be
thought of as a triple $(U, \overline{u}, \sigma)$ where
$\sigma \in \mathcal{F}(U)$. Two triples
$(U, \overline{u}, \sigma)$, $(U', \overline{u}', \sigma')$
define the same element of the stalk if there exists a third
\'etale neighbourhood $(U'', \overline{u}'')$ and morphisms of \'etale
neighbourhoods $h : (U'', \overline{u}'') \to (U, \overline{u})$,
$h' : (U'', \overline{u}'') \to (U', \overline{u}')$ such that
$h^*\sigma = (h')^*\sigma'$ in $\mathcal{F}(U'')$. See
Categories, Section \ref{categories-section-directed-colimits}.
\begin{lemma}
\label{lemma-stalk-gives-point}
Let $S$ be a scheme. Let $\overline{s}$ be a geometric point of $S$.
Consider the functor
\begin{align*}
u : S_\etale & \longrightarrow \textit{Sets}, \\
U & \longmapsto
|U_{\overline{s}}|
=
\{\overline{u} \text{ such that }(U, \overline{u})
\text{ is an \'etale neighbourhood of }\overline{s}\}.
\end{align*}
Here $|U_{\overline{s}}|$ denotes the underlying set of the geometric fibre.
Then $u$ defines a point $p$ of the site $S_\etale$
(Sites, Definition \ref{sites-definition-point})
and its associated stalk functor $\mathcal{F} \mapsto \mathcal{F}_p$
(Sites, Equation \ref{sites-equation-stalk})
is the functor $\mathcal{F} \mapsto \mathcal{F}_{\overline{s}}$
defined above.
\end{lemma}
\begin{proof}
In the proof of
Lemma \ref{lemma-geometric-lift-to-cover}
we have seen that the scheme $U_{\overline{s}}$ is a disjoint union of
schemes isomorphic to $\overline{s}$. Thus we can also think of
$|U_{\overline{s}}|$ as the set of geometric points of $U$ lying over
$\overline{s}$, i.e., as the collection of morphisms
$\overline{u} : \overline{s} \to U$ fitting into the diagram of
Definition \ref{definition-geometric-point}.
From this it follows that $u(S)$ is a singleton, and that
$u(U \times_V W) = u(U) \times_{u(V)} u(W)$
whenever $U \to V$ and $W \to V$ are morphisms in $S_\etale$.
And, given a covering $\{U_i \to U\}_{i \in I}$ in $S_\etale$
we see that $\coprod u(U_i) \to u(U)$ is surjective by
Lemma \ref{lemma-geometric-lift-to-cover}.
Hence
Sites, Proposition \ref{sites-proposition-point-limits}
applies, so $p$ is a point of the site $S_\etale$.
Finally, the our functor $\mathcal{F} \mapsto \mathcal{F}_{\overline{s}}$
is given by exactly the same colimit as the functor
$\mathcal{F} \mapsto \mathcal{F}_p$ associated to $p$ in
Sites, Equation \ref{sites-equation-stalk}
which proves the final assertion.
\end{proof}
\begin{remark}
\label{remark-map-stalks}
Let $S$ be a scheme and let $\overline{s} : \Spec(k) \to S$
and $\overline{s}' : \Spec(k') \to S$ be two geometric points of
$S$. A {\it morphism $a : \overline{s} \to \overline{s}'$ of geometric points}
is simply a morphism $a : \Spec(k) \to \Spec(k')$ such that
$a \circ \overline{s}' = \overline{s}$. Given such a morphism we obtain
a functor from the category of \'etale neighbourhoods of $\overline{s}'$
to the category of \'etale neighbourhoods of $\overline{s}$ by the rule
$(U, \overline{u}') \mapsto (U, \overline{u}' \circ a)$. Hence we obtain
a canonical map
$$
\mathcal{F}_{\overline{s}'}
=
\colim_{(U, \overline{u}')} \mathcal{F}(U)
\longrightarrow
\colim_{(U, \overline{u})} \mathcal{F}(U)
=
\mathcal{F}_{\overline{s}}
$$
from Categories, Lemma \ref{categories-lemma-functorial-colimit}. Using the
description of elements of stalks as triples this maps the element of
$\mathcal{F}_{\overline{s}'}$ represented by the triple
$(U, \overline{u}', \sigma)$ to the element of $\mathcal{F}_{\overline{s}}$
represented by the triple $(U, \overline{u}' \circ a, \sigma)$.
Since the functor above is clearly an equivalence we conclude that this
canonical map is an isomorphism of stalk functors.
\medskip\noindent
Let us make sure we have the map of stalks corresponding to $a$ pointing
in the correct direction. Note that the above means, according to
Sites, Definition \ref{sites-definition-morphism-points},
that $a$ defines a morphism $a : p \to p'$ between the points $p, p'$ of
the site $S_\etale$ associated to $\overline{s}, \overline{s}'$ by
Lemma \ref{lemma-stalk-gives-point}. There are more general morphisms of
points (corresponding to specializations of points of $S$) which we will
describe later, and which will not be isomorphisms (insert future
reference here).
\end{remark}
\begin{lemma}
\label{lemma-stalk-exact}
Let $S$ be a scheme. Let $\overline{s}$ be a geometric point of $S$.
\begin{enumerate}
\item The stalk functor
$\textit{PAb}(S_\etale) \to \textit{Ab}$,
$\mathcal{F} \mapsto \mathcal{F}_{\overline{s}}$
is exact.
\item We have $(\mathcal{F}^\#)_{\overline{s}} = \mathcal{F}_{\overline{s}}$
for any presheaf of sets $\mathcal{F}$ on $S_\etale$.
\item The functor
$\textit{Ab}(S_\etale) \to \textit{Ab}$,
$\mathcal{F} \mapsto \mathcal{F}_{\overline{s}}$ is exact.
\item Similarly the functors
$\textit{PSh}(S_\etale) \to \textit{Sets}$ and
$\Sh(S_\etale) \to \textit{Sets}$ given by the stalk functor
$\mathcal{F} \mapsto \mathcal{F}_{\overline{x}}$ are exact (see
Categories, Definition \ref{categories-definition-exact})
and commute with arbitrary colimits.
\end{enumerate}
\end{lemma}
\begin{proof}
Before we indicate how to prove this by direct arguments
we note that the result follows from the general material in
Modules on Sites, Section \ref{sites-modules-section-stalks}.
This is true because $\mathcal{F} \mapsto \mathcal{F}_{\overline{s}}$
comes from a point of the small \'etale site of $S$, see
Lemma \ref{lemma-stalk-gives-point}.
We will only give a direct proof of (1), (2) and (3), and omit
a direct proof of (4).
\medskip\noindent
Exactness as a functor on $\textit{PAb}(S_\etale)$ is formal from the
fact that directed colimits commute with all colimits and with finite
limits. The identification of the stalks in (2) is via the map
$$
\kappa :
\mathcal{F}_{\overline{s}}
\longrightarrow
(\mathcal{F}^\#)_{\overline{s}}
$$
induced by the natural morphism $\mathcal{F}\to \mathcal{F}^\#$, see
Theorem \ref{theorem-sheafification}.
We claim that this map is an isomorphism of abelian groups. We will show
injectivity and omit the proof of surjectivity.