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spaces-more-cohomology.tex
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\input{preamble}
% OK, start here.
%
\begin{document}
\title{More on Cohomology of Spaces}
\maketitle
\phantomsection
\label{section-phantom}
\tableofcontents
\section{Introduction}
\label{section-introduction}
\noindent
In this chapter continues the discussion started in
Cohomology of Spaces, Section \ref{spaces-cohomology-section-introduction}.
One can also view this chapter as the analogue for algebraic spaces
of the chapter on \'etale cohomology for schemes, see
\'Etale Cohomology, Section \ref{etale-cohomology-section-introduction}.
\medskip\noindent
In fact, we intend this chapter to be mainly a translation of the
results already proved for schemes into the language of algebraic
spaces. Some of our results can be found in \cite{Kn}.
\section{Conventions}
\label{section-conventions}
\noindent
The standing assumption is that all schemes are contained in
a big fppf site $\Sch_{fppf}$. And all rings $A$ considered
have the property that $\Spec(A)$ is (isomorphic) to an
object of this big site.
\medskip\noindent
Let $S$ be a scheme and let $X$ be an algebraic space over $S$.
In this chapter and the following we will write $X \times_S X$
for the product of $X$ with itself (in the category of algebraic
spaces over $S$), instead of $X \times X$.
\section{Transporting results from schemes}
\label{section-api}
\noindent
In this section we explain briefly how results for schemes
imply results for (representable) algebraic spaces and
(representable) morphisms of algebraic spaces.
For quasi-coherent modules more is true
(because \'etale cohomology of a quasi-coherent module
over a scheme agrees with Zariski cohomology) and this
has already been discussed in Cohomology of Spaces, Section
\ref{spaces-cohomology-section-higher-direct-image}.
\medskip\noindent
Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.
Now suppose that $X$ is representable by the scheme $X_0$
(awkward but temporary notation; we usually just say ``$X$
is a scheme''). In this case $X$ and $X_0$ have the same small
\'etale sites:
$$
X_\etale = (X_0)_\etale
$$
This is pointed out in
Properties of Spaces, Section \ref{spaces-properties-section-etale-site}.
Moreover, if $f : X \to Y$ is a morphism of representable algebraic spaces
over $S$ and if $f_0 : X_0 \to Y_0$ is a morphism of
schemes representing $f$, then the induced morphisms of small
\'etale topoi agree:
$$
\xymatrix{
\Sh(X_\etale) \ar[rr]_{f_{small}} \ar@{=}[d] & &
\Sh(Y_\etale) \ar@{=}[d] \\
\Sh((X_0)_\etale) \ar[rr]^{(f_0)_{small}} & &
\Sh((Y_0)_\etale)
}
$$
See Properties of Spaces, Lemma
\ref{spaces-properties-lemma-functoriality-etale-site} and
Topologies, Lemma \ref{topologies-lemma-morphism-big-small-etale}.
\medskip\noindent
Thus there is absolutely no difference between \'etale cohomology
of a scheme and the \'etale cohomology of the corresponding algebraic space.
Similarly for higher direct images along morphisms of schemes.
In fact, if $f : X \to Y$ is a morphism of algebraic spaces over $S$
which is representable (by schemes), then the higher direct images
$R^if_*\mathcal{F}$ of a sheaf $\mathcal{F}$ on $X_\etale$
can be computed \'etale locally on $Y$ (Cohomology on Sites,
Lemma \ref{sites-cohomology-lemma-higher-direct-images})
hence this often reduces computations and proofs to the case
where $Y$ and $X$ are schemes.
\medskip\noindent
We will use the above without further mention in this chapter.
For other topologies the same thing is true; we state it
explicitly as a lemma for cohomology here.
\begin{lemma}
\label{lemma-compare-cohomology-other-topologies}
Let $S$ be a scheme. Let $\tau \in \{\etale, fppf, ph\}$ (add more here).
The inclusion functor
$$
(\Sch/S)_\tau \longrightarrow (\textit{Spaces}/S)_\tau
$$
is a special cocontinuous functor
(Sites, Definition \ref{sites-definition-special-cocontinuous-functor})
and hence identifies topoi.
\end{lemma}
\begin{proof}
The conditions of Sites, Lemma \ref{sites-lemma-equivalence}
are immediately verified as our functor is fully faithful
and as every algebraic space has an \'etale covering by schemes.
\end{proof}
\section{Proper base change}
\label{section-proper-base-change}
\noindent
The proper base change theorem for algebraic spaces follows from
the proper base change theorem for schemes and Chow's lemma
with a little bit of work.
\begin{lemma}
\label{lemma-surjective-proper}
Let $S$ be a scheme. Let $f : Y \to X$ be a surjective proper morphism
of algebraic spaces over $S$. Let $\mathcal{F}$ be a sheaf on $X_\etale$.
Then $\mathcal{F} \to f_*f^{-1}\mathcal{F}$ is injective with
image the equalizer of the two maps
$f_*f^{-1}\mathcal{F} \to g_*g^{-1}\mathcal{F}$ where
$g$ is the structure morphism $g : Y \times_X Y \to X$.
\end{lemma}
\begin{proof}
For any surjective morphism $f : Y \to X$ of algebraic spaces over $S$,
the map $\mathcal{F} \to f_*f^{-1}\mathcal{F}$ is injective.
Namely, if $\overline{x}$ is a geometric point of $X$, then we
choose a geometric point $\overline{y}$ of $Y$ lying over $\overline{x}$
and we consider
$$
\mathcal{F}_{\overline{x}} \to
(f_*f^{-1}\mathcal{F})_{\overline{x}} \to
(f^{-1}\mathcal{F})_{\overline{y}} = \mathcal{F}_{\overline{x}}
$$
See Properties of Spaces, Lemma \ref{spaces-properties-lemma-stalk-pullback}
for the last equality.
\medskip\noindent
The second statement is local on $X$ in the \'etale topology, hence we may
and do assume $Y$ is an affine scheme.
\medskip\noindent
Choose a surjective proper morphism $Z \to Y$ where $Z$ is a scheme, see
Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-weak-chow}.
The result for $Z \to X$ implies the result for $Y \to X$.
Since $Z \to X$ is a surjective proper morphism of schemes
and hence a ph covering
(Topologies, Lemma \ref{topologies-lemma-surjective-proper-ph})
the result for $Z \to X$ follows from
\'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-describe-pullback-pi-ph}
(in fact it is in some sense equivalent to this lemma).
\end{proof}
\begin{lemma}
\label{lemma-h0-proper-over-henselian-pair}
Let $(A, I)$ be a henselian pair. Let $X$ be an algebraic space over $A$
such that the structure morphism $f : X \to \Spec(A)$ is proper.
Let $i : X_0 \to X$ be the inclusion of $X \times_{\Spec(A)} \Spec(A/I)$.
For any sheaf $\mathcal{F}$ on $X_\etale$ we
have $\Gamma(X, \mathcal{F}) = \Gamma(Z, i^{-1}\mathcal{F})$.
\end{lemma}
\begin{proof}
Choose a surjective proper morphism $Y \to X$ where $Y$ is a scheme, see
Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-weak-chow}.
Consider the diagram
$$
\xymatrix{
\Gamma(X_0, \mathcal{F}_0) \ar[r] \ar[d] &
\Gamma(Y_0, \mathcal{G}_0) \ar@<1ex>[r] \ar@<-1ex>[r] \ar[d] &
\Gamma((Y \times_X Y)_0, \mathcal{H}_0) \ar[d] \\
\Gamma(X, \mathcal{F}) \ar[r] &
\Gamma(Y, \mathcal{G}) \ar@<1ex>[r] \ar@<-1ex>[r] &
\Gamma(Y \times_X Y, \mathcal{H})
}
$$
Here $\mathcal{G}$, resp.\ $\mathcal{H}$ is the pullbackf or
$\mathcal{F}$ to $Y$, resp.\ $Y \times_X Y$ and the index $0$
indicates base change to $\Spec(A/I)$. By the case of schemes
(\'Etale Cohomology, Lemma
\ref{etale-cohomology-lemma-h0-proper-over-henselian-pair})
we see that the middle and right vertical arrows are bijective.
By Lemma \ref{lemma-surjective-proper} it follows that the left one is too.
\end{proof}
\begin{lemma}
\label{lemma-h0-proper-over-henselian-local}
Let $A$ be a henselian local ring. Let $X$ be an algebraic space
over $A$ such that $f : X \to \Spec(A)$
be a proper morphism. Let $X_0 \subset X$ be the fibre of
$f$ over the closed point. For any sheaf $\mathcal{F}$ on $X_\etale$ we
have $\Gamma(X, \mathcal{F}) = \Gamma(X_0, \mathcal{F}|_{X_0})$.
\end{lemma}
\begin{proof}
This is a special case of Lemma \ref{lemma-h0-proper-over-henselian-pair}.
\end{proof}
\begin{lemma}
\label{lemma-proper-base-change-f-star}
Let $S$ be a scheme. Let $f : X \to Y$ and $g : Y' \to Y$
be a morphisms of algebraic spaces over $S$. Assume $f$ is proper.
Set $X' = Y' \times_Y X$ with projections $f' : X' \to Y'$ and $g' : X' \to X$.
Let $\mathcal{F}$ be any sheaf on $X_\etale$. Then
$g^{-1}f_*\mathcal{F} = f'_*(g')^{-1}\mathcal{F}$.
\end{lemma}
\begin{proof}
The question is \'etale local on $Y'$. Choose a scheme $V$ and a surjective
\'etale morphism $V \to Y$. Choose a scheme $V'$ and a surjective \'etale
morphism $V' \to V \times_Y Y'$. Then we may replace $Y'$ by $V'$ and
$Y$ by $V$. Hence we may assume $Y$ and $Y'$ are schemes.
Then we may work Zariski locally on $Y$ and $Y'$ and hence we may
assume $Y$ and $Y'$ are affine schemes.
\medskip\noindent
Assume $Y$ and $Y'$ are affine schemes. Choose a surjective proper morphism
$h_1 : X_1 \to X$ where $X_1$ is a scheme, see
Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-weak-chow}.
Set $X_2 = X_1 \times_X X_1$ and denote
$h_2 : X_2 \to X$ the structure morphism. Observe this is a scheme.
By the case of schemes
(\'Etale Cohomology, Lemma
\ref{etale-cohomology-lemma-proper-base-change-f-star})
we know the lemma is true for the cartesian diagrams
$$
\vcenter{
\xymatrix{
X'_1 \ar[r] \ar[d] & X_1 \ar[d] \\
Y' \ar[r] & Y
}
}
\quad\text{and}\quad
\vcenter{
\xymatrix{
X'_2 \ar[r] \ar[d] & X_2 \ar[d] \\
Y' \ar[r] & Y
}
}
$$
and the sheaves $\mathcal{F}_i = (X_i \to X)^{-1}\mathcal{F}$.
By Lemma \ref{lemma-surjective-proper} we have an exact sequence
$0 \to \mathcal{F} \to h_{1, *}\mathcal{F}_1 \to h_{2, *}\mathcal{F}_2$
and similarly for $(g')^{-1}\mathcal{F}$ because
$X'_2 = X'_1 \times_{X'} X'_1$. Hence we conlude that the
lemma is true (some details omitted).
\end{proof}
\noindent
Let $S$ be a scheme.
Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. Let
$\overline{x} : \Spec(k) \to S$ be a geometric point. The fibre
of $f$ at $\overline{x}$ is the algebraic space
$Y_{\overline{x}} = \Spec(k) \times_{\overline{x}, X} Y$ over $\Spec(k)$.
If $\mathcal{F}$ is a sheaf on $Y_\etale$, then denote
$\mathcal{F}_{\overline{x}} = p^{-1}\mathcal{F}$
the pullback of $\mathcal{F}$ to $(Y_{\overline{x}})_\etale$.
Here $p : Y_{\overline{x}} \to Y$ is the projection.
In the following we will consider the set
$\Gamma(Y_{\overline{x}}, \mathcal{F}_{\overline{x}})$.
\begin{lemma}
\label{lemma-proper-pushforward-stalk}
Let $S$ be a scheme.
Let $f : Y \to X$ be a proper morphism of algebraic spaces over $S$. Let
$\overline{x} \to X$ be a geometric point.
For any sheaf $\mathcal{F}$ on $Y_\etale$
the canonical map
$$
(f_*\mathcal{F})_{\overline{x}} \longrightarrow
\Gamma(Y_{\overline{x}}, \mathcal{F}_{\overline{x}})
$$
is bijective.
\end{lemma}
\begin{proof}
This is a special case of Lemma \ref{lemma-proper-base-change-f-star}.
\end{proof}
\begin{theorem}
\label{theorem-proper-base-change}
Let $S$ be a scheme. Let
$$
\xymatrix{
X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\
Y' \ar[r]^g & Y
}
$$
be a cartesian square of algebraic spaces over $S$.
Assume $f$ is proper.
Let $\mathcal{F}$ be an abelian torsion sheaf on $X_\etale$.
Then the base change map
$$
g^{-1}Rf_*\mathcal{F} \longrightarrow Rf'_*(g')^{-1}\mathcal{F}
$$
is an isomorphism.
\end{theorem}
\begin{proof}
This proof repeats a few of the arguments given in the proof of the
proper base change theorem for schemes. See
\'Etale Cohomology, Section \ref{etale-cohomology-section-proper-base-change}
for more details.
\medskip\noindent
The statement is \'etale local on $Y'$ and $Y$, hence we may assume
both $Y$ and $Y'$ are affine schemes. Observe that this in particular
proves the theorem in case $f$ is representable (we will use this
below).
\medskip\noindent
For every $n \geq 1$ let $\mathcal{F}[n]$ be the subsheaf of sections
of $\mathcal{F}$ annihilated by $n$. Then
$\mathcal{F} = \colim \mathcal{F}[n]$. By
Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-colimit-cohomology}
the functors $g^{-1}R^pf_*$ and $R^pf'_*(g')^{-1}$ commute
with filtered colimits. Hence it suffices to prove the theorem
if $\mathcal{F}$ is killed by $n$.
\medskip\noindent
Let $\mathcal{F} \to \mathcal{I}^\bullet$ be a resolution by
injective sheaves of $\mathbf{Z}/n\mathbf{Z}$-modules.
Observe that
$g^{-1}f_*\mathcal{I}^\bullet = f'_*(g')^{-1}\mathcal{I}^\bullet$
by Lemma \ref{lemma-proper-base-change-f-star}.
Applying Leray's acyclicity lemma
(Derived Categories, Lemma \ref{derived-lemma-leray-acyclicity})
we conclude it suffices to prove
$R^pf'_*(g')^{-1}\mathcal{I}^m = 0$ for $p > 0$ and $m \in \mathbf{Z}$.
\medskip\noindent
Choose a surjective proper morphism
$h : Z \to X$ where $Z$ is a scheme, see
Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-weak-chow}.
Choose an injective map $h^{-1}\mathcal{I}^m \to \mathcal{J}$
where $\mathcal{J}$ is an injective sheaf of
$\mathbf{Z}/n\mathbf{Z}$-modules on $Z_\etale$.
Since $h$ is surjective the map $\mathcal{I}^m \to h_*\mathcal{J}$
is injective (see Lemma \ref{lemma-surjective-proper}).
Since $\mathcal{I}^m$ is injective we see that $\mathcal{I}^m$
is a direct summand of $h_*\mathcal{J}$. Thus it suffices
to prove the desired vanishing for $h_*\mathcal{J}$.
\medskip\noindent
Denote $h'$ the base change by $g$ and denote $g'' : Z' \to Z$
the projection. There is a spectral sequence
$$
E_2^{p, q} = R^pf'_* R^qh'_* (g'')^{-1}\mathcal{J}
$$
converging to $R^{p + q}(f' \circ h')_*(g'')^{-1}\mathcal{J}$.
Since $h$ and $f \circ h$ are representable (by schemes)
we know the result we want holds for them. Thus in the
spectral sequence we see that $E_2^{p, q} = 0$ for $q > 0$
and $R^{p + q}(f' \circ h')_*(g'')^{-1}\mathcal{J} = 0$
for $p + q > 0$. It follows that $E_2^{p, 0} = 0$ for $p > 0$.
Now
$$
E_2^{p, 0} = R^pf'_* h'_* (g'')^{-1}\mathcal{J} =
R^pf'_* (g')^{-1}h_*\mathcal{J}
$$
by Lemma \ref{lemma-proper-base-change-f-star}. This finishes the proof.
\end{proof}
\begin{lemma}
\label{lemma-proper-base-change}
Let $S$ be a scheme. Let
$$
\xymatrix{
X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\
Y' \ar[r]^g & Y
}
$$
be a cartesian square of algebraic spaces over $S$. Assume $f$ is proper.
Let $E \in D^+(X_\etale)$ have torsion cohomology sheaves.
Then the base change map $g^{-1}Rf_*E \to Rf'_*(g')^{-1}E$
is an isomorphism.
\end{lemma}
\begin{proof}
This is a simple consequence of the proper base change theorem
(Theorem \ref{theorem-proper-base-change}) using the spectral
sequences
$$
E_2^{p, q} = R^pf_*H^q(E)
\quad\text{and}\quad
{E'}_2^{p, q} = R^pf'_*(g')^{-1}H^q(E)
$$
converging to $R^nf_*E$ and $R^nf'_*(g')^{-1}E$.
The spectral sequences are constructed in
Derived Categories, Lemma \ref{derived-lemma-two-ss-complex-functor}.
Some details omitted.
\end{proof}
\begin{lemma}
\label{lemma-proper-base-change-stalk}
Let $S$ be a scheme.
Let $f : X \to Y$ be a proper morphism of algebraic spaces.
Let $\overline{y} \to Y$ be a geometric point.
\begin{enumerate}
\item For a torsion abelian sheaf $\mathcal{F}$ on $X_\etale$ we have
$(R^nf_*\mathcal{F})_{\overline{y}} =
H^n_\etale(X_{\overline{y}}, \mathcal{F}_{\overline{y}})$.
\item For $E \in D^+(X_\etale)$ with torsion cohomology sheaves we have
$(R^nf_*E)_{\overline{y}} = H^n_\etale(X_{\overline{y}}, E_{\overline{y}})$.
\end{enumerate}
\end{lemma}
\begin{proof}
In the statement, $\mathcal{F}_{\overline{y}}$ denotes the pullback
of $\mathcal{F}$ to $X_{\overline{y}} = \overline{y} \times_Y X$.
Since pulling back by $\overline{y} \to Y$ produces the
stalk of $\mathcal{F}$, the first statement of the lemma
is a special case of Theorem \ref{theorem-proper-base-change}.
The second one is a special case of Lemma \ref{lemma-proper-base-change}.
\end{proof}
\begin{lemma}
\label{lemma-base-change-separably-closed}
Let $k \subset k'$ be an extension of separably closed fields.
Let $X$ be a proper algebraic space over $k$.
Let $\mathcal{F}$ be a torsion abelian sheaf on $X$.
Then the map $H^q_\etale(X, \mathcal{F}) \to
H^q_\etale(X_{k'}, \mathcal{F}|_{X_{k'}})$ is an isomorphism
for $q \geq 0$.
\end{lemma}
\begin{proof}
This is a special case of Theorem \ref{theorem-proper-base-change}.
\end{proof}
\section{Comparing big and small topoi}
\label{section-compare}
\noindent
Let $S$ be a scheme and let $X$ be an algebraic space over $S$.
In Topologies on Spaces, Lemma
\ref{spaces-topologies-lemma-at-the-bottom-etale}
we have introduced comparison morphisms
$\pi_X : (\textit{Spaces}/X)_\etale \to X_{spaces, \etale}$ and
$i_X : \Sh(X_\etale) \to \Sh((\textit{Spaces}/X)_\etale)$
with $\pi_X \circ i_X = \text{id}$ as morphisms of topoi and
$\pi_{X, *} = i_X^{-1}$.
More generally, if $f : Y \to X$ is an object of $(\textit{Spaces}/X)_\etale$,
then there is a morphism
$i_f : \Sh(Y_\etale) \to \Sh((\textit{Spaces}/X)_\etale)$
such that $f_{small} = \pi_X \circ i_f$, see
Topologies on Spaces, Lemmas \ref{spaces-topologies-lemma-put-in-T-etale} and
\ref{spaces-topologies-lemma-morphism-big-small-etale}. In
Topologies on Spaces, Remark
\ref{spaces-topologies-remark-change-topologies-ringed}
we have extended these to a morphism of ringed sites
$$
\pi_X :
((\textit{Spaces}/X)_\etale, \mathcal{O})
\to
(X_{spaces, \etale}, \mathcal{O}_X)
$$
and morphisms of ringed topoi
$$
i_X :
(\Sh(X_\etale), \mathcal{O}_X)
\to
(\Sh((\textit{Spaces}/X)_\etale), \mathcal{O})
$$
and
$$
i_f :
(\Sh(Y_\etale), \mathcal{O}_Y)
\to
(\Sh((\textit{Spaces}/X)_\etale, \mathcal{O}))
$$
Note that the restriction $i_X^{-1} = \pi_{X, *}$ (see
Topologies, Definition \ref{topologies-definition-restriction-small-etale})
transforms $\mathcal{O}$ into $\mathcal{O}_X$.
Similarly, $i_f^{-1}$ transforms $\mathcal{O}$ into $\mathcal{O}_Y$.
See Topologies on Spaces, Remark
\ref{spaces-topologies-remark-change-topologies-ringed}.
Hence $i_X^*\mathcal{F} = i_X^{-1}\mathcal{F}$ and
$i_f^*\mathcal{F} = i_f^{-1}\mathcal{F}$ for any $\mathcal{O}$-module
$\mathcal{F}$ on $(\textit{Spaces}/X)_\etale$. In particular $i_X^*$ and $i_f^*$
are exact functors. The functor $i_X^*$ is often denoted
$\mathcal{F} \mapsto \mathcal{F}|_{X_\etale}$ (and this does not
conflict with the notation in
Topologies on Spaces, Definition
\ref{spaces-topologies-definition-restriction-small-etale}).
\begin{lemma}
\label{lemma-describe-pullback}
Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.
Let $\mathcal{F}$ be a sheaf on $X_\etale$. Then
$\pi_X^{-1}\mathcal{F}$ is given by the rule
$$
(\pi_X^{-1}\mathcal{F})(Y) = \Gamma(Y_\etale, f_{small}^{-1}\mathcal{F})
$$
for $f : Y \to X$ in $(\textit{Spaces}/X)_\etale$.
Moreover, $\pi_Y^{-1}\mathcal{F}$ satisfies the
sheaf condition with respect to smooth, syntomic, fppf, fpqc, and ph coverings.
\end{lemma}
\begin{proof}
Since pullback is transitive and $f_{small} = \pi_X \circ i_f$
(see above) we see that
$i_f^{-1} \pi_X^{-1}\mathcal{F} = f_{small}^{-1}\mathcal{F}$.
This shows that $\pi_X^{-1}$ has the description given in the lemma.
\medskip\noindent
To prove that $\pi_X^{-1}\mathcal{F}$ is a sheaf for the ph topology
it suffices by Topologies on Spaces, Lemma
\ref{spaces-topologies-lemma-characterize-sheaf}
to show that for a surjective proper morphism
$V \to U$ of algebraic spaces over $X$ we have
$(\pi_X^{-1}\mathcal{F})(U)$ is the equalizer of the two maps
$(\pi_X^{-1}\mathcal{F})(V) \to (\pi_X^{-1}\mathcal{F})(V \times_U V)$.
This we have seen in Lemma \ref{lemma-surjective-proper}.
\medskip\noindent
The case of smooth, syntomic, fppf coverings follows from the case
of ph coverings by Topologies on Spaces, Lemma
\ref{spaces-topologies-lemma-zariski-etale-smooth-syntomic-fppf-ph}.
\medskip\noindent
Let $\mathcal{U} = \{U_i \to U\}_{i \in I}$ be an fpqc covering of algebraic
spaces over $X$. Let $s_i \in (\pi_X^{-1}\mathcal{F})(U_i)$ be sections
which agree over $U_i \times_U U_j$. We have to prove there exists a unique
$s \in (\pi_X^{-1}\mathcal{F})(U)$ restricting to $s_i$ over $U_i$.
Case I: $U$ and $U_i$ are schemes. This case follows from
\'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-describe-pullback}.
Case II: $U$ is a scheme. Here we choose surjective \'etale morphisms
$T_i \to U_i$ where $T_i$ is a scheme. Then $\mathcal{T} = \{T_i \to U\}$ is an
fpqc covering by schemes and by case I the result holds for $\mathcal{T}$.
We omit the verification that this implies the result for $\mathcal{U}$.
Case III: general case. Let $W \to U$ be a surjective \'etale
morphism, where $W$ is a scheme. Then $\mathcal{W} = \{U_i \times_U W \to W\}$
is an fpqc covering (by algebraic spaces) of the scheme $W$.
By case II the result hold for $\mathcal{W}$.
We omit the verification that this implies the result for $\mathcal{U}$.
\end{proof}
\begin{lemma}
\label{lemma-compare-injectives}
Let $S$ be a scheme.
Let $Y \to X$ be a morphism of $(\textit{Spaces}/S)_\etale$.
\begin{enumerate}
\item If $\mathcal{I}$ is injective in
$\textit{Ab}((\textit{Spaces}/X)_\etale)$, then
\begin{enumerate}
\item $i_f^{-1}\mathcal{I}$ is injective in $\textit{Ab}(Y_\etale)$,
\item $\mathcal{I}|_{X_\etale}$ is injective in $\textit{Ab}(X_\etale)$,
\end{enumerate}
\item If $\mathcal{I}^\bullet$ is a K-injective complex
in $\textit{Ab}((\textit{Spaces}/X)_\etale)$, then
\begin{enumerate}
\item $i_f^{-1}\mathcal{I}^\bullet$ is a K-injective complex in
$\textit{Ab}(Y_\etale)$,
\item $\mathcal{I}^\bullet|_{X_\etale}$ is a K-injective complex in
$\textit{Ab}(X_\etale)$,
\end{enumerate}
\end{enumerate}
The corresponding statements for modules do not hold.
\end{lemma}
\begin{proof}
Parts (1)(b) and (2)(b)
follow formally from the fact that the restriction functor
$\pi_{X, *} = i_X^{-1}$ is a right adjoint of the exact functor
$\pi_X^{-1}$, see
Homology, Lemma \ref{homology-lemma-adjoint-preserve-injectives} and
Derived Categories, Lemma \ref{derived-lemma-adjoint-preserve-K-injectives}.
\medskip\noindent
Parts (1)(a) and (2)(a) can be seen in two ways. First proof: We can use
that $i_f^{-1}$ is a right adjoint of the exact functor $i_{f, !}$.
This functor is constructed in
Topologies, Lemma \ref{topologies-lemma-put-in-T-etale}
for sheaves of sets and for abelian sheaves in
Modules on Sites, Lemma \ref{sites-modules-lemma-g-shriek-adjoint}.
It is shown in Modules on Sites, Lemma
\ref{sites-modules-lemma-exactness-lower-shriek} that it is exact.
Second proof. We can use that $i_f = i_Y \circ f_{big}$ as is shown
in Topologies, Lemma \ref{topologies-lemma-morphism-big-small-etale}.
Since $f_{big}$ is a localization, we see that pullback by it
preserves injectives and K-injectives, see
Cohomology on Sites, Lemmas \ref{sites-cohomology-lemma-cohomology-of-open} and
\ref{sites-cohomology-lemma-restrict-K-injective-to-open}.
Then we apply the already proved parts (1)(b) and (2)(b)
to the functor $i_Y^{-1}$ to conclude.
\medskip\noindent
To see a counter example for the case of modules we refer to
\'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-compare-injectives}.
\end{proof}
\noindent
Let $S$ be a scheme.
Let $f : Y \to X$ be a morphism of algebraic spaces over $S$.
The commutative diagram of
Topologies on Spaces, Lemma
\ref{spaces-topologies-lemma-morphism-big-small-etale} (3)
leads to a commutative diagram of ringed sites
$$
\xymatrix{
(Y_{spaces, \etale}, \mathcal{O}_Y) \ar[d]_{f_{spaces, \etale}} &
((\textit{Spaces}/Y)_\etale, \mathcal{O}) \ar[d]^{f_{big}} \ar[l]^{\pi_Y} \\
(X_{spaces, \etale}, \mathcal{O}_X) &
((\textit{Spaces}/X)_\etale, \mathcal{O}) \ar[l]_{\pi_X}
}
$$
as one easily sees by writing out the definitions of
$f_{small}^\sharp$, $f_{big}^\sharp$, $\pi_X^\sharp$, and $\pi_Y^\sharp$.
In particular this means that
\begin{equation}
\label{equation-compare-big-small}
(f_{big, *}\mathcal{F})|_{X_\etale} =
f_{small, *}(\mathcal{F}|_{Y_\etale})
\end{equation}
for any sheaf $\mathcal{F}$ on $(\textit{Spaces}/Y)_\etale$ and if
$\mathcal{F}$ is a sheaf of $\mathcal{O}$-modules, then
(\ref{equation-compare-big-small})
is an isomorphism of $\mathcal{O}_X$-modules on $X_\etale$.
\begin{lemma}
\label{lemma-compare-higher-direct-image}
Let $S$ be a scheme.
Let $f : Y \to X$ be a morphism of algebraic spaces over $S$.
\begin{enumerate}
\item For $K$ in $D((\textit{Spaces}/Y)_\etale)$ we have
$
(Rf_{big, *}K)|_{X_\etale} = Rf_{small, *}(K|_{Y_\etale})
$
in $D(X_\etale)$.
\item For $K$ in $D((\textit{Spaces}/Y)_\etale, \mathcal{O})$ we have
$
(Rf_{big, *}K)|_{X_\etale} = Rf_{small, *}(K|_{Y_\etale})
$
in $D(\textit{Mod}(X_\etale, \mathcal{O}_X))$.
\end{enumerate}
More generally, let $g : X' \to X$ be an object of
$(\textit{Spaces}/X)_\etale$. Consider the fibre product
$$
\xymatrix{
Y' \ar[r]_{g'} \ar[d]_{f'} & Y \ar[d]^f \\
X' \ar[r]^g & X
}
$$
Then
\begin{enumerate}
\item[(3)] For $K$ in $D((\textit{Spaces}/Y)_\etale)$ we have
$i_g^{-1}(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^{-1}K)$
in $D(X'_\etale)$.
\item[(4)] For $K$ in $D((\textit{Spaces}/Y)_\etale, \mathcal{O})$ we have
$i_g^*(Rf_{big, *}K) = Rf'_{small, *}(i_{g'}^*K)$
in $D(\textit{Mod}(X'_\etale, \mathcal{O}_{X'}))$.
\item[(5)] For $K$ in $D((\textit{Spaces}/Y)_\etale)$ we have
$g_{big}^{-1}(Rf_{big, *}K) = Rf'_{big, *}((g'_{big})^{-1}K)$
in $D((\textit{Spaces}/X')_\etale)$.
\item[(6)] For $K$ in $D((\textit{Spaces}/Y)_\etale, \mathcal{O})$ we have
$g_{big}^*(Rf_{big, *}K) = Rf'_{big, *}((g'_{big})^*K)$
in $D(\textit{Mod}(X'_\etale, \mathcal{O}_{X'}))$.
\end{enumerate}
\end{lemma}
\begin{proof}
Part (1) follows from
Lemma \ref{lemma-compare-injectives}
and (\ref{equation-compare-big-small})
on choosing a K-injective complex of abelian sheaves representing $K$.
\medskip\noindent
Part (3) follows from Lemma \ref{lemma-compare-injectives}
and Topologies, Lemma
\ref{topologies-lemma-morphism-big-small-cartesian-diagram-etale}
on choosing a K-injective complex of abelian sheaves representing $K$.
\medskip\noindent
Part (5) follows from
Cohomology on Sites, Lemmas \ref{sites-cohomology-lemma-cohomology-of-open} and
\ref{sites-cohomology-lemma-restrict-K-injective-to-open}
and Topologies, Lemma
\ref{topologies-lemma-morphism-big-small-cartesian-diagram-etale}
on choosing a K-injective complex of abelian sheaves representing $K$.
\medskip\noindent
Part (6): Observe that $g_{big}$ and $g'_{big}$ are localizations
and hence $g_{big}^{-1} = g_{big}^*$ and $(g'_{big})^{-1} = (g'_{big})^*$
are the restriction functors. Hence (6) follows from
Cohomology on Sites, Lemmas \ref{sites-cohomology-lemma-cohomology-of-open} and
\ref{sites-cohomology-lemma-restrict-K-injective-to-open}
and Topologies, Lemma
\ref{topologies-lemma-morphism-big-small-cartesian-diagram-etale}
on choosing a K-injective complex of modules representing $K$.
\medskip\noindent
Part (2) can be proved as follows. Above we have seen
that $\pi_X \circ f_{big} = f_{small} \circ \pi_Y$ as morphisms
of ringed sites. Hence we obtain
$R\pi_{X, *} \circ Rf_{big, *} = Rf_{small, *} \circ R\pi_{Y, *}$
by Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-derived-pushforward-composition}.
Since the restriction functors $\pi_{X, *}$ and $\pi_{Y, *}$
are exact, we conclude.
\medskip\noindent
Part (4) follows from part (6) and part (2) applied to $f' : Y' \to X'$.
\end{proof}
\noindent
Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.
Let $\mathcal{H}$ be an abelian sheaf on
$(\textit{Spaces}/X)_\etale$. Recall that $H^n_\etale(U, \mathcal{H})$
denotes the cohomology of $\mathcal{H}$ over an object
$U$ of $(\textit{Spaces}/X)_\etale$.
\begin{lemma}
\label{lemma-compare-cohomology}
Let $S$ be a scheme.
Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. Then
\begin{enumerate}
\item For $K$ in $D(X_\etale)$ we have
$H^n_\etale(X, \pi_X^{-1}K) = H^n(X_\etale, K)$.
\item For $K$ in $D(X_\etale, \mathcal{O}_X)$ we have
$H^n_\etale(X, L\pi_X^*K) = H^n(X_\etale, K)$.
\item For $K$ in $D(X_\etale)$ we have
$H^n_\etale(Y, \pi_X^{-1}K) = H^n(Y_\etale, f_{small}^{-1}K)$.
\item For $K$ in $D(X_\etale, \mathcal{O}_X)$ we have
$H^n_\etale(Y, L\pi_X^*K) = H^n(Y_\etale, Lf_{small}^*K)$.
\item For $M$ in $D((\textit{Spaces}/X)_\etale)$ we have
$H^n_\etale(Y, M) = H^n(Y_\etale, i_f^{-1}M)$.
\item For $M$ in $D((\textit{Spaces}/X)_\etale, \mathcal{O})$ we have
$H^n_\etale(Y, M) = H^n(Y_\etale, i_f^*M)$.
\end{enumerate}
\end{lemma}
\begin{proof}
To prove (5) represent $M$ by a K-injective complex of abelian sheaves
and apply Lemma \ref{lemma-compare-injectives}
and work out the definitions. Part (3) follows from
this as $i_f^{-1}\pi_X^{-1} = f_{small}^{-1}$. Part (1) is a special
case of (3).
\medskip\noindent
Part (6) follows from the very general Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-pullback-same-cohomology}. Then part
(4) follows because $Lf_{small}^* = i_f^* \circ L\pi_X^*$.
Part (2) is a special case of (4).
\end{proof}
\begin{lemma}
\label{lemma-cohomological-descent-etale}
Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.
For $K \in D(X_\etale)$ the map
$$
K \longrightarrow R\pi_{X, *}\pi_X^{-1}K
$$
is an isomorphism where
$\pi_X : \Sh((\textit{Spaces}/X)_\etale) \to \Sh(X_\etale)$ is as above.
\end{lemma}
\begin{proof}
This is true because both $\pi_X^{-1}$ and $\pi_{X, *} = i_X^{-1}$
are exact functors and the composition $\pi_{X, *} \circ \pi_X^{-1}$
is the identity functor.
\end{proof}
\begin{lemma}
\label{lemma-compare-higher-direct-image-proper}
Let $S$ be a scheme.
Let $f : Y \to X$ be a proper morphism of algebraic spaces over $S$.
Then we have
\begin{enumerate}
\item $\pi_X^{-1} \circ f_{small, *} = f_{big, *} \circ \pi_Y^{-1}$
as functors $\Sh(Y_\etale) \to \Sh((\textit{Spaces}/X)_\etale)$,
\item $\pi_X^{-1}Rf_{small, *}K = Rf_{big, *}\pi_Y^{-1}K$
for $K$ in $D^+(Y_\etale)$ whose cohomology sheaves are torsion, and
\item $\pi_X^{-1}Rf_{small, *}K = Rf_{big, *}\pi_Y^{-1}K$
for all $K$ in $D(Y_\etale)$ if $f$ is finite.
\end{enumerate}
\end{lemma}
\begin{proof}
Proof of (1). Let $\mathcal{F}$ be a sheaf on $Y_\etale$.
Let $g : X' \to X$ be an object of $(\textit{Spaces}/X)_\etale$.
Consider the fibre product
$$
\xymatrix{
Y' \ar[r]_{f'} \ar[d]_{g'} & X' \ar[d]^g \\
Y \ar[r]^f & X
}
$$
Then we have
$$
(f_{big, *}\pi_Y^{-1}\mathcal{F})(X') =
(\pi_Y^{-1}\mathcal{F})(Y') =
((g'_{small})^{-1}\mathcal{F})(Y') =
(f'_{small, *}(g'_{small})^{-1}\mathcal{F})(X')
$$
the second equality by Lemma \ref{lemma-describe-pullback}.
On the other hand
$$
(\pi_X^{-1}f_{small, *}\mathcal{F})(X') =
(g_{small}^{-1}f_{small, *}\mathcal{F})(X')
$$
again by Lemma \ref{lemma-describe-pullback}.
Hence by proper base change for sheaves of sets
(Lemma \ref{lemma-proper-base-change-f-star})
we conclude the two sets are canonically isomorphic.
The isomorphism is compatible with restriction mappings
and defines an isomorphism
$\pi_X^{-1}f_{small, *}\mathcal{F} = f_{big, *}\pi_Y^{-1}\mathcal{F}$.
Thus an isomorphism of functors
$\pi_X^{-1} \circ f_{small, *} = f_{big, *} \circ \pi_Y^{-1}$.
\medskip\noindent
Proof of (2). There is a canonical base change map
$\pi_X^{-1}Rf_{small, *}K \to Rf_{big, *}\pi_Y^{-1}K$
for any $K$ in $D(Y_\etale)$, see
Cohomology on Sites, Remark \ref{sites-cohomology-remark-base-change}.
To prove it is an isomorphism, it suffices to prove the pull back of
the base change map by $i_g : \Sh(X'_\etale) \to \Sh((\Sch/X)_\etale)$
is an isomorphism for any object $g : X' \to X$ of $(\Sch/X)_\etale$.
Let $T', g', f'$ be as in the previous paragraph.
The pullback of the base change map is
\begin{align*}
g_{small}^{-1}Rf_{small, *}K
& =
i_g^{-1}\pi_X^{-1}Rf_{small, *}K \\
& \to
i_g^{-1}Rf_{big, *}\pi_Y^{-1}K \\
& =
Rf'_{small, *}(i_{g'}^{-1}\pi_Y^{-1}K) \\
& =
Rf'_{small, *}((g'_{small})^{-1}K)
\end{align*}
where we have used $\pi_X \circ i_g = g_{small}$,
$\pi_Y \circ i_{g'} = g'_{small}$, and
Lemma \ref{lemma-compare-higher-direct-image}.
This map is an isomorphism by the proper base change theorem
(Lemma \ref{lemma-proper-base-change}) provided $K$ is bounded
below and the cohomology sheaves of $K$ are torsion.
\medskip\noindent
Proof of (3). If $f$ is finite, then the functors
$f_{small, *}$ and $f_{big, *}$ are exact. This follows
from Cohomology of Spaces, Lemma
\ref{spaces-cohomology-lemma-finite-higher-direct-image-zero}
for $f_{small}$. Since any base change $f'$ of $f$ is finite too,
we conclude from Lemma \ref{lemma-compare-higher-direct-image} part (3)
that $f_{big, *}$ is exact too (as the higher derived functors are zero).
Thus this case follows from part (1).
\end{proof}
\section{Comparing fppf and \'etale topologies}
\label{section-fppf-etale}
\noindent
This section is the analogue of
\'Etale Cohomology, Section \ref{etale-cohomology-section-fppf-etale}.
\medskip\noindent
Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.
On the category $\textit{Spaces}/X$ we consider the fppf
and \'etale topologies. The identity functor
$(\textit{Spaces}/X)_\etale \to (\textit{Spaces}/X)_{fppf}$
is continuous and defines a morphism of sites
$$
\epsilon_X :
(\textit{Spaces}/X)_{fppf} \longrightarrow (\textit{Spaces}/X)_\etale
$$
by an application of Sites, Proposition \ref{sites-proposition-get-morphism}.
Please note that $\epsilon_{X, *}$ is the identity functor on underlying
presheaves and that $\epsilon_X^{-1}$ associates to an \'etale sheaf the
fppf sheafification.
Consider the morphism of sites
$$
\pi_X : (\textit{Spaces}/X)_\etale \longrightarrow X_{spaces, \etale}
$$
comparing big and small \'etale sites, see Section \ref{section-compare}.
The composition determines a morphism of sites
$$
a_X = \pi_X \circ \epsilon_X :
(\textit{Spaces}/X)_{fppf}
\longrightarrow
X_{spaces, \etale}
$$
If $\mathcal{H}$ is an abelian sheaf on $(\textit{Spaces}/X)_{fppf}$,
then we will write $H^n_{fppf}(U, \mathcal{H})$ for the cohomology
of $\mathcal{H}$ over an object $U$ of $(\textit{Spaces}/X)_{fppf}$.
\begin{lemma}
\label{lemma-comparison-fppf-etale}
Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.
\begin{enumerate}
\item For $\mathcal{F} \in \Sh(X_\etale)$ we have
$\epsilon_{X, *}a_X^{-1}\mathcal{F} = \pi_X^{-1}\mathcal{F}$
and $a_{X, *}a_X^{-1}\mathcal{F} = \mathcal{F}$.
\item For $\mathcal{F} \in \textit{Ab}(X_\etale)$ we have
$R^i\epsilon_{X, *}(a_X^{-1}\mathcal{F}) = 0$ for $i > 0$.
\end{enumerate}
\end{lemma}
\begin{proof}
We have $a_X^{-1}\mathcal{F} = \epsilon_X^{-1} \pi_X^{-1}\mathcal{F}$.
By Lemma \ref{lemma-describe-pullback} the \'etale sheaf
$\pi_X^{-1}\mathcal{F}$ is a sheaf for the fppf topology
and therefore is equal to $a_X^{-1}\mathcal{F}$ (as pulling
back by $\epsilon_X$ is given by fppf sheafification).
Recall moreover that $\epsilon_{X, *}$ is the identity
on underlying presheaves.
Now part (1) is immediate from the explicit description of $\pi_X^{-1}$
in Lemma \ref{lemma-describe-pullback}.
\medskip\noindent
We will prove part (2) by reducing it to the case of schemes --
see part (1) of
\'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-V-C-all-n-etale-fppf}.
This will ``clearly work'' as every algebraic space is
\'etale locally a scheme. The details are given below but we urge
the reader to skip the proof.
\medskip\noindent
For an abelian sheaf $\mathcal{H}$ on $(\textit{Spaces}/X)_{fppf}$ the
higher direct image $R^p\epsilon_{X, *}\mathcal{H}$ is the sheaf
associated to the presheaf $U \mapsto H^p_{fppf}(U, \mathcal{H})$
on $(\textit{Spaces}/X)_\etale$. See
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-higher-direct-images}.
Since every object of $(\textit{Spaces}/X)_\etale$ has a covering
by schemes, it suffices to prove that given $U/X$ a scheme and
$\xi \in H^p_{fppf}(U, a_X^{-1}\mathcal{F})$ we can find
an \'etale covering $\{U_i \to U\}$ such that $\xi$
restricts to zero on $U_i$. We have
\begin{align*}
H^p_{fppf}(U, a_X^{-1}\mathcal{F})
& =
H^p((\textit{Spaces}/U)_{fppf}, (a_X^{-1}\mathcal{F})|_{\textit{Spaces}/U}) \\
& =
H^p((\Sch/U)_{fppf}, (a_X^{-1}\mathcal{F})|_{\Sch/U})
\end{align*}
where the second identification is
Lemma \ref{lemma-compare-cohomology-other-topologies}
and the first is a general fact about restriction
(Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cohomology-of-open}).
Looking at the first paragraph and the corresponding result in the
case of schemes (\'Etale Cohomology, Lemma
\ref{etale-cohomology-lemma-describe-pullback-pi-fppf})
we conclude that the sheaf $(a_X^{-1}\mathcal{F})|_{\Sch/U}$
matches the pullback by the ``schemes version of $a_U$''.
Therefore we can find an \'etale covering
$\{U_i \to U\}$ such that our class dies in
$H^p((\Sch/U_i)_{fppf}, (a_X^{-1}\mathcal{F})|_{\Sch/U_i})$
for each $i$, see
\'Etale Cohomology, Lemma \ref{etale-cohomology-lemma-V-C-all-n-etale-fppf}
(the precise statement one should use here is that $V_n$ holds for all $n$