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spaces-perfect.tex
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\input{preamble}
% OK, start here.
%
\begin{document}
\title{Derived Categories of Spaces}
\maketitle
\phantomsection
\label{section-phantom}
\tableofcontents
\section{Introduction}
\label{section-introduction}
\noindent
In this chapter we discuss derived categories of modules on algebraic spaces.
There do not seem to be good introductory references addressing this topic;
it is covered in the literature by referring to papers dealing with derived
categories of modules on algebraic stacks, for example see
\cite{olsson_sheaves}.
\section{Conventions}
\label{section-conventions}
\noindent
If $\mathcal{A}$ is an abelian category and $M$ is an object
of $\mathcal{A}$ then we also denote $M$ the object of $K(\mathcal{A})$
and/or $D(\mathcal{A})$ corresponding to the complex which has
$M$ in degree $0$ and is zero in all other degrees.
\medskip\noindent
If we have a ring $A$, then $K(A)$ denotes the homotopy category
of complexes of $A$-modules and $D(A)$ the associated derived category.
Similarly, if we have a ringed space $(X, \mathcal{O}_X)$ the symbol
$K(\mathcal{O}_X)$ denotes the homotopy category of complexes of
$\mathcal{O}_X$-modules and $D(\mathcal{O}_X)$ the associated derived
category.
\section{Generalities}
\label{section-generalities}
\noindent
In this section we put some general results on cohomology of unbounded
complexes of modules on algebraic spaces.
\begin{lemma}
\label{lemma-restrict-direct-image-open}
Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces
over $S$. Given an \'etale morphism $V \to Y$, set $U = V \times_Y X$
and denote $g : U \to V$ the projection morphism. Then
$(Rf_*E)|_V = Rg_*(E|_U)$ for $E$ in $D(\mathcal{O}_X)$.
\end{lemma}
\begin{proof}
Represent $E$ by a K-injective complex $\mathcal{I}^\bullet$ of
$\mathcal{O}_X$-modules. Then $Rf_*(E) = f_*\mathcal{I}^\bullet$
and $Rg_*(E|_U) = g_*(\mathcal{I}^\bullet|_U)$ by
Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-restrict-K-injective-to-open}.
Hence the result follows from
Properties of Spaces,
Lemma \ref{spaces-properties-lemma-pushforward-etale-base-change-modules}.
\end{proof}
\begin{definition}
\label{definition-supported-on}
Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.
Let $E$ be an object of $D(\mathcal{O}_X)$.
Let $T \subset |X|$ be a closed subset.
We say $E$ is {\it supported on $T$} if the
cohomology sheaves $H^i(E)$ are supported on $T$.
\end{definition}
\section{Derived category of quasi-coherent modules on the small \'etale site}
\label{section-derived-quasi-coherent-etale}
\noindent
Let $X$ be a scheme. In this section we show that
$D_\QCoh(\mathcal{O}_X)$
can be defined in terms of the small \'etale site $X_\etale$ of $X$.
Denote $\mathcal{O}_\etale$ the structure sheaf on
$X_\etale$.
Consider the morphism of ringed sites
\begin{equation}
\label{equation-epsilon}
\epsilon :
(X_\etale, \mathcal{O}_\etale)
\longrightarrow
(X_{Zar}, \mathcal{O}_X).
\end{equation}
denoted $\text{id}_{small, \etale, Zar}$ in
Descent, Lemma \ref{descent-lemma-compare-sites}.
\begin{lemma}
\label{lemma-epsilon-flat}
The morphism $\epsilon$ of (\ref{equation-epsilon})
is a flat morphism of ringed sites. In particular the functor
$\epsilon^* : \textit{Mod}(\mathcal{O}_X) \to
\textit{Mod}(\mathcal{O}_\etale)$ is exact.
Moreover, if $\epsilon^*\mathcal{F} = 0$, then $\mathcal{F} = 0$.
\end{lemma}
\begin{proof}
The second assertion follows from the first by
Modules on Sites, Lemma \ref{sites-modules-lemma-flat-pullback-exact}.
To prove the first assertion we have to show that
$\mathcal{O}_\etale$ is a flat $\epsilon^{-1}\mathcal{O}_X$-module.
To do this it suffices to check
$\mathcal{O}_{X, x} \to \mathcal{O}_{\etale, \overline{x}}$
is flat for any geometric point $\overline{x}$ of $X$, see
Modules on Sites, Lemma
\ref{sites-modules-lemma-check-flat-stalks},
Sites, Lemma
\ref{sites-lemma-point-morphism-sites},
and
\'Etale Cohomology, Remarks
\ref{etale-cohomology-remarks-enough-points}.
By \'Etale Cohomology, Lemma
\ref{etale-cohomology-lemma-describe-etale-local-ring}
we see that $\mathcal{O}_{\etale, \overline{x}}$ is the
strict henselization of $\mathcal{O}_{X, x}$. Thus
$\mathcal{O}_{X, x} \to \mathcal{O}_{\etale, \overline{x}}$
is faithfully flat by More on Algebra,
Lemma \ref{more-algebra-lemma-dumb-properties-henselization}.
The final statement follows also: if $\epsilon^*\mathcal{F} = 0$, then
$$
0 = \epsilon^*\mathcal{F}_{\overline{x}} =
\mathcal{F}_x \otimes_{\mathcal{O}_{X, x}} \mathcal{O}_\etale
$$
(Modules on Sites, Lemma \ref{sites-modules-lemma-pullback-stalk})
for all geometric points $\overline{x}$. By faithful flatness of
$\mathcal{O}_{X, x} \to \mathcal{O}_{\etale, \overline{x}}$
we conclude $\mathcal{F}_x = 0$ for all $x \in X$.
\end{proof}
\noindent
Let $X$ be a scheme. Notation as in (\ref{equation-epsilon}).
Recall that $\epsilon^* : \QCoh(\mathcal{O}_X)
\to \QCoh(\mathcal{O}_\etale)$
is an equivalence by
Descent, Proposition \ref{descent-proposition-equivalence-quasi-coherent} and
Remark \ref{descent-remark-change-topologies-ringed-sites}.
Moreover, $\QCoh(\mathcal{O}_\etale)$ forms a
Serre subcategory of
$\textit{Mod}(\mathcal{O}_\etale)$ by
Descent, Lemma \ref{descent-lemma-equivalence-quasi-coherent-limits}.
Hence we can let $D_\QCoh(\mathcal{O}_\etale)$ be the triangulated
subcategory of $D(\mathcal{O}_\etale)$ whose objects are the
complexes with quasi-coherent cohomology sheaves, see
Derived Categories, Section \ref{derived-section-triangulated-sub}.
The functor $\epsilon^*$ is exact (Lemma \ref{lemma-epsilon-flat})
hence induces
$\epsilon^* : D(\mathcal{O}_X) \to D(\mathcal{O}_\etale)$
and since pullbacks of quasi-coherent modules are quasi-coherent
also $\epsilon^* : D_\QCoh(\mathcal{O}_X) \to
D_\QCoh(\mathcal{O}_\etale)$.
\begin{lemma}
\label{lemma-derived-quasi-coherent-small-etale-site}
Let $X$ be a scheme. The functor
$\epsilon^* : D_\QCoh(\mathcal{O}_X) \to
D_\QCoh(\mathcal{O}_\etale)$
defined above is an equivalence.
\end{lemma}
\begin{proof}
We will prove this by showing the functor
$R\epsilon_* : D(\mathcal{O}_\etale) \to D(\mathcal{O}_X)$
induces a quasi-inverse. We will use freely that $\epsilon_*$
is given by restriction to $X_{Zar} \subset X_\etale$ and the description of
$\epsilon^* = \text{id}_{small, \etale, Zar}^*$
in Descent, Lemma \ref{descent-lemma-compare-sites}.
\medskip\noindent
For a quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ the adjunction map
$\mathcal{F} \to \epsilon_*\epsilon^*\mathcal{F}$ is an isomorphism by
the fact that $\mathcal{F}^a$
(Descent, Definition \ref{descent-definition-structure-sheaf})
is a sheaf as proved in
Descent, Lemma \ref{descent-lemma-sheaf-condition-holds}.
Conversely, every quasi-coherent $\mathcal{O}_\etale$-module
$\mathcal{H}$ is of the form $\epsilon^*\mathcal{F}$ for some quasi-coherent
$\mathcal{O}_X$-module $\mathcal{F}$, see
Descent, Proposition \ref{descent-proposition-equivalence-quasi-coherent}.
Then $\mathcal{F} = \epsilon_*\mathcal{H}$ by what we just said and
we conclude that the adjunction map
$\epsilon^*\epsilon_*\mathcal{H} \to \mathcal{H}$ is an isomorphism for all
quasi-coherent $\mathcal{O}_\etale$-modules $\mathcal{H}$.
\medskip\noindent
Let $E$ be an object of $D_\QCoh(\mathcal{O}_\etale)$
and denote $\mathcal{H}^q = H^q(E)$ its $q$th cohomology
sheaf. Let $\mathcal{B}$ be the set of affine objects of $X_\etale$.
Then $H^p(U, \mathcal{H}^q) = 0$ for all $p > 0$, all $q \in \mathbf{Z}$,
and all $U \in \mathcal{B}$, see
Descent, Proposition \ref{descent-proposition-same-cohomology-quasi-coherent}
and
Cohomology of Schemes, Lemma
\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}.
By Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-cohomology-over-U-trivial}
this means that
$$
H^q(U, E) = H^0(U, \mathcal{H}^q)
$$
for all $U \in \mathcal{B}$. In particular, we find that this holds
for affine opens $U \subset X$. It follows that the $q$th cohomology of
$R\epsilon_*E$ over $U$ is the value of the sheaf $\epsilon_*\mathcal{H}^q$
over $U$. Applying sheafification we obtain
$$
H^q(R\epsilon_*E) = \epsilon_*\mathcal{H}^q
$$
which in particular shows that $R\epsilon_*$ induces a functor
$D_\QCoh(\mathcal{O}_\etale) \to D_\QCoh(\mathcal{O}_X)$.
Since $\epsilon^*$ is exact we then obtain
$H^q(\epsilon^*R\epsilon_*E) = \epsilon^*\epsilon_*\mathcal{H}^q =
\mathcal{H}^q$ (by discussion above). Thus the adjunction map
$\epsilon^*R\epsilon_*E \to E$ is an isomorphism.
\medskip\noindent
Conversely, for $F \in D_\QCoh(\mathcal{O}_X)$ the
adjunction map $F \to R\epsilon_*\epsilon^*F$
is an isomorphism for the same reason, i.e., because
the cohomology sheaves of $R\epsilon_*\epsilon^*F$
are isomorphic to
$\epsilon_*H^m(\epsilon^*F) = \epsilon_*\epsilon^*H^m(F) = H^m(F)$.
\end{proof}
\section{Derived category of quasi-coherent modules}
\label{section-derived-quasi-coherent}
\noindent
Let $S$ be a scheme. Lemma
\ref{lemma-derived-quasi-coherent-small-etale-site}
shows that the category $D_\QCoh(\mathcal{O}_S)$ can be defined
in terms of complexes of $\mathcal{O}_S$-modules on the scheme $S$
or by complexes of $\mathcal{O}$-modules on the small \'etale site
of $S$. Hence the following definition is compatible with the definition
in the case of schemes.
\begin{definition}
\label{definition-derived-quasi-coherent}
Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.
The {\it derived category of $\mathcal{O}_X$-modules with
quasi-coherent cohomology sheaves} is denoted
$D_\QCoh(\mathcal{O}_X)$.
\end{definition}
\noindent
This makes sense by
Properties of Spaces, Lemma
\ref{spaces-properties-lemma-properties-quasi-coherent}
and
Derived Categories, Section \ref{derived-section-triangulated-sub}.
Thus we obtain a canonical functor
\begin{equation}
\label{equation-compare}
D(\QCoh(\mathcal{O}_X))
\longrightarrow
D_\QCoh(\mathcal{O}_X)
\end{equation}
see Derived Categories, Equation (\ref{derived-equation-compare}).
\medskip\noindent
Observe that a flat morphism $f : Y \to X$ of algebraic spaces
induces an exact functor
$f^* : \textit{Mod}(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}_Y)$,
see
Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-flat-morphism-sites}
and
Modules on Sites, Lemma \ref{sites-modules-lemma-flat-pullback-exact}.
In particular $Lf^* : D(\mathcal{O}_X) \to D(\mathcal{O}_Y)$
is computed on any representative complex
(Derived Categories, Lemma \ref{derived-lemma-right-derived-exact-functor}).
We will write $Lf^* = f^*$ when $f$ is flat and we have
$H^i(f^*E) = f^*H^i(E)$ for $E$ in $D(\mathcal{O}_X)$ in this case.
We will use this often when $f$ is \'etale. Of course in the \'etale
case the pullback functor is just the restriction to $Y_\etale$,
see Properties of Spaces, Equation
(\ref{spaces-properties-equation-restrict-modules}).
\begin{lemma}
\label{lemma-check-quasi-coherence-on-covering}
Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.
Let $E$ be an object of $D(\mathcal{O}_X)$. The following are equivalent
\begin{enumerate}
\item $E$ is in $D_\QCoh(\mathcal{O}_X)$,
\item for every \'etale morphism $\varphi : U \to X$ where $U$ is an
affine scheme $\varphi^*E$ is an object of
$D_\QCoh(\mathcal{O}_U)$,
\item for every \'etale morphism $\varphi : U \to X$ where $U$ is a scheme
$\varphi^*E$ is an object of
$D_\QCoh(\mathcal{O}_U)$,
\item there exists a surjective \'etale morphism $\varphi : U \to X$
where $U$ is a scheme such that $\varphi^*E$ is an object of
$D_\QCoh(\mathcal{O}_U)$, and
\item there exists a surjective \'etale morphism of algebraic spaces
$f : Y \to X$ such that $Lf^*E$ is an object of
$D_\QCoh(\mathcal{O}_Y)$.
\end{enumerate}
\end{lemma}
\begin{proof}
This follows immediately from the discussion preceding the lemma and
Properties of Spaces, Lemma
\ref{spaces-properties-lemma-characterize-quasi-coherent}.
\end{proof}
\begin{lemma}
\label{lemma-quasi-coherence-direct-sums}
Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.
Then $D_\QCoh(\mathcal{O}_X)$ has direct sums.
\end{lemma}
\begin{proof}
By Injectives, Lemma \ref{injectives-lemma-derived-products}
the derived category $D(\mathcal{O}_X)$ has direct sums and
they are computed by taking termwise direct sums of any representatives.
Thus it is clear that the cohomology sheaf of a direct sum is the
direct sum of the cohomology sheaves as taking direct sums is
an exact functor (in any Grothendieck abelian category). The lemma
follows as the direct sum of quasi-coherent sheaves is quasi-coherent, see
Properties of Spaces, Lemma
\ref{spaces-properties-lemma-properties-quasi-coherent}.
\end{proof}
\noindent
We will need some information on derived limits. We warn the reader
that in the lemma below the derived limit will typically not be
an object of $D_\QCoh$.
\begin{lemma}
\label{lemma-Rlim-quasi-coherent}
Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.
Let $(K_n)$ be an inverse system of
$D_\QCoh(\mathcal{O}_X)$ with derived limit
$K = R\lim K_n$ in $D(\mathcal{O}_X)$. Assume $H^q(K_{n + 1}) \to H^q(K_n)$
is surjective for all $q \in \mathbf{Z}$ and $n \geq 1$.
Then
\begin{enumerate}
\item $H^q(K) = \lim H^q(K_n)$,
\item $R\lim H^q(K_n) = \lim H^q(K_n)$, and
\item for every affine open $U \subset X$ we have
$H^p(U, \lim H^q(K_n)) = 0$ for $p > 0$.
\end{enumerate}
\end{lemma}
\begin{proof}
Let $\mathcal{B} \subset \Ob(X_\etale)$ be the set of affine objects.
Since $H^q(K_n)$ is quasi-coherent we have $H^p(U, H^q(K_n)) = 0$
for $U \in \mathcal{B}$ by the discussion in
Cohomology of Spaces, Section
\ref{spaces-cohomology-section-higher-direct-image}
and
Cohomology of Schemes, Lemma
\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}.
Moreover, the maps $H^0(U, H^q(K_{n + 1})) \to H^0(U, H^q(K_n))$
are surjective for $U \in \mathcal{B}$ by similar reasoning.
Part (1) follows from Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-derived-limit-suitable-system}
whose conditions we have just verified.
Parts (2) and (3) follow from
Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-inverse-limit-is-derived-limit}.
\end{proof}
\begin{lemma}
\label{lemma-quasi-coherence-pullback}
Let $S$ be a scheme.
Let $f : Y \to X$ be a morphism of algebraic spaces over $S$.
The functor $Lf^*$ sends $D_\QCoh(\mathcal{O}_X)$
into $D_\QCoh(\mathcal{O}_Y)$.
\end{lemma}
\begin{proof}
Choose a diagram
$$
\xymatrix{
U \ar[d]_a \ar[r]_h & V \ar[d]^b \\
X \ar[r]^f & Y
}
$$
where $U$ and $V$ are schemes, the vertical arrows are \'etale, and
$a$ is surjective. Since $a^* \circ Lf^* = Lh^* \circ b^*$ the result
follows from
Lemma \ref{lemma-check-quasi-coherence-on-covering}
and the case of schemes which is
Derived Categories of Schemes, Lemma
\ref{perfect-lemma-quasi-coherence-pullback}.
\end{proof}
\begin{lemma}
\label{lemma-quasi-coherence-tensor-product}
Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.
For objects $K, L$ of $D_\QCoh(\mathcal{O}_X)$
the derived tensor product $K \otimes^\mathbf{L} L$ is in
$D_\QCoh(\mathcal{O}_X)$.
\end{lemma}
\begin{proof}
Let $\varphi : U \to X$ be a surjective \'etale morphism from a scheme $U$.
Since
$\varphi^*(K \otimes_{\mathcal{O}_X}^\mathbf{L} L) =
\varphi^*K \otimes_{\mathcal{O}_U}^\mathbf{L} \varphi^*L$
we see from
Lemma \ref{lemma-check-quasi-coherence-on-covering}
that this follows from the case of schemes which is
Derived Categories of Schemes, Lemma
\ref{perfect-lemma-quasi-coherence-tensor-product}.
\end{proof}
\noindent
The following lemma will help us to ``compute'' a right derived functor
on an object of $D_\QCoh(\mathcal{O}_X)$.
\begin{lemma}
\label{lemma-nice-K-injective}
Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $E$ be an
object of $D_\QCoh(\mathcal{O}_X)$. Then the canonical map
$E \to R\lim \tau_{\geq -n}E$ is an isomorphism\footnote{In particular,
$E$ has a K-injective representative as in
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-K-injective}.}.
\end{lemma}
\begin{proof}
Denote $\mathcal{H}^i = H^i(E)$ the $i$th cohomology sheaf of $E$.
Let $\mathcal{B}$ be the set of affine objects of $X_\etale$.
Then $H^p(U, \mathcal{H}^i) = 0$ for all $p > 0$, all $i \in \mathbf{Z}$,
and all $U \in \mathcal{B}$ as $U$ is an affine scheme.
See discussion in
Cohomology of Spaces, Section
\ref{spaces-cohomology-section-higher-direct-image}
and
Cohomology of Schemes, Lemma
\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}.
Thus the lemma follows from
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-is-limit-dimension}
with $d = 0$.
\end{proof}
\begin{lemma}
\label{lemma-application-nice-K-injective}
Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.
Let $F : \textit{Mod}(\mathcal{O}_X) \to \textit{Ab}$
be a functor and $N \geq 0$ an integer. Assume that
\begin{enumerate}
\item $F$ is left exact,
\item $F$ commutes with countable direct products,
\item $R^pF(\mathcal{F}) = 0$ for all $p \geq N$ and $\mathcal{F}$
quasi-coherent.
\end{enumerate}
Then for $E \in D_\QCoh(\mathcal{O}_X)$ the maps
$R^pF(E) \to R^pF(\tau_{\geq p - N + 1}E)$ are isomorphisms.
\end{lemma}
\begin{proof}
By shifting the complex we see it suffices to prove the assertion for $p = 0$.
Write $E_n = \tau_{\geq -n}E$. We have $E = R\lim E_n$, see
Lemma \ref{lemma-nice-K-injective}. Thus
$RF(E) = R\lim RF(E_n)$ in $D(\textit{Ab})$ by Injectives, Lemma
\ref{injectives-lemma-RF-commutes-with-Rlim}. Thus we have a short
exact sequence
$$
0 \to R^1\lim R^{-1}F(E_n) \to R^0F(E) \to \lim R^0F(E_n) \to 0
$$
see More on Algebra, Remark
\ref{more-algebra-remark-compare-derived-limit}.
To finish the proof we will show that the term on the left is zero
and that the term on the right equals $R^0F(E_{N - 1})$.
\medskip\noindent
We have a distinguished triangle
$$
H^{-n}(E)[n] \to E_n \to E_{n - 1} \to H^{-n}(E)[n + 1]
$$
(Derived Categories, Remark
\ref{derived-remark-truncation-distinguished-triangle})
in $D(\mathcal{O}_X)$. Since $H^{-n}(E)$ is quasi-coherent we have
$$
R^pF(H^{-n}(E)[n]) = R^{p + n}F(H^{-n}(E)) = 0
$$
for $p + n \geq N$ and
$$
R^pF(H^{-n}(E)[n + 1]) = R^{p + n + 1}F(H^{-n}(E)) = 0
$$
for $p + n + 1 \geq N$. We conclude that
$$
R^pF(E_n) \to R^pF(E_{n - 1})
$$
is an isomorphism for all $n \gg p$ and an isomorphism for
$n \geq N$ for $p = 0$. Thus the systems $R^pF(E_n)$ all
satisfy the ML condition and $R^1\lim$ gives zero (see discussion
in More on Algebra, Section \ref{more-algebra-section-Rlim}).
Moreover, the system $R^0F(\tau_{\geq - n}E)$ is constant starting
with $n = N - 1$ as desired.
\end{proof}
\section{Total direct image}
\label{section-total-direct-image}
\noindent
The following lemma is the analogue of
Cohomology of Spaces, Lemma
\ref{spaces-cohomology-lemma-vanishing-higher-direct-images}.
\begin{lemma}
\label{lemma-quasi-coherence-direct-image}
Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-separated and quasi-compact
morphism of algebraic spaces over $S$.
\begin{enumerate}
\item The functor $Rf_*$ sends $D_\QCoh(\mathcal{O}_X)$
into $D_\QCoh(\mathcal{O}_Y)$.
\item If $Y$ is quasi-compact, there exists an integer $N = N(X, Y, f)$
such that for an object $E$ of $D_\QCoh(\mathcal{O}_X)$
with $H^m(E) = 0$ for $m > 0$ we have
$H^m(Rf_*E) = 0$ for $m \geq N$.
\item In fact, if $Y$ is quasi-compact we can find $N = N(X, Y, f)$
such that for every morphism of algebraic spaces $Y' \to Y$
the same conclusion holds for the functor $R(f')_*$
where $f' : X' \to Y'$ is the base change of $f$.
\end{enumerate}
\end{lemma}
\begin{proof}
Let $E$ be an object of $D_\QCoh(\mathcal{O}_X)$.
To prove (1) we have to show that $Rf_*E$ has quasi-coherent
cohomology sheaves. This question is local on $Y$, hence we may
assume $Y$ is quasi-compact. Pick $N = N(X, Y, f)$ as in
Cohomology of Spaces, Lemma
\ref{spaces-cohomology-lemma-vanishing-higher-direct-images}.
Thus $R^pf_*\mathcal{F} = 0$ for all quasi-coherent $\mathcal{O}_X$-modules
$\mathcal{F}$ and all $p \geq N$. Moreover $R^pf_*\mathcal{F}$
is quasi-coherent for all $p$ by
Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-higher-direct-image}.
These statements remain true after base change.
\medskip\noindent
First, assume $E$ is bounded below. We will show (1) and (2) and (3) hold
for such $E$ with our choice of $N$. In this case we can for example use the
spectral sequence
$$
R^pf_*H^q(E) \Rightarrow R^{p + q}f_*E
$$
(Derived Categories, Lemma \ref{derived-lemma-two-ss-complex-functor}),
the quasi-coherence of $R^pf_*H^q(E)$, and the vanishing of $R^pf_*H^q(E)$
for $p \geq N$ to see that (1), (2), and (3) hold in this case.
\medskip\noindent
Next we prove (2) and (3). Say $H^m(E) = 0$ for $m > 0$.
Let $V$ be an affine object of $Y_\etale$.
We have $H^p(V \times_Y X, \mathcal{F}) = 0$ for $p \geq N$, see
Cohomology of Spaces, Lemma
\ref{spaces-cohomology-lemma-quasi-coherence-higher-direct-images-application}.
Hence we may apply Lemma \ref{lemma-application-nice-K-injective}
to the functor $\Gamma(V \times_Y X, -)$ to see that
$$
R\Gamma(V, Rf_*E) = R\Gamma(V \times_Y X, E)
$$
has vanishing cohomology in degrees $\geq N$. Since this holds for
all $V$ affine in $Y_\etale$ we conclude that $H^m(Rf_*E) = 0$
for $m \geq N$.
\medskip\noindent
Next, we prove (1) in the general case. Recall that there is a
distinguished triangle
$$
\tau_{\leq -n - 1}E \to E \to \tau_{\geq -n}E \to
(\tau_{\leq -n - 1}E)[1]
$$
in $D(\mathcal{O}_X)$, see Derived Categories, Remark
\ref{derived-remark-truncation-distinguished-triangle}.
By (2) we see that $Rf_*\tau_{\leq -n - 1}E$
has vanishing cohomology sheaves in degrees $\geq -n + N$.
Thus, given an integer $q$ we see that $R^qf_*E$ is equal
to $R^qf_*\tau_{\geq -n}E$ for some $n$ and the result
above applies.
\end{proof}
\begin{lemma}
\label{lemma-quasi-coherence-pushforward-direct-sums}
Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-separated and
quasi-compact morphism of algebraic spaces over $S$. Then
$Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_S)$
commutes with direct sums.
\end{lemma}
\begin{proof}
Let $E_i$ be a family of objects of $D_\QCoh(\mathcal{O}_X)$
and set $E = \bigoplus E_i$. We want to show that the map
$$
\bigoplus Rf_*E_i \longrightarrow Rf_*E
$$
is an isomorphism. We will show it induces an isomorphism on
cohomology sheaves in degree $0$ which will imply the lemma.
Choose an integer $N$ as in Lemma \ref{lemma-quasi-coherence-direct-image}.
Then $R^0f_*E = R^0f_*\tau_{\geq -N}E$ and
$R^0f_*E_i = R^0f_*\tau_{\geq -N}E_i$ by the lemma cited. Observe that
$\tau_{\geq -N}E = \bigoplus \tau_{\geq -N}E_i$.
Thus we may assume all of the $E_i$ have vanishing cohomology
sheaves in degrees $< -N$. Next we use the spectral sequences
$$
R^pf_*H^q(E) \Rightarrow R^{p + q}f_*E
\quad\text{and}\quad
R^pf_*H^q(E_i) \Rightarrow R^{p + q}f_*E_i
$$
(Derived Categories, Lemma \ref{derived-lemma-two-ss-complex-functor})
to reduce to the case of a direct sum of quasi-coherent sheaves.
This case is handled by
Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-colimit-cohomology}.
\end{proof}
\begin{remark}
\label{remark-match-total-direct-images}
Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of representable
algebraic spaces $X$ and $Y$ over $S$. Let $f_0 : X_0 \to Y_0$ be a
morphism of schemes representing $f$ (awkward but temporary notation).
Then the diagram
$$
\xymatrix{
D_\QCoh(\mathcal{O}_{X_0})
\ar@{=}[rrrrrr]_{\text{Lemma
\ref{lemma-derived-quasi-coherent-small-etale-site}}}
& & & & & &
D_\QCoh(\mathcal{O}_X) \\
D_\QCoh(\mathcal{O}_{Y_0})
\ar[u]^{Lf^*_0}
\ar@{=}[rrrrrr]^{\text{Lemma
\ref{lemma-derived-quasi-coherent-small-etale-site}}}
& & & & & &
D_\QCoh(\mathcal{O}_Y) \ar[u]_{Lf^*}
}
$$
(Lemma \ref{lemma-quasi-coherence-pullback} and
Derived Categories of Schemes, Lemma
\ref{perfect-lemma-quasi-coherence-pullback})
is commutative. This follows as the
equivalences
$D_\QCoh(\mathcal{O}_{X_0}) \to D_\QCoh(\mathcal{O}_X)$
and
$D_\QCoh(\mathcal{O}_{Y_0}) \to D_\QCoh(\mathcal{O}_Y)$
of Lemma \ref{lemma-derived-quasi-coherent-small-etale-site}
come from pulling back by the (flat) morphisms of ringed sites
$\epsilon : X_\etale \to X_{0, Zar}$ and
$\epsilon : Y_\etale \to Y_{0, Zar}$
and the diagram of ringed sites
$$
\xymatrix{
X_{0, Zar} \ar[d]_{f_0} & X_\etale \ar[l]^\epsilon \ar[d]^f \\
Y_{0, Zar} & Y_\etale \ar[l]_\epsilon
}
$$
is commutative (details omitted). If $f$ is quasi-compact and
quasi-separated, equivalently if $f_0$ is quasi-compact and
quasi-separated, then we claim
$$
\xymatrix{
D_\QCoh(\mathcal{O}_{X_0})
\ar[d]_{Rf_{0, *}} \ar@{=}[rrrrrr]_{\text{Lemma
\ref{lemma-derived-quasi-coherent-small-etale-site}}}
& & & & & &
D_\QCoh(\mathcal{O}_X) \ar[d]^{Rf_*} \\
D_\QCoh(\mathcal{O}_{Y_0})
\ar@{=}[rrrrrr]^{\text{Lemma
\ref{lemma-derived-quasi-coherent-small-etale-site}}}
& & & & & &
D_\QCoh(\mathcal{O}_Y)
}
$$
(Lemma \ref{lemma-quasi-coherence-direct-image} and
Derived Categories of Schemes, Lemma
\ref{perfect-lemma-quasi-coherence-direct-image})
is commutative as well. This also follows from the commutative
diagram of sites displayed above as the proof of Lemma
\ref{lemma-derived-quasi-coherent-small-etale-site}
shows that the functor $R\epsilon_*$ gives the equivalences
$D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_{X_0})$
and
$D_\QCoh(\mathcal{O}_Y) \to D_\QCoh(\mathcal{O}_{Y_0})$.
\end{remark}
\begin{lemma}
\label{lemma-affine-morphism}
Let $S$ be a scheme. Let $f : X \to Y$ be an affine morphism of algebraic
spaces over $S$. Then
$Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$
reflects isomorphisms.
\end{lemma}
\begin{proof}
The statement means that a morphism $\alpha : E \to F$ of
$D_\QCoh(\mathcal{O}_X)$ is an isomorphism if
$Rf_*\alpha$ is an isomorphism. We may check this on cohomology sheaves.
In particular, the question is \'etale local on $Y$. Hence we may assume
$Y$ and therefore $X$ is affine. In this case the problem reduces to the
case of schemes
(Derived Categories of Schemes, Lemma \ref{perfect-lemma-affine-morphism})
via Lemma \ref{lemma-derived-quasi-coherent-small-etale-site} and
Remark \ref{remark-match-total-direct-images}.
\end{proof}
\begin{lemma}
\label{lemma-affine-morphism-pull-push}
Let $S$ be a scheme. Let $f : X \to Y$ be an affine morphism of algebraic
spaces over $S$. For $E$ in $D_\QCoh(\mathcal{O}_Y)$ we have
$Rf_* Lf^* E = E \otimes^\mathbf{L}_{\mathcal{O}_Y} f_*\mathcal{O}_X$.
\end{lemma}
\begin{proof}
Since $f$ is affine the map $f_*\mathcal{O}_X \to Rf_*\mathcal{O}_X$
is an isomorphism (Cohomology of Spaces, Lemma
\ref{spaces-cohomology-lemma-affine-vanishing-higher-direct-images}).
There is a canonical map
$E \otimes^\mathbf{L} f_*\mathcal{O}_X =
E \otimes^\mathbf{L} Rf_*\mathcal{O}_X \to Rf_* Lf^* E$
adjoint to the map
$$
Lf^*(E \otimes^\mathbf{L} Rf_*\mathcal{O}_X) =
Lf^*E \otimes^\mathbf{L} Lf^*Rf_*\mathcal{O}_X \longrightarrow
Lf^* E \otimes^\mathbf{L} \mathcal{O}_X = Lf^* E
$$
coming from $1 : Lf^*E \to Lf^*E$ and the canonical map
$Lf^*Rf_*\mathcal{O}_X \to \mathcal{O}_X$. To check the map so constructed
is an isomorphism we may work locally on $Y$. Hence we may assume
$Y$ and therefore $X$ is affine. In this case the problem reduces to the
case of schemes
(Derived Categories of Schemes, Lemma
\ref{perfect-lemma-affine-morphism-pull-push})
via Lemma \ref{lemma-derived-quasi-coherent-small-etale-site} and
Remark \ref{remark-match-total-direct-images}.
\end{proof}
\section{Being proper over a base}
\label{section-proper-over-base}
\noindent
This section is the analogue of Cohomology of Schemes, Section
\ref{coherent-section-proper-over-base}.
As usual with material having to do with topology on the sets of points,
we have to be careful translating the material to algebraic spaces.
\begin{lemma}
\label{lemma-closed-proper-over-base}
Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces
over $S$ which is locally of finite type. Let $T \subset |X|$ be a closed
subset. The following are equivalent
\begin{enumerate}
\item the morphism $Z \to Y$ is proper if $Z$ is the reduced
induced algebraic space structure on $T$
(Properties of Spaces, Definition
\ref{spaces-properties-definition-reduced-induced-space}),
\item for some closed subspace $Z \subset X$ with $|Z| = T$
the morphism $Z \to Y$ is proper, and
\item for any closed subspace $Z \subset X$ with $|Z| = T$ the morphism
$Z \to Y$ is proper.
\end{enumerate}
\end{lemma}
\begin{proof}
The implications (3) $\Rightarrow$ (1) and (1) $\Rightarrow$ (2)
are immediate. Thus it suffices to prove that (2) implies (3).
We urge the reader to find their own proof of this fact.
Let $Z'$ and $Z''$ be closed subspaces with $T = |Z'| = |Z''|$
such that $Z' \to Y$ is a proper morphism of algebraic spaces.
We have to show that $Z'' \to Y$ is proper too.
Let $Z''' = Z' \cup Z''$ be the scheme theoretic union, see
Morphisms of Spaces, Definition
\ref{spaces-morphisms-definition-scheme-theoretic-intersection-union}.
Then $Z'''$ is another closed subspace with $|Z'''| = T$.
This follows for example from the description of scheme theoretic unions in
Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-scheme-theoretic-union}.
Since $Z'' \to Z'''$ is a closed immersion it suffices to prove
that $Z''' \to Y$ is proper (see
Morphisms of Spaces, Lemmas
\ref{spaces-morphisms-lemma-closed-immersion-proper} and
\ref{spaces-morphisms-lemma-composition-proper}).
The morphism $Z' \to Z'''$ is a bijective closed immersion
and in particular surjective and universally closed.
Then the fact that $Z' \to Y$ is separated implies that
$Z''' \to Y$ is separated, see
Morphisms of Spaces, Lemma
\ref{spaces-morphisms-lemma-image-universally-closed-separated}.
Moreover $Z''' \to Y$ is locally of finite type
as $X \to Y$ is locally of finite type
(Morphisms of Spaces, Lemmas
\ref{spaces-morphisms-lemma-immersion-locally-finite-type} and
\ref{spaces-morphisms-lemma-composition-finite-type}).
Since $Z' \to Y$ is quasi-compact and $Z' \to Z'''$ is a
universal homeomorphism we see that $Z''' \to Y$ is quasi-compact.
Finally, since $Z' \to Y$ is universally closed, we see that
the same thing is true for $Z''' \to Y$ by
Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-image-proper-is-proper}.
This finishes the proof.
\end{proof}
\begin{definition}
\label{definition-proper-over-base}
Let $S$ be a scheme.
Let $f : X \to Y$ be a morphism of algebraic spaces over $S$
which is locally of finite type.
Let $T \subset |X|$ be a closed subset.
We say {\it $T$ is proper over $Y$}
if the equivalent conditions of Lemma \ref{lemma-closed-proper-over-base}
are satisfied.
\end{definition}
\noindent
The lemma used in the definition above is false if the morphism
$f : X \to Y$ is not locally of finite type. Therefore we urge
the reader not to use this terminology if $f$ is not locally of
finite type.
\begin{lemma}
\label{lemma-closed-closed-proper-over-base}
Let $S$ be a scheme.
Let $f : X \to Y$ be a morphism of algebraic spaces over $S$
which is locally of finite type.
Let $T' \subset T \subset |X|$ be closed subsets.
If $T$ is proper over $Y$, then the same is true for $T'$.
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\begin{lemma}
\label{lemma-base-change-closed-proper-over-base}
Let $S$ be a scheme.
Consider a cartesian diagram of algebraic spaces over $S$
$$
\xymatrix{
X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^f \\
Y' \ar[r]^g & Y
}
$$
with $f$ locally of finite type.
If $T$ is a closed subset of $|X|$ proper over $Y$, then
$|g'|^{-1}(T)$ is a closed subset of $|X'|$ proper over $Y'$.
\end{lemma}
\begin{proof}
Observe that the statement makes sense as $f'$ is locally of
finite type by Morphisms of Spaces, Lemma
\ref{spaces-morphisms-lemma-base-change-finite-type}.
Let $Z \subset X$ be the reduced induced closed subspace structure on $T$.
Denote $Z' = (g')^{-1}(Z)$ the scheme theoretic inverse image.
Then $Z' = X' \times_X Z = (Y' \times_Y X) \times_X Z = Y' \times_Y Z$
is proper over $Y'$ as a base change of $Z$ over $Y$
(Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-proper}).
On the other hand, we have $T' = |Z'|$. Hence the lemma holds.
\end{proof}
\begin{lemma}
\label{lemma-functoriality-closed-proper-over-base}
Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.
Let $f : X \to Y$ be a morphism of algebraic spaces which
are locally of finite type over $B$.
\begin{enumerate}
\item If $Y$ is separated over $B$ and $T \subset |X|$ is a closed subset
proper over $B$, then $|f|(T)$ is a closed subset of $|Y|$ proper over $B$.
\item If $f$ is universally closed and $T \subset |X|$ is a
closed subset proper over $B$, then $|f|(T)$ is a closed subset
of $Y$ proper over $B$.
\item If $f$ is proper and $T \subset |Y|$ is a closed subset
proper over $B$, then $|f|^{-1}(T)$ is a closed subset of $|X|$
proper over $B$.
\end{enumerate}
\end{lemma}
\begin{proof}
Proof of (1). Assume $Y$ is separated over $B$ and $T \subset |X|$
is a closed subset proper over $B$. Let $Z$ be the reduced induced
closed subspace structure on $T$ and apply
Morphisms of Spaces, Lemma
\ref{spaces-morphisms-lemma-scheme-theoretic-image-is-proper}
to $Z \to Y$ over $B$ to conclude.
\medskip\noindent
Proof of (2). Assume $f$ is universally closed and $T \subset |X|$ is a
closed subset proper over $B$. Let $Z$ be the reduced induced
closed subspace structure on $T$ and let $Z'$ be the reduced
induced closed subspace structure on $|f|(T)$. We obtain an induced
morphism $Z \to Z'$.
Denote $Z'' = f^{-1}(Z')$ the scheme theoretic inverse image.
Then $Z'' \to Z'$ is universally closed as a base change of $f$
(Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-proper}).
Hence $Z \to Z'$ is universally closed as a composition of
the closed immersion $Z \to Z''$ and $Z'' \to Z'$
(Morphisms of Spaces, Lemmas
\ref{spaces-morphisms-lemma-closed-immersion-proper} and
\ref{spaces-morphisms-lemma-composition-proper}).
We conclude that $Z' \to B$ is separated by
Morphisms of Spaces, Lemma
\ref{spaces-morphisms-lemma-image-universally-closed-separated}.
Since $Z \to B$ is quasi-compact and $Z \to Z'$ is surjective
we see that $Z' \to B$ is quasi-compact.
Since $Z' \to B$ is the composition of $Z' \to Y$ and $Y \to B$
we see that $Z' \to B$ is locally of finite type
(Morphisms of Spaces, Lemmas
\ref{spaces-morphisms-lemma-immersion-locally-finite-type} and
\ref{spaces-morphisms-lemma-composition-finite-type}).
Finally, since $Z \to B$ is universally closed, we see that
the same thing is true for $Z' \to B$ by
Morphisms of Spaces, Lemma
\ref{spaces-morphisms-lemma-image-proper-is-proper}.
This finishes the proof.
\medskip\noindent
Proof of (3). Assume $f$ is proper and $T \subset |Y|$ is a closed subset
proper over $B$. Let $Z$ be the reduced induced closed subspace
structure on $T$. Denote $Z' = f^{-1}(Z)$ the scheme theoretic inverse image.
Then $Z' \to Z$ is proper as a base change of $f$
(Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-proper}).
Whence $Z' \to B$ is proper as the composition of $Z' \to Z$
and $Z \to B$
(Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-composition-proper}).
This finishes the proof.
\end{proof}
\begin{lemma}
\label{lemma-union-closed-proper-over-base}
Let $S$ be a scheme.
Let $f : X \to Y$ be a morphism of algebraic spaces over $S$
which is locally of finite type.
Let $T_i \subset |X|$, $i = 1, \ldots, n$ be closed subsets.
If $T_i$, $i = 1, \ldots, n$ are proper over $Y$, then the same is
true for $T_1 \cup \ldots \cup T_n$.
\end{lemma}
\begin{proof}
Let $Z_i$ be the reduced induced closed subscheme structure on $T_i$.
The morphism
$$
Z_1 \amalg \ldots \amalg Z_n \longrightarrow X
$$
is finite by Morphisms of Spaces, Lemmas
\ref{spaces-morphisms-lemma-closed-immersion-finite} and
\ref{spaces-morphisms-lemma-finite-union-finite}.
As finite morphisms are universally closed
(Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-finite-proper})
and since $Z_1 \amalg \ldots \amalg Z_n$ is proper over $S$
we conclude by
Lemma \ref{lemma-functoriality-closed-proper-over-base} part (2)
that the image $Z_1 \cup \ldots \cup Z_n$ is proper over $S$.
\end{proof}
\noindent
Let $S$ be a scheme.
Let $f : X \to Y$ be a morphism of algebraic spaces over $S$
which is locally
of finite type. Let $\mathcal{F}$ be a finite type, quasi-coherent
$\mathcal{O}_X$-module. Then the support $\text{Supp}(\mathcal{F})$
of $\mathcal{F}$ is a closed subset of $|X|$, see
Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-support-finite-type}.
Hence it makes sense to say
``the support of $\mathcal{F}$ is proper over $Y$''.
\begin{lemma}
\label{lemma-module-support-proper-over-base}