spaces-perfect.tex

\input{preamble}

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\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 $\epsilon^* : \textit{QCoh}(\mathcal{O}_X)
\to \textit{QCoh}(\mathcal{O}_\etale)$
is an equivalence by
Descent, Proposition \ref{descent-proposition-equivalence-quasi-coherent} and
Remark \ref{descent-remark-change-topologies-ringed-sites}.
Moreover, $\textit{QCoh}(\mathcal{O}_\etale)$ 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 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.

\medskip\noindent
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}.
Since $\mathcal{F} = \epsilon_*\mathcal{H}$ in this case
(as $\epsilon_*$ is the restriction to $X_{Zar} \subset X_\etale$)
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 $D_{\textit{QCoh}}(\mathcal{O}_\etale)$
and denote $\mathcal{H}^i = H^i(E)$ its $i$th cohomology
sheaf. 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 $U \in \mathcal{B}$, see
Descent, Proposition \ref{descent-proposition-same-cohomology-quasi-coherent}
and
Cohomology of Schemes, Lemma
\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}.
According to
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-K-injective}
this implies $E$ is represented by a K-injective complex
$\mathcal{I}^\bullet$ and
$\mathcal{I}^\bullet = \lim \mathcal{I}_n^\bullet$ where
each $\mathcal{I}_n^\bullet$ is a bounded below complex of injectives,
the maps in the system
$\ldots \to \mathcal{I}_2^\bullet \to \mathcal{I}_1^\bullet$
are termwise split surjections, and each $\mathcal{I}_n^\bullet$ is
quasi-isomorphic to $\tau_{\geq -n}E$.
In particular,
$$
R\epsilon_*E = \epsilon_*\mathcal{I}^\bullet =
\lim \epsilon_*\mathcal{I}_n^\bullet
$$
For every $U \in \mathcal{B}$ we have
$$
H^m(\mathcal{I}_n^\bullet(U)) =
\left\{
\begin{matrix}
\mathcal{H}^m(U) & \text{if} & m \geq -n \\
0 & \text{if} & m < n
\end{matrix}
\right.
$$
by the vanishing of $H^p(U, \mathcal{H}^i)$ for $p > 0$,
the spectral sequence
Derived Categories, Lemma \ref{derived-lemma-two-ss-complex-functor}, and
the fact that $\tau_{\geq -n}E \cong \mathcal{I}_n^\bullet$. Hence we can apply
Homology, Lemma \ref{homology-lemma-apply-Mittag-Leffler-again}
to the sequence of complexes
$$
\lim_n \mathcal{I}_n^{m - 2}(U) \to
\lim_n \mathcal{I}_n^{m - 1}(U) \to
\lim_n \mathcal{I}_n^m(U) \to
\lim_n \mathcal{I}_n^{m + 1}(U)
$$
to conclude that $H^m(\mathcal{I}^\bullet(U)) = \mathcal{H}^m(U)$ for
$U \in \mathcal{B}$. Since $\epsilon_*$ is restriction
to $X_{Zar}$ we see, on applying the above to $U \subset X$
affine open, that
$H^m(\epsilon_*\mathcal{I}^\bullet) = \epsilon_*\mathcal{H}^m$.
Thus $R\epsilon_*$ indeed gives rise to a functor
$$
R\epsilon_* :
D_{QCoh}(\mathcal{O}_\etale)
\longrightarrow
D_{QCoh}(\mathcal{O}_X)
$$
For our object $E$ of $D_{\textit{QCoh}}(\mathcal{O}_\etale)$ above
the adjunction map $\epsilon^*R\epsilon_*E \to E$
is an isomorphism as we've seen that the cohomology sheaves
of $R\epsilon_*E$ are $\epsilon_*\mathcal{H}^m$ and we have
$\epsilon^*\epsilon_*\mathcal{H}^m = \mathcal{H}^m$ (see above).
For $F \in D_{\textit{QCoh}}(\mathcal{O}_X)$ 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(\textit{QCoh}(\mathcal{O}_X))
\longrightarrow
D_{\textit{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 $D_{\textit{QCoh}}(\mathcal{O}_X)$,
\item for every \'etale morphism $\varphi : U \to X$ where $U$ is an
affine scheme $\varphi^*E$ is an object of
$D_{\textit{QCoh}}(\mathcal{O}_U)$,
\item for every \'etale morphism $\varphi : U \to X$ where $U$ is a scheme
$\varphi^*E$ is an object of
$D_{\textit{QCoh}}(\mathcal{O}_U)$,
\item there exists a surjective \'etale morphism $\varphi : U \to X$
where $U$ is a scheme such that $\varphi^*E$ is an object of
$D_{\textit{QCoh}}(\mathcal{O}_U)$, and
\item there exists a surjective \'etale morphism of algebraic spaces
$f : Y \to X$ such that $Lf^*E$ is an object of
$D_{\textit{QCoh}}(\mathcal{O}_Y)$.
\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 $D_{\textit{QCoh}}(\mathcal{O}_X)$ 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}

\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 $D_{\textit{QCoh}}(\mathcal{O}_X)$
into $D_{\textit{QCoh}}(\mathcal{O}_Y)$.
\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 $D_{\textit{QCoh}}(\mathcal{O}_X)$
the derived tensor product $K \otimes^\mathbf{L} L$ is in
$D_{\textit{QCoh}}(\mathcal{O}_X)$.
\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 $D_{\textit{QCoh}}(\mathcal{O}_X)$.

\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 $D_{\textit{QCoh}}(\mathcal{O}_X)$. Then there exists an inverse
system $\mathcal{I}_n^\bullet$ of complexes of $\mathcal{O}_X$-modules
such that
\begin{enumerate}
\item $\mathcal{I}^\bullet = \lim_n \mathcal{I}_n^\bullet$ represents $E$,
\item $\mathcal{I}_n^\bullet$ is a bounded below complex of injectives,
\item $\mathcal{I}^\bullet \to \mathcal{I}_n^\bullet$ induces an
identification $\tau_{\geq -n}E \to \mathcal{I}_n^\bullet$
in $D(\mathcal{O}_X)$,
\item the transition maps
$\mathcal{I}_{n + 1}^\bullet \to \mathcal{I}_n^\bullet$
are termwise split surjections, and
\item $\mathcal{I}^\bullet$ is a K-injective complex of
$\mathcal{O}_X$-modules.
\end{enumerate}
Moreover, $E$ is the derived limit of the inverse system of
its canonical truncations $\tau_{\geq -n}E$.
\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, Lemmas \ref{sites-cohomology-lemma-K-injective} and
\ref{sites-cohomology-lemma-is-limit}.
\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 $E \in D_{\textit{QCoh}}(\mathcal{O}_X)$ the maps
$R^pF(E) \to R^pF(\tau_{\geq p - N + 1}E)$ are isomorphisms.
\end{lemma}

\begin{proof}
Let $E$ be an object of $D_{\textit{QCoh}}(\mathcal{O}_X)$.
By shifting the complex we see it suffices to prove the assertion
for $p = 0$. Choose $\mathcal{I}^\bullet = \lim \mathcal{I}_n^\bullet$
as in Lemma \ref{lemma-nice-K-injective}.
As $\mathcal{I}^\bullet$ is K-injective $RF(E)$ is represented by
$F(\mathcal{I}^\bullet)$. As $F$ commutes with countable direct
products, and since the maps $\mathcal{I}_n^m \to \mathcal{I}_{n - 1}^m$
are split surjections, we get
$F(\mathcal{I}^\bullet) = \lim F(\mathcal{I}_n^\bullet)$.
The cohomology of
\begin{equation}
\label{equation-system-F}
F(\mathcal{I}_n^{-2}) \to
F(\mathcal{I}_n^{-1}) \to
F(\mathcal{I}_n^0) \to
F(\mathcal{I}_n^1)
\end{equation}
in degree $0$, resp.\ $-1$ is equal to $R^0F(\tau_{\geq -n}E)$,
resp.\ $R^{-1}F(\tau_{\geq -n}E)$ because $\mathcal{I}_n^\bullet$
is a bounded below complex of injectives representing $\tau_{\geq -n}E$.
We have a distinguished triangle
$$
H^{-n}(E)[n] \to \tau_{\geq -n}E \to \tau_{\geq - n + 1}E \to H^{-n}(E)[n + 1]
$$
(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
$$
R^pF(\tau_{\geq -n}E) \to R^pF(\tau_{\geq - n + 1}E)
$$
is an isomorphism for all $n \gg p$ and an isomorphism for
$n \geq N$ for $p = 0$. Thus
Homology, Lemma \ref{homology-lemma-apply-Mittag-Leffler-again}
applies to the system of sequences (\ref{equation-system-F})
and we conclude that $R^0F(E) = \lim R^0F(\tau_{\geq - n}E)$.
By the above 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 $D_{\textit{QCoh}}(\mathcal{O}_X)$
into $D_{\textit{QCoh}}(\mathcal{O}_Y)$.
\item If $Y$ is quasi-compact, there exists an integer $N = N(X, Y, f)$
such that for an object $E$ of $D_{\textit{QCoh}}(\mathcal{O}_X)$
with $H^m(E) = 0$ for $m > 0$ we have
$H^m(Rf_*E) = 0$ for $m > N$.
\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 $D_{\textit{QCoh}}(\mathcal{O}_X)$.
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 $p \geq N$. In particular, for any affine
object $V$ 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}.

\medskip\noindent
Let $E$ be an object of $D_{\textit{QCoh}}(\mathcal{O}_X)$.
Choose $\mathcal{I}^\bullet = \lim \mathcal{I}_n^\bullet$
as in Lemma \ref{lemma-nice-K-injective}.
As $\mathcal{I}^\bullet$ is K-injective $Rf_*E$ is represented by
$f_*\mathcal{I}^\bullet = \lim f_*\mathcal{I}_n^\bullet$.
Let $V$ be an affine object of $Y_\etale$. The cohomology
$H^m(f_*\mathcal{I}_n^\bullet(V))$ of
$$
f_*\mathcal{I}_n^{m - 1}(V) \to
f_*\mathcal{I}_n^m(V) \to
f_*\mathcal{I}_n^{m + 1}(V)
$$
is equal to $H^m(V \times_Y X, \tau_{\geq -n}E)$ because
$\mathcal{I}_n^\bullet$ is a bounded below complex of injectives
representing $\tau_{\geq -n}E$. We have a distinguished triangle
$$
H^{-n}(E)[n] \to \tau_{\geq -n}E \to \tau_{\geq - n + 1}E \to H^{-n}(E)[n + 1]
$$
in $D(\mathcal{O}_X)$. Since $H^{-n}(E)$ is quasi-coherent we have
$H^m(V \times_Y X, H^{-n}(E)[n]) = 0$ for $n + m \geq N$ by
our choice of $N$. Similarly, $H^m(V \times_Y X, H^{-n}(E)[n + 1]) = 0$
for $n + m + 1 \geq N$. We conclude that
$$
H^m(f_*\mathcal{I}_n^\bullet(V)) \to H^m(f_*\mathcal{I}_{n - 1}^\bullet(V))
$$
is an isomorphism for all $n \geq N - m$. Thus
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-inverse-limit-complexes}
applies to show that the $m$th cohomology sheaf of
$\lim f_*\mathcal{I}_n^\bullet$ agrees with the $m$th cohomology
sheaf of $f_*\mathcal{I}_n^\bullet$ for $n \geq N - m$. Since these
cohomology sheaves are quasi-coherent by
Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-higher-direct-image}
we get (1).

\medskip\noindent
Finally, we show that (2) and (3) hold with our choice of $N$.
Namely, the stabilization proven above gives that $H^m(Rf_*E)$ is equal
to $H^m(Rf_*(\tau_{\geq -n}E))$ for all $n$ large enough which means we
can work with objects in $D^+(\mathcal{O}_X)$ in order to prove (2) and (3).
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})
and the vanishing of $R^pf_*H^q(E)$ for $p \geq N$ to conclude.
Some details omitted.
\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
$Rf_* : D_{\textit{QCoh}}(\mathcal{O}_X) \to D_{\textit{QCoh}}(\mathcal{O}_S)$
commutes with direct sums.
\end{lemma}

\begin{proof}
Let $E_i$ be a family of objects of $D_{\textit{QCoh}}(\mathcal{O}_X)$
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 quasi-compact and quasi-separated
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 we claim the diagrams
$$
\xymatrix{
D_{\textit{QCoh}}(\mathcal{O}_{X_0})
\ar[d]_{Rf_{0, *}} \ar@{=}[rrrrrr]_{\text{Lemma
\ref{lemma-derived-quasi-coherent-small-etale-site}}}
& & & & & &
D_{\textit{QCoh}}(\mathcal{O}_X) \ar[d]^{Rf_*} \\
D_{\textit{QCoh}}(\mathcal{O}_{Y_0})
\ar@{=}[rrrrrr]^{\text{Lemma
\ref{lemma-derived-quasi-coherent-small-etale-site}}}
& & & & & &
D_{\textit{QCoh}}(\mathcal{O}_Y)
}
$$
(Lemma \ref{lemma-quasi-coherence-direct-image} and
Derived Categories of Schemes, Lemma
\ref{perfect-lemma-quasi-coherence-direct-image})
and
$$
\xymatrix{
D_{\textit{QCoh}}(\mathcal{O}_{X_0})
\ar@{=}[rrrrrr]_{\text{Lemma
\ref{lemma-derived-quasi-coherent-small-etale-site}}}
& & & & & &
D_{\textit{QCoh}}(\mathcal{O}_X) \\
D_{\textit{QCoh}}(\mathcal{O}_{Y_0})
\ar[u]^{Lf^*_0}
\ar@{=}[rrrrrr]^{\text{Lemma
\ref{lemma-derived-quasi-coherent-small-etale-site}}}
& & & & & &
D_{\textit{QCoh}}(\mathcal{O}_Y) \ar[u]_{Lf^*}
}
$$
(Lemma \ref{lemma-quasi-coherence-pullback}
and Derived Categories of Schemes, Lemma
\ref{perfect-lemma-quasi-coherence-pullback})
are commutative. The result for $Lf^*$ and $Lf_0^*$ follows as the
equivalences
$D_{\textit{QCoh}}(\mathcal{O}_{X_0}) \to D_{\textit{QCoh}}(\mathcal{O}_X)$
and
$D_{\textit{QCoh}}(\mathcal{O}_{Y_0}) \to D_{\textit{QCoh}}(\mathcal{O}_Y)$
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). In fact the commutativity of the
first diagram also follows as the proof of Lemma
\ref{lemma-derived-quasi-coherent-small-etale-site}
shows that the functor $R\epsilon_*$ gives the equivalences
$D_{\textit{QCoh}}(\mathcal{O}_X) \to D_{\textit{QCoh}}(\mathcal{O}_{X_0})$
and
$D_{\textit{QCoh}}(\mathcal{O}_Y) \to D_{\textit{QCoh}}(\mathcal{O}_{Y_0})$.
\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
$Rf_* : D_{\textit{QCoh}}(\mathcal{O}_X) \to D_{\textit{QCoh}}(\mathcal{O}_Y)$
reflects isomorphisms.
\end{lemma}

\begin{proof}
The statement means that a morphism $\alpha : E \to F$ of
$D_{\textit{QCoh}}(\mathcal{O}_X)$ 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 $D_{\textit{QCoh}}(\mathcal{O}_Y)$ 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{Derived category of coherent modules}
\label{section-derived-coherent}

\noindent
Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$.
In this case the category
$\textit{Coh}(\mathcal{O}_X) \subset \textit{Mod}(\mathcal{O}_X)$
of coherent $\mathcal{O}_X$-modules is a weak Serre subcategory, see
Homology, Section \ref{homology-section-serre-subcategories}
and
Cohomology of Spaces, Lemma
\ref{spaces-cohomology-lemma-coherent-abelian-Noetherian}.
Denote
$$
D_{\textit{Coh}}(\mathcal{O}_X) \subset D(\mathcal{O}_X)
$$
the subcategory of complexes whose cohomology sheaves are coherent, see
Derived Categories, Section \ref{derived-section-triangulated-sub}.
Thus we obtain a canonical functor
\begin{equation}
\label{equation-compare-coherent}
D(\textit{Coh}(\mathcal{O}_X))
\longrightarrow
D_{\textit{Coh}}(\mathcal{O}_X)
\end{equation}
see Derived Categories, Equation (\ref{derived-equation-compare}).

\begin{lemma}
\label{lemma-direct-image-coherent}
Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces
over $S$. Assume $f$ is locally of finite type and $Y$ is Noetherian.
Let $E$ be an object of $D^b_{\textit{Coh}}(\mathcal{O}_X)$ such that the
scheme theoretic support of $H^i(E)$ is proper over $Y$ for all $i$.
Then $Rf_*E$ is an object of $D^b_{\textit{Coh}}(\mathcal{O}_Y)$.
\end{lemma}

\begin{proof}
Consider the spectral sequence
$$
R^pf_*H^q(E) \Rightarrow R^{p + q}f_*E
$$
see Derived Categories, Lemma \ref{derived-lemma-two-ss-complex-functor}.
By assumption and
Cohomology of Spaces, Remark
\ref{spaces-cohomology-remark-scheme-theoretic-support-proper}
the sheaves $R^pf_*H^q(E)$ are coherent. Hence
$R^{p + q}f_*E$ is coherent, i.e., $E \in D_{\textit{Coh}}(\mathcal{O}_S)$.
Boundedness from below is trivial. Boundedness from above
follows from
Cohomology of Spaces, Lemma
\ref{spaces-cohomology-lemma-vanishing-higher-direct-images}
or from
Lemma \ref{lemma-quasi-coherence-direct-image}.
\end{proof}




\section{Induction principle}
\label{section-induction}

\noindent
In this section we discuss an induction principle for algebraic spaces
analogues to what is
Cohomology of Schemes, Lemma \ref{lemma-induction-principle}
for schemes. To formulate it we introduce the notion of an
{\it elementary distinguished square}; this terminology is borrowed
from \cite{MV}.
The principle as formulated here is implicit in the paper \cite{GruRay}
by Raynaud and Gruson.
A related principle for algebraic stacks is
\cite[Theorem D]{rydh_etale_devissage} by David Rydh.

\begin{definition}
\label{definition-elementary-distinguished-square}
Let $S$ be a scheme. A commutative diagram
$$
\xymatrix{
U \times_W V \ar[r] \ar[d] & V \ar[d]^f \\
U \ar[r]^j & W
}
$$
of algebraic spaces over $S$ is called an {\it elementary distinguished square}
if
\begin{enumerate}
\item $U$ is an open subspace of $W$ and $j$ is the inclusion morphism,
\item $f$ is \'etale, and
\item setting $T = W \setminus U$ (with reduced induced
subspace structure) the morphism $f^{-1}(T) \to T$ is an isomorphism.
\end{enumerate}
We will indicate this by saying: ``Let $(U \subset W, f : V \to W)$
be an elementary distinguished square.''
\end{definition}

\noindent
Note that if $(U \subset W, f : V \to W)$ is an elementary distinguished
square, then we have $W = U \cup f(V)$. Thus $\{U \to W, V \to W\}$ is
an \'etale covering of $W$. It turns out that these \'etale coverings
have nice properties and that in some sense
there are ``enough'' of them.

\begin{lemma}
\label{lemma-make-more-elementary-distinguished-squares}
Let $S$ be a scheme. Let $(U \subset W, f : V \to W)$ be an elementary
distinguished square of algebraic spaces over $S$.
\begin{enumerate}
\item If $V' \subset V$ and
$U \subset U' \subset W$ are open subspaces and $W' = U' \cup f(V')$
then $(U' \subset W', f|_{V'} : V' \to W')$ is an elementary distinguished
square.
\item If $p : W' \to W$ is a morphism of algebraic spaces, then
$(p^{-1}(U) \subset W', V \times_W W' \to W')$ is an elementary distinguished
square.
\end{enumerate}
\end{lemma}

\begin{proof}
Omitted.
\end{proof}

\begin{lemma}
\label{lemma-induction-principle}
Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated
algebraic space over $S$. Let $P$ be a property of the quasi-compact
and quasi-separated objects of $X_{spaces, \etale}$. Assume that
\begin{enumerate}
\item $P$ holds for every affine object of $X_{spaces, \etale}$,
\item for every elementary distinguished square $(U \subset W, f : V \to W)$
such that
\begin{enumerate}
\item $W$ is a quasi-compact and quasi-separated object of
$X_{spaces, \etale}$,
\item $U$ is quasi-compact,
\item $V$ is affine, and
\item $P$ holds for $U$, $V$, and $U \times_W V$,
\end{enumerate}
then $P$ holds for $W$.
\end{enumerate}
Then $P$ holds for every quasi-compact and quasi-separated object
of $X_{spaces, \etale}$ and in particular for $X$.
\end{lemma}

\begin{proof}
We first claim that $P$ holds for every representable
quasi-compact and quasi-separated object of $X_{spaces, \etale}$.
Namely, suppose that $U \to X$ is \'etale and $U$ is a
quasi-compact and quasi-separated scheme. By assumption (1)
property $P$ holds for every affine open of $U$. Moreover, if
$W, V \subset U$ are quasi-compact open with $V$ affine and $P$ holds for 
$W$, $V$, and $W \cap V$, then $P$ holds for $W \cup V$ by (2)
(as the pair $(W \subset W \cup V, V \to W \cup V)$ is an elementary
distinguished square). Thus $P$ holds for $U$ by the induction
principle for schemes, see
Cohomology of Schemes, Lemma \ref{coherent-lemma-induction-principle}.

\medskip\noindent
To finish the proof it suffices to prove $P$ holds for $X$
(because we can simply replace $X$ by any quasi-compact and quasi-separated
object of $X_{spaces, \etale}$ we want to prove the result for).
We will use the filtration
$$
\emptyset = U_{n + 1} \subset
U_n \subset U_{n - 1} \subset \ldots \subset U_1 = X
$$
and the morphisms $f_p : V_p \to U_p$ of
Limits of Spaces, Lemma
\ref{spaces-limits-lemma-filter-quasi-compact-quasi-separated}.
We will prove that $P$ holds for $U_p$ by descending induction on $p$.
Note that $P$ holds for $U_{n + 1}$ by (1)
as an empty algebraic space is affine. Assume $P$ holds for $U_{p + 1}$.
Note that $(U_{p + 1} \subset U_p, f_p : V_p \to U_p)$ is an elementary
distinguished square, but (2) may not apply as $V_p$ may not be affine.
However, as $V_p$ is a quasi-compact scheme we may choose a finite affine open
covering $V_p = V_{p, 1} \cup \ldots \cup V_{p, m}$.
Set $W_{p, 0} = U_{p + 1}$ and
$$
W_{p, i} = U_{p + 1} \cup f_p(V_{p, 1} \cup \ldots \cup V_{p, i})
$$
for $i = 1, \ldots, m$. These are quasi-compact open subspaces of $X$.
Then we have
$$
U_{p + 1} = W_{p, 0} \subset
W_{p, 1} \subset \ldots \subset
W_{p, m} = U_p
$$
and the pairs
$$
(W_{p, 0} \subset W_{p, 1}, f_p|_{V_{p, 1}}),
(W_{p, 1} \subset W_{p, 2}, f_p|_{V_{p, 2}}),\ldots,
(W_{p, m - 1} \subset W_{p, m}, f_p|_{V_{p, m}})
$$
are elementary distinguished squares by
Lemma \ref{lemma-make-more-elementary-distinguished-squares}.
Note that $P$ holds for each $V_{p, 1}$ (as affine schemes) and for
$W_{p, i} \times_{W_{p, i + 1}} V_{p, i + 1}$ as this is a quasi-compact
open of $V_{p, i + 1}$ and hence $P$ holds for it by the first paragraph
of this proof. Thus (2) applies to each of these and we inductively
conclude $P$ holds for $W_{p, 1}, \ldots, W_{p, m} = U_p$.
\end{proof}

\begin{lemma}
\label{lemma-induction-principle-separated}
Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated
algebraic space over $S$. Let
$\mathcal{B} \subset \Ob(X_{spaces, \etale})$.
Let $P$ be a property of the elements of $\mathcal{B}$.
Assume that
\begin{enumerate}
\item every $W \in \mathcal{B}$ is quasi-compact and quasi-separated,
\item if $W \in \mathcal{B}$ and $U \subset W$ is quasi-compact open, then
$U \in \mathcal{B}$,
\item if $V \in \Ob(X_{spaces, \etale})$ is affine, then
(a) $V \in \mathcal{B}$ and (b) $P$ holds for $V$,
\item for every elementary distinguished square $(U \subset W, f : V \to W)$
such that
\begin{enumerate}
\item $W \in \mathcal{B}$,
\item $U$ is quasi-compact,
\item $V$ is affine, and
\item $P$ holds for $U$, $V$, and $U \times_W V$,
\end{enumerate}
then $P$ holds for $W$.
\end{enumerate}
Then $P$ holds for every $W \in \mathcal{B}$.
\end{lemma}

\begin{proof}
This is proved in exactly the same manner as the proof of
Lemma \ref{lemma-induction-principle}.
(We remark that (4)(d) makes sense as $U \times_W V$ is a quasi-compact
open of $V$ hence an element of $\mathcal{B}$ by conditions
(2) and (3).)
\end{proof}

\begin{remark}
\label{remark-how-to}
How to choose the collection $\mathcal{B}$ in
Lemma \ref{lemma-induction-principle-separated}?
Here are some examples:
\begin{enumerate}
\item If $X$ is quasi-compact and separated, then we can choose
$\mathcal{B}$ to be the set of quasi-compact and separated objects
of $X_{spaces, \etale}$. Then $X \in \mathcal{B}$ and $\mathcal{B}$
satisfies (1), (2), and (3)(a). With this choice of $\mathcal{B}$
Lemma \ref{lemma-induction-principle-separated} reproduces
Lemma \ref{lemma-induction-principle}.
\item If $X$ is quasi-compact with affine diagonal, then we can choose
$\mathcal{B}$ to be the set of objects
of $X_{spaces, \etale}$ which are quasi-compact and have affine
diagonal. Again $X \in \mathcal{B}$ and $\mathcal{B}$
satisfies (1), (2), and (3)(a).
\item If $X$ is quasi-compact and quasi-separated, then the
smallest subset $\mathcal{B}$ which contains $X$ and satisfies
(1), (2), and (3)(a) is given by the rule $W \in \mathcal{B}$ if and only
if either $W$ is a quasi-compact open subspace of $X$, or
$W$ is a quasi-compact open of an affine object of $X_{spaces, \etale}$.
\end{enumerate}
\end{remark}

\noindent
Here is a variant where we extend the truth from an open to larger opens.

\begin{lemma}
\label{lemma-induction-principle-enlarge}
Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated
algebraic space over $S$. Let $W \subset X$ be a quasi-compact open
subspace. Let $P$ be a property of quasi-compact open subspaces of $X$.
Assume that
\begin{enumerate}
\item $P$ holds for $W$, and
\item for every elementary distinguished square
$(W_1 \subset W_2, f : V \to W_2)$ where 
such that
\begin{enumerate}
\item $W_1$, $W_2$ are quasi-compact open subspaces of $X$,
\item $W \subset W_1$,
\item $V$ is affine, and
\item $P$ holds for $W_1$,
\end{enumerate}
then $P$ holds for $W_2$.
\end{enumerate}
Then $P$ holds for $X$.
\end{lemma}

\begin{proof}
We can deduce this from Lemma \ref{lemma-induction-principle-separated},
but instead we will give a direct argument by eplicitly redoing the proof of
Lemma \ref{lemma-induction-principle}. We will use the filtration
$$
\emptyset = U_{n + 1} \subset
U_n \subset U_{n - 1} \subset \ldots \subset U_1 = X
$$
and the morphisms $f_p : V_p \to U_p$ of
Limits of Spaces, Lemma
\ref{spaces-limits-lemma-filter-quasi-compact-quasi-separated}.
We will prove that $P$ holds for $W_p = W \cup U_p$ by descending
induction on $p$. This will finish the proof as $W_1 = X$.
Note that $P$ holds for $W_{n + 1} = W \cap U_{n + 1} = W$
by (1). Assume $P$ holds for $W_{p + 1}$. Observe that
$W_p \setminus W_{p + 1}$ (with reduced induced subspace structure)
is a closed subspace of $U_p \setminus U_{p + 1}$.
Since $(U_{p + 1} \subset U_p, f_p : V_p \to U_p)$ is an elementary
distinguished square, the same is true for
$(W_{p + 1} \subset W_p, f_p : V_p \to W_p)$.
However (2) may not apply as $V_p$ may not be affine.
However, as $V_p$ is a quasi-compact scheme we may choose
a finite affine open covering $V_p = V_{p, 1} \cup \ldots \cup V_{p, m}$.
Set $W_{p, 0} = W_{p + 1}$ and
$$
W_{p, i} = W_{p + 1} \cup f_p(V_{p, 1} \cup \ldots \cup V_{p, i})
$$
for $i = 1, \ldots, m$. These are quasi-compact open subspaces of $X$
containing $W$. Then we have
$$
W_{p + 1} = W_{p, 0} \subset
W_{p, 1} \subset \ldots \subset
W_{p, m} = W_p
$$
and the pairs
$$
(W_{p, 0} \subset W_{p, 1}, f_p|_{V_{p, 1}}),
(W_{p, 1} \subset W_{p, 2}, f_p|_{V_{p, 2}}),\ldots,
(W_{p, m - 1} \subset W_{p, m}, f_p|_{V_{p, m}})
$$
are elementary distinguished squares by
Lemma \ref{lemma-make-more-elementary-distinguished-squares}.
Now (2) applies to each of these and we inductively
conclude $P$ holds for $W_{p, 1}, \ldots, W_{p, m} = W_p$.
\end{proof}



\section{Mayer-Vietoris}
\label{section-mayer-vietoris}

\noindent
In this section we prove that an elementary distinguished triangle
gives rise to various Mayer-Vietoris sequences.

\medskip\noindent
Let $S$ be a scheme. Let $U \to X$ be an \'etale morphism of algebraic
spaces over $S$. In
Properties of Spaces, Section \ref{spaces-properties-section-localize}
it was shown that
$U_{spaces, \etale} = X_{spaces, \etale}/U$
compatible with structure sheaves. Hence in this situation we
often think of the morphism $j_U : U \to X$ as a localization morphism
(see Modules on Sites, Definition
\ref{sites-modules-definition-localize-ringed-site}).
In particular we think of pullback $j_U^*$ as restriction to $U$
and we often denote it by ${}|_U$; this is compatible with
Properties of Spaces, Equation
(\ref{spaces-properties-equation-restrict-modules}).
In particular we see that
\begin{equation}
\label{equation-stalk-restriction}
(\mathcal{F}|_U)_{\overline{u}} = \mathcal{F}_{\overline{x}}
\end{equation}
if $\overline{u}$ is a geometric point of $U$ and $\overline{x}$
the image of $\overline{u}$ in $X$.
Moreover, restriction has an exact left adjoint $j_{U!}$, see
Modules on Sites, Lemmas \ref{sites-modules-lemma-extension-by-zero} and
\ref{sites-modules-lemma-extension-by-zero-exact}.
Finally, recall that if $\mathcal{G}$ is an $\mathcal{O}_X$-module,
then
\begin{equation}
\label{equation-stalk-j-shriek}
(j_{U!}\mathcal{G})_{\overline{x}} =
\bigoplus\nolimits_{\overline{u}} \mathcal{G}_{\overline{u}}
\end{equation}
for any geometric point $\overline{x} : \Spec(k) \to X$ where the
direct sum is over those morphism $\overline{u} : \Spec(k) \to U$
such that $j_U \circ \overline{u} = \overline{x}$, see
Modules on Sites, Lemma \ref{sites-modules-lemma-stalk-j-shriek}
and
Properties of Spaces, Lemma
\ref{spaces-properties-lemma-points-small-etale-site}.

\begin{lemma}
\label{lemma-exact-sequence-lower-shriek}
Let $S$ be a scheme. Let $(U \subset X, V \to X)$ be an elementary
distinguished square of algebraic spaces over $S$.
\begin{enumerate}
\item For a sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$
we have a short exact sequence
$$
0 \to j_{U \times_X V!}\mathcal{F}|_{U \times_X V} \to
j_{U!}\mathcal{F}|_U \oplus j_{V!}\mathcal{F}|_V \to \mathcal{F} \to 0
$$
\item For an object $E$ of $D(\mathcal{O}_X)$ we have a distinguished
triangle
$$
j_{U \times_X V!}E|_{U \times_X V} \to
j_{U!}E|_U \oplus j_{V!}E|_V \to E \to 
j_{U \times_X V!}E|_{U \times_X V}[1]
$$
in $D(\mathcal{O}_X)$.
\end{enumerate}
\end{lemma}

\begin{proof}
To show the sequence of (1) is exact we may check on stalks at
geometric points by
Properties of Spaces, Theorem
\ref{spaces-properties-theorem-exactness-stalks}.
Let $\overline{x}$ be a geometric point of $X$. By Equations
(\ref{equation-stalk-restriction}) and (\ref{equation-stalk-j-shriek})
taking stalks at $\overline{x}$ we obtain the sequence
$$
0 \to
\bigoplus\nolimits_{(\overline{u}, \overline{v})} \mathcal{F}_{\overline{x}}
\to
\bigoplus\nolimits_{\overline{u}} \mathcal{F}_{\overline{x}}
\oplus
\bigoplus\nolimits_{\overline{v}} \mathcal{F}_{\overline{x}}
\to
\mathcal{F}_{\overline{x}} \to 0
$$
This sequence is exact because for every $\overline{x}$
there either is exactly one $\overline{u}$ mapping to $\overline{x}$,
or there is no $\overline{u}$ and exactly one $\overline{v}$
mapping to $\overline{x}$.

\medskip\noindent
Proof of (2). We have seen in Cohomology on Sites, Section
\ref{sites-cohomology-section-properties-K-injective}
that the restriction functors and the extension by zero functors
on derived categories are computed by just applying the functor
to any complex. Let $\mathcal{E}^\bullet$ be a complex
of $\mathcal{O}_X$-modules representing $E$.
The distinguished triangle of the lemma is the
distinguished triangle associated (by
Derived Categories, Section
\ref{derived-section-canonical-delta-functor} and especially
Lemma \ref{derived-lemma-derived-canonical-delta-functor})
to the short exact sequence of complexes of $\mathcal{O}_X$-modules
$$
0 \to j_{U \times_X V!}\mathcal{E}^\bullet|_{U \times_X V} \to
j_{U!}\mathcal{E}^\bullet|_U \oplus j_{V!}\mathcal{E}^\bullet|_V
\to \mathcal{E}^\bullet \to 0
$$
which is short exact by (1).
\end{proof}

\begin{lemma}
\label{lemma-exact-sequence-j-star}
Let $S$ be a scheme. Let $(U \subset X, V \to X)$ be an elementary
distinguished square of algebraic spaces over $S$.
\begin{enumerate}
\item For every sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$
we have a short exact sequence
$$
0 \to \mathcal{F} \to
j_{U, *}\mathcal{F}|_U \oplus j_{V, *}\mathcal{F}|_V \to
j_{U \times_X V, *}\mathcal{F}|_{U \times_X V} \to 0
$$
\item For any object $E$ of $D(\mathcal{O}_X)$ we have a distinguished
triangle
$$
E \to 
Rj_{U, *}E|_U \oplus Rj_{V, *}E|_V \to
Rj_{U \times_X V, *}E|_{U \times_X V} \to
E[1]
$$
in $D(\mathcal{O}_X)$.
\end{enumerate}
\end{lemma}

\begin{proof}
Let $W$ be an object of $X_\etale$. We claim the sequence
$$
0 \to
\mathcal{F}(W) \to
\mathcal{F}(W \times_X U) \oplus \mathcal{F}(W \times_X V) \to
\mathcal{F}(W \times_X U \times_X V)
$$
is exact and that an element of the last group can locally on $W$
be lifted to the middle one.
By Lemma \ref{lemma-make-more-elementary-distinguished-squares}
the pair $(W \times_X U \subset W, V \times_X W \to W)$ is an elementary
distinguished square. Thus we may assume $W = X$ and it suffices
to prove the same thing for
$$
0 \to
\mathcal{F}(X) \to
\mathcal{F}(U) \oplus \mathcal{F}(V) \to
\mathcal{F}(U \times_X V)
$$
We have seen that
$$
0 \to j_{U \times_X V!}\mathcal{O}_{U \times_X V}
\to j_{U!}\mathcal{O}_U \oplus
j_{V!}\mathcal{O}_V \to
\mathcal{O}_X \to 0
$$
is a exact sequence of $\mathcal{O}_X$-modules in
Lemma \ref{lemma-exact-sequence-lower-shriek} and applying
the right exact functor $\Hom_{\mathcal{O}_X}(- , \mathcal{F})$
gives the sequence above. This also means that the obstruction
to lifting $s \in \mathcal{F}(U \times_X V)$ to
an element of $\mathcal{F}(U) \oplus \mathcal{F}(V)$ lies in
$\text{Ext}^1_{\mathcal{O}_X}(\mathcal{O}_X, \mathcal{F}) =
H^1(X, \mathcal{F})$. By locality of cohomology
(Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-kill-cohomology-class-on-covering})
this obstruction vanishes \'etale locally on $X$ and the proof
of (1) is complete.

\medskip\noindent
Proof of (2).
Choose a K-injective complex $\mathcal{I}^\bullet$ representing $E$
whose terms $\mathcal{I}^n$ are injective objects of
$\textit{Mod}(\mathcal{O}_X)$, see Injectives, Theorem
\ref{injectives-theorem-K-injective-embedding-grothendieck}.
Then $\mathcal{I}^\bullet|U$ is a K-injective complex
(Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-restrict-K-injective-to-open}).
Hence $Rj_{U, *}E|_U$ is represented by $j_{U, *}\mathcal{I}^\bullet|_U$.
Similarly for $V$ and $U \times_X V$. Hence the distinguished triangle
of the lemma is the distinguished triangle associated (by
Derived Categories, Section
\ref{derived-section-canonical-delta-functor} and especially
Lemma \ref{derived-lemma-derived-canonical-delta-functor})
to the short exact sequence of complexes
$$
0 \to
\mathcal{I}^\bullet \to
j_{U, *}\mathcal{I}^\bullet|_U \oplus j_{V, *}\mathcal{I}^\bullet|_V \to
j_{U \times_X V, *}\mathcal{I}^\bullet|_{U \times_X V} \to
0.
$$
This sequence is exact by (1).
\end{proof}

\begin{lemma}
\label{lemma-unbounded-relative-mayer-vietoris}
Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces
over $S$. Let $(U \subset X, V \to X)$ be an elementary distinguished square.
Denote $a = f|_U : U \to Y$, $b = f|_V : V \to Y$, and
$c = f|_{U \times_X V} : U \times_X V \to Y$ the restrictions.
For every object $E$ of $D(\mathcal{O}_X)$ there exists a
distinguished triangle
$$
Rf_*E \to
Ra_*(E|_U) \oplus Rb_*(E|_V) \to
Rc_*(E|_{U \times_X V}) \to
Rf_*E[1]
$$
in $D(\mathcal{O}_Y)$. This triangle is functorial in $E$.
\end{lemma}

\begin{proof}
Choose a K-injective complex $\mathcal{I}^\bullet$
representing $E$. We may assume $\mathcal{I}^n$ is an injective
object of $\textit{Mod}(\mathcal{O}_X)$ for all $n$, see
Injectives, Theorem
\ref{injectives-theorem-K-injective-embedding-grothendieck}.
Then $Rf_*E$ is computed by $f_*\mathcal{I}^\bullet$.
Similarly for $U$, $V$, and $U \cap V$ by
Cohomology on Sites,
Lemma \ref{sites-cohomology-lemma-restrict-K-injective-to-open}.
Hence the distinguished triangle of the lemma is the distinguished
triangle associated (by
Derived Categories, Section
\ref{derived-section-canonical-delta-functor} and especially
Lemma \ref{derived-lemma-derived-canonical-delta-functor})
to the short exact sequence of complexes
$$
0 \to
f_*\mathcal{I}^\bullet \to
a_*\mathcal{I}^\bullet|_U \oplus b_*\mathcal{I}^\bullet|_V \to
c_*\mathcal{I}^\bullet|_{U \times_X V} \to
0.
$$
To see this is a short exact sequence of complexes we argue as
follows. Pick an injective object $\mathcal{I}$ of
$\textit{Mod}(\mathcal{O}_X)$. Apply $f_*$ to the short exact sequence
$$
0 \to \mathcal{I} \to
j_{U, *}\mathcal{I}|_U \oplus j_{V, *}\mathcal{I}|_V \to
j_{U \times_X V, *}\mathcal{I}|_{U \times_X V} \to 0
$$
of Lemma \ref{lemma-exact-sequence-j-star}
and use that $R^1f_*\mathcal{I} = 0$ to get a short exact sequence
$$
0 \to f_*\mathcal{I} \to
f_*j_{U, *}\mathcal{I}|_U \oplus f_*j_{V, *}\mathcal{I}|_V \to
f_*j_{U \times_X V, *}\mathcal{I}|_{U \times_X V} \to 0
$$
The proof is finished by observing that $a_* = f_*j_{U, *}$ and similarly
for $b_*$ and $c_*$.
\end{proof}

\begin{lemma}
\label{lemma-mayer-vietoris-hom}
Let $S$ be a scheme. Let $(U \subset X, V \to X)$ be an elementary
distinguished square of algebraic spaces over $S$.
For objects $E$, $F$ of $D(\mathcal{O}_X)$ we have a
Mayer-Vietoris sequence
$$
\xymatrix{
& \ldots \ar[r] &
\text{Ext}^{-1}(E_{U \times_X V}, F_{U \times_X V}) \ar[lld] \\
\Hom(E, F) \ar[r] &
\Hom(E_U, F_U) \oplus
\Hom(E_V, F_V) \ar[r] &
\Hom(E_{U \times_X V}, F_{U \times_X V})
}
$$
where the subscripts denote restrictions to the relevant opens
and the $\Hom$'s are taken in the relevant derived categories.
\end{lemma}

\begin{proof}
Use the distinguished triangle of
Lemma \ref{lemma-exact-sequence-lower-shriek}
to obtain a long exact sequence of $\Hom$'s
(from Derived Categories, Lemma \ref{derived-lemma-representable-homological})
and use that $\Hom(j_{U!}E|_U, F) = \Hom(E|_U, F|_U)$
by Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-adjoint-lower-shriek-restrict}.
\end{proof}

\begin{lemma}
\label{lemma-restrict-lower-shriek}
Let $S$ be a scheme. Let $j : U \to X$ be a \'etale morphism of algebraic
spaces over $S$. Given an \'etale morphism $V \to Y$, set $W = V \times_X U$
and denote $j_W : W \to V$ the projection morphism. Then
$(j_!E)|_V = j_{W!}(E|_W)$ for $E$ in $D(\mathcal{O}_U)$.
\end{lemma}

\begin{proof}
This is true because
$(j_!\mathcal{F})|_V = j_{W!}(\mathcal{F}|_W)$
for an $\mathcal{O}_X$-module $\mathcal{F}$ as follows immediately 
from the construction of the functors $j_!$ and $j_{W!}$, see
Modules on Sites, Lemma \ref{sites-modules-lemma-extension-by-zero}.
\end{proof}

\begin{lemma}
\label{lemma-pushforward-with-support-in-open}
Let $S$ be a scheme. Let $(U \subset X, j : V \to X)$ be an elementary
distinguished square of algebraic spaces over $S$. Set
$T = |X| \setminus |U|$.
\begin{enumerate}
\item If $E$ is an object of $D(\mathcal{O}_X)$ supported on $T$, then
(a) $E \to Rj_*(E|_V)$ and (b) $j_!(E|_V) \to E$ are isomorphisms.
\item If $F$ is an object of $D(\mathcal{O}_V)$ supported on $j^{-1}T$, then
(a) $F \to (j_!F)|_V$, (b) $(Rj_*F)|_V \to F$, and (c)
$j_!F \to Rj_*F$ are isomorphisms.
\end{enumerate}
\end{lemma}

\begin{proof}
Let $E$ be an object of $D(\mathcal{O}_X)$ whose cohomology sheaves are
supported on $T$. Then we see that $E|_U = 0$ and $E|_{U \times_X V} = 0$
as $T$ doesn't meet $U$ and $j^{-1}T$ doesn't meet $U \times_X V$.
Thus (1)(a) follows from Lemma \ref{lemma-exact-sequence-j-star}.
In exactly the same way (1)(b) follows from
Lemma \ref{lemma-exact-sequence-lower-shriek}.

\medskip\noindent
Let $F$ be an object of $D(\mathcal{O}_V)$ whose cohomology sheaves
are supported on $j^{-1}T$. By
Lemma \ref{lemma-restrict-direct-image-open} we have
$(Rj_*F)|_U = Rj_{W, *}(F|_W) = 0$ because $F|_W = 0$ by our assumption.
Similarly $(j_!F)|_U = j_{W!}(F|_W) = 0$ by
Lemma \ref{lemma-restrict-lower-shriek}.
Thus $j_!F$ and $Rj_*F$ are
supported on $T$ and $(j_!F)|_V$ and $(Rj_*F)|_V$ are supported on
$j^{-1}(T)$. To check that the maps (2)(a), (b), (c) are isomorphisms
in the derived category, it suffices to check that these map induce
isomorphisms on stalks of cohomology sheaves at geometric points of $T$
and $j^{-1}(T)$ by
Properties of Spaces, Theorem
\ref{spaces-properties-theorem-exactness-stalks}.
This we may do after replacing $X$ by $V$, $U$ by $U \times_X V$,
$V$ by $V \times_X V$ and $F$ by $F|_{V \times_X V}$ (restriction via
first projection), see
Lemmas \ref{lemma-restrict-direct-image-open},
\ref{lemma-restrict-lower-shriek}, and
\ref{lemma-make-more-elementary-distinguished-squares}.
Since $V \times_X V \to V$ has a section this
reduces (2) to the case that $j : V \to X$ has a section.

\medskip\noindent
Assume $j$ has a section $\sigma : X \to V$.
Set $V' = \sigma(X)$. This is an open subspace of $V$.
Set $U' = j^{-1}(U)$. This is another open subspace of $V$.
Then $(U' \subset V, V' \to V)$ is an elementary distinguished
square. Observe that $F|_{U'} = 0$ and $F|_{V' \cap U'} = 0$
because $F$ is supported on $j^{-1}(T)$. Denote $j' : V' \to V$
the open immersion and $j_{V'} : V' \to X$ the composition
$V' \to V \to X$ which is the inverse of $\sigma$.
Set $F' = \sigma^*F$. The distinguished triangles of
Lemmas \ref{lemma-exact-sequence-lower-shriek} and
\ref{lemma-exact-sequence-j-star} show that
$F = j'_!(F|_{V'})$ and $F = Rj'_*(F|_{V'})$.
It follows that $j_!F = j_!j'_!(F|_{V'}) = j_{V'!}F = F'$
because $j_{V'} : V' \to X$ is an isomorphism and the inverse
of $\sigma$. Similarly, $Rj_*F = Rj_*Rj'_*F = Rj_{V', *}F = F'$.
This proves (2)(c). To prove (2)(a) and (2)(b) it suffices
to show that $F = F'|_V$. This is clear because both $F$ and $F'|_V$
restrict to zero on $U'$ and $U' \cap V'$ and the same object
on $V'$.
\end{proof}

\noindent
We can glue complexes!

\begin{lemma}
\label{lemma-glue}
Let $S$ be a scheme. Let $(U \subset X, V \to X)$ be an elementary
distinguished square of algebraic spaces over $S$. Suppose given
\begin{enumerate}
\item an object $E$ of $D(\mathcal{O}_X)$,
\item a morphism $a : A \to E|_U$ of $D(\mathcal{O}_U)$,
\item a morphism $b : B \to E|_V$ of $D(\mathcal{O}_V)$,
\item an isomorphism $c : A|_{U \times_X V} \to B|_{U \times_X V}$
\end{enumerate}
such that
$$
a|_{U \times_X V}  = b|_{U \times_X V} \circ c.
$$
Then there exists a morphism $F \to E$ in $D(\mathcal{O}_X)$
whose restriction to $U$ is isomorphic to $a$
and whose restriction to $V$ is isomorphic to $b$.
\end{lemma}

\begin{proof}
Denote $j_U$, $j_V$, $j_{U \times_X V}$ the corresponding morphisms towards
$X$. Choose a distinguished triangle
$$
F \to Rj_{U, *}A \oplus Rj_{V, *}B \to
Rj_{U \times_X V, *}(B|_{U \times_X V}) \to F[1]
$$
Here the map $Rj_{V, *}B \to Rj_{U \times_X V, *}(B|_{U \times_X V})$
is the obvious one. The map
$Rj_{U, *}A \to Rj_{U \times_X V, *}(B|_{U \times_X V})$
is the composition of
$Rj_{U, *}A \to Rj_{U \times_X V, *}(A|_{U \times_X V})$
with $Rj_{U \times_X V, *}c$. Restricting to $U$ we obtain
$$
F|_U \to A \oplus (Rj_{V, *}B)|_U \to
(Rj_{U \times_X V, *}(B|_{U \times_X V}))|_U \to F|_U[1]
$$
Denote $j : U \times_X V \to U$. Compatibility of restriction and
total direct image (Lemma \ref{lemma-restrict-direct-image-open})
shows that both $(Rj_{V, *}B)|_U$ and
$(Rj_{U \times_X V, *}(B|_{U \times_X V}))|_U$
are canonically isomorphic to $Rj_*(B|_{U \times_X V})$.
Hence the second arrow of the last displayed equation has
a section, and we conclude that the morphism $F|_U \to A$ is
an isomorphism.

\medskip\noindent
To see that the morphism $F|_V \to B$ is an isomorphism we will use a trick.
Namely, choose a distinguished triangle
$$
F|_V \to B \to B' \to F[1]|_V
$$
in $D(\mathcal{O}_V)$. Since $F|_U \to A$ is an isomorphism, and since
we have the isomorphism $c : A|_{U \times_X V} \to B|_{U \times_X V}$
the restriction of $F|_V \to B$ is an isomorphism over $U \times_X V$.
Thus $B'$ is supported on $j_V^{-1}(T)$ where $T = |X| \setminus |U|$.
On the other hand, there is a morphism of distinguished triangles
$$
\xymatrix{
F \ar[r] \ar[d] &
Rj_{U, *}F|_U \oplus Rj_{V, *}F|_V \ar[r] \ar[d] &
Rj_{U \times_X V, *}F|_{U \times_X V} \ar[r] \ar[d] &
F[1] \ar[d] \\
F \ar[r] &
Rj_{U, *}A \oplus Rj_{V, *}B \ar[r] &
Rj_{U \times_X V, *}(B|_{U \times_X V}) \ar[r] &
F[1]
}
$$
The all of the vertical maps in this diagram are isomorphisms, except
for the map $Rj_{V, *}F|_V \to Rj_{V, *}B$, hence that is an isomorphism too
(Derived Categories, Lemma \ref{derived-lemma-third-isomorphism-triangle}).
This implies that $Rj_{V, *}B' = 0$. Hence $B' = 0$ by
Lemma \ref{lemma-pushforward-with-support-in-open}.

\medskip\noindent
The existence of the morphism $F \to E$ follows
from the Mayer-Vietoris sequence for $\Hom$, see
Lemma \ref{lemma-mayer-vietoris-hom}.
\end{proof}







\section{The coherator}
\label{section-coherator}

\noindent
Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.
The {\it coherator} is a functor
$$
Q_X :
\textit{Mod}(\mathcal{O}_X)
\longrightarrow
\textit{QCoh}(\mathcal{O}_X)
$$
which is right adjoint to the inclusion functor
$\textit{QCoh}(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}_X)$.
It exists for any algebraic space $X$ and moreover the adjunction mapping
$Q_X(\mathcal{F}) \to \mathcal{F}$ is an isomorphism for every
quasi-coherent module $\mathcal{F}$, see
Properties of Spaces, Proposition
\ref{spaces-properties-proposition-coherator}.
Since $Q_X$ is left exact (as a right adjoint) we can consider its
right derived extension
$$
RQ_X :
D(\mathcal{O}_X)
\longrightarrow
D(\textit{QCoh}(\mathcal{O}_X)).
$$
As this functor is constructed by applying $Q_X$ to a K-injective replacement
we see that $RQ_X$ is a right adjoint to the canonical functor
$D(\textit{QCoh}(\mathcal{O}_X)) \to D(\mathcal{O}_X)$.

\begin{lemma}
\label{lemma-affine-pushforward}
Let $S$ be a scheme. Let $f : X \to Y$ be an affine morphism of
algebraic spaces over $S$. Then $f_*$ defines a derived functor
$f_* : D(\textit{QCoh}(\mathcal{O}_X)) \to D(\textit{QCoh}(\mathcal{O}_Y))$.
This functor has the property that
$$
\xymatrix{
D(\textit{QCoh}(\mathcal{O}_X)) \ar[d]_{f_*} \ar[r] &
D_{\textit{QCoh}}(\mathcal{O}_X) \ar[d]^{Rf_*} \\
D(\textit{QCoh}(\mathcal{O}_Y)) \ar[r] &
D_{\textit{QCoh}}(\mathcal{O}_Y)
}
$$
commutes.
\end{lemma}

\begin{proof}
The functor
$f_* : \textit{QCoh}(\mathcal{O}_X) \to \textit{QCoh}(\mathcal{O}_Y)$
is exact, see
Cohomology of Spaces, Lemma
\ref{spaces-cohomology-lemma-affine-vanishing-higher-direct-images}.
Hence $f_*$ defines a derived functor
$f_* : D(\textit{QCoh}(\mathcal{O}_X)) \to D(\textit{QCoh}(\mathcal{O}_Y))$
by simply applying $f_*$ to any representative complex, see
Derived Categories, Lemma \ref{derived-lemma-right-derived-exact-functor}.
For any complex of $\mathcal{O}_X$-modules
$\mathcal{F}^\bullet$ there is a canonical map
$f_*\mathcal{F}^\bullet \to Rf_*\mathcal{F}^\bullet$.
To finish the proof we show this is a quasi-isomorphism when
$\mathcal{F}^\bullet$ is a complex with each $\mathcal{F}^n$
quasi-coherent. The statement is \'etale local on $Y$ hence we
may assume $Y$ affine. As an affine morphism is representable
we reduce to the case of schemes by the compatibility of
Remark \ref{remark-match-total-direct-images}. The case of schemes is
Derived Categories of Schemes, Lemma \ref{perfect-lemma-affine-pushforward}.
\end{proof}

\begin{lemma}
\label{lemma-affine-flat-coherator}
Let $S$ be a scheme. Let $f : X \to Y$ be an affine flat morphism of
algebraic spaces over $S$.
Then $RQ_Y \circ Rf_* = f_* \circ RQ_X$ where $f_*$ is as
in Lemma \ref{lemma-affine-pushforward}.
\end{lemma}

\begin{proof}
Since $f$ is flat, then functor $f^*$ is exact hence defines
$f^* : D(\mathcal{O}_Y) \to D(\mathcal{O}_X)$ and also
$f^* : D(\textit{QCoh}(\mathcal{O}_Y)) \to D(\textit{QCoh}(\mathcal{O}_X))$.
The functor $f^* = Lf^* : D(\mathcal{O}_Y) \to D(\mathcal{O}_X)$
is left adjoint to
$Rf_* : D(\mathcal{O}_X) \to D(\mathcal{O}_Y)$,
see Cohomology, Lemma \ref{cohomology-lemma-adjoint}.
On the other hand, the functor
$f^* : D(\textit{QCoh}(\mathcal{O}_Y)) \to D(\textit{QCoh}(\mathcal{O}_X))$
is left adjoint to
$f_* : D(\textit{QCoh}(\mathcal{O}_X)) \to D(\textit{QCoh}(\mathcal{O}_Y))$
just because these are constructed from an adjoint pair of exact functors.
Now let $A$ be an object of $D(\textit{QCoh}(\mathcal{O}_Y))$ and
$E$ an object of $D(\mathcal{O}_X)$. Then
\begin{align*}
\Hom_{D(\textit{QCoh}(\mathcal{O}_Y))}(A, RQ_Y(Rf_*E))
& =
\Hom_{D(\mathcal{O}_Y)}(A, Rf_*E) \\
& =
\Hom_{D(\mathcal{O}_X)}(f^*A, E) \\
& =
\Hom_{D(\textit{QCoh}(\mathcal{O}_X))}(f^*A, RQ_X(E)) \\
& =
\Hom_{D(\textit{QCoh}(\mathcal{O}_Y))}(A, f_*RQ_X(E))
\end{align*}
This implies what we want.
\end{proof}

\begin{lemma}
\label{lemma-affine-coherator}
Let $S$ be a scheme. Let $X$ be an affine algebraic space over $S$.
Set $A = \Gamma(X, \mathcal{O}_X)$. Then
\begin{enumerate}
\item $Q_X : \textit{Mod}(\mathcal{O}_X) \to \textit{QCoh}(\mathcal{O}_X)$
is the functor
which sends $\mathcal{F}$ to the quasi-coherent $\mathcal{O}_X$-module
associated to the $A$-module $\Gamma(X, \mathcal{F})$,
\item $RQ_X : D(\mathcal{O}_X) \to D(\textit{QCoh}(\mathcal{O}_X))$
is the functor which sends $E$ to the complex of quasi-coherent
$\mathcal{O}_X$-modules associated to the object $R\Gamma(X, E)$ of $D(A)$,
\item restricted to $D_{\textit{QCoh}}(\mathcal{O}_X)$ the functor
$RQ_X$ defines a quasi-inverse to (\ref{equation-compare}).
\end{enumerate}
\end{lemma}

\begin{proof}
Let $X_0 = \Spec(A)$ be the affine scheme representing $X$.
Recall that there is a morphism of ringed sites
$\epsilon : X_\etale \to X_{0, Zar}$
which induces equivalences
$$
\xymatrix{
\textit{QCoh}(\mathcal{O}_X) \ar@<1ex>[r]^{\epsilon_*} &
\textit{QCoh}(\mathcal{O}_{X_0}) \ar@<1ex>[l]^{\epsilon^*},
}
$$
see Lemma
\ref{lemma-derived-quasi-coherent-small-etale-site}.
Hence we see that $Q_X = \epsilon^* \circ Q_{X_0} \circ \epsilon_*$
by uniqueness of adjoint functors. Hence (1) follows from
the description of $Q_{X_0}$ in
Derived Categories of Schemes, Lemma \ref{perfect-lemma-affine-coherator}
and the fact that
$\Gamma(X_0, \epsilon_*\mathcal{F}) = \Gamma(X, \mathcal{F})$.
Part (2) follows from (1) and the fact that the functor
from $A$-modules to quasi-coherent $\mathcal{O}_X$-modules is exact.
The third assertion now follows from the result for schemes
(Derived Categories of Schemes, Lemma \ref{perfect-lemma-affine-coherator})
and Lemma
\ref{lemma-derived-quasi-coherent-small-etale-site}.
\end{proof}

\begin{proposition}
\label{proposition-quasi-compact-affine-diagonal}
Let $S$ be a scheme. Let $X$ be a quasi-compact algebraic space with affine
diagonal. Then the functor (\ref{equation-compare})
$$
D(\textit{QCoh}(\mathcal{O}_X))
\longrightarrow
D_{\textit{QCoh}}(\mathcal{O}_X)
$$
is an equivalence with quasi-inverse given by $RQ_X$.
\end{proposition}

\begin{proof}
Note that it is enough to prove that (\ref{equation-compare}) is
an equivalence, since then it will be the case that $RQ_X$ defines
the quasi-inverse for formal reasons (by uniqueness of adjoints and
the fact that the quasi-inverse is an adjoint).
We will use the induction principle to prove this. Let
$\mathcal{B} \subset \Ob(X_{spaces, \etale})$ be the set of
objects which are quasi-compact and have affine diagonal.
For $U \in \mathcal{B}$ let $P(U) =$ ``the functor
$D(\textit{QCoh}(\mathcal{O}_U)) \to D_{\textit{QCoh}}(\mathcal{O}_U)$
is an equivalence''.
By Remark \ref{remark-how-to} conditions (1), (2), and (3)(a) of
Lemma \ref{lemma-induction-principle-separated} hold and we are
left with proving (3)(b) and (4). Condition (3)(b) holds by
Lemma \ref{lemma-affine-coherator}.

\medskip\noindent
Let $(U \subset W, V \to W)$ be an elementary distinguished square
with $V$ affine. Assume that $P$ holds for $U$, $V$, and $U \times_W V$.
We have to show that $P$ holds for $W$. We may replace $X$ by $W$, i.e.,
we may assume $W = X$ (we do this just to simplify the notation).

\medskip\noindent
We first prove that the functor
$D(\textit{QCoh}(\mathcal{O}_X)) \to D(\mathcal{O}_X)$
is fully faithful. Suppose that $A, B$ are objects of
$D(\textit{QCoh}(\mathcal{O}_X))$, i.e., $A$ and $B$ are given to us
as complexes of quasi-coherent $\mathcal{O}_X$-modules.
We want to show that
$$
\Hom_{D(\textit{QCoh}(\mathcal{O}_X))}(A, B)
\longrightarrow
\Hom_{D(\mathcal{O}_X)}(A, B)
$$
is bijective. Let $T = |X| \setminus |U|$. If $B$ is supported on $T$, then
the maps $B \to Rj_{V, *}(B|_V)$ is a quasi-isomorphism
(Lemma \ref{lemma-pushforward-with-support-in-open}).
The morphism $V \to X$ is affine as $V$ is affine and $X$ has affine diagonal
(Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-affine-permanence}).
Thus we have an object $j_{V, *}(B|_V)$ in $\textit{QCoh}(\mathcal{O}_X)$
and a quasi-isomorphism $j_{V, *}(B|_V) \to Rj_{V, *}B$ in $D(\mathcal{O}_X)$
(Lemma \ref{lemma-affine-pushforward}). Moreover, $j_{V, *}$ and $j_V^*$ are
adjoint functors on the derived categories of quasi-coherent modules
(see proof of Lemma \ref{lemma-affine-flat-coherator}). Hence
\begin{align*}
\Mor_{D(\textit{QCoh}(\mathcal{O}_X))}(A, B)
& =
\Mor_{D(\textit{QCoh}(\mathcal{O}_V))}(A|_V, B|_V) \\
& =
\Mor_{D(\mathcal{O}_V)}(A|_V, B|_V) \\
& =
\Mor_{D(\mathcal{O}_X)}(A, Rj_{V, *}B|_V) \\
& =
\Mor_{D(\mathcal{O}_X)}(A, B)
\end{align*}
as desired. In general, choose any quasi-isomorphism $s : B|_U \to C$ of
complexes of quasi-coherent $\mathcal{O}_U$-modules. We obtain a map
$B \to j_{U, *}B|_U \to j_{U, *}C$ and fit it into a
distinguished triangle
$$
B \to j_{U, *}C \to B' \to B[1]
$$
in $D(\textit{QCoh}(\mathcal{O}_X))$. This makes sense because $j_U : U \to X$
is quasi-compact and quasi-separated, hence $j_{U, *}$ transforms
quasi-coherent modules into quasi-coherent modules
(Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-pushforward}).
Since $s$ is a quasi-isomorphism
we see that $B'$ is supported on $T$. Hence in the diagram
$$
\xymatrix{
\Hom_{D(\textit{QCoh}(\mathcal{O}_X))}(A, B'[-1]) \ar[r] \ar[d] &
\Hom_{D(\mathcal{O}_X)}(A, B'[-1]) \ar[d] \\
\Hom_{D(\textit{QCoh}(\mathcal{O}_X))}(A, B) \ar[r] \ar[d] &
\Hom_{D(\mathcal{O}_X)}(A, B) \ar[d] \\
\Hom_{D(\textit{QCoh}(\mathcal{O}_X))}(A, j_{U, *}C) \ar[r] \ar[d] &
\Hom_{D(\mathcal{O}_X)}(A, j_{U, *}C) \ar[d] \\
\Hom_{D(\textit{QCoh}(\mathcal{O}_X))}(A, B') \ar[r] &
\Hom_{D(\mathcal{O}_X)}(A, B')
}
$$
we have exact columns and the top and bottom horizontal arrows are
bijective. Let $\alpha : A \to B$ be a morphism between $A$ and $B$
in $D(\mathcal{O}_X)$. The restriction
$\alpha|_U$ comes from a morphism in $D(\textit{QCoh}(\mathcal{O}_U))$
by assumption. Hence there exists a choice of $s : B|_U \to C$ as above
such that $\alpha|_U$ is represented by an actual map of complexes
$A|_U \to C$. This corresponds to a map of complexes $A \to j_{U, *}C$.
In other words, the image of $\alpha$ in
$\Hom_{D(\mathcal{O}_X)}(A, j_{U, *}C)$ comes from
an element of $\Hom_{D(\textit{QCoh}(\mathcal{O}_X))}(A, j_{U, *}C)$.
A diagram chase then shows that $\alpha$ comes from a morphism
$A \to B$ in $D(\textit{QCoh}(\mathcal{O}_X))$. Finally, suppose
that $a : A \to B$ is a morphism of $D(\textit{QCoh}(\mathcal{O}_X))$
which becomes zero in $D(\mathcal{O}_X)$. Then by the same reasoning
as above we can pick an $s$ as above such that $\alpha$ maps to
zero in $\Hom_{D(\textit{QCoh}(\mathcal{O}_X))}(A, j_{U, *}C)$.
Thus we may assume that $a$ is the image of an element
of $b \in \Hom_{D(\textit{QCoh}(\mathcal{O}_X))}(A, B'[-1])$.
The image of $b$ in $\Hom_{D(\mathcal{O}_X)}(A, B'[-1])$
comes from a $c' \in \Hom_{D(\mathcal{O}_X)}(A, j_{U, *}C[-1])$
(as $a$ maps to zero in the group on the right). Since we've
seen above the horizontal arrows are surjective, we see
that $c'$ comes from a $c$ in
$\Hom_{D(\textit{QCoh}(\mathcal{O}_X))}(A, j_{U, *}C)$
which implies $a = 0$ as desired.

\medskip\noindent
To finish the proof we show that the functor
$D(\textit{QCoh}(\mathcal{O}_X)) \to D_{\textit{QCoh}}(\mathcal{O}_X)$
is essentially surjective. Let $E$ be an object of
$D_{\textit{QCoh}}(\mathcal{O}_X)$. Choose a quasi-isomorphism
$E|_U \to C$ with $C$ a complex of quasi-coherent $\mathcal{O}_U$-modules.
Choose a distinguised triangle
$$
E \to j_{U, *}C \to E' \to E[1]
$$
Observe that $E'$ is supported on $T$. By
Lemma \ref{lemma-pushforward-with-support-in-open}
we see that $E' = Rj_{V, *}(E'|_V)$. Choose a quasi-isomorphism
$E'|_V \to C'$ with $C'$ in $D(\textit{QCoh}(\mathcal{O}_V))$.
Then $C'' = j_{V, *}C'$ is an object of $\textit{QCoh}(\mathcal{O}_X)$
representing $E'$ (as $V \to X$ is affine and by
Lemma \ref{lemma-affine-pushforward}).
Hence we found an isomorphism $E' \to C''$
in $D(\mathcal{O}_X)$ with $C''$ a complex of quasi-coherent
$\mathcal{O}_X$-modules. Because we've seen above the functor is
fully faithful we can pick a morphism $j_{U, *}C \to C''$ in
$D(\textit{QCoh}(\mathcal{O}_X))$ representing the composition
$j_{U, *}C \to E' \to C''$.
Choose a distinguished triangle
$$
A \to j_{U, *}C \to C'' \to A[1]
$$
in $D(\textit{QCoh}(\mathcal{O}_X))$.
Using the axioms of triangulated categories we obtain a quasi-isomorphism
$E \to A$ in $D(\mathcal{O}_X)$ which is what we had to show.
\end{proof}






\section{Pseudo-coherent and perfect complexes}
\label{section-spell-out}

\noindent
In this section we study the general notions defined in
Cohomology on Sites, Sections
\ref{sites-cohomology-section-strictly-perfect},
\ref{sites-cohomology-section-pseudo-coherent},
\ref{sites-cohomology-section-tor}, and
\ref{sites-cohomology-section-perfect}
for the \'etale site of an algebraic space. In particular we
match this with what happens for schemes.

\medskip\noindent
First we compare the notion of a pseudo-coherent complex on a
scheme and on its associated small \'etale site.

\begin{lemma}
\label{lemma-descend-finite-type}
Let $X$ be a scheme. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module.
The following are equivalent
\begin{enumerate}
\item $\mathcal{F}$ is of finite type as an $\mathcal{O}_X$-module, and
\item $\epsilon^*\mathcal{F}$ is of finite type as an
$\mathcal{O}_\etale$-module on the small \'etale site of $X$.
\end{enumerate}
Here $\epsilon$ is as in (\ref{equation-epsilon}).
\end{lemma}

\begin{proof}
The implication (1) $\Rightarrow$ (2) is a general fact, see
Modules on Sites, Lemma \ref{sites-modules-lemma-local-pullback}.
Assume (2). By assumption there exists an \'etale covering
$\{f_i : X_i \to X\}$ such that
$\epsilon^*\mathcal{F}|_{(X_i)_\etale}$ is generated by
finitely many sections. Let $x \in X$. We will show that $\mathcal{F}$
is generated by finitely many sections in a neighbourhood of $x$.
Say $x$ is in the image of $X_i \to X$ and denote $X' = X_i$. Let
$s_1, \ldots, s_n \in
\Gamma(X', \epsilon^*\mathcal{F}|_{X'_\etale})$
be generating sections. As
$\epsilon^*\mathcal{F} =
\epsilon^{-1}\mathcal{F} \otimes_{\epsilon^{-1}\mathcal{O}_X}
\mathcal{O}_\etale$
we can find an \'etale morphism $X'' \to X'$ such that $x$ is
in the image of $X' \to X$ and such that
$s_i|_{X''} = \sum s_{ij} \otimes a_{ij}$ for some sections
$s_{ij} \in \epsilon^{-1}\mathcal{F}(X'')$ and
$a_{ij} \in \mathcal{O}_\etale(X'')$. Denote $U \subset X$ the image
of $X'' \to X$. This is an open subscheme as $f'' : X'' \to X$ is \'etale
(Morphisms, Lemma \ref{morphisms-lemma-etale-open}). After possibly
shrinking $X''$ more we may assume $s_{ij}$ come from elements
$t_{ij} \in \mathcal{F}(U)$ as follows from the construction of
the inverse image functor $\epsilon^{-1}$. Now we claim that
$t_{ij}$ generate $\mathcal{F}|_U$ which finishes the proof
of the lemma. Namely, the corresponding map
$\mathcal{O}_U^{\oplus N} \to \mathcal{F}|_U$ has the property
that its pullback by $f''$ to $X''$ is surjective. Since $f'' : X'' \to U$
is a surjective flat morphism of schemes, this implies that
$\mathcal{O}_U^{\oplus N} \to \mathcal{F}|_U$ is surjective by
looking at stalks and using that
$\mathcal{O}_{U, f''(z)} \to \mathcal{O}_{X'', z}$
is faithfully flat for all $z \in X''$.
\end{proof}

\noindent
In the situation above the morphism of sites $\epsilon$ is flat
hence defines a pullback on complexes of modules.

\begin{lemma}
\label{lemma-descend-pseudo-coherent}
Let $X$ be a scheme. Let $E$ be an object of $D(\mathcal{O}_X)$.
The following are equivalent
\begin{enumerate}
\item $E$ is $m$-pseudo-coherent, and
\item $\epsilon^*E$ is $m$-pseudo-coherent on the small \'etale site of $X$.
\end{enumerate}
Here $\epsilon$ is as in (\ref{equation-epsilon}).
\end{lemma}

\begin{proof}
The implication (1) $\Rightarrow$ (2) is a general fact, see
Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-pseudo-coherent-pullback}.
Assume $\epsilon^*E$ is $m$-pseudo-coherent.
We will use without further mention that $\epsilon^*$ is
an exact functor and that therefore
$$
\epsilon^*H^i(E) = H^i(\epsilon^*E).
$$
To show that $E$ is $m$-pseudo-coherent we may work locally on $X$,
hence we may assume that $X$ is quasi-compact (for example affine).
Since $X$ is quasi-compact every \'etale covering $\{U_i \to X\}$
has a finite refinement. Thus we see that $\epsilon^*E$ is
an object of $D^{-}(\mathcal{O}_\etale)$, see
comments following
Cohomology on Sites, Definition
\ref{sites-cohomology-definition-pseudo-coherent}.
By Lemma \ref{lemma-epsilon-flat} it follows that $E$ is an object of
$D^-(\mathcal{O}_X)$.

\medskip\noindent
Let $n \in \mathbf{Z}$ be the largest integer such that
$H^n(E)$ is nonzero; then $n$ is also the largest integer
such that $H^n(\epsilon^*E)$ is nonzero.
We will prove the lemma by induction on $n - m$.
If $n < m$, then the lemma is clearly true.
If $n \geq m$, then $H^n(\epsilon^*E)$ is a finite
$\mathcal{O}_\etale$-module, see
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-finite-cohomology}.
Hence $H^n(E)$ is a finite $\mathcal{O}_X$-module, see
Lemma \ref{lemma-descend-finite-type}.
After replacing $X$ by the members of an open covering, we may
assume there exists a surjection $\mathcal{O}_X^{\oplus t} \to H^n(E)$.
We may locally on $X$ lift this to a map of complexes
$\alpha : \mathcal{O}_X^{\oplus t}[-n] \to E$ (details omitted).
Choose a distinguished triangle
$$
\mathcal{O}_X^{\oplus t}[-n] \to E \to C \to \mathcal{O}_X^{\oplus t}[-n + 1]
$$
Then $C$ has vanishing cohomology in degrees $\geq n$. On the other hand, the
complex $\epsilon^*C$ is $m$-pseudo-coherent, see
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cone-pseudo-coherent}.
Hence by induction we see that $C$ is $m$-pseudo-coherent. Applying
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cone-pseudo-coherent}
once more we conclude.
\end{proof}

\begin{lemma}
\label{lemma-descend-tor-amplitude}
Let $X$ be a scheme. Let $E$ be an object of $D(\mathcal{O}_X)$.
Then
\begin{enumerate}
\item $E$ has tor amplitude in $[a, b]$ if and only if
$\epsilon^*E$ has tor amplitude in $[a, b]$.
\item $E$ has finite tor dimension if and only if $\epsilon^*E$ has finite
tor dimension.
\end{enumerate}
Here $\epsilon$ is as in (\ref{equation-epsilon}).
\end{lemma}

\begin{proof}
The easy implication follows from the general result contained in
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-tor-amplitude-pullback}
(and the fact that the small \'etale site of $X$ has enough points, see
\'Etale Cohomology, Remarks \ref{etale-cohomology-remarks-enough-points}).
For the converse, assume that $\epsilon^*E$ has tor amplitude in $[a, b]$
Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. As $\epsilon$ is a flat
morphism of ringed sites (Lemma \ref{lemma-epsilon-flat})
we have
$$
\epsilon^*(E \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{F})
=
\epsilon^*E
\otimes^\mathbf{L}_{\mathcal{O}_\etale}
\epsilon^*\mathcal{F}
$$
Thus the (assumed) vanishing of cohomology sheaves on the right hand side
implies the desired vanishing of the cohomology sheaves of
$E \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{F}$ via
Lemma \ref{lemma-epsilon-flat}.
\end{proof}

\begin{lemma}
\label{lemma-descend-perfect}
Let $X$ be a scheme. Let $E$ be an object of $D(\mathcal{O}_X)$.
Then $E$ is a perfect object of $D(\mathcal{O}_X)$ if and only if
$\epsilon^*E$ is a perfect object of $D(\mathcal{O}_\etale)$.
Here $\epsilon$ is as in (\ref{equation-epsilon}).
\end{lemma}

\begin{proof}
The easy implication follows from the general result contained in
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-perfect-pullback}
(and the fact that the small \'etale site of $X$ has enough points, see
\'Etale Cohomology, Remarks \ref{etale-cohomology-remarks-enough-points}).
For the converse, we can use the equivalence of
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-perfect}
and the corresponding results for pseudo-coherent and complexes of
finite tor dimension, namely
Lemmas \ref{lemma-descend-pseudo-coherent} and
\ref{lemma-descend-tor-amplitude}.
Some details omitted.
\end{proof}

\begin{lemma}
\label{lemma-pseudo-coherent}
Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.
If $E$ is an $m$-pseudo-coherent object of $D(\mathcal{O}_X)$,
then $H^i(E)$ is a quasi-coherent $\mathcal{O}_X$-module for $i > m$.
If $E$ is pseudo-coherent, then $E$ is an object of
$D_{\textit{QCoh}}(\mathcal{O}_X)$.
\end{lemma}

\begin{proof}
Locally $H^i(E)$ is isomorphic to $H^i(\mathcal{E}^\bullet)$
with $\mathcal{E}^\bullet$ strictly perfect. The sheaves
$\mathcal{E}^i$ are direct summands of finite free modules,
hence quasi-coherent. The lemma follows.
\end{proof}

\begin{lemma}
\label{lemma-identify-pseudo-coherent-noetherian}
Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$.
Let $E$ be an object of $D_{\textit{QCoh}}(\mathcal{O}_X)$. For
$m \in \mathbf{Z}$ the following are equivalent
\begin{enumerate}
\item $H^i(E)$ is coherent for $i \geq m$ and zero for $i \gg 0$, and
\item $E$ is $m$-pseudo-coherent.
\end{enumerate}
In particular, $E$ is pseudo-coherent if and only if $E$ is an object
of $D^-_{\textit{Coh}}(\mathcal{O}_X)$.
\end{lemma}

\begin{proof}
As $X$ is quasi-compact we can find an affine scheme $U$ and a surjective
\'etale morphism $U \to X$ (Properties of Spaces, Lemma
\ref{spaces-properties-lemma-quasi-compact-affine-cover}).
Observe that $U$ is Noetherian.
Note that $E$ is $m$-pseudo-coherent if and only if $E|_U$ is
$m$-pseudo-coherent (follows from the definition or from
Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-pseudo-coherent-independent-representative}).
Similarly, $H^i(E)$ is coherent if and only if $H^i(E)|_U = H^i(E|_U)$
is coherent (see Cohomology of Spaces, Lemma
\ref{spaces-cohomology-lemma-coherent-Noetherian}).
Thus we may assume that $X$ is representable.

\medskip\noindent
If $X$ is representable by a scheme $X_0$ then
(Lemma \ref{lemma-derived-quasi-coherent-small-etale-site})
we can write $E = \epsilon^*E_0$ where $E_0$ is an object of
$D_{\textit{QCoh}}(\mathcal{O}_{X_0})$ and
$\epsilon : X_\etale \to (X_0)_{Zar}$ is as in
(\ref{equation-epsilon}).
In this case $E$ is $m$-pseudo-coherent
if and only if $E_0$ is by Lemma \ref{lemma-descend-pseudo-coherent}.
Similarly, $H^i(E_0)$ is of finite type (i.e., coherent) if and only if
$H^i(E)$ is by Lemma \ref{lemma-descend-finite-type}.
Finally, $H^i(E_0) = 0$ if and only if $H^i(E) = 0$ by
Lemma \ref{lemma-epsilon-flat}.
Thus we reduce to the case of schemes which is
Derived Categories of Schemes, Lemma
\ref{perfect-lemma-identify-pseudo-coherent-noetherian}.
\end{proof}

\begin{lemma}
\label{lemma-tor-qc-qs}
Let $S$ be a scheme. Let $X$ be a quasi-separated algebraic space over $S$.
Let $E$ be an object of $D_{\textit{QCoh}}(\mathcal{O}_X)$. Let $a \leq b$.
The following are equivalent
\begin{enumerate}
\item $E$ has tor amplitude in $[a, b]$, and
\item for all $\mathcal{F}$ in $\textit{QCoh}(\mathcal{O}_X)$
we have $H^i(E \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{F}) = 0$
for $i \not \in [a, b]$.
\end{enumerate}
\end{lemma}

\begin{proof}
It is clear that (1) implies (2). Assume (2). Let $j : U \to X$ be
an \'etale morphism with $U$ affine. As $X$ is quasi-separated $j : U \to X$
is quasi-compact and separated, hence $j_*$ transforms quasi-coherent
modules into quasi-coherent modules (Morphisms of Spaces, Lemma
\ref{spaces-morphisms-lemma-pushforward}).
Thus the functor
$\textit{QCoh}(\mathcal{O}_X) \to \textit{QCoh}(\mathcal{O}_U)$
is essentially surjective. It follows that condition (2)
implies the vanishing of
$H^i(E|_U \otimes_{\mathcal{O}_U}^\mathbf{L} \mathcal{G})$
for $i \not \in [a, b]$ for all quasi-coherent $\mathcal{O}_U$-modules
$\mathcal{G}$. Since it suffices to prove that $E|_U$ has tor amplitude
in $[a, b]$ we reduce to the case where $X$ is representable.

\medskip\noindent
If $X$ is representable by a scheme $X_0$ then
(Lemma \ref{lemma-derived-quasi-coherent-small-etale-site})
we can write $E = \epsilon^*E_0$ where $E_0$ is an object of
$D_{\textit{QCoh}}(\mathcal{O}_{X_0})$ and
$\epsilon : X_\etale \to (X_0)_{Zar}$ is as in
(\ref{equation-epsilon}). For every quasi-coherent module
$\mathcal{F}_0$ on $X_0$ the module $\epsilon^*\mathcal{F}_0$
is quasi-coherent on $X$ and
$$
H^i(E \otimes_{\mathcal{O}_X}^\mathbf{L} \epsilon^*\mathcal{F}_0)
=
\epsilon^*H^i(E_0 \otimes_{\mathcal{O}_{X_0}}^\mathbf{L} \mathcal{F}_0)
$$
as $\epsilon$ is flat (Lemma \ref{lemma-epsilon-flat}).
Moreover, the vanishing of these sheaves for $i \not \in [a, b]$
implies the same thing for
$H^i(E_0 \otimes_{\mathcal{O}_{X_0}}^\mathbf{L} \mathcal{F}_0)$
by the same lemma. Thus we've reduced the problem to the case
of schemes which is treated in
Derived Categories of Schemes, Lemma \ref{perfect-lemma-tor-qc-qs}.
\end{proof}









\section{Approximation by perfect complexes}
\label{section-approximation}

\noindent
In this section we continue the discussion started in
Derived Categories of Schemes, Section \ref{perfect-section-approximation}.

\begin{definition}
\label{definition-approximation-holds}
Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.
Consider triples $(T, E, m)$ where
\begin{enumerate}
\item $T \subset |X|$ is a closed subset,
\item $E$ is an object of $D_{\textit{QCoh}}(\mathcal{O}_X)$, and
\item $m \in \mathbf{Z}$.
\end{enumerate}
We say {\it approximation holds for the triple} $(T, E, m)$ if
there exists a perfect object $P$ of $D(\mathcal{O}_X)$ supported on $T$
and a map $\alpha : P \to E$ which induces isomorphisms $H^i(P) \to H^i(E)$
for $i > m$ and a surjection $H^m(P) \to H^m(E)$.
\end{definition}

\noindent
Approximation cannot hold for every triple. Please read the remarks following
Derived Categories of Schemes, Definition
\ref{perfect-definition-approximation-holds} to see why.

\begin{definition}
\label{definition-approximation}
Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.
We say {\it approximation by perfect complexes holds}
on $X$ if for any closed subset $T \subset |X|$ such that
the morphism $X \setminus T \to X$ is quasi-compact
there exists an integer $r$ such that for every triple $(T, E, m)$ as in
Definition \ref{definition-approximation-holds} with
\begin{enumerate}
\item $E$ is $(m - r)$-pseudo-coherent, and
\item $H^i(E)$ is supported on $T$ for $i \geq m - r$
\end{enumerate}
approximation holds.
\end{definition}

\begin{lemma}
\label{lemma-pushforward-perfect}
Let $S$ be a scheme. Let $(U \subset X, j : V \to X)$ be an
elementary distinguished square of algebraic space over $S$.
Let $E$ be a perfect object of $D(\mathcal{O}_V)$ supported on
$j^{-1}(T)$ where $T = |X| \setminus |U|$. Then $Rj_*E$ is a
perfect object of $D(\mathcal{O}_X)$.
\end{lemma}

\begin{proof}
Being perfect is local on $X_\etale$. Thus it suffices to
check that $Rj_*E$ is perfect when restricted to $U$ and $V$.
We have $Rj_*E|_V = E$ by Lemma \ref{lemma-pushforward-with-support-in-open}
which is perfect. We have $Rj_*E|_U = 0$ because
$E|_{V \setminus j^{-1}(T)} = 0$ (use
Lemma \ref{lemma-restrict-direct-image-open}).
\end{proof}

\begin{lemma}
\label{lemma-open}
Let $S$ be a scheme. Let $(U \subset X, j : V \to X)$ be an elementary
distinguished square of algebraic spaces over $S$. Let $T$ be a closed
subset of $|X| \setminus |U|$ and let $(T, E, m)$ be a triple as in
Definition \ref{definition-approximation-holds}. If
\begin{enumerate}
\item approximation holds for $(j^{-1}T, E|_V, m)$, and
\item the sheaves $H^i(E)$ for $i \geq m$ are supported on $T$,
\end{enumerate}
then approximation holds for $(T, E, m)$.
\end{lemma}

\begin{proof}
Let $P \to E|_V$ be an approximation of the triple $(j^{-1}T, E|_V, m)$
over $V$. Then $Rj_*P$ is a perfect object of $D(\mathcal{O}_X)$ by
Lemma \ref{lemma-pushforward-perfect}. On the other hand,
$Rj_*P = j_!P$ by Lemma \ref{lemma-pushforward-with-support-in-open}.
We see that $j_!P$ is supported on $T$ for example by
(\ref{equation-stalk-j-shriek}).
Hence we obtain an approximation $Rj_*P = j_!P \to j_!(E|_V) \to E$.
\end{proof}

\begin{lemma}
\label{lemma-approximation-affine}
Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ which is
representable by an affine scheme. Then approximation holds for every
triple $(T, E, m)$ as in Definition \ref{definition-approximation-holds}
such that there exists an integer $r \geq 0$ with
\begin{enumerate}
\item $E$ is $m$-pseudo-coherent,
\item $H^i(E)$ is supported on $T$ for $i \geq m - r + 1$,
\item $X \setminus T$ is the union of $r$ affine opens.
\end{enumerate}
In particular, approximation by perfect complexes holds for affine schemes.
\end{lemma}

\begin{proof}
Let $X_0$ be an affine scheme representing $X$. Let $T_0 \subset X_0$
by the closed subset corresponding to $T$. Let
$\epsilon : X_\etale \to X_{0, Zar}$ be the morphism
(\ref{equation-epsilon}). We may write $E = \epsilon^*E_0$ for some object
$E_0$ of $D_{\textit{QCoh}}(\mathcal{O}_{X_0})$, see
Lemma \ref{lemma-derived-quasi-coherent-small-etale-site}.
Then $E_0$ is $m$-pseudo-coherent, see
Lemma \ref{lemma-descend-pseudo-coherent}.
Comparing stalks of cohomology sheaves (see proof of
Lemma \ref{lemma-epsilon-flat})
we see that $H^i(E_0)$ is supported on $T_0$ for $i \geq m - r + 1$. By
Derived Categories of Schemes, Lemma \ref{perfect-lemma-approximation-affine}
there exists an approximation $P_0 \to E_0$ of
$(T_0, E_0, m)$. By Lemma \ref{lemma-descend-perfect}
we see that $P = \epsilon^*P_0$ is a perfect object of $D(\mathcal{O}_X)$.
Pulling back we obtain an approximation
$P = \epsilon^*P_0 \to \epsilon^*E_0 = E$ as desired.
\end{proof}

\begin{lemma}
\label{lemma-induction-step}
Let $S$ be a scheme. Let $(U \subset X, j : V \to X)$ be an
elementary distinguished square of algebraic spaces over $S$.
Assume $U$ quasi-compact, $V$ affine, and $U \times_X V$ quasi-compact.
If approximation by perfect complexes holds on $U$,
then approximation by perfect complexes holds on $X$.
\end{lemma}

\begin{proof}
Let $T \subset |X|$ be a closed subset with $X \setminus T \to X$
quasi-compact. Let $r_U$ be the integer of
Definition \ref{definition-approximation}
adapted to the pair $(U, T \cap |U|)$.
Set $T' = T \setminus |U|$. Endow $T'$ with the induced reduced
subspace structure. Since $|T'|$ is contained in $|X| \setminus |U|$
we see that $j^{-1}(T') \to T'$ is an isomorphism. Moreover,
$V \setminus j^{-1}(T')$ is quasi-compact as it is the fibre product
of $U \times_X V$ with $X \setminus T$ over $X$ and we've assumed
$U \times_X V$ quasi-compact and $X \setminus T \to X$ quasi-compact.
Let $r'$ be the number of affines needed to cover $V \setminus j^{-1}(T')$.
We claim that $r = \max(r_U, r')$ works for the pair $(X, T)$.

\medskip\noindent
To see this choose a triple $(T, E, m)$ such that $E$ is
$(m - r)$-pseudo-coherent and $H^i(E)$ is supported on $T$ for
$i \geq m - r$. Let $t$ be the largest integer such that
$H^t(E)|_U$ is nonzero. (Such an integer exists as $U$ is quasi-compact
and $E|_U$ is $(m - r)$-pseudo-coherent.)
We will prove that $E$ can be approximated by induction on $t$.

\medskip\noindent
Base case: $t \leq m - r'$. This means that $H^i(E)$ is supported
on $T'$ for $i \geq m - r'$. Hence
Lemma \ref{lemma-approximation-affine}
guarantees the existence of an approximation
$P \to E|_V$ of $(T', E|_V, m)$ on $V$.
Applying Lemma \ref{lemma-open} we see that
$(T', E, m)$ can be approximated. Such an approximation
is also an approximation of $(T, E, m)$.

\medskip\noindent
Induction step. Choose an approximation $P \to E|_U$ of
$(T \cap |U|, E|_U, m)$. This in particular gives a surjection
$H^t(P) \to H^t(E|_U)$.
In the rest of the proof we will use the equivalence of
Lemma \ref{lemma-derived-quasi-coherent-small-etale-site}
(and the compatibilities of Remark \ref{remark-match-total-direct-images})
for the representable algebraic spaces $V$ and $U \times_X V$.
We will also use the fact that $(m - r)$-pseudo-coherence,
resp.\ perfectness on the Zariski site and \'etale site agree, see
Lemmas \ref{lemma-descend-pseudo-coherent} and
\ref{lemma-descend-perfect}.
Thus we can use the results of
Derived Categories of Schemes, Section \ref{perfect-section-lift}
for the open immersion $U \times_X V \subset V$. In this way
Derived Categories of Schemes,
Lemma \ref{perfect-lemma-lift-perfect-complex-plus-shift-support}
implies there exists a perfect object $Q$ in $D(\mathcal{O}_V)$
supported on $j^{-1}(T)$ and an isomorphism
$Q|_{U \times_X V} \to (P \oplus P[1])|_{U \times_X V}$. By
Derived Categories of Schemes, Lemma \ref{perfect-lemma-lift-map}
we can replace $Q$ by $Q \otimes^\mathbf{L} I$
and assume that the map
$$
Q|_{U \times_X V} \longrightarrow
(P \oplus P[1])|_{U \times_X V} \longrightarrow
P|_{U \times_X V} \longrightarrow E|_{U \times_X V}
$$
lifts to $Q \to E|_V$. By Lemma \ref{lemma-glue}
we find an morphism $a : R \to E$ of $D(\mathcal{O}_X)$
such that $a|_U$ is isomorphic to $P \oplus P[1] \to E|_U$
and $a|_V$ isomorphic to $Q \to E|_V$.
Thus $R$ is perfect and supported on $T$
and the map $H^t(R) \to H^t(E)$ is surjective on restriction to $U$.
Choose a distinguised triangle
$$
R \to E \to E' \to R[1]
$$
Then $E'$ is $(m - r)$-pseudo-coherent
(Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cone-pseudo-coherent}),
$H^i(E')|_U = 0$ for $i \geq t$, and
$H^i(E')$ is supported on $T$ for $i \geq m - r$.
By induction we find an approximation $R' \to E'$
of $(T, E', m)$. Fit the composition $R' \to E' \to R[1]$
into a distringuished triangle $R \to R'' \to R' \to R[1]$
and extend the morphisms $R' \to E'$ and $R[1] \to R[1]$ into
a morphism of distinguished triangles
$$
\xymatrix{
R \ar[r] \ar[d] & R'' \ar[d] \ar[r] & R' \ar[d] \ar[r] & R[1] \ar[d] \\
R \ar[r] &  E \ar[r] & E' \ar[r] & R[1]
}
$$
using TR3. Then $R''$ is a perfect complex
(Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-two-out-of-three-perfect})
supported on $T$. An easy diagram chase shows that $R'' \to E$ is the desired
approximation.
\end{proof}

\begin{theorem}
\label{theorem-approximation}
Let $S$ be a scheme.
Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$.
Then approximation by perfect complexes holds on $X$.
\end{theorem}

\begin{proof}
This follows from the induction principle of
Lemma \ref{lemma-induction-principle}
and Lemmas \ref{lemma-induction-step} and \ref{lemma-approximation-affine}.
\end{proof}






\section{Generating derived categories}
\label{section-generating}

\noindent
This section is the analogue of
Derived Categories of Schemes, Section \ref{perfect-section-generating}.
However, we first prove the following lemma which is the
analogue of Derived Categories of Schemes, Lemma
\ref{perfect-lemma-lift-map-from-perfect-complex-with-support}.

\begin{lemma}
\label{lemma-lift-map-from-perfect-complex-with-support}
Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated
algebraic space over $S$. Let $W \subset X$ be a quasi-compact open.
Let $T \subset |X|$ be a closed subset such that
$X \setminus T \to X$ is a quasi-compact morphism.
Let $E$ be an object of $D_{\textit{QCoh}}(\mathcal{O}_X)$.
Let $\alpha : P \to E|_W$ be a map where $P$ is a perfect object of
$D(\mathcal{O}_W)$ supported on $T \cap W$. Then there exists a map
$\beta : R \to E$ where $R$ is a perfect object of $D(\mathcal{O}_X)$
supported on $T$ such that $P$ is a direct summand of $R|_W$ in
$D(\mathcal{O}_W)$ compatible $\alpha$ and $\beta|_W$.
\end{lemma}

\begin{proof}
We will use the induction principle of
Lemma \ref{lemma-induction-principle-enlarge} to prove this.
Thus we immediately reduce to the case where we have an
elementary distinguished square $(W \subset X, f : V \to X)$
with $V$ affine and $P \to E|_W$ as in the statement of the lemma.
In the rest of the proof we will use
Lemma \ref{lemma-derived-quasi-coherent-small-etale-site}
(and the compatibilities of Remark \ref{remark-match-total-direct-images})
for the representable algebraic spaces $V$ and $W \times_X V$.
We will also use the fact that perfectness on the Zariski site
and \'etale site agree, see Lemma \ref{lemma-descend-perfect}.

\medskip\noindent
By Derived Categories of Schemes,
Lemma \ref{perfect-lemma-lift-perfect-complex-plus-shift-support}
we can choose a perfect object $Q$ in $D(\mathcal{O}_V)$
supported on $f^{-1}T$ and an isomorphism
$Q|_{W \times_X V} \to (P \oplus P[1])|_{W \times_X V}$. By
Derived Categories of Schemes, Lemma \ref{perfect-lemma-lift-map}
we can replace $Q$ by $Q \otimes^\mathbf{L} I$ (still supported on $f^{-1}T$)
and assume that the map
$$
Q|_{W \times_X V} \to (P \oplus P[1])|_{W \times V}
\longrightarrow P|_{W \times_X V}
\longrightarrow
E|_{W \times_X V}
$$
lifts to $Q \to E|_V$. By Lemma \ref{lemma-glue}
we find an morphism $a : R \to E$ of $D(\mathcal{O}_X)$
such that $a|_W$ is isomorphic to $P \oplus P[1] \to E|_W$
and $a|_V$ isomorphic to $Q \to E|_V$.
Thus $R$ is perfect and supported on $T$ as desired.
\end{proof}

\begin{remark}
\label{remark-addendum}
The proof of Lemma \ref{lemma-lift-map-from-perfect-complex-with-support}
shows that
$$
R|_W = P \oplus P^{\oplus n_1}[1] \oplus \ldots \oplus P^{\oplus n_m}[m]
$$
for some $m \geq 0$ and $n_j \geq 0$. Thus the highest degree cohomology sheaf
of $R|_W$ equals that of $P$. By repeating the construction for the map
$P^{\oplus n_1}[1] \oplus \ldots \oplus P^{\oplus n_m}[m] \to R|_W$, taking
cones, and using induction we can achieve equality of cohomology sheaves
of $R|_W$ and $P$ above any given degree.
\end{remark}

\begin{lemma}
\label{lemma-direct-summand-of-a-restriction}
Let $S$ be a scheme.
Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$.
Let $W$ be a quasi-compact open subspace of $X$.
Let $P$ be a perfect object of $D(\mathcal{O}_W)$.
Then $P$ is a direct summand of the restriction of a perfect
object of $D(\mathcal{O}_X)$.
\end{lemma}

\begin{proof}
Special case of Lemma \ref{lemma-lift-map-from-perfect-complex-with-support}.
\end{proof}

\begin{theorem}
\label{theorem-bondal-van-den-Bergh}
Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated
algebraic space over $S$. The category
$D_{\textit{QCoh}}(\mathcal{O}_X)$ can be generated by a single
perfect object. More precisely, there exists a perfect object
$P$ of $D(\mathcal{O}_X)$ such that for 
$E \in D_{\textit{QCoh}}(\mathcal{O}_X)$ the following are equivalent
\begin{enumerate}
\item $E = 0$, and
\item $\Hom_{D(\mathcal{O}_X)}(P[n], E) = 0$ for all $n \in \mathbf{Z}$.
\end{enumerate}
\end{theorem}

\begin{proof}
We will prove this using the induction principle of
Lemma \ref{lemma-induction-principle}

\medskip\noindent
If $X$ is affine, then $\mathcal{O}_X$ is a perfect generator.
This follows from Lemma \ref{lemma-derived-quasi-coherent-small-etale-site}
and
Derived Categories of Schemes,
Lemma \ref{perfect-lemma-affine-compare-bounded}.

\medskip\noindent
Assume that $(U \subset X, f : V \to X)$ is an elementary distinguished
square with $U$ quasi-compact such that the theorem holds for $U$ and $V$
is an affine scheme.
Let $P$ be a perfect object of $D(\mathcal{O}_U)$ which is a generator
for $D_{\textit{QCoh}}(\mathcal{O}_U)$. Using
Lemma \ref{lemma-direct-summand-of-a-restriction} we may
choose a perfect object
$Q$ of $D(\mathcal{O}_X)$ whose restriction to $U$ is a direct sum one
of whose summands is $P$. Say $V = \Spec(A)$. Let $Z \subset V$
be the reduced closed subscheme which is the inverse image of
$X \setminus U$ and maps isomorphically to it
(see Definition \ref{definition-elementary-distinguished-square}).
This is a retrocompact closed subset of $V$.
Choose $f_1, \ldots, f_r \in A$ such that
$Z = V(f_1, \ldots, f_r)$. Let $K \in D(\mathcal{O}_V)$ be the perfect
object corresponding to the Koszul complex on $f_1, \ldots, f_r$ over $A$.
Note that since $K$ is supported on $Z$, the pushforward
$K' = Rf_*K$ is a perfect object of $D(\mathcal{O}_X)$ whose
restriction to $V$ is $K$ (see Lemmas \ref{lemma-pushforward-perfect}
and \ref{lemma-pushforward-with-support-in-open}).
We claim that $Q \oplus K'$ is a generator for
$D_{\textit{QCoh}}(\mathcal{O}_X)$.

\medskip\noindent
Namely, by Lemma \ref{lemma-pushforward-with-support-in-open}
we have $K' =  f_! K$ and hence
$$
\Hom_{D(\mathcal{O}_X)}(K'[n], E) = \Hom_{D(\mathcal{O}_V)}(K[n], E|_V)
$$
Thus by
Derived Categories of Schemes,
Lemma \ref{perfect-lemma-orthogonal-koszul-complex}
(using also
Lemma \ref{lemma-derived-quasi-coherent-small-etale-site})
the vanishing of these groups implies that $E|_V$ is isomorphic to
$R(U \times_X V \to V)_*E|_{U \times_X V}$. This implies that
$E = R(U \to X)_*E|_U$ (small detail omitted). If this is the case then
$$
\Hom_{D(\mathcal{O}_X)}(Q[n], E) = \Hom_{D(\mathcal{O}_U)}(Q|_U[n], E|_U)
$$
which contains $\Hom_{D(\mathcal{O}_U)}(P[n], E|_U)$ as a direct summand.
Thus by our choice of $P$ the vanishing of these groups implies that $E|_U$
is zero. Whence $E$ is zero.
\end{proof}








\section{Compact and perfect objects}
\label{section-compact}

\noindent
This section is the analogue of
Derived Categories of Schemes, Section \ref{perfect-section-compact}.

\begin{proposition}
\label{proposition-compact-is-perfect}
Let $S$ be a scheme.
Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$.
An object of $D_{\textit{QCoh}}(\mathcal{O}_X)$ is compact
if and only if it is perfect.
\end{proposition}

\begin{proof}
By Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-perfect-is-compact}
the perfect objects even define compact objects of $D(\mathcal{O}_X)$.
Conversely, let $K$ be a compact object of $D_{\textit{Qcoh}}(\mathcal{O}_X)$.
To show that $K$ is perfect, it suffices to show that
$K|_U$ is perfect for every affine scheme $U$ \'etale over $X$, see
Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-perfect-independent-representative}.
Observe that $j : U \to X$ is a quasi-compact and separated morphism.
Hence
$Rj_* : D_{\textit{QCoh}}(\mathcal{O}_U) \to D_{\textit{QCoh}}(\mathcal{O}_X)$
commutes with direct sums, see
Lemma \ref{lemma-quasi-coherence-pushforward-direct-sums}.
Thus the adjointness of restriction to $U$ and $Rj_*$ implies that
$K|_U$ is a perfect object of $D_{\textit{QCoh}}(\mathcal{O}_U)$.
Hence we reduce to the case that $X$ is affine, in particular a
quasi-compact and quasi-separated scheme. Via
Lemma \ref{lemma-derived-quasi-coherent-small-etale-site} and
\ref{lemma-descend-perfect}
we reduce to the case of schemes, i.e., to
Derived Categories of Schemes, Proposition
\ref{perfect-proposition-compact-is-perfect}.
\end{proof}











\section{Derived categories as module categories}
\label{section-derived-is-dga}

\noindent
The section is the analogue of
Derived Categories of Schemes, Section \ref{perfect-section-derived-is-dga}.

\begin{lemma}
\label{lemma-tensor-with-QCoh-complex}
Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $K^\bullet$
be a complex of $\mathcal{O}_X$-modules whose cohomology sheaves are
quasi-coherent. Let
$(E, d) = \Hom_{\text{Comp}^{dg}(\mathcal{O}_X)}(K^\bullet, K^\bullet)$
be the endomorphism differential graded algebra. Then the functor
$$
- \otimes_E^\mathbf{L} K^\bullet :
D(E, \text{d}) \longrightarrow D(\mathcal{O}_X)
$$
of
Differential Graded Algebra, Lemma
\ref{dga-lemma-tensor-with-complex-derived}
has image contained in $D_{\textit{QCoh}}(\mathcal{O}_X)$.
\end{lemma}

\begin{proof}
Let $P$ be a differential graded $E$-module with property $P$.
Let $F_\bullet$ be a filtration on $P$ as in
Differential Graded Algebra, Section \ref{dga-section-P-resolutions}.
Then we have
$$
P \otimes_E K^\bullet = \text{hocolim}\ F_iP \otimes_E K^\bullet
$$
Each of the $F_iP$ has a finite filtration whose graded pieces
are direct sums of $E[k]$. The result follows easily.
\end{proof}

\noindent
The following lemma can be strengthened (there is a uniformity
in the vanishing over all $L$ with nonzero cohomology sheaves
only in a fixed range).

\begin{lemma}
\label{lemma-ext-from-perfect-into-bounded-QCoh}
Let $S$ be a scheme.
Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$.
Let $K$, $L$ be objects of $D(\mathcal{O}_X)$ with
$K$ perfect and $L$ in $D^b_{\textit{QCoh}}(\mathcal{O}_X)$.
Then $\text{Ext}^n_{D(\mathcal{O}_X)}(K, L)$ is nonzero
for only a finite number of $n$.
\end{lemma}

\begin{proof}
Since $K$ is perfect we have
$$
\text{Ext}^i_{D(\mathcal{O}_X)}(K, L) =
H^i(X, K^\wedge \otimes_{\mathcal{O}_X}^\mathbf{L} L)
$$
where $K^\wedge$ is the ``dual'' perfect complex to $K$, see
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-dual-perfect-complex}.
Note that $P = K^\wedge \otimes_{\mathcal{O}_X}^\mathbf{L} L$
is in $D_{\textit{QCoh}}(X)$ by
Lemmas \ref{lemma-quasi-coherence-tensor-product} and
\ref{lemma-pseudo-coherent} (to see that a perfect complex
has quasi-coherent cohomology sheaves). On the other hand,
the spectral sequence
$$
E_1^{p, q} = H^p(K^\wedge \otimes_{\mathcal{O}_X}^\mathbf{L} H^q(L))
\Rightarrow
H^{p + q}(K^\wedge \otimes_{\mathcal{O}_X}^\mathbf{L} L) = H^{p + q}(P),
$$
the boundedness of $L$, and the finite tor amplitude of $K^\wedge$
show that $P$ has only finitely many nonzero cohomology sheaves.
It follows that $H^n(X, P) = 0$ for $n \ll 0$.
But also $H^n(X, P) = 0$ for $n \gg 0$ by
Cohomology of Spaces, Lemma
\ref{spaces-cohomology-lemma-vanishing-quasi-separated}
and the spectral sequence expressing $H^n(X, P^\bullet)$
in terms of $H^p(X, H^q(P^\bullet))$ using that the cohomology
sheaves of $P$ are quasi-coherent.
\end{proof}

\noindent
The following is the analogue of
Derived Categories of Schemes, Theorem \ref{perfect-theorem-DQCoh-is-Ddga}.

\begin{theorem}
\label{theorem-DQCoh-is-Ddga}
Let $S$ be a scheme.
Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$.
Then there exist a differential graded algebra $(E, \text{d})$
with only a finite number of nonzero cohomology groups $H^i(E)$
such that $D_{\textit{QCoh}}(\mathcal{O}_X)$ is equivalent
to $D(E, \text{d})$.
\end{theorem}

\begin{proof}
Let $K^\bullet$ be a K-injective complex of $\mathcal{O}$-modules which
is perfect and generates $D_{\textit{QCoh}}(\mathcal{O}_X)$. Such a
thing exists by Theorem \ref{theorem-bondal-van-den-Bergh}
and the existence of K-injective resolutions. We will show the
theorem holds with
$$
(E, \text{d}) = \Hom_{\text{Comp}^{dg}(\mathcal{O}_X)}(K^\bullet, K^\bullet)
$$
where $\text{Comp}^{dg}(\mathcal{O}_X)$ is the differential graded category
of complexes of $\mathcal{O}$-modules. Please see
Differential Graded Algebra, Section \ref{dga-section-variant-base-change}.
Since $K^\bullet$ is K-injective we
have
\begin{equation}
\label{equation-E-is-OK}
H^n(E) = \text{Ext}^n_{D(\mathcal{O}_X)}(K^\bullet, K^\bullet)
\end{equation}
for all $n \in \mathbf{Z}$. Only a finite number of these Exts
are nonzero by Lemma \ref{lemma-ext-from-perfect-into-bounded-QCoh}.
Consider the functor
$$
- \otimes_E^\mathbf{L} K^\bullet :
D(E, \text{d}) \longrightarrow D(\mathcal{O}_X)
$$
of
Differential Graded Algebra, Lemma
\ref{dga-lemma-tensor-with-complex-derived}.
Since $K^\bullet$ is perfect, it defines a compact object of
$D(\mathcal{O}_X)$, see Proposition \ref{proposition-compact-is-perfect}.
Combined with (\ref{equation-E-is-OK}) the functor above is fully
faithful as follows from
Differential Graded Algebra, Lemmas
\ref{dga-lemma-fully-faithful-in-compact-case}. It has a right adjoint
$$
R\Hom(K^\bullet, - ) : D(\mathcal{O}_X) \longrightarrow D(E, \text{d})
$$
by Differential Graded Algebra, Lemmas
\ref{dga-lemma-tensor-with-complex-hom-adjoint}
which is a left quasi-inverse functor by generalities on adjoint
functors. On the other hand, it follows from
Lemma \ref{lemma-tensor-with-QCoh-complex} that we obtain
$$
- \otimes_E^\mathbf{L} K^\bullet :
D(E, \text{d}) \longrightarrow D_{\textit{QCoh}}(\mathcal{O}_X)
$$
and by our choice of $K^\bullet$ as a generator of
$D_{\textit{QCoh}}(\mathcal{O}_X)$ the kernel of the adjoint
restricted to $D_{\textit{QCoh}}(\mathcal{O}_X)$ is zero.
A formal argument shows that we obtain the desired equivalence, see
Derived Categories, Lemma
\ref{derived-lemma-fully-faithful-adjoint-kernel-zero}.
\end{proof}










\section{Cohomology and base change, IV}
\label{section-cohomology-and-base-change-perfect}

\noindent
This section is the analogue of Derived Categories of Schemes, Section
\ref{perfect-section-cohomology-and-base-change-perfect}.

\begin{lemma}
\label{lemma-cohomology-base-change}
Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-compact and quasi-separated
morphism of algebraic spaces over $S$. For $E$ in
$D_{\textit{QCoh}}(\mathcal{O}_X)$ and
$K$ in $D_{\textit{QCoh}}(\mathcal{O}_Y)$ we have
$$
Rf_*(E) \otimes_{\mathcal{O}_Y}^\mathbf{L} K =
Rf_*(E \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*K)
$$
\end{lemma}

\begin{proof}
Without any assumptions there is a map
$Rf_*(E) \otimes_{\mathcal{O}_Y}^\mathbf{L} K \to
Rf_*(E \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*K)$.
Namely, it is the adjoint to the canonical map
$$
Lf^*(Rf_*(E) \otimes_{\mathcal{O}_Y}^\mathbf{L} K) =
Lf^*(Rf_*(E)) \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*K
\longrightarrow
E \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*K
$$
coming from the map $Lf^*Rf_*E \to E$. See
Cohomology on Sites, Lemmas
\ref{sites-cohomology-lemma-pullback-tensor-product} and
\ref{sites-cohomology-lemma-adjoint}.
To check it is an isomorphism we may work \'etale locally on $Y$.
Hence we reduce to the case that $Y$ is an affine scheme.

\medskip\noindent
Suppose that $K = \bigoplus K_i$ is a direct
sum of some complexes $K_i \in D_{\textit{QCoh}}(\mathcal{O}_Y)$.
If the statement holds for each $K_i$, then it holds for $K$.
Namely, the functors $Lf^*$ and $\otimes^\mathbf{L}$ preserve
direct sums by construction and $Rf_*$ commutes with direct sums
(for complexes with quasi-coherent cohomology sheaves) by
Lemma \ref{lemma-quasi-coherence-pushforward-direct-sums}.
Moreover, suppose that $K \to L \to M \to K[1]$ is a distinguished
triangle in $D_{\textit{QCoh}}(Y)$. Then if the statement of the
lemma holds for two of $K, L, M$, then it holds for the third
(as the functors involved are exact functors of triangulated categories).

\medskip\noindent
Assume $Y$ affine, say $Y = \Spec(A)$. The functor
$\widetilde{\ } : D(A) \to D_\textit{QCoh}(\mathcal{O}_Y)$ is an equivalence
by
Lemma \ref{lemma-derived-quasi-coherent-small-etale-site} and
Derived Categories of Schemes,
Lemma \ref{perfect-lemma-affine-compare-bounded}.
Let $T$ be the property for $K \in D(A)$ that
the statement of the lemma holds for $\widetilde{K}$.
The discussion above and
More on Algebra, Remark \ref{more-algebra-remark-P-resolution}
shows that it suffices to prove $T$ holds for $A[k]$.
This finishes the proof, as the statement of the lemma
is clear for shifts of the structure sheaf.
\end{proof}

\begin{definition}
\label{definition-tor-independent}
Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.
Let $X$, $Y$ be algebraic spaces over $B$. We say $X$ and
$Y$ are {\it Tor independent over $B$} if and only if for every
commutative diagram
$$
\xymatrix{
\Spec(k) \ar[d]_{\overline{y}} \ar[dr]_{\overline{b}} \ar[r]_-{\overline{x}} &
X \ar[d] \\
Y \ar[r] & B
}
$$
of geometric points the rings
$\mathcal{O}_{X, \overline{x}}$ and $\mathcal{O}_{Y, \overline{y}}$
are Tor independent over $\mathcal{O}_{B, \overline{b}}$ (see
More on Algebra, Definition \ref{more-algebra-definition-tor-independent}).
\end{definition}

\noindent
The following lemma shows in particular that this definition agrees
with our definition in the case of representable algebraic spaces.

\begin{lemma}
\label{lemma-tor-independent}
Let $S$ be a scheme. Let $B$ be an algebraic space over $S$.
Let $X$, $Y$ be algebraic spaces over $B$. The following are equivalent
\begin{enumerate}
\item $X$ and $Y$ are Tor independent over $B$,
\item for every commutative diagram
$$
\xymatrix{
U \ar[d] \ar[r] & W \ar[d] & V \ar[d] \ar[l] \\
X \ar[r] & B & Y \ar[l]
}
$$
with \'etale vertical arrows $U$ and $V$ are Tor independent over $W$,
\item for some commutative diagram as in (2) with (a) $W \to B$ \'etale
surjective, (b) $U \to X \times_B W$ \'etale surjective, (c)
$V \to Y \times_B W$ \'etale surjective, the spaces $U$ and $V$ are Tor
independent over $W$, and
\item for some commutative diagram as in (3) with $U$, $V$, $W$ schemes,
the schemes $U$ and $V$ are Tor independent over $W$ in the sense of
Derived Categories of Schemes, Definition
\ref{perfect-definition-tor-independent}.
\end{enumerate}
\end{lemma}

\begin{proof}
For an \'etale morphism $\varphi : U \to X$ of algebraic spaces
and geometric point $\overline{u}$ the map of local rings
$\mathcal{O}_{X, \varphi(\overline{u})} \to \mathcal{O}_{U, \overline{u}}$
is an isomorphism. Hence the equivalence of (1) and (2) follows.
So does the implication (1) $\Rightarrow$ (3). Assume (3) and
pick a diagram of geometric points as in
Definition \ref{definition-tor-independent}.
The assumptions imply that we can first lift $\overline{b}$ to a geometric
point $\overline{w}$ of $W$, then lift the geometric point
$(\overline{x}, \overline{b})$ to a geometric point $\overline{u}$ of $U$,
and finally lift the geometric point
$(\overline{y}, \overline{b})$ to a geometric point $\overline{v}$ of $V$.
Use Properties of Spaces, Lemma
\ref{spaces-properties-lemma-geometric-lift-to-usual}
to find the lifts.
Using the remark on local rings above we conclude that the condition
of the definition is satisfied for the given diagram.

\medskip\noindent
Having made these initial points, it is clear that (4) comes down to the
statement that
Definition \ref{definition-tor-independent}
agrees with
Derived Categories of Schemes, Definition
\ref{perfect-definition-tor-independent}
when $X$, $Y$, and $B$ are schemes.

\medskip\noindent
Let $\overline{x}, \overline{b}, \overline{y}$ be as in
Definition \ref{definition-tor-independent} lying over the points
$x, y, b$. Recall that
$\mathcal{O}_{X, \overline{x}} = \mathcal{O}_{X, x}^{sh}$
(Properties of Spaces, Lemma
\ref{spaces-properties-lemma-describe-etale-local-ring}) and similarly
for the other two. By Algebra, Lemma
\ref{algebra-lemma-strictly-henselian-functorial-improve} we see that
$\mathcal{O}_{X, \overline{x}}$ is a strict henselization of
$\mathcal{O}_{X, x} \otimes_{\mathcal{O}_{B, b}} \mathcal{O}_{B, \overline{b}}$.
In particular, the ring map
$$
\mathcal{O}_{X, x} \otimes_{\mathcal{O}_{B, b}} \mathcal{O}_{B, \overline{b}}
\longrightarrow
\mathcal{O}_{X, \overline{x}}
$$
is flat (More on Algebra, Lemma
\ref{more-algebra-lemma-dumb-properties-henselization}). By
More on Algebra, Lemma \ref{more-algebra-lemma-tor-independent-flat}
we see that
$$
\text{Tor}_i^{\mathcal{O}_{B, b}}(\mathcal{O}_{X, x}, \mathcal{O}_{Y, y})
\otimes_{\mathcal{O}_{X, x} \otimes_{\mathcal{O}_{B, b}} \mathcal{O}_{Y, y}}
(\mathcal{O}_{X, \overline{x}} \otimes_{\mathcal{O}_{B, \overline{b}}}
\mathcal{O}_{Y, \overline y})
=
\text{Tor}_i^{\mathcal{O}_{B, \overline{b}}}(
\mathcal{O}_{X, \overline{x}}, \mathcal{O}_{Y, \overline{y}})
$$
Hence it follows that if $X$ and $Y$ are Tor independent over $B$
as schemes, then $X$ and $Y$ are Tor independent as algebraic spaces over $B$.

\medskip\noindent
For the converse, we may assume $X$, $Y$, and $B$ are affine.
Observe that the ring map
$$
\mathcal{O}_{X, x} \otimes_{\mathcal{O}_{B, b}} \mathcal{O}_{Y, y}
\longrightarrow
\mathcal{O}_{X, \overline{x}} \otimes_{\mathcal{O}_{B, \overline{b}}}
\mathcal{O}_{Y, \overline y}
$$
is flat by the observations given above. Moreover, the image of the map
on spectra includes all primes
$\mathfrak s \subset
\mathcal{O}_{X, x} \otimes_{\mathcal{O}_{B, b}} \mathcal{O}_{Y, y}$
lying over $\mathfrak m_x$ and $\mathfrak m_y$.
Hence from this and the displayed formula of Tor's above we see that if
$X$ and $Y$ are Tor independent over $B$ as algebraic spaces, then
$$
\text{Tor}_i^{\mathcal{O}_{B, b}}
(\mathcal{O}_{X, x}, \mathcal{O}_{Y, y})_\mathfrak s = 0
$$
for all $i > 0$ and all $\mathfrak s$ as above. By
More on Algebra, Lemma \ref{more-algebra-lemma-tor-independent}
applied to the ring maps
$\Gamma(B, \mathcal{O}_B) \to \Gamma(X, \mathcal{O}_X)$
and
$\Gamma(B, \mathcal{O}_B) \to \Gamma(X, \mathcal{O}_X)$
this implies that $X$ and $Y$ are Tor independent over $B$.
\end{proof}

\begin{lemma}
\label{lemma-compare-base-change}
Let $S$ be a scheme. Let $g : Y' \to Y$ be a morphism of algebraic spaces over
$S$. Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of
algebraic spaces over $S$. Consider the base change diagram
$$
\xymatrix{
X' \ar[r]_h \ar[d]_{f'} &
X \ar[d]^f \\
Y' \ar[r]^g &
Y
}
$$
If $X$ and $Y'$ are Tor independent over $Y$, then for all
$E \in D_{\textit{QCoh}}(\mathcal{O}_X)$ we have
$Rf'_*Lh^*E = Lg^*Rf_*E$.
\end{lemma}

\begin{proof}
For any object $E$ of $D(\mathcal{O}_X)$ we can use
Cohomology on Sites, Remark \ref{sites-cohomology-remark-base-change}
to get a canonical base change map $Lg^*Rf_*E \to Rf'_*Lh^*E$. To check this
is an isomorphism we may work \'etale locally on $Y'$. Hence we may assume
$g : Y' \to Y$ is a morphism of affine schemes. In particular, $g$
is affine and it suffices to show that
$$
Rg_*Lg^*Rf_*E \to Rg_*Rf'_*Lh^*E = Rf_*(Rh_* Lh^* E)
$$
is an isomorphism, see Lemma \ref{lemma-affine-morphism}
(and use Lemmas \ref{lemma-quasi-coherence-pullback},
\ref{lemma-quasi-coherence-tensor-product}, and
\ref{lemma-quasi-coherence-direct-image}
to see that the objects $Rf'_*Lh^*E$ and $Lg^*Rf_*E$
have quasi-coherent cohomology sheaves). Note that $h$ is
affine as well (Morphisms of Spaces, Lemma
\ref{spaces-morphisms-lemma-base-change-affine}).
By Lemma \ref{lemma-affine-morphism-pull-push} the map becomes a map
$$
Rf_*E \otimes_{\mathcal{O}_Y}^\mathbf{L} g_*\mathcal{O}_{Y'}
\longrightarrow
Rf_*(E \otimes_{\mathcal{O}_X}^\mathbf{L} h_*\mathcal{O}_{X'})
$$
Observe that $h_*\mathcal{O}_{X'} = f^*g_*\mathcal{O}_{Y'}$. Thus by
Lemma \ref{lemma-cohomology-base-change} it suffices to prove that
$Lf^*g_*\mathcal{O}_{Y'} = f^*g_*\mathcal{O}_{Y'}$. This follows from our
assumption that $X$ and $Y'$ are Tor independent over $Y$. Namely, to
check it we may work \'etale locally on $X$, hence we may also assume $X$
is affine. Say $X = \Spec(A)$, $Y = \Spec(R)$ and $Y' = \Spec(R')$.
Our assumption implies that $A$ and $R'$ are Tor independent over $R$
(see
Lemma \ref{lemma-tor-independent}
and
More on Algebra, Lemma \ref{more-algebra-lemma-tor-independent}), i.e.,
$\text{Tor}_i^R(A, R') = 0$ for $i > 0$. In other words
$A \otimes_R^\mathbf{L} R' = A \otimes_R R'$ which exactly means
that $Lf^*g_*\mathcal{O}_{Y'} = f^*g_*\mathcal{O}_{Y'}$.
\end{proof}

\begin{lemma}
\label{lemma-perfect-direct-image}
Let $S$ be a scheme. Let $Y$ be a Noetherian algebraic space over $S$.
Let $f : X \to Y$ be a morphism of algebraic spaces which is locally of
finite type and quasi-separated. Let $E \in D(\mathcal{O}_X)$ such that
\begin{enumerate}
\item $E \in D^b_{\textit{Coh}}(\mathcal{O}_X)$,
\item the scheme theoretic support of $H^i(E)$ is proper over $Y$ for all $i$,
\item $E$ has finite tor dimension as an object of $D(f^{-1}\mathcal{O}_Y)$.
\end{enumerate}
Then $Rf_*E$ is a perfect object of $D(\mathcal{O}_Y)$.
\end{lemma}

\begin{proof}
By Lemma \ref{lemma-direct-image-coherent} we see that $Rf_*E$ is an object of
$D^b_{\textit{Coh}}(\mathcal{O}_Y)$. Hence $Rf_*E$ is pseudo-coherent
(Lemma \ref{lemma-identify-pseudo-coherent-noetherian}).
Hence it suffices to show that $Rf_*E$ has finite tor dimension, see
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-perfect}.
By Lemma \ref{lemma-tor-qc-qs} it suffices to check that
$Rf_*(E) \otimes_{\mathcal{O}_Y}^\mathbf{L} \mathcal{F}$
has universally bounded cohomology for all quasi-coherent
sheaves $\mathcal{F}$ on $Y$. Bounded from above is clear as $Rf_*(E)$
is bounded from above. Let $T \subset X$ be the union of the supports
of $H^i(E)$ for all $i$. Then $T$ is proper over $Y$ by assumptions (1)
and (2). In particular there exists a quasi-compact open subspace
$X' \subset X$ containing $T$. Setting $f' = f|_{X'}$ we have
$Rf_*(E) = Rf'_*(E|_{X'})$ because $E$ restricts to zero on $X \setminus T$.
Thus we may replace $X$ by $X'$ and assume $f$ is quasi-compact.
We have assumed $f$ is quasi-separated. Thus
$$
Rf_*(E) \otimes_{\mathcal{O}_Y}^\mathbf{L} \mathcal{F} =
Rf_*\left(E \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*\mathcal{F}\right) =
Rf_*\left(E \otimes_{f^{-1}\mathcal{O}_Y}^\mathbf{L} f^{-1}\mathcal{F}\right)
$$
by
Lemma \ref{lemma-cohomology-base-change}
and
Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-variant-derived-pullback}.
By assumption (3) the complex
$E \otimes_{f^{-1}\mathcal{O}_Y}^\mathbf{L} f^{-1}\mathcal{F}$
has cohomology sheaves in a
given finite range, say $[a, b]$. Then $Rf_*$ of it
has cohomology in the range $[a, \infty)$ and we win.
\end{proof}






\section{Computing Ext groups and base change}
\label{section-ext}

\noindent
The results in this section will be used to verify one of Artin's criteria
for Quot functors, Hilbert schemes, and other moduli problems.

\begin{lemma}
\label{lemma-compute-ext-perfect}
Let $S$ be a scheme. Let $B$ be a Noetherian algebraic space over $S$.
Let $f : X \to B$ be a morphism of algebraic spaces which is locally of
finite type and quasi-separated. Let $E \in D(\mathcal{O}_X)$
and $\mathcal{G}$ an $\mathcal{O}_X$-module. Assume
\begin{enumerate}
\item $E$ is perfect, and
\item $\mathcal{G}$ is a coherent $\mathcal{O}_X$-module flat over $B$
with scheme theoretic support proper over $B$.
\end{enumerate}
Then
$$
K = Rf_*R\SheafHom(E, \mathcal{G})
$$
is a perfect object of $D(\mathcal{O}_B)$ and there are functorial isomorphisms
$$
H^i(B, K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F})
\longrightarrow
\text{Ext}^i_{\mathcal{O}_X}(E,
\mathcal{G} \otimes_{\mathcal{O}_X} f^*\mathcal{F})
$$
for $\mathcal{F}$ quasi-coherent on $B$
compatible with boundary maps (see proof).
\end{lemma}

\begin{proof}
Since $E$ is a perfect complex there exists a dual perfect complex
$E'$, see
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-dual-perfect-complex}.
Observe that
$$
\text{Ext}^i_{\mathcal{O}_X}(E,
\mathcal{G} \otimes_{\mathcal{O}_X} f^*\mathcal{F})
=
H^i(X, E' \otimes^\mathbf{L}_{\mathcal{O}_X}
(\mathcal{G} \otimes_{\mathcal{O}_X} f^*\mathcal{F}))
$$
by construction of $E'$. We have
$$
\mathcal{G} \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*\mathcal{F} =
\mathcal{G} \otimes_{f^{-1}\mathcal{O}_B}^\mathbf{L} f^{-1}\mathcal{F} =
\mathcal{G} \otimes_{f^{-1}\mathcal{O}_B} f^{-1}\mathcal{F} =
\mathcal{G} \otimes_{\mathcal{O}_X} f^*\mathcal{F}
$$
the first equality by
Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-variant-derived-pullback},
the second as $\mathcal{G}$ is a flat $f^{-1}\mathcal{O}_B$-module, and
the third by definition of pullbacks. Hence we obtain
\begin{align*}
H^i(X, E' \otimes^\mathbf{L}_{\mathcal{O}_X}
(\mathcal{G} \otimes_{\mathcal{O}_X} f^*\mathcal{F}))
& =
H^i(X, E' \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{G}
\otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*\mathcal{F}) \\
& =
H^i(B,
Rf_*(E' \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{G}
\otimes^\mathbf{L}_{\mathcal{O}_X} Lf^*\mathcal{F})) \\
& =
H^i(B,
Rf_*(E' \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{G})
\otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F})
\end{align*}
The first equality by the above, the second by Leray
(Cohomology on Sites, Remark \ref{sites-cohomology-remark-before-Leray}), and
the third equality by Lemma \ref{lemma-cohomology-base-change}.
The object
$$
K = Rf_*(E' \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{G}) =
Rf_*R\SheafHom(E, \mathcal{G})
$$
(second equality by
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-dual-perfect-complex})
is perfect by Lemma \ref{lemma-perfect-direct-image}.
This lemma applies. Namely, the hypotheses on the cohomology sheaves
$H^i(E' \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{G})$
are clear and the hypothesis on finite tor dimension follows as $\mathcal{G}$
is flat over $f^{-1}\mathcal{O}_B$ and locally $E'$ is isomorphic
to a finite complex of free $\mathcal{O}_X$-modules.

\medskip\noindent
The statement on boundary maps means the following: Given a short
exact sequence $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$
then the isomorphisms fit into commutative diagrams
$$
\xymatrix{
H^i(B, K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}_3)
\ar[r] \ar[d]_\delta &
\text{Ext}^i_{\mathcal{O}_X}(E,
\mathcal{G} \otimes_{\mathcal{O}_X} f^*\mathcal{F}_3) \ar[d]^\delta \\
H^{i + 1}(B, K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}_1)
\ar[r] &
\text{Ext}^{i + 1}_{\mathcal{O}_X}(E,
\mathcal{G} \otimes_{\mathcal{O}_X} f^*\mathcal{F}_1)
}
$$
where the boundary maps come from the distinguished triangle
$$
K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}_1 \to
K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}_2 \to
K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}_3 \to
K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F}_1[1]
$$
and the distinguished triangle in $D(\mathcal{O}_X)$ associated to
the short exact sequence
$$
0 \to
\mathcal{G} \otimes_{\mathcal{O}_X} f^*\mathcal{F}_1 \to
\mathcal{G} \otimes_{\mathcal{O}_X} f^*\mathcal{F}_2 \to
\mathcal{G} \otimes_{\mathcal{O}_X} f^*\mathcal{F}_3 \to 0
$$
This sequence is exact because $\mathcal{G}$ is flat over $B$.
We omit the verification of the commutativity of the displayed diagram.
\end{proof}

\begin{lemma}
\label{lemma-descend-RHom}
Let $X$ be a scheme. Let $E, F$ be objects of $D(\mathcal{O}_X)$.
If $E$ is pseudo-coherent and $F$ lies in $D^+(\mathcal{O}_X)$, then
there is a canonical isomorphism
$$
\epsilon^*R\SheafHom(E, F) \longrightarrow R\SheafHom(\epsilon^*E, \epsilon^*F)
$$
Here $\epsilon$ is as in (\ref{equation-epsilon}).
\end{lemma}

\begin{proof}
Recall that $\epsilon$ is flat (Lemma \ref{lemma-epsilon-flat}) and
hence $\epsilon^* = L\epsilon^*$. There is a canonical map
from left to right by
Cohomology on Sites, Remark
\ref{sites-cohomology-remark-prepare-fancy-base-change}.
To see this is an isomorphism we can work locally, i.e., we may
assume $X$ is an affine scheme. In this case we can represent
$E$ by a bounded above complex $\mathcal{E}^\bullet$
of finite free $\mathcal{O}_X$-modules, see
Derived Categories of Schemes, Lemma \ref{perfect-lemma-lift-pseudo-coherent}.
We may also represent $F$ by a bounded below complex $\mathcal{F}^\bullet$
of $\mathcal{O}_X$-modules. Applying
Cohomology, Lemma
\ref{cohomology-lemma-Rhom-complex-of-direct-summands-finite-free}
we see that $R\SheafHom(E, F)$ is represented by the complex with terms
$$
\bigoplus\nolimits_{n = - p + q}
\SheafHom_{\mathcal{O}_X}(\mathcal{E}^p, \mathcal{F}^q)
$$
Applying Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-Rhom-complex-of-direct-summands-finite-free}
we see that $R\SheafHom(\epsilon^*E, \epsilon^*F)$ is represented by the
complex with terms
$$
\bigoplus\nolimits_{n = - p + q}
\SheafHom_{\mathcal{O}_\etale}
(\epsilon^*\mathcal{E}^p, \epsilon^*\mathcal{F}^q)
$$
Thus the statement of the lemma boils down to the true fact
that the canonical map
$$
\epsilon^*\SheafHom_{\mathcal{O}_X}(\mathcal{E}, \mathcal{F})
\longrightarrow
\SheafHom_{\mathcal{O}_\etale}
(\epsilon^*\mathcal{E}, \epsilon^*\mathcal{F})
$$
is an isomorphism for any $\mathcal{O}_X$-module $\mathcal{F}$ and
finite free $\mathcal{O}_X$-module $\mathcal{E}$.
\end{proof}

\begin{lemma}
\label{lemma-base-change-RHom}
Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-compact and
quasi-separated morphism of algebraic spaces over $S$.
Let $E \in D(\mathcal{O}_X)$ be pseudo-coherent.
Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_X$-module
flat over $Y$. Then formation of
$$
Rf_*R\SheafHom(E, \mathcal{G})
$$
commutes with arbitrary base change (see proof for precise statement).
\end{lemma}

\begin{proof}
The statement means the following. Let $g : Y' \to Y$ be a morphism of
algebraic spaces and consider the base change diagram
$$
\xymatrix{
X' \ar[r]_h \ar[d]_{f'} &
X \ar[d]^f \\
Y' \ar[r]^g &
Y
}
$$
in other words $X' = Y' \times_Y X$. Set $E' = Lh^*E$ and
$\mathcal{G}' = h^*\mathcal{G}$ (here we do {\bf not} use the derived
pullback). The lemma asserts that we have
$$
Lg^*Rf_*R\SheafHom(E, \mathcal{G}) = Rf'_*R\SheafHom(E', \mathcal{G}')
$$
To prove this, note that in
Cohomology on Sites, Remark \ref{sites-cohomology-remark-fancy-base-change}
we have constructed an arrow
$$
Lg^*Rf_*R\SheafHom(E, \mathcal{G})
\longrightarrow
R(f')_*R\SheafHom(Lh^*E, Lh^*\mathcal{G})
$$
which we can compose with the map $Lh^*\mathcal{G} \to h^*\mathcal{G}$
to get a canonical map
$$
Lg^*Rf_*R\SheafHom(E, \mathcal{G}) \to Rf'_*R\SheafHom(E', \mathcal{G}')
$$
To check this map is an isomorphism we may work \'etale locally on $Y'$.
Hence we may assume $g : Y' \to Y$ is a morphism of affine schemes.
In this case, we will use the induction principle to prove this map
is always an isomorphism for any quasi-compact and quasi-separated
algebraic space $X$ over $Y$ (Lemma
\ref{lemma-induction-principle}).

\medskip\noindent
If $X$ is a scheme (for example affine), then the result holds.
Namely, $E \in D_{\textit{QCoh}}(\mathcal{O}_X)$ by
Lemma \ref{lemma-pseudo-coherent}.
Hence $E$ comes from an object of the derived category of the
underlying scheme by Lemma \ref{lemma-derived-quasi-coherent-small-etale-site}.
This complex is pseudo-coherent by Lemma \ref{lemma-descend-pseudo-coherent}.
Furthermore, the constructions $R\SheafHom$ (internal hom),
$Rf_*$ (derived pushforward), and $Lg^*$ (derived pullback)
are (in the current situation) compatible with pulling back
from the Zariski site
(Lemma \ref{lemma-descend-RHom} and
Remark \ref{remark-match-total-direct-images}).
Thus in this case the result follows from the case of schemes which is
Derived Categories of Schemes, Lemma \ref{perfect-lemma-base-change-RHom}.

\medskip\noindent
The induction step. Let $(U \subset X, f : V \to X)$ be an elementary
distinguished square with $U$, $V$, $U \times_X V$ quasi-compact such that
the result holds for the restriction of $E$ and
$\mathcal{G}$ to $U$, $V$, and $U \times_X V$. Denote $a = f|_U$,
$b = f|_V$ and $c = f|_{U \times_X V}$. Let $a' : U' \to Y'$, $b' : V' \to Y'$
and $c' : U' \times_{X'} V' \to Y'$ be the base changes of $a$, $b$, and $c$.
Note that formation of $R\SheafHom$ commutes with restriction
(Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-restriction-RHom-to-U}).
Set $H = R\SheafHom(E, \mathcal{G})$ and $H' = R\SheafHom(E', \mathcal{G}')$.
Using the distinguished triangles from relative Mayer-Vietoris
(Lemma \ref{lemma-unbounded-relative-mayer-vietoris})
we obtain a commutative diagram
$$
\xymatrix{
Lg^*Rf_* H \ar[r] \ar[d] &
Rf'_* H' \ar[d] \\
Lg^*Ra_* H|_U \oplus
Lg^*Rb_* H|_V \ar[r] \ar[d] &
Ra'_* H'|_{U'} \oplus
Rb'_* H'|_{V'} \ar[d] \\
Lg^*Rc_* H|_{U \times_X V} \ar[r] \ar[d] &
Rc'_* H'|_{U' \times_{X'} V'} \ar[d] \\
Lg^*Rf_* H[1] \ar[r] &
Rf'_* H'[1]
}
$$
Since the 2nd and 3rd horizontal arrows are isomorphisms so is the first
(Derived Categories, Lemma \ref{derived-lemma-third-isomorphism-triangle})
and the proof of the lemma is finished.
\end{proof}

\begin{lemma}
\label{lemma-compute-ext}
Let $S$ be a scheme. Let $B$ be a Noetherian algebraic space over $S$.
Let $f : X \to B$ be a morphism of algebraic spaces which is locally of
finite type and quasi-separated. Let $E \in D(\mathcal{O}_X)$
and $\mathcal{G}$ an $\mathcal{O}_X$-module. Assume
\begin{enumerate}
\item $E \in D^-_{\textit{Coh}}(\mathcal{O}_X)$, and
\item $\mathcal{G}$ is a coherent $\mathcal{O}_X$-module flat over $B$
with scheme theoretic support proper over $B$.
\end{enumerate}
Then for every $m \in \mathbf{Z}$ there exists a perfect object $K$
of $D(\mathcal{O}_B)$ and functorial maps
$$
\alpha^i_\mathcal{F} :
\text{Ext}^i_{\mathcal{O}_X}(E,
\mathcal{G} \otimes_{\mathcal{O}_X} f^*\mathcal{F})
\longrightarrow
H^i(B, K \otimes^\mathbf{L}_{\mathcal{O}_B} \mathcal{F})
$$
for $\mathcal{F}$ quasi-coherent on $B$
compatible with boundary maps (see proof)
such that $\alpha^i_\mathcal{F}$ is an isomorphism for $i \leq m$.
\end{lemma}

\begin{proof}
We may replace $X$ by a quasi-compact open neighbourhood of
the support of $\mathcal{G}$, hence we may assume $X$ is Noetherian.
In this case $X$ and $f$ are quasi-compact and quasi-separated.
Choose an approximation $P \to E$ by a perfect complex $P$ of $(X, E, -m - 1)$
(possible by Theorem \ref{theorem-approximation}).
Then the induced map
$$
\text{Ext}^i_{\mathcal{O}_X}(E,
\mathcal{G} \otimes_{\mathcal{O}_X} f^*\mathcal{F})
\longrightarrow
\text{Ext}^i_{\mathcal{O}_X}(P,
\mathcal{G} \otimes_{\mathcal{O}_X} f^*\mathcal{F})
$$
is an isomorphism for $i \leq m$. Namely, the kernel, resp.\ cokernel of this
map is a quotient, resp.\ submodule of
$$
\text{Ext}^i_{\mathcal{O}_X}(C,
\mathcal{G} \otimes_{\mathcal{O}_X} f^*\mathcal{F})
\quad\text{resp.}\quad
\text{Ext}^{i + 1}_{\mathcal{O}_X}(C,
\mathcal{G} \otimes_{\mathcal{O}_X} f^*\mathcal{F})
$$
where $C$ is the cone of $P \to E$. Since $C$ has vanishing cohomology
sheaves in degrees $\geq -m - 1$ these $\text{Ext}$-groups are zero
for $i \leq m + 1$ by
Derived Categories, Lemma \ref{derived-lemma-negative-exts}.
This reduces us to the case that
$E$ is a perfect complex which is Lemma \ref{lemma-compute-ext-perfect}.

\medskip\noindent
The statement on boundaries is explained in the proof of
Lemma \ref{lemma-compute-ext-perfect}.
\end{proof}











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