Permalink
Find file
Fetching contributors…
Cannot retrieve contributors at this time
6121 lines (5574 sloc) 225 KB
\input{preamble}
% OK, start here.
%
\begin{document}
\title{Derived Categories of Schemes}
\maketitle
\phantomsection
\label{section-phantom}
\tableofcontents
\section{Introduction}
\label{section-introduction}
\noindent
In this chapter we discuss derived categories of modules on schemes.
Most of the material discussed here can be found in
\cite{TT}, \cite{Bokstedt-Neeman}, \cite{BvdB}, and \cite{LN}.
Of course there are many other references.
\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{Derived category of quasi-coherent modules}
\label{section-derived-quasi-coherent}
\noindent
In this section we discuss the relationship between quasi-coherent
modules and all modules on a scheme $X$. A reference is
\cite[Appendix B]{TT}. By the discussion in
Schemes, Section \ref{schemes-section-quasi-coherent}
the embedding
$\QCoh(\mathcal{O}_X) \subset \textit{Mod}(\mathcal{O}_X)$
exhibits $\QCoh(\mathcal{O}_X)$ as a weak Serre subcategory of
the category of $\mathcal{O}_X$-modules. Denote
$$
D_\QCoh(\mathcal{O}_X) \subset D(\mathcal{O}_X)
$$
the subcategory of complexes whose cohomology sheaves are quasi-coherent, see
Derived Categories, Section \ref{derived-section-triangulated-sub}.
Thus we obtain a canonical functor
\begin{equation}
\label{equation-compare}
D(\QCoh(\mathcal{O}_X))
\longrightarrow
D_\QCoh(\mathcal{O}_X)
\end{equation}
see Derived Categories, Equation (\ref{derived-equation-compare}).
\begin{lemma}
\label{lemma-quasi-coherence-direct-sums}
Let $X$ be a scheme. Then $D_\QCoh(\mathcal{O}_X)$
has direct sums.
\end{lemma}
\begin{proof}
By Injectives, Lemma \ref{injectives-lemma-derived-products}
the derived category $D(\mathcal{O}_X)$ has direct sums and
they are computed by taking termwise direct sums of any representatives.
Thus it is clear that the cohomology sheaf of a direct sum is the
direct sum of the cohomology sheaves as taking direct sums is
an exact functor (in any Grothendieck abelian category). The lemma
follows as the direct sum of quasi-coherent sheaves is quasi-coherent, see
Schemes, Section \ref{schemes-section-quasi-coherent}.
\end{proof}
\noindent
We will need some information on derived limits. We warn the reader
that in the lemma below the derived limit will typically not be
an object of $D_\QCoh$.
\begin{lemma}
\label{lemma-Rlim-quasi-coherent}
Let $X$ be a scheme. Let $(K_n)$ be an inverse system of
$D_\QCoh(\mathcal{O}_X)$ with derived limit
$K = R\lim K_n$ in $D(\mathcal{O}_X)$. Assume $H^q(K_{n + 1}) \to H^q(K_n)$
is surjective for all $q \in \mathbf{Z}$ and $n \geq 1$.
Then
\begin{enumerate}
\item $H^q(K) = \lim H^q(K_n)$,
\item $R\lim H^q(K_n) = \lim H^q(K_n)$, and
\item for every affine open $U \subset X$ we have
$H^p(U, \lim H^q(K_n)) = 0$ for $p > 0$.
\end{enumerate}
\end{lemma}
\begin{proof}
Let $\mathcal{B}$ be the set of affine opens of $X$.
Since $H^q(K_n)$ is quasi-coherent we have $H^p(U, H^q(K_n)) = 0$
for $U \in \mathcal{B}$ by Cohomology of Schemes, Lemma
\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}.
Moreover, the maps $H^0(U, H^q(K_{n + 1})) \to H^0(U, H^q(K_n))$
are surjective for $U \in \mathcal{B}$ by
Schemes, Lemma \ref{schemes-lemma-equivalence-quasi-coherent}.
Part (1) follows from Cohomology, Lemma
\ref{cohomology-lemma-derived-limit-suitable-system}
whose conditions we have just verified.
Parts (2) and (3) follow from
Cohomology, Lemma \ref{cohomology-lemma-inverse-limit-is-derived-limit}.
\end{proof}
\noindent
The following lemma will help us to ``compute'' a right derived functor
on an object of $D_\QCoh(\mathcal{O}_X)$.
\begin{lemma}
\label{lemma-nice-K-injective}
Let $X$ be a scheme. Let $E$ be an object of
$D_\QCoh(\mathcal{O}_X)$. Then the canonical map
$E \to R\lim \tau_{\geq -n}E$ is an isomorphism\footnote{In particular,
$E$ has a K-injective representative as in
Cohomology, Lemma \ref{cohomology-lemma-K-injective}.}.
\end{lemma}
\begin{proof}
Denote $\mathcal{H}^i = H^i(E)$ the $i$th cohomology sheaf of $E$.
Let $\mathcal{B}$ be the set of affine open subsets of $X$. Then
$H^p(U, \mathcal{H}^i) = 0$ for all $p > 0$, all $i \in \mathbf{Z}$,
and all $U \in \mathcal{B}$, see
Cohomology of Schemes, Lemma
\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}.
Thus the lemma follows from
Cohomology, Lemma \ref{cohomology-lemma-is-limit-dimension}.
\end{proof}
\begin{lemma}
\label{lemma-application-nice-K-injective}
Let $X$ be a scheme. Let $F : \textit{Mod}(\mathcal{O}_X) \to \textit{Ab}$
be an additive functor and $N \geq 0$ an integer. Assume that
\begin{enumerate}
\item $F$ commutes with countable direct products,
\item $R^pF(\mathcal{F}) = 0$ for all $p \geq N$ and $\mathcal{F}$
quasi-coherent.
\end{enumerate}
Then for $E \in D_\QCoh(\mathcal{O}_X)$ the maps
$R^pF(E) \to R^pF(\tau_{\geq p - N + 1}E)$ are isomorphisms.
\end{lemma}
\begin{proof}
By shifting the complex we see it suffices to prove the assertion for $p = 0$.
Write $E_n = \tau_{\geq -n}E$. We have $E = R\lim E_n$, see
Lemma \ref{lemma-nice-K-injective}. Thus
$RF(E) = R\lim RF(E_n)$ in $D(\textit{Ab})$ by Injectives, Lemma
\ref{injectives-lemma-RF-commutes-with-Rlim}. Thus we have a short
exact sequence
$$
0 \to R^1\lim R^{-1}F(E_n) \to R^0F(E) \to \lim R^0F(E_n) \to 0
$$
see More on Algebra, Remark
\ref{more-algebra-remark-compare-derived-limit}.
To finish the proof we will show that the term on the left is zero
and that the term on the right equals $R^0F(E_{N - 1})$.
\medskip\noindent
We have a distinguished triangle
$$
H^{-n}(E)[n] \to E_n \to E_{n - 1} \to H^{-n}(E)[n + 1]
$$
(Derived Categories, Remark
\ref{derived-remark-truncation-distinguished-triangle})
in $D(\mathcal{O}_X)$. Since $H^{-n}(E)$ is quasi-coherent we have
$$
R^pF(H^{-n}(E)[n]) = R^{p + n}F(H^{-n}(E)) = 0
$$
for $p + n \geq N$ and
$$
R^pF(H^{-n}(E)[n + 1]) = R^{p + n + 1}F(H^{-n}(E)) = 0
$$
for $p + n + 1 \geq N$. We conclude that
$$
R^pF(E_n) \to R^pF(E_{n - 1})
$$
is an isomorphism for all $n \gg p$ and an isomorphism for
$n \geq N$ for $p = 0$. Thus the systems $R^pF(E_n)$ all
satisfy the ML condition and $R^1\lim$ gives zero (see discussion
in More on Algebra, Section \ref{more-algebra-section-Rlim}).
Moreover, the system $R^0F(\tau_{\geq - n}E)$ is constant starting
with $n = N - 1$ as desired.
\end{proof}
\noindent
The following lemma is the key ingredient to many of the
results in this chapter.
\begin{lemma}
\label{lemma-affine-compare-bounded}
Let $X = \Spec(A)$ be an affine scheme. All the functors in the diagram
$$
\xymatrix{
D(\QCoh(\mathcal{O}_X)) \ar[rr]_{(\ref{equation-compare})}
& &
D_\QCoh(\mathcal{O}_X) \ar[ld]^{R\Gamma(X, -)} \\
& D(A) \ar[lu]^{\widetilde{\ \ }}
}
$$
are equivalences of triangulated categories. Moreover, for $E$ in
$D_\QCoh(\mathcal{O}_X)$ we have $H^0(X, E) = H^0(X, H^0(E))$.
\end{lemma}
\begin{proof}
The functor $R\Gamma(X, -)$ gives a functor
$D(\mathcal{O}_X) \to D(A)$ and hence by restriction a functor
\begin{equation}
\label{equation-back}
R\Gamma(X, -) : D_\QCoh(\mathcal{O}_X) \longrightarrow D(A).
\end{equation}
We will show this functor is quasi-inverse to (\ref{equation-compare})
via the equivalence between quasi-coherent modules on $X$ and
the category of $A$-modules.
\medskip\noindent
Elucidation. Denote $(Y, \mathcal{O}_Y)$ the one point space with sheaf
of rings given by $A$. Denote
$\pi : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$
the obvious morphism of ringed spaces.
Then $R\Gamma(X, -)$ can be identified with $R\pi_*$ and the functor
(\ref{equation-compare}) via the equivalence
$\textit{Mod}(\mathcal{O}_Y) = \text{Mod}_A = \QCoh(\mathcal{O}_X)$
can be identified with $L\pi^* = \pi^* = \widetilde{\ }$ (see
Modules, Lemma \ref{modules-lemma-construct-quasi-coherent-sheaves} and
Schemes, Lemmas \ref{schemes-lemma-compare-constructions} and
\ref{schemes-lemma-equivalence-quasi-coherent}). Thus the functors
$$
\xymatrix{
D(A) \ar@<1ex>[r] & D_\QCoh(\mathcal{O}_X) \ar@<1ex>[l]
}
$$
are adjoint (by Cohomology, Lemma \ref{cohomology-lemma-adjoint}). In
particular we obtain canonical adjunction mappings
$$
a : \widetilde{R\Gamma(X, E)} \longrightarrow E
$$
for $E$ in $D(\mathcal{O}_X)$ and
$$
b : M^\bullet \longrightarrow R\Gamma(X, \widetilde{M^\bullet})
$$
for $M^\bullet$ a complex of $A$-modules.
\medskip\noindent
Let $E$ be an object of $D_\QCoh(\mathcal{O}_X)$. We may apply
Lemma \ref{lemma-application-nice-K-injective}
to the functor $F(-) = \Gamma(X, -)$
with $N = 1$ by Cohomology of Schemes, Lemma
\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}.
Hence
$$
R^0\Gamma(X, E) = R^0\Gamma(X, \tau_{\geq 0}E) = \Gamma(X, H^0(E))
$$
(the last equality by definition of the canonical truncation).
Using this we will show that the adjunction mappings $a$ and $b$
induce isomorphisms $H^0(a)$ and $H^0(b)$. Thus $a$ and $b$
are quasi-isomorphisms (as the statement is invariant under shifts)
and the lemma is proved.
\medskip\noindent
In both cases we use that $\widetilde{\ }$ is an exact functor
(Schemes, Lemma \ref{schemes-lemma-spec-sheaves}). Namely, this
implies that
$$
H^0\left(\widetilde{R\Gamma(X, E)}\right) = \widetilde{R^0\Gamma(X, E)}
= \widetilde{\Gamma(X, H^0(E))}
$$
which is equal to $H^0(E)$ because $H^0(E)$ is quasi-coherent. Thus
$H^0(a)$ is an isomorphism. For the other direction we have
$$
H^0(R\Gamma(X, \widetilde{M^\bullet})) =
R^0\Gamma(X, \widetilde{M^\bullet}) =
\Gamma(X, H^0(\widetilde{M^\bullet})) =
\Gamma(X, \widetilde{H^0(M^\bullet)}) = H^0(M^\bullet)
$$
which proves that $H^0(b)$ is an isomorphism.
\end{proof}
\begin{lemma}
\label{lemma-affine-K-flat}
Let $X = \Spec(A)$ be an affine scheme. If $K^\bullet$ is a K-flat
complex of $A$-modules, then $\widetilde{K^\bullet}$ is a K-flat
complex of $\mathcal{O}_X$-modules.
\end{lemma}
\begin{proof}
By More on Algebra, Lemma \ref{more-algebra-lemma-base-change-K-flat}
we see that $K^\bullet \otimes_A A_\mathfrak p$ is a K-flat complex
of $A_\mathfrak p$-modules for every $\mathfrak p \in \Spec(A)$.
Hence we conclude from
Cohomology, Lemma \ref{cohomology-lemma-check-K-flat-stalks}
(and
Schemes, Lemma \ref{schemes-lemma-spec-sheaves})
that $\widetilde{K^\bullet}$ is K-flat.
\end{proof}
\begin{lemma}
\label{lemma-quasi-coherence-pullback}
Let $f : Y \to X$ be a morphism of schemes.
\begin{enumerate}
\item The functor $Lf^*$ sends $D_\QCoh(\mathcal{O}_X)$
into $D_\QCoh(\mathcal{O}_Y)$.
\item If $X$ and $Y$ are affine and $f$ is given by the ring map
$A \to B$, then the diagram
$$
\xymatrix{
D(B) \ar[r] & D_\QCoh(\mathcal{O}_Y) \\
D(A) \ar[r] \ar[u]^{- \otimes_A^\mathbf{L} B} &
D_\QCoh(\mathcal{O}_X) \ar[u]_{Lf^*}
}
$$
commutes.
\end{enumerate}
\end{lemma}
\begin{proof}
We first prove the diagram
$$
\xymatrix{
D(B) \ar[r] & D(\mathcal{O}_Y) \\
D(A) \ar[r] \ar[u]^{- \otimes_A^\mathbf{L} B} &
D(\mathcal{O}_X) \ar[u]_{Lf^*}
}
$$
commutes. This is clear from Lemma \ref{lemma-affine-K-flat} and
the constructions of the functors in question. To see (1) let
$E$ be an object of $D_\QCoh(\mathcal{O}_X)$. To see that
$Lf^*E$ has quasi-coherent cohomology sheaves we may work locally on $X$.
Note that $Lf^*$ is compatible with restricting to open subschemes.
Hence we can assume that $f$ is a morphism of affine schemes as in (2).
Then we can apply Lemma \ref{lemma-affine-compare-bounded} to see that
$E$ comes from a complex of $A$-modules. By the commutativity of the first
diagram of the proof the same holds for $Lf^*E$ and we conclude (1) is true.
\end{proof}
\begin{lemma}
\label{lemma-quasi-coherence-tensor-product}
Let $X$ be a scheme.
\begin{enumerate}
\item For objects $K, L$ of $D_\QCoh(\mathcal{O}_X)$
the derived tensor product $K \otimes^\mathbf{L}_{\mathcal{O}_X} L$ is in
$D_\QCoh(\mathcal{O}_X)$.
\item If $X = \Spec(A)$ is affine then
$$
\widetilde{M^\bullet} \otimes_{\mathcal{O}_X}^\mathbf{L} \widetilde{K^\bullet}
=
\widetilde{M^\bullet \otimes_A^\mathbf{L} K^\bullet}
$$
for any pair of complexes of $A$-modules $K^\bullet$, $M^\bullet$.
\end{enumerate}
\end{lemma}
\begin{proof}
The equality of (2) follows immediately from Lemma \ref{lemma-affine-K-flat}
and the construction of the derived tensor product.
To see (1) let $K, L$ be objects of $D_\QCoh(\mathcal{O}_X)$.
To check that $K \otimes^\mathbf{L} L$ is in
$D_\QCoh(\mathcal{O}_X)$ we may work locally on $X$, hence
we may assume $X = \Spec(A)$ is affine. By
Lemma \ref{lemma-affine-compare-bounded} we may represent
$K$ and $L$ by complexes of $A$-modules. Then part (2) implies
the result.
\end{proof}
\section{Total direct image}
\label{section-total-direct-image}
\noindent
The following lemma is the analogue of
Cohomology of Schemes, Lemma
\ref{coherent-lemma-quasi-coherence-higher-direct-images}.
\begin{lemma}
\label{lemma-quasi-coherence-direct-image}
Let $f : X \to S$ be a morphism of schemes.
Assume that $f$ is quasi-separated and quasi-compact.
\begin{enumerate}
\item The functor $Rf_*$ sends $D_\QCoh(\mathcal{O}_X)$
into $D_\QCoh(\mathcal{O}_S)$.
\item If $S$ is quasi-compact, there exists an integer $N = N(X, S, f)$
such that for an object $E$ of $D_\QCoh(\mathcal{O}_X)$
with $H^m(E) = 0$ for $m > 0$ we have
$H^m(Rf_*E) = 0$ for $m \geq N$.
\item In fact, if $S$ is quasi-compact we can find $N = N(X, S, f)$
such that for every morphism of schemes $S' \to S$
the same conclusion holds for the functor $R(f')_*$
where $f' : X' \to S'$ is the base change of $f$.
\end{enumerate}
\end{lemma}
\begin{proof}
Let $E$ be an object of $D_\QCoh(\mathcal{O}_X)$. To prove (1) we have to
show that $Rf_*E$ has quasi-coherent cohomology sheaves. The question is local
on $S$, hence we may assume $S$ is quasi-compact. Pick $N = N(X, S, f)$ as in
Cohomology of Schemes, Lemma
\ref{coherent-lemma-quasi-coherence-higher-direct-images}.
Thus $R^pf_*\mathcal{F} = 0$ for all quasi-coherent $\mathcal{O}_X$-modules
$\mathcal{F}$ and all $p \geq N$ and the same remains true after base change.
\medskip\noindent
First, assume $E$ is bounded below. We will show (1) and (2) and (3) hold
for such $E$ with our choice of $N$. In this case we can for example use the
spectral sequence
$$
R^pf_*H^q(E) \Rightarrow R^{p + q}f_*E
$$
(Derived Categories, Lemma \ref{derived-lemma-two-ss-complex-functor}),
the quasi-coherence of $R^pf_*H^q(E)$, and the vanishing of $R^pf_*H^q(E)$
for $p \geq N$ to see that (1), (2), and (3) hold in this case.
\medskip\noindent
Next we prove (2) and (3). Say $H^m(E) = 0$ for $m > 0$.
Let $U \subset S$ be affine open. By Cohomology of Schemes, Lemma
\ref{coherent-lemma-quasi-coherence-higher-direct-images-application}
and our choice of $N$
we have $H^p(f^{-1}(U), \mathcal{F}) = 0$ for $p \geq N$
and any quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$.
Hence we may apply Lemma \ref{lemma-application-nice-K-injective}
to the functor $\Gamma(f^{-1}(U), -)$ to see that
$$
R\Gamma(U, Rf_*E) = R\Gamma(f^{-1}(U), E)
$$
has vanishing cohomology in degrees $\geq N$. Since this holds for
all $U \subset S$ affine open we conclude that $H^m(Rf_*E) = 0$
for $m \geq N$.
\medskip\noindent
Next, we prove (1) in the general case. Recall that there is a
distinguished triangle
$$
\tau_{\leq -n - 1}E \to E \to \tau_{\geq -n}E \to
(\tau_{\leq -n - 1}E)[1]
$$
in $D(\mathcal{O}_X)$, see Derived Categories, Remark
\ref{derived-remark-truncation-distinguished-triangle}.
By (2) we see that $Rf_*\tau_{\leq -n - 1}E$
has vanishing cohomology sheaves in degrees $\geq -n + N$.
Thus, given an integer $q$ we see that $R^qf_*E$ is equal
to $R^qf_*\tau_{\geq -n}E$ for some $n$ and the result
above applies.
\end{proof}
\begin{lemma}
\label{lemma-quasi-coherence-pushforward-direct-sums}
Let $f : X \to S$ be a quasi-separated and quasi-compact morphism of
schemes. Then
$Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_S)$
commutes with direct sums.
\end{lemma}
\begin{proof}
Let $E_i$ be a family of objects of $D_\QCoh(\mathcal{O}_X)$
and set $E = \bigoplus E_i$. We want to show that the map
$$
\bigoplus Rf_*E_i \longrightarrow Rf_*E
$$
is an isomorphism. We will show it induces an isomorphism on
cohomology sheaves in degree $0$ which will imply the lemma.
Choose an integer $N$ as in Lemma \ref{lemma-quasi-coherence-direct-image}.
Then $R^0f_*E = R^0f_*\tau_{\geq -N}E$ and
$R^0f_*E_i = R^0f_*\tau_{\geq -N}E_i$ by the lemma cited. Observe that
$\tau_{\geq -N}E = \bigoplus \tau_{\geq -N}E_i$.
Thus we may assume all of the $E_i$ have vanishing cohomology
sheaves in degrees $< -N$. Next we use the spectral sequences
$$
R^pf_*H^q(E) \Rightarrow R^{p + q}f_*E
\quad\text{and}\quad
R^pf_*H^q(E_i) \Rightarrow R^{p + q}f_*E_i
$$
(Derived Categories, Lemma \ref{derived-lemma-two-ss-complex-functor})
to reduce to the case of a direct sum of quasi-coherent sheaves.
This case is handled by
Cohomology of Schemes, Lemma \ref{coherent-lemma-colimit-cohomology}.
\end{proof}
\section{Affine morphisms}
\label{section-affine-morphisms}
\noindent
In this section we collect some information about pushforward
along an affine morphism of schemes.
\begin{lemma}
\label{lemma-affine-morphism}
Let $f : X \to S$ be an affine morphism of schemes.
Then
$Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_S)$
reflects isomorphisms.
\end{lemma}
\begin{proof}
The statement means that a morphism $\alpha : E \to F$ of
$D_\QCoh(\mathcal{O}_X)$ is an isomorphism if
$Rf_*\alpha$ is an isomorphism. We may check this on cohomology sheaves.
In particular, the question is local on $S$. Hence we may assume $S$
and therefore $X$ is affine. In this case the statement is clear from
the description of the derived categories
$D_\QCoh(\mathcal{O}_X)$ and
$D_\QCoh(\mathcal{O}_S)$ given in
Lemma \ref{lemma-affine-compare-bounded}.
Some details omitted.
\end{proof}
\begin{lemma}
\label{lemma-affine-morphism-pull-push}
Let $f : X \to S$ be an affine morphism of schemes.
For $E$ in $D_\QCoh(\mathcal{O}_S)$ we have
$Rf_* Lf^* E = E \otimes^\mathbf{L}_{\mathcal{O}_S} f_*\mathcal{O}_X$.
\end{lemma}
\begin{proof}
Since $f$ is affine the map $f_*\mathcal{O}_X \to Rf_*\mathcal{O}_X$
is an isomorphism
(Cohomology of Schemes, Lemma \ref{coherent-lemma-relative-affine-vanishing}).
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 $S$. Hence we may assume
$S$ and therefore $X$ is affine. In this case the statement is clear from
the description of the derived categories
$D_\QCoh(\mathcal{O}_X)$ and
$D_\QCoh(\mathcal{O}_S)$ and the functor $Lf^*$ given in
Lemmas \ref{lemma-affine-compare-bounded} and
\ref{lemma-quasi-coherence-pullback}.
Some details omitted.
\end{proof}
\noindent
Let $Y$ be a scheme. Let $\mathcal{A}$ be a sheaf of $\mathcal{O}_Y$-algebras.
We will denote $D_\QCoh(\mathcal{A})$ the inverse image of
$D_\QCoh(\mathcal{O}_X)$ under the restriction functor
$D(\mathcal{A}) \to D(\mathcal{O}_X)$. In other words, $K \in D(\mathcal{A})$
is in $D_\QCoh(\mathcal{A})$ if and only if its cohomology sheaves are
quasi-coherent as $\mathcal{O}_X$-modules. If $\mathcal{A}$ is quasi-coherent
itself this is the same as asking the cohomology sheaves to be quasi-coherent
as $\mathcal{A}$-modules, see
Morphisms, Lemma \ref{morphisms-lemma-affine-equivalence-modules}.
\begin{lemma}
\label{lemma-affine-morphism-equivalence}
Let $f : X \to Y$ be an affine morphism of schemes. Then $f_*$ induces
an equivalence
$$
\Phi : D_\QCoh(\mathcal{O}_X) \longrightarrow D_\QCoh(f_*\mathcal{O}_X)
$$
whose composition with $D_\QCoh(f_*\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$
is $Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$.
\end{lemma}
\begin{proof}
Recall that $Rf_*$ is computed on an object $K \in D_\QCoh(\mathcal{O}_X)$
by choosing a K-injective complex $\mathcal{I}^\bullet$ of
$\mathcal{O}_X$-modules representing $K$ and taking $f_*\mathcal{I}^\bullet$.
Thus we let $\Phi(K)$ be the complex $f_*\mathcal{I}^\bullet$
viewed as a complex of $f_*\mathcal{O}_X$-modules.
Denote $g : (X, \mathcal{O}_X) \to (Y, f_*\mathcal{O}_X)$ the
obvious morphism of ringed spaces. Then $g$ is a flat morphism of
ringed spaces (see below for a description of the stalks) and
$\Phi$ is the restriction of $Rg_*$ to $D_\QCoh(\mathcal{O}_X)$.
We claim that $Lg^*$ is a quasi-inverse. First, observe that
$Lg^*$ sends $D_\QCoh(f_*\mathcal{O}_X)$ into $D_\QCoh(\mathcal{O}_X)$
because $g^*$ transforms quasi-coherent modules into quasi-coherent
modules (Modules, Lemma \ref{modules-lemma-pullback-quasi-coherent}).
To finish the proof it suffices to show that
the adjunction mappings
$$
Lg^*\Phi(K) = Lg^*Rg_*K \to K
\quad\text{and}\quad
M \to Rg_*Lg^*M = \Phi(Lg^*M)
$$
are isomorphisms for $K \in D_\QCoh(\mathcal{O}_X)$ and
$M \in D_\QCoh(f_*\mathcal{O}_X)$. This is a local question, hence
we may assume $Y$ and therefore $X$ are affine.
\medskip\noindent
Assume $Y = \Spec(B)$ and $X = \Spec(A)$. Let
$\mathfrak p = x \in \Spec(A) = X$ be a point mapping to
$\mathfrak q = y \in \Spec(B) = Y$. Then
$(f_*\mathcal{O}_X)_y = A_\mathfrak q$ and $\mathcal{O}_{X, x} = A_\mathfrak p$
hence $g$ is flat. Hence $g^*$ is exact and $H^i(Lg^*M) = g^*H^i(M)$
for any $M$ in $D(f_*\mathcal{O}_X)$.
For $K \in D_\QCoh(\mathcal{O}_X)$ we see that
$$
H^i(\Phi(K)) = H^i(Rf_*K) = f_*H^i(K)
$$
by the vanishing of higher direct images
(Cohomology of Schemes, Lemma \ref{coherent-lemma-relative-affine-vanishing})
and Lemma \ref{lemma-application-nice-K-injective}.
Thus it suffice to show that
$$
g^*g_*\mathcal{F} \to \mathcal{F}
\quad\text{and}\quad
\mathcal{G} \to g_*g^*\mathcal{F}
$$
are isomorphisms where $\mathcal{F}$ is
a quasi-coherent $\mathcal{O}_X$-module and $\mathcal{G}$ is
a quasi-coherent $f_*\mathcal{O}_X$-module. This follows from
Morphisms, Lemma \ref{morphisms-lemma-affine-equivalence-modules}.
\end{proof}
\section{The coherator}
\label{section-coherator}
\noindent
Let $X$ be a scheme. The {\it coherator} is a functor
$$
Q_X :
\textit{Mod}(\mathcal{O}_X)
\longrightarrow
\QCoh(\mathcal{O}_X)
$$
which is right adjoint to the inclusion functor
$\QCoh(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}_X)$.
It exists for any scheme $X$ and moreover the adjunction mapping
$Q_X(\mathcal{F}) \to \mathcal{F}$ is an isomorphism for every
quasi-coherent module $\mathcal{F}$, see
Properties, Proposition \ref{properties-proposition-coherator}.
Since $Q_X$ is left exact (as a right adjoint) we can consider its
right derived extension
$$
RQ_X :
D(\mathcal{O}_X)
\longrightarrow
D(\QCoh(\mathcal{O}_X)).
$$
Since $Q_X$ is right adjoint to the inclusion functor
$\QCoh(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}_X)$
we see that $RQ_X$ is right adjoint to the canonical functor
$D(\QCoh(\mathcal{O}_X)) \to D(\mathcal{O}_X)$ by
Derived Categories, Lemma \ref{derived-lemma-derived-adjoint-functors}.
\medskip\noindent
In this section we will study the functor $RQ_X$. In
Section \ref{section-better-coherator}
we will study the (closely related) right adjoint to the inclusion functor
$D_\QCoh(\mathcal{O}_X) \to D(\mathcal{O}_X)$ (when it exists).
\begin{lemma}
\label{lemma-affine-pushforward}
Let $f : X \to Y$ be an affine morphism of schemes.
Then $f_*$ defines a derived functor
$f_* : D(\QCoh(\mathcal{O}_X)) \to D(\QCoh(\mathcal{O}_Y))$.
This functor has the property that
$$
\xymatrix{
D(\QCoh(\mathcal{O}_X)) \ar[d]_{f_*} \ar[r] &
D_\QCoh(\mathcal{O}_X) \ar[d]^{Rf_*} \\
D(\QCoh(\mathcal{O}_Y)) \ar[r] &
D_\QCoh(\mathcal{O}_Y)
}
$$
commutes.
\end{lemma}
\begin{proof}
The functor
$f_* : \QCoh(\mathcal{O}_X) \to \QCoh(\mathcal{O}_Y)$
is exact, see
Cohomology of Schemes, Lemma \ref{coherent-lemma-relative-affine-vanishing}.
Hence $f_*$ defines a derived functor
$f_* : D(\QCoh(\mathcal{O}_X)) \to D(\QCoh(\mathcal{O}_Y))$
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. As the statement is invariant under shifts it
suffices to show that
$H^0(f_*(\mathcal{F}^\bullet)) \to R^0f_*\mathcal{F}^\bullet$
is an isomorphism. The statement is local on $Y$ hence we
may assume $Y$ affine. By
Lemma \ref{lemma-quasi-coherence-direct-image}
we have $R^0f_*\mathcal{F}^\bullet = R^0f_*\tau_{\geq -n}\mathcal{F}^\bullet$
for all sufficiently large $n$. Thus we may assume $\mathcal{F}^\bullet$
bounded below. As each $\mathcal{F}^n$ is $f_*$-acyclic by
Cohomology of Schemes, Lemma \ref{coherent-lemma-relative-affine-vanishing}
we see that
$f_*\mathcal{F}^\bullet \to Rf_*\mathcal{F}^\bullet$
is a quasi-isomorphism by
Leray's acyclicity lemma (Derived Categories, Lemma
\ref{derived-lemma-leray-acyclicity}).
\end{proof}
\begin{lemma}
\label{lemma-flat-pushforward-coherator}
Let $f : X \to Y$ be a morphism of schemes. Assume $f$ is
quasi-compact, quasi-separated, and flat. Then, denoting
$$
\Phi : D(\QCoh(\mathcal{O}_X)) \to D(\QCoh(\mathcal{O}_Y))
$$
the right derived functor of
$f_* : \QCoh(\mathcal{O}_X) \to \QCoh(\mathcal{O}_Y)$
we have $RQ_Y \circ Rf_* = \Phi \circ RQ_X$.
\end{lemma}
\begin{proof}
We will prove this by showing that $RQ_Y \circ Rf_*$ and $\Phi \circ RQ_X$
are right adjoint to the same functor
$D(\QCoh(\mathcal{O}_Y)) \to D(\mathcal{O}_X)$.
\medskip\noindent
Since $f$ is quasi-compact and quasi-separated, we see that
$f_*$ preserves quasi-coherence, see
Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent}.
Recall that $\QCoh(\mathcal{O}_X)$ is a Grothendieck abelian category
(Properties, Proposition \ref{properties-proposition-coherator}).
Hence any $K$ in $D(\QCoh(\mathcal{O}_X))$
can be represented by a K-injective complex $\mathcal{I}^\bullet$
of $\QCoh(\mathcal{O}_X)$, see
Injectives, Theorem
\ref{injectives-theorem-K-injective-embedding-grothendieck}.
Then we can define $\Phi(K) = f_*\mathcal{I}^\bullet$.
\medskip\noindent
Since $f$ is flat, the functor $f^*$ is exact. Hence $f^*$ defines
$f^* : D(\mathcal{O}_Y) \to D(\mathcal{O}_X)$ and also
$f^* : D(\QCoh(\mathcal{O}_Y)) \to D(\QCoh(\mathcal{O}_X))$.
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}.
Similarly, the functor
$f^* : D(\QCoh(\mathcal{O}_Y)) \to D(\QCoh(\mathcal{O}_X))$
is left adjoint to
$\Phi : D(\QCoh(\mathcal{O}_X)) \to D(\QCoh(\mathcal{O}_Y))$
by Derived Categories, Lemma \ref{derived-lemma-derived-adjoint-functors}.
\medskip\noindent
Let $A$ be an object of $D(\QCoh(\mathcal{O}_Y))$ and
$E$ an object of $D(\mathcal{O}_X)$. Then
\begin{align*}
\Hom_{D(\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(\QCoh(\mathcal{O}_X))}(f^*A, RQ_X(E)) \\
& =
\Hom_{D(\QCoh(\mathcal{O}_Y))}(A, \Phi(RQ_X(E)))
\end{align*}
This implies what we want.
\end{proof}
\begin{lemma}
\label{lemma-affine-coherator}
Let $X = \Spec(A)$ be an affine scheme. Then
\begin{enumerate}
\item $Q_X : \textit{Mod}(\mathcal{O}_X) \to \QCoh(\mathcal{O}_X)$
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 $RQ_X : D(\mathcal{O}_X) \to D(\QCoh(\mathcal{O}_X))$
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 $D_\QCoh(\mathcal{O}_X)$ the functor
$RQ_X$ defines a quasi-inverse to (\ref{equation-compare}).
\end{enumerate}
\end{lemma}
\begin{proof}
The functor $Q_X$ is the functor
$$
\mathcal{F} \mapsto \widetilde{\Gamma(X, \mathcal{F})}
$$
by Schemes, Lemma \ref{schemes-lemma-compare-constructions}.
This immediately implies (1) and (2). The third assertion
follows from (the proof of)
Lemma \ref{lemma-affine-compare-bounded}.
\end{proof}
\begin{definition}
\label{definition-supported-on}
Let $X$ be a scheme. 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}
\begin{proposition}
\label{proposition-quasi-compact-affine-diagonal}
Let $X$ be a quasi-compact scheme with affine diagonal.
Then the functor (\ref{equation-compare})
$$
D(\QCoh(\mathcal{O}_X))
\longrightarrow
D_\QCoh(\mathcal{O}_X)
$$
is an equivalence with quasi-inverse given by $RQ_X$.
\end{proposition}
\begin{proof}
In this proof we will denote $i_X : D(\QCoh(\mathcal{O}_X))
\to D_\QCoh(\mathcal{O}_X)$ the functor of the lemma.
Let $E$ be an object of $D_\QCoh(\mathcal{O}_X)$ and
let $A$ be an object of $D(\QCoh(\mathcal{O}_X))$.
We have to show that the adjunction maps
$$
RQ_X(i_X(A)) \to A
\quad\text{and}\quad
E \to i_X(RQ_X(E))
$$
are isomorphisms. We will prove this by induction on $n$:
the smallest integer $n \geq 0$ such that $E$ and $i_X(A)$
are supported on a closed subset of $X$ which
is contained in the union of $n$ affine opens of $X$.
\medskip\noindent
Base case: $n = 0$. In this case $E = 0$, hence the map
$E \to i_X(RQ_X(E))$ is an isomorphism. Similarly $i_X(A) = 0$.
Thus the cohomology sheaves of $i_X(A)$ are zero. Since the inclusion
functor $\QCoh(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}_X)$
is fully faithful and exact, we conclude that the cohomology
objects of $A$ are zero, i.e., $A = 0$ and
$RQ_X(i_X(A)) \to A$ is an isomorphism as well.
\medskip\noindent
Induction step. Suppose that $E$ and $i_X(A)$ are supported on a
closed subset $T$ of $X$ contained in $U_1 \cup \ldots \cup U_n$
with $U_i \subset X$ affine open. Set $U = U_n$. The inclusion morphism
$j : U \to X$ is flat and affine
(Morphisms, Lemma \ref{morphisms-lemma-affine-permanence}).
Consider the distinguished triangles
$$
A \to j_*(A|_U) \to A' \to A[1]
\quad\text{and}\quad
E \to Rj_*(E|_U) \to E' \to E[1]
$$
where $j_*$ is as in Lemma \ref{lemma-affine-pushforward}.
Note that $E \to Rj_*(E|_U)$ is a quasi-isomorphism over $U = U_n$.
Since $i_X \circ j_* = Rj_* \circ i_U$ by Lemma \ref{lemma-affine-pushforward}
and since $i_X(A)|_U = i_U(A|_U)$
we see that $i_X(A) \to i_X(j_*(A|_U))$ is a quasi-isomorphism over $U$.
Hence $i_X(A')$ and $E'$ are supported on the closed
subset $T \setminus U$ of $X$ which is contained in
$U_1 \cup \ldots \cup U_{n - 1}$.
By induction hypothesis the statement is true for $A'$ and $E'$. By
Derived Categories, Lemma \ref{derived-lemma-third-isomorphism-triangle}
it suffices to prove the maps
$$
RQ_X(i_X(j_*(A|_U))) \to j_*(A|_U)
\quad\text{and}\quad
Rj_*(E|_U) \to i_X(RQ_X(Rj_*E|_U))
$$
are isomorphisms. By
Lemmas \ref{lemma-affine-pushforward} and
\ref{lemma-flat-pushforward-coherator} we have
$$
RQ_X(i_X(j_*(A|_U))) = RQ_X(Rj_*(i_U(A|_U))) = j_*RQ_U(i_U(A|_U))
$$
and
$$
i_X(RQ_X(Rj_*(E|_U))) = i_X(j_*RQ_U(E|_U)) = Rj_*(i_U(RQ_U(E|_U)))
$$
Finally, the maps
$$
RQ_U(i_U(A|_U)) \to A|_U
\quad\text{and}\quad
E|_U \to i_U(RQ_U(E|_U))
$$
are isomorphisms by Lemma \ref{lemma-affine-coherator}. The result follows.
\end{proof}
\begin{remark}
\label{remark-argument-proves}
Analyzing the proof of
Proposition \ref{proposition-quasi-compact-affine-diagonal}
we see that we have shown the following.
Let $X$ be a quasi-compact and quasi-separated scheme. Suppose that
for every affine open $U \subset X$ the right derived functor
$$
\Phi : D(\QCoh(\mathcal{O}_U)) \to D(\QCoh(\mathcal{O}_X))
$$
of the left exact functor
$j_* : \QCoh(\mathcal{O}_U) \to \QCoh(\mathcal{O}_X)$
fits into a commutative diagram
$$
\xymatrix{
D(\QCoh(\mathcal{O}_U)) \ar[d]_\Phi \ar[r]_{i_U} &
D_\QCoh(\mathcal{O}_U) \ar[d]^{Rj_*} \\
D(\QCoh(\mathcal{O}_X)) \ar[r]^{i_X} &
D_\QCoh(\mathcal{O}_X)
}
$$
Then the functor (\ref{equation-compare})
$$
D(\QCoh(\mathcal{O}_X))
\longrightarrow
D_\QCoh(\mathcal{O}_X)
$$
is an equivalence with quasi-inverse given by $RQ_X$.
\end{remark}
\begin{lemma}
\label{lemma-direct-image-coherator}
Let $f : X \to Y$ be a morphism of schemes.
Assume $X$ and $Y$ are quasi-compact and have affine diagonal.
Then, denoting
$$
\Phi : D(\QCoh(\mathcal{O}_X)) \to D(\QCoh(\mathcal{O}_Y))
$$
the right derived functor of
$f_* : \QCoh(\mathcal{O}_X) \to \QCoh(\mathcal{O}_Y)$
the diagram
$$
\xymatrix{
D(\QCoh(\mathcal{O}_X)) \ar[d]_\Phi \ar[r] &
D_\QCoh(\mathcal{O}_X) \ar[d]^{Rf_*} \\
D(\QCoh(\mathcal{O}_Y)) \ar[r] &
D_\QCoh(\mathcal{O}_Y)
}
$$
is commutative.
\end{lemma}
\begin{proof}
Observe that the horizontal arrows in the diagram are
equivalences of categories by
Proposition \ref{proposition-quasi-compact-affine-diagonal}.
Hence we can identify these categories (and similarly for
other quasi-compact schemes with affine diagonal).
The statement of the lemma is that the canonical map
$\Phi(K) \to Rf_*(K)$ is an isomorphism for all $K$ in
$D(\QCoh(\mathcal{O}_X))$. Note that if $K_1 \to K_2 \to K_3 \to K_1[1]$
is a distinguished triangle in $D(\QCoh(\mathcal{O}_X))$ and
the statement is true for two-out-of-three, then it is true
for the third.
\medskip\noindent
Let $U \subset X$ be an affine open. Since the diagonal of $X$ is affine,
the inclusion morphism $j : U \to X$
is affine (Morphisms, Lemma \ref{morphisms-lemma-affine-permanence}).
Similarly, the composition $g = f \circ j : U \to Y$ is affine.
Let $\mathcal{I}^\bullet$ be a K-injective complex in $\QCoh(\mathcal{O}_U)$.
Since $j_* : \QCoh(\mathcal{O}_U) \to \QCoh(\mathcal{O}_X)$
has an exact left adjoint
$j^* : \QCoh(\mathcal{O}_X) \to \QCoh(\mathcal{O}_U)$
we see that $j_*\mathcal{I}^\bullet$ is a K-injective complex
in $\QCoh(\mathcal{O}_X)$, see
Derived Categories, Lemma \ref{derived-lemma-adjoint-preserve-K-injectives}.
It follows that
$$
\Phi(j_*\mathcal{I}^\bullet) =
f_*j_*\mathcal{I}^\bullet =
g_*\mathcal{I}^\bullet
$$
By Lemma \ref{lemma-affine-pushforward} we see that
$j_*\mathcal{I}^\bullet$ represents $Rj_*\mathcal{I}^\bullet$ and
$g_*\mathcal{I}^\bullet$ represents $Rg_*\mathcal{I}^\bullet$.
On the other hand, we have $Rf_* \circ Rj_* = Rg_*$.
Hence $f_*j_*\mathcal{I}^\bullet$ represents $Rf_*(j_*\mathcal{I}^\bullet)$.
We conclude that the lemma is true for any complex
of the form $j_*\mathcal{G}^\bullet$ with $\mathcal{G}^\bullet$
a complex of quasi-coherent modules on $U$. (Note that if
$\mathcal{G}^\bullet \to \mathcal{I}^\bullet$ is a quasi-isomorphism,
then $j_*\mathcal{G}^\bullet \to j_*\mathcal{I}^\bullet$ is a
quasi-isomorphism as well since $j_*$ is an exact functor
on quasi-coherent modules.)
\medskip\noindent
Let $\mathcal{F}^\bullet$ be a complex of quasi-coherent
$\mathcal{O}_X$-modules. Let $T \subset X$ be a closed subset
such that the support of $\mathcal{F}^p$ is contained in $T$
for all $p$. We will use induction on the minimal number $n$
of affine opens $U_1, \ldots, U_n$ such that
$T \subset U_1 \cup \ldots \cup U_n$. The base case $n = 0$ is trivial.
If $n \geq 1$, then set $U = U_1$ and denote $j : U \to X$ the
open immersion as above. We consider the map of complexes
$c : \mathcal{F}^\bullet \to j_*j^*\mathcal{F}^\bullet$.
We obtain two short exact sequences of complexes:
$$
0 \to \Ker(c) \to \mathcal{F}^\bullet \to \Im(c) \to 0
$$
and
$$
0 \to \Im(c) \to j_*j^*\mathcal{F}^\bullet \to \Coker(c) \to 0
$$
The complexes $\Ker(c)$ and $\Coker(c)$ are supported
on $T \setminus U \subset U_2 \cup \ldots \cup U_n$ and the result
holds for them by induction. The result holds for
$j_*j^*\mathcal{F}^\bullet$ by the discussion in the preceding
paragraph. We conclude by looking at the distinguished triangles
associated to the short exact sequences and using the initial
remark of the proof.
\end{proof}
\begin{remark}[Warning]
\label{remark-warning-coherator}
Let $X$ be a quasi-compact scheme with affine diagonal. Even though we know
that $D(\QCoh(\mathcal{O}_X)) = D_\QCoh(\mathcal{O}_X)$ by
Proposition \ref{proposition-quasi-compact-affine-diagonal}
strange things can
happen and it is easy to make mistakes with this material. One pitfall
is to carelessly assume that this equality means derived functors are the same.
For example, suppose we have a quasi-compact open $U \subset X$. Then we can
consider the higher right derived functors
$$
R^i(\QCoh)\Gamma(U, -) : \QCoh(\mathcal{O}_X) \to \textit{Ab}
$$
of the left exact functor $\Gamma(U, -)$. Since this is a universal
$\delta$-functor, and since the functors $H^i(U, -)$ (defined for all
abelian sheaves on $X$) restricted to $\QCoh(\mathcal{O}_X)$ form
a $\delta$-functor, we obtain canonical tranformations
$$
t^i : R^i(\QCoh)\Gamma(U, -) \to H^i(U, -).
$$
These transformations aren't in general isomorphisms even if $X = \Spec(A)$
is affine! Namely, we have $R^1(\QCoh)\Gamma(U, \widetilde{I}) = 0$
if $I$ an injective $A$-module by construction of right derived functors
and the equivalence of $\QCoh(\mathcal{O}_X)$ and $\text{Mod}_A$.
But Examples, Lemma \ref{examples-lemma-nonvanishing}
shows there exists $A$, $I$, and $U$ such that
$H^1(U, \widetilde{I}) \not = 0$.
\end{remark}
\section{The coherator for Noetherian schemes}
\label{section-coherator-Noetherian}
\noindent
In the case of Noetherian schemes we can use the following lemma.
\begin{lemma}
\label{lemma-injective-quasi-coherent-sheaf-Noetherian}
Let $X$ be a Noetherian scheme. Let $\mathcal{J}$ be an injective
object of $\QCoh(\mathcal{O}_X)$. Then $\mathcal{J}$
is a flasque sheaf of $\mathcal{O}_X$-modules.
\end{lemma}
\begin{proof}
Let $U \subset X$ be an open subset and let $s \in \mathcal{J}(U)$
be a section. Let $\mathcal{I} \subset X$ be the quasi-coherent sheaf
of ideals defining the reduced induced scheme structure on $X \setminus U$
(see Schemes, Definition \ref{schemes-definition-reduced-induced-scheme}).
By Cohomology of Schemes, Lemma \ref{coherent-lemma-homs-over-open}
the section $s$ corresponds to a map $\sigma : \mathcal{I}^n \to \mathcal{J}$
for some $n$. As $\mathcal{J}$ is an injective object of
$\QCoh(\mathcal{O}_X)$ we can extend $\sigma$ to a map
$\tilde s : \mathcal{O}_X \to \mathcal{J}$. Then $\tilde s$ corresponds
to a global section of $\mathcal{J}$ restricting to $s$.
\end{proof}
\begin{lemma}
\label{lemma-Noetherian-pushforward}
Let $f : X \to Y$ be a morphism of Noetherian schemes.
Then $f_*$ on quasi-coherent sheaves has a right derived
extension
$\Phi : D(\QCoh(\mathcal{O}_X)) \to D(\QCoh(\mathcal{O}_Y))$
such that the diagram
$$
\xymatrix{
D(\QCoh(\mathcal{O}_X)) \ar[d]_{\Phi} \ar[r] &
D_\QCoh(\mathcal{O}_X) \ar[d]^{Rf_*} \\
D(\QCoh(\mathcal{O}_Y)) \ar[r] &
D_\QCoh(\mathcal{O}_Y)
}
$$
commutes.
\end{lemma}
\begin{proof}
Since $X$ and $Y$ are Noetherian schemes the morphism is quasi-compact
and quasi-separated (see
Properties, Lemma \ref{properties-lemma-locally-Noetherian-quasi-separated}
and
Schemes, Remark \ref{schemes-remark-quasi-compact-and-quasi-separated}).
Thus $f_*$ preserve quasi-coherence, see
Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent}.
Next, Let $K$ be an object of $D(\QCoh(\mathcal{O}_X))$.
Since $\QCoh(\mathcal{O}_X)$ is a Grothendieck abelian category
(Properties, Proposition \ref{properties-proposition-coherator}), we can
represent $K$ by a K-injective complex $\mathcal{I}^\bullet$
such that each $\mathcal{I}^n$ is an injective object of
$\QCoh(\mathcal{O}_X)$, see
Injectives, Theorem
\ref{injectives-theorem-K-injective-embedding-grothendieck}.
Thus we see that the functor $\Phi$ is defined by setting
$$
\Phi(K) = f_*\mathcal{I}^\bullet
$$
where the right hand side is viewed as an object of
$D(\QCoh(\mathcal{O}_Y))$. To finish the proof of the lemma
it suffices to show that the canonical map
$$
f_*\mathcal{I}^\bullet \longrightarrow Rf_*\mathcal{I}^\bullet
$$
is an isomorphism in $D(\mathcal{O}_Y)$. To see this it suffices to
prove the map induces an isomorphism on cohomology sheaves. Pick any
$m \in \mathbf{Z}$. Let $N = N(X, Y, f)$ be as in
Lemma \ref{lemma-quasi-coherence-direct-image}.
Consider the short exact sequence
$$
0 \to \sigma_{\geq m - N - 1}\mathcal{I}^\bullet \to
\mathcal{I}^\bullet \to \sigma_{\leq m - N - 2}\mathcal{I}^\bullet \to 0
$$
of complexes of quasi-coherent sheaves on $X$. By
Lemma \ref{lemma-quasi-coherence-direct-image}
we see that the cohomology sheaves of
$Rf_*\sigma_{\leq m - N - 2}\mathcal{I}^\bullet$
are zero in degrees $\geq m - 1$. Thus we see that
$R^mf_*\mathcal{I}^\bullet$ is isomorphic to
$R^mf_*\sigma_{\geq m - N - 1}\mathcal{I}^\bullet$.
In other words, we may assume that $\mathcal{I}^\bullet$
is a bounded below complex of injective objects of
$\QCoh(\mathcal{O}_X)$.
This follows from
Leray's acyclicity lemma
(Derived Categories, Lemma \ref{derived-lemma-leray-acyclicity})
via
Cohomology, Lemma \ref{cohomology-lemma-flasque-acyclic-pushforward}
and
Lemma \ref{lemma-injective-quasi-coherent-sheaf-Noetherian}.
\end{proof}
\begin{proposition}
\label{proposition-Noetherian}
Let $X$ be a Noetherian scheme. Then the functor (\ref{equation-compare})
$$
D(\QCoh(\mathcal{O}_X))
\longrightarrow
D_\QCoh(\mathcal{O}_X)
$$
is an equivalence with quasi-inverse given by $RQ_X$.
\end{proposition}
\begin{proof}
This follows using the exact same argument as in the proof of
Proposition \ref{proposition-quasi-compact-affine-diagonal}
using Lemma \ref{lemma-Noetherian-pushforward}.
See discussion in Remark \ref{remark-argument-proves}.
\end{proof}
\section{Koszul complexes}
\label{section-koszul}
\noindent
Let $A$ be a ring and let $f_1, \ldots, f_r$ be a sequence of elements
of $A$. We have defined the Koszul complex
$K_\bullet(f_1, \ldots, f_r)$ in
More on Algebra, Definition \ref{more-algebra-definition-koszul-complex}.
It is a chain complex sitting in degrees $r, \ldots, 0$.
We turn this into a cochain complex $K^\bullet(f_1, \ldots, f_r)$
by setting $K^{-n}(f_1, \ldots, f_r) = K_n(f_1, \ldots, f_r)$
and using the same differentials. In the rest of this section all
the complexes will be cochain complexes.
\medskip\noindent
We define a complex $I^\bullet(f_1, \ldots, f_r)$
such that we have a distinguished triangle
$$
I^\bullet(f_1, \ldots, f_r) \to
A \to
K^\bullet(f_1, \ldots, f_r) \to
I^\bullet(f_1, \ldots, f_r)[1]
$$
in $K(A)$.
In other words, we set
$$
I^i(f_1, \ldots, f_r) =
\left\{
\begin{matrix}
K^{i - 1}(f_1, \ldots, f_r) & \text{if } i \leq 0 \\
0 & \text{else}
\end{matrix}
\right.
$$
and we use the negative of the differential on $K^\bullet(f_1, \ldots, f_r)$.
The maps in the distinguished triangle are the obvious ones. Note that
$I^0(f_1, \ldots, f_r) = A^{\oplus r} \to A$ is given by
multiplication by $f_i$ on the $i$th factor.
Hence $I^\bullet(f_1, \ldots, f_r) \to A$ factors as
$$
I^\bullet(f_1, \ldots, f_r) \to I \to A
$$
where $I = (f_1, \ldots, f_r)$. In fact, there is a short exact sequence
$$
0 \to H^{-1}(K^\bullet(f_1, \ldots, f_s)) \to
H^0(I^\bullet(f_1, \ldots, f_s)) \to I \to 0
$$
and for every $i < 0$ we have
$H^i(I^\bullet(f_1, \ldots, f_r)) = H^{i - 1}(K^\bullet(f_1, \ldots, f_r)$.
Observe that given a second sequence $g_1, \ldots, g_r$ of elements of $A$
there are canonical maps
$$
I^\bullet(f_1g_1, \ldots, f_rg_r) \to I^\bullet(f_1, \ldots, f_r)
\quad\text{and}\quad
K^\bullet(f_1g_1, \ldots, f_rg_r) \to K^\bullet(f_1, \ldots, f_r)
$$
compatible with the maps described above. The first of these maps is
given by multiplication by $g_i$ on the $i$th summand of
$I^0(f_1g_1, \ldots, f_rg_r) = A^{\oplus r}$. In particular, given
$f_1, \ldots, f_r$ we obtain an inverse system of complexes
\begin{equation}
\label{equation-system}
I^\bullet(f_1, \ldots, f_r) \leftarrow
I^\bullet(f_1^2, \ldots, f_r^2) \leftarrow
I^\bullet(f_1^3, \ldots, f_r^3) \leftarrow \ldots
\end{equation}
which will play an important role in that which is to follow.
To easily formulate the following lemmas we fix some notation.
\begin{situation}
\label{situation-complex}
Here $A$ is a ring and $f_1, \ldots, f_r$ is a sequence of elements of $A$.
We set $X = \Spec(A)$ and $U = D(f_1) \cup \ldots \cup D(f_r) \subset X$.
We denote $\mathcal{U} : U = \bigcup_{i = 1, \ldots, r} D(f_i)$ the
given open covering of $U$.
\end{situation}
\noindent
Our first lemma is that the complexes above can be used to compute
the cohomology of quasi-coherent sheaves on $U$. Suppose given a
complex $I^\bullet$ of $A$-modules and an $A$-module $M$. Then we
define $\Hom_A(I^\bullet, M)$ to be the complex with $n$th
term $\Hom_A(I^{-n}, M)$ and differentials given as the contragredients
of the differentials on $I^\bullet$.
\begin{lemma}
\label{lemma-alternating-cech-complex}
In Situation \ref{situation-complex}. Let $M$ be an $A$-module and
denote $\mathcal{F}$ the associated $\mathcal{O}_X$-module. Then
there is a canonical isomorphism of complexes
$$
\colim_e \Hom_A(I^\bullet(f_1^e, \ldots, f_r^e), M)
\longrightarrow
\check{\mathcal{C}}_{alt}^\bullet(\mathcal{U}, \mathcal{F})
$$
functorial in $M$.
\end{lemma}
\begin{proof}
Recall that the alternating {\v C}ech complex is the subcomplex
of the usual {\v C}ech complex given by alternating cochains, see
Cohomology, Section \ref{cohomology-section-alternating-cech}.
As usual we view a $p$-cochain in
$\check{\mathcal{C}}_{alt}^\bullet(\mathcal{U}, \mathcal{F})$
as an alternating function $s$ on $\{1, \ldots, r\}^{p + 1}$
whose value $s_{i_0\ldots i_p}$ at $(i_0, \ldots, i_p)$ lies in
$M_{f_{i_0}\ldots f_{i_p}} = \mathcal{F}(U_{i_0\ldots i_p})$.
On the other hand, a $p$-cochain $t$ in
$\Hom_A(I^\bullet(f_1^e, \ldots, f_r^e), M)$
is given by a map $t : \wedge^{p + 1}(A^{\oplus r}) \to M$.
Write $[i] \in A^{\oplus r}$ for the $i$th basis element and
write
$$
[i_0, \ldots, i_p] = [i_0] \wedge \ldots \wedge [i_p]
\in \wedge^{p + 1}(A^{\oplus r})
$$
Then we send $t$ as above to $s$ with
$$
s_{i_0\ldots i_p} = \frac{t([i_0, \ldots, i_p])}{f_{i_0}^e\ldots f_{i_p}^e}
$$
It is clear that $s$ so defined is an alternating cochain.
The construction of this map is compatible with the transition maps
of the system as the transition map
$$
I^\bullet(f_1^e, \ldots, f_r^e) \leftarrow
I^\bullet(f_1^{e + 1}, \ldots, f_r^{e + 1}),
$$
of the (\ref{equation-system}) sends $[i_0, \ldots, i_p]$
to $f_{i_0}\ldots f_{i_p}[i_0, \ldots, i_p]$.
It is clear from the description of the localizations
$M_{f_{i_0}\ldots f_{i_p}}$ in
Algebra, Lemma \ref{algebra-lemma-localization-colimit}
that these maps define an isomorphism of cochain modules in degree $p$
in the limit. To finish the proof we have to show that the map
is compatible with differentials. To see this recall that
\begin{align*}
d(s)_{i_0\ldots i_{p + 1}}
& =
\sum\nolimits_{j = 0}^{p + 1} (-1)^j
s_{i_0\ldots \hat i_j \ldots i_p} \\
& =
\sum\nolimits_{j = 0}^{p + 1} (-1)^j
\frac{t([i_0, \ldots, \hat i_j, \ldots i_{p + 1}])}
{f_{i_0}^e\ldots \hat f_{i_j}^e \ldots f_{i_{p + 1}}^e}
\end{align*}
On the other hand, we have
\begin{align*}
\frac{d(t)([i_0, \ldots, i_{p + 1}])}{f_{i_0}^e\ldots f_{i_{p + 1}}^e}
& =
\frac{t(d[i_0, \ldots, i_{p + 1}])}{f_{i_0}^e\ldots f_{i_{p + 1}}^e} \\
& =
\frac{\sum_j (-1)^j f_{i_j}^e t([i_0, \ldots, \hat i_j, \ldots i_{p + 1}])}
{f_{i_0}^e \ldots f_{i_{p + 1}}^e}
\end{align*}
The two formulas agree by inspection.
\end{proof}
\noindent
Suppose given a finite complex $I^\bullet$ of $A$-modules and a
complex of $A$-modules $M^\bullet$. We obtain a double complex
$H^{\bullet, \bullet} = \Hom_A(I^\bullet, M^\bullet)$ where
$H^{p, q} = \Hom_A(I^p, M^q)$. The first differential comes from
the differential on $\Hom_A(I^\bullet, M^q)$ and the second
from the differential on $M^\bullet$. Associated to this double
complex is the total complex with degree $n$ term given by
$$
\bigoplus\nolimits_{p + q = n} \Hom_A(I^p, M^q)
$$
and differential as in
Homology, Definition \ref{homology-definition-associated-simple-complex}.
As our complex $I^\bullet$ has only finitely many nonzero terms, the
direct sum displayed above is finite.
The conventions for taking the total complex associated to a
{\v C}ech complex of a complex are as in
Cohomology, Section \ref{cohomology-section-cech-cohomology-of-complexes}.
\begin{lemma}
\label{lemma-alternating-cech-complex-complex}
In Situation \ref{situation-complex}. Let $M^\bullet$ be a
complex of $A$-modules and
denote $\mathcal{F}^\bullet$ the associated complex of
$\mathcal{O}_X$-modules. Then
there is a canonical isomorphism of complexes
$$
\colim_e \text{Tot}(\Hom_A(I^\bullet(f_1^e, \ldots, f_r^e), M^\bullet))
\longrightarrow
\text{Tot}(\check{\mathcal{C}}_{alt}^\bullet(\mathcal{U}, \mathcal{F}^\bullet))
$$
functorial in $M^\bullet$.
\end{lemma}
\begin{proof}
Immediate from Lemma \ref{lemma-alternating-cech-complex}
and our conventions for taking associated total complexes.
\end{proof}
\begin{lemma}
\label{lemma-alternating-cech-complex-complex-computes-cohomology}
In Situation \ref{situation-complex}. Let $\mathcal{F}^\bullet$
be a complex of quasi-coherent $\mathcal{O}_X$-modules. Then
there is a canonical isomorphism
$$
\text{Tot}(\check{\mathcal{C}}_{alt}^\bullet(\mathcal{U}, \mathcal{F}^\bullet))
\longrightarrow
R\Gamma(U, \mathcal{F}^\bullet)
$$
in $D(A)$ functorial in $\mathcal{F}^\bullet$.
\end{lemma}
\begin{proof}
Let $\mathcal{B}$ be the set of affine opens of $U$. Since the higher
cohomology groups of a quasi-coherent module on an affine scheme are zero
(Cohomology of Schemes, Lemma
\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero})
this is a special case of
Cohomology, Lemma \ref{cohomology-lemma-alternating-cech-complex-complex-ss}.
\end{proof}
\noindent
In Situation \ref{situation-complex} denote $I_e$ the object of
$D(\mathcal{O}_X)$ corresponding to the complex of $A$-modules
$I^\bullet(f_1^e, \ldots, f_r^e)$ via the equivalence of
Lemma \ref{lemma-affine-compare-bounded}. The maps
(\ref{equation-system}) give a system
$$
I_1 \leftarrow
I_2 \leftarrow
I_3 \leftarrow \ldots
$$
Moreover, there is a compatible system of maps $I_e \to \mathcal{O}_X$
which become isomorphisms when restricted to $U$. Thus we see that for
every object $E$ of $D(\mathcal{O}_X)$ there is a canonical map
\begin{equation}
\label{equation-comparison}
\colim_e \Hom_{D(\mathcal{O}_X)}(I_e, E) \longrightarrow H^0(U, E)
\end{equation}
constructed by sending a map $I_e \to E$ to its restriction to $U$
and using that
$\Hom_{D(\mathcal{O}_U)}(\mathcal{O}_U, E|_U) = H^0(U, E)$.
\begin{proposition}
\label{proposition-represent-cohomology-class-on-open}
In Situation \ref{situation-complex}. For every object $E$
of $D_\QCoh(\mathcal{O}_X)$ the map
(\ref{equation-comparison}) is an isomorphism.
\end{proposition}
\begin{proof}
By Lemma \ref{lemma-affine-compare-bounded} we may assume that $E$
is given by a complex of quasi-coherent sheaves $\mathcal{F}^\bullet$.
Let $M^\bullet = \Gamma(X, \mathcal{F}^\bullet)$ be the corresponding
complex of $A$-modules. By
Lemmas \ref{lemma-alternating-cech-complex-complex} and
\ref{lemma-alternating-cech-complex-complex-computes-cohomology}
we have quasi-isomorphisms
$$
\colim_e \text{Tot}(\Hom_A(I^\bullet(f_1^e, \ldots, f_r^e), M^\bullet))
\longrightarrow
\text{Tot}(\check{\mathcal{C}}_{alt}^\bullet(\mathcal{U}, \mathcal{F}^\bullet))
\longrightarrow
R\Gamma(U, \mathcal{F}^\bullet)
$$
Taking $H^0$ on both sides we obtain
$$
\colim_e \Hom_{D(A)}(I^\bullet(f_1^e, \ldots, f_r^e), M^\bullet)
=
H^0(U, E)
$$
Since $\Hom_{D(A)}(I^\bullet(f_1^e, \ldots, f_r^e), M^\bullet) =
\Hom_{D(\mathcal{O}_X)}(I_e, E)$ by
Lemma \ref{lemma-affine-compare-bounded} the lemma follows.
\end{proof}
\noindent
In Situation \ref{situation-complex} denote $K_e$ the object of
$D(\mathcal{O}_X)$ corresponding to the complex of $A$-modules
$K^\bullet(f_1^e, \ldots, f_r^e)$ via the equivalence of
Lemma \ref{lemma-affine-compare-bounded}. Thus we have distinguished
triangles
$$
I_e \to \mathcal{O}_X \to K_e \to I_e[1]
$$
and a system
$$
K_1 \leftarrow
K_2 \leftarrow
K_3 \leftarrow \ldots
$$
compatible with the system $(I_e)$.
Moreover, there is a compatible system of maps
$$
K_e \to H^0(K_e) = \mathcal{O}_X/(f_1^e, \ldots, f_r^e)
$$
\begin{lemma}
\label{lemma-represent-cohomology-class-on-closed}
In Situation \ref{situation-complex}. Let $E$ be an object of
$D_\QCoh(\mathcal{O}_X)$.
Assume that $H^i(E)|_U = 0$ for $i = - r + 1, \ldots, 0$.
Then given $s \in H^0(X, E)$ there exists an $e \geq 0$ and
a morphism $K_e \to E$ such that $s$ is in the image of
$H^0(X, K_e) \to H^0(X, E)$.
\end{lemma}
\begin{proof}
Since $U$ is covered by $r$ affine opens we have $H^j(U, \mathcal{F}) = 0$
for $j \geq r$ and any quasi-coherent module
(Cohomology of Schemes, Lemma \ref{coherent-lemma-vanishing-nr-affines}).
By Lemma \ref{lemma-application-nice-K-injective} we see that $H^0(U, E)$
is equal to $H^0(U, \tau_{\geq -r + 1}E)$. There is
a spectral sequence
$$
H^j(U, H^i(\tau_{\geq -r + 1}E)) \Rightarrow H^{i + j}(U, \tau_{\geq -N}E)
$$
see Derived Categories, Lemma \ref{derived-lemma-two-ss-complex-functor}.
Hence $H^0(U, E) = 0$ by our assumed vanishing of cohomology sheaves of $E$.
We conclude that $s|_U = 0$.
Think of $s$ as a morphism $\mathcal{O}_X \to E$ in $D(\mathcal{O}_X)$.
By Proposition \ref{proposition-represent-cohomology-class-on-open}
the composition $I_e \to \mathcal{O}_X \to E$ is zero for some $e$.
By the distinguished triangle $I_e \to \mathcal{O}_X \to K_e \to I_e[1]$
we obtain a morphism $K_e \to E$ such that $s$ is the composition
$\mathcal{O}_X \to K_e \to E$.
\end{proof}
\section{Pseudo-coherent and perfect complexes}
\label{section-spell-out}
\noindent
In this section we make the connection between the general
notions defined in
Cohomology, Sections \ref{cohomology-section-strictly-perfect},
\ref{cohomology-section-pseudo-coherent},
\ref{cohomology-section-tor}, and
\ref{cohomology-section-perfect}
and the corresponding notions for complexes of modules in
More on Algebra, Sections
\ref{more-algebra-section-pseudo-coherent},
\ref{more-algebra-section-tor}, and
\ref{more-algebra-section-perfect}.
\begin{lemma}
\label{lemma-pseudo-coherent}
Let $X$ be a scheme. 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 $i > m$ and $H^m(E)$ is a quotient
of a quasi-coherent $\mathcal{O}_X$-module.
If $E$ is pseudo-coherent, then $E$ is an object of
$D_\QCoh(\mathcal{O}_X)$.
\end{lemma}
\begin{proof}
Locally on $X$ there exists a strictly perfect complex $\mathcal{E}^\bullet$
such that $H^i(E)$ is isomorphic to $H^i(\mathcal{E}^\bullet)$ for $i > m$
and $H^m(E)$ is a quotient of $H^m(\mathcal{E}^\bullet)$. The sheaves
$\mathcal{E}^i$ are direct summands of finite free modules,
hence quasi-coherent. The lemma follows.
\end{proof}
\begin{lemma}
\label{lemma-pseudo-coherent-affine}
Let $X = \Spec(A)$ be an affine scheme. Let $M^\bullet$ be a
complex of $A$-modules and let $E$ be the corresponding object
of $D(\mathcal{O}_X)$. Then $E$ is an $m$-pseudo-coherent
(resp.\ pseudo-coherent) as an object of $D(\mathcal{O}_X)$
if and only if $M^\bullet$ is $m$-pseudo-coherent (resp.\ pseudo-coherent)
as a complex of $A$-modules.
\end{lemma}
\begin{proof}
It is immediate from the definitions that if $M^\bullet$ is
$m$-pseudo-coherent, so is $E$. To prove the converse, assume
$E$ is $m$-pseudo-coherent. As $X = \Spec(A)$ is quasi-compact with
a basis for the topology given by standard opens, we can find a standard
open covering $X = D(f_1) \cup \ldots \cup D(f_n)$ and strictly
perfect complexes $\mathcal{E}_i^\bullet$ on $D(f_i)$ and
maps $\alpha_i : \mathcal{E}_i^\bullet \to E|_{U_i}$ inducing
isomorphisms on $H^j$ for $j > m$ and surjections on $H^m$.
By Cohomology, Lemma \ref{cohomology-lemma-local-actual}
after refining the open covering
we may assume $\alpha_i$ is given by a map of complexes
$\mathcal{E}_i^\bullet \to \widetilde{M^\bullet}|_{U_i}$
for each $i$. By Modules, Lemma
\ref{modules-lemma-direct-summand-of-locally-free-is-locally-free}
the terms $\mathcal{E}_i^n$ are finite locally free modules.
Hence after refining the open covering we may assume each
$\mathcal{E}_i^n$ is a finite free $\mathcal{O}_{U_i}$-module.
From the definition it follows that $M^\bullet_{f_i}$ is
an $m$-pseudo-coherent complex of $A_{f_i}$-modules.
We conclude by applying
More on Algebra, Lemma \ref{more-algebra-lemma-glue-pseudo-coherent}.
\medskip\noindent
The case ``pseudo-coherent'' follows from the fact that $E$ is
pseudo-coherent if and only if $E$ is $m$-pseudo-coherent for
all $m$ (by definition) and the same is true for $M^\bullet$
by More on Algebra, Lemma \ref{more-algebra-lemma-pseudo-coherent}.
\end{proof}
\begin{lemma}
\label{lemma-identify-pseudo-coherent-noetherian}
Let $X$ be a Noetherian scheme. Let $E$ be an object of
$D_\QCoh(\mathcal{O}_X)$. 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 see that in both (1) and (2) the object $E$
is bounded above. Thus the question is local on $X$ and we may assume
$X$ is affine. Say $X = \Spec(A)$ for some Noetherian ring $A$.
In this case $E$ corresponds to a complex of $A$-modules $M^\bullet$
by Lemma \ref{lemma-affine-compare-bounded}. By
Lemma \ref{lemma-pseudo-coherent-affine}
we see that $E$ is $m$-pseudo-coherent if and only if $M^\bullet$
is $m$-pseudo-coherent. On the other hand, $H^i(E)$ is coherent
if and only if $H^i(M^\bullet)$ is a finite $A$-module
(Properties, Lemma \ref{properties-lemma-finite-type-module}).
Thus the result follows from More on Algebra, Lemma
\ref{more-algebra-lemma-Noetherian-pseudo-coherent}.
\end{proof}
\begin{lemma}
\label{lemma-tor-dimension-affine}
Let $X = \Spec(A)$ be an affine scheme. Let $M^\bullet$ be a
complex of $A$-modules and let $E$ be the corresponding object
of $D(\mathcal{O}_X)$. Then
\begin{enumerate}
\item $E$ has tor amplitude in $[a, b]$ if and only if $M^\bullet$
has tor amplitude in $[a, b]$.
\item $E$ has finite tor dimension if and only if $M^\bullet$
has finite tor dimension.
\end{enumerate}
\end{lemma}
\begin{proof}
Part (2) follows trivially from part (1). In the proof of (1) we will
use the equivalence $D(A) = D_\QCoh(X)$ of
Lemma \ref{lemma-affine-compare-bounded}
without further mention.
Assume $M^\bullet$ has tor amplitude in $[a, b]$. Then $K^\bullet$
is isomorphic in $D(A)$ to a complex $K^\bullet$ of flat $A$-modules
with $K^i = 0$ for $i \not \in [a, b]$, see
More on Algebra, Lemma \ref{more-algebra-lemma-tor-amplitude}.
Then $E$ is isomorphic to $\widetilde{K^\bullet}$. Since each
$\widetilde{K^i}$ is a flat $\mathcal{O}_X$-module, we see
that $E$ has tor amplitude in $[a, b]$ by
Cohomology, Lemma \ref{cohomology-lemma-tor-amplitude}.
\medskip\noindent
Assume that $E$ has tor amplitude in $[a, b]$. Then $E$ is bounded
whence $M^\bullet$ is in $K^-(A)$. Thus we may replace $M^\bullet$
by a bounded above complex of $A$-modules. We may even choose
a projective resolution and assume that $M^\bullet$ is a bounded above
complex of free $A$-modules. Then for any $A$-module $N$ we have
$$
E \otimes_{\mathcal{O}_X}^\mathbf{L} \widetilde{N}
\cong
\widetilde{M^\bullet} \otimes_{\mathcal{O}_X}^\mathbf{L} \widetilde{N}
\cong
\widetilde{M^\bullet \otimes_A N}
$$
in $D(\mathcal{O}_X)$. Thus the vanishing of cohomology sheaves of
the left hand side implies $M^\bullet$ has tor amplitude in $[a, b]$.
\end{proof}
\begin{lemma}
\label{lemma-tor-qc-qs}
Let $X$ be a quasi-separated scheme. Let $E$ be an object
of $D_\QCoh(\mathcal{O}_X)$. 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 $\QCoh(\mathcal{O}_X)$
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 $U \subset X$ be
an affine open. As $X$ is quasi-separated the morphism $j : U \to X$
is quasi-compact and separated, hence $j_*$ transforms quasi-coherent
modules into quasi-coherent modules
(Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent}).
Thus the functor
$\QCoh(\mathcal{O}_X) \to \QCoh(\mathcal{O}_U)$
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
$\mathcal{G}$. Write $U = \Spec(A)$ and let $M^\bullet$ be the
complex of $A$-modules corresponding to $E|_U$ by
Lemma \ref{lemma-affine-compare-bounded}.
We have just shown that $M^\bullet \otimes_A^\mathbf{L} N$
has vanishing cohomology groups outside the range $[a, b]$,
in other words $M^\bullet$ has tor amplitude in $[a, b]$.
By Lemma \ref{lemma-tor-dimension-affine}
we conclude that $E|_U$ has tor amplitude in $[a, b]$.
This proves the lemma.
\end{proof}
\begin{lemma}
\label{lemma-perfect-affine}
Let $X = \Spec(A)$ be an affine scheme. Let $M^\bullet$ be a
complex of $A$-modules and let $E$ be the corresponding object
of $D(\mathcal{O}_X)$. Then $E$ is a perfect object of $D(\mathcal{O}_X)$
if and only if $M^\bullet$ is perfect as an object of $D(A)$.
\end{lemma}
\begin{proof}
This is a logical consequence of
Lemmas \ref{lemma-pseudo-coherent-affine} and
\ref{lemma-tor-dimension-affine},
Cohomology, Lemma \ref{cohomology-lemma-perfect}, and
More on Algebra, Lemma \ref{more-algebra-lemma-perfect}.
\end{proof}
\noindent
As a consequence of our description of pseudo-coherent
complexes on schemes we can prove certain internal homs
are quasi-coherent.
\begin{lemma}
\label{lemma-quasi-coherence-internal-hom}
Let $X$ be a scheme.
\begin{enumerate}
\item If $L$ is in $D^+_\QCoh(\mathcal{O}_X)$ and
$K$ in $D(\mathcal{O}_X)$ is pseudo-coherent, then
$R\SheafHom(K, L)$ is in $D_\QCoh^+(\mathcal{O}_X)$.
\item If $L$ is in $D_\QCoh(\mathcal{O}_X)$ and
$K$ in $D(\mathcal{O}_X)$ is perfect, then
$R\SheafHom(K, L)$ is in $D_\QCoh(\mathcal{O}_X)$.
\item If $X = \Spec(A)$ is affine and $K, L \in D(A)$ then
$$
R\SheafHom(\widetilde{K}, \widetilde{L}) = \widetilde{R\Hom(K, L)}
$$
in the following two cases
\begin{enumerate}
\item $K$ is pseudo-coherent and $L$ is bounded below,
\item $K$ is perfect and $L$ arbitrary.
\end{enumerate}
\item If $X = \Spec(A)$ and $K, L$ are in $D(A)$, then the $n$th
cohomology sheaf of $R\SheafHom(\widetilde{K}, \widetilde{L})$
is the sheaf associated to the presheaf
$$
X \supset D(f) \longmapsto \text{Ext}^n_{A_f}(K \otimes_A A_f, L \otimes_A A_f)
$$
for $f \in A$.
\end{enumerate}
\end{lemma}
\begin{proof}
The construction of the internal hom in the derived category of
$\mathcal{O}_X$ commutes with localization (see
Cohomology, Section \ref{cohomology-section-internal-hom}).
Hence to prove (1) and (2) we may replace $X$ by an affine open.
By Lemmas \ref{lemma-affine-compare-bounded},
\ref{lemma-pseudo-coherent-affine}, and
\ref{lemma-perfect-affine}
in order to prove (1) and (2) it suffices to prove (3).
\medskip\noindent
Part (3) follows from the computation of the
internal hom of Cohomology, Lemma
\ref{cohomology-lemma-Rhom-complex-of-direct-summands-finite-free}
by representing $K$ by a bounded above (resp.\ finite) complex of
finite projective $A$-modules and $L$ by a bounded below
(resp.\ arbitrary) complex of $A$-modules.
\medskip\noindent
To prove (4) recall that on any ringed space the $n$th cohomology sheaf of
$R\SheafHom(A, B)$ is the sheaf associated to the presheaf
$$
U \mapsto \Hom_{D(U)}(A|_U, B|_U[n]) =
\text{Ext}^n_{D(\mathcal{O}_U)}(A|_U, B|_U)
$$
See Cohomology, Section \ref{cohomology-section-internal-hom}.
On the other hand, the restriction of $\widetilde{K}$ to a principal
open $D(f)$ is the image of $K \otimes_A A_f$ and similarly for $L$.
Hence (4) follows from the equivalence of categories of
Lemma \ref{lemma-affine-compare-bounded}.
\end{proof}
\begin{lemma}
\label{lemma-internal-hom-evaluate-tensor-isomorphism}
Let $X$ be a scheme. Let $K, L, M$ be objects of $D_\QCoh(\mathcal{O}_X)$.
The map
$$
K \otimes_{\mathcal{O}_X}^\mathbf{L} R\SheafHom(M, L)
\longrightarrow
R\SheafHom(M, K \otimes_{\mathcal{O}_X}^\mathbf{L} L)
$$
of Cohomology, Lemma \ref{cohomology-lemma-internal-hom-diagonal-better}
is an isomorphism in the following cases
\begin{enumerate}
\item $M$ perfect, or
\item $K$ is perfect, or
\item $M$ is pseudo-coherent, $L \in D^+(\mathcal{O}_X)$, and $K$ has finite
tor dimension.
\end{enumerate}
\end{lemma}
\begin{proof}
Lemma \ref{lemma-quasi-coherence-internal-hom}
reduces cases (1) and (3) to the affine case which is treated in
More on Algebra, Lemma
\ref{more-algebra-lemma-internal-hom-evaluate-tensor-isomorphism}.
(You also have to use Lemmas \ref{lemma-pseudo-coherent-affine},
\ref{lemma-perfect-affine}, and \ref{lemma-tor-dimension-affine}
to do the translation into algebra.)
If $K$ is perfect but no other assumptions are made, then we
do not know that either side of the arrow is in $D_\QCoh(\mathcal{O}_X)$
but the result is still true because we can work locally and reduce
to the case that $K$ is a finite complex of finite free modules
in which case it is clear.
\end{proof}
\section{Derived category of coherent modules}
\label{section-derived-coherent}
\noindent
Let $X$ be a locally Noetherian scheme. 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 Schemes, Lemma \ref{coherent-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 Noetherian scheme. Let $f : X \to S$ be a morphism of schemes
which is locally of finite type. Let $E$ be an object of
$D^b_{\textit{Coh}}(\mathcal{O}_X)$ such that the support of $H^i(E)$
is proper over $S$ for all $i$.
Then $Rf_*E$ is an object of $D^b_{\textit{Coh}}(\mathcal{O}_S)$.
\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 Schemes, Lemma
\ref{coherent-lemma-support-proper-over-base-pushforward}
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 Schemes, Lemma
\ref{coherent-lemma-quasi-coherence-higher-direct-images}
or from
Lemma \ref{lemma-quasi-coherence-direct-image}.
\end{proof}
\begin{lemma}
\label{lemma-direct-image-coherent-bdd-below}
Let $S$ be a Noetherian scheme. Let $f : X \to S$ be a morphism of schemes
which is locally of finite type. Let $E$ be an object of
$D^+_{\textit{Coh}}(\mathcal{O}_X)$ such that the support of $H^i(E)$
is proper over $S$ for all $i$.
Then $Rf_*E$ is an object of $D^+_{\textit{Coh}}(\mathcal{O}_S)$.
\end{lemma}
\begin{proof}
The proof is the same as the proof of
Lemma \ref{lemma-direct-image-coherent}.
You can also deduce it from
Lemma \ref{lemma-direct-image-coherent}
by considering what the exact functor $Rf_*$ does to
the distinguished triangles
$\tau_{\leq a}E \to E \to \tau_{\geq a + 1}E \to \tau_{\leq a}E[1]$.
\end{proof}
\begin{lemma}
\label{lemma-coherent-internal-hom}
Let $X$ be a locally Noetherian scheme. If $L$ is in
$D^+_{\textit{Coh}}(\mathcal{O}_X)$ and $K$ in
$D^-_{\textit{Coh}}(\mathcal{O}_X)$, then
$R\SheafHom(K, L)$ is in $D^+_{\textit{Coh}}(\mathcal{O}_X)$.
\end{lemma}
\begin{proof}
It suffices to prove this when $X$ is the spectrum of
a Noetherian ring $A$.
By Lemma \ref{lemma-identify-pseudo-coherent-noetherian}
we see that $K$ is pseudo-coherent.
Then we can use Lemma \ref{lemma-quasi-coherence-internal-hom}
to translate the problem into the following algebra problem:
for $L \in D^+_{\textit{Coh}}(A)$ and $K$ in $D^-_{\textit{Coh}}(A)$, then
$R\Hom(K, L)$ is in $D^+_{\textit{Coh}}(A)$.
Since $L$ is bounded below and $K$ is bounded below there is a
convergent spectral sequence
$$
\text{Ext}^p_A(K, H^q(L)) \Rightarrow \text{Ext}^{p + q}_A(K, L)
$$
and there are convergent spectral sequences
$$
\text{Ext}^i_A(H^{-j}(K), H^q(L)) \Rightarrow \text{Ext}^{i + j}_A(K, H^q(L))
$$
This finishes the proof as the modules $\text{Ext}^p_A(M, N)$
are finite for finite $A$-modules $M$, $N$ by
Algebra, Lemma \ref{algebra-lemma-ext-noetherian}.
\end{proof}
\begin{lemma}
\label{lemma-ext-finite}
Let $A$ be a Noetherian ring. Let $X$ be a proper scheme over $A$.
For $L$ in
$D^+_{\textit{Coh}}(\mathcal{O}_X)$ and $K$ in
$D^-_{\textit{Coh}}(\mathcal{O}_X)$, the $A$-modules
$\text{Ext}_{\mathcal{O}_X}^n(K, L)$ are finite.
\end{lemma}
\begin{proof}
Recall that
$$
\text{Ext}_{\mathcal{O}_X}^n(K, L) =
H^n(X, R\SheafHom_{\mathcal{O}_X}(K, L)) =
H^n(\Spec(A), Rf_*R\SheafHom_{\mathcal{O}_X}(K, L))
$$
see Cohomology, Lemma \ref{cohomology-lemma-section-RHom-over-U}
and Cohomology, Section \ref{cohomology-section-Leray}.
Thus the result follows from
Lemmas \ref{lemma-coherent-internal-hom} and
\ref{lemma-direct-image-coherent-bdd-below}.
\end{proof}
\section{Descent finiteness properties of complexes}
\label{section-descent-finiteness}
\noindent
This section is the analogue of
Descent, Section \ref{descent-section-descent-finiteness}
for objects of the derived category of a scheme.
The easiest such result is probably the following.
\begin{lemma}
\label{lemma-tor-amplitude-descends}
Let $f : X \to Y$ be a surjective flat morphism of schemes
(or more generally locally ringed spaces).
Let $E \in D(\mathcal{O}_Y)$. Let $a, b \in \mathbf{Z}$.
Then $E$ has tor-amplitude in $[a, b]$ if and only if
$Lf^*E$ has tor-amplitude in $[a, b]$.
\end{lemma}
\begin{proof}
Pullback always preserves tor-amplitude, see
Cohomology, Lemma \ref{cohomology-lemma-tor-amplitude-pullback}.
We may check tor-amplitude in $[a, b]$ on stalks, see
Cohomology, Lemma \ref{cohomology-lemma-tor-amplitude-stalk}.
A flat local ring homomorphism is faithfully flat by
Algebra, Lemma \ref{algebra-lemma-local-flat-ff}.
Thus the result follows from
More on Algebra, Lemma
\ref{more-algebra-lemma-flat-descent-tor-amplitude}.
\end{proof}
\begin{lemma}
\label{lemma-pseudo-coherent-descends-fpqc}
Let $\{f_i : X_i \to X\}$ be an fpqc covering of schemes. Let
$E \in D_\QCoh(\mathcal{O}_X)$. Let $m \in \mathbf{Z}$.
Then $E$ is $m$-pseudo-coherent if and only if each
$Lf_i^*E$ is $m$-pseudo-coherent.
\end{lemma}
\begin{proof}
Pullback always preserves $m$-pseudo-coherence, see
Cohomology, Lemma \ref{cohomology-lemma-pseudo-coherent-pullback}.
Conversely, assume that $Lf_i^*E$ is $m$-pseudo-coherent for all $i$.
Let $U \subset X$ be an affine open. It suffices to prove that
$E|_U$ is $m$-pseudo-coherent. Since $\{f_i : X_i \to X\}$ is an
fpqc covering, we can find finitely many affine open $V_j \subset X_{a(j)}$
such that $f_{a(j)}(V_j) \subset U$ and $U = \bigcup f_{a(j)}(V_j)$.
Set $V = \coprod V_i$.
Thus we may replace $X$ by $U$ and $\{f_i : X_i \to X\}$ by
$\{V \to U\}$ and assume that $X$ is affine and our covering
is given by a single surjective flat morphism $\{f : Y \to X\}$
of affine schemes. In this case the result follows from
More on Algebra, Lemma \ref{more-algebra-lemma-flat-descent-pseudo-coherent}
via Lemmas \ref{lemma-affine-compare-bounded} and
\ref{lemma-pseudo-coherent-affine}.
\end{proof}
\begin{lemma}
\label{lemma-pseudo-coherent-descends-fppf}
Let $\{f_i : X_i \to X\}$ be an fppf covering of schemes. Let
$E \in D(\mathcal{O}_X)$. Let $m \in \mathbf{Z}$.
Then $E$ is $m$-pseudo-coherent if and only if each
$Lf_i^*E$ is $m$-pseudo-coherent.
\end{lemma}
\begin{proof}
Pullback always preserves $m$-pseudo-coherence, see
Cohomology, Lemma \ref{cohomology-lemma-pseudo-coherent-pullback}.
Conversely, assume that $Lf_i^*E$ is $m$-pseudo-coherent for all $i$.
Let $U \subset X$ be an affine open. It suffices to prove that
$E|_U$ is $m$-pseudo-coherent. Since $\{f_i : X_i \to X\}$ is an
fppf covering, we can find finitely many affine open $V_j \subset X_{a(j)}$
such that $f_{a(j)}(V_j) \subset U$ and $U = \bigcup f_{a(j)}(V_j)$.
Set $V = \coprod V_i$.
Thus we may replace $X$ by $U$ and $\{f_i : X_i \to X\}$ by
$\{V \to U\}$ and assume that $X$ is affine and our covering
is given by a single surjective flat morphism $\{f : Y \to X\}$
of finite presentation.
\medskip\noindent
Since $f$ is flat the derived functor $Lf^*$ is just given by $f^*$ and $f^*$
is exact. Hence $H^i(Lf^*E) = f^*H^i(E)$. Since $Lf^*E$ is $m$-pseudo-coherent,
we see that $Lf^*E \in D^-(\mathcal{O}_Y)$. Since $f$ is surjective and flat,
we see that $E \in D^-(\mathcal{O}_X)$. Let $i \in \mathbf{Z}$ be the largest
integer such that $H^i(E)$ is nonzero. If $i < m$, then we are done. Otherwise,
$f^*H^i(E)$ is a finite type $\mathcal{O}_Y$-module by
Cohomology, Lemma \ref{cohomology-lemma-finite-cohomology}.
Then by Descent, Lemma \ref{descent-lemma-finite-type-descends-fppf}
the $\mathcal{O}_X$-module $H^i(E)$ is of finite type.
Thus, after replacing $X$ by the members of a finite affine open covering,
we may assume there exists a map
$$
\alpha : \mathcal{O}_X^{\oplus n}[-i] \longrightarrow E
$$
such that $H^i(\alpha)$ is a surjection. Let $C$ be the cone of $\alpha$
in $D(\mathcal{O}_X)$. Pulling back to $Y$ and using
Cohomology, Lemma \ref{cohomology-lemma-cone-pseudo-coherent}
we find that $Lf^*C$ is $m$-pseudo-coherent. Moreover $H^j(C) = 0$
for $j \geq i$. Thus by induction on $i$ we see that $C$ is
$m$-pseudo-coherent. Using
Cohomology, Lemma \ref{cohomology-lemma-cone-pseudo-coherent}
again we conclude.
\end{proof}
\begin{lemma}
\label{lemma-perfect-descends-fpqc}
Let $\{f_i : X_i \to X\}$ be an fpqc covering of schemes. Let
$E \in D(\mathcal{O}_X)$. Then $E$ is perfect
if and only if each $Lf_i^*E$ is perfect.
\end{lemma}
\begin{proof}
Pullback always preserves perfect complexes, see
Cohomology, Lemma \ref{cohomology-lemma-perfect-pullback}.
Conversely, assume that $Lf_i^*E$ is perfect for all $i$.
Then the cohomology sheaves of each $Lf_i^*E$ are quasi-coherent, see
Lemma \ref{lemma-pseudo-coherent}
and
Cohomology, Lemma \ref{cohomology-lemma-perfect}.
Since the morphisms $f_i$ is flat we see that $H^p(Lf_i^*E) = f_i^*H^p(E)$.
Thus the cohomology sheaves of $E$ are quasi-coherent by
Descent, Proposition \ref{descent-proposition-fpqc-descent-quasi-coherent}.
Having said this the lemma follows formally from
Cohomology, Lemma \ref{cohomology-lemma-perfect}
and
Lemmas \ref{lemma-tor-amplitude-descends} and
\ref{lemma-pseudo-coherent-descends-fpqc}.
\end{proof}
\begin{lemma}
\label{lemma-closed-push-pseudo-coherent}
Let $i : Z \to X$ be a morphism of ringed spaces such that
$i$ is a closed immersion of underlying topological spaces and such that
$i_*\mathcal{O}_Z$ is pseudo-coherent as an $\mathcal{O}_X$-module.
Let $E \in D(\mathcal{O}_Z)$. Then $E$ is $m$-pseudo-coherent
if and only if $Ri_*E$ is $m$-pseudo-coherent.
\end{lemma}
\begin{proof}
Throughout this proof we will use that $i_*$ is an exact functor, and
hence that $Ri_* = i_*$, see Modules, Lemma \ref{modules-lemma-i-star-exact}.
\medskip\noindent
Assume $E$ is $m$-pseudo-coherent. Let $x \in X$. We will find a neighbourhood
of $x$ such that $i_*E$ is $m$-pseudo-coherent on it. If $x \not \in Z$
then this is clear. Thus we may assume $x \in Z$. We will use
that $U \cap Z$ for $x \in U \subset X$ open form a fundamental system of
neighbourhoods of $x$ in $Z$. After shrinking $X$ we may assume $E$ is
bounded above. We will argue by induction on
the largest integer $p$ such that $H^p(E)$ is nonzero. If $p < m$, then
there is nothing to prove. If $p \geq m$, then $H^p(E)$ is an
$\mathcal{O}_Z$-module of finite type, see
Cohomology, Lemma \ref{cohomology-lemma-finite-cohomology}.
Thus we may choose, after shrinking $X$, a map
$\mathcal{O}_Z^{\oplus n}[-p] \to E$ which induces a surjection
$\mathcal{O}_Z^{\oplus n} \to H^p(E)$. Choose a distinguished triangle
$$
\mathcal{O}_Z^{\oplus n}[-p] \to E \to C \to \mathcal{O}_Z^{\oplus n}[-p + 1]
$$
We see that $H^j(C) = 0$ for $j \geq p$ and that $C$ is $m$-pseudo-coherent
by Cohomology, Lemma \ref{cohomology-lemma-cone-pseudo-coherent}.
By induction we see that $i_*C$ is $m$-pseudo-coherent on $X$.
Since $i_*\mathcal{O}_Z$ is $m$-pseudo-coherent on $X$ as well, we conclude
from the distinguished triangle
$$
i_*\mathcal{O}_Z^{\oplus n}[-p] \to i_*E \to i_*C \to
i_*\mathcal{O}_Z^{\oplus n}[-p + 1]
$$
and
Cohomology, Lemma \ref{cohomology-lemma-cone-pseudo-coherent}
that $i_*E$ is $m$-pseudo-coherent.
\medskip\noindent
Assume that $i_*E$ is $m$-pseudo-coherent. Let $z \in Z$.
We will find a neighbourhood of $z$ such that $E$
is $m$-pseudo-coherent on it. We will use
that $U \cap Z$ for $z \in U \subset X$ open form a fundamental system of
neighbourhoods of $z$ in $Z$. After shrinking $X$ we may assume $i_*E$
and hence $E$ is bounded above. We will argue by induction on
the largest integer $p$ such that $H^p(E)$ is nonzero. If $p < m$, then
there is nothing to prove. If $p \geq m$, then $H^p(i_*E) = i_*H^p(E)$
is an $\mathcal{O}_X$-module of finite type, see
Cohomology, Lemma \ref{cohomology-lemma-finite-cohomology}.
Choose a complex $\mathcal{E}^\bullet$ of $\mathcal{O}_Z$-modules
representing $E$. We may choose, after shrinking $X$,
a map $\alpha : \mathcal{O}_X^{\oplus n}[-p] \to i_*\mathcal{E}^\bullet$
which induces a surjection
$\mathcal{O}_X^{\oplus n} \to i_*H^p(\mathcal{E}^\bullet)$.
By adjunction we find a map
$\alpha : \mathcal{O}_Z^{\oplus n}[-p] \to \mathcal{E}^\bullet$
which induces a surjection
$\mathcal{O}_Z^{\oplus n} \to H^p(\mathcal{E}^\bullet)$.
Choose a distinguished triangle
$$
\mathcal{O}_Z^{\oplus n}[-p] \to E \to C \to \mathcal{O}_Z^{\oplus n}[-p + 1]
$$
We see that $H^j(C) = 0$ for $j \geq p$. From the distinguished triangle
$$
i_*\mathcal{O}_Z^{\oplus n}[-p] \to i_*E \to i_*C \to
i_*\mathcal{O}_Z^{\oplus n}[-p + 1]
$$
the fact that $i_*\mathcal{O}_Z$ is pseudo-coherent
and
Cohomology, Lemma \ref{cohomology-lemma-cone-pseudo-coherent}
we conclude that $i_*C$ is $m$-pseudo-coherent.
By induction we conclude that $C$ is $m$-pseudo-coherent.
By Cohomology, Lemma \ref{cohomology-lemma-cone-pseudo-coherent}
again we conclude that $E$ is $m$-pseudo-coherent.
\end{proof}
\begin{lemma}
\label{lemma-finite-push-pseudo-coherent}
Let $f : X \to Y$ be a finite morphism of schemes such that
$f_*\mathcal{O}_X$ is pseudo-coherent as an
$\mathcal{O}_Y$-module\footnote{This means that $f$ is pseudo-coherent, see
More on Morphisms, Lemma
\ref{more-morphisms-lemma-finite-pseudo-coherent}.}.
Let $E \in D_\QCoh(\mathcal{O}_X)$. Then $E$ is $m$-pseudo-coherent
if and only if $Rf_*E$ is $m$-pseudo-coherent.
\end{lemma}
\begin{proof}
This is a translation of
More on Algebra, Lemma \ref{more-algebra-lemma-finite-push-pseudo-coherent}
into the language of schemes. To do the translation, use
Lemmas \ref{lemma-affine-compare-bounded} and
\ref{lemma-pseudo-coherent-affine}.
\end{proof}
\section{Lifting complexes}
\label{section-lift}
\noindent
Let $U \subset X$ be an open subspace of a ringed space
and denote $j : U \to X$ the inclusion morphism. The functor
$D(\mathcal{O}_X) \to D(\mathcal{O}_U)$ is essentially surjective as
$Rj_*$ is a right inverse to restriction.
In this section we extend this to complexes with quasi-coherent cohomology
sheaves, etc.
\begin{lemma}
\label{lemma-lift-quasi-coherent}
Let $X$ be a scheme and let $j : U \to X$ be a quasi-compact
open immersion. The functors
$$
D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_U)
\quad\text{and}\quad
D^+_\QCoh(\mathcal{O}_X) \to D^+_\QCoh(\mathcal{O}_U)
$$
are essentially surjective. If $X$ is quasi-compact, then the functors
$$
D^-_\QCoh(\mathcal{O}_X) \to D^-_\QCoh(\mathcal{O}_U)
\quad\text{and}\quad
D^b_\QCoh(\mathcal{O}_X) \to D^b_\QCoh(\mathcal{O}_U)
$$
are essentially surjective.
\end{lemma}
\begin{proof}
The argument preceding the lemma applies for the first case because $Rj_*$
maps $D_\QCoh(\mathcal{O}_U)$ into $D_\QCoh(\mathcal{O}_X)$
by Lemma \ref{lemma-quasi-coherence-direct-image}.
It is clear that $Rj_*$ maps
$D^+_\QCoh(\mathcal{O}_U)$ into
$D^+_\QCoh(\mathcal{O}_X)$
which implies the statement on bounded below complexes.
Finally, Lemma \ref{lemma-quasi-coherence-direct-image}
guarantees that $Rj_*$ maps
$D^-_\QCoh(\mathcal{O}_U)$ into
$D^-_\QCoh(\mathcal{O}_X)$
if $X$ is quasi-compact. Combining these two we obtain the last statement.
\end{proof}
\begin{lemma}
\label{lemma-lift-pseudo-coherent}
Let $X$ be an affine scheme and let $U \subset X$ be a quasi-compact
open subscheme. For any pseudo-coherent object $E$ of $D(\mathcal{O}_U)$
there exists a bounded above complex of finite free $\mathcal{O}_X$-modules
whose restriction to $U$ is isomorphic to $E$.
\end{lemma}
\begin{proof}
By Lemma \ref{lemma-pseudo-coherent} we see that $E$ is an object of
$D_\QCoh(\mathcal{O}_U)$. By
Lemma \ref{lemma-lift-quasi-coherent}
we may assume $E = E'|U$ for some object $E'$ of
$D_\QCoh(\mathcal{O}_X)$.
Write $X = \Spec(A)$. By Lemma \ref{lemma-affine-compare-bounded}
we can find a complex $M^\bullet$ of $A$-modules whose associated
complex of $\mathcal{O}_X$-modules is a representative of $E'$.
\medskip\noindent
Choose $f_1, \ldots, f_r \in A$ such that $U = D(f_1) \cup \ldots \cup D(f_r)$.
By Lemma \ref{lemma-pseudo-coherent-affine} the complexes
$M^\bullet_{f_j}$ are pseudo-coherent complexes of $A_{f_j}$-modules.
Let $n$ be an integer. Assume we have a map of complexes
$\alpha : F^\bullet \to M^\bullet$ where $F^\bullet$ is
bounded above, $F^i = 0$ for $i < n$, each $F^i$ is a finite free
$R$-module, such that
$$
H^i(\alpha_{f_j}) : H^i(F^\bullet_{f_j}) \to H^i(M^\bullet_{f_j})
$$
is an isomorphism for $i > n$ and surjective for $i = n$. Picture
$$
\xymatrix{
& F^n \ar[r] \ar[d]^\alpha & F^{n + 1} \ar[d]^\alpha \ar[r] & \ldots \\
M^{n-1} \ar[r] & M^n \ar[r] & M^{n + 1} \ar[r] & \ldots
}
$$
Since each $M^\bullet_{f_j}$ has vanishing cohomology
in large degrees we can find such a map for $n \gg 0$.
By induction on $n$ we are going to extend this to a map
of complexes $F^\bullet \to M^\bullet$
such that $H^i(\alpha_{f_j})$ is an isomorphism
for all $i$. The lemma will follow by taking $\widetilde{F^\bullet}$.
\medskip\noindent
The induction step will be to extend the diagram
above by adding $F^{n - 1}$. Let $C^\bullet$ be the cone on $\alpha$
(Derived Categories, Definition \ref{derived-definition-cone}).
The long exact sequence of cohomology shows that
$H^i(C^\bullet_{f_j}) = 0$ for $i \geq n$. By
More on Algebra, Lemma \ref{more-algebra-lemma-cone-pseudo-coherent}
we see that $C^\bullet_{f_j}$ is $(n - 1)$-pseudo-coherent. By
More on Algebra, Lemma \ref{more-algebra-lemma-finite-cohomology}
we see that $H^{-1}(C^\bullet_{f_j})$ is a finite $A_{f_j}$-module.
Choose a finite free $A$-module $F^{n - 1}$ and an $A$-module
$\beta : F^{n - 1} \to C^{-1}$ such that the composition
$F^{n - 1} \to C^{n - 1} \to C^n$ is zero and such that
$F^{n - 1}_{f_j}$ surjects onto $H^{n - 1}(C^\bullet_{f_j})$.
(Some details omitted; hint: clear denominators.)
Since $C^{n - 1} = M^{n - 1} \oplus F^n$
we can write $\beta = (\alpha^{n - 1}, -d^{n - 1})$. The vanishing of the
composition $F^{n - 1} \to C^{n - 1} \to C^n$ implies
these maps fit into a morphism of complexes
$$
\xymatrix{
& F^{n - 1} \ar[d]^{\alpha^{n - 1}} \ar[r]_{d^{n - 1}} &
F^n \ar[r] \ar[d]^\alpha &
F^{n + 1} \ar[d]^\alpha \ar[r] & \ldots \\
\ldots \ar[r] &
M^{n - 1} \ar[r] & M^n \ar[r] & M^{n + 1} \ar[r] & \ldots
}
$$
Moreover, these maps define a morphism of distinguished triangles
$$
\xymatrix{
(F^n \to \ldots) \ar[r] \ar[d] &
(F^{n-1} \to \ldots) \ar[r] \ar[d] &
F^{n-1} \ar[r] \ar[d]_\beta &
(F^n \to \ldots)[1] \ar[d] \\
(F^n \to \ldots) \ar[r] &
M^\bullet \ar[r] &
C^\bullet \ar[r] &
(F^n \to \ldots)[1]
}
$$
Hence our choice of $\beta$ implies that the map of complexes
$(F^{-1} \to \ldots) \to M^\bullet$ induces an isomorphism on
cohomology localized at $f_j$ in degrees $\geq n$ and a surjection
in degree $-1$. This finishes the proof of the lemma.
\end{proof}
\begin{lemma}
\label{lemma-vanishing-ext}
Let $X$ be a quasi-compact and quasi-separated scheme.
Let $E \in D^b_\QCoh(\mathcal{O}_X)$.
There exists an integer $n_0 > 0$ such that
$\text{Ext}^n_{D(\mathcal{O}_X)}(\mathcal{E}, E) = 0$
for every finite locally free
$\mathcal{O}_X$-module $\mathcal{E}$ and every $n \geq n_0$.
\end{lemma}
\begin{proof}
Recall that $\text{Ext}^n_{D(\mathcal{O}_X)}(\mathcal{E}, E) =
\Hom_{D(\mathcal{O}_X)}(\mathcal{E}, E[n])$. We have
Mayer-Vietoris for morphisms in the derived category, see
Cohomology, Lemma \ref{cohomology-lemma-mayer-vietoris-hom}.
Thus if $X = U \cup V$ and the result of the lemma holds
for $E|_U$, $E|_V$, and $E|_{U \cap V}$ for some bound $n_0$,
then the result holds for $E$ with bound $n_0 + 1$.
Thus it suffices to prove the lemma when $X$ is affine, see
Cohomology of Schemes, Lemma \ref{coherent-lemma-induction-principle}.
\medskip\noindent
Assume $X = \Spec(A)$ is affine. Choose a complex of $A$-modules
$M^\bullet$ whose associated complex of quasi-coherent modules
represents $E$, see Lemma \ref{lemma-affine-compare-bounded}.
Write $\mathcal{E} = \widetilde{P}$ for some $A$-module $P$.
Since $\mathcal{E}$ is finite locally free, we see that $P$
is a finite projective $A$-module. We have
\begin{align*}
\Hom_{D(\mathcal{O}_X)}(\mathcal{E}, E[n])
& =
\Hom_{D(A)}(P, M^\bullet[n]) \\
& =
\Hom_{K(A)}(P, M^\bullet[n]) \\
& =
\Hom_A(P, H^n(M^\bullet))
\end{align*}
The first equality by Lemma \ref{lemma-affine-compare-bounded},
the second equality by
Derived Categories, Lemma
\ref{derived-lemma-morphisms-from-projective-complex}, and
the final equality because $\Hom_A(P, -)$ is an exact functor.
As $E$ and hence $M^\bullet$ is bounded
we get zero for all sufficiently large $n$.
\end{proof}
\begin{lemma}
\label{lemma-lift-perfect-complex-plus-locally-free}
Let $X$ be an affine scheme. Let $U \subset X$ be a quasi-compact open.
For every perfect object $E$ of $D(\mathcal{O}_U)$ there exists an integer
$r$ and a finite locally free sheaf $\mathcal{F}$ on $U$ such that
$\mathcal{F}[-r] \oplus E$ is the restriction of a perfect object of
$D(\mathcal{O}_X)$.
\end{lemma}
\begin{proof}
Say $X = \Spec(A)$. Recall that a perfect complex is
pseudo-coherent, see
Cohomology, Lemma \ref{cohomology-lemma-perfect}.
By Lemma \ref{lemma-lift-pseudo-coherent} we can find a bounded above complex
$\mathcal{F}^\bullet$ of finite free $A$-modules such that $E$ is
isomorphic to $\mathcal{F}^\bullet|_U$ in $D(\mathcal{O}_U)$.
By Cohomology, Lemma \ref{cohomology-lemma-perfect} and since
$U$ is quasi-compact, we see that $E$ has finite tor dimension, say
$E$ has tor amplitude in $[a, b]$. Pick $r < a$ and set
$$
\mathcal{F} = \Ker(\mathcal{F}^{r} \to \mathcal{F}^{r + 1})
= \Im(\mathcal{F}^{r - 1} \to \mathcal{F}^r).
$$
Since $E$ has tor amplitude in $[a, b]$ we see that $\mathcal{F}|_U$ is
flat (Cohomology, Lemma \ref{cohomology-lemma-last-one-flat}).
Hence $\mathcal{F}|_U$ is flat and of finite presentation, thus finite
locally free (Properties, Lemma \ref{properties-lemma-finite-locally-free}).
It follows that
$$
(\mathcal{F} \to \mathcal{F}^r \to \mathcal{F}^{r + 1} \to \ldots )|_U
$$
is a strictly perfect complex on $U$ representing $E$.
We obtain a distinguished triangle
$$
\mathcal{F}|_U[- r - 1] \to E \to
(\mathcal{F}^r \to \mathcal{F}^{r + 1} \to \ldots )|_U \to
\mathcal{F}|_U[- r]
$$
Note that $(\mathcal{F}^r \to \mathcal{F}^{r + 1} \to \ldots )$ is
a perfect complex on $X$. To finish the proof it suffices to pick $r$
such that the map
$\mathcal{F}|_U[- r - 1] \to E$ is zero in $D(\mathcal{O}_U)$, see
Derived Categories, Lemma \ref{derived-lemma-split}. By
Lemma \ref{lemma-vanishing-ext} this holds if $r \ll 0$.
\end{proof}
\begin{lemma}
\label{lemma-lift-map}
Let $X$ be an affine scheme. Let $U \subset X$ be a quasi-compact open.
Let $E, E'$ be objects of $D_\QCoh(\mathcal{O}_X)$ with $E$ perfect.
For every map $\alpha : E|_U \to E'|_U$ there exist maps
$$
E \xleftarrow{\beta} E_1 \xrightarrow{\gamma} E'
$$
of perfect complexes on $X$ such that $\beta : E_1 \to E$ restricts to an
isomorphism on $U$ and such that $\alpha = \gamma|_U \circ \beta|_U^{-1}$.
Moreover we can assume $E_1 = E \otimes_{\mathcal{O}_X}^\mathbf{L} I$
for some perfect complex $I$ on $X$.
\end{lemma}
\begin{proof}
Write $X = \Spec(A)$. Write $U = D(f_1) \cup \ldots \cup D(f_r)$. Choose
finite complex of finite projective $A$-modules $M^\bullet$ representing
$E$ (Lemma \ref{lemma-perfect-affine}). Choose a complex of $A$-modules
$(M')^\bullet$ representing $E'$ (Lemma \ref{lemma-affine-compare-bounded}).
In this case the complex $H^\bullet = \Hom_A(M^\bullet, (M')^\bullet)$
is a complex of $A$-modules whose associated complex of quasi-coherent
$\mathcal{O}_X$-modules represents $R\SheafHom(E, E')$, see
Cohomology, Lemma \ref{cohomology-lemma-Rhom-strictly-perfect}.
Then $\alpha$ determines an element $s$ of $H^0(U, R\SheafHom(E, E'))$, see
Cohomology, Lemma \ref{cohomology-lemma-section-RHom-over-U}.
There exists an $e$ and a map
$$
\xi : I^\bullet(f_1^e, \ldots, f_r^e) \to \Hom_A(M^\bullet, (M')^\bullet)
$$
corresponding to $s$, see
Proposition \ref{proposition-represent-cohomology-class-on-open}.
Letting $E_1$ be the object corresponding to
complex of quasi-coherent $\mathcal{O}_X$-modules
associated to
$$
\text{Tot}(I^\bullet(f_1^e, \ldots, f_r^e) \otimes_A M^\bullet)
$$
we obtain $E_1 \to E$ using the canonical map
$I^\bullet(f_1^e, \ldots, f_r^e) \to A$ and $E_1 \to E'$
using $\xi$ and
Cohomology, Lemma \ref{cohomology-lemma-section-RHom-over-U}.
\end{proof}
\begin{lemma}
\label{lemma-lift-perfect-complex-plus-shift}
Let $X$ be an affine scheme. Let $U \subset X$ be a quasi-compact open.
For every perfect object $F$ of $D(\mathcal{O}_U)$
the object $F \oplus F[1]$ is the restriction of
a perfect object of $D(\mathcal{O}_X)$.
\end{lemma}
\begin{proof}
By Lemma \ref{lemma-lift-perfect-complex-plus-locally-free}
we can find a perfect object $E$ of $D(\mathcal{O}_X)$
such that $E|_U = \mathcal{F}[r] \oplus F$ for some finite locally
free $\mathcal{O}_U$-module $\mathcal{F}$.
By Lemma \ref{lemma-lift-map} we can find a morphism of
perfect complexes $\alpha : E_1 \to E$ such that $(E_1)|_U \cong E|_U$
and such that $\alpha|_U$ is the map
$$
\left(
\begin{matrix}
\text{id}_{\mathcal{F}[r]} & 0 \\
0 & 0
\end{matrix}
\right)
:
\mathcal{F}[r] \oplus F \to \mathcal{F}[r] \oplus F
$$
Then the cone on $\alpha$ is a solution.
\end{proof}
\begin{lemma}
\label{lemma-perfect-into-support-on-T}
Let $X$ be a quasi-compact and quasi-separated scheme.
Let $f \in \Gamma(X, \mathcal{O}_X)$.
For any morphism $\alpha : E \to E'$ in
$D_\QCoh(\mathcal{O}_X)$ such that
\begin{enumerate}
\item $E$ is perfect, and
\item $E'$ is supported on $T = V(f)$
\end{enumerate}
there exists an $n \geq 0$ such that $f^n \alpha = 0$.
\end{lemma}
\begin{proof}
We have Mayer-Vietoris for morphisms in the derived category, see
Cohomology, Lemma \ref{cohomology-lemma-mayer-vietoris-hom}.
Thus if $X = U \cup V$ and the result of the lemma holds
for $f|_U$, $f|_V$, and $f|_{U \cap V}$, then the result holds for $f$.
Thus it suffices to prove the lemma when $X$ is affine, see
Cohomology of Schemes, Lemma \ref{coherent-lemma-induction-principle}.
\medskip\noindent
Let $X = \Spec(A)$. Then $f \in A$. We will
use the equivalence $D(A) = D_\QCoh(X)$ of
Lemma \ref{lemma-affine-compare-bounded}
without further mention.
Represent $E$ by a finite complex of finite projective $A$-modules
$P^\bullet$. This is possible by Lemma \ref{lemma-perfect-affine}.
Let $t$ be the largest integer such that $P^t$ is nonzero.
The distinguished triangle
$$
P^t[-t] \to P^\bullet \to \sigma_{\leq t - 1}P^\bullet \to P^t[-t + 1]
$$
shows that by induction on the length of the complex $P^\bullet$
we can reduce to the case where $P^\bullet$ has a single nonzero term.
This and the shift functor reduces us to the case where $P^\bullet$
consists of a single finite projective $A$-module $P$ in degree $0$.
Represent $E'$ by a complex $M^\bullet$ of $A$-modules.
Then $\alpha$ corresponds to a map $P \to H^0(M^\bullet)$.
Since the module $H^0(M^\bullet)$ is supported on $V(f)$ by assumption (2)
we see that every element of $H^0(M^\bullet)$ is annihilated by a power
of $f$. Since $P$ is a finite $A$-module the map
$f^n\alpha : P \to H^0(M^\bullet)$ is zero for some $n$ as desired.
\end{proof}
\begin{lemma}
\label{lemma-lift-perfect-complex-plus-shift-support}
Let $X$ be an affine scheme. Let $T \subset X$ be a closed subset
such that $X \setminus T$ is quasi-compact. Let $U \subset X$ be a
quasi-compact open. For every perfect object $F$ of $D(\mathcal{O}_U)$
supported on $T \cap U$ the object $F \oplus F[1]$ is the restriction of
a perfect object $E$ of $D(\mathcal{O}_X)$ supported in $T$.
\end{lemma}
\begin{proof}
Say $T = V(g_1, \ldots, g_s)$. After replacing $g_j$ by a power we
may assume multiplication by $g_j$ is zero on $F$, see
Lemma \ref{lemma-perfect-into-support-on-T}. Choose $E$ as in
Lemma \ref{lemma-lift-perfect-complex-plus-shift}.
Note that $g_j : E \to E$ restricts to zero on $U$.
Choose a distinguished triangle
$$
E \xrightarrow{g_1} E \to C_1 \to E[1]
$$
By Derived Categories, Lemma \ref{derived-lemma-split}
the object $C_1$ restricts to
$F \oplus F[1] \oplus F[1] \oplus F[2]$ on $U$.
Moreover, $g_1 : C_1 \to C_1$ has square zero by
Derived Categories, Lemma \ref{derived-lemma-third-map-square-zero}.
Namely, the diagram
$$
\xymatrix{
E \ar[r] \ar[d]_0 & C_1 \ar[d]_{g_1} \ar[r] & E[1] \ar[d]_0 \\
E \ar[r] & C_1 \ar[r] & E[1]
}
$$
is commutative since the compositions $E \xrightarrow{g_1} E \to C_1$ and
$C_1 \to E[1] \xrightarrow{g_1} E[1]$ are zero. Continuing, setting
$C_{i + 1}$ equal to the cone of the map $g_i : C_i \to C_i$ we obtain
a perfect complex $C_s$ on $X$ supported on $T$
whose restriction to $U$ gives
$$
F \oplus F[1]^{\oplus s} \oplus F[2]^{\oplus {s \choose 2}}
\oplus \ldots \oplus F[s]
$$
Choose morphisms of perfect complexes $\beta : C' \to C_s$
and $\gamma : C' \to C_s$ as in Lemma \ref{lemma-lift-map}
such that $\beta|_U$ is an isomorphism and such that
$\gamma|_U \circ \beta|_U^{-1}$ is the morphism
$$
F \oplus F[1]^{\oplus s} \oplus F[2]^{\oplus {s \choose 2}}
\oplus \ldots \oplus F[s]
\to
F \oplus F[1]^{\oplus s} \oplus F[2]^{\oplus {s \choose 2}}
\oplus \ldots \oplus F[s]
$$
which is the identity on all summands except for $F$ where it is zero.
By Lemma \ref{lemma-lift-map} we also have
$C' = C_s \otimes^\mathbf{L} I$ for some perfect complex
$I$ on $X$. Hence the nullity of $g_j^2\text{id}_{C_s}$ implies the
same thing for $C'$. Thus $C'$ is supported on $T$ as well.
Then $\text{Cone}(\gamma)$ is a solution.
\end{proof}
\noindent
A special case of the following lemma can be found in
\cite{Neeman-Grothendieck}.
\begin{lemma}
\label{lemma-lift-map-from-perfect-complex-with-support}
Let $X$ be a quasi-compact and quasi-separated scheme.
Let $U \subset X$ be a quasi-compact open. Let $T \subset X$
be a closed subset with $X \setminus T$ retro-compact in $X$.
Let $E$ be an object of $D_\QCoh(\mathcal{O}_X)$.
Let $\alpha : P \to E|_U$ be a map where $P$ is a perfect object of
$D(\mathcal{O}_U)$ supported on $T \cap U$. 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 $R|_U$ in
$D(\mathcal{O}_U)$ compatible $\alpha$ and $\beta|_U$.
\end{lemma}
\begin{proof}
Since $X$ is quasi-compact there exists an integer $m$ such that
$X = U \cup V_1 \cup \ldots \cup V_m$ for some affine opens $V_j$ of $X$.
Arguing by induction on $m$ we see that we may assume $m = 1$. In other
words, we may assume that $X = U \cup V$ with $V$ affine. By
Lemma \ref{lemma-lift-perfect-complex-plus-shift-support}
we can choose a perfect object $Q$ in $D(\mathcal{O}_V)$
supported on $T \cap V$ and an isomorphism
$Q|_{U \cap V} \to (P \oplus P[1])|_{U \cap V}$.
By Lemma \ref{lemma-lift-map} we can replace $Q$ by
$Q \otimes^\mathbf{L} I$ (still supported on $T \cap V$)
and assume that the map
$$
Q|_{U \cap V} \to (P \oplus P[1])|_{U \cap V}
\longrightarrow P|_{U \cap V}
\longrightarrow
E|_{U \cap V}
$$
lifts to $Q \to E|_V$. By
Cohomology, Lemma \ref{cohomology-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 $T$ as desired.
\end{proof}
\begin{remark}
\label{remark-addendum}
The proof of Lemma \ref{lemma-lift-map-from-perfect-complex-with-support}
shows that
$$
R|_U = P \oplus P^{\oplus n_1}[1] \oplus \ldots \oplus P^{\oplus n_m}[m]
$$
for some $m \geq 0$ and $n_j \geq 0$. Thus the highest degree cohomology sheaf
of $R|_U$ equals that of $P$. By repeating the construction for the map
$P^{\oplus n_1}[1] \oplus \ldots \oplus P^{\oplus n_m}[m] \to R|_U$, taking
cones, and using induction we can achieve equality of cohomology sheaves
of $R|_U$ and $P$ above any given degree.
\end{remark}
\section{Approximation by perfect complexes}
\label{section-approximation}
\noindent
In this section we discuss the observation, due to Neeman and Lipman,
that a pseudo-coherent complex can be ``approximated'' by perfect complexes.
\begin{definition}
\label{definition-approximation-holds}
Let $X$ be a scheme. Consider triples $(T, E, m)$ where
\begin{enumerate}
\item $T \subset X$ is a closed subset,
\item $E$ is an object of $D_\QCoh(\mathcal{O}_X)$, 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. Namely, it is clear that if
approximation holds for the triple $(T, E, m)$, then
\begin{enumerate}
\item $E$ is $m$-pseudo-coherent, see
Cohomology, Definition \ref{cohomology-definition-pseudo-coherent}, and
\item the cohomology sheaves $H^i(E)$ are supported on $T$ for $i \geq m$.
\end{enumerate}
Moreover, the ``support'' of a perfect complex is a closed subscheme
whose complement is retrocompact in $X$ (details omitted). Hence we cannot
expect approximation to hold without this assumption on $T$.
This partly explains the conditions in the following definition.
\begin{definition}
\label{definition-approximation}
Let $X$ be a scheme. We say {\it approximation by perfect complexes holds}
on $X$ if for any closed subset $T \subset X$ with $X \setminus T$
retro-compact in $X$ 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}
\noindent
We will prove that approximation by perfect complexes holds for
quasi-compact and quasi-separated schemes. It seems that the second
condition is necessary for our method of proof. It is possible that the
first condition may be weakened to ``$E$ is $m$-pseudo-coherent''
by carefuly analyzing the arguments below.
\begin{lemma}
\label{lemma-open}
Let $X$ be a scheme. Let $U \subset X$ be an open subscheme.
Let $(T, E, m)$ be a triple as in
Definition \ref{definition-approximation-holds}.
If
\begin{enumerate}
\item $T \subset U$,
\item approximation holds for $(T, E|_U, m)$, 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 $j : U \to X$ be the inclusion morphism.
If $P \to E|_U$ is an approximation of the triple $(T, E|_U, m)$
over $U$, then $j_!P = Rj_*P \to j_!(E|_U) \to E$ is an approximation
of $(T, E, m)$ over $X$.
See Cohomology, Lemmas \ref{cohomology-lemma-pushforward-restriction} and
\ref{cohomology-lemma-pushforward-perfect}.
\end{proof}
\begin{lemma}
\label{lemma-approximation-affine}
Let $X$ be 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}
Say $X = \Spec(A)$. Write $T = V(f_1, \ldots, f_r)$.
(The case $r = 0$, i.e., $T = X$ follows immediately from
Lemma \ref{lemma-pseudo-coherent-affine} and the definitions.)
Let $(T, E, m)$ be a triple as in the lemma.
Let $t$ be the largest integer such that $H^t(E)$ is nonzero.
We will proceed by induction on $t$. The base case is $t < m$; in
this case the result is trivial. Now suppose that $t \geq m$. By
Cohomology, Lemma \ref{cohomology-lemma-finite-cohomology}
the sheaf $H^t(E)$ is of finite type. Since it is quasi-coherent
it is generated by finitely many sections
(Properties, Lemma \ref{properties-lemma-finite-type-module}).
For every $s \in \Gamma(X, H^t(E)) = H^t(X, E)$
(see proof of Lemma \ref{lemma-affine-compare-bounded})
we can find an $e > 0$ and a morphism $K_e[-t] \to E$
such that $s$ is in the image of
$H^0(K_e) = H^t(K_e[-t]) \to H^t(E)$, see
Lemma \ref{lemma-represent-cohomology-class-on-closed}.
Taking a finite direct sum of these maps we obtain a map
$P \to E$ where $P$ is a perfect complex supported on $T$,
where $H^i(P) = 0$ for $i > t$, and where $H^t(P) \to E$ is
surjective. Choose a distinguished triangle
$$
P \to E \to E' \to P[1]
$$
Then $E'$ is $m$-pseudo-coherent
(Cohomology, Lemma \ref{cohomology-lemma-cone-pseudo-coherent}),
$H^i(E') = 0$ for $i \geq t$, and
$H^i(E')$ is supported on $T$ for $i \geq m - r + 1$.
By induction we find an approximation $P' \to E'$
of $(T, E', m)$. Fit the composition $P' \to E' \to P[1]$
into a distinguished triangle $P \to P'' \to P' \to P[1]$
and extend the morphisms $P' \to E'$ and $P[1] \to P[1]$ into
a morphism of distinguished triangles
$$
\xymatrix{
P \ar[r] \ar[d] & P'' \ar[d] \ar[r] & P' \ar[d] \ar[r] & P[1] \ar[d] \\
P \ar[r] & E \ar[r] & E' \ar[r] & P[1]
}
$$
using TR3. Then $P''$ is a perfect complex
(Cohomology, Lemma \ref{cohomology-lemma-two-out-of-three-perfect})
supported on $T$.
An easy diagram chase shows that $P'' \to E$ is the desired
approximation.
\end{proof}
\begin{lemma}
\label{lemma-induction-step}
Let $X$ be a scheme. Let $X = U \cup V$ be an open covering
with $U$ quasi-compact, $V$ affine, and $U \cap V$ quasi-compact.
If approximation by perfect complexes holds on $U$,
then approximation holds on $X$.
\end{lemma}
\begin{proof}
Let $T \subset X$ be a closed subset with $X \setminus T$ retro-compact
in $X$. Let $r_U$ be the integer of Definition \ref{definition-approximation}
adapted to the pair $(U, T \cap U)$.
Set $T' = T \setminus U$. Note that
$T' \subset V$ and that $V \setminus T' = (X \setminus T) \cap U \cap V$
is quasi-compact by our assumption on $T$.
Let $r'$ be the number of affines needed to cover $V \setminus T'$.
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
$H^t(P) \to H^t(E|_U)$. By
Lemma \ref{lemma-lift-perfect-complex-plus-shift-support}
we can choose a perfect object $Q$ in $D(\mathcal{O}_V)$
supported on $T \cap V$ and an isomorphism
$Q|_{U \cap V} \to (P \oplus P[1])|_{U \cap V}$.
By Lemma \ref{lemma-lift-map} we can replace $Q$ by
$Q \otimes^\mathbf{L} I$
and assume that the map
$$
Q|_{U \cap V} \to (P \oplus P[1])|_{U \cap V}
\longrightarrow P|_{U \cap V}
\longrightarrow
E|_{U \cap V}
$$
lifts to $Q \to E|_V$. By
Cohomology, Lemma \ref{cohomology-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 distinguished triangle
$$
R \to E \to E' \to R[1]
$$
Then $E'$ is $(m - r)$-pseudo-coherent
(Cohomology, Lemma \ref{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 distinguished 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, Lemma \ref{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 $X$ be a quasi-compact and quasi-separated scheme.
Then approximation by perfect complexes holds on $X$.
\end{theorem}
\begin{proof}
This follows from the induction principle of
Cohomology of Schemes, Lemma \ref{coherent-lemma-induction-principle}
and Lemmas \ref{lemma-induction-step} and \ref{lemma-approximation-affine}.
\end{proof}
\section{Generating derived categories}
\label{section-generating}
\noindent
In this section we prove that the derived category
$D_\QCoh(\mathcal{O}_X)$ of a quasi-compact
and quasi-separated scheme can be generated by a single perfect object.
We urge the reader to read the proof of this result in the wonderful paper by
Bondal and van den Bergh, see \cite{BvdB}.
\begin{lemma}
\label{lemma-direct-summand-of-a-restriction}
Let $X$ be a quasi-compact and quasi-separated scheme.
Let $U$ be a quasi-compact open subscheme.
Let $P$ be a perfect object of $D(\mathcal{O}_U)$.
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{lemma}
\label{lemma-orthogonal-koszul-complex}
\begin{reference}
\cite[Proposition 6.1]{Bokstedt-Neeman}
\end{reference}
In Situation \ref{situation-complex} denote $j : U \to X$ the open
immersion and let $K$ be the perfect object of $D(\mathcal{O}_X)$
corresponding to the Koszul complex on $f_1, \ldots, f_r$ over $A$.
For $E \in D_\QCoh(\mathcal{O}_X)$ the following are equivalent
\begin{enumerate}
\item $E = Rj_*(E|_U)$, and
\item $\Hom_{D(\mathcal{O}_X)}(K[n], E) = 0$ for all $n \in \mathbf{Z}$.
\end{enumerate}
\end{lemma}
\begin{proof}
Choose a distinguished triangle $E \to Rj_*(E|_U) \to N \to E[1]$.
Observe that
$$
\Hom_{D(\mathcal{O}_X)}(K[n], Rj_*(E|_U)) =
\Hom_{D(\mathcal{O}_U)}(K|_U[n], E) = 0
$$
for all $n$ as $K|_U = 0$. Thus it suffices to prove the result for
$N$. In other words, we may assume that $E$ restricts to zero on $U$.
Observe that there are distinguished triangles
$$
K^\bullet(f_1^{e_1}, \ldots, f_i^{e'_i}, \ldots, f_r^{e_r}) \to
K^\bullet(f_1^{e_1}, \ldots, f_i^{e'_i + e''_i}, \ldots, f_r^{e_r}) \to
K^\bullet(f_1^{e_1}, \ldots, f_i^{e''_i}, \ldots, f_r^{e_r}) \to \ldots
$$
of Koszul complexes, see
More on Algebra, Lemma \ref{more-algebra-lemma-koszul-mult}.
Hence if $\Hom_{D(\mathcal{O}_X)}(K[n], E) = 0$ for all $n \in \mathbf{Z}$
then the same thing is true for the $K$ replaced by
$K_e$ as in Lemma \ref{lemma-represent-cohomology-class-on-closed}.
Thus our lemma follows immediately from that one and the fact that $E$
is determined by the complex of $A$-modules $R\Gamma(X, E)$, see
Lemma \ref{lemma-affine-compare-bounded}.
\end{proof}
\begin{theorem}
\label{theorem-bondal-van-den-Bergh}
Let $X$ be a quasi-compact and quasi-separated scheme. The category
$D_\QCoh(\mathcal{O}_X)$ can be generated by a single
perfect object. More precisely, there exists a perfect object
$P$ of $D(\mathcal{O}_X)$ such that for
$E \in D_\QCoh(\mathcal{O}_X)$ 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
Cohomology of Schemes, Lemma \ref{coherent-lemma-induction-principle}.
\medskip\noindent
If $X$ is affine, then $\mathcal{O}_X$ is a perfect generator.
This follows from Lemma \ref{lemma-affine-compare-bounded}.
\medskip\noindent
Assume that $X = U \cup V$ is an open covering with $U$ quasi-compact
such that the theorem holds for $U$ and $V$ is an affine open.
Let $P$ be a perfect object of $D(\mathcal{O}_U)$ which is a generator
for $D_\QCoh(\mathcal{O}_U)$. 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 $Z = X \setminus U$.
This is a closed subset of $V$ with $V \setminus Z$ quasi-compact.
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 $Z \subset V$ closed, the pushforward
$K' = R(V \to X)_*K$ is a perfect object of $D(\mathcal{O}_X)$ whose
restriction to $V$ is $K$ (see
Cohomology, Lemma \ref{cohomology-lemma-pushforward-perfect}).
We claim that $Q \oplus K'$ is a generator for
$D_\QCoh(\mathcal{O}_X)$.
\medskip\noindent
Let $E$ be an object of $D_\QCoh(\mathcal{O}_X)$ such that
there are no nontrivial maps from any shift of $Q \oplus K'$ into $E$.
By Cohomology, Lemma \ref{cohomology-lemma-pushforward-restriction}
we have $K' = R(V \to X)_! K$ and hence
$$
\Hom_{D(\mathcal{O}_X)}(K'[n], E) = \Hom_{D(\mathcal{O}_V)}(K[n], E|_V)
$$
Thus by Lemma \ref{lemma-orthogonal-koszul-complex} the vanishing of
these groups implies that $E|_V$ is isomorphic to
$R(U \cap V \to V)_*E|_{U \cap V}$. 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}
\noindent
The following result is an strengthening of
Theorem \ref{theorem-bondal-van-den-Bergh}
proved using exactly the same methods.
Let $T \subset X$ be a closed subset of a scheme $X$.
Let's denote $D_T(\mathcal{O}_X)$ the strictly full, saturated,
triangulated subcategory consisting of complexes whose
cohomology sheaves are supported on $T$.
\begin{lemma}
\label{lemma-generator-with-support}
\begin{reference}
\cite[Theorem 6.8]{Rouquier-dimensions}
\end{reference}
Let $X$ be a quasi-compact and quasi-separated scheme. Let $T \subset X$ be a
closed subset such that $X \setminus T$ is quasi-compact. With notation
as above, the category $D_{\QCoh, T}(\mathcal{O}_X)$ is generated by a
single perfect object.
\end{lemma}
\begin{proof}
We will prove this using the induction principle of
Cohomology of Schemes, Lemma \ref{coherent-lemma-induction-principle}.
\medskip\noindent
Assume $X = \Spec(A)$ is affine. In this case there exist
$f_1, \ldots, f_r \in A$ such that $T = V(f_1, \ldots, f_r)$.
Let $K$ be the Koszul complex on $f_1, \ldots, f_r$ as in
Lemma \ref{lemma-orthogonal-koszul-complex}.
Then $K$ is a perfect object with cohomology supported on
$T$ and hence a perfect object of $D_{\QCoh, T}(\mathcal{O}_X)$.
On the other hand, if $E \in D_{\QCoh, T}(\mathcal{O}_X)$ and
$\Hom(K, E[n]) = 0$ for all $n$, then
Lemma \ref{lemma-orthogonal-koszul-complex}
tells us that $E = Rj_*(E|_{X \setminus T}) = 0$.
Hence $K$ generates $D_{\QCoh, T}(\mathcal{O}_X)$,
(by our definition of generators of triangulated categories in
Derived Categories, Definition \ref{derived-definition-generators}).
\medskip\noindent
Assume that $X = U \cup V$ is an open covering with $U$ quasi-compact
such that the lemma holds for $U$ and $V$ is an affine open.
Let $P$ be a perfect object of $D(\mathcal{O}_U)$ supported on $T \cap U$
which is a generator for $D_{\QCoh, T \cap U}(\mathcal{O}_U)$. Using
Lemma \ref{lemma-lift-map-from-perfect-complex-with-support}
we may choose a perfect object $Q$ of $D(\mathcal{O}_X)$ supported on $T$
whose restriction to $U$ is a direct sum one of whose summands is $P$.
Write $V = \Spec(B)$. Let $Z = X \setminus U$. Then $Z$ is a closed subset
of $V$ such that $V \setminus Z$ is quasi-compact. As $X$ is quasi-separated,
it follows that $Z \cap T$ is a closed subset of $V$ such that
$W = V \setminus (Z \cap T)$ is quasi-compact. Thus we can choose
$g_1, \ldots, g_s \in B$ such that $Z \cap T = V(g_1, \ldots, g_r)$.
Let $K \in D(\mathcal{O}_V)$ be the perfect object corresponding to the
Koszul complex on $g_1, \ldots, g_s$ over $B$. Note that since $K$ is
supported on $(Z \cap T) \subset V$ closed, the pushforward
$K' = R(V \to X)_*K$ is a perfect object of $D(\mathcal{O}_X)$ whose
restriction to $V$ is $K$ (see
Cohomology, Lemma \ref{cohomology-lemma-pushforward-perfect}).
We claim that $Q \oplus K'$ is a generator for
$D_{\QCoh, T}(\mathcal{O}_X)$.
\medskip\noindent
Let $E$ be an object of $D_{\QCoh, T}(\mathcal{O}_X)$ such that
there are no nontrivial maps from any shift of $Q \oplus K'$ into $E$.
By Cohomology, Lemma \ref{cohomology-lemma-pushforward-restriction}
we have $K' = R(V \to X)_! K$ and hence
$$
\Hom_{D(\mathcal{O}_X)}(K'[n], E) = \Hom_{D(\mathcal{O}_V)}(K[n], E|_V)
$$
Thus by Lemma \ref{lemma-orthogonal-koszul-complex} we have
$E|_V = Rj_*E|_W$ where $j : W \to V$ is the inclusion. Picture
$$
\xymatrix{
W \ar[r]_j & V & Z \cap T \ar[l] \ar[d] \\
U \cap V \ar[u]^{j'} \ar[ru]_{j''} & & Z \ar[lu]
}
$$
Since $E$ is supported on $T$ we see that $E|_W$ is supported on
$T \cap W = T \cap U \cap V$ which is closed in $W$.
We conclude that
$$
E|_V = Rj_*(E|_W) = Rj_*(Rj'_*(E|_{U \cap V})) = Rj''_*(E|_{U \cap V})
$$
where the second equality is part (1) of
Cohomology, Lemma \ref{cohomology-lemma-pushforward-restriction}.
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{An example generator}
\label{section-example-generator}
\noindent
In this section we prove that the derived category of projective
space over a ring is generated by a vector bundle, in fact a direct
sum of shifts of the structure sheaf.
\medskip\noindent
The following lemma says that $\bigoplus_{n \geq 0} \mathcal{L}^{\otimes -n}$
is a generator if $\mathcal{L}$ is ample.
\begin{lemma}
\label{lemma-nonzero-some-cohomology}
Let $X$ be a scheme and $\mathcal{L}$ an ample invertible
$\mathcal{O}_X$-module. If $K$ is a nonzero object of
$D_\QCoh(\mathcal{O}_X)$, then for some $n \geq 0$ and $p \in \mathbf{Z}$
the cohomology group
$H^p(X, K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{L}^{\otimes n})$
is nonzero.
\end{lemma}
\begin{proof}
Recall that as $X$ has an ample invertible sheaf, it is quasi-compact
and separated (Properties, Definition \ref{properties-definition-ample} and
Lemma \ref{properties-lemma-affine-s-opens-cover-quasi-separated}).
Thus we may apply
Proposition \ref{proposition-quasi-compact-affine-diagonal}
and represent $K$ by a complex $\mathcal{F}^\bullet$ of
quasi-coherent modules. Pick any $p$ such that
$\mathcal{H}^p = \Ker(\mathcal{F}^p \to \mathcal{F}^{p + 1})/
\Im(\mathcal{F}^{p - 1} \to \mathcal{F}^p)$ is nonzero.
Choose a point $x \in X$ such that the stalk $\mathcal{H}^p_x$ is
nonzero. Choose an $n \geq 0$ and $s \in \Gamma(X, \mathcal{L}^{\otimes n})$
such that $X_s$ is an affine open neighbourhood of $x$.
Choose $\tau \in \mathcal{H}^p(X_s)$ which maps to a nonzero
element of the stalk $\mathcal{H}^p_x$; this is possible
as $\mathcal{H}^p$ is quasi-coherent and $X_s$ is affine.
Since taking sections over $X_s$ is an exact functor on
quasi-coherent modules, we can find a section $\tau' \in \mathcal{F}^p(X_s)$
mapping to zero in $\mathcal{F}^{p + 1}(X_s)$ and mapping to
$\tau$ in $\mathcal{H}^p(X_s)$. By
Properties, Lemma \ref{properties-lemma-invert-s-sections}
there exists an $m$ such that $\tau' \otimes s^{\otimes m}$
is the image of a section
$\tau'' \in \Gamma(X, \mathcal{F}^p \otimes \mathcal{L}^{\otimes mn})$.
Applying the same lemma once more, we find $l \geq 0$ such that
$\tau'' \otimes s^{\otimes l}$ maps to zero in
$\mathcal{F}^{p + 1} \otimes \mathcal{L}^{\otimes (m + l)n}$.
Then $\tau''$ gives a nonzero class in
$H^p(X, K \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{L}^{(m + l)n})$
as desired.
\end{proof}
\begin{lemma}
\label{lemma-construct-the-next-one}
Let $A$ be a ring. Let $X = \mathbf{P}^n_A$. For every $a \in \mathbf{Z}$
there exists an exact complex
$$
0 \to \mathcal{O}_X(a) \to \ldots
\to \mathcal{O}_X(a + i)^{\oplus {n + 1 \choose i}} \to
\ldots \to \mathcal{O}_X(a + n + 1) \to 0
$$
of vectorbundles on $X$.
\end{lemma}
\begin{proof}
Recall that $\mathbf{P}^n_A$ is $\text{Proj}(A[X_0, \ldots, X_n])$, see
Constructions, Definition \ref{constructions-definition-projective-space}.
Consider the Koszul complex
$$
K_\bullet = K_\bullet(A[X_0, \ldots, X_n], X_0, \ldots, X_n)
$$
over $S = A[X_0, \ldots, X_n]$ on $X_0, \ldots, X_n$.
Since $X_0, \ldots, X_n$ is clearly a regular sequence in the
polynomial ring $S$, we see that
(More on Algebra, Lemma \ref{more-algebra-lemma-regular-koszul-regular})
that the Koszul complex $K_\bullet$ is exact, except in degree $0$
where the cohomology is $S/(X_0, \ldots, X_n)$.
Note that $K_\bullet$ becomes a complex of graded modules if we
put the generators of $K_i$ in degree $+i$. In other words an
exact complex
$$
0 \to S(-n - 1) \to \ldots \to S(-n - 1 + i)^{\oplus {n \choose i}} \to \ldots
\to S \to S/(X_0, \ldots, X_n) \to 0
$$
Applying the exact functor $\tilde{\ }$ functor of Constructions,
Lemma \ref{constructions-lemma-proj-sheaves} and using that
the last term is in the kernel of this functor,
we obtain the exact complex
$$
0 \to \mathcal{O}_X(-n - 1) \to \ldots
\to \mathcal{O}_X(-n - 1 + i)^{\oplus {n + 1 \choose i}} \to
\ldots \to \mathcal{O}_X \to 0
$$
Twisting by the invertible sheaves $\mathcal{O}_X(n + a)$
we get the exact complexes of the lemma.
\end{proof}
\begin{lemma}
\label{lemma-generator-P1}
Let $A$ be a ring. Let $X = \mathbf{P}^n_A$. Then
$$
E =
\mathcal{O}_X \oplus \mathcal{O}_X(-1) \oplus \ldots \oplus \mathcal{O}_X(-n)
$$
is a generator
(Derived Categories, Definition \ref{derived-definition-generators})
of $D_\QCoh(X)$.
\end{lemma}
\begin{proof}
Let $K \in D_\QCoh(\mathcal{O}_X)$. Assume
$\Hom(E, K[p]) = 0$ for all $p \in \mathbf{Z}$.
We have to show that $K = 0$.
By Derived Categories, Lemma
\ref{derived-lemma-right-orthogonal}
we see that $\Hom(E', K[p])$ is zero for all $E' \in \langle E \rangle$
and $p \in \mathbf{Z}$.
By Lemma \ref{lemma-construct-the-next-one}
applied with $a = -n - 1$
we see that $\mathcal{O}_X(-n - 1) \in \langle E \rangle$
because it is quasi-isomorphic to a finite complex
whose terms are finite direct sums of summands of $E$.
Repeating the argument with $a = -n - 2$ we see that
$\mathcal{O}_X(-n - 2) \in \langle E \rangle$.
Arguing by induction we find that $\mathcal{O}_X(-m) \in \langle E \rangle$
for all $m \geq 0$.
Since
$$
\Hom(\mathcal{O}_X(-m), K[p]) =
H^p(X, K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{O}_X(m)) =
H^p(X, K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{O}_X(1)^{\otimes m})
$$
we conclude that $K = 0$ by Lemma \ref{lemma-nonzero-some-cohomology}.
(This also uses that $\mathcal{O}_X(1)$ is an ample
invertible sheaf on $X$ which follows from
Properties, Lemma \ref{properties-lemma-open-in-proj-ample}.)
\end{proof}
\begin{remark}
\label{remark-pullback-generator}
Let $f : X \to Y$ be a morphism of quasi-compact and quasi-separated schemes.
Let $E \in D_\QCoh(\mathcal{O}_Y)$ be a generator
(see Theorem \ref{theorem-bondal-van-den-Bergh}).
Then the following are equivalent
\begin{enumerate}
\item for $K \in D_\QCoh(\mathcal{O}_X)$ we have
$Rf_*K = 0$ if and only if $K = 0$,
\item $Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$
reflects isomorphisms, and
\item $Lf^*E$ is a generator for $D_\QCoh(\mathcal{O}_X)$.
\end{enumerate}
The equivalence between (1) and (2) is a formal consequence of the fact that
$Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$ is an
exact functor of triangulated categories. Similarly, the equivalence
between (1) and (3) follows formally from the fact that $Lf^*$
is the left adjoint to $Rf_*$.
These conditions hold if $f$ is affine (Lemma \ref{lemma-affine-morphism})
or if $f$ is an open immersion, or if $f$ is a composition of such.
We conclude that
\begin{enumerate}
\item if $X$ is a quasi-affine scheme then $\mathcal{O}_X$ is a generator
for $D_\QCoh(\mathcal{O}_X)$,
\item if $X \subset \mathbf{P}^n_A$ is a quasi-compact
locally closed subscheme, then
$\mathcal{O}_X \oplus \mathcal{O}_X(-1) \oplus \ldots \oplus \mathcal{O}_X(-n)$
is a generator for $D_\QCoh(\mathcal{O}_X)$ by
Lemma \ref{lemma-generator-P1}.
\end{enumerate}
\end{remark}
\section{Compact and perfect objects}
\label{section-compact}
\noindent
Let $X$ be a Noetherian scheme of finite dimension. By
Cohomology, Proposition \ref{cohomology-proposition-vanishing-Noetherian}
and
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-when-jshriek-compact}
the sheaves of modules $j_!\mathcal{O}_U$ are compact objects
of $D(\mathcal{O}_X)$ for all opens $U \subset X$.
These sheaves are typically not quasi-coherent, hence these
do not give perfect object of the derived category $D(\mathcal{O}_X)$.
However, if we restrict ourselves to complexes with quasi-coherent
cohomology sheaves, then this does not happen.
Here is the precise statement.
\begin{proposition}
\label{proposition-compact-is-perfect}
Let $X$ be a quasi-compact and quasi-separated scheme.
An object of $D_\QCoh(\mathcal{O}_X)$ is compact
if and only if it is perfect.
\end{proposition}
\begin{proof}
By Cohomology, Lemma \ref{cohomology-lemma-perfect-is-compact}
the perfect objects define compact objects of $D(\mathcal{O}_X)$.
Conversely, let $K$ be a compact object of $D_\QCoh(\mathcal{O}_X)$.
To show that $K$ is perfect, it suffices to show that
$K|_U$ is perfect for every affine open $U \subset X$, see
Cohomology, Lemma \ref{cohomology-lemma-perfect-independent-representative}.
Observe that $j : U \to X$ is a quasi-compact and separated morphism.
Hence
$Rj_* : D_\QCoh(\mathcal{O}_U) \to D_\QCoh(\mathcal{O}_X)$
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 compact object of $D_\QCoh(\mathcal{O}_U)$.
Hence we reduce to the case that $X$ is affine.
\medskip\noindent
Assume $X = \Spec(A)$ is affine. By Lemma \ref{lemma-affine-compare-bounded}
the problem is translated into the same problem for $D(A)$.
For $D(A)$ the result is
More on Algebra, Proposition \ref{more-algebra-proposition-perfect-is-compact}.
\end{proof}
\noindent
The following result is a strengthening of
Proposition \ref{proposition-compact-is-perfect}.
Let $T \subset X$ be a closed subset of a scheme $X$. As before
$D_T(\mathcal{O}_X)$ denotes the the strictly full, saturated,
triangulated subcategory consisting of complexes whose
cohomology sheaves are supported on $T$. Since taking direct
sums commutes with taking cohomology sheaves, it follows
that $D_T(\mathcal{O}_X)$ has direct sums and that they are equal
to direct sums in $D(\mathcal{O}_X)$.
\begin{lemma}