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 \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}