\input{preamble} % OK, start here. % \begin{document} \title{Derived Categories of Varieties} \maketitle \phantomsection \label{section-phantom} \tableofcontents \section{Introduction} \label{section-introduction} \noindent In this chapter we continue the discussion started in Derived Categories of Schemes, Section \ref{perfect-section-introduction}. We will discuss Fourier-Mukai transforms, first studied by Mukai in \cite{Mukai}. We will prove Orlov's theorem on derived equivalences (\cite{Orlov-K3}). We also discuss the countability of derived equivalence classes proved by Anel and To\"en in \cite{AT}. \medskip\noindent A good introduction to this material is the book \cite{Huybrechts} by Daniel Huybrechts. Some other papers which helped popularize this topic are \begin{enumerate} \item the paper by Bondal and Kapranov, see \cite{Bondal-Kapranov} \item the paper by Bondal and Orlov, see \cite{Bondal-Orlov} \item the paper by Bondal and Van den Bergh, see \cite{BvdB} \item the papers by Beilinson, see \cite{Beilinson} and \cite{Beilinson-derived} \item the paper by Orlov, see \cite{Orlov-AV} \item the paper by Orlov, see \cite{Orlov-motives} \item the paper by Rouquier, see \cite{Rouquier-dimensions} \item there are many more we could mention here. \end{enumerate} \section{Conventions and notation} \label{section-conventions} \noindent Let $k$ be a field. A $k$-linear triangulated category $\mathcal{T}$ is a triangulated category (Derived Categories, Section \ref{derived-section-triangulated-definitions}) which is endowed with a $k$-linear structure (Differential Graded Algebra, Section \ref{dga-section-linear}) such that the translation functors $[n] : \mathcal{T} \to \mathcal{T}$ are $k$-linear for all $n \in \mathbf{Z}$. \medskip\noindent Let $k$ be a field. We denote $\text{Vect}_k$ the category of $k$-vector spaces. For a $k$-vector space $V$ we denote $V^\vee$ the $k$-linear dual of $V$, i.e., $V^\vee = \Hom_k(V, k)$. \medskip\noindent Let $X$ be a scheme. We denote $D_{perf}(\mathcal{O}_X)$ the full subcategory of $D(\mathcal{O}_X)$ consisting of perfect complexes (Cohomology, Section \ref{cohomology-section-perfect}). If $X$ is Noetherian then $D_{perf}(\mathcal{O}_X) \subset D^b_{\textit{Coh}}(\mathcal{O}_X)$, see Derived Categories of Schemes, Lemma \ref{perfect-lemma-perfect-on-noetherian}. If $X$ is Noetherian and regular, then $D_{perf}(\mathcal{O}_X) = D^b_{\textit{Coh}}(\mathcal{O}_X)$, see Derived Categories of Schemes, Lemma \ref{perfect-lemma-perfect-on-regular}. \medskip\noindent Let $k$ be a field. Let $X$ and $Y$ be schemes over $k$. In this situation we will write $X \times Y$ instead of $X \times_{\Spec(k)} Y$. \medskip\noindent Let $S$ be a scheme. Let $X$, $Y$ be schemes over $S$. Let $\mathcal{F}$ be a $\mathcal{O}_X$-module and let $\mathcal{G}$ be a $\mathcal{O}_Y$-module. We set $$ \mathcal{F} \boxtimes \mathcal{G} = \text{pr}_1^*\mathcal{F} \otimes_{\mathcal{O}_{X \times_S Y}} \text{pr}_2^*\mathcal{G} $$ as $\mathcal{O}_{X \times_S Y}$-modules. If $K \in D(\mathcal{O}_X)$ and $M \in D(\mathcal{O}_Y)$ then we set $$ K \boxtimes M = L\text{pr}_1^*K \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} L\text{pr}_2^*M $$ as an object of $D(\mathcal{O}_{X \times_S Y})$. Thus our notation is potentially ambiguous, but context should make it clear which of the two is meant. \section{Serre functors} \label{section-Serre-functors} \noindent The material in this section is taken from \cite{Bondal-Kapranov}. \begin{lemma} \label{lemma-Serre-functor-exists} Let $k$ be a field. Let $\mathcal{T}$ be a $k$-linear triangulated category such that $\dim_k \Hom_\mathcal{T}(X, Y) < \infty$ for all $X, Y \in \Ob(\mathcal{T})$. The following are equivalent \begin{enumerate} \item there exists a $k$-linear equivalence $S : \mathcal{T} \to \mathcal{T}$ and $k$-linear isomorphisms $c_{X, Y} : \Hom_\mathcal{T}(X, Y) \to \Hom_\mathcal{T}(Y, S(X))^\vee$ functorial in $X, Y \in \Ob(\mathcal{T})$, \item for every $X \in \Ob(\mathcal{T})$ the functor $Y \mapsto \Hom_\mathcal{T}(X, Y)^\vee$ is representable and the functor $Y \mapsto \Hom_\mathcal{T}(Y, X)^\vee$ is corepresentable. \end{enumerate} \end{lemma} \begin{proof} Condition (1) implies (2) since given $(S, c)$ and $X \in \Ob(\mathcal{T})$ the object $S(X)$ represents the functor $Y \mapsto \Hom_\mathcal{T}(X, Y)^\vee$ and the object $S^{-1}(X)$ corepresents the functor $Y \mapsto \Hom_\mathcal{T}(Y, X)^\vee$. \medskip\noindent Assume (2). We will repeatedly use the Yoneda lemma, see Categories, Lemma \ref{categories-lemma-yoneda}. For every $X$ denote $S(X)$ the object representing the functor $Y \mapsto \Hom_\mathcal{T}(X, Y)^\vee$. Given $\varphi : X \to X'$, we obtain a unique arrow $S(\varphi) : S(X) \to S(X')$ determined by the corresponding transformation of functors $\Hom_\mathcal{T}(X, -)^\vee \to \Hom_\mathcal{T}(X', -)^\vee$. Thus $S$ is a functor and we obtain the isomorphisms $c_{X, Y}$ by construction. It remains to show that $S$ is an equivalence. For every $X$ denote $S'(X)$ the object corepresenting the functor $Y \mapsto \Hom_\mathcal{T}(Y, X)^\vee$. Arguing as above we find that $S'$ is a functor. We claim that $S'$ is quasi-inverse to $S$. To see this observe that $$ \Hom_\mathcal{T}(X, Y) = \Hom_\mathcal{T}(Y, S(X))^\vee = \Hom_\mathcal{T}(S'(S(X)), Y) $$ bifunctorially, i.e., we find $S' \circ S \cong \text{id}_\mathcal{T}$. Similarly, we have $$ \Hom_\mathcal{T}(Y, X) = \Hom_\mathcal{T}(S'(X), Y)^\vee = \Hom_\mathcal{T}(Y, S(S'(X))) $$ and we find $S \circ S' \cong \text{id}_\mathcal{T}$. \end{proof} \begin{definition} \label{definition-Serre-functor} Let $k$ be a field. Let $\mathcal{T}$ be a $k$-linear triangulated category such that $\dim_k \Hom_\mathcal{T}(X, Y) < \infty$ for all $X, Y \in \Ob(\mathcal{T})$. We say {\it a Serre functor exists} if the equivalent conditions of Lemma \ref{lemma-Serre-functor-exists} are satisfied. In this case a {\it Serre functor} is a $k$-linear equivalence $S : \mathcal{T} \to \mathcal{T}$ endowed with $k$-linear isomorphisms $c_{X, Y} : \Hom_\mathcal{T}(X, Y) \to \Hom_\mathcal{T}(Y, S(X))^\vee$ functorial in $X, Y \in \Ob(\mathcal{T})$. \end{definition} \begin{lemma} \label{lemma-Serre-functor} In the situation of Definition \ref{definition-Serre-functor}. If a Serre functor exists, then it is unique up to unique isomorphism and it is an exact functor of triangulated categories. \end{lemma} \begin{proof} Given a Serre functor $S$ the object $S(X)$ represents the functor $Y \mapsto \Hom_\mathcal{T}(X, Y)^\vee$. Thus the object $S(X)$ together with the functorial identification $\Hom_\mathcal{T}(X, Y)^\vee = \Hom_\mathcal{T}(Y, S(X))$ is determined up to unique isomorphism by the Yoneda lemma (Categories, Lemma \ref{categories-lemma-yoneda}). Moreover, for $\varphi : X \to X'$, the arrow $S(\varphi) : S(X) \to S(X')$ is uniquely determined by the corresponding transformation of functors $\Hom_\mathcal{T}(X, -)^\vee \to \Hom_\mathcal{T}(X', -)^\vee$. \medskip\noindent For objects $X, Y$ of $\mathcal{T}$ we have \begin{align*} \Hom(Y, S(X)[1])^\vee & = \Hom(Y[-1], S(X))^\vee \\ & = \Hom(X, Y[-1]) \\ & = \Hom(X[1], Y) \\ & = \Hom(Y, S(X[1]))^\vee \end{align*} By the Yoneda lemma we conclude that there is a unique isomorphism $S(X[1]) \to S(X)[1]$ inducing the isomorphism from top left to bottom right. Since each of the isomorphisms above is functorial in both $X$ and $Y$ we find that this defines an isomorphism of functors $S \circ [1] \to [1] \circ S$. \medskip\noindent Let $(A, B, C, f, g, h)$ be a distinguished triangle in $\mathcal{T}$. We have to show that the triangle $(S(A), S(B), S(C), S(f), S(g), S(h))$ is distinguished. Here we use the canonical isomorphism $S(A[1]) \to S(A)[1]$ constructed above to identify the target $S(A[1])$ of $S(h)$ with $S(A)[1]$. We first observe that for any $X$ in $\mathcal{T}$ the triangle $(S(A), S(B), S(C), S(f), S(g), S(h))$ induces a long exact sequence $$ \ldots \to \Hom(X, S(A)) \to \Hom(X, S(B)) \to \Hom(X, S(C)) \to \Hom(X, S(A)[1]) \to \ldots $$ of finite dimensional $k$-vector spaces. Namely, this sequence is $k$-linear dual of the sequence $$ \ldots \leftarrow \Hom(A, X) \leftarrow \Hom(B, X) \leftarrow \Hom(C, X) \leftarrow \Hom(A[1], X) \leftarrow \ldots $$ which is exact by Derived Categories, Lemma \ref{derived-lemma-representable-homological}. Next, we choose a distinguished triangle $(S(A), E, S(C), i, p, S(h))$ which is possible by axioms TR1 and TR2. We want to construct the dotted arrow making following diagram commute $$ \xymatrix{ S(C)[-1] \ar[r]_-{S(h[-1])} & S(A) \ar[r]_{S(f)} & S(B) \ar[r]_{S(g)} & S(C) \ar[r]_{S(h)} & S(A)[1] \\ S(C)[-1] \ar[r]^-{S(h[-1])} \ar@{=}[u] & S(A) \ar[r]^i \ar@{=}[u] & E \ar[r]^p \ar@{..>}[u]^\varphi & S(C) \ar[r]^{S(h)} \ar@{=}[u] & S(A)[1] \ar@{=}[u] } $$ Namely, if we have $\varphi$, then we claim for any $X$ the resulting map $\Hom(X, E) \to \Hom(X, S(B))$ will be an isomorphism of $k$-vector spaces. Namely, we will obtain a commutative diagram $$ \xymatrix{ \Hom(X, S(C)[-1]) \ar[r] & \Hom(X, S(A)) \ar[r] & \Hom(X, S(B)) \ar[r] & \Hom(X, S(C)) \ar[r] & \Hom(X, S(A)[1]) \\ \Hom(X, S(C)[-1]) \ar[r] \ar@{=}[u] & \Hom(X, S(A)) \ar[r] \ar@{=}[u] & \Hom(X, E) \ar[r] \ar[u]^\varphi & \Hom(X, S(C)) \ar[r] \ar@{=}[u] & \Hom(X, S(A)[1]) \ar@{=}[u] } $$ with exact rows (see above) and we can apply the 5 lemma (Homology, Lemma \ref{homology-lemma-five-lemma}) to see that the middle arrow is an isomorphism. By the Yoneda lemma we conclude that $\varphi$ is an isomorphism. To find $\varphi$ consider the following diagram $$ \xymatrix{ \Hom(E, S(C)) \ar[r] & \Hom(S(A), S(C)) \\ \Hom(E, S(B)) \ar[u] \ar[r] & \Hom(S(A), S(B)) \ar[u] } $$ The elements $p$ and $S(f)$ in positions $(0, 1)$ and $(1, 0)$ define a cohomology class $\xi$ in the total complex of this double complex. The existence of $\varphi$ is equivalent to whether $\xi$ is zero. If we take $k$-linear duals of this and we use the defining property of $S$ we obtain $$ \xymatrix{ \Hom(C, E) \ar[d] & \Hom(C, S(A)) \ar[l] \ar[d] \\ \Hom(B, E) & \Hom(B, S(A)) \ar[l] } $$ Since both $A \to B \to C$ and $S(A) \to E \to S(C)$ are distinguished triangles, we know by TR3 that given elements $\alpha \in \Hom(C, E)$ and $\beta \in \Hom(B, S(A))$ mapping to the same element in $\Hom(B, E)$, there exists an element in $\Hom(C, S(A))$ mapping to both $\alpha$ and $\beta$. In other words, the cohomology of the total complex associated to this double complex is zero in degree $1$, i.e., the degree corresponding to $\Hom(C, E) \oplus \Hom(B, S(A))$. Taking duals the same must be true for the previous one which concludes the proof. \end{proof} \section{Examples of Serre functors} \label{section-examples-Serre-functors} \noindent The lemma below is the standard example. \begin{lemma} \label{lemma-Serre-functor-Gorenstein-proper} Let $k$ be a field. Let $X$ be a proper scheme over $k$ which is Gorenstein. Consider the complex $\omega_X^\bullet$ of Duality for Schemes, Lemmas \ref{duality-lemma-duality-proper-over-field}. Then the functor $$ S : D_{perf}(\mathcal{O}_X) \longrightarrow D_{perf}(\mathcal{O}_X),\quad K \longmapsto S(K) = \omega_X^\bullet \otimes_{\mathcal{O}_X}^\mathbf{L} K $$ is a Serre functor. \end{lemma} \begin{proof} The statement make sense because $\dim \Hom_X(K, L) < \infty$ for $K, L \in D_{perf}(\mathcal{O}_X)$ by Derived Categories of Schemes, Lemma \ref{perfect-lemma-ext-finite}. Since $X$ is Gorenstein the dualizing complex $\omega_X^\bullet$ is an invertible object of $D(\mathcal{O}_X)$, see Duality for Schemes, Lemma \ref{duality-lemma-gorenstein}. In particular, locally on $X$ the complex $\omega_X^\bullet$ has one nonzero cohomology sheaf which is an invertible module, see Cohomology, Lemma \ref{cohomology-lemma-invertible-derived}. Thus $S(K)$ lies in $D_{perf}(\mathcal{O}_X)$. On the other hand, the invertibility of $\omega_X^\bullet$ clearly implies that $S$ is a self-equivalence of $D_{perf}(\mathcal{O}_X)$. Finally, we have to find an isomorphism $$ c_{K, L} : \Hom_X(K, L) \longrightarrow \Hom_X(L, \omega_X^\bullet \otimes_{\mathcal{O}_X}^\mathbf{L} K)^\vee $$ bifunctorially in $K, L$. To do this we use the canonical isomorphisms $$ \Hom_X(K, L) = H^0(X, L \otimes_{\mathcal{O}_X}^\mathbf{L} K^\vee) $$ and $$ \Hom_X(L, \omega_X^\bullet \otimes_{\mathcal{O}_X}^\mathbf{L} K) = H^0(X, \omega_X^\bullet \otimes_{\mathcal{O}_X}^\mathbf{L} K \otimes_{\mathcal{O}_X}^\mathbf{L} L^\vee) $$ given in Cohomology, Lemma \ref{cohomology-lemma-dual-perfect-complex}. Since $(L \otimes_{\mathcal{O}_X}^\mathbf{L} K^\vee)^\vee = (K^\vee)^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} L^\vee$ and since there is a canonical isomorphism $K \to (K^\vee)^\vee$ we find these $k$-vector spaces are canonically dual by Duality for Schemes, Lemma \ref{duality-lemma-duality-proper-over-field-perfect}. This produces the isomorphisms $c_{K, L}$. We omit the proof that these isomorphisms are functorial. \end{proof} \section{Characterizing coherent modules} \label{section-coherent} \noindent This section is in some sense a continuation of the discussion in Derived Categories of Schemes, Section \ref{perfect-section-pseudo-coherent} and More on Morphisms, Section \ref{more-morphisms-section-characterize-pseudo-coherent}. \medskip\noindent Before we can state the result we need some notation. Let $k$ be a field. Let $n \geq 0$ be an integer. Let $S = k[X_0, \ldots, X_n]$. For an integer $e$ denote $S_e \subset S$ the homogeneous polynomials of degree $e$. Consider the (noncommutative) $k$-algebra $$ R = \left( \begin{matrix} S_0 & S_1 & S_2 & \ldots & \ldots \\ 0 & S_0 & S_1 & \ldots & \ldots\\ 0 & 0 & S_0 & \ldots & \ldots \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & \ldots & \ldots & \ldots & S_0 \end{matrix} \right) $$ (with $n + 1$ rows and columns) with obvious multiplication and addition. \begin{lemma} \label{lemma-perfect-for-R} With $k$, $n$, and $R$ as above, for an object $K$ of $D(R)$ the following are equivalent \begin{enumerate} \item $\sum_{i \in \mathbf{Z}} \dim_k H^i(K) < \infty$, and \item $K$ is a compact object. \end{enumerate} \end{lemma} \begin{proof} If $K$ is a compact object, then $K$ can be represented by a complex $M^\bullet$ which is finite projective as a graded $R$-module, see Differential Graded Algebra, Lemma \ref{dga-lemma-compact}. Since $\dim_k R < \infty$ we conclude $\sum \dim_k M^i < \infty$ and a fortiori $\sum \dim_k H^i(M^\bullet) < \infty$. (One can also easily deduce this implication from the easier Differential Graded Algebra, Proposition \ref{dga-proposition-compact}.) \medskip\noindent Assume $K$ satisfies (1). Consider the distinguished triangle of trunctions $\tau_{\leq m}K \to K \to \tau_{\geq m + 1}K$, see Derived Categories, Remark \ref{derived-remark-truncation-distinguished-triangle}. It is clear that both $\tau_{\leq m}K$ and $\tau_{\geq m + 1} K$ satisfy (1). If we can show both are compact, then so is $K$, see Derived Categories, Lemma \ref{derived-lemma-compact-objects-subcategory}. Hence, arguing on the number of nonzero cohomology modules of $K$ we may assume $H^i(K)$ is nonzero only for one $i$. Shifting, we may assume $K$ is given by the complex consisting of a single finite dimensional $R$-module $M$ sitting in degree $0$. \medskip\noindent Since $\dim_k(M) < \infty$ we see that $M$ is Artinian as an $R$-module. Thus it suffices to show that every simple $R$-module represents a compact object of $D(R)$. Observe that $$ I = \left( \begin{matrix} 0 & S_1 & S_2 & \ldots & \ldots \\ 0 & 0 & S_1 & \ldots & \ldots\\ 0 & 0 & 0 & \ldots & \ldots \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & \ldots & \ldots & \ldots & 0 \end{matrix} \right) $$ is a nilpotent two sided ideal of $R$ and that $R/I$ is a commutative $k$-algebra isomorphic to a product of $n + 1$ copies of $k$ (placed along the diagonal in the matrix, i.e., $R/I$ can be lifted to a $k$-subalgebra of $R$). It follows that $R$ has exactly $n + 1$ isomorphism classes of simple modules $M_0, \ldots, M_n$ (sitting along the diagonal). Consider the right $R$-module $P_i$ of row vectors $$ P_i = \left( \begin{matrix} 0 & \ldots & 0 & S_0 & \ldots & S_{i - 1} & S_i \end{matrix} \right) $$ with obvious multiplication $P_i \times R \to P_i$. Then we see that $R \cong P_0 \oplus \ldots \oplus P_n$ as a right $R$-module. Since clearly $R$ is a compact object of $D(R)$, we conclude each $P_i$ is a compact object of $D(R)$. (We of course also conclude each $P_i$ is projective as an $R$-module, but this isn't what we have to show in this proof.) Clearly, $P_0 = M_0$ is the first of our simple $R$-modules. For $P_1$ we have a short exact sequence $$ 0 \to P_0^{\oplus n + 1} \to P_1 \to M_1 \to 0 $$ which proves that $M_1$ fits into a distinguished triangle whose other members are compact objects and hence $M_1$ is a compact object of $D(R)$. More generally, there exists a short exact sequence $$ 0 \to C_i \to P_i \to M_i \to 0 $$ where $C_i$ is a finite dimensional $R$-module whose simple constituents are isomorphic to $M_j$ for $j < i$. By induction, we first conclude that $C_i$ determines a compact object of $D(R)$ whereupon we conclude that $M_i$ does too as desired. \end{proof} \begin{lemma} \label{lemma-coherent-on-projective-space} Let $k$ be a field. Let $n \geq 0$. Let $K \in D_\QCoh(\mathcal{O}_{\mathbf{P}^n_k})$. The following are equivalent \begin{enumerate} \item $K$ is in $D^b_{\textit{Coh}}(\mathcal{O}_{\mathbf{P}^n_k})$, \item $\sum_{i \in \mathbf{Z}} \dim_k H^i(\mathbf{P}^n_k, E \otimes^\mathbf{L} K) < \infty$ for each perfect object $E$ of $D(\mathcal{O}_{\mathbf{P}^n_k})$, \item $\sum_{i \in \mathbf{Z}} \dim_k \Ext^i_{\mathbf{P}^n_k}(E, K) < \infty$ for each perfect object $E$ of $D(\mathcal{O}_{\mathbf{P}^n_k})$, \item $\sum_{i \in \mathbf{Z}} \dim_k H^i(\mathbf{P}^n_k, K \otimes^\mathbf{L} \mathcal{O}_{\mathbf{P}^n_k}(d)) < \infty$ for $d = 0, 1, \ldots, n$. \end{enumerate} \end{lemma} \begin{proof} Parts (2) and (3) are equivalent by Cohomology, Lemma \ref{cohomology-lemma-dual-perfect-complex}. If (1) is true, then for $E$ perfect the derived tensor product $E \otimes^\mathbf{L} K$ is in $D^b_{\textit{Coh}}(\mathcal{O}_{\mathbf{P}^n_k})$ and we see that (2) holds by Derived Categories of Schemes, Lemma \ref{perfect-lemma-direct-image-coherent}. It is clear that (2) implies (4) as $\mathcal{O}_{\mathbf{P}^n_k}(d)$ can be viewed as a perfect object of the derived category of $\mathbf{P}^n_k$. Thus it suffices to prove that (4) implies (1). \medskip\noindent Assume (4). Let $R$ be as in Lemma \ref{lemma-perfect-for-R}. Let $P = \bigoplus_{d = 0, \ldots, n} \mathcal{O}_{\mathbf{P}^n_k}(-d)$. Recall that $R = \text{End}_{\mathbf{P}^n_k}(P)$ whereas all other self-Exts of $P$ are zero and that $P$ determines an equivalence $- \otimes^\mathbf{L} P : D(R) \to D_\QCoh(\mathcal{O}_{\mathbf{P}^n_k})$ by Derived Categories of Schemes, Lemma \ref{perfect-lemma-Pn-module-category}. Say $K$ corresponds to $L$ in $D(R)$. Then \begin{align*} H^i(L) & = \Ext^i_{D(R)}(R, L) \\ & = \Ext^i_{\mathbf{P}^n_k}(P, K) \\ & = H^i(\mathbf{P}^n_k, K \otimes P^\vee) \\ & = \bigoplus\nolimits_{d = 0, \ldots, n} H^i(\mathbf{P}^n_k, K \otimes \mathcal{O}(d)) \end{align*} by Differential Graded Algebra, Lemma \ref{dga-lemma-upgrade-tensor-with-complex-derived} (and the fact that $- \otimes^\mathbf{L} P$ is an equivalence) and Cohomology, Lemma \ref{cohomology-lemma-dual-perfect-complex}. Thus our assumption (4) implies that $L$ satisfies condition (2) of Lemma \ref{lemma-perfect-for-R} and hence is a compact object of $D(R)$. Therefore $K$ is a compact object of $D_\QCoh(\mathcal{O}_{\mathbf{P}^n_k})$. Thus $K$ is perfect by Derived Categories of Schemes, Proposition \ref{perfect-proposition-compact-is-perfect}. Since $D_{perf}(\mathcal{O}_{\mathbf{P}^n_k}) = D^b_{\textit{Coh}}(\mathcal{O}_{\mathbf{P}^n_k})$ by Derived Categories of Schemes, Lemma \ref{perfect-lemma-perfect-on-regular} we conclude (1) holds. \end{proof} \begin{lemma} \label{lemma-finiteness} Let $X$ be a scheme proper over a field $k$. Let $K \in D^b_{\textit{Coh}}(\mathcal{O}_X)$ and let $E$ in $D(\mathcal{O}_X)$ be perfect. Then $\sum_{i \in \mathbf{Z}} \dim_k \Ext^i_X(E, K) < \infty$. \end{lemma} \begin{proof} This follows for example by combining Derived Categories of Schemes, Lemmas \ref{perfect-lemma-ext-finite} and \ref{perfect-lemma-ext-from-perfect-into-bounded-QCoh}. Alternative proof: combine Derived Categories of Schemes, Lemmas \ref{perfect-lemma-perfect-on-noetherian} and \ref{perfect-lemma-direct-image-coherent}. \end{proof} \begin{lemma} \label{lemma-characterize-dbcoh-projective} \begin{reference} \cite[Lemma 7.46]{Rouquier-dimensions} and implicit in \cite[Theorem A.1]{BvdB} \end{reference} Let $X$ be a projective scheme over a field $k$. Let $K \in \Ob(D_\QCoh(\mathcal{O}_X))$. The following are equivalent \begin{enumerate} \item $K \in D^b_{\textit{Coh}}(\mathcal{O}_X)$, and \item $\sum_{i \in \mathbf{Z}} \dim_k \Ext^i_X(E, K) < \infty$ for all perfect $E$ in $D(\mathcal{O}_X)$. \end{enumerate} \end{lemma} \begin{proof} The implication (1) $\Rightarrow$ (2) follows from Lemma \ref{lemma-finiteness}. \medskip\noindent Assume (2). Choose a closed immersion $i : X \to \mathbf{P}^n_k$. It suffices to show that $Ri_*K$ is in $D^b_{\textit{Coh}}(\mathbf{P}^n_k)$ since a quasi-coherent module $\mathcal{F}$ on $X$ is coherent, resp.\ zero if and only if $i_*\mathcal{F}$ is coherent, resp.\ zero. For a perfect object $E$ of $D(\mathcal{O}_{\mathbf{P}^n_k})$, $Li^*E$ is a perfect object of $D(\mathcal{O}_X)$ and $$ \Ext^q_{\mathbf{P}^n_k}(E, Ri_*K) = \Ext^q_X(Li^*E, K) $$ Hence by our assumption we see that $\sum_{q \in \mathbf{Z}} \dim_k \Ext^q_{\mathbf{P}^n_k}(E, Ri_*K) < \infty$. We conclude by Lemma \ref{lemma-coherent-on-projective-space}. \end{proof} \section{A representability theorem} \label{section-bondal-van-den-bergh} \noindent The material in this section is taken from \cite{BvdB}. \medskip\noindent Let $\mathcal{T}$ be a $k$-linear triangulated category. In this section we consider $k$-linear cohomological functors $H$ from $\mathcal{T}$ to the category of $k$-vector spaces. This will mean $H$ is a functor $$ H : \mathcal{T}^{opp} \longrightarrow \text{Vect}_k $$ which is $k$-linear such that for any distinguished triangle $X \to Y \to Z$ in $\mathcal{T}$ the sequence $H(Z) \to H(Y) \to H(X)$ is an exact sequence of $k$-vector spaces. See Derived Categories, Definition \ref{derived-definition-homological} and Differential Graded Algebra, Section \ref{dga-section-linear}. \begin{lemma} \label{lemma-maps-from-compact-filtered} Let $\mathcal{D}$ be a triangulated category. Let $\mathcal{D}' \subset \mathcal{D}$ be a full triangulated subcategory. Let $X \in \Ob(\mathcal{D})$. The category of arrows $E \to X$ with $E \in \Ob(\mathcal{D}')$ is filtered. \end{lemma} \begin{proof} We check the conditions of Categories, Definition \ref{categories-definition-directed}. The category is nonempty because it contains $0 \to X$. If $E_i \to X$, $i = 1, 2$ are objects, then $E_1 \oplus E_2 \to X$ is an object and there are morphisms $(E_i \to X) \to (E_1 \oplus E_2 \to X)$. Finally, suppose that $a, b : (E \to X) \to (E' \to X)$ are morphisms. Choose a distinguished triangle $E \xrightarrow{a - b} E' \to E''$ in $\mathcal{D}'$. By Axiom TR3 we obtain a morphism of triangles $$ \xymatrix{ E \ar[r]_{a - b} \ar[d] & E' \ar[d] \ar[r] & E'' \ar[d] \\ 0 \ar[r] & X \ar[r] & X } $$ and we find that the resulting arrow $(E' \to X) \to (E'' \to X)$ equalizes $a$ and $b$. \end{proof} \begin{lemma} \label{lemma-van-den-bergh} \begin{reference} \cite[Lemma 2.14]{CKN} \end{reference} Let $k$ be a field. Let $\mathcal{D}$ be a $k$-linear triangulated category which has direct sums and is compactly generated. Denote $\mathcal{D}_c$ the full subcategory of compact objects. Let $H : \mathcal{D}_c^{opp} \to \text{Vect}_k$ be a $k$-linear cohomological functor such that $\dim_k H(X) < \infty$ for all $X \in \Ob(\mathcal{D}_c)$. Then $H$ is isomorphic to the functor $X \mapsto \Hom(X, Y)$ for some $Y \in \Ob(\mathcal{D})$. \end{lemma} \begin{proof} We will use Derived Categories, Lemma \ref{derived-lemma-compact-objects-subcategory} without further mention. Denote $G : \mathcal{D}_c \to \text{Vect}_k$ the $k$-linear homological functor which sends $X$ to $H(X)^\vee$. For any object $Y$ of $\mathcal{D}$ we set $$ G'(Y) = \colim_{X \to Y, X \in \Ob(\mathcal{D}_c)} G(X) $$ The colimit is filtered by Lemma \ref{lemma-maps-from-compact-filtered}. We claim that $G'$ is a $k$-linear homological functor, the restriction of $G'$ to $\mathcal{D}_c$ is $G$, and $G'$ sends direct sums to direct sums. \medskip\noindent Namely, suppose that $Y_1 \to Y_2 \to Y_3$ is a distinguished triangle. Let $\xi \in G'(Y_2)$ map to zero in $G'(Y_3)$. Since the colimit is filtered $\xi$ is represented by some $X \to Y_2$ with $X \in \Ob(\mathcal{D}_c)$ and $g \in G(X)$. The fact that $\xi$ maps to zero in $G'(Y_3)$ means the composition $X \to Y_2 \to Y_3$ factors as $X \to X' \to Y_3$ with $X' \in \mathcal{D}_c$ and $g$ mapping to zero in $G(X')$. Choose a distinguished triangle $X'' \to X \to X'$. Then $X'' \in \Ob(\mathcal{D}_c)$. Since $G$ is homological we find that $g$ is the image of some $g'' \in G'(X'')$. By Axiom TR3 the maps $X \to Y_2$ and $X' \to Y_3$ fit into a morphism of distinguished triangles $(X'' \to X \to X') \to (Y_1 \to Y_2 \to Y_3)$ and we find that indeed $\xi$ is the image of the element of $G'(Y_1)$ represented by $X'' \to Y_1$ and $g'' \in G(X'')$. \medskip\noindent If $Y \in \Ob(\mathcal{D}_c)$, then $\text{id} : Y \to Y$ is the final object in the category of arrows $X \to Y$ with $X \in \Ob(\mathcal{D}_c)$. Hence we see that $G'(Y) = G(Y)$ in this case and the statement on restriction holds. Let $Y = \bigoplus_{i \in I} Y_i$ be a direct sum. Let $a : X \to Y$ with $X \in \Ob(\mathcal{D}_c)$ and $g \in G(X)$ represent an element $\xi$ of $G'(Y)$. The morphism $a : X \to Y$ can be uniquely written as a sum of morphisms $a_i : X \to Y_i$ almost all zero as $X$ is a compact object of $\mathcal{D}$. Let $I' = \{i \in I \mid a_i \not = 0\}$. Then we can factor $a$ as the composition $$ X \xrightarrow{(1, \ldots, 1)} \bigoplus\nolimits_{i \in I'} X \xrightarrow{\bigoplus_{i \in I'} a_i} \bigoplus\nolimits_{i \in I} Y_i = Y $$ We conclude that $\xi = \sum_{i \in I'} \xi_i$ is the sum of the images of the elements $\xi_i \in G'(Y_i)$ corresponding to $a_i : X \to Y_i$ and $g \in G(X)$. Hence $\bigoplus G'(Y_i) \to G'(Y)$ is surjective. We omit the (trivial) verification that it is injective. \medskip\noindent It follows that the functor $Y \mapsto G'(Y)^\vee$ is cohomological and sends direct sums to direct products. Hence by Brown representability, see Derived Categories, Proposition \ref{derived-proposition-brown} we conclude that there exists a $Y \in \Ob(\mathcal{D})$ and an isomorphism $G'(Z)^\vee = \Hom(Z, Y)$ functorially in $Z$. For $X \in \Ob(\mathcal{D}_c)$ we have $G'(X)^\vee = G(X)^\vee = (H(X)^\vee)^\vee = H(X)$ because $\dim_k H(X) < \infty$ and the proof is complete. \end{proof} \begin{theorem} \label{theorem-bondal-van-den-bergh} \begin{reference} \cite[Theorem A.1]{BvdB} \end{reference} Let $X$ be a projective scheme over a field $k$. Let $F : D_{perf}(\mathcal{O}_X)^{opp} \to \text{Vect}_k$ be a $k$-linear cohomological functor such that $$ \sum\nolimits_{n \in \mathbf{Z}} \dim_k F(E[n]) < \infty $$ for all $E \in D_{perf}(\mathcal{O}_X)$. Then $F$ is isomorphic to a functor of the form $E \mapsto \Hom_X(E, K)$ for some $K \in D^b_{\textit{Coh}}(\mathcal{O}_X)$. \end{theorem} \begin{proof} The derived category $D_\QCoh(\mathcal{O}_X)$ has direct sums, is compactly generated, and $D_{perf}(\mathcal{O}_X)$ is the full subcategory of compact objects, see Derived Categories of Schemes, Lemma \ref{perfect-lemma-quasi-coherence-direct-sums}, Theorem \ref{perfect-theorem-bondal-van-den-Bergh}, and Proposition \ref{perfect-proposition-compact-is-perfect}. By Lemma \ref{lemma-van-den-bergh} we may assume $F(E) = \Hom_X(E, K)$ for some $K \in \Ob(D_\QCoh(\mathcal{O}_X))$. Then it follows that $K$ is in $D^b_{\textit{Coh}}(\mathcal{O}_X)$ by Lemma \ref{lemma-characterize-dbcoh-projective}. \end{proof} \section{Representability in the regular proper case} \label{section-regular-proper} \noindent Theorem \ref{theorem-bondal-van-den-bergh} also holds for regular (for example smooth) proper varieties. This is proven in \cite{BvdB} using their general characterization of ``right saturated'' $k$-linear triangulated categories. In this section we give a quick and dirty proof of this result using a little bit of duality. \begin{lemma} \label{lemma-trace-map} \begin{reference} The proof given here follows the argument given in \cite[Remark 3.4]{MS} \end{reference} Let $f : X' \to X$ be a proper birational morphism of integral Noetherian schemes with $X$ regular. The map $\mathcal{O}_X \to Rf_*\mathcal{O}_{X'}$ canonically splits in $D(\mathcal{O}_X)$. \end{lemma} \begin{proof} Set $E = Rf_*\mathcal{O}_{X'}$ in $D(\mathcal{O}_X)$. Observe that $E$ is in $D^b_{\textit{Coh}}(\mathcal{O}_X)$ by Derived Categories of Schemes, Lemma \ref{perfect-lemma-direct-image-coherent}. By Derived Categories of Schemes, Lemma \ref{perfect-lemma-perfect-on-regular} we find that $E$ is a perfect object of $D(\mathcal{O}_X)$. Since $\mathcal{O}_{X'}$ is a sheaf of algebras, we have the relative cup product $\mu : E \otimes_{\mathcal{O}_X}^\mathbf{L} E \to E$ by Cohomology, Remark \ref{cohomology-remark-cup-product}. Let $\sigma : E \otimes E^\vee \to E^\vee \otimes E$ be the commutativity constraint on the symmetric monoidal category $D(\mathcal{O}_X)$ (Cohomology, Lemma \ref{cohomology-lemma-symmetric-monoidal-derived}). Denote $\eta : \mathcal{O}_X \to E \otimes E^\vee$ and $\epsilon : E^\vee \otimes E \to \mathcal{O}_X$ the maps constructed in Cohomology, Example \ref{cohomology-example-dual-derived}. Then we can consider the map $$ E \xrightarrow{\eta \otimes 1} E \otimes E^\vee \otimes E \xrightarrow{\sigma \otimes 1} E^\vee \otimes E \otimes E \xrightarrow{1 \otimes \mu} E^\vee \otimes E \xrightarrow{\epsilon} \mathcal{O}_X $$ We claim that this map is a one sided inverse to the map in the statement of the lemma. To see this it suffices to show that the composition $\mathcal{O}_X \to \mathcal{O}_X$ is the identity map. This we may do in the generic point of $X$ (or on an open subscheme of $X$ over which $f$ is an isomorphism). In this case $E = \mathcal{O}_X$ and $\mu$ is the usual multiplication map and the result is clear. \end{proof} \begin{lemma} \label{lemma-characterize-dbcoh-proper-regular} Let $X$ be a proper scheme over a field $k$ which is regular. Let $K \in \Ob(D_\QCoh(\mathcal{O}_X))$. The following are equivalent \begin{enumerate} \item $K \in D^b_{\textit{Coh}}(\mathcal{O}_X) = D_{perf}(\mathcal{O}_X)$, and \item $\sum_{i \in \mathbf{Z}} \dim_k \Ext^i_X(E, K) < \infty$ for all perfect $E$ in $D(\mathcal{O}_X)$. \end{enumerate} \end{lemma} \begin{proof} The equality in (1) holds by Derived Categories of Schemes, Lemma \ref{perfect-lemma-perfect-on-regular}. The implication (1) $\Rightarrow$ (2) follows from Lemma \ref{lemma-finiteness}. The implication (2) $\Rightarrow$ (1) follows from More on Morphisms, Lemma \ref{more-morphisms-lemma-characterize-relatively-perfect}. \end{proof} \begin{lemma} \label{lemma-bondal-van-den-bergh} Let $X$ be a proper scheme over a field $k$ which is regular. \begin{enumerate} \item Let $F : D_{perf}(\mathcal{O}_X)^{opp} \to \text{Vect}_k$ be a $k$-linear cohomological functor such that $$ \sum\nolimits_{n \in \mathbf{Z}} \dim_k F(E[n]) < \infty $$ for all $E \in D_{perf}(\mathcal{O}_X)$. Then $F$ is isomorphic to a functor of the form $E \mapsto \Hom_X(E, K)$ for some $K \in D_{perf}(\mathcal{O}_X)$. \item Let $G : D_{perf}(\mathcal{O}_X) \to \text{Vect}_k$ be a $k$-linear homological functor such that $$ \sum\nolimits_{n \in \mathbf{Z}} \dim_k G(E[n]) < \infty $$ for all $E \in D_{perf}(\mathcal{O}_X)$. Then $G$ is isomorphic to a functor of the form $E \mapsto \Hom_X(K, E)$ for some $K \in D_{perf}(\mathcal{O}_X)$. \end{enumerate} \end{lemma} \begin{proof} Proof of (1). The derived category $D_\QCoh(\mathcal{O}_X)$ has direct sums, is compactly generated, and $D_{perf}(\mathcal{O}_X)$ is the full subcategory of compact objects, see Derived Categories of Schemes, Lemma \ref{perfect-lemma-quasi-coherence-direct-sums}, Theorem \ref{perfect-theorem-bondal-van-den-Bergh}, and Proposition \ref{perfect-proposition-compact-is-perfect}. By Lemma \ref{lemma-van-den-bergh} we may assume $F(E) = \Hom_X(E, K)$ for some $K \in \Ob(D_\QCoh(\mathcal{O}_X))$. Then it follows that $K$ is in $D^b_{\textit{Coh}}(\mathcal{O}_X)$ by Lemma \ref{lemma-characterize-dbcoh-proper-regular}. \medskip\noindent Proof of (2). Consider the contravariant functor $E \mapsto E^\vee$ on $D_{perf}(\mathcal{O}_X)$, see Cohomology, Lemma \ref{cohomology-lemma-dual-perfect-complex}. This functor is an exact anti-self-equivalence of $D_{perf}(\mathcal{O}_X)$. Hence we may apply part (1) to the functor $F(E) = G(E^\vee)$ to find $K \in D_{perf}(\mathcal{O}_X)$ such that $G(E^\vee) = \Hom_X(E, K)$. It follows that $G(E) = \Hom_X(E^\vee, K) = \Hom_X(K^\vee, E)$ and we conclude that taking $K^\vee$ works. \end{proof} \section{Existence of adjoints} \label{section-adjoints} \noindent As a consequence of the results in the paper of Bondal and van den Bergh we get the following automatic existence of adjoints. \begin{lemma} \label{lemma-always-right-adjoints} Let $k$ be a field. Let $X$ and $Y$ be proper schemes over $k$. If $X$ is regular, then $k$-linear any exact functor $F : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)$ has an exact right adjoint and an exact left adjoint. \end{lemma} \begin{proof} If an adjoint exists it is an exact functor by the very general Derived Categories, Lemma \ref{derived-lemma-adjoint-is-exact}. \medskip\noindent Let us prove the existence of a right adjoint. To see existence, it suffices to show that for $M \in D_{perf}(\mathcal{O}_Y)$ the contravariant functor $K \mapsto \Hom_Y(F(K), M)$ is representable. This functor is contravariant, $k$-linear, and cohomological. Hence by Lemma \ref{lemma-bondal-van-den-bergh} part (1) it suffices to show that $$ \sum\nolimits_{i \in \mathbf{Z}} \dim_k \Ext^i_Y(F(K), M) < \infty $$ This follows from Lemma \ref{lemma-finiteness}. \medskip\noindent For the existence of the left adjoint we argue in the same manner using part (2) of Lemma \ref{lemma-bondal-van-den-bergh}. \end{proof} \section{Fourier-Mukai functors} \label{section-fourier-mukai} \noindent These functors were first introduced in \cite{Mukai}. \begin{definition} \label{definition-fourier-mukai-functor} Let $S$ be a scheme. Let $X$ and $Y$ be schemes over $S$. Let $K \in D(\mathcal{O}_{X \times_S Y})$. The exact functor $$ \Phi_K : D(\mathcal{O}_X) \longrightarrow D(\mathcal{O}_Y),\quad M \longmapsto R\text{pr}_{2, *}( L\text{pr}_1^*M \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} K) $$ of triangulated categories is called a {\it Fourier-Mukai functor} and $K$ is called a {\it Fourier-Mukai kernel} for this functor. Moreover, \begin{enumerate} \item if $\Phi_K$ sends $D_\QCoh(\mathcal{O}_X)$ into $D_\QCoh(\mathcal{O}_Y)$ then the resulting exact functor $\Phi_K : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$ is called a Fourier-Mukai functor, \item if $\Phi_K$ sends $D_{perf}(\mathcal{O}_X)$ into $D_{perf}(\mathcal{O}_Y)$ then the resulting exact functor $\Phi_K : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)$ is called a Fourier-Mukai functor, and \item if $X$ and $Y$ are Noetherian and $\Phi_K$ sends $D^b_{\textit{Coh}}(\mathcal{O}_X)$ into $D^b_{\textit{Coh}}(\mathcal{O}_Y)$ then the resulting exact functor $\Phi_K : D^b_{\textit{Coh}}(\mathcal{O}_X) \to D^b_{\textit{Coh}}(\mathcal{O}_Y)$ is called a Fourier-Mukai functor. Similarly for $D_{\textit{Coh}}$, $D^+_{\textit{Coh}}$, $D^-_{\textit{Coh}}$. \end{enumerate} \end{definition} \begin{lemma} \label{lemma-fourier-Mukai-QCoh} Let $S$ be a scheme. Let $X$ and $Y$ be schemes over $S$. Let $K \in D(\mathcal{O}_{X \times_S Y})$. The corresponding Fourier-Mukai functor $\Phi_K$ sends $D_\QCoh(\mathcal{O}_X)$ into $D_\QCoh(\mathcal{O}_Y)$ if $K$ is in $D_\QCoh(\mathcal{O}_{X \times_S Y})$ and $X \to S$ is quasi-compact and quasi-separated. \end{lemma} \begin{proof} This follows from the fact that derived pullback preserves $D_\QCoh$ (Derived Categories of Schemes, Lemma \ref{perfect-lemma-quasi-coherence-pullback}), derived tensor products preserve $D_\QCoh$ (Derived Categories of Schemes, Lemma \ref{perfect-lemma-quasi-coherence-tensor-product}), the projection $\text{pr}_2 : X \times_S Y \to Y$ is quasi-compact and quasi-separated (Schemes, Lemmas \ref{schemes-lemma-quasi-compact-preserved-base-change} and \ref{schemes-lemma-separated-permanence}), and total direct image along a quasi-separated and quasi-compact morphism preserves $D_\QCoh$ (Derived Categories of Schemes, Lemma \ref{perfect-lemma-quasi-coherence-direct-image}). \end{proof} \begin{lemma} \label{lemma-compose-fourier-mukai} Let $S$ be a scheme. Let $X, Y, Z$ be schemes over $S$. Assume $X \to S$, $Y \to S$, and $Z \to S$ are quasi-compact and quasi-separated. Let $K \in D_\QCoh(\mathcal{O}_{X \times_S Y})$. Let $K' \in D_\QCoh(\mathcal{O}_{Y \times_S Z})$. Consider the Fourier-Mukai functors $\Phi_K : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$ and $\Phi_{K'} : D_\QCoh(\mathcal{O}_Y) \to D_\QCoh(\mathcal{O}_Z)$. If $X$ and $Z$ are tor independent over $S$ and $Y \to S$ is flat, then $$ \Phi_{K'} \circ \Phi_K = \Phi_{K''} : D_\QCoh(\mathcal{O}_X) \longrightarrow D_\QCoh(\mathcal{O}_Z) $$ where $$ K'' = R\text{pr}_{13, *}( L\text{pr}_{12}^*K \otimes_{\mathcal{O}_{X \times_S Y \times_S Z}}^\mathbf{L} L\text{pr}_{23}^*K') $$ in $D_\QCoh(\mathcal{O}_{X \times_S Z})$. \end{lemma} \begin{proof} The statement makes sense by Lemma \ref{lemma-fourier-Mukai-QCoh}. We are going to use Derived Categories of Schemes, Lemmas \ref{perfect-lemma-quasi-coherence-pullback}, \ref{perfect-lemma-quasi-coherence-tensor-product}, and \ref{perfect-lemma-quasi-coherence-direct-image} and Schemes, Lemmas \ref{schemes-lemma-quasi-compact-preserved-base-change} and \ref{schemes-lemma-separated-permanence} without further mention. By Derived Categories of Schemes, Lemma \ref{perfect-lemma-flat-base-change-tor-independent} we see that $X \times_S Y$ and $Y \times_S Z$ are tor independent over $Y$. This means that we have base change for the cartesian diagram $$ \xymatrix{ X \times_S Y \times_S Z \ar[d] \ar[r] & Y \times_S Z \ar[d]^{p^{YZ}_Y} \\ X \times_S Y \ar[r]^{p^{XY}_Y} & Y } $$ for complexes with quasi-coherent cohomology sheaves, see Derived Categories of Schemes, Lemma \ref{perfect-lemma-compare-base-change}. Abbreviating $p^* = Lp^*$, $p_* = Rp_*$ and $\otimes = \otimes^\mathbf{L}$ we have for $M \in D_\QCoh(\mathcal{O}_X)$ the sequence of equalities \begin{align*} \Phi_{K'}(\Phi_K(M)) & = p^{YZ}_{Z, *}(p^{YZ, *}_Y p^{XY}_{Y, *}(p^{XY, *}_X M \otimes K) \otimes K') \\ & = p^{YZ}_{Z, *}(\text{pr}_{23, *} \text{pr}_{12}^*(p^{XY, *}_X M \otimes K) \otimes K') \\ & = p^{YZ}_{Z, *}(\text{pr}_{23, *}(\text{pr}_1^*M \otimes \text{pr}_{12}^*K) \otimes K') \\ & = p^{YZ}_{Z, *}(\text{pr}_{23, *}(\text{pr}_1^*M \otimes \text{pr}_{12}^*K \otimes \text{pr}_{23}^*K')) \\ & = \text{pr}_{3, *}(\text{pr}_1^*M \otimes \text{pr}_{12}^*K \otimes \text{pr}_{23}^*K') \\ & = p^{XZ}_{Z, *}\text{pr}_{13, *}(\text{pr}_1^*M \otimes \text{pr}_{12}^*K \otimes \text{pr}_{23}^*K') \\ & = p^{XZ}_{Z, *} (p^{XZ, *}_X M \otimes \text{pr}_{13, *}(\text{pr}_{12}^*K \otimes \text{pr}_{23}^*K')) \end{align*} as desired. Here we have used the remark on base change in the second equality and we have use Derived Categories of Schemes, Lemma \ref{perfect-lemma-cohomology-base-change} in the $4$th and last equality. \end{proof} \begin{lemma} \label{lemma-fourier-mukai} Let $S$ be a scheme. Let $X$ and $Y$ be schemes over $S$. Let $K \in D(\mathcal{O}_{X \times_S Y})$. The corresponding Fourier-Mukai functor $\Phi_K$ sends $D_{perf}(\mathcal{O}_X)$ into $D_{perf}(\mathcal{O}_Y)$ if at least one of the following conditions is satisfied: \begin{enumerate} \item $S$ is Noetherian, $X \to S$ and $Y \to S$ are of finite type, $K \in D^b_{\textit{Coh}}(\mathcal{O}_{X \times_S Y})$, the support of $H^i(K)$ is proper over $Y$ for all $i$, and $K$ has finite tor dimension as an object of $D(\text{pr}_2^{-1}\mathcal{O}_Y)$, \item $X \to S$ is of finite presentation and $K$ can be represented by a bounded complex $\mathcal{K}^\bullet$ of finitely presented $\mathcal{O}_{X \times_S Y}$-modules, flat over $Y$, with support proper over $Y$, \item $X \to S$ is a proper flat morphism of finite presentation and $K$ is perfect, \item $S$ is Noetherian, $X \to S$ is flat and proper, and $K$ is perfect \item $X \to S$ is a proper flat morphism of finite presentation and $K$ is $Y$-perfect, \item $S$ is Noetherian, $X \to S$ is flat and proper, and $K$ is $Y$-perfect. \end{enumerate} \end{lemma} \begin{proof} If $M$ is perfect on $X$, then $L\text{pr}_1^*M$ is perfect on $X \times_S Y$, see Cohomology, Lemma \ref{cohomology-lemma-perfect-pullback}. We will use this without further mention below. We will also use that if $X \to S$ is of finite type, or proper, or flat, or of finite presentation, then the same thing is true for the base change $\text{pr}_2 : X \times_S Y \to Y$, see Morphisms, Lemmas \ref{morphisms-lemma-base-change-finite-type}, \ref{morphisms-lemma-base-change-proper}, \ref{morphisms-lemma-base-change-flat}, and \ref{morphisms-lemma-base-change-finite-presentation}. \medskip\noindent Part (1) follows from Derived Categories of Schemes, Lemma \ref{perfect-lemma-perfect-direct-image} combined with Derived Categories of Schemes, Lemma \ref{perfect-lemma-perfect-on-noetherian}. \medskip\noindent Part (2) follows from Derived Categories of Schemes, Lemma \ref{perfect-lemma-base-change-tensor-perfect}. \medskip\noindent Part (3) follows from Derived Categories of Schemes, Lemma \ref{perfect-lemma-flat-proper-perfect-direct-image-general}. \medskip\noindent Part (4) follows from part (3) and the fact that a finite type morphism of Noetherian schemes is of finite presentation by Morphisms, Lemma \ref{morphisms-lemma-noetherian-finite-type-finite-presentation}. \medskip\noindent Part (5) follows from Derived Categories of Schemes, Lemma \ref{perfect-lemma-derived-pushforward-rel-perfect} combined with Derived Categories of Schemes, Lemma \ref{perfect-lemma-perfect-relatively-perfect}. \medskip\noindent Part (6) follows from part (5) in the same way that part (4) follows from part (3). \end{proof} \begin{lemma} \label{lemma-fourier-mukai-Coh} Let $S$ be a Noetherian scheme. Let $X$ and $Y$ be schemes of finite type over $S$. Let $K \in D^b_{\textit{Coh}}(\mathcal{O}_{X \times_S Y})$. The corresponding Fourier-Mukai functor $\Phi_K$ sends $D^b_{\textit{Coh}}(\mathcal{O}_X)$ into $D^b_{\textit{Coh}}(\mathcal{O}_Y)$ if at least one of the following conditions is satisfied: \begin{enumerate} \item the support of $H^i(K)$ is proper over $Y$ for all $i$, and $K$ has finite tor dimension as an object of $D(\text{pr}_1^{-1}\mathcal{O}_X)$, \item $K$ can be represented by a bounded complex $\mathcal{K}^\bullet$ of coherent $\mathcal{O}_{X \times_S Y}$-modules, flat over $X$, with support proper over $Y$, \item the support of $H^i(K)$ is proper over $Y$ for all $i$ and $X$ is a regular scheme, \item $K$ is perfect, the support of $H^i(K)$ is proper over $Y$ for all $i$, and $Y \to S$ is flat. \end{enumerate} Furthermore in each case the support condition is automatic if $X \to S$ is proper. \end{lemma} \begin{proof} Let $M$ be an object of $D^b_{\textit{Coh}}(\mathcal{O}_X)$. In each case we will use Derived Categories of Schemes, Lemma \ref{perfect-lemma-direct-image-coherent} to show that $$ \Phi_K(M) = R\text{pr}_{2, *}( L\text{pr}_1^*M \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} K) $$ is in $D^b_{\textit{Coh}}(\mathcal{O}_Y)$. The derived tensor product $L\text{pr}_1^*M \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} K$ is a pseudo-coherent object of $D(\mathcal{O}_{X \times_S Y})$ (by Cohomology, Lemma \ref{cohomology-lemma-pseudo-coherent-pullback}, Derived Categories of Schemes, Lemma \ref{perfect-lemma-identify-pseudo-coherent-noetherian}, and Cohomology, Lemma \ref{cohomology-lemma-tensor-pseudo-coherent}) whence has coherent cohomology sheaves (by Derived Categories of Schemes, Lemma \ref{perfect-lemma-identify-pseudo-coherent-noetherian} again). In each case the supports of the cohomology sheaves $H^i(L\text{pr}_1^*M \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} K)$ is proper over $Y$ as these supports are contained in the union of the supports of the $H^i(K)$. Hence in each case it suffices to prove that this tensor product is bounded below. \medskip\noindent Case (1). By Cohomology, Lemma \ref{cohomology-lemma-variant-derived-pullback} we have $$ L\text{pr}_1^*M \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} K \cong \text{pr}_1^{-1}M \otimes_{\text{pr}_1^{-1}\mathcal{O}_X}^\mathbf{L} K $$ with obvious notation. Hence the assumption on tor dimension and the fact that $M$ has only a finite number of nonzero cohomology sheaves, implies the bound we want. \medskip\noindent Case (2) follows because here the assumption implies that $K$ has finite tor dimension as an object of $D(\text{pr}_1^{-1}\mathcal{O}_X)$ hence the argument in the previous paragraph applies. \medskip\noindent In Case (3) it is also the case that $K$ has finite tor dimension as an object of $D(\text{pr}_1^{-1}\mathcal{O}_X)$. Namely, choose affine opens $U = \Spec(A)$ and $V = \Spec(B)$ of $X$ and $Y$ mapping into the affine open $W = \Spec(R)$ of $S$. Then $K|_{U \times V}$ is given by a bounded complex of finite $A \otimes_R B$-modules $M^\bullet$. Since $A$ is a regular ring of finite dimension we see that each $M^i$ has finite projective dimension as an $A$-module (Algebra, Lemma \ref{algebra-lemma-finite-gl-dim-finite-dim-regular}) and hence finite tor dimension as an $A$-module. Thus $M^\bullet$ has finite tor dimension as a complex of $A$-modules (More on Algebra, Lemma \ref{more-algebra-lemma-complex-finite-tor-dimension-modules}). Since $X \times Y$ is quasi-compact we conclude there exist $[a, b]$ such that for every point $z \in X \times Y$ the stalk $K_z$ has tor amplitude in $[a, b]$ over $\mathcal{O}_{X, \text{pr}_1(z)}$. This implies $K$ has bounded tor dimension as an object of $D(\text{pr}_1^{-1}\mathcal{O}_X)$, see Cohomology, Lemma \ref{cohomology-lemma-tor-amplitude-stalk}. We conclude as in the previous to paragraphs. \medskip\noindent Case (4). With notation as above, the ring map $R \to B$ is flat. Hence the ring map $A \to A \otimes_R B$ is flat. Hence any projective $A \otimes_R B$-module is $A$-flat. Thus any perfect complex of $A \otimes_R B$-modules has finite tor dimension as a complex of $A$-modules and we conclude as before. \end{proof} \begin{example} \label{example-diagonal-fourier-mukai} Let $X \to S$ be a separated morphism of schemes. Then the diagonal $\Delta : X \to X \times_S X$ is a closed immersion and hence $\mathcal{O}_\Delta = \Delta_*\mathcal{O}_X = R\Delta_*\mathcal{O}_X$ is a quasi-coherent $\mathcal{O}_{X \times_S X}$-module of finite type which is flat over $X$ (under either projection). The Fourier-Mukai functor $\Phi_{\mathcal{O}_\Delta}$ is equal to the identity in this case. Namely, for any $M \in D(\mathcal{O}_X)$ we have \begin{align*} L\text{pr}_1^*M \otimes_{\mathcal{O}_{X \times_S X}}^\mathbf{L} \mathcal{O}_\Delta & = L\text{pr}_1^*M \otimes_{\mathcal{O}_{X \times_S X}}^\mathbf{L} R\Delta_*\mathcal{O}_X \\ & = R\Delta_*( L\Delta^*L\text{pr}_1^*M \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{O}_X) \\ & = R\Delta_*(M) \end{align*} The first equality we discussed above. The second equality is Cohomology, Lemma \ref{cohomology-lemma-projection-formula-closed-immersion}. The third because $\text{pr}_1 \circ \Delta = \text{id}_X$ and we have Cohomology, Lemma \ref{cohomology-lemma-derived-pullback-composition}. If we push this to $X$ using $R\text{pr}_{2, *}$ we obtain $M$ by Cohomology, Lemma \ref{cohomology-lemma-derived-pushforward-composition} and the fact that $\text{pr}_2 \circ \Delta = \text{id}_X$. \end{example} \begin{lemma} \label{lemma-fourier-mukai-right-adjoint} \begin{reference} Compare with discussion in \cite{Rizzardo}. \end{reference} Let $X \to S$ and $Y \to S$ be morphisms of quasi-compact and quasi-separated schemes. Let $\Phi : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$ be a Fourier-Mukai functor with pseudo-coherent kernel $K \in D_\QCoh(\mathcal{O}_{X \times_S Y})$. Let $a : D_\QCoh(\mathcal{O}_Y) \to D_\QCoh(\mathcal{O}_{X \times_S Y})$ be the right adjoint to $R\text{pr}_{2, *}$, see Duality for Schemes, Lemma \ref{duality-lemma-twisted-inverse-image}. Denote $$ K' = (Y \times_S X \to X \times_S Y)^* R\SheafHom_{\mathcal{O}_{X \times_S Y}}(K, a(\mathcal{O}_Y)) \in D_\QCoh(\mathcal{O}_{Y \times_S X}) $$ and denote $\Phi' : D_\QCoh(\mathcal{O}_Y) \to D_\QCoh(\mathcal{O}_X)$ the corresponding Fourier-Mukai transform. There is a canonical map $$ \Hom_X(M, \Phi'(N)) \longrightarrow \Hom_Y(\Phi(M), N) $$ functorial in $M$ in $D_\QCoh(\mathcal{O}_X)$ and $N$ in $D_\QCoh(\mathcal{O}_Y)$ which is an isomorphism if \begin{enumerate} \item $N$ is perfect, or \item $K$ is perfect and $X \to S$ is proper flat and of finite presentation. \end{enumerate} \end{lemma} \begin{proof} By Lemma \ref{lemma-fourier-Mukai-QCoh} we obtain a functor $\Phi$ as in the statement. Observe that $a(\mathcal{O}_Y)$ is in $D^+_\QCoh(\mathcal{O}_{X \times_S Y})$ by Duality for Schemes, Lemma \ref{duality-lemma-twisted-inverse-image-bounded-below}. Hence for $K$ pseudo-coherent we have $K' \in D_\QCoh(\mathcal{O}_{Y \times_S X})$ by Derived Categories of Schemes, Lemma \ref{perfect-lemma-quasi-coherence-internal-hom} we we obtain $\Phi'$ as indicated. \medskip\noindent We abbreviate $\otimes^\mathbf{L} = \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L}$ and $\SheafHom = R\SheafHom_{\mathcal{O}_{X \times_S Y}}$. Let $M$ be in $D_\QCoh(\mathcal{O}_X)$ and let $N$ be in $D_\QCoh(\mathcal{O}_Y)$. We have \begin{align*} \Hom_Y(\Phi(M), N) & = \Hom_Y(R\text{pr}_{2, *}(L\text{pr}_1^*M \otimes^\mathbf{L} K), N) \\ & = \Hom_{X \times_S Y}(L\text{pr}_1^*M \otimes^\mathbf{L} K, a(N)) \\ & = \Hom_{X \times_S Y}(L\text{pr}_1^*M, R\SheafHom(K, a(N))) \\ & = \Hom_X(M, R\text{pr}_{1, *}R\SheafHom(K, a(N))) \end{align*} where we have used Cohomology, Lemmas \ref{cohomology-lemma-internal-hom} and \ref{cohomology-lemma-adjoint}. There are canonical maps $$ L\text{pr}_2^*N \otimes^\mathbf{L} R\SheafHom(K, a(\mathcal{O}_Y)) \xrightarrow{\alpha} R\SheafHom(K, L\text{pr}_2^*N \otimes^\mathbf{L} a(\mathcal{O}_Y)) \xrightarrow{\beta} R\SheafHom(K, a(N)) $$ Here $\alpha$ is Cohomology, Lemma \ref{cohomology-lemma-internal-hom-diagonal-better} and $\beta$ is Duality for Schemes, Equation (\ref{duality-equation-compare-with-pullback}). Combining all of these arrows we obtain the functorial displayed arrow in the statement of the lemma. \medskip\noindent The arrow $\alpha$ is an isomorphism by Derived Categories of Schemes, Lemma \ref{perfect-lemma-internal-hom-evaluate-tensor-isomorphism} as soon as either $K$ or $N$ is perfect. The arrow $\beta$ is an isomorphism if $N$ is perfect by Duality for Schemes, Lemma \ref{duality-lemma-compare-with-pullback-perfect} or in general if $X \to S$ is flat proper of finite presentation by Duality for Schemes, Lemma \ref{duality-lemma-compare-with-pullback-flat-proper}. \end{proof} \begin{lemma} \label{lemma-fourier-mukai-left-adjoint} \begin{reference} Compare with discussion in \cite{Rizzardo}. \end{reference} Let $S$ be a Noetherian scheme. Let $Y \to S$ be a flat proper Gorenstein morphism and let $X \to S$ be a finite type morphism. Denote $\omega^\bullet_{Y/S}$ the relative dualizing complex of $Y$ over $S$. Let $\Phi : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$ be a Fourier-Mukai functor with perfect kernel $K \in D_\QCoh(\mathcal{O}_{X \times_S Y})$. Denote $$ K' = (Y \times_S X \to X \times_S Y)^*(K^\vee \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} L\text{pr}_2^*\omega^\bullet_{Y/S}) \in D_\QCoh(\mathcal{O}_{Y \times_S X}) $$ and denote $\Phi' : D_\QCoh(\mathcal{O}_Y) \to D_\QCoh(\mathcal{O}_X)$ the corresponding Fourier-Mukai transform. There is a canonical isomorphism $$ \Hom_Y(N, \Phi(M)) \longrightarrow \Hom_X(\Phi'(N), M) $$ functorial in $M$ in $D_\QCoh(\mathcal{O}_X)$ and $N$ in $D_\QCoh(\mathcal{O}_Y)$. \end{lemma} \begin{proof} By Lemma \ref{lemma-fourier-Mukai-QCoh} we obtain a functor $\Phi$ as in the statement. \medskip\noindent Observe that formation of the relative dualizing complex commutes with base change in our setting, see Duality for Schemes, Remark \ref{duality-remark-relative-dualizing-complex}. Thus $L\text{pr}_2^*\omega^\bullet_{Y/S} = \omega^\bullet_{X \times_S Y/X}$. Moreover, we observe that $\omega^\bullet_{Y/S}$ is an invertible object of the derived category, see Duality for Schemes, Lemma \ref{duality-lemma-affine-flat-Noetherian-gorenstein}, and a fortiori perfect. \medskip\noindent To actually prove the lemma we're going to cheat. Namely, we will show that if we replace the roles of $X$ and $Y$ and $K$ and $K'$ then these are as in Lemma \ref{lemma-fourier-mukai-right-adjoint} and we get the result. It is clear that $K'$ is perfect as a tensor product of perfect objects so that the discussion in Lemma \ref{lemma-fourier-mukai-right-adjoint} applies to it. To show that the procedure of Lemma \ref{lemma-fourier-mukai-right-adjoint} applied to $K'$ on $Y \times_S X$ produces a complex isomorphic to $K$ it suffices (details omitted) to show that $$ R\SheafHom(R\SheafHom(K, \omega^\bullet_{X \times_S Y/X}), \omega^\bullet_{X \times_S Y/X}) = K $$ This is clear because $K$ is perfect and $\omega^\bullet_{X \times_S Y/X}$ is invertible; details omitted. Thus Lemma \ref{lemma-fourier-mukai-right-adjoint} produces a map $$ \Hom_Y(N, \Phi(M)) \longrightarrow \Hom_X(\Phi'(N), M) $$ functorial in $M$ in $D_\QCoh(\mathcal{O}_X)$ and $N$ in $D_\QCoh(\mathcal{O}_Y)$ which is an isomorphism because $K'$ is perfect. This finishes the proof. \end{proof} \begin{lemma} \label{lemma-fourier-mukai-flat-proper-over-noetherian} Let $S$ be a Noetherian scheme. \begin{enumerate} \item For $X$, $Y$ proper and flat over $S$ and $K$ in $D_{perf}(\mathcal{O}_{X \times_S Y})$ we obtain a Fourier-Mukai functor $\Phi_K : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)$. \item For $X$, $Y$, $Z$ proper and flat over $S$, $K \in D_{perf}(\mathcal{O}_{X \times_S Y})$, $K' \in D_{perf}(\mathcal{O}_{Y \times_S Z})$ the composition $\Phi_{K'} \circ \Phi_K : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Z)$ is equal to $\Phi_{K''}$ with $K'' \in D_{perf}(\mathcal{O}_{X \times_S Z})$ computed as in Lemma \ref{lemma-compose-fourier-mukai}, \item For $X$, $Y$, $K$, $\Phi_K$ as in (1) if $X \to S$ is Gorenstein, then $\Phi_{K'} : D_{perf}(\mathcal{O}_Y) \to D_{perf}(\mathcal{O}_X)$ is a right adjoint to $\Phi_K$ where $K' \in D_{perf}(\mathcal{O}_{Y \times_S X})$ is the pullback of $L\text{pr}_1^*\omega_{X/S}^\bullet \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} K^\vee$ by $Y \times_S X \to X \times_S Y$. \item For $X$, $Y$, $K$, $\Phi_K$ as in (1) if $Y \to S$ is Gorenstein, then $\Phi_{K''} : D_{perf}(\mathcal{O}_Y) \to D_{perf}(\mathcal{O}_X)$ is a left adjoint to $\Phi_K$ where $K'' \in D_{perf}(\mathcal{O}_{Y \times_S X})$ is the pullback of $L\text{pr}_2^*\omega_{Y/S}^\bullet \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} K^\vee$ by $Y \times_S X \to X \times_S Y$. \end{enumerate} \end{lemma} \begin{proof} Part (1) is immediate from Lemma \ref{lemma-fourier-mukai} part (4). \medskip\noindent Part (2) follows from Lemma \ref{lemma-compose-fourier-mukai} and the fact that $K'' = R\text{pr}_{13, *}( L\text{pr}_{12}^*K \otimes_{\mathcal{O}_{X \times_S Y \times_S Z}}^\mathbf{L} L\text{pr}_{23}^*K')$ is perfect for example by Derived Categories of Schemes, Lemma \ref{perfect-lemma-flat-proper-perfect-direct-image}. \medskip\noindent The adjointness in part (3) on all complexes with quasi-coherent cohomology sheaves follows from Lemma \ref{lemma-fourier-mukai-right-adjoint} with $K'$ equal to the pullback of $R\SheafHom_{\mathcal{O}_{X \times_S Y}}(K, a(\mathcal{O}_Y))$ by $Y \times_S X \to X \times_S Y$ where $a$ is the right adjoint to $R\text{pr}_{2, *} : D_\QCoh(\mathcal{O}_{X \times_S Y}) \to D_\QCoh(\mathcal{O}_Y)$. Denote $f : X \to S$ the structure morphism of $X$. Since $f$ is proper the functor $f^! : D_\QCoh^+(\mathcal{O}_S) \to D_\QCoh^+(\mathcal{O}_X)$ is the restriction to $D_\QCoh^+(\mathcal{O}_S)$ of the right adjoint to $Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_S)$, see Duality for Schemes, Section \ref{duality-section-upper-shriek}. Hence the relative dualizing complex $\omega_{X/S}^\bullet$ as defined in Duality for Schemes, Remark \ref{duality-remark-relative-dualizing-complex} is equal to $\omega_{X/S}^\bullet = f^!\mathcal{O}_S$. Since formation of the relative dualizing complex commutes with base change (see Duality for Schemes, Remark \ref{duality-remark-relative-dualizing-complex}) we see that $a(\mathcal{O}_Y) = L\text{pr}_1^*\omega_{X/S}^\bullet$. Thus $$ R\SheafHom_{\mathcal{O}_{X \times_S Y}}(K, a(\mathcal{O}_Y)) \cong L\text{pr}_1^*\omega_{X/S}^\bullet \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} K^\vee $$ by Cohomology, Lemma \ref{cohomology-lemma-dual-perfect-complex}. Finally, since $X \to S$ is assumed Gorenstein the relative dualizing complex is invertible: this follows from Duality for Schemes, Lemma \ref{duality-lemma-affine-flat-Noetherian-gorenstein}. We conclude that $\omega_{X/S}^\bullet$ is perfect (Cohomology, Lemma \ref{cohomology-lemma-invertible-derived}) and hence $K'$ is perfect. Therefore $\Phi_{K'}$ does indeed map $D_{perf}(\mathcal{O}_Y)$ into $D_{perf}(\mathcal{O}_X)$ which finishes the proof of (3). \medskip\noindent The proof of (4) is the same as the proof of (3) except one uses Lemma \ref{lemma-fourier-mukai-left-adjoint} instead of Lemma \ref{lemma-fourier-mukai-right-adjoint}. \end{proof} \section{Resolutions and bounds} \label{section-tricks-smooth} \noindent The diagonal of a smooth proper scheme has a nice resolution. \begin{lemma} \label{lemma-on-product} Let $R$ be a Noetherian ring. Let $X$, $Y$ be finite type schemes over $R$ having the resolution property. For any coherent $\mathcal{O}_{X \times_R Y}$-module $\mathcal{F}$ there exist a surjection $\mathcal{E} \boxtimes \mathcal{G} \to \mathcal{F}$ where $\mathcal{E}$ is a finite locally free $\mathcal{O}_X$-module and $\mathcal{G}$ is a finite locally free $\mathcal{O}_Y$-module. \end{lemma} \begin{proof} Let $U \subset X$ and $V \subset Y$ be affine open subschemes. Let $\mathcal{I} \subset \mathcal{O}_X$ be the ideal sheaf of the reduced induced closed subscheme structure on $X \setminus U$. Similarly, let $\mathcal{I}' \subset \mathcal{O}_Y$ be the ideal sheaf of the reduced induced closed subscheme structure on $Y \setminus V$. Then the ideal sheaf $$ \mathcal{J} = \Im(\text{pr}_1^*\mathcal{I} \otimes_{\mathcal{O}_{X \times_R Y}} \text{pr}_2^*\mathcal{I}' \to \mathcal{O}_{X \times_R Y}) $$ satisfies $V(\mathcal{J}) = X \times_R Y \setminus U \times_R V$. For any section $s \in \mathcal{F}(U \times_R V)$ we can find an integer $n > 0$ and a map $\mathcal{J}^n \to \mathcal{F}$ whose restriction to $U \times_R V$ gives $s$, see Cohomology of Schemes, Lemma \ref{coherent-lemma-homs-over-open}. By assumption we can choose surjections $\mathcal{E} \to \mathcal{I}$ and $\mathcal{G} \to \mathcal{I}'$. These produce corresponding surjections $$ \mathcal{E} \boxtimes \mathcal{G} \to \mathcal{J} \quad\text{and}\quad \mathcal{E}^{\otimes n} \boxtimes \mathcal{G}^{\otimes n} \to \mathcal{J}^n $$ and hence a map $\mathcal{E}^{\otimes n} \boxtimes \mathcal{G}^{\otimes n} \to \mathcal{F}$ whose image contains the section $s$ over $U \times_R V$. Since we can cover $X \times_R Y$ by a finite number of affine opens of the form $U \times_R V$ and since $\mathcal{F}|_{U \times_R V}$ is generated by finitely many sections (Properties, Lemma \ref{properties-lemma-finite-type-module}) we conclude that there exists a surjection $$ \bigoplus\nolimits_{j = 1, \ldots, N} \mathcal{E}_j^{\otimes n_j} \boxtimes \mathcal{G}_j^{\otimes n_j} \to \mathcal{F} $$ where $\mathcal{E}_j$ is finite locally free on $X$ and $\mathcal{G}_j$ is finite locally free on $Y$. Setting $\mathcal{E} = \bigoplus \mathcal{E}_j^{\otimes n_j}$ and $\mathcal{G} = \bigoplus \mathcal{G}_j^{\otimes n_j}$ we conclude that the lemma is true. \end{proof} \begin{lemma} \label{lemma-on-product-general} Let $R$ be a ring. Let $X$, $Y$ be quasi-compact and quasi-separated schemes over $R$ having the resolution property. For any finite type quasi-coherent $\mathcal{O}_{X \times_R Y}$-module $\mathcal{F}$ there exist a surjection $\mathcal{E} \boxtimes \mathcal{G} \to \mathcal{F}$ where $\mathcal{E}$ is a finite locally free $\mathcal{O}_X$-module and $\mathcal{G}$ is a finite locally free $\mathcal{O}_Y$-module. \end{lemma} \begin{proof} Follows from Lemma \ref{lemma-on-product} by a limit argument. We urge the reader to skip the proof. Since $X \times_R Y$ is a closed subscheme of $X \times_\mathbf{Z} Y$ it is harmless if we replace $R$ by $\mathbf{Z}$. We can write $\mathcal{F}$ as the quotient of a finitely presented $\mathcal{O}_{X \times_R Y}$-module by Properties, Lemma \ref{properties-lemma-finite-directed-colimit-surjective-maps}. Hence we may assume $\mathcal{F}$ is of finite presentation. Next we can write $X = \lim X_i$ with $X_i$ of finite presentation over $\mathbf{Z}$ and similarly $Y = \lim Y_j$, see Limits, Proposition \ref{limits-proposition-approximate}. Then $\mathcal{F}$ will descend to $\mathcal{F}_{ij}$ on some $X_i \times_R Y_j$ (Limits, Lemma \ref{limits-lemma-descend-modules-finite-presentation}) and so does the property of having the resolution property (Derived Categories of Schemes, Lemma \ref{perfect-lemma-resolution-property-descends}). Then we apply Lemma \ref{lemma-on-product} to $\mathcal{F}_{ij}$ and we pullback. \end{proof} \begin{lemma} \label{lemma-diagonal-resolution} Let $R$ be a Noetherian ring. Let $X$ be a separated finite type scheme over $R$ which has the resolution property. Set $\mathcal{O}_\Delta = \Delta_*(\mathcal{O}_X)$ where $\Delta : X \to X \times_R X$ is the diagonal of $X/k$. There exists a resolution $$ \ldots \to \mathcal{E}_2 \boxtimes \mathcal{G}_2 \to \mathcal{E}_1 \boxtimes \mathcal{G}_1 \to \mathcal{E}_0 \boxtimes \mathcal{G}_0 \to \mathcal{O}_\Delta \to 0 $$ where each $\mathcal{E}_i$ and $\mathcal{G}_i$ is a finite locally free $\mathcal{O}_X$-module. \end{lemma} \begin{proof} Since $X$ is separated, the diagonal morphism $\Delta$ is a closed immersion and hence $\mathcal{O}_\Delta$ is a coherent $\mathcal{O}_{X \times_R X}$-module (Cohomology of Schemes, Lemma \ref{coherent-lemma-i-star-equivalence}). Thus the lemma follows immediately from Lemma \ref{lemma-on-product}. \end{proof} \begin{lemma} \label{lemma-Ext-0-regular} Let $X$ be a regular Noetherian scheme of dimension $d < \infty$. Then \begin{enumerate} \item for $\mathcal{F}$, $\mathcal{G}$ coherent $\mathcal{O}_X$-modules we have $\Ext^n_X(\mathcal{F}, \mathcal{G}) = 0$ for $n > d$, and \item for $K, L \in D^b_{\textit{Coh}}(\mathcal{O}_X)$ and $a \in \mathbf{Z}$ if $H^i(K) = 0$ for $i < a + d$ and $H^i(L) = 0$ for $i \geq a$ then $\Hom_X(K, L) = 0$. \end{enumerate} \end{lemma} \begin{proof} To prove (1) we use the spectral sequence $$ H^p(X, \SheafExt^q(\mathcal{F}, \mathcal{G})) \Rightarrow \Ext^{p + q}_X(\mathcal{F}, \mathcal{G}) $$ of Cohomology, Section \ref{cohomology-section-ext}. Let $x \in X$. We have $$ \SheafExt^q(\mathcal{F}, \mathcal{G})_x = \SheafExt^q_{\mathcal{O}_{X, x}}(\mathcal{F}_x, \mathcal{G}_x) $$ see Cohomology, Lemma \ref{cohomology-lemma-stalk-internal-hom} (this also uses that $\mathcal{F}$ is pseudo-coherent by Derived Categories of Schemes, Lemma \ref{perfect-lemma-identify-pseudo-coherent-noetherian}). Set $d_x = \dim(\mathcal{O}_{X, x})$. Since $\mathcal{O}_{X, x}$ is regular the ring $\mathcal{O}_{X, x}$ has global dimension $d_x$, see Algebra, Proposition \ref{algebra-proposition-regular-finite-gl-dim}. Thus $\SheafExt^q_{\mathcal{O}_{X, x}}(\mathcal{F}_x, \mathcal{G}_x)$ is zero for $q > d_x$. It follows that the modules $\SheafExt^q(\mathcal{F}, \mathcal{G})$ have support of dimension at most $d - q$. Hence we have $H^p(X, \SheafExt^q(\mathcal{F}, \mathcal{G})) = 0$ for $p > d - q$ by Cohomology, Proposition \ref{cohomology-proposition-vanishing-Noetherian}. This proves (1). \medskip\noindent Proof of (2). We may use induction on the number of nonzero cohomology sheaves of $K$ and $L$. The case where these numbers are $0, 1$ follows from (1). If the number of nonzero cohomology sheaves of $K$ is $> 1$, then we let $i \in \mathbf{Z}$ be minimal such that $H^i(K)$ is nonzero. We obtain a distinguished triangle $$ H^i(K)[-i] \to K \to \tau_{\geq i + 1}K $$ (Derived Categories, Remark \ref{derived-remark-truncation-distinguished-triangle}) and we get the vanishing of $\Hom(K, L)$ from the vanishing of $\Hom(H^i(K)[-i], L)$ and $\Hom(\tau_{\geq i + 1}K, L)$ by Derived Categories, Lemma \ref{derived-lemma-representable-homological}. Simlarly if $L$ has more than one nonzero cohomology sheaf. \end{proof} \begin{lemma} \label{lemma-split-complex-regular} Let $X$ be a regular Noetherian scheme of dimension $d < \infty$. Let $K \in D^b_{\textit{Coh}}(\mathcal{O}_X)$ and $a \in \mathbf{Z}$. If $H^i(K) = 0$ for $a < i < a + d$, then $K = \tau_{\leq a}K \oplus \tau_{\geq a + d}K$. \end{lemma} \begin{proof} We have $\tau_{\leq a}K = \tau_{\leq a + d - 1}K$ by the assumed vanishing of cohomology sheaves. By Derived Categories, Remark \ref{derived-remark-truncation-distinguished-triangle} we have a distinguished triangle $$ \tau_{\leq a}K \to K \to \tau_{\geq a + d}K \xrightarrow{\delta} (\tau_{\leq a}K)[1] $$ By Derived Categories, Lemma \ref{derived-lemma-split} it suffices to show that the morphism $\delta$ is zero. This follows from Lemma \ref{lemma-Ext-0-regular}. \end{proof} \begin{lemma} \label{lemma-diagonal-trick} Let $k$ be a field. Let $X$ be a quasi-compact separated smooth scheme over $k$. There exist finite locally free $\mathcal{O}_X$-modules $\mathcal{E}$ and $\mathcal{G}$ such that $$ \mathcal{O}_\Delta \in \langle \mathcal{E} \boxtimes \mathcal{G} \rangle $$ in $D(\mathcal{O}_{X \times X})$ where the notation is as in Derived Categories, Section \ref{derived-section-generators}. \end{lemma} \begin{proof} Recall that $X$ is regular by Varieties, Lemma \ref{varieties-lemma-smooth-regular}. Hence $X$ has the resolution property by Derived Categories of Schemes, Lemma \ref{perfect-lemma-regular-resolution-property}. Hence we may choose a resolution as in Lemma \ref{lemma-diagonal-resolution}. Say $\dim(X) = d$. Since $X \times X$ is smooth over $k$ it is regular. Hence $X \times X$ is a regular Noetherian scheme with $\dim(X \times X) = 2d$. The object $$ K = (\mathcal{E}_{2d} \boxtimes \mathcal{G}_{2d} \to \ldots \to \mathcal{E}_0 \boxtimes \mathcal{G}_0) $$ of $D_{perf}(\mathcal{O}_{X \times X})$ has cohomology sheaves $\mathcal{O}_\Delta$ in degree $0$ and $\Ker(\mathcal{E}_{2d} \boxtimes \mathcal{G}_{2d} \to \mathcal{E}_{2d-1} \boxtimes \mathcal{G}_{2d-1})$ in degree $-2d$ and zero in all other degrees. Hence by Lemma \ref{lemma-split-complex-regular} we see that $\mathcal{O}_\Delta$ is a summand of $K$ in $D_{perf}(\mathcal{O}_{X \times X})$. Clearly, the object $K$ is in $$ \left\langle \bigoplus\nolimits_{i = 0, \ldots, 2d} \mathcal{E}_i \boxtimes \mathcal{G}_i \right\rangle \subset \left\langle \left(\bigoplus\nolimits_{i = 0, \ldots, 2d} \mathcal{E}_i\right) \boxtimes \left(\bigoplus\nolimits_{i = 0, \ldots, 2d} \mathcal{G}_i\right) \right\rangle $$ which finishes the proof. (The reader may consult Derived Categories, Lemmas \ref{derived-lemma-generated-by-E-explicit} and \ref{derived-lemma-in-cone-n} to see that our object is contained in this category.) \end{proof} \begin{lemma} \label{lemma-smooth-proper-strong-generator} Let $k$ be a field. Let $X$ be a scheme proper and smooth over $k$. Then $D_{perf}(\mathcal{O}_X)$ has a strong generator. \end{lemma} \begin{proof} Using Lemma \ref{lemma-diagonal-trick} choose finite locally free $\mathcal{O}_X$-modules $\mathcal{E}$ and $\mathcal{G}$ such that $\mathcal{O}_\Delta \in \langle \mathcal{E} \boxtimes \mathcal{G} \rangle$ in $D(\mathcal{O}_{X \times X})$. We claim that $\mathcal{G}$ is a strong generator for $D_{perf}(\mathcal{O}_X)$. With notation as in Derived Categories, Section \ref{derived-section-operate-on-full} choose $m, n \geq 1$ such that $$ \mathcal{O}_\Delta \in smd(add(\mathcal{E} \boxtimes \mathcal{G}[-m, m])^{\star n}) $$ This is possible by Derived Categories, Lemma \ref{derived-lemma-find-smallest-containing-E}. Let $K$ be an object of $D_{perf}(\mathcal{O}_X)$. Since $L\text{pr}_1^*K \otimes_{\mathcal{O}_{X \times X}}^\mathbf{L} -$ is an exact functor and since $$ L\text{pr}_1^*K \otimes_{\mathcal{O}_{X \times X}}^\mathbf{L} (\mathcal{E} \boxtimes \mathcal{G}) = (K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{E}) \boxtimes \mathcal{G} $$ we conclude from Derived Categories, Remark \ref{derived-remark-operations-functor} that $$ L\text{pr}_1^*K \otimes_{\mathcal{O}_{X \times X}}^\mathbf{L} \mathcal{O}_\Delta \in smd(add( (K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{E}) \boxtimes \mathcal{G}[-m, m])^{\star n}) $$ Applying the exact functor $R\text{pr}_{2, *}$ and observing that $$ R\text{pr}_{2, *} \left((K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{E}) \boxtimes \mathcal{G}\right) = R\Gamma(X, K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{E}) \otimes_k \mathcal{G} $$ by Derived Categories of Schemes, Lemma \ref{perfect-lemma-cohomology-base-change} we conclude that $$ K = R\text{pr}_{2, *}(L\text{pr}_1^*K \otimes_{\mathcal{O}_{X \times X}}^\mathbf{L} \mathcal{O}_\Delta) \in smd(add(R\Gamma(X, K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{E}) \otimes_k \mathcal{G}[-m, m])^{\star n}) $$ The equality follows from the discussion in Example \ref{example-diagonal-fourier-mukai}. Since $K$ is perfect, there exist $a \leq b$ such that $H^i(X, K)$ is nonzero only for $i \in [a, b]$. Since $X$ is proper, each $H^i(X, K)$ is finite dimensional. We conclude that the right hand side is contained in $smd(add(\mathcal{G}[-m + a, m + b])^{\star n})$ which is itself contained in $\langle \mathcal{G} \rangle_n$ by one of the references given above. This finishes the proof. \end{proof} \begin{lemma} \label{lemma-diagonal-trick-proper} Let $k$ be a field. Let $X$ be a proper smooth scheme over $k$. There exists integers $m, n \geq 1$ and a finite locally free $\mathcal{O}_X$-module $\mathcal{G}$ such that every coherent $\mathcal{O}_X$-module is contained in $smd(add(\mathcal{G}[-m, m])^{\star n})$ with notation as in Derived Categories, Section \ref{derived-section-operate-on-full}. \end{lemma} \begin{proof} In the proof of Lemma \ref{lemma-smooth-proper-strong-generator} we have shown that there exist $m', n \geq 1$ such that for any coherent $\mathcal{O}_X$-module $\mathcal{F}$, $$ \mathcal{F} \in smd(add(\mathcal{G}[-m' + a, m' + b])^{\star n}) $$ for any $a \leq b$ such that $H^i(X, \mathcal{F})$ is nonzero only for $i \in [a, b]$. Thus we can take $a = 0$ and $b = \dim(X)$. Taking $m = \max(m', m' + b)$ finishes the proof. \end{proof} \noindent The following lemma is the boundedness result referred to in the title of this section. \begin{lemma} \label{lemma-boundedness} Let $k$ be a field. Let $X$ be a smooth proper scheme over $k$. Let $\mathcal{A}$ be an abelian category. Let $H : D_{perf}(\mathcal{O}_X) \to \mathcal{A}$ be a homological functor (Derived Categories, Definition \ref{derived-definition-homological}) such that for all $K$ in $D_{perf}(\mathcal{O}_X)$ the object $H^i(K)$ is nonzero for only a finite number of $i \in \mathbf{Z}$. Then there exists an integer $m \geq 1$ such that $H^i(\mathcal{F}) = 0$ for any coherent $\mathcal{O}_X$-module $\mathcal{F}$ and $i \not \in [-m, m]$. Similarly for cohomological functors. \end{lemma} \begin{proof} Combine Lemma \ref{lemma-diagonal-trick-proper} with Derived Categories, Lemma \ref{derived-lemma-forward-cone-n}. \end{proof} \begin{lemma} \label{lemma-bounded-fibres} Let $k$ be a field. Let $X$, $Y$ be finite type schemes over $k$. Let $K_0 \to K_1 \to K_2 \to \ldots$ be a system of objects of $D_{perf}(\mathcal{O}_{X \times Y})$ and $m \geq 0$ an integer such that \begin{enumerate} \item $H^q(K_i)$ is nonzero only for $q \leq m$, \item for every coherent $\mathcal{O}_X$-module $\mathcal{F}$ with $\dim(\text{Supp}(\mathcal{F})) = 0$ the object $$ R\text{pr}_{2, *}( \text{pr}_1^*\mathcal{F} \otimes_{\mathcal{O}_{X \times Y}}^\mathbf{L} K_n) $$ has vanishing cohomology sheaves in degrees outside $[-m, m] \cup [-m - n, m - n]$ and for $n > 2m$ the transition maps induce isomorphisms on cohomology sheaves in degrees in $[-m, m]$. \end{enumerate} Then $K_n$ has vanishing cohomology sheaves in degrees outside $[-m, m] \cup [-m - n, m - n]$ and for $n > 2m$ the transition maps induce isomorphisms on cohomology sheaves in degrees in $[-m, m]$. Moreover, if $X$ and $Y$ are smooth over $k$, then for $n$ large enough we find $K_n = K \oplus C_n$ in $D_{perf}(\mathcal{O}_{X \times Y})$ where $K$ has cohomology only indegrees $[-m, m]$ and $C_n$ only in degrees $[-m - n, m - n]$ and the transition maps define isomorphisms between various copies of $K$. \end{lemma} \begin{proof} Let $Z$ be the scheme theoretic support of an $\mathcal{F}$ as in (2). Then $Z \to \Spec(k)$ is finite, hence $Z \times Y \to Y$ is finite. It follows that for an object $M$ of $D_\QCoh(\mathcal{O}_{X \times Y})$ with cohomology sheaves supported on $Z \times Y$ we have $H^i(R\text{pr}_{2, *}(M)) = \text{pr}_{2, *}H^i(M)$ and the functor $\text{pr}_{2, *}$ is faithful on quasi-coherent modules supported on $Z \times Y$; details omitted. Hence we see that the objects $$ \text{pr}_1^*\mathcal{F} \otimes_{\mathcal{O}_{X \times Y}}^\mathbf{L} K_n $$ in $D_{perf}(\mathcal{O}_{X \times Y})$ have vanishing cohomology sheaves outside $[-m, m] \cup [-m - n, m - n]$ and for $n > 2m$ the transition maps induce isomorphisms on cohomology sheaves in $[-m, m]$. Let $z \in X \times Y$ be a closed point mapping to the closed point $x \in X$. Then we know that $$ K_{n, z} \otimes_{\mathcal{O}_{X \times Y, z}}^\mathbf{L} \mathcal{O}_{X \times Y, z}/\mathfrak m_x^t\mathcal{O}_{X \times Y, z} $$ has nonzero cohomology only in the intervals $[-m, m] \cup [-m - n, m - n]$. We conclude by More on Algebra, Lemma \ref{more-algebra-lemma-kollar-kovacs-pseudo-coherent} that $K_{n, z}$ only has nonzero cohomology in degrees $[-m, m] \cup [-m - n, m - n]$. Since this holds for all closed points of $X \times Y$, we conclude $K_n$ only has nonzero cohomology sheaves in degrees $[-m, m] \cup [-m - n, m - n]$. In exactly the same way we see that the maps $K_n \to K_{n + 1}$ are isomorphisms on cohomology sheaves in degrees $[-m, m]$ for $n > 2m$. \medskip\noindent If $X$ and $Y$ are smooth over $k$, then $X \times Y$ is smooth over $k$ and hence regular by Varieties, Lemma \ref{varieties-lemma-smooth-regular}. Thus we will obtain the direct sum decomposition of $K_n$ as soon as $n > 2m + \dim(X \times Y)$ from Lemma \ref{lemma-split-complex-regular}. The final statement is clear from this. \end{proof} \section{Sibling functors} \label{section-sibling} \noindent In this section we prove some categorical result on the following notion. \begin{definition} \label{definition-siblings} Let $\mathcal{A}$ be an abelian category. Let $\mathcal{D}$ be a triangulated category. We say two exact functors of triangulated categories $$ F, F' : D^b(\mathcal{A}) \longrightarrow \mathcal{D} $$ are {\it siblings}, or we say $F'$ is a {\it sibling} of $F$, if the following two conditions are satisfied \begin{enumerate} \item the functors $F \circ i$ and $F' \circ i$ are isomorphic where $i : \mathcal{A} \to D^b(\mathcal{A})$ is the inclusion functor, and \item $F(K) \cong F'(K)$ for any $K$ in $D^b(\mathcal{A})$. \end{enumerate} \end{definition} \noindent Sometimes the second condition is a consequence of the first. \begin{lemma} \label{lemma-sibling-fully-faithful} Let $\mathcal{A}$ be an abelian category. Let $\mathcal{D}$ be a triangulated category. Let $F, F' : D^b(\mathcal{A}) \longrightarrow \mathcal{D}$ be exact functors of triangulated categories. Assume \begin{enumerate} \item the functors $F \circ i$ and $F' \circ i$ are isomorphic where $i : \mathcal{A} \to D^b(\mathcal{A})$ is the inclusion functor, and \item for all $X, Y \in \Ob(\mathcal{A})$ we have $\Ext^q_\mathcal{D}(F(X), F(Y)) = 0$ for $q < 0$ (for example if $F$ is fully faithful). \end{enumerate} Then $F$ and $F'$ are siblings. \end{lemma} \begin{proof} Let $K \in D^b(\mathcal{A})$. We will show $F(K)$ is isomorphic to $F'(K)$. We can represent $K$ by a bounded complex $A^\bullet$ of objects of $\mathcal{A}$. After replacing $K$ by a translation we may assume $A^i = 0$ for $i > 0$. Choose $n \geq 0$ such that $A^{-i} = 0$ for $i > n$. The objects $$ M_i = (A^{-i} \to \ldots \to A^0)[-i],\quad i = 0, \ldots, n $$ form a Postnikov system in $D^b(\mathcal{A})$ for the complex $A^\bullet = A^{-n} \to \ldots \to A^0$ in $D^b(\mathcal{A})$. See Derived Categories, Example \ref{derived-example-key-postnikov}. Since both $F$ and $F'$ are exact functors of triangulated categories both $$ F(M_i) \quad\text{and}\quad F'(M_i) $$ form a Postnikov system in $\mathcal{D}$ for the complex $$ F(A^{-n}) \to \ldots \to F(A^0) = F'(A^{-n}) \to \ldots \to F'(A^0) $$ Since all negative $\Ext$s between these objects vanish by assumption we conclude by uniqueness of Postnikov systems (Derived Categories, Lemma \ref{derived-lemma-existence-postnikov-system}) that $F(K) = F(M_n[n]) \cong F'(M_n[n]) = F'(K)$. \end{proof} \begin{lemma} \label{lemma-sibling-faithful} Let $F$ and $F'$ be siblings as in Definition \ref{definition-siblings}. Then \begin{enumerate} \item if $F$ is essentially surjective, then $F'$ is essentially surjective, \item if $F$ is fully faithful, then $F'$ is fully faithful. \end{enumerate} \end{lemma} \begin{proof} Part (1) is immediate from property (2) for siblings. \medskip\noindent Assume $F$ is fully faithful. Denote $\mathcal{D}' \subset \mathcal{D}$ the essential image of $F$ so that $F : D^b(\mathcal{A}) \to \mathcal{D}'$ is an equivalence. Since the functor $F'$ factors through $\mathcal{D}'$ by property (2) for siblings, we can consider the functor $H = F^{-1} \circ F' : D^b(\mathcal{A}) \to D^b(\mathcal{A})$. Observe that $H$ is a sibling of the identity functor. Since it suffices to prove that $H$ is fully faithful, we reduce to the problem discussed in the next paragraph. \medskip\noindent Set $\mathcal{D} = D^b(\mathcal{A})$. We have to show a sibling $F : \mathcal{D} \to \mathcal{D}$ of the identity functor is fully faithful. Denote $a_X : X \to F(X)$ the functorial isomorphism for $X \in \Ob(\mathcal{A})$ given to us by Definition \ref{definition-siblings}. For any $K$ in $\mathcal{D}$ and distinguished triangle $K_1 \to K_2 \to K_3$ of $\mathcal{D}$ if the maps $$ F : \Hom(K, K_i[n]) \to \Hom(F(K), F(K_i[n])) $$ are isomorphisms for all $n \in \mathbf{Z}$ and $i = 1, 3$, then the same is true for $i = 2$ and all $n \in \mathbf{Z}$. This uses the $5$-lemma Homology, Lemma \ref{homology-lemma-five-lemma} and Derived Categories, Lemma \ref{derived-lemma-representable-homological}; details omitted. Similarly, if the maps $$ F : \Hom(K_i[n], K) \to \Hom(F(K_i[n]), F(K)) $$ are isomorphisms for all $n \in \mathbf{Z}$ and $i = 1, 3$, then the same is true for $i = 2$ and all $n \in \mathbf{Z}$. Using the canonical truncations and induction on the number of nonzero cohomology objects, we see that it is enough to show $$ F : \Ext^q(X, Y) \to \Ext^q(F(X), F(Y)) $$ is bijective for all $X, Y \in \Ob(\mathcal{A})$ and all $q \in \mathbf{Z}$. Since $F$ is a sibling of $\text{id}$ we have $F(X) \cong X$ and $F(Y) \cong Y$ hence the right hand side is zero for $q < 0$. The case $q = 0$ is OK by our assumption that $F$ is a sibling of the identity functor. It remains to prove the cases $q > 0$. \medskip\noindent The case $q = 1$: Injectivity. An element $\xi$ of $\Ext^1(X, Y)$ gives rise to a distinguished triangle $$ Y \to E \to X \xrightarrow{\xi} Y[1] $$ Observe that $E \in \Ob(\mathcal{A})$. Since $F$ is a sibling of the identity functor we obtain a commutative diagram $$ \xymatrix{ E \ar[d] \ar[r] & X \ar[d] \\ F(E) \ar[r] & F(X) } $$ whose vertical arrows are the isomorphisms $a_E$ and $a_X$. By TR3 the distinguished triangle associated to $\xi$ we started with is isomorphic to the distinguished triangle $$ F(Y) \to F(E) \to F(X) \xrightarrow{F(\xi)} F(Y[1]) = F(Y)[1] $$ Thus $\xi = 0$ if and only if $F(\xi)$ is zero, i.e., we see that $F : \Ext^1(X, Y) \to \Ext^1(F(X), F(Y))$ is injective. \medskip\noindent The case $q = 1$: Surjectivity. Let $\theta$ be an element of $\Ext^1(F(X), F(Y))$. This defines an extension of $F(X)$ by $F(Y)$ in $\mathcal{A}$ which we may write as $F(E)$ as $F$ is a sibling of the identity functor. We thus get a distinguished triangle $$ F(Y) \xrightarrow{F(\alpha)} F(E) \xrightarrow{F(\beta)} F(X) \xrightarrow{\theta} F(Y[1]) = F(Y)[1] $$ for some morphisms $\alpha : Y \to E$ and $\beta : E \to X$. Since $F$ is a sibling of the identity functor, the sequence $0 \to Y \to E \to X \to 0$ is a short exact sequence in $\mathcal{A}$! Hence we obtain a distinguished triangle $$ Y \xrightarrow{\alpha} E \xrightarrow{\beta} X \xrightarrow{\delta} Y[1] $$ for some morphism $\delta : X \to Y[1]$. Applying the exact functor $F$ we obtain the distinguished triangle $$ F(Y) \xrightarrow{F(\alpha)} F(E) \xrightarrow{F(\beta)} F(X) \xrightarrow{F(\delta)} F(Y)[1] $$ Arguing as above, we see that these triangles are isomorphic. Hence there exists a commutative diagram $$ \xymatrix{ F(X) \ar[d]^\gamma \ar[r]_{F(\delta)} & F(Y[1]) \ar[d]_\epsilon \\ F(X) \ar[r]^\theta & F(Y[1]) } $$ for some isomorphisms $\gamma$, $\epsilon$ (we can say more but we won't need more information). We may write $\gamma = F(\gamma')$ and $\epsilon = F(\epsilon')$. Then we have $\theta = F(\epsilon' \circ \delta \circ (\gamma')^{-1})$ and we see the surjectivity holds. \medskip\noindent The case $q > 1$: surjectivity. Using Yoneda extensions, see Derived Categories, Section \ref{derived-section-ext}, we find that for any element $\xi$ in $\Ext^q(F(X), F(Y))$ we can find $F(X) = B_0, B_1, \ldots, B_{q - 1}, B_q = F(Y) \in \Ob(\mathcal{A})$ and elements $$ \xi_i \in \Ext^1(B_{i - 1}, B_i) $$ such that $\xi$ is the composition $\xi_q \circ \ldots \circ \xi_1$. Write $B_i = F(A_i)$ (of course we have $A_i = B_i$ but we don't need to use this) so that $$ \xi_i = F(\eta_i) \in \Ext^1(F(A_{i - 1}), F(A_i)) \quad\text{with}\quad \eta_i \in \Ext^1(A_{i - 1}, A_i) $$ by surjectivity for $q = 1$. Then $\eta = \eta_q \circ \ldots \circ \eta_1$ is an element of $\Ext^q(X, Y)$ with $F(\eta) = \xi$. \medskip\noindent The case $q > 1$: injectivity. An element $\xi$ of $\Ext^q(X, Y)$ gives rise to a distinguished triangle $$ Y[q - 1] \to E \to X \xrightarrow{\xi} Y[q] $$ Applying $F$ we obtain a distinguished triangle $$ F(Y)[q - 1] \to F(E) \to F(X) \xrightarrow{F(\xi)} F(Y)[q] $$ If $F(\xi) = 0$, then $F(E) \cong F(Y)[q - 1] \oplus F(X)$ in $\mathcal{D}$, see Derived Categories, Lemma \ref{derived-lemma-split}. Since $F$ is a sibling of the identity functor we have $E \cong F(E)$ and hence $$ E \cong F(E) \cong F(Y)[q - 1] \oplus F(X) \cong Y[q - 1] \oplus X $$ In other words, $E$ is isomorphic to the direct sum of its cohomology objects. This implies that the initial distinguished triangle is split, i.e., $\xi = 0$. \end{proof} \noindent Let us make a nonstandard definition. Let $\mathcal{A}$ be an abelian category. Let us say $\mathcal{A}$ {\it has enough negative objects} if given any $X \in \Ob(\mathcal{A})$ there exists an object $N$ such that \begin{enumerate} \item there is a surjection $N \to X$ and \item $\Hom(X, N) = 0$. \end{enumerate} Let us prove a couple of lemmas about this notion in order to help with the proof of Proposition \ref{proposition-siblings-isomorphic}. \begin{lemma} \label{lemma-good-map} Let $\mathcal{A}$ be an abelian category with enough negative objects. Let $X \in D^b(\mathcal{A})$. Let $b \in \mathbf{Z}$ with $H^i(X) = 0$ for $i > b$. Then there exists a map $N[-b] \to X$ such that the induced map $N \to H^b(X)$ is surjective and $\Hom(H^b(X), N) = 0$. \end{lemma} \begin{proof} Using the truncation functors we can represent $X$ by a complex $A^a \to A^{a + 1} \to \ldots \to A^b$ of objects of $\mathcal{A}$. Choose $N$ in $\mathcal{A}$ such that there exists a surjection $t : N \to A^b$ and such that $\Hom(A^b, N) = 0$. Then the surjection $t$ defines a map $N[-b] \to X$ as desired. \end{proof} \begin{lemma} \label{lemma-good-map-zero} Let $\mathcal{A}$ be an abelian category with enough negative objects. Let $f : X \to X'$ be a morphism of $D^b(\mathcal{A})$. Let $b \in \mathbf{Z}$ such that $H^i(X) = 0$ for $i > b$ and $H^i(X') = 0$ for $i \geq b$. Then there exists a map $N[-b] \to X$ such that the induced map $N \to H^b(X)$ is surjective, such that $\Hom(H^b(X), N) = 0$, and such that the composition $N[-b] \to X \to X'$ is zero. \end{lemma} \begin{proof} We can represent $f$ by a map $f^\bullet : A^\bullet \to B^\bullet$ of bounded complexes of objects of $\mathcal{A}$, see for example Derived Categories, Lemma \ref{derived-lemma-bounded-derived}. Consider the object $$ C = \Ker(A^b \to A^{b + 1}) \times_{\Ker(B^b \to B^{b + 1})} B^{b - 1} $$ of $\mathcal{A}$. Since $H^b(B^\bullet) = 0$ we see that $C \to H^b(A^\bullet)$ is surjective. On the other hand, the map $C \to A^b \to B^b$ is the same as the map $C \to B^{b - 1} \to B^b$ and hence the composition $C[-b] \to X \to X'$ is zero. Since $\mathcal{A}$ has enough negative objects, we can find an object $N$ which has a surjection $N \to C \oplus H^b(X)$ such that $\Hom(C \oplus H^b(X), N) = 0$. Then $N$ together with the map $N[-b] \to X$ is a solution to the problem posed by the lemma. \end{proof} \noindent We encourage the reader to read the original \cite[Proposition 2.16]{Orlov-K3} for the marvellous ideas that go into the proof of the following proposition. \begin{proposition} \label{proposition-siblings-isomorphic} \begin{reference} \cite[Proposition 2.16]{Orlov-K3}; the fact that we do not need to assume vanishing of $\Ext^q(N, X)$ for $q > 0$ in the definition of negative objects above is due to \cite{Canonaco-Stellari}. \end{reference} Let $F$ and $F'$ be siblings as in Definition \ref{definition-siblings}. Assume that $F$ is fully faithful and that $\mathcal{A}$ has enough negative objects (see above). Then $F$ and $F'$ are isomorphic functors. \end{proposition} \begin{proof} By part (2) of Definition \ref{definition-siblings} the image of the functor $F'$ is contained in the essential image of the functor $F$. Hence the functor $H = F^{-1} \circ F'$ is a sibling of the identity functor. This reduces us to the case described in the next paragraph. \medskip\noindent Let $\mathcal{D} = D^b(\mathcal{A})$. We have to show a sibling $F : \mathcal{D} \to \mathcal{D}$ of the identity functor is isomorphic to the identity functor. Given an object $X$ of $\mathcal{D}$ let us say $X$ has {\it width} $w = w(X)$ if $w \geq 0$ is minimal such that there exists an integer $a \in \mathbf{Z}$ with $H^i(X) = 0$ for $i \not \in [a, a + w - 1]$. Since $F$ is a sibling of the identity and since $F \circ [n] = [n] \circ F$ we are aready given isomorphisms $$ c_X : X \to F(X) $$ for $w(X) \leq 1$ compatible with shifts. Moreover, if $X = A[-a]$ and $X' = A'[-a]$ for some $A, A' \in \Ob(\mathcal{A})$ then for any morphism $f : X \to X'$ the diagram \begin{equation} \label{equation-to-show} \vcenter{ \xymatrix{ X \ar[d]_{c_X} \ar[r]_f & X' \ar[d]^{c_{X'}} \\ F(X) \ar[r]^{F(f)} & F(X') } } \end{equation} is commutative. \medskip\noindent Next, let us show that for any morphism $f : X \to X'$ with $w(X), w(X') \leq 1$ the diagram (\ref{equation-to-show}) commutes. If $X$ or $X'$ is zero, this is clear. If not then we can write $X = A[-a]$ and $X' = A'[-a']$ for unique $A, A'$ in $\mathcal{A}$ and $a, a' \in \mathbf{Z}$. The case $a = a'$ was discussed above. If $a' > a$, then $f = 0$ (Derived Categories, Lemma \ref{derived-lemma-negative-exts}) and the result is clear. If $a' < a$ then $f$ corresponds to an element $\xi \in \Ext^q(A, A')$ with $q = a - a'$. Using Yoneda extensions, see Derived Categories, Section \ref{derived-section-ext}, we can find $A = A_0, A_1, \ldots, A_{q - 1}, A_q = A' \in \Ob(\mathcal{A})$ and elements $$ \xi_i \in \Ext^1(A_{i - 1}, A_i) $$ such that $\xi$ is the composition $\xi_q \circ \ldots \circ \xi_1$. In other words, setting $X_i = A_i[-a + i]$ we obtain morphisms $$ X = X_0 \xrightarrow{f_1} X_1 \to \ldots \to X_{q - 1} \xrightarrow{f_q} X_q = X' $$ whose compostion is $f$. Since the commutativity of (\ref{equation-to-show}) for $f_1, \ldots, f_q$ implies it for $f$, this reduces us to the case $q = 1$. In this case after shifting we may assume we have a distinguished triangle $$ A' \to E \to A \xrightarrow{f} A'[1] $$ Observe that $E$ is an object of $\mathcal{A}$. Consider the following diagram $$ \xymatrix{ E \ar[d]_{c_E} \ar[r] & A \ar[d]_{c_A} \ar[r]_f & A'[1] \ar[d]^{c_{A'}[1]} \ar@{..>}@<-1ex>[d]_\gamma \ar@{..>}[ld]^\epsilon \ar[r] & E[1] \ar[d]^{c_E[1]} \\ F(E) \ar[r] & F(A) \ar[r]^{F(f)} & F(A')[1] \ar[r] & F(E)[1] } $$ whose rows are distinguished triangles. The square on the right commutes already but we don't yet know that the middle square does. By the axioms of a triangulated category we can find a morphism $\gamma$ which does make the diagram commute. Then $\gamma - c_{A'}[1]$ composed with $F(A')[1] \to F(E)[1]$ is zero hence we can find $\epsilon : A'[1] \to F(A)$ such that $\gamma - c_{A'}[1] = F(f) \circ \epsilon$. However, any arrow $A'[1] \to F(A)$ is zero as it is a negative ext class between objects of $\mathcal{A}$. Hence $\gamma = c_{A'}[1]$ and we conclude the middle square commutes too which is what we wanted to show. \medskip\noindent To finish the proof we are going to argue by induction on $w$ that there exist isomorphisms $c_X : X \to F(X)$ for all $X$ with $w(X) \leq w$ compatible with all morphisms between such objects. The base case $w = 1$ was shown above. Assume we know the result for some $w \geq 1$. \medskip\noindent Let $X$ be an object with $w(X) = w + 1$. Pick $a \in \mathbf{Z}$ with $H^i(X) = 0$ for $i \not \in [a, a + w]$. Set $b = a + w$ so that $H^b(X)$ is nonzero. Choose $N[-b] \to X$ as in Lemma \ref{lemma-good-map}. Choose a distinguished diagram $$ N[-b] \to X \to Y \to N[-b + 1] $$ Computing the long exact cohomology sequence we find $w(Y) \leq w$. Hence by induction we find the solid arrows in the following diagram $$ \xymatrix{ N[-b] \ar[r] \ar[d]_{c_N[-b]} & X \ar[r] \ar@{..>}[d]_{c_{N[-b] \to X}} & Y \ar[r] \ar[d]^{c_Y} & N[-b + 1] \ar[d]^{c_N[-b + 1]} \\ F(N)[-b] \ar[r] & F(X) \ar[r] & F(Y) \ar[r] & F(N)[-b + 1] } $$ We obtain the dotted arrow $c_{N[-b] \to X}$. By Derived Categories, Lemma \ref{derived-lemma-uniqueness-third-arrow} the dotted arrow is unique because $\Hom(X, F(N)[-b]) \cong \Hom(X, N[-b]) = 0$ by our choice of $N$. In fact, $c_{N[-b] \to X}$ is the unique dotted arrow making the square with vertices $X, Y, F(X), F(Y)$ commute. \medskip\noindent Let $N'[-b] \to X$ be another map as in Lemma \ref{lemma-good-map} and let us prove that $c_{N[-b] \to X} = c_{N'[-b] \to X}$. Observe that the map $(N \oplus N')[-b] \to X$ also satisfies the conditions of Lemma \ref{lemma-good-map}. Thus we may assume $N'[-b] \to X$ factors as $N'[-b] \to N[-b] \to X$ for some morphism $N' \to N$. Choose distinguished triangles $N[-b] \to X \to Y \to N[-b + 1]$ and $N'[-b] \to X \to Y' \to N'[-b + 1]$. By axiom TR3 we can find a morphism $g : Y' \to Y$ which joint with $\text{id}_X$ and $N' \to N$ forms a morphism of triangles. Since we have (\ref{equation-to-show}) for $g$ we conclude that $$ (F(X) \to F(Y)) \circ c_{N'[-b] \to X} = (F(X) \to F(Y)) \circ c_{N[-b] \to X} $$ The uniqueness of $c_{N[-b] \to X}$ pointed out in the construction above now shows that $c_{N'[-b] \to X} = c_{N[-b] \to X}$. \medskip\noindent Thus we can now define for $X$ of width $w + 1$ the isomorphism $c_X : X \to F(X)$ as the common value of the maps $c_{N[-b] \to X}$ where $N[-b] \to X$ is as in Lemma \ref{lemma-good-map}. To finish the proof, we have to show that the diagrams (\ref{equation-to-show}) commute for all morphisms $f : X \to X'$ between objects with $w(X) \leq w + 1$ and $w(X') \leq w + 1$. Choose $a \leq b \leq a + w$ such that $H^i(X) = 0$ for $i \not \in [a, b]$ and $a' \leq b' \leq a' + w$ such that $H^i(X') = 0$ for $i \not \in [a', b']$. We will use induction on $(b' - a') + (b - a)$ to show the claim. (The base case is when this number is zero which is OK because $w \geq 1$.) We distinguish two cases. \medskip\noindent Case I: $b' < b$. In this case, by Lemma \ref{lemma-good-map-zero} we may choose $N[-b] \to X$ as in Lemma \ref{lemma-good-map} such that the composition $N[-b] \to X \to X'$ is zero. Choose a distuiguished triangle $N[-b] \to X \to Y \to N[-b + 1]$. Since $N[-b] \to X'$ is zero, we find that $f$ factors as $X \to Y \to X'$. Since $H^i(Y)$ is nonzero only for $i \in [a, b - 1]$ we see by induction that (\ref{equation-to-show}) commutes for $Y \to X'$. The diagram (\ref{equation-to-show}) commutes for $X \to Y$ by construction if $w(X) = w + 1$ and by our first induction hypothesis if $w(X) \leq w$. Hence (\ref{equation-to-show}) commutes for $f$. \medskip\noindent Case II: $b' \geq b$. In this case we choose $N'[-b'] \to X'$ as in Lemma \ref{lemma-good-map}. We may also assume that $\Hom(H^{b'}(X), N') = 0$ (this is relevant only if $b' = b$), for example because we can replace $N'$ by an object $N''$ which surjects onto $N' \oplus H^{b'}(X)$ and such that $\Hom(N' \oplus H^{b'}(X), N'') = 0$. We choose a distinguished triangle $N'[-b'] \to X' \to Y' \to N'[-b' + 1]$. Since $\Hom(X, X') \to \Hom(X, Y')$ is injective by our choice of $N'$ (details omitted) the same is true for $\Hom(X, F(X')) \to \Hom(X, F(Y'))$. Hence it suffices in this case to check that (\ref{equation-to-show}) commutes for the composition $X \to Y'$ of the morphisms $X \to X' \to Y'$. Since $H^i(Y')$ is nonzero only for $i \in [a', b' - 1]$ we conclude by induction hypothesis. \end{proof} \section{Deducing fully faithfulness} \label{section-get-fully-faithful} \noindent It will be useful for us to know when a functor is fully faithful we offer the following variant of \cite[Lemma 2.15]{Orlov-K3}. \begin{lemma} \label{lemma-get-fully-faithful} \begin{reference} Variant of \cite[Lemma 2.15]{Orlov-K3} \end{reference} Let $F : \mathcal{D} \to \mathcal{D}'$ be an exact functor of triangulated categories. Let $S \subset \Ob(\mathcal{D})$ be a set of objects. Assume \begin{enumerate} \item $F$ has both right and left adjoints, \item for $K \in \mathcal{D}$ if $\Hom(E, K[i]) = 0$ for all $E \in S$ and $i \in \mathbf{Z}$ then $K = 0$, \item for $K \in \mathcal{D}$ if $\Hom(K, E[i]) = 0$ for all $E \in S$ and $i \in \mathbf{Z}$ then $K = 0$, \item the map $\Hom(E, E'[i]) \to \Hom(F(E), F(E')[i])$ induced by $F$ is bijective for all $E, E' \in S$ and $i \in \mathbf{Z}$. \end{enumerate} Then $F$ is fully faithful. \end{lemma} \begin{proof} Denote $F_r$ and $F_l$ the right and left adjoints of $F$. For $E \in S$ choose a distinguished triangle $$ E \to F_r(F(E)) \to C \to E[1] $$ where the first arrow is the unit of the adjunction. For $E' \in S$ we have $$ \Hom(E', F_r(F(E))[i]) = \Hom(F(E'), F(E)[i]) = \Hom(E', E[i]) $$ The last equality holds by assumption (4). Hence applying the homological functor $\Hom(E', -)$ (Derived Categories, Lemma \ref{derived-lemma-representable-homological}) to the distinguished triangle above we conclude that $\Hom(E', C[i]) = 0$ for all $i \in \mathbf{Z}$ and $E' \in S$. By assumption (2) we conclude that $C = 0$ and $E = F_r(F(E))$. \medskip\noindent For $K \in \Ob(\mathcal{D})$ choose a distinguished triangle $$ F_l(F(K)) \to K \to C \to F_l(F(K))[1] $$ where the first arrow is the counit of the adjunction. For $E \in S$ we have $$ \Hom(F_l(F(K)), E[i]) = \Hom(F(K), F(E)[i]) = \Hom(K, F_r(F(E))[i]) = \Hom(K, E[i]) $$ where the last equality holds by the result of the first paragraph. Thus we conclude as before that $\Hom(C, E[i]) = 0$ for all $E \in S$ and $i \in \mathbf{Z}$. Hence $C = 0$ by assumption (3). Thus $F$ is fully faithful by Categories, Lemma \ref{categories-lemma-adjoint-fully-faithful}. \end{proof} \begin{lemma} \label{lemma-duality-at-point} Let $k$ be a field. Let $X$ be a scheme of finite type over $k$ which is regular. Let $x \in X$ be a closed point. For a coherent $\mathcal{O}_X$-module $\mathcal{F}$ supported at $x$ choose a coherent $\mathcal{O}_X$-module $\mathcal{F}'$ supported at $x$ such that $\mathcal{F}_x$ and $\mathcal{F}'_x$ are Matlis dual. Then there is an isomorphism $$ \Hom_X(\mathcal{F}, M) = H^0(X, M \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{F}'[-d_x]) $$ where $d_x = \dim(\mathcal{O}_{X, x})$ functorial in $M$ in $D_{perf}(\mathcal{O}_X)$. \end{lemma} \begin{proof} Since $\mathcal{F}$ is supported at $x$ we have $$ \Hom_X(\mathcal{F}, M) = \Hom_{\mathcal{O}_{X, x}}(\mathcal{F}_x, M_x) $$ and similarly we have $$ H^0(X, M \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{F}'[-d_x]) = \text{Tor}^{\mathcal{O}_{X, x}}_{d_x}(M_x, \mathcal{F}'_x) $$ Thus it suffices to show that given a Noetherian regular local ring $A$ of dimension $d$ and a finite length $A$-module $N$, if $N'$ is the Matlis dual to $N$, then there exists a functorial isomorphism $$ \Hom_A(N, K) = \text{Tor}^A_d(K, N') $$ for $K$ in $D_{perf}(A)$. We can write the left hand side as $H^0(R\Hom_A(N, A) \otimes_A^\mathbf{L} K)$ by More on Algebra, Lemma \ref{more-algebra-lemma-dual-perfect-complex} and the fact that $N$ determines a perfect object of $D(A)$. Hence the formula holds because $$ R\Hom_A(N, A) = R\Hom_A(N, A[d])[-d] = N'[-d] $$ by Dualizing Complexes, Lemma \ref{dualizing-lemma-dualizing-finite-length} and the fact that $A[d]$ is a normalized dualizing complex over $A$ ($A$ is Gorenstein by Dualizing Complexes, Lemma \ref{dualizing-lemma-regular-gorenstein}). \end{proof} \begin{lemma} \label{lemma-orthogonal-point-sheaf} Let $k$ be a field. Let $X$ be a scheme of finite type over $k$ which is regular. Let $x \in X$ be a closed point and denote $\mathcal{O}_x$ the skyscraper sheaf at $x$ with value $\kappa(x)$. Let $K$ in $D_{perf}(\mathcal{O}_X)$. \begin{enumerate} \item If $\Ext^i_X(\mathcal{O}_x, K) = 0$ then there exists an open neighbourhood $U$ of $x$ such that $H^{i - d_x}(K)|_U = 0$ where $d_x = \dim(\mathcal{O}_{X, x})$. \item If $\Hom_X(\mathcal{O}_x, K[i]) = 0$ for all $i \in \mathbf{Z}$, then $K$ is zero in an open neighbourhood of $x$. \item If $\Ext^i_X(K, \mathcal{O}_x) = 0$ then there exists an open neighbourhood $U$ of $x$ such that $H^i(K^\vee)|_U = 0$. \item If $\Hom_X(K, \mathcal{O}_x[i]) = 0$ for all $i \in \mathbf{Z}$, then $K$ is zero in an open neighbourhood of $x$. \item If $H^i(X, K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{O}_x) = 0$ then there exists an open neighbourhood $U$ of $x$ such that $H^i(K)|_U = 0$. \item If $H^i(X, K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{O}_x) = 0$ for $i \in \mathbf{Z}$ then $K$ is zero in an open neighbourhood of $x$. \end{enumerate} \end{lemma} \begin{proof} Observe that $H^i(X, K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{O}_x)$ is equal to $K_x \otimes_{\mathcal{O}_{X, x}}^\mathbf{L} \kappa(x)$. Hence part (5) follows from More on Algebra, Lemma \ref{more-algebra-lemma-cut-complex-in-two}. Part (6) follows from part (5). Part (1) follows from part (5), Lemma \ref{lemma-duality-at-point}, and the fact that the Matlis dual of $\kappa(x)$ is $\kappa(x)$. Part (2) follows from part (1). Part (3) follows from part (5) and the fact that $\Ext^i(K, \mathcal{O}_x) = H^i(X, K^\vee \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{O}_x)$ by Cohomology, Lemma \ref{cohomology-lemma-dual-perfect-complex}. Part (4) follows from part (3) and the fact that $K \cong (K^\vee)^\vee$ by the lemma just cited. \end{proof} \begin{lemma} \label{lemma-hom-into-point-sheaf} Let $X$ be a Noetherian scheme. Let $x \in X$ be a closed point and denote $\mathcal{O}_x$ the skyscraper sheaf at $x$ with value $\kappa(x)$. Let $K$ in $D^b_{\textit{Coh}}(\mathcal{O}_X)$. Let $b \in \mathbf{Z}$. The following are equivalent \begin{enumerate} \item $H^i(K)_x = 0$ for all $i > b$ and \item $\Hom_X(K, \mathcal{O}_x[-i]) = 0$ for all $i > b$. \end{enumerate} \end{lemma} \begin{proof} Consider the complex $K_x$ in $D^b_{\textit{Coh}}(\mathcal{O}_{X, x})$. There exist an integer $b_x \in \mathbf{Z}$ such that $K_x$ can be represented by a bounded above complex $$ \ldots \to \mathcal{O}_{X, x}^{\oplus n_{b_x - 2}} \to \mathcal{O}_{X, x}^{\oplus n_{b_x - 1}} \to \mathcal{O}_{X, x}^{\oplus n_{b_x}} \to 0 \to \ldots $$ with $\mathcal{O}_{X, x}^{\oplus n_i}$ sitting in degree $i$ where all the transition maps are given by matrices whose coefficients are in $\mathfrak m_x$. See More on Algebra, Lemma \ref{more-algebra-lemma-lift-pseudo-coherent-from-residue-field}. The result follows easily from this (and the equivalent conditions hold if and only if $b \geq b_x$). \end{proof} \begin{lemma} \label{lemma-get-fully-faithful-geometric} Let $k$ be a field. Let $X$ and $Y$ be proper schemes over $k$. Assume $X$ is regular. Then a $k$-linear exact functor $F : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)$ is fully faithful if and only if for any closed points $x, x' \in X$ the maps $$ F : \Ext^i_X(\mathcal{O}_x, \mathcal{O}_{x'}) \longrightarrow \Ext^i_Y(F(\mathcal{O}_x), F(\mathcal{O}_{x'})) $$ are isomorphisms for all $i \in \mathbf{Z}$. Here $\mathcal{O}_x$ is the skyscraper sheaf at $x$ with value $\kappa(x)$. \end{lemma} \begin{proof} By Lemma \ref{lemma-always-right-adjoints} the functor $F$ has both a left and a right adjoint. Thus we may apply the criterion of Lemma \ref{lemma-get-fully-faithful} because assumptions (2) and (3) of that lemma follow from Lemma \ref{lemma-orthogonal-point-sheaf}. \end{proof} \begin{lemma} \label{lemma-noah-pre} \begin{reference} Email from Noah Olander of Jun 9, 2020 \end{reference} Let $k$ be a field. Let $X$ be a proper scheme over $k$ which is regular. Let $F : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_X)$ be a $k$-linear exact functor. Assume for every coherent $\mathcal{O}_X$-module $\mathcal{F}$ with $\dim(\text{Supp}(\mathcal{F})) = 0$ there is an isomorphism $\mathcal{F} \cong F(\mathcal{F})$. Then $F$ is fully faithful. \end{lemma} \begin{proof} By Lemma \ref{lemma-get-fully-faithful-geometric} it suffices to show that the maps $$ F : \Ext^i_X(\mathcal{O}_x, \mathcal{O}_{x'}) \longrightarrow \Ext^i_X(F(\mathcal{O}_x), F(\mathcal{O}_{x'})) $$ are isomorphisms for all $i \in \mathbf{Z}$ and all closed points $x, x' \in X$. By assumption, the source and the target are isomorphic. If $x \not = x'$, then both sides are zero and the result is true. If $x = x'$, then it suffices to prove that the map is either injective or surjective. For $i < 0$ both sides are zero and the result is true. For $i = 0$ any nonzero map $\alpha : \mathcal{O}_x \to \mathcal{O}_x$ of $\mathcal{O}_X$-modules is an isomorphism. Hence $F(\alpha)$ is an isomorphism too and so $F(\alpha)$ is nonzero. Thus the result for $i = 0$. For $i = 1$ a nonzero element $\xi$ in $\Ext^1(\mathcal{O}_x, \mathcal{O}_x)$ corresponds to a nonsplit short exact sequence $$ 0 \to \mathcal{O}_x \to \mathcal{F} \to \mathcal{O}_x \to 0 $$ Since $F(\mathcal{F}) \cong \mathcal{F}$ we see that $F(\mathcal{F})$ is a nonsplit extension of $\mathcal{O}_x$ by $\mathcal{O}_x$ as well. Since $\mathcal{O}_x \cong F(\mathcal{O}_x)$ is a simple $\mathcal{O}_X$-module and $\mathcal{F} \cong F(\mathcal{F})$ has length $2$, we see that in the distinguished triangle $$ F(\mathcal{O}_x) \to F(\mathcal{F}) \to F(\mathcal{O}_x) \xrightarrow{F(\xi)} F(\mathcal{O}_x)[1] $$ the first two arrows must form a short exact sequence which must be isomorphic to the above short exact sequence and hence is nonsplit. It follows that $F(\xi)$ is nonzero and we conclude for $i = 1$. For $i > 1$ composition of ext classes defines a surjection $$ \Ext^1(F(\mathcal{O}_x), F(\mathcal{O}_x)) \otimes \ldots \otimes \Ext^1(F(\mathcal{O}_x), F(\mathcal{O}_x)) \longrightarrow \Ext^i(F(\mathcal{O}_x), F(\mathcal{O}_x)) $$ See Duality for Schemes, Lemma \ref{duality-lemma-regular-ideal-ext}. Hence surjectivity in degree $1$ implies surjectivity for $i > 0$. This finishes the proof. \end{proof} \section{Special functors} \label{section-special-functors} \noindent In this section we prove some results on functors of a special type that we will use later in this chapter. \begin{definition} \label{definition-siblings-geometric} Let $k$ be a field. Let $X$, $Y$ be finite type schemes over $k$. Recall that $D^b_{\textit{Coh}}(\mathcal{O}_X) = D^b(\textit{Coh}(\mathcal{O}_X))$ by Derived Categories of Schemes, Proposition \ref{perfect-proposition-DCoh}. We say two $k$-linear exact functors $$ F, F' : D^b_{\textit{Coh}}(\mathcal{O}_X) = D^b(\textit{Coh}(\mathcal{O}_X)) \longrightarrow D^b_{\textit{Coh}}(\mathcal{O}_Y) $$ are {\it siblings}, or we say $F'$ is a {\it sibling} of $F$ if $F$ and $F'$ are siblings in the sense of Definition \ref{definition-siblings} with abelian category being $\textit{Coh}(\mathcal{O}_X)$. If $X$ is regular then $D_{perf}(\mathcal{O}_X) = D^b_{\textit{Coh}}(\mathcal{O}_X)$ by Derived Categories of Schemes, Lemma \ref{perfect-lemma-perfect-on-noetherian} and we use the same terminology for $k$-linear exact functors $F, F' : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)$. \end{definition} \begin{lemma} \label{lemma-exact-functor-preserving-Coh} Let $k$ be a field. Let $X$, $Y$ be finite type schemes over $k$ with $X$ separated. Let $F : D^b_{\textit{Coh}}(\mathcal{O}_X) \to D^b_{\textit{Coh}}(\mathcal{O}_Y)$ be a $k$-linear exact functor sending $\textit{Coh}(\mathcal{O}_X) \subset D^b_{\textit{Coh}}(\mathcal{O}_X)$ into $\textit{Coh}(\mathcal{O}_Y) \subset D^b_{\textit{Coh}}(\mathcal{O}_Y)$. Then there exists a Fourier-Mukai functor $F' : D^b_{\textit{Coh}}(\mathcal{O}_X) \to D^b_{\textit{Coh}}(\mathcal{O}_Y)$ whose kernel is a coherent $\mathcal{O}_{X \times Y}$-module $\mathcal{K}$ flat over $X$ and with support finite over $Y$ which is a sibling of $F$. \end{lemma} \begin{proof} Denote $H : \textit{Coh}(\mathcal{O}_X) \to \textit{Coh}(\mathcal{O}_Y)$ the restriction of $F$. Since $F$ is an exact functor of triangulated categories, we see that $H$ is an exact functor of abelian categories. Of course $H$ is $k$-linear as $F$ is. By Functors and Morphisms, Lemma \ref{functors-lemma-functor-coherent-over-field} we obtain a coherent $\mathcal{O}_{X \times Y}$-module $\mathcal{K}$ which is flat over $X$ and has support finite over $Y$. Let $F'$ be the Fourier-Mukai functor defined using $\mathcal{K}$ so that $F'$ restricts to $H$ on $ \textit{Coh}(\mathcal{O}_X)$. The functor $F'$ sends $D^b_{\textit{Coh}}(\mathcal{O}_X)$ into $D^b_{\textit{Coh}}(\mathcal{O}_Y)$ by Lemma \ref{lemma-fourier-mukai-Coh}. Observe that $F$ and $F'$ satisfy the first and second condition of Lemma \ref{lemma-sibling-fully-faithful} and hence are siblings. \end{proof} \begin{remark} \label{remark-difficult} If $F, F' : D^b_{\textit{Coh}}(\mathcal{O}_X) \to \mathcal{D}$ are siblings, $F$ is fully faithful, and $X$ is reduced and projective over $k$ then $F \cong F'$; this follows from Proposition \ref{proposition-siblings-isomorphic} via the argument given in the proof of Theorem \ref{theorem-fully-faithful}. However, in general we do not know whether siblings are isomorphic. Even in the situation of Lemma \ref{lemma-exact-functor-preserving-Coh} it seems difficult to prove that the siblings $F$ and $F'$ are isomorphic functors. If $X$ is smooth and proper over $k$ and $F$ is fully faithful, then $F \cong F'$ as is shown in \cite{Noah}. If you have a proof or a counter example in more general situations, please email \href{mailto:stacks.project@gmail.com}{stacks.project@gmail.com}. \end{remark} \begin{lemma} \label{lemma-two-functors-pre} Let $k$ be a field. Let $X$, $Y$ be proper schemes over $k$. Assume $X$ is regular. Let $F, G : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)$ be $k$-linear exact functors such that \begin{enumerate} \item $F(\mathcal{F}) \cong G(\mathcal{F})$ for any coherent $\mathcal{O}_X$-module $\mathcal{F}$ with $\dim(\text{Supp}(\mathcal{F})) = 0$, \item $F$ is fully faithful. \end{enumerate} Then the essential image of $G$ is contained in the essential image of $F$. \end{lemma} \begin{proof} Recall that $F$ and $G$ have both adjoints, see Lemma \ref{lemma-always-right-adjoints}. In particular the essential image $\mathcal{A} \subset D_{perf}(\mathcal{O}_Y)$ of $F$ satisfies the equivalent conditions of Derived Categories, Lemma \ref{derived-lemma-right-adjoint}. We claim that $G$ factors through $\mathcal{A}$. Since $\mathcal{A} = {}^\perp(\mathcal{A}^\perp)$ by Derived Categories, Lemma \ref{derived-lemma-right-adjoint} it suffices to show that $\Hom_Y(G(M), N) = 0$ for all $M$ in $D_{perf}(\mathcal{O}_X)$ and $N \in \mathcal{A}^\perp$. We have $$ \Hom_Y(G(M), N) = \Hom_X(M, G_r(N)) $$ where $G_r$ is the right adjoint to $G$. Thus it suffices to prove that $G_r(N) = 0$. Since $G(\mathcal{F}) \cong F(\mathcal{F})$ for $\mathcal{F}$ as in (1) we see that $$ \Hom_X(\mathcal{F}, G_r(N)) = \Hom_Y(G(\mathcal{F}), N) = \Hom_Y(F(\mathcal{F}), N) = 0 $$ as $N$ is in the right orthogonal to the essential image $\mathcal{A}$ of $F$. Of course, the same vanishing holds for $\Hom_X(\mathcal{F}, G_r(N)[i])$ for any $i \in \mathbf{Z}$. Thus $G_r(N) = 0$ by Lemma \ref{lemma-orthogonal-point-sheaf} and we win. \end{proof} \begin{lemma} \label{lemma-noah} \begin{reference} Email from Noah Olander of Jun 8, 2020 \end{reference} Let $k$ be a field. Let $X$ be a proper scheme over $k$ which is regular. Let $F : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_X)$ be a $k$-linear exact functor. Assume for every coherent $\mathcal{O}_X$-module $\mathcal{F}$ with $\dim(\text{Supp}(\mathcal{F})) = 0$ there is an isomorphism $\mathcal{F} \cong F(\mathcal{F})$. Then there exists an automorphism $f : X \to X$ over $k$ which induces the identity on the underlying topological space\footnote{This often forces $f$ to be the identity, see Varieties, Lemma \ref{varieties-lemma-automorphism}.} and an invertible $\mathcal{O}_X$-module $\mathcal{L}$ such that $F$ and $F'(M) = f^*M \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{L}$ are siblings. \end{lemma} \begin{proof} By Lemma \ref{lemma-noah-pre} the functor $F$ is fully faithful. By Lemma \ref{lemma-two-functors-pre} the essential image of the identity functor is contained in the essential image of $F$, i.e., we see that $F$ is essentially surjective. Thus $F$ is an equivalence. Observe that the quasi-inverse $F^{-1}$ satisfies the same assumptions as $F$. \medskip\noindent Let $M \in D_{perf}(\mathcal{O}_X)$ and say $H^i(M) = 0$ for $i > b$. Since $F$ is fully faithful, we see that $$ \Hom_X(M, \mathcal{O}_x[-i]) = \Hom_X(F(M), F(\mathcal{O}_x)[-i]) \cong \Hom_X(F(M), \mathcal{O}_x[-i]) $$ for any $i \in \mathbf{Z}$ for any closed point $x$ of $X$. Thus by Lemma \ref{lemma-hom-into-point-sheaf} we see that $F(M)$ has vanishing cohomology sheaves in degrees $> b$. \medskip\noindent Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. By the above $F(\mathcal{F})$ has nonzero cohomology sheaves only in degrees $\leq 0$. Set $\mathcal{G} = H^0(F(\mathcal{F}))$. Choose a distinguished triangle $$ K \to F(\mathcal{F}) \to \mathcal{G} \to K[1] $$ Then $K$ has nonvanishing cohomology sheaves only in degrees $\leq -1$. Applying $F^{-1}$ we obtain a distinguished triangle $$ F^{-1}(K) \to \mathcal{F} \to F^{-1}(\mathcal{G}) \to F^{-1}(K')[1] $$ Since $F^{-1}(K)$ has nonvanishing cohomology sheaves only in degrees $\leq -1$ (by the previous paragraph applied to $F^{-1}$) we see that the arrow $F^{-1}(K) \to \mathcal{F}$ is zero (Derived Categories, Lemma \ref{derived-lemma-negative-exts}). Hence $K \to F(\mathcal{F})$ is zero, which implies that $F(\mathcal{F}) = \mathcal{G}$ by our choice of the first distinguished triangle. \medskip\noindent From the preceding paragraph, we deduce that $F$ preserves $\textit{Coh}(\mathcal{O}_X)$ and indeed defines an equivalence $H : \textit{Coh}(\mathcal{O}_X) \to \textit{Coh}(\mathcal{O}_X)$. By Functors and Morphisms, Lemma \ref{functors-lemma-equivalence-coherent-over-field} we get an automorphism $f : X \to X$ over $k$ and an invertible $\mathcal{O}_X$-module $\mathcal{L}$ such that $H(\mathcal{F}) = f^*\mathcal{F} \otimes \mathcal{L}$. Set $F'(M) = f^*M \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{L}$. Using Lemma \ref{lemma-sibling-fully-faithful} we see that $F$ and $F'$ are siblings. To see that $f$ is the identity on the underlying topological space of $X$, we use that $F(\mathcal{O}_x) \cong \mathcal{O}_x$ and that the support of $\mathcal{O}_x$ is $\{x\}$. This finishes the proof. \end{proof} \begin{lemma} \label{lemma-two-functors} Let $k$ be a field. Let $X$, $Y$ be proper schemes over $k$. Assume $X$ regular. Let $F, G : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)$ be $k$-linear exact functors such that \begin{enumerate} \item $F(\mathcal{F}) \cong G(\mathcal{F})$ for any coherent $\mathcal{O}_X$-module $\mathcal{F}$ with $\dim(\text{Supp}(\mathcal{F})) = 0$, \item $F$ is fully faithful, and \item $G$ is a Fourier-Mukai functor whose kernel is in $D_{perf}(\mathcal{O}_{X \times Y})$. \end{enumerate} Then there exists a Fourier-Mukai functor $F' : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)$ whose kernel is in $D_{perf}(\mathcal{O}_{X \times Y})$ such that $F$ and $F'$ are siblings. \end{lemma} \begin{proof} The essential image of $G$ is contained in the essential image of $F$ by Lemma \ref{lemma-two-functors-pre}. Consider the functor $H = F^{-1} \circ G$ which makes sense as $F$ is fully faithful. By Lemma \ref{lemma-noah} we obtain an automorphism $f : X \to X$ and an invertible $\mathcal{O}_X$-module $\mathcal{L}$ such that the functor $H' : K \mapsto f^*K \otimes \mathcal{L}$ is a sibling of $H$. In particular $H$ is an auto-equivalence by Lemma \ref{lemma-sibling-faithful} and $H$ induces an auto-equivalence of $\textit{Coh}(\mathcal{O}_X)$ (as this is true for its sibling functor $H'$). Thus the quasi-inverses $H^{-1}$ and $(H')^{-1}$ exist, are siblings (small detail omitted), and $(H')^{-1}$ sends $M$ to $(f^{-1})^*(M \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{L}^{\otimes -1})$ which is a Fourier-Mukai functor (details omitted). Then of course $F = G \circ H^{-1}$ is a sibling of $G \circ (H')^{-1}$. Since compositions of Fourier-Mukai functors are Fourier-Mukai by Lemma \ref{lemma-compose-fourier-mukai} we conclude. \end{proof} \section{Fully faithful functors} \label{section-fully-faithful} \noindent Our goal is to prove fully faithful functors between derived categories are siblings of Fourier-Mukai functors, following \cite{Orlov-K3} and \cite{Ballard}. \begin{situation} \label{situation-fully-faithful} Here $k$ is a field. We have proper smooth schemes $X$ and $Y$ over $k$. We have a $k$-linear, exact, fully faithful functor $F : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)$. \end{situation} \noindent Before reading on, it makes sense to read at least some of Derived Categories, Section \ref{derived-section-postnikov}. \medskip\noindent Recall that $X$ is regular and hence has the resolution property (Varieties, Lemma \ref{varieties-lemma-smooth-regular} and Derived Categories of Schemes, Lemma \ref{perfect-lemma-regular-resolution-property}). Thus on $X \times X$ we may choose a resolution $$ \ldots \to \mathcal{E}_2 \boxtimes \mathcal{G}_2 \to \mathcal{E}_1 \boxtimes \mathcal{G}_1 \to \mathcal{E}_0 \boxtimes \mathcal{G}_0 \to \mathcal{O}_\Delta \to 0 $$ where each $\mathcal{E}_i$ and $\mathcal{G}_i$ is a finite locally free $\mathcal{O}_X$-module, see Lemma \ref{lemma-diagonal-resolution}. Using the complex \begin{equation} \label{equation-original-complex} \ldots \to \mathcal{E}_2 \boxtimes \mathcal{G}_2 \to \mathcal{E}_1 \boxtimes \mathcal{G}_1 \to \mathcal{E}_0 \boxtimes \mathcal{G}_0 \end{equation} in $D_{perf}(\mathcal{O}_{X \times X})$ as in Derived Categories, Example \ref{derived-example-key-postnikov} if for each $n$ we denote $$ M_n = (\mathcal{E}_n \boxtimes \mathcal{G}_n \to \ldots \to \mathcal{E}_0 \boxtimes \mathcal{G}_0)[-n] $$ we obtain an infinite Postnikov system for the complex (\ref{equation-original-complex}). This means the morphisms $M_0 \to M_1[1] \to M_2[2] \to \ldots$ and $M_n \to \mathcal{E}_n \boxtimes \mathcal{G}_n$ and $\mathcal{E}_n \boxtimes \mathcal{G}_n \to M_{n - 1}$ satisfy certain conditions documented in Derived Categories, Definition \ref{derived-definition-postnikov-system}. Set $$ \mathcal{F}_n = \Ker(\mathcal{E}_n \boxtimes \mathcal{G}_n \to \mathcal{E}_{n - 1} \boxtimes \mathcal{G}_{n - 1}) $$ Observe that since $\mathcal{O}_\Delta$ is flat over $X$ via $\text{pr}_1$ the same is true for $\mathcal{F}_n$ for all $n$ (this is a convenient though not essential observation). We have $$ H^q(M_n[n]) = \left\{ \begin{matrix} \mathcal{O}_\Delta & \text{if} & q = 0 \\ \mathcal{F}_n & \text{if} & q = -n \\ 0 & \text{if} & q \not = 0, -n \end{matrix} \right. $$ Thus for $n \geq \dim(X \times X)$ we have $$ M_n[n] \cong \mathcal{O}_\Delta \oplus \mathcal{F}_n[n] $$ in $D_{perf}(\mathcal{O}_{X \times X})$ by Lemma \ref{lemma-split-complex-regular}. \medskip\noindent We are interested in the complex \begin{equation} \label{equation-complex} \ldots \to \mathcal{E}_2 \boxtimes F(\mathcal{G}_2) \to \mathcal{E}_1 \boxtimes F(\mathcal{G}_1) \to \mathcal{E}_0 \boxtimes F(\mathcal{G}_0) \end{equation} in $D_{perf}(\mathcal{O}_{X \times Y})$ as the ``totalization'' of this complex should give us the kernel of the Fourier-Mukai functor we are trying to construct. For all $i, j \geq 0$ we have \begin{align*} \Ext^q_{X \times Y}(\mathcal{E}_i \boxtimes F(\mathcal{G}_i), \mathcal{E}_j \boxtimes F(\mathcal{G}_j)) & = \bigoplus\nolimits_p \Ext^{q + p}_X(\mathcal{E}_i, \mathcal{E}_j) \otimes_k \Ext^{-p}_Y(F(\mathcal{G}_i), F(\mathcal{G}_j)) \\ & = \bigoplus\nolimits_p \Ext^{q + p}_X(\mathcal{E}_i, \mathcal{E}_j) \otimes_k \Ext^{-p}_X(\mathcal{G}_i, \mathcal{G}_j) \end{align*} The second equality holds because $F$ is fully faithful and the first by Derived Categories of Schemes, Lemma \ref{perfect-lemma-kunneth-Ext}. We find these $\Ext^q$ are zero for $q < 0$. Hence by Derived Categories, Lemma \ref{derived-lemma-existence-postnikov-system} we can build an infinite Postnikov system $K_0, K_1, K_2, \ldots$ in $D_{perf}(\mathcal{O}_{X \times Y})$ for the complex (\ref{equation-complex}). Parallel to what happens with $M_0, M_1, M_2, \ldots$ this means we obtain morphisms $K_0 \to K_1[1] \to K_2[2] \to \ldots$ and $K_n \to \mathcal{E}_n \boxtimes F(\mathcal{G}_n)$ and $\mathcal{E}_n \boxtimes F(\mathcal{G}_n) \to K_{n - 1}$ in $D_{perf}(\mathcal{O}_{X \times Y})$ satisfying certain conditions documented in Derived Categories, Definition \ref{derived-definition-postnikov-system}. \medskip\noindent Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module whose support has a finite number of points, i.e., with $\dim(\text{Supp}(\mathcal{F})) = 0$. Consider the exact functor of triangulated categories $$ D_{perf}(\mathcal{O}_{X \times Y}) \longrightarrow D_{perf}(\mathcal{O}_Y),\quad N \longmapsto R\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes^\mathbf{L}_{\mathcal{O}_{X \times Y}} N) $$ It follows that the objects $R\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes^\mathbf{L}_{\mathcal{O}_{X \times Y}} K_i)$ form a Postnikov system for the complex in $D_{perf}(\mathcal{O}_Y)$ with terms $$ R\text{pr}_{2, *}( (\mathcal{F} \otimes \mathcal{E}_i) \boxtimes F(\mathcal{G}_i)) = \Gamma(X, \mathcal{F} \otimes \mathcal{E}_i) \otimes_k F(\mathcal{G}_i) = F(\Gamma(X, \mathcal{F} \otimes \mathcal{E}_i) \otimes_k \mathcal{G}_i) $$ Here we have used that $\mathcal{F} \otimes \mathcal{E}_i$ has vanishing higher cohomology as its support has dimension $0$. On the other hand, applying the exact functor $$ D_{perf}(\mathcal{O}_{X \times X}) \longrightarrow D_{perf}(\mathcal{O}_Y),\quad N \longmapsto F(R\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes^\mathbf{L}_{\mathcal{O}_{X \times X}} N)) $$ we find that the objects $F(R\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes^\mathbf{L}_{\mathcal{O}_{X \times X}} M_n))$ form a second infinite Postnikov system for the complex in $D_{perf}(\mathcal{O}_Y)$ with terms $$ F(R\text{pr}_{2, *}( (\mathcal{F} \otimes \mathcal{E}_i) \boxtimes \mathcal{G}_i)) = F(\Gamma(X, \mathcal{F} \otimes \mathcal{E}_i) \otimes_k \mathcal{G}_i) $$ This is the same as before! By uniqueness of Postnikov systems (Derived Categories, Lemma \ref{derived-lemma-existence-postnikov-system}) which applies because $$ \Ext^q_Y( F(\Gamma(X, \mathcal{F} \otimes \mathcal{E}_i) \otimes_k \mathcal{G}_i), F(\Gamma(X, \mathcal{F} \otimes \mathcal{E}_j) \otimes_k \mathcal{G}_j)) = 0, \quad q < 0 $$ as $F$ is fully faithful, we find a system of isomorphisms $$ F(R\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes^\mathbf{L}_{\mathcal{O}_{X \times X}} M_n[n])) \cong R\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes^\mathbf{L}_{\mathcal{O}_{X \times Y}} K_n[n]) $$ in $D_{perf}(\mathcal{O}_Y)$ compatible with the morphisms in $D_{perf}(\mathcal{O}_Y)$ induced by the morphisms $$ M_{n - 1}[n - 1] \to M_n[n] \quad\text{and}\quad K_{n - 1}[n - 1] \to K_n[n] $$ $$ M_n \to \mathcal{E}_n \boxtimes \mathcal{G}_n \quad\text{and}\quad K_n \to \mathcal{E}_n \boxtimes F(\mathcal{G}_n) $$ $$ \mathcal{E}_n \boxtimes \mathcal{G}_n \to M_{n - 1} \quad\text{and}\quad \mathcal{E}_n \boxtimes F(\mathcal{G}_n) \to K_{n - 1} $$ which are part of the structure of Postnikov systems. For $n$ sufficiently large we obtain a direct sum decomposition $$ F(R\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes^\mathbf{L}_{\mathcal{O}_{X \times X}} M_n[n])) = F(\mathcal{F}) \oplus F(R\text{pr}_{2, *}( \text{pr}_1^*\mathcal{F} \otimes_{\mathcal{O}_{X \times Y}} \mathcal{F}_n ))[n] $$ corresponding to the direct sum decomposition of $M_n$ constructed above (we are using the flatness of $\mathcal{F}_n$ over $X$ via $\text{pr}_1$ to write a usual tensor product in the formula above, but this isn't essential for the argument). By Lemma \ref{lemma-boundedness} we find there exists an integer $m \geq 0$ such that the first summand in this direct sum decomposition has nonzero cohomology sheaves only in the interval $[-m, m]$ and the second summand in this direct sum decomposition has nonzero cohomology sheaves only in the interval $[-m - n, m + \dim(X) - n]$. We conclude the system $K_0 \to K_1[1] \to K_2[2] \to \ldots$ in $D_{perf}(\mathcal{O}_{X \times Y})$ satisfies the assumptions of Lemma \ref{lemma-bounded-fibres} after possibly replacing $m$ by a larger integer. We conclude we can write $$ K_n[n] = K \oplus C_n $$ for $n \gg 0$ compatible with transition maps and with $C_n$ having nonzero cohomology sheaves only in the range $[-m - n, m - n]$. Denote $G$ the Fourier-Mukai functor corresponding to $K$. Putting everything together we find $$ \begin{matrix} G(\mathcal{F}) \oplus R\text{pr}_{2, *}( \text{pr}_1^*\mathcal{F} \otimes_{\mathcal{O}_{X \times Y}}^\mathbf{L} C_n) \cong \\ R\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes^\mathbf{L}_{\mathcal{O}_{X \times Y}} K_n[n]) \cong \\ F(R\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes^\mathbf{L}_{\mathcal{O}_{X \times X}} M_n[n])) \cong \\ F(\mathcal{F}) \oplus F(R\text{pr}_{2, *}( \text{pr}_1^*\mathcal{F} \otimes_{\mathcal{O}_{X \times Y}} \mathcal{F}_n ))[n] \end{matrix} $$ Looking at the degrees that objects live in we conclude that for $n \gg m$ we obtain an isomorphism $$ F(\mathcal{F}) \cong G(\mathcal{F}) $$ Moreover, recall that this holds for every coherent $\mathcal{F}$ on $X$ whose support has dimension $0$. \begin{lemma} \label{lemma-fully-faithful} Let $k$ be a field. Let $X$ and $Y$ be smooth proper schemes over $k$. Given a $k$-linear, exact, fully faithful functor $F : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)$ there exists a Fourier-Mukai functor $F' : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)$ whose kernel is in $D_{perf}(\mathcal{O}_{X \times Y})$ which is a sibling to $F$. \end{lemma} \begin{proof} Apply Lemma \ref{lemma-two-functors} to $F$ and the functor $G$ constructed above. \end{proof} \noindent The following theorem is also true without assuming $X$ is projective, see \cite{Noah}. \begin{theorem}[Orlov] \label{theorem-fully-faithful} \begin{reference} \cite[Theorem 2.2]{Orlov-K3}; this is shown in \cite{Noah} without the assumption that $X$ be projective \end{reference} Let $k$ be a field. Let $X$ and $Y$ be smooth proper schemes over $k$ with $X$ projective over $k$. Any $k$-linear fully faithful exact functor $F : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)$ is a Fourier-Mukai functor for some kernel in $D_{perf}(\mathcal{O}_{X \times Y})$. \end{theorem} \begin{proof} Let $F'$ be the Fourier-Mukai functor which is a sibling of $F$ as in Lemma \ref{lemma-fully-faithful}. By Proposition \ref{proposition-siblings-isomorphic} we have $F \cong F'$ provided we can show that $\textit{Coh}(\mathcal{O}_X)$ has enough negative objects. However, if $X = \Spec(k)$ for example, then this isn't true. Thus we first decompose $X = \coprod X_i$ into its connected (and irreducible) components and we argue that it suffices to prove the result for each of the (fully faithful) composition functors $$ F_i : D_{perf}(\mathcal{O}_{X_i}) \to D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y) $$ Details omitted. Thus we may assume $X$ is irreducible. \medskip\noindent The case $\dim(X) = 0$. Here $X$ is the spectrum of a finite (separable) extension $k'/k$ and hence $D_{perf}(\mathcal{O}_X)$ is equivalent to the category of graded $k'$-vector spaces such that $\mathcal{O}_X$ corresponds to the trivial $1$-dimensional vector space in degree $0$. It is straightforward to see that any two siblings $F, F' : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)$ are isomorphic. Namely, we are given an isomorphism $F(\mathcal{O}_X) \cong F'(\mathcal{O}_X)$ compatible the action of the $k$-algebra $k' = \text{End}_{D_{perf}(\mathcal{O}_X)}(\mathcal{O}_X)$ which extends canonically to an isomorphism on any graded $k'$-vector space. \medskip\noindent The case $\dim(X) > 0$. Here $X$ is a projective smooth variety of dimension $> 1$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. We have to show there exists a coherent module $\mathcal{N}$ such that \begin{enumerate} \item there is a surjection $\mathcal{N} \to \mathcal{F}$ and \item $\Hom(\mathcal{F}, \mathcal{N}) = 0$. \end{enumerate} Choose an ample invertible $\mathcal{O}_X$-module $\mathcal{L}$. We claim that $\mathcal{N} = (\mathcal{L}^{\otimes n})^{\oplus r}$ will work for $n \ll 0$ and $r$ large enough. Condition (1) follows from Properties, Proposition \ref{properties-proposition-characterize-ample}. Finally, we have $$ \Hom(\mathcal{F}, \mathcal{L}^{\otimes n}) = H^0(X, \SheafHom(\mathcal{F}, \mathcal{L}^{\otimes n})) = H^0(X, \SheafHom(\mathcal{F}, \mathcal{O}_X) \otimes \mathcal{L}^{\otimes n}) $$ Since the dual $\SheafHom(\mathcal{F}, \mathcal{O}_X)$ is torsion free, this vanishes for $n \ll 0$ by Varieties, Lemma \ref{varieties-lemma-vanishin-h0-negative}. This finishes the proof. \end{proof} \begin{proposition} \label{proposition-equivalence} Let $k$ be a field. Let $X$ and $Y$ be smooth proper schemes over $k$. If $F : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)$ is a $k$-linear exact equivalence of triangulated categories then there exists a Fourier-Mukai functor $F' : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)$ whose kernel is in $D_{perf}(\mathcal{O}_{X \times Y})$ which is an equivalence and a sibling of $F$. \end{proposition} \begin{proof} The functor $F'$ of Lemma \ref{lemma-fully-faithful} is an equivalence by Lemma \ref{lemma-sibling-faithful}. \end{proof} \begin{lemma} \label{lemma-uniqueness} Let $k$ be a field. Let $X$ be a smooth proper scheme over $k$. Let $K \in D_{perf}(\mathcal{O}_{X \times X})$. If the Fourier-Mukai functor $\Phi_K : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_X)$ is isomorphic to the identity functor, then $K \cong \Delta_*\mathcal{O}_X$ in $_{perf}(\mathcal{O}_{X \times X})$. \end{lemma} \begin{proof} Let $i$ be the minimal integer such that the cohomology sheaf $H^i(K)$ is nonzero. Let $\mathcal{E}$ and $\mathcal{G}$ be finite locally free $\mathcal{O}_X$-modules. Then \begin{align*} H^i(X \times X, K \otimes_{\mathcal{O}_{X \times X}}^\mathbf{L} (\mathcal{E} \boxtimes \mathcal{G})) & = H^i(X, R\text{pr}_{2, *}(K \otimes_{\mathcal{O}_{X \times X}}^\mathbf{L} (\mathcal{E} \boxtimes \mathcal{G}))) \\ & = H^i(X, \Phi_K(\mathcal{E}) \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{G}) \\ & \cong H^i(X, \mathcal{E} \otimes \mathcal{G}) \end{align*} which is zero if $i < 0$. On the other hand, we can choose $\mathcal{E}$ and $\mathcal{G}$ such that there is a surjection $\mathcal{E}^\vee \boxtimes \mathcal{G}^\vee \to H^i(K)$ by Lemma \ref{lemma-on-product}. In this case the left hand side of the equalities is nonzero. Hence we conclude that $H^i(K) = 0$ for $i < 0$. \medskip\noindent Let $i$ be the maximal integer such that $H^i(K)$ is nonzero. The same argument with $\mathcal{E}$ and $\mathcal{G}$ support of dimension $0$ shows that $i \leq 0$. Hence we conclude that $K$ is given by a single coherent $\mathcal{O}_{X \times X}$-module $\mathcal{K}$ sitting in degree $0$. \medskip\noindent Since $R\text{pr}_{2, *}(\text{pr}_1^*\mathcal{F} \otimes \mathcal{K})$ is $\mathcal{F}$, by taking $\mathcal{F}$ supported at closed points we see that the support of $\mathcal{K}$ is finite over $X$ via $\text{pr}_2$. Since $R\text{pr}_{2, *}(\mathcal{K}) \cong \mathcal{O}_X$ we conclude by Functors and Morphisms, Lemma \ref{functors-lemma-pushforward-invertible-pre} that $\mathcal{K} = s_*\mathcal{O}_X$ for some section $s : X \to X \times X$ of the second projection. Then $\Phi_K(M) = f^*M$ where $f = \text{pr}_1 \circ s$ and this can happen only if $s$ is the diagonal morphism as desired. \end{proof} \section{A category of Fourier-Mukai kernels} \label{section-category-Fourier-Mukai-kernels} \noindent Let $S$ be a scheme. We claim there is a category with \begin{enumerate} \item Objects are proper smooth schemes over $S$. \item Morphisms from $X$ to $Y$ are isomorphism classes of objects of $D_{perf}(\mathcal{O}_{X \times_S Y})$. \item Composition of the isomorphism class of $K \in D_{perf}(\mathcal{O}_{X \times_S Y})$ and the isomorphism class of $K'$ in $D_{perf}(\mathcal{O}_{Y \times_S Z})$ is the isomorphism class of $$ R\text{pr}_{13, *}( L\text{pr}_{12}^*K \otimes_{\mathcal{O}_{X \times_S Y \times_S Z}}^\mathbf{L} L\text{pr}_{23}^*K') $$ which is in $D_{perf}(\mathcal{O}_{X \times_S Z})$ by Derived Categories of Schemes, Lemma \ref{perfect-lemma-flat-proper-perfect-direct-image-general}. \item The identity morphism from $X$ to $X$ is the isomorphism class of $\Delta_{X/S, *}\mathcal{O}_X$ which is in $D_{perf}(\mathcal{O}_{X \times_S X})$ by More on Morphisms, Lemma \ref{more-morphisms-lemma-perfect-closed-immersion-perfect-direct-image} and the fact that $\Delta_{X/S}$ is a perfect morphism by Divisors, Lemma \ref{divisors-lemma-immersion-smooth-into-smooth-regular-immersion} and More on Morphisms, Lemma \ref{more-morphisms-lemma-regular-immersion-perfect}. \end{enumerate} Let us check that associativity of composition of morphisms holds; we omit verifying that the identity morphisms are indeed identities. To see this suppose we have $X, Y, Z, W$ and $c \in D_{perf}(\mathcal{O}_{X \times_S Y})$, $c' \in D_{perf}(\mathcal{O}_{Y \times_S Z})$, and $c'' \in D_{perf}(\mathcal{O}_{Z \times_S W})$. Then we have \begin{align*} c'' \circ (c' \circ c) & \cong \text{pr}^{134}_{14, *}( \text{pr}^{134, *}_{13} \text{pr}^{123}_{13, *}(\text{pr}^{123, *}_{12}c \otimes \text{pr}^{123, *}_{23}c') \otimes \text{pr}^{134, *}_{34}c'') \\ & \cong \text{pr}^{134}_{14, *}( \text{pr}^{1234}_{134, *} \text{pr}^{1234, *}_{123}(\text{pr}^{123, *}_{12}c \otimes \text{pr}^{123, *}_{23}c') \otimes \text{pr}^{134, *}_{34}c'') \\ & \cong \text{pr}^{134}_{14, *}( \text{pr}^{1234}_{134, *} (\text{pr}^{1234, *}_{12}c \otimes \text{pr}^{1234, *}_{23}c') \otimes \text{pr}^{134, *}_{34}c'') \\ & \cong \text{pr}^{134}_{14, *} \text{pr}^{1234}_{134, *} ((\text{pr}^{1234, *}_{12}c \otimes \text{pr}^{1234, *}_{23}c') \otimes \text{pr}^{1234, *}_{34}c'') \\ & \cong \text{pr}^{1234}_{14, *}( (\text{pr}^{1234, *}_{12}c \otimes \text{pr}^{1234, *}_{23}c') \otimes \text{pr}^{1234, *}_{34}c'') \end{align*} Here we use the notation $$ p^{1234}_{134} : X \times_S Y \times_S Z \times_S W \to X \times_S Z \times_S W \quad\text{and}\quad p^{134}_{14} : X \times_S Z \times_S W \to X \times_S W $$ the projections and similarly for other indices. We also write $\text{pr}_*$ instead of $R\text{pr}_*$ and $\text{pr}^*$ instead of $L\text{pr}^*$ and we drop all super and sub scripts on $\otimes$. The first equality is the definition of the composition. The second equality holds because $\text{pr}^{134, *}_{13} \text{pr}^{123}_{13, *} = \text{pr}^{1234}_{134, *} \text{pr}^{1234, *}_{123}$ by base change (Derived Categories of Schemes, Lemma \ref{perfect-lemma-compare-base-change}). The third equality holds because pullbacks compose correctly and pass through tensor products, see Cohomology, Lemmas \ref{cohomology-lemma-derived-pullback-composition} and \ref{cohomology-lemma-pullback-tensor-product}. The fourth equality follows from the ``projection formula'' for $p^{1234}_{134}$, see Derived Categories of Schemes, Lemma \ref{perfect-lemma-cohomology-base-change}. The fifth equality is that proper pushforward is compatible with composition, see Cohomology, Lemma \ref{cohomology-lemma-derived-pushforward-composition}. Since tensor product is associative this concludes the proof of associativity of composition. \begin{lemma} \label{lemma-base-change-is-functor} Let $S' \to S$ be a morphism of schemes. The rule which sends \begin{enumerate} \item a smooth proper scheme $X$ over $S$ to $X' = S' \times_S X$, and \item the isomorphism class of an object $K$ of $D_{perf}(\mathcal{O}_{X \times_S Y})$ to the isomorphism class of $L(X' \times_{S'} Y' \to X \times_S Y)^*K$ in $D_{perf}(\mathcal{O}_{X' \times_{S'} Y'})$ \end{enumerate} is a functor from the category defined for $S$ to the category defined for $S'$. \end{lemma} \begin{proof} To see this suppose we have $X, Y, Z$ and $K \in D_{perf}(\mathcal{O}_{X \times_S Y})$ and $M \in D_{perf}(\mathcal{O}_{Y \times_S Z})$. Denote $K' \in D_{perf}(\mathcal{O}_{X' \times_{S'} Y'})$ and $M' \in D_{perf}(\mathcal{O}_{Y' \times_{S'} Z'})$ their pullbacks as in the statement of the lemma. The diagram $$ \xymatrix{ X' \times_{S'} Y' \times_{S'} Z' \ar[r] \ar[d]_{\text{pr}'_{13}} & X \times_S Y \times_S Z \ar[d]^{\text{pr}_{13}} \\ X' \times_{S'} Z' \ar[r] & X \times_S Z } $$ is cartesian and $\text{pr}_{13}$ is proper and smooth. By Derived Categories of Schemes, Lemma \ref{perfect-lemma-flat-proper-perfect-direct-image-general} we see that the derived pullback by the lower horizontal arrow of the composition $$ R\text{pr}_{13, *}( L\text{pr}_{12}^*K \otimes_{\mathcal{O}_{X \times_S Y \times_S Z}}^\mathbf{L} L\text{pr}_{23}^*M) $$ indeed is (canonically) isomorphic to $$ R\text{pr}'_{13, *}( L(\text{pr}'_{12})^*K' \otimes_{\mathcal{O}_{X' \times_{S'} Y' \times_{S'} Z'}}^\mathbf{L} L(\text{pr}'_{23})^*M') $$ as desired. Some details omitted. \end{proof} \section{Relative equivalences} \label{section-relative-equivalences} \noindent In this section we prove some lemmas about the following concept. \begin{definition} \label{definition-relative-equivalence-kernel} Let $S$ be a scheme. Let $X \to S$ and $Y \to S$ be smooth proper morphisms. An object $K \in D_{perf}(\mathcal{O}_{X \times_S Y})$ is said to be {\it the Fourier-Mukai kernel of a relative equivalence from $X$ to $Y$ over $S$} if there exist an object $K' \in D_{perf}(\mathcal{O}_{X \times_S Y})$ such that $$ \Delta_{X/S, *}\mathcal{O}_X \cong R\text{pr}_{13, *}(L\text{pr}_{12}^*K \otimes_{\mathcal{O}_{X \times_S Y \times_S X}}^\mathbf{L} L\text{pr}_{23}^*K') $$ in $D(\mathcal{O}_{X \times_S X})$ and $$ \Delta_{Y/S, *}\mathcal{O}_Y \cong R\text{pr}_{13, *}(L\text{pr}_{12}^*K' \otimes_{\mathcal{O}_{Y \times_S X \times_S Y}}^\mathbf{L} L\text{pr}_{23}^*K) $$ in $D(\mathcal{O}_{Y \times_S Y})$. In other words, the isomorphism class of $K$ defines an invertible arrow in the category defined in Section \ref{section-category-Fourier-Mukai-kernels}. \end{definition} \noindent The language is intentionally cumbersome. \begin{lemma} \label{lemma-equivalences-rek} With notation as in Definition \ref{definition-relative-equivalence-kernel} let $K$ be the Fourier-Mukai kernel of a relative equivalence from $X$ to $Y$ over $S$. Then the corresponding Fourier-Mukai functors $\Phi_K : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$ (Lemma \ref{lemma-fourier-Mukai-QCoh}) and $\Phi_K : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)$ (Lemma \ref{lemma-fourier-mukai}) are equivalences. \end{lemma} \begin{proof} Immediate from Lemma \ref{lemma-compose-fourier-mukai} and Example \ref{example-diagonal-fourier-mukai}. \end{proof} \begin{lemma} \label{lemma-base-change-rek} With notation as in Definition \ref{definition-relative-equivalence-kernel} let $K$ be the Fourier-Mukai kernel of a relative equivalence from $X$ to $Y$ over $S$. Let $S_1 \to S$ be a morphism of schemes. Let $X_1 = S_1 \times_S X$ and $Y_1 = S_1 \times_S Y$. Then the pullback $K_1 = L(X_1 \times_{S_1} Y_1 \to X \times_S Y)^*K$ is the Fourier-Mukai kernel of a relative equivalence from $X_1$ to $Y_1$ over $S_1$. \end{lemma} \begin{proof} Let $K' \in D_{perf}(\mathcal{O}_{Y \times_S X})$ be the object assumed to exist in Definition \ref{definition-relative-equivalence-kernel}. Denote $K'_1$ the pullback of $K'$ by $Y_1 \times_{S_1} X_1 \to Y \times_S X$. Then it suffices to prove that we have $$ \Delta_{X_1/S_1, *}\mathcal{O}_X \cong R\text{pr}_{13, *}(L\text{pr}_{12}^*K_1 \otimes_{\mathcal{O}_{X_1 \times_{S_1} Y_1 \times_{S_1} X_1}}^\mathbf{L} L\text{pr}_{23}^*K_1') $$ in $D(\mathcal{O}_{X_1 \times_{S_1} X_1})$ and similarly for the other condition. Since $$ \xymatrix{ X_1 \times_{S_1} Y_1 \times_{S_1} X_1 \ar[r] \ar[d]_{\text{pr}_{13}} & X \times_S Y \times_S X \ar[d]^{\text{pr}_{13}} \\ X_1 \times_{S_1} X_1 \ar[r] & X \times_S X } $$ is cartesian it suffices by Derived Categories of Schemes, Lemma \ref{perfect-lemma-flat-proper-perfect-direct-image-general} to prove that $$ \Delta_{X_1/S_1, *}\mathcal{O}_{X_1} \cong L(X_1 \times_{S_1} X_1 \to X \times_S X)^*\Delta_{X/S, *}\mathcal{O}_X $$ This in turn will be true if $X$ and $X_1 \times_{S_1} X_1$ are tor independent over $X \times_S X$, see Derived Categories of Schemes, Lemma \ref{perfect-lemma-compare-base-change}. This tor independence can be seen directly but also follows from the more general More on Morphisms, Lemma \ref{more-morphisms-lemma-case-of-tor-independence} applied to the square with corners $X, X, X, S$ and its base change by $S_1 \to S$. \end{proof} \begin{lemma} \label{lemma-descend-rek} Let $S = \lim_{i \in I} S_i$ be a limit of a directed system of schemes with affine transition morphisms $g_{i'i} : S_{i'} \to S_i$. We assume that $S_i$ is quasi-compact and quasi-separated for all $i \in I$. Let $0 \in I$. Let $X_0 \to S_0$ and $Y_0 \to S_0$ be smooth proper morphisms. We set $X_i = S_i \times_{S_0} X_0$ for $i \geq 0$ and $X = S \times_{S_0} X_0$ and similarly for $Y_0$. If $K$ is the Fourier-Mukai kernel of a relative equivalence from $X$ to $Y$ over $S$ then for some $i \geq 0$ there exists a Fourier-Mukai kernel of a relative equivalence from $X_i$ to $Y_i$ over $S_i$. \end{lemma} \begin{proof} Let $K' \in D_{perf}(\mathcal{O}_{Y \times_S X})$ be the object assumed to exist in Definition \ref{definition-relative-equivalence-kernel}. Since $X \times_S Y = \lim X_i \times_{S_i} Y_i$ there exists an $i$ and objects $K_i$ and $K'_i$ in $D_{perf}(\mathcal{O}_{Y_i \times_{S_i} X_i})$ whose pullbacks to $Y \times_S X$ give $K$ and $K'$. See Derived Categories of Schemes, Lemma \ref{perfect-lemma-descend-perfect}. By Derived Categories of Schemes, Lemma \ref{perfect-lemma-flat-proper-perfect-direct-image-general} the object $$ R\text{pr}_{13, *}(L\text{pr}_{12}^*K_i \otimes_{\mathcal{O}_{X_i \times_{S_i} Y_i \times_{S_i} X_i}}^\mathbf{L} L\text{pr}_{23}^*K_i') $$ is perfect and its pullback to $X \times_S X$ is equal to $$ R\text{pr}_{13, *}(L\text{pr}_{12}^*K \otimes_{\mathcal{O}_{X \times_S Y \times_S X}}^\mathbf{L} L\text{pr}_{23}^*K') \cong \Delta_{X/S, *}\mathcal{O}_X $$ See proof of Lemma \ref{lemma-base-change-rek}. On the other hand, since $X_i \to S$ is smooth and separated the object $$ \Delta_{i, *}\mathcal{O}_{X_i} $$ of $D(\mathcal{O}_{X_i \times_{S_i} X_i})$ is also perfect (by More on Morphisms, Lemmas \ref{more-morphisms-lemma-smooth-diagonal-perfect} and \ref{more-morphisms-lemma-perfect-proper-perfect-direct-image}) and its pullback to $X \times_S X$ is equal to $$ \Delta_{X/S, *}\mathcal{O}_X $$ See proof of Lemma \ref{lemma-base-change-rek}. Thus by Derived Categories of Schemes, Lemma \ref{perfect-lemma-descend-perfect} after increasing $i$ we may assume that $$ \Delta_{i, *}\mathcal{O}_{X_i} \cong R\text{pr}_{13, *}(L\text{pr}_{12}^*K_i \otimes_{\mathcal{O}_{X_i \times_{S_i} Y_i \times_{S_i} X_i}}^\mathbf{L} L\text{pr}_{23}^*K_i') $$ as desired. The same works for the roles of $K$ and $K'$ reversed. \end{proof} \section{No deformations} \label{section-no-deformations} \noindent The title of this section refers to Lemma \ref{lemma-no-deformations} \begin{lemma} \label{lemma-deform-koszul} Let $(R, \mathfrak m, \kappa) \to (A, \mathfrak n, \lambda)$ be a flat local ring homorphism of local rings which is essentially of finite presentation. Let $\overline{f}_1, \ldots, \overline{f}_r \in \mathfrak n/\mathfrak m A \subset A/\mathfrak m A$ be a regular sequence. Let $K \in D(A)$. Assume \begin{enumerate} \item $K$ is perfect, \item $K \otimes_A^\mathbf{L} A/\mathfrak m A$ is isomorphic in $D(A/\mathfrak m A)$ to the Koszul complex on $\overline{f}_1, \ldots, \overline{f}_r$. \end{enumerate} Then $K$ is isomorphic in $D(A)$ to a Koszul complex on a regular sequence $f_1, \ldots, f_r \in A$ lifting the given elements $\overline{f}_1, \ldots, \overline{f}_r$. Moreover, $A/(f_1, \ldots, f_r)$ is flat over $R$. \end{lemma} \begin{proof} Let us use chain complexes in the proof of this lemma. The Koszul complex $K_\bullet(\overline{f}_1, \ldots, \overline{f}_r)$ is defined in More on Algebra, Definition \ref{more-algebra-definition-koszul-complex}. By More on Algebra, Lemma \ref{more-algebra-lemma-lift-complex-stably-frees} we can represent $K$ by a complex $$ K_\bullet : A \to A^{\oplus r} \to \ldots \to A^{\oplus r} \to A $$ whose tensor product with $A/\mathfrak mA$ is equal (!) to $K_\bullet(\overline{f}_1, \ldots, \overline{f}_r)$. Denote $f_1, \ldots, f_r \in A$ the components of the arrow $A^{\oplus r} \to A$. These $f_i$ are lifts of the $\overline{f}_i$. By Algebra, Lemma \ref{algebra-lemma-grothendieck-regular-sequence-general} $f_1, \ldots, f_r$ form a regular sequence in $A$ and $A/(f_1, \ldots, f_r)$ is flat over $R$. Let $J = (f_1, \ldots, f_r) \subset A$. Consider the diagram $$ \xymatrix{ K_\bullet \ar[rd] \ar@{..>}[rr]_{\varphi_\bullet} & & K_\bullet(f_1, \ldots, f_r) \ar[ld] \\ & A/J } $$ Since $f_1, \ldots, f_r$ is a regular sequence the south-west arrow is a quasi-isomorphism (see More on Algebra, Lemma \ref{more-algebra-lemma-regular-koszul-regular}). Hence we can find the dotted arrow making the diagram commute for example by Algebra, Lemma \ref{algebra-lemma-compare-resolutions}. Reducing modulo $\mathfrak m$ we obtain a commutative diagram $$ \xymatrix{ K_\bullet(\overline{f}_1, \ldots, \overline{f}_r) \ar[rd] \ar[rr]_{\overline{\varphi}_\bullet} & & K_\bullet(\overline{f}_1, \ldots, \overline{f}_r) \ar[ld] \\ & (A/\mathfrak m A)/(\overline{f}_1, \ldots, \overline{f}_r) } $$ by our choice of $K_\bullet$. Thus $\overline{\varphi}$ is an isomorphism in the derived category $D(A/\mathfrak m A)$. It follows that $\overline{\varphi} \otimes_{A/\mathfrak m A}^\mathbf{L} \lambda$ is an isomorphism. Since $\overline{f}_i \in \mathfrak n / \mathfrak m A$ we see that $$ \text{Tor}_i^{A/\mathfrak m A}( K_\bullet(\overline{f}_1, \ldots, \overline{f}_r), \lambda) = K_i(\overline{f}_1, \ldots, \overline{f}_r) \otimes_{A/\mathfrak m A} \lambda $$ Hence $\varphi_i \bmod \mathfrak n$ is invertible. Since $A$ is local this means that $\varphi_i$ is an isomorphism and the proof is complete. \end{proof} \begin{lemma} \label{lemma-limit-arguments} Let $R \to S$ be a finite type flat ring map of Noetherian rings. Let $\mathfrak q \subset S$ be a prime ideal lying over $\mathfrak p \subset R$. Let $K \in D(S)$ be perfect. Let $f_1, \ldots, f_r \in \mathfrak q S_\mathfrak q$ be a regular sequence such that $S_\mathfrak q/(f_1, \ldots, f_r)$ is flat over $R$ and such that $K \otimes_S^\mathbf{L} S_\mathfrak q$ is isomorphic to the Koszul complex on $f_1, \ldots, f_r$. Then there exists a $g \in S$, $g \not \in \mathfrak q$ such that \begin{enumerate} \item $f_1, \ldots, f_r$ are the images of $f'_1, \ldots, f'_r \in S_g$, \item $f'_1, \ldots, f'_r$ form a regular sequence in $S_g$, \item $S_g/(f'_1, \ldots, f'_r)$ is flat over $R$, \item $K \otimes_S^\mathbf{L} S_g$ is isomorphic to the Koszul complex on $f_1, \ldots, f_r$. \end{enumerate} \end{lemma} \begin{proof} We can find $g \in S$, $g \not \in \mathfrak q$ with property (1) by the definition of localizations. After replacing $g$ by $gg'$ for some $g' \in S$, $g' \not \in \mathfrak q$ we may assume (2) holds, see Algebra, Lemma \ref{algebra-lemma-regular-sequence-in-neighbourhood}. By Algebra, Theorem \ref{algebra-theorem-openness-flatness} we find that $S_g/(f'_1, \ldots, f'_r)$ is flat over $R$ in an open neighbourhood of $\mathfrak q$. Hence after once more replacing $g$ by $gg'$ for some $g' \in S$, $g' \not \in \mathfrak q$ we may assume (3) holds as well. Finally, we get (4) for a further replacement by More on Algebra, Lemma \ref{more-algebra-lemma-colimit-perfect-complexes}. \end{proof} \noindent For a generalization of the following lemma, please see More on Morphisms of Spaces, Lemma \ref{spaces-more-morphisms-lemma-where-isomorphism}. \begin{lemma} \label{lemma-isomorphism-in-neighbourhood} Let $S$ be a Noetherian scheme. Let $s \in S$. Let $p : X \to Y$ be a morphism of schemes over $S$. Assume \begin{enumerate} \item $Y \to S$ and $X \to S$ proper, \item $X$ is flat over $S$, \item $X_s \to Y_s$ an isomorphism. \end{enumerate} Then there exists an open neighbourhood $U \subset S$ of $s$ such that the base change $X_U \to Y_U$ is an isomorphism. \end{lemma} \begin{proof} The morphism $p$ is proper by Morphisms, Lemma \ref{morphisms-lemma-closed-immersion-proper}. By Cohomology of Schemes, Lemma \ref{coherent-lemma-proper-finite-fibre-finite-in-neighbourhood} there is an open $Y_s \subset V \subset Y$ such that $p|_{p^{-1}(V)} : p^{-1}(V) \to V$ is finite. By More on Morphisms, Theorem \ref{more-morphisms-theorem-criterion-flatness-fibre-Noetherian} there is an open $X_s \subset U \subset X$ such that $p|_U : U \to Y$ is flat. After removing the images of $X \setminus U$ and $Y \setminus V$ (which are closed subsets not containing $s$) we may assume $p$ is flat and finite. Then $p$ is open (Morphisms, Lemma \ref{morphisms-lemma-fppf-open}) and $Y_s \subset p(X) \subset Y$ hence after shrinking $S$ we may assume $p$ is surjective. As $p_s : X_s \to Y_s$ is an isomorphism, the map $$ p^\sharp : \mathcal{O}_Y \longrightarrow p_*\mathcal{O}_X $$ of coherent $\mathcal{O}_Y$-modules ($p$ is finite) becomes an isomorphism after pullback by $i : Y_s \to Y$ (by Cohomology of Schemes, Lemma \ref{coherent-lemma-affine-base-change} for example). By Nakayama's lemma, this implies that $\mathcal{O}_{Y, y} \to (p_*\mathcal{O}_X)_y$ is surjective for all $y \in Y_s$. Hence there is an open $Y_s \subset V \subset Y$ such that $p^\sharp|_V$ is surjective (Modules, Lemma \ref{modules-lemma-finite-type-surjective-on-stalk}). Hence after shrinking $S$ once more we may assume $p^\sharp$ is surjective which means that $p$ is a closed immersion (as $p$ is already finite). Thus now $p$ is a surjective flat closed immersion of Noetherian schemes and hence an isomorphism, see Morphisms, Section \ref{morphisms-section-flat-closed-immersions}. \end{proof} \begin{lemma} \label{lemma-no-deformations} Let $k$ be a field. Let $S$ be a finite type scheme over $k$ with $k$-rational point $s$. Let $Y \to S$ be a smooth proper morphism. Let $X = Y_s \times S \to S$ be the constant family with fibre $Y_s$. Let $K$ be the Fourier-Mukai kernel of a relative equivalence from $X$ to $Y$ over $S$. Assume the restriction $$ L(Y_s \times_S Y_s \to X \times_S Y)^*K \cong \Delta_{Y_s/k, *} \mathcal{O}_{Y_s} $$ in $D(\mathcal{O}_{Y_s \times Y_s})$. Then there is an open neighbourhood $s \in U \subset S$ such that $Y|_U$ is isomorphic to $Y_s \times U$ over $U$. \end{lemma} \begin{proof} Denote $i : Y_s \times Y_s = X_s \times Y_s \to X \times_S Y$ the natural closed immersion. (We will write $Y_s$ and not $X_s$ for the fibre of $X$ over $s$ from now on.) Let $z \in Y_s \times Y_s = (X \times_S Y)_s \subset X \times_S Y$ be a closed point. As indicated we think of $z$ both as a closed point of $Y_s \times Y_s$ as well as a closed point of $X \times_S Y$. \medskip\noindent Case I: $z \not \in \Delta_{Y_s/k}(Y_s)$. Denote $\mathcal{O}_z$ the coherent $\mathcal{O}_{Y_s \times Y_s}$-module supported at $z$ whose value is $\kappa(z)$. Then $i_*\mathcal{O}_z$ is the coherent $\mathcal{O}_{X \times_S Y}$-module supported at $z$ whose value is $\kappa(z)$. Our assumption means that $$ K \otimes_{\mathcal{O}_{X \times_S Y}}^\mathbf{L} i_*\mathcal{O}_z = Li^*K \otimes_{\mathcal{O}_{Y_s \times Y_s}}^\mathbf{L} \mathcal{O}_z = 0 $$ Hence by Lemma \ref{lemma-orthogonal-point-sheaf} we find an open neighbourhood $U(z) \subset X \times_S Y$ of $z$ such that $K|_{U(z)} = 0$. In this case we set $Z(z) = \emptyset$ as closed subscheme of $U(z)$. \medskip\noindent Case II: $z \in \Delta_{Y_s/k}(Y_s)$. Since $Y_s$ is smooth over $k$ we know that $\Delta_{Y_s/k} : Y_s \to Y_s \times Y_s$ is a regular immersion, see More on Morphisms, Lemma \ref{more-morphisms-lemma-smooth-diagonal-perfect}. Choose a regular sequence $\overline{f}_1, \ldots, \overline{f}_r \in \mathcal{O}_{Y_s \times Y_s, z}$ cutting out the ideal sheaf of $\Delta_{Y_s/k}(Y_s)$. Since a regular sequence is Koszul-regular (More on Algebra, Lemma \ref{more-algebra-lemma-regular-koszul-regular}) our assumption means that $$ K_z \otimes_{\mathcal{O}_{X \times_S Y, z}}^\mathbf{L} \mathcal{O}_{Y_s \times Y_s, z} \in D(\mathcal{O}_{Y_s \times Y_s, z}) $$ is represented by the Koszul complex on $\overline{f}_1, \ldots, \overline{f}_r$ over $\mathcal{O}_{Y_s \times Y_s, z}$. By Lemma \ref{lemma-deform-koszul} applied to $\mathcal{O}_{S, s} \to \mathcal{O}_{X \times_S Y, z}$ we conclude that $K_z \in D(\mathcal{O}_{X \times_S Y, z})$ is represented by the Koszul complex on a regular sequence $f_1, \ldots, f_r \in \mathcal{O}_{X \times_S Y, z}$ lifting the regular sequence $\overline{f}_1, \ldots, \overline{f}_r$ such that moreover $\mathcal{O}_{X \times_S Y}/(f_1, \ldots, f_r)$ is flat over $\mathcal{O}_{S, s}$. By some limit arguments (Lemma \ref{lemma-limit-arguments}) we conclude that there exists an affine open neighbourhood $U(z) \subset X \times_S Y$ of $z$ and a closed subscheme $Z(z) \subset U(z)$ such that \begin{enumerate} \item $Z(z) \to U(z)$ is a regular closed immersion, \item $K|_{U(z)}$ is quasi-isomorphic to $\mathcal{O}_{Z(z)}$, \item $Z(z) \to S$ is flat, \item $Z(z)_s = \Delta_{Y_s/k}(Y_s) \cap U(z)_s$ as closed subschemes of $U(z)_s$. \end{enumerate} \noindent By property (2), for $z, z' \in Y_s \times Y_s$, we find that $Z(z) \cap U(z') = Z(z') \cap U(z)$ as closed subschemes. Hence we obtain an open neighbourhood $$ U = \bigcup\nolimits_{z \in Y_s \times Y_s\text{ closed}} U(z) $$ of $Y_s \times Y_s$ in $X \times_S Y$ and a closed subscheme $Z \subset U$ such that (1) $Z \to U$ is a regular closed immersion, (2) $Z \to S$ is flat, and (3) $Z_s = \Delta_{Y_s/k}(Y_s)$. Since $X \times_S Y \to S$ is proper, after replacing $S$ by an open neighbourhood of $s$ we may assume $U = X \times_S Y$. Since the projections $Z_s \to Y_s$ and $Z_s \to X_s$ are isomorphisms, we conclude that after shrinking $S$ we may assume $Z \to Y$ and $Z \to X$ are isomorphisms, see Lemma \ref{lemma-isomorphism-in-neighbourhood}. This finishes the proof. \end{proof} \begin{lemma} \label{lemma-no-deformations-better} Let $k$ be an algebraically closed field. Let $X$ be a smooth proper scheme over $k$. Let $f : Y \to S$ be a smooth proper morphism with $S$ of finite type over $k$. Let $K$ be the Fourier-Mukai kernel of a relative equivalence from $X \times S$ to $Y$ over $S$. Then $S$ can be covered by open subschemes $U$ such that there is a $U$-isomorphism $f^{-1}(U) \cong Y_0 \times U$ for some $Y_0$ proper and smooth over $k$. \end{lemma} \begin{proof} Choose a closed point $s \in S$. Since $k$ is algebraically closed this is a $k$-rational point. Set $Y_0 = Y_s$. The restriction $K_0$ of $K$ to $X \times Y_0$ is the Fourier-Mukai kernel of a relative equivalence from $X$ to $Y_0$ over $\Spec(k)$ by Lemma \ref{lemma-base-change-rek}. Let $K'_0$ in $D_{perf}(\mathcal{O}_{Y_0 \times X})$ be the object assumed to exist in Definition \ref{definition-relative-equivalence-kernel}. Then $K'_0$ is the Fourier-Mukai kernel of a relative equivalence from $Y_0$ to $X$ over $\Spec(k)$ by the symmetry inherent in Definition \ref{definition-relative-equivalence-kernel}. Hence by Lemma \ref{lemma-base-change-rek} we see that the pullback $$ M = (Y_0 \times X \times S \to Y_0 \times X)^*K'_0 $$ on $(Y_0 \times S) \times_S (X \times S) = Y_0 \times X \times S$ is the Fourier-Mukai kernel of a relative equivalence from $Y_0 \times S$ to $X \times S$ over $S$. Now consider the kernel $$ K_{new} = R\text{pr}_{13, *}(L\text{pr}_{12}^*M \otimes_{\mathcal{O}_{(Y_0 \times S) \times_S (X \times S) \times_S Y}}^\mathbf{L} L\text{pr}_{23}^*K) $$ on $(Y_0 \times S) \times_S Y$. This is the Fourier-Mukai kernel of a relative equivalence from $Y_0 \times S$ to $Y$ over $S$ since it is the composition of two invertible arrows in the category constructed in Section \ref{section-category-Fourier-Mukai-kernels}. Moreover, this composition passes through base change (Lemma \ref{lemma-base-change-is-functor}). Hence we see that the pullback of $K_{new}$ to $((Y_0 \times S) \times_S Y)_s = Y_0 \times Y_0$ is equal to the composition of $K_0$ and $K'_0$ and hence equal to the identity in this category. In other words, we have $$ L(Y_0 \times Y_0 \to (Y_0 \times S) \times_S Y)^*K_{new} \cong \Delta_{Y_0/k, *}\mathcal{O}_{Y_0} $$ Thus by Lemma \ref{lemma-no-deformations} we conclude that $Y \to S$ is isomorphic to $Y_0 \times S$ in an open neighbourhood of $s$. This finishes the proof. \end{proof} \section{Countability} \label{section-countability} \noindent In this section we prove some elementary lemmas about countability of certain sets. Let $\mathcal{C}$ be a category. In this section we will say that $\mathcal{C}$ is {\it countable} if \begin{enumerate} \item for any $X, Y \in \Ob(\mathcal{C})$ the set $\Mor_\mathcal{C}(X, Y)$ is countable, and \item the set of isomorphism classes of objects of $\mathcal{C}$ is countable. \end{enumerate} \begin{lemma} \label{lemma-countable-finite-type} Let $R$ be a countable Noetherian ring. Then the category of schemes of finite type over $R$ is countable. \end{lemma} \begin{proof} Omitted. \end{proof} \begin{lemma} \label{lemma-countable-abelian} Let $\mathcal{A}$ be a countable abelian category. Then $D^b(\mathcal{A})$ is countable. \end{lemma} \begin{proof} It suffices to prove the statement for $D(\mathcal{A})$ as the others are full subcategories of this one. Since every object in $D(\mathcal{A})$ is a complex of objects of $\mathcal{A}$ it is immediate that the set of isomorphism classes of objects of $D^b(\mathcal{A})$ is countable. Moreover, for bounded complexes $A^\bullet$ and $B^\bullet$ of $\mathcal{A}$ it is clear that $\Hom_{K^b(\mathcal{A})}(A^\bullet, B^\bullet)$ is countable. We have $$ \Hom_{D^b(\mathcal{A})}(A^\bullet, B^\bullet) = \colim_{s : (A')^\bullet \to A^\bullet \text{ qis and }(A')^\bullet\text{ bounded}} \Hom_{K^b(\mathcal{A})}((A')^\bullet, B^\bullet) $$ by Derived Categories, Lemma \ref{derived-lemma-bounded-derived}. Thus this is a countable set as a countable colimit of \end{proof} \begin{lemma} \label{lemma-countable-perfect} Let $X$ be a scheme of finite type over a countable Noetherian ring. Then the categories $D_{perf}(\mathcal{O}_X)$ and $D^b_{\textit{Coh}}(\mathcal{O}_X)$ are countable. \end{lemma} \begin{proof} Observe that $X$ is Noetherian by Morphisms, Lemma \ref{morphisms-lemma-finite-type-noetherian}. Hence $D_{perf}(\mathcal{O}_X)$ is a full subcategory of $D^b_{\textit{Coh}}(\mathcal{O}_X)$ by Derived Categories of Schemes, Lemma \ref{perfect-lemma-perfect-on-noetherian}. Thus it suffices to prove the result for $D^b_{\textit{Coh}}(\mathcal{O}_X)$. Recall that $D^b_{\textit{Coh}}(\mathcal{O}_X) = D^b(\textit{Coh}(\mathcal{O}_X))$ by Derived Categories of Schemes, Proposition \ref{perfect-proposition-DCoh}. Hence by Lemma \ref{lemma-countable-abelian} it suffices to prove that $\textit{Coh}(\mathcal{O}_X)$ is countable. This we omit. \end{proof} \begin{lemma} \label{lemma-countable-isos} Let $K$ be an algebraically closed field. Let $S$ be a finite type scheme over $K$. Let $X \to S$ and $Y \to S$ be finite type morphisms. There exists a countable set $I$ and for $i \in I$ a pair $(S_i \to S, h_i)$ with the following properties \begin{enumerate} \item $S_i \to S$ is a morphism of finite type, set $X_i = X \times_S S_i$ and $Y_i = Y \times_S S_i$, \item $h_i : X_i \to Y_i$ is an isomorphism over $S_i$, and \item for any closed point $s \in S(K)$ if $X_s \cong Y_s$ over $K = \kappa(s)$ then $s$ is in the image of $S_i \to S$ for some $i$. \end{enumerate} \end{lemma} \begin{proof} The field $K$ is the filtered union of its countable subfields. Dually, $\Spec(K)$ is the cofiltered limit of the spectra of the countable subfields of $K$. Hence Limits, Lemma \ref{limits-lemma-descend-finite-presentation} guarantees that we can find a countable subfield $k$ and morphisms $X_0 \to S_0$ and $Y_0 \to S_0$ of schemes of finite type over $k$ such that $X \to S$ and $Y \to S$ are the base changes of these. \medskip\noindent By Lemma \ref{lemma-countable-finite-type} there is a countable set $I$ and pairs $(S_{0, i} \to S_0, h_{0, i})$ such that \begin{enumerate} \item $S_{0, i} \to S_0$ is a morphism of finite type, set $X_{0, i} = X_0 \times_{S_0} S_{0, i}$ and $Y_{0, i} = Y_0 \times_{S_0} S_{0, i}$, \item $h_{0, i} : X_{0, i} \to Y_{0, i}$ is an isomorphism over $S_{0, i}$. \end{enumerate} such that every pair $(T \to S_0, h_T)$ with $T \to S_0$ of finite type and $h_T : X_0 \times_{S_0} T \to Y_0 \times_{S_0} T$ an isomorphism is isomorphic to one of these. Denote $(S_i \to S, h_i)$ the base change of $(S_{0, i} \to S_0, h_{0, i})$ by $\Spec(K) \to \Spec(k)$. We claim this works. \medskip\noindent Let $s \in S(K)$ and let $h_s : X_s \to Y_s$ be an isomorphism over $K = \kappa(s)$. We can write $K$ as the filtered union of its finitely generated $k$-subalgebras. Hence by Limits, Proposition \ref{limits-proposition-characterize-locally-finite-presentation} and Lemma \ref{limits-lemma-descend-finite-presentation} we can find such a finitely generated $k$-subalgebra $K \supset A \supset k$ such that \begin{enumerate} \item there is a commutative diagram $$ \xymatrix{ \Spec(K) \ar[d]_s \ar[r] & \Spec(A) \ar[d]^{s'} \\ S \ar[r] & S_0} $$ for some morphism $s' : \Spec(A) \to S_0$ over $k$, \item $h_s$ is the base change of an isomorphism $h_{s'} : X_0 \times_{S_0, s'} \Spec(A) \to X_0 \times_{S_0, s'} \Spec(A)$ over $A$. \end{enumerate} Of course, then $(s' : \Spec(A) \to S_0, h_{s'})$ is isomorphic to the pair $(S_{0, i} \to S_0, h_{0, i})$ for some $i \in I$. This concludes the proof because the commutative diagram in (1) shows that $s$ is in the image of the base change of $s'$ to $\Spec(K)$. \end{proof} \begin{lemma} \label{lemma-countable-equivs} Let $K$ be an algebraically closed field. There exists a countable set $I$ and for $i \in I$ a pair $(S_i/K, X_i \to S_i, Y_i \to S_i, M_i)$ with the following properties \begin{enumerate} \item $S_i$ is a scheme of finite type over $K$, \item $X_i \to S_i$ and $Y_i \to S_i$ are proper smooth morphisms of schemes, \item $M_i \in D_{perf}(\mathcal{O}_{X_i \times_{S_i} Y_i})$ is the Fourier-Mukai kernel of a relative equivalence from $X_i$ to $Y_i$ over $S_i$, and \item for any smooth proper schemes $X$ and $Y$ over $K$ such that there is a $K$-linear exact equivalence $D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)$ there exists an $i \in I$ and a $s \in S_i(K)$ such that $X \cong (X_i)_s$ and $Y \cong (Y_i)_s$. \end{enumerate} \end{lemma} \begin{proof} Choose a countable subfield $k \subset K$ for example the prime field. By Lemmas \ref{lemma-countable-finite-type} and \ref{lemma-countable-perfect} there exists a countable set of isomorphism classes of systems over $k$ satisfying parts (1), (2), (3) of the lemma. Thus we can choose a countable set $I$ and for each $i \in I$ such a system $$ (S_{0, i}/k, X_{0, i} \to S_{0, i}, Y_{0, i} \to S_{0, i}, M_{0, i}) $$ over $k$ such that each isomorphism class occurs at least once. Denote $(S_i/K, X_i \to S_i, Y_i \to S_i, M_i)$ the base change of the displayed system to $K$. This system has properties (1), (2), (3), see Lemma \ref{lemma-base-change-rek}. Let us prove property (4). \medskip\noindent Consider smooth proper schemes $X$ and $Y$ over $K$ such that there is a $K$-linear exact equivalence $F : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)$. By Proposition \ref{proposition-equivalence} we may assume that there exists an object $M \in D_{perf}(\mathcal{O}_{X \times Y})$ such that $F = \Phi_M$ is the corresponding Fourier-Mukai functor. By Lemma \ref{lemma-fourier-mukai-flat-proper-over-noetherian} there is an $M'$ in $D_{perf}(\mathcal{O}_{Y \times X})$ such that $\Phi_{M'}$ is the right adjoint to $\Phi_M$. Since $\Phi_M$ is an equivalence, this means that $\Phi_{M'}$ is the quasi-inverse to $\Phi_M$. By Lemma \ref{lemma-fourier-mukai-flat-proper-over-noetherian} we see that the Fourier-Mukai functors defined by the objects $$ A = R\text{pr}_{13, *}( L\text{pr}_{12}^*M \otimes_{\mathcal{O}_{X \times Y \times X}}^\mathbf{L} L\text{pr}_{23}^*M') $$ in $D_{perf}(\mathcal{O}_{X \times X})$ and $$ B = R\text{pr}_{13, *}( L\text{pr}_{12}^*M' \otimes_{\mathcal{O}_{Y \times X \times Y}}^\mathbf{L} L\text{pr}_{23}^*M) $$ in $D_{perf}(\mathcal{O}_{Y \times Y})$ are isomorphic to $\text{id} : D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_X)$ and $\text{id} : D_{perf}(\mathcal{O}_Y) \to D_{perf}(\mathcal{O}_Y)$ Hence $A \cong \Delta_{X/K, *}\mathcal{O}_X$ and $B \cong \Delta_{Y/K, *}\mathcal{O}_Y$ by Lemma \ref{lemma-uniqueness}. Hence we see that $M$ is the Fourier-Mukai kernel of a relative equivalence from $X$ to $Y$ over $K$ by definition. \medskip\noindent We can write $K$ as the filtered colimit of its finite type $k$-subalgebras $A \subset K$. By Limits, Lemma \ref{limits-lemma-descend-finite-presentation} we can find $X_0, Y_0$ of finite type over $A$ whose base changes to $K$ produces $X$ and $Y$. By Limits, Lemmas \ref{limits-lemma-eventually-proper} and \ref{limits-lemma-descend-smooth} after enlarging $A$ we may assume $X_0$ and $Y_0$ are smooth and proper over $A$. By Lemma \ref{lemma-descend-rek} after enlarging $A$ we may assume $M$ is the pullback of some $M_0 \in D_{perf}(\mathcal{O}_{X_0 \times_{\Spec(A)} Y_0})$ which is the Fourier-Mukai kernel of a relative equivalence from $X_0$ to $Y_0$ over $\Spec(A)$. Thus we see that $(S_0/k, X_0 \to S_0, Y_0 \to S_0, M_0)$ is isomorphic to $(S_{0, i}/k, X_{0, i} \to S_{0, i}, Y_{0, i} \to S_{0, i}, M_{0, i})$ for some $i \in I$. Since $S_i = S_{0, i} \times_{\Spec(k)} \Spec(K)$ we conclude that (4) is true with $s : \Spec(K) \to S_i$ induced by the morphism $\Spec(K) \to \Spec(A) \cong S_{0, i}$ we get from $A \subset K$. \end{proof} \section{Countability of derived equivalent varieties} \label{section-countable-derived-equivalent} \noindent In this section we prove a result of Anel and To\"en, see \cite{AT}. \begin{definition} \label{definition-derived-equivalent} Let $k$ be a field. Let $X$ and $Y$ be smooth projective schemes over $k$. We say $X$ and $Y$ are {\it derived equivalent} if there exists a $k$-linear exact equivalence $D_{perf}(\mathcal{O}_X) \to D_{perf}(\mathcal{O}_Y)$. \end{definition} \noindent Here is the result \begin{theorem} \label{theorem-countable} \begin{reference} Slight improvement of \cite{AT} \end{reference} Let $K$ be an algebraically closed field. Let $\mathbf{X}$ be a smooth proper scheme over $K$. There are at most countably many isomorphism classes of smooth proper schemes $\mathbf{Y}$ over $K$ which are derived equivalent to $\mathbf{X}$. \end{theorem} \begin{proof} Choose a countable set $I$ and for $i \in I$ systems $(S_i/K, X_i \to S_i, Y_i \to S_i, M_i)$ satisfying properties (1), (2), (3), and (4) of Lemma \ref{lemma-countable-equivs}. Pick $i \in I$ and set $S = S_i$, $X = X_i$, $Y = Y_i$, and $M = M_i$. Clearly it suffice to show that the set of isomorphism classes of fibres $Y_s$ for $s \in S(K)$ such that $X_s \cong \mathbf{X}$ is countable. This we prove in the next paragraph. \medskip\noindent Let $S$ be a finite type scheme over $K$, let $X \to S$ and $Y \to S$ be proper smooth morphisms, and let $M \in D_{perf}(\mathcal{O}_{X \times_S Y})$ be the Fourier-Mukai kernel of a relative equivalence from $X$ to $Y$ over $S$. We will show the set of isomorphism classes of fibres $Y_s$ for $s \in S(K)$ such that $X_s \cong \mathbf{X}$ is countable. By Lemma \ref{lemma-countable-isos} applied to the families $\mathbf{X} \times S \to S$ and $X \to S$ there exists a countable set $I$ and for $i \in I$ a pair $(S_i \to S, h_i)$ with the following properties \begin{enumerate} \item $S_i \to S$ is a morphism of finite type, set $X_i = X \times_S S_i$, \item $h_i : \mathbf{X} \times S_i \to X_i$ is an isomorphism over $S_i$, and \item for any closed point $s \in S(K)$ if $\mathbf{X} \cong X_s$ over $K = \kappa(s)$ then $s$ is in the image of $S_i \to S$ for some $i$. \end{enumerate} Set $Y_i = Y \times_S S_i$. Denote $M_i \in D_{perf}(\mathcal{O}_{X_i \times_{S_i} Y_i})$ the pullback of $M$. By Lemma \ref{lemma-base-change-rek} $M_i$ is the Fourier-Mukai kernel of a relative equivalence from $X_i$ to $Y_i$ over $S_i$. Since $I$ is countable, by property (3) it suffices to prove that the set of isomorphism classes of fibres $Y_{i, s}$ for $s \in S_i(K)$ is countable. In fact, this number is finite by Lemma \ref{lemma-no-deformations-better} and the proof is complete. \end{proof} \input{chapters} \bibliography{my} \bibliographystyle{amsalpha} \end{document}