Lemma 56.15.7. Let $k$ be a field. Let $X$, $Y$ be smooth proper schemes over $k$. Let $F, G : D_{perf}(\mathcal{O}_ X) \to D_{perf}(\mathcal{O}_ Y)$ be $k$-linear exact functors such that

$F(\mathcal{F}) \cong G(\mathcal{F})$ for any coherent $\mathcal{O}_ X$-module $\mathcal{F}$ with $\dim (\text{Supp}(\mathcal{F})) = 0$,

$F$ is fully faithful, and

$G$ is a Fourier-Mukai functor whose kernel is in $D_{perf}(\mathcal{O}_{X \times Y})$.

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.

**Proof.**
Recall that $F$ has both adjoints, see Lemma 56.8.1. In particular the essential image $\mathcal{A} \subset D_{perf}(\mathcal{O}_ Y)$ of $F$ satisfies the equivalent conditions of Derived Categories, Lemma 13.39.5. We claim that $G$ factors through $\mathcal{A}$. Since $\mathcal{A} = {}^\perp (\mathcal{A}^\perp )$ by Derived Categories, Lemma 13.39.5 it suffices to show that $\mathop{\mathrm{Hom}}\nolimits _ Y(G(M), N) = 0$ for all $M$ in $D_{perf}(\mathcal{O}_ X)$ and $N \in \mathcal{A}^\perp $. We have

\[ \mathop{\mathrm{Hom}}\nolimits _ Y(G(M), N) = \mathop{\mathrm{Hom}}\nolimits _ X(M, G_ r(N)) \]

where $G_ r$ is the right adjoint to $G$. Since $G(\mathcal{F}) \cong F(\mathcal{F})$ for $\mathcal{F}$ as in (1) we see that $\mathop{\mathrm{Hom}}\nolimits _ X(\mathcal{F}, G_ r(N)) = 0$ by the same formula and the fact that $N$ is in the right orthogonal to the essential image $\mathcal{A}$ of $F$. Of course, the same vanishing holds for $\mathop{\mathrm{Hom}}\nolimits _ X(\mathcal{F}, G_ r(N)[i])$ for any $i \in \mathbf{Z}$. Thus $G_ r(N) = 0$ by Lemma 56.14.3 and the claim holds.

Apply Lemma 56.15.6 to the functor $H = F^{-1} \circ G$ which makes sense because the essential image of $G$ is contained in the essential image of $F$ by the previous paragraph and because $F$ is fully faithful. 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 56.13.3 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 56.9.3 we conclude.
$\square$

**Second proof.**
Recall that $F$ has both adjoints, see Lemma 56.8.1. In particular the essential image $\mathcal{A} \subset D_{perf}(\mathcal{O}_ Y)$ of $F$ satisfies the equivalent conditions of Derived Categories, Lemma 13.39.5. We claim that $G$ factors through $\mathcal{A}$. Since $\mathcal{A} = {}^\perp (\mathcal{A}^\perp )$ by Derived Categories, Lemma 13.39.5 it suffices to show that $\mathop{\mathrm{Hom}}\nolimits _ Y(G(M), N) = 0$ for all $M$ in $D_{perf}(\mathcal{O}_ X)$ and $N \in \mathcal{A}^\perp $. We have

\[ \mathop{\mathrm{Hom}}\nolimits _ Y(G(M), N) = \mathop{\mathrm{Hom}}\nolimits _ X(M, G_ r(N)) \]

where $G_ r$ is the right adjoint to $G$. Since $G(\mathcal{F}) \cong F(\mathcal{F})$ for $\mathcal{F}$ as in (1) we see that $\mathop{\mathrm{Hom}}\nolimits _ X(\mathcal{F}, G_ r(N)) = 0$ by the same formula and the fact that $N$ is in the right orthogonal to the essential image $\mathcal{A}$ of $F$. Of course, the same vanishing holds for $\mathop{\mathrm{Hom}}\nolimits _ X(\mathcal{F}, G_ r(N)[i])$ for any $i \in \mathbf{Z}$. Thus $G_ r(N) = 0$ by Lemma 56.14.3 and the claim holds.

Apply Lemma 56.15.6 to the functor $H = F^{-1} \circ G$ which makes sense because the essential image of $G$ is contained in the essential image of $F$ by the previous paragraph and because $F$ is fully faithful. 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 56.13.3 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 56.9.3 we conclude.
$\square$

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