The Stacks project

Lemma 57.13.6. 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

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

  2. $F$ is fully faithful, and

  3. $G$ 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. The essential image of $G$ is contained in the essential image of $F$ by Lemma 57.13.4. Consider the functor $H = F^{-1} \circ G$ which makes sense as $F$ is fully faithful. By Lemma 57.13.5 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 57.11.3 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 57.9.3 we conclude. $\square$

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