The Stacks project

Compare with discussion in [Rizzardo].

Lemma 57.8.7. Let $X \to S$ and $Y \to S$ be morphisms of quasi-compact and quasi-separated schemes. Let $\Phi : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$ be a Fourier-Mukai functor with pseudo-coherent kernel $K \in D_\mathit{QCoh}(\mathcal{O}_{X \times _ S Y})$. Let $a : D_\mathit{QCoh}(\mathcal{O}_ Y) \to D_\mathit{QCoh}(\mathcal{O}_{X \times _ S Y})$ be the right adjoint to $R\text{pr}_{2, *}$, see Duality for Schemes, Lemma 48.3.1. Denote

\[ K' = (Y \times _ S X \to X \times _ S Y)^* R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_{X \times _ S Y}}(K, a(\mathcal{O}_ Y)) \in D_\mathit{QCoh}(\mathcal{O}_{Y \times _ S X}) \]

and denote $\Phi ' : D_\mathit{QCoh}(\mathcal{O}_ Y) \to D_\mathit{QCoh}(\mathcal{O}_ X)$ the corresponding Fourier-Mukai transform. There is a canonical map

\[ \mathop{\mathrm{Hom}}\nolimits _ X(M, \Phi '(N)) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _ Y(\Phi (M), N) \]

functorial in $M$ in $D_\mathit{QCoh}(\mathcal{O}_ X)$ and $N$ in $D_\mathit{QCoh}(\mathcal{O}_ Y)$ which is an isomorphism if

  1. $N$ is perfect, or

  2. $K$ is perfect and $X \to S$ is proper flat and of finite presentation.

Proof. By Lemma 57.8.2 we obtain a functor $\Phi $ as in the statement. Observe that $a(\mathcal{O}_ Y)$ is in $D^+_\mathit{QCoh}(\mathcal{O}_{X \times _ S Y})$ by Duality for Schemes, Lemma 48.3.5. Hence for $K$ pseudo-coherent we have $K' \in D_\mathit{QCoh}(\mathcal{O}_{Y \times _ S X})$ by Derived Categories of Schemes, Lemma 36.10.8 we we obtain $\Phi '$ as indicated.

We abbreviate $\otimes ^\mathbf {L} = \otimes _{\mathcal{O}_{X \times _ S Y}}^\mathbf {L}$ and $\mathop{\mathcal{H}\! \mathit{om}}\nolimits = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_{X \times _ S Y}}$. Let $M$ be in $D_\mathit{QCoh}(\mathcal{O}_ X)$ and let $N$ be in $D_\mathit{QCoh}(\mathcal{O}_ Y)$. We have

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _ Y(\Phi (M), N) & = \mathop{\mathrm{Hom}}\nolimits _ Y(R\text{pr}_{2, *}(L\text{pr}_1^*M \otimes ^\mathbf {L} K), N) \\ & = \mathop{\mathrm{Hom}}\nolimits _{X \times _ S Y}(L\text{pr}_1^*M \otimes ^\mathbf {L} K, a(N)) \\ & = \mathop{\mathrm{Hom}}\nolimits _{X \times _ S Y}(L\text{pr}_1^*M, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, a(N))) \\ & = \mathop{\mathrm{Hom}}\nolimits _ X(M, R\text{pr}_{1, *}R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, a(N))) \end{align*}

where we have used Cohomology, Lemmas 20.42.2 and 20.28.1. There are canonical maps

\[ L\text{pr}_2^*N \otimes ^\mathbf {L} R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, a(\mathcal{O}_ Y)) \xrightarrow {\alpha } R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L\text{pr}_2^*N \otimes ^\mathbf {L} a(\mathcal{O}_ Y)) \xrightarrow {\beta } R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, a(N)) \]

Here $\alpha $ is Cohomology, Lemma 20.42.6 and $\beta $ is Duality for Schemes, Equation ( Combining all of these arrows we obtain the functorial displayed arrow in the statement of the lemma.

The arrow $\alpha $ is an isomorphism by Derived Categories of Schemes, Lemma 36.10.9 as soon as either $K$ or $N$ is perfect. The arrow $\beta $ is an isomorphism if $N$ is perfect by Duality for Schemes, Lemma 48.8.1 or in general if $X \to S$ is flat proper of finite presentation by Duality for Schemes, Lemma 48.12.3. $\square$

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