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56.9 Fourier-Mukai functors

These functors were first introduced in [Mukai].

Definition 56.9.1. 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 Fourier-Mukai functor and $K$ is called a Fourier-Mukai kernel for this functor. Moreover,

  1. if $\Phi _ K$ sends $D_\mathit{QCoh}(\mathcal{O}_ X)$ into $D_\mathit{QCoh}(\mathcal{O}_ Y)$ then the resulting exact functor $\Phi _ K : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$ is called a Fourier-Mukai functor,

  2. 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

  3. 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 $D_{\textit{Coh}}$, $D^+_{\textit{Coh}}$, $D^-_{\textit{Coh}}$.

Lemma 56.9.2. 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 $D_\mathit{QCoh}(\mathcal{O}_ X)$ into $D_\mathit{QCoh}(\mathcal{O}_ Y)$ if $K$ is in $D_\mathit{QCoh}(\mathcal{O}_{X \times _ S Y})$ and $X \to S$ is quasi-compact and quasi-separated.

Proof. This follows from the fact that derived pullback preserves $D_\mathit{QCoh}$ (Derived Categories of Schemes, Lemma 36.3.8), derived tensor products preserve $D_\mathit{QCoh}$ (Derived Categories of Schemes, Lemma 36.3.9), the projection $\text{pr}_2 : X \times _ S Y \to Y$ is quasi-compact and quasi-separated (Schemes, Lemmas 26.19.3 and 26.21.12), and total direct image along a quasi-separated and quasi-compact morphism preserves $D_\mathit{QCoh}$ (Derived Categories of Schemes, Lemma 36.4.1). $\square$

Lemma 56.9.3. 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 $K \in D_\mathit{QCoh}(\mathcal{O}_{X \times _ S Y})$. Let $K' \in D_\mathit{QCoh}(\mathcal{O}_{Y \times _ S Z})$. Consider the Fourier-Mukai functors $\Phi _ K : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$ and $\Phi _{K'} : D_\mathit{QCoh}(\mathcal{O}_ Y) \to D_\mathit{QCoh}(\mathcal{O}_ Z)$. If $X$ and $Z$ are tor independent over $S$ and $Y \to S$ is flat, then

\[ \Phi _{K'} \circ \Phi _ K = \Phi _{K''} : D_\mathit{QCoh}(\mathcal{O}_ X) \longrightarrow D_\mathit{QCoh}(\mathcal{O}_ Z) \]

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 $D_\mathit{QCoh}(\mathcal{O}_{X \times _ S Z})$.

Proof. The statement makes sense by Lemma 56.9.2. We are going to use Derived Categories of Schemes, Lemmas 36.3.8, 36.3.9, and 36.4.1 and Schemes, Lemmas 26.19.3 and 26.21.12 without further mention. By Derived Categories of Schemes, Lemma 36.22.4 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 36.22.5. Abbreviating $p^* = Lp^*$, $p_* = Rp_*$ and $\otimes = \otimes ^\mathbf {L}$ we have for $M \in D_\mathit{QCoh}(\mathcal{O}_ X)$ 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 36.22.1 in the $4$th and last equality. $\square$

Lemma 56.9.4. 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:

  1. $S$ 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 $D(\text{pr}_2^{-1}\mathcal{O}_ Y)$,

  2. $X \to S$ 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 $Y$,

  3. $X \to S$ is a proper flat morphism of finite presentation and $K$ is perfect,

  4. $S$ is Noetherian, $X \to S$ is flat and proper, and $K$ is perfect

  5. $X \to S$ is a proper flat morphism of finite presentation and $K$ is $Y$-perfect,

  6. $S$ is Noetherian, $X \to S$ is flat and proper, and $K$ is $Y$-perfect.

Proof. If $M$ is perfect on $X$, then $L\text{pr}_1^*M$ is perfect on $X \times _ S Y$, see Cohomology, Lemma 20.46.6. 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 29.15.4, 29.41.5, 29.25.8, and 29.21.4.

Part (1) follows from Derived Categories of Schemes, Lemma 36.27.1 combined with Derived Categories of Schemes, Lemma 36.11.6.

Part (2) follows from Derived Categories of Schemes, Lemma 36.30.1.

Part (3) follows from Derived Categories of Schemes, Lemma 36.30.4.

Part (4) follows from part (3) and the fact that a finite type morphism of Noetherian schemes is of finite presentation by Morphisms, Lemma 29.21.9.

Part (5) follows from Derived Categories of Schemes, Lemma 36.35.10 combined with Derived Categories of Schemes, Lemma 36.35.5.

Part (6) follows from part (5) in the same way that part (4) follows from part (3). $\square$

Lemma 56.9.5. 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:

  1. the support of $H^ i(K)$ is proper over $Y$ for all $i$, and $K$ has finite tor dimension as an object of $D(\text{pr}_1^{-1}\mathcal{O}_ X)$,

  2. $K$ 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 $Y$,

  3. the support of $H^ i(K)$ is proper over $Y$ for all $i$ and $X$ is a regular scheme,

  4. $K$ is perfect, the support of $H^ i(K)$ is proper over $Y$ for all $i$, and $Y \to S$ is flat.

Furthermore in each case the support condition is automatic if $X \to S$ is proper.

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 36.11.3 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 20.44.3, Derived Categories of Schemes, Lemma 36.10.3, and Cohomology, Lemma 20.44.5) whence has coherent cohomology sheaves (by Derived Categories of Schemes, Lemma 36.10.3 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.

Case (1). By Cohomology, Lemma 20.27.4 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.

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.

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 $U = \mathop{\mathrm{Spec}}(A)$ and $V = \mathop{\mathrm{Spec}}(B)$ of $X$ and $Y$ mapping into the affine open $W = \mathop{\mathrm{Spec}}(R)$ 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 10.110.8) 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 15.65.8). 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 20.45.5. We conclude as in the previous to paragraphs.

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. $\square$

Example 56.9.6. 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 20.51.4. The third because $\text{pr}_1 \circ \Delta = \text{id}_ X$ and we have Cohomology, Lemma 20.27.2. If we push this to $X$ using $R\text{pr}_{2, *}$ we obtain $M$ by Cohomology, Lemma 20.28.2 and the fact that $\text{pr}_2 \circ \Delta = \text{id}_ X$.

reference

Lemma 56.9.7. 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 56.9.2 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 20.39.2 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 20.39.6 and $\beta $ is Duality for Schemes, Equation (48.8.0.1). 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. $\square$

reference

Lemma 56.9.8. 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 $\Phi : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$ be a Fourier-Mukai functor with perfect kernel $K \in D_\mathit{QCoh}(\mathcal{O}_{X \times _ S Y})$. Denote

\[ K' = (Y \times _ S X \to X \times _ S Y)^*(K^\vee \otimes _{\mathcal{O}_{X \times _ S Y}}^\mathbf {L} L\text{pr}_2^*\omega ^\bullet _{Y/S}) \in D_\mathit{QCoh}(\mathcal{O}_{Y \times _ S X}) \]

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 isomorphism

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

functorial in $M$ in $D_\mathit{QCoh}(\mathcal{O}_ X)$ and $N$ in $D_\mathit{QCoh}(\mathcal{O}_ Y)$.

Proof. By Lemma 56.9.2 we obtain a functor $\Phi $ as in the statement.

Observe that formation of the relative dualizing complex commutes with base change in our setting, see Duality for Schemes, Remark 48.12.5. 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 48.25.10, and a fortiori perfect.

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 56.9.7 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 56.9.7 applies to it. To show that the procedure of Lemma 56.9.7 applied to $K'$ on $Y \times _ S X$ produces a complex isomorphic to $K$ it suffices (details omitted) to show that

\[ R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, \omega ^\bullet _{X \times _ S Y/X}), \omega ^\bullet _{X \times _ S Y/X}) = K \]

This is clear because $K$ is perfect and $\omega ^\bullet _{X \times _ S Y/X}$ is invertible; details omitted. Thus Lemma 56.9.7 produces a map

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

functorial in $M$ in $D_\mathit{QCoh}(\mathcal{O}_ X)$ and $N$ in $D_\mathit{QCoh}(\mathcal{O}_ Y)$ which is an isomorphism because $K'$ is perfect. This finishes the proof. $\square$

Lemma 56.9.9. Let $S$ be a Noetherian scheme.

  1. 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 $\Phi _ K : D_{perf}(\mathcal{O}_ X) \to D_{perf}(\mathcal{O}_ Y)$.

  2. 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 56.9.3,

  3. 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 $Y \times _ S X \to X \times _ S Y$.

  4. 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 $Y \times _ S X \to X \times _ S Y$.

Proof. Part (1) is immediate from Lemma 56.9.4 part (4).

Part (2) follows from Lemma 56.9.3 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 36.27.4.

The adjointness in part (3) on all complexes with quasi-coherent cohomology sheaves follows from Lemma 56.9.7 with $K'$ equal to the pullback of $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_{X \times _ S Y}}(K, a(\mathcal{O}_ Y))$ by $Y \times _ S X \to X \times _ S Y$ where $a$ is the right adjoint to $R\text{pr}_{2, *} : D_\mathit{QCoh}(\mathcal{O}_{X \times _ S Y}) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$. Denote $f : X \to S$ the structure morphism of $X$. Since $f$ is proper the functor $f^! : D_\mathit{QCoh}^+(\mathcal{O}_ S) \to D_\mathit{QCoh}^+(\mathcal{O}_ X)$ is the restriction to $D_\mathit{QCoh}^+(\mathcal{O}_ S)$ of the right adjoint to $Rf_* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ S)$, see Duality for Schemes, Section 48.16. Hence the relative dualizing complex $\omega _{X/S}^\bullet $ as defined in Duality for Schemes, Remark 48.12.5 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 48.12.5) we see that $a(\mathcal{O}_ Y) = L\text{pr}_1^*\omega _{X/S}^\bullet $. Thus

\[ R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_{X \times _ S Y}}(K, a(\mathcal{O}_ Y)) \cong L\text{pr}_1^*\omega _{X/S}^\bullet \otimes _{\mathcal{O}_{X \times _ S Y}}^\mathbf {L} K^\vee \]

by Cohomology, Lemma 20.47.5. Finally, since $X \to S$ is assumed Gorenstein the relative dualizing complex is invertible: this follows from Duality for Schemes, Lemma 48.25.10. We conclude that $\omega _{X/S}^\bullet $ is perfect (Cohomology, Lemma 20.49.2) 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).

The proof of (4) is the same as the proof of (3) except one uses Lemma 56.9.8 instead of Lemma 56.9.7. $\square$


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