57.15 Relative equivalences
In this section we prove some lemmas about the following concept.
Definition 57.15.1. 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 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 57.14.
The language is intentionally cumbersome.
Lemma 57.15.2. With notation as in Definition 57.15.1 let $K$ be the Fourier-Mukai kernel of a relative equivalence from $X$ to $Y$ over $S$. Then the corresponding Fourier-Mukai functors $\Phi _ K : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$ (Lemma 57.8.2) and $\Phi _ K : D_{perf}(\mathcal{O}_ X) \to D_{perf}(\mathcal{O}_ Y)$ (Lemma 57.8.4) are equivalences.
Proof.
Immediate from Lemma 57.8.3 and Example 57.8.6.
$\square$
Lemma 57.15.3. With notation as in Definition 57.15.1 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 $S_1$.
Proof.
Let $K' \in D_{perf}(\mathcal{O}_{Y \times _ S X})$ be the object assumed to exist in Definition 57.15.1. 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 36.30.4 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 36.22.5. This tor independence can be seen directly but also follows from the more general More on Morphisms, Lemma 37.69.1 applied to the square with corners $X, X, X, S$ and its base change by $S_1 \to S$.
$\square$
Lemma 57.15.4. Let $S = \mathop{\mathrm{lim}}\nolimits _{i \in I} S_ i$ 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 $S_ i$.
Proof.
Let $K' \in D_{perf}(\mathcal{O}_{Y \times _ S X})$ be the object assumed to exist in Definition 57.15.1. Since $X \times _ S Y = \mathop{\mathrm{lim}}\nolimits X_ i \times _{S_ i} Y_ i$ 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 36.29.3. By Derived Categories of Schemes, Lemma 36.30.4 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 57.15.3. 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 37.62.18 and 37.61.13) and its pullback to $X \times _ S X$ is equal to
\[ \Delta _{X/S, *}\mathcal{O}_ X \]
See proof of Lemma 57.15.3. Thus by Derived Categories of Schemes, Lemma 36.29.3 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.
$\square$
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