The Stacks project

Lemma 57.16.4. 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 57.16.1. 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 57.16.3. 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 37.60.18 and 37.59.13) and its pullback to $X \times _ S X$ is equal to

\[ \Delta _{X/S, *}\mathcal{O}_ X \]

See proof of Lemma 57.16.3. 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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