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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