Lemma 36.35.9. In Situation 36.35.7 the category of S-perfect objects of D(\mathcal{O}_ X) is the colimit of the categories of S_ i-perfect objects of D(\mathcal{O}_{X_ i}).
Proof. For every quasi-compact open U_0 \subset X_0 consider the condition P that the functor
\mathop{\mathrm{colim}}\nolimits _{i \geq 0} D_{S_ i\text{-perfect}}(\mathcal{O}_{U_ i}) \longrightarrow D_{S\text{-perfect}}(\mathcal{O}_ U)
is an equivalence where U = f_0^{-1}(U_0) and U_ i = f_{i0}^{-1}(U_0). We observe that we already know this functor is fully faithful by Lemma 36.35.8. Thus it suffices to prove essential surjectivity.
Suppose that P holds for quasi-compact opens U_0, V_0 of X_0. We claim that P holds for U_0 \cup V_0. We will use the notation U_ i = f_{i0}^{-1}U_0, U = f_0^{-1}U_0, V_ i = f_{i0}^{-1}V_0, and V = f_0^{-1}V_0 and we will abusively use the symbol f_ i for all the morphisms U \to U_ i, V \to V_ i, U \cap V \to U_ i \cap V_ i, and U \cup V \to U_ i \cup V_ i. Suppose E is an S-perfect object of D(\mathcal{O}_{U \cup V}). Goal: show E is in the essential image of the functor. By assumption, we can find i \geq 0, an S_ i-perfect object E_{U, i} on U_ i, an S_ i-perfect object E_{V, i} on V_ i, and isomorphisms Lf_ i^*E_{U, i} \to E|_ U and Lf_ i^*E_{V, i} \to E|_ V. Let
a : E_{U, i} \to (Rf_{i, *}E)|_{U_ i} \quad \text{and}\quad b : E_{V, i} \to (Rf_{i, *}E)|_{V_ i}
the maps adjoint to the isomorphisms Lf_ i^*E_{U, i} \to E|_ U and Lf_ i^*E_{V, i} \to E|_ V. By fully faithfulness, after increasing i, we can find an isomorphism c : E_{U, i}|_{U_ i \cap V_ i} \to E_{V, i}|_{U_ i \cap V_ i} which pulls back to the identifications
Lf_ i^*E_{U, i}|_{U \cap V} \to E|_{U \cap V} \to Lf_ i^*E_{V, i}|_{U \cap V}.
Apply Cohomology, Lemma 20.45.1 to get an object E_ i on U_ i \cup V_ i and a map d : E_ i \to Rf_{i, *}E which restricts to the maps a and b over U_ i and V_ i. Then it is clear that E_ i is S_ i-perfect (because being relatively perfect is a local property) and that d is adjoint to an isomorphism Lf_ i^*E_ i \to E.
By exactly the same argument as used in the proof of Lemma 36.35.8 using the induction principle (Cohomology of Schemes, Lemma 30.4.1) we reduce to the case where both X_0 and S_0 are affine. (First work with opens in S_0 to reduce to S_0 affine, then work with opens in X_0 to reduce to X_0 affine.) In the affine case the result follows from More on Algebra, Lemma 15.83.7. The translation into algebra is done by Lemma 36.35.3. \square
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