Lemma 76.52.9. In Situation 76.52.7 the category of $Y$-perfect objects of $D(\mathcal{O}_ X)$ is the colimit of the categories of $Y_ i$-perfect objects of $D(\mathcal{O}_{X_ i})$.

Proof. For every quasi-compact and quasi-separated object $U_0$ of $(X_0)_{spaces, {\acute{e}tale}}$ consider the condition $P$ that the functor

$\mathop{\mathrm{colim}}\nolimits _{i \geq 0} D_{Y_ i\text{-perfect}}(\mathcal{O}_{U_ i}) \longrightarrow D_{Y\text{-perfect}}(\mathcal{O}_ U)$

is an equivalence where $U = X \times _{X_0} U_0$ and $U_ i = X_ i \times _{X_0} U_0$. We observe that we already know this functor is fully faithful by Lemma 76.52.8. Thus it suffices to prove essential surjectivity.

Suppose that $(U_0 \subset W_0, V_0 \to W_0)$ is an elementary distinguished square in $(X_0)_{spaces, {\acute{e}tale}}$ and $P$ holds for $U_0, V_0, U_0 \times _{W_0} V_0$. We claim that $P$ holds for $W_0$. We will use the notation $U_ i = X_ i \times _{X_0} U_0$, $U = X \times _{X_0} U_0$, and similarly for $V_0$ and $W_0$. We will abusively use the symbol $f_ i$ for all the morphisms $U \to U_ i$, $V \to V_ i$, $U \times _ W V \to U_ i \times _{W_ i} V_ i$, and $W \to W_ i$. Suppose $E$ is an $Y$-perfect object of $D(\mathcal{O}_ W)$. Goal: show $E$ is in the essential image of the functor. By assumption, we can find $i \geq 0$, an $Y_ i$-perfect object $E_{U, i}$ on $U_ i$, an $Y_ i$-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 $c : E_{U, i}|_{U_ i \times _{W_ i} V_ i} \to E_{V, i}|_{U_ i \times _{W_ i} V_ i}$ which pulls back to the identifications

$Lf_ i^*E_{U, i}|_{U \times _ W V} \to E|_{U \times _ W V} \to Lf_ i^*E_{V, i}|_{U \times _ W V}.$

Apply Derived Categories of Spaces, Lemma 75.10.8 to get an object $E_ i$ on $W_ i$ 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 $Y_ i$-perfect (because being relatively perfect is an étale 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 76.52.8 using the induction principle (Derived Categories of Spaces, Lemma 75.9.3) we reduce to the case where both $X_0$ and $Y_0$ are affine: first work with quasi-compact and quasi-separated objects in $(Y_0)_{spaces, {\acute{e}tale}}$ to reduce to $Y_0$ affine, then work with quasi-compact and quasi-separated object in $(X_0)_{spaces, {\acute{e}tale}}$ to reduce to $X_0$ affine. In the affine case the result follows from the case of schemes which is Derived Categories of Schemes, Lemma 36.35.9. The translation into the case for schemes is done by Lemma 76.52.3. $\square$

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