The Stacks project

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

is an equivalence where ${}_{perf}$ indicates the full subcategory of perfect objects and where $U = X \times _{X_0} U_0$ and $U_ i = X_ i \times _{X_0} U_0$. We will prove $P$ holds for every $U_0$ by the induction principle of Lemma 75.9.3. First, we observe that we already know the functor is fully faithful by Lemma 75.24.2. Thus it suffices to prove essential surjectivity.

We first check condition (2) of the induction principle. Thus suppose that we have an elementary distinguished square $(U_0 \subset X_0, V_0 \to X_0)$ and that $P$ holds for $U_0$, $V_0$, and $U_0 \times _{X_0} V_0$. Let $E$ be a perfect object of $D(\mathcal{O}_ X)$. We can find $i \geq 0$ and $E_{U, i}$ perfect on $U_ i$ and $E_{V, i}$ perfect on $V_ i$ whose pullback to $U$ and $V$ are isomorphic to $E|_ U$ and $E|_ V$. Denote

the maps adjoint to the isomorphisms $L(U \to U_ i)^*E_{U, i} \to E|_ U$ and $L(V \to V_ i)^*E_{V, i} \to E|_ V$. By fully faithfulness, after increasing $i$, we can find an isomorphism $c : E_{U, i}|_{U_ i \times _{X_ i} V_ i} \to E_{V, i}|_{U_ i \times _{X_ i} V_ i}$ which pulls back to the identifications

Apply Lemma 75.10.8 to get an object $E_ i$ on $X_ i$ and a map $d : E_ i \to R(X \to X_ i)_*E$ which restricts to the maps $a$ and $b$ over $U_ i$ and $V_ i$. Then it is clear that $E_ i$ is perfect and that $d$ is adjoint to an isomorphism $L(X \to X_ i)^*E_ i \to E$.

Finally, we check condition (1) of the induction principle, in other words, we check the lemma holds when $X_0$ is affine. This follows from the case of schemes, see Derived Categories of Schemes, Lemma 36.29.3. To see this use the equivalence of Lemma 75.4.2 and use the translation of Lemma 75.13.5. $\square$