The Stacks project

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 20.42.1 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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