Lemma 75.24.2. In Situation 75.24.1. Let $E_0$ and $K_0$ be objects of $D(\mathcal{O}_{X_0})$. Set $E_ i = Lf_{i0}^*E_0$ and $K_ i = Lf_{i0}^*K_0$ for $i \geq 0$ and set $E = Lf_0^*E_0$ and $K = Lf_0^*K_0$. Then the map
\[ \mathop{\mathrm{colim}}\nolimits _{i \geq 0} \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_{X_ i})}(E_ i, K_ i) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(E, K) \]
is an isomorphism if either
$E_0$ is perfect and $K_0 \in D_\mathit{QCoh}(\mathcal{O}_{X_0})$, or
$E_0$ is pseudo-coherent and $K_0 \in D_\mathit{QCoh}(\mathcal{O}_{X_0})$ has finite tor dimension.
Proof.
For every quasi-compact and quasi-separated object $U_0$ of $(X_0)_{spaces, {\acute{e}tale}}$ consider the condition $P$ that the canonical map
\[ \mathop{\mathrm{colim}}\nolimits _{i \geq 0} \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_{U_ i})}(E_ i|_{U_ i}, K_ i|_{U_ i}) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(E|_ U, K|_ U) \]
is an isomorphism, where $U = X \times _{X_0} U_0$ and $U_ i = X_ i \times _{X_0} U_0$. We will prove $P$ holds for each $U_0$ by the induction principle of Lemma 75.9.3. Condition (2) of this lemma follows immediately from Mayer-Vietoris for hom in the derived category, see Lemma 75.10.4. Thus it suffices to prove the lemma when $X_0$ is affine.
If $X_0$ is affine, then the result follows from the case of schemes, see Derived Categories of Schemes, Lemma 36.29.2. To see this use the equivalence of Lemma 75.4.2 and use the translation of properties explained in Lemmas 75.13.2, 75.13.3, and 75.13.5.
$\square$
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