The Stacks project

Lemma 36.35.8. In Situation 36.35.7. Let $K_0$ and $L_0$ be objects of $D(\mathcal{O}_{X_0})$. Set $K_ i = Lf_{i0}^*K_0$ and $L_ i = Lf_{i0}^*L_0$ for $i \geq 0$ and set $K = Lf_0^*K_0$ and $L = Lf_0^*L_0$. Then the map

\[ \mathop{\mathrm{colim}}\nolimits _{i \geq 0} \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_{X_ i})}(K_ i, L_ i) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(K, L) \]

is an isomorphism if $K_0$ is pseudo-coherent and $L_0 \in D_\mathit{QCoh}(\mathcal{O}_{X_0})$ has (locally) finite tor dimension as an object of $D((X_0 \to S_0)^{-1}\mathcal{O}_{S_0})$

Proof. For every quasi-compact open $U_0 \subset X_0$ consider the condition $P$ that

\[ \mathop{\mathrm{colim}}\nolimits _{i \geq 0} \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_{U_ i})}(K_ i|_{U_ i}, L_ i|_{U_ i}) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(K|_ U, L|_ U) \]

is an isomorphism where $U = f_0^{-1}(U_0)$ and $U_ i = f_{i0}^{-1}(U_0)$. If $P$ holds for $U_0$, $V_0$ and $U_0 \cap V_0$, then it holds for $U_0 \cup V_0$ by Mayer-Vietoris for hom in the derived category, see Cohomology, Lemma 20.33.3.

Denote $\pi _0 : X_0 \to S_0$ the given morphism. Then we can first consider $U_0 = \pi _0^{-1}(W_0)$ with $W_0 \subset S_0$ quasi-compact open. By the induction principle of Cohomology of Schemes, Lemma 30.4.1 applied to quasi-compact opens of $S_0$ and the remark above, we find that it is enough to prove $P$ for $U_0 = \pi _0^{-1}(W_0)$ with $W_0$ affine. In other words, we have reduced to the case where $S_0$ is affine. Next, we apply the induction principle again, this time to all quasi-compact and quasi-separated opens of $X_0$, to reduce to the case where $X_0$ is affine as well.

If $X_0$ and $S_0$ are affine, the result follows from More on Algebra, Lemma 15.83.7. Namely, by Lemmas 36.10.1 and 36.3.5 the statement is translated into computations of homs in the derived categories of modules. Then Lemma 36.10.2 shows that the complex of modules corresponding to $K_0$ is pseudo-coherent. And Lemma 36.10.5 shows that the complex of modules corresponding to $L_0$ has finite tor dimension over $\mathcal{O}_{S_0}(S_0)$. Thus the assumptions of More on Algebra, Lemma 15.83.7 are satisfied and we win. $\square$

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