The Stacks project

Proof. We may replace $X$ by a quasi-compact open neighbourhood of the support of $\mathcal{G}$, hence we may assume $X$ is Noetherian. In this case $X$ and $f$ are quasi-compact and quasi-separated. Choose an approximation $P \to \mathcal{F}$ by a perfect complex $P$ of the triple $(X, \mathcal{F}, -1)$, see Derived Categories of Spaces, Definition 75.14.1 and Theorem 75.14.7). Then the induced map

is an isomorphism because $P \to \mathcal{F}$ induces an isomorphism $H^0(P) \to \mathcal{F}$ and $H^ i(P) = 0$ for $i > 0$. Moreover, for any morphism $g : T \to B$ denote $h : X_ T = T \times _ B X \to X$ the projection and set $P_ T = Lh^*P$. Then it is equally true that

is an isomorphism, as $P_ T = Lh^*P \to Lh^*\mathcal{F} \to \mathcal{F}_ T$ induces an isomorphism $H^0(P_ T) \to \mathcal{F}_ T$ (because $h^*$ is right exact and $H^ i(P) = 0$ for $i > 0$). Thus it suffices to prove the result for the functor

By the Leray spectral sequence (see Cohomology on Sites, Remark 21.14.4) we have

where $f_ T : X_ T \to T$ is the base change of $f$. By Derived Categories of Spaces, Lemma 75.21.5 we have

By Derived Categories of Spaces, Lemma 75.22.3 the object $K = Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (P, \mathcal{G})$ of $D(\mathcal{O}_ B)$ is perfect. This means we can apply Lemma 99.3.8 as long as we can prove that the cohomology sheaf $H^ i(Lg^*K)$ is $0$ for all $i < 0$ and $g : T \to B$ as above. This is clear from the last displayed formula as the cohomology sheaves of $Rf_{T, *}R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (P_ T, \mathcal{G}_ T)$ are zero in negative degrees due to the fact that $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (P_ T, \mathcal{G}_ T)$ has vanishing cohomology sheaves in negative degrees as $P_ T$ is perfect with vanishing cohomology sheaves in positive degrees. $\square$