The Stacks project

Proof. Let $E$ be an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$. To prove (1) we have to show that $Rf_*E$ has quasi-coherent cohomology sheaves. This question is local on $Y$, hence we may assume $Y$ is quasi-compact. Pick $N = N(X, Y, f)$ as in Cohomology of Spaces, Lemma 69.8.1. Thus $R^ pf_*\mathcal{F} = 0$ for all quasi-coherent $\mathcal{O}_ X$-modules $\mathcal{F}$ and all $p \geq N$. Moreover $R^ pf_*\mathcal{F}$ is quasi-coherent for all $p$ by Cohomology of Spaces, Lemma 69.3.1. These statements remain true after base change.

First, assume $E$ is bounded below. We will show (1) and (2) and (3) hold for such $E$ with our choice of $N$. In this case we can for example use the spectral sequence

(Derived Categories, Lemma 13.21.3), the quasi-coherence of $R^ pf_*H^ q(E)$, and the vanishing of $R^ pf_*H^ q(E)$ for $p \geq N$ to see that (1), (2), and (3) hold in this case.

Next we prove (2) and (3). Say $H^ m(E) = 0$ for $m > 0$. Let $V$ be an affine object of $Y_{\acute{e}tale}$. We have $H^ p(V \times _ Y X, \mathcal{F}) = 0$ for $p \geq N$, see Cohomology of Spaces, Lemma 69.3.2. Hence we may apply Lemma 75.5.8 to the functor $\Gamma (V \times _ Y X, -)$ to see that

has vanishing cohomology in degrees $\geq N$. Since this holds for all $V$ affine in $Y_{\acute{e}tale}$ we conclude that $H^ m(Rf_*E) = 0$ for $m \geq N$.

Next, we prove (1) in the general case. Recall that there is a distinguished triangle

in $D(\mathcal{O}_ X)$, see Derived Categories, Remark 13.12.4. By (2) we see that $Rf_*\tau _{\leq -n - 1}E$ has vanishing cohomology sheaves in degrees $\geq -n + N$. Thus, given an integer $q$ we see that $R^ qf_*E$ is equal to $R^ qf_*\tau _{\geq -n}E$ for some $n$ and the result above applies. $\square$