The Stacks project

Lemma 73.6.1. Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-separated and quasi-compact morphism of algebraic spaces over $S$.

  1. The functor $Rf_*$ sends $D_\mathit{QCoh}(\mathcal{O}_ X)$ into $D_\mathit{QCoh}(\mathcal{O}_ Y)$.

  2. If $Y$ is quasi-compact, there exists an integer $N = N(X, Y, f)$ such that for an object $E$ of $D_\mathit{QCoh}(\mathcal{O}_ X)$ with $H^ m(E) = 0$ for $m > 0$ we have $H^ m(Rf_*E) = 0$ for $m \geq N$.

  3. In fact, if $Y$ is quasi-compact we can find $N = N(X, Y, f)$ such that for every morphism of algebraic spaces $Y' \to Y$ the same conclusion holds for the functor $R(f')_*$ where $f' : X' \to Y'$ is the base change of $f$.

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 67.8.1. 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 67.3.1. 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

\[ R^ pf_*H^ q(E) \Rightarrow R^{p + q}f_*E \]

(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 67.3.2. Hence we may apply Lemma 73.5.8 to the functor $\Gamma (V \times _ Y X, -)$ to see that

\[ R\Gamma (V, Rf_*E) = R\Gamma (V \times _ Y X, E) \]

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

\[ \tau _{\leq -n - 1}E \to E \to \tau _{\geq -n}E \to (\tau _{\leq -n - 1}E)[1] \]

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. $\square$


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