75.6 Total direct image
The following lemma is the analogue of Cohomology of Spaces, Lemma 69.8.1.
Lemma 75.6.1. Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-separated and quasi-compact morphism of algebraic spaces over $S$.
The functor $Rf_*$ sends $D_\mathit{QCoh}(\mathcal{O}_ X)$ into $D_\mathit{QCoh}(\mathcal{O}_ Y)$.
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 $m \geq N$.
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 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
\[ R^ pf_*H^ q(E) \Rightarrow R^{p + q}f_*E \]
(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 69.3.2. Hence we may apply Lemma 75.5.8 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.
$\square$
Lemma 75.6.2. Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-separated and quasi-compact morphism of algebraic spaces over $S$. Then $Rf_* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$ commutes with direct sums.
Proof.
Let $E_ i$ be a family of objects of $D_\mathit{QCoh}(\mathcal{O}_ X)$ and set $E = \bigoplus E_ i$. We want to show that the map
\[ \bigoplus Rf_*E_ i \longrightarrow Rf_*E \]
is an isomorphism. We will show it induces an isomorphism on cohomology sheaves in degree $0$ which will imply the lemma. Choose an integer $N$ as in Lemma 75.6.1. Then $R^0f_*E = R^0f_*\tau _{\geq -N}E$ and $R^0f_*E_ i = R^0f_*\tau _{\geq -N}E_ i$ by the lemma cited. Observe that $\tau _{\geq -N}E = \bigoplus \tau _{\geq -N}E_ i$. Thus we may assume all of the $E_ i$ have vanishing cohomology sheaves in degrees $< -N$. Next we use the spectral sequences
\[ R^ pf_*H^ q(E) \Rightarrow R^{p + q}f_*E \quad \text{and}\quad R^ pf_*H^ q(E_ i) \Rightarrow R^{p + q}f_*E_ i \]
(Derived Categories, Lemma 13.21.3) to reduce to the case of a direct sum of quasi-coherent sheaves. This case is handled by Cohomology of Spaces, Lemma 69.5.2.
$\square$
Lemma 75.6.4. Let $S$ be a scheme. Let $f : X \to Y$ be an affine morphism of algebraic spaces over $S$. Then $Rf_* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$ reflects isomorphisms.
Proof.
The statement means that a morphism $\alpha : E \to F$ of $D_\mathit{QCoh}(\mathcal{O}_ X)$ is an isomorphism if $Rf_*\alpha $ is an isomorphism. We may check this on cohomology sheaves. In particular, the question is étale local on $Y$. Hence we may assume $Y$ and therefore $X$ is affine. In this case the problem reduces to the case of schemes (Derived Categories of Schemes, Lemma 36.5.2) via Lemma 75.4.2 and Remark 75.6.3.
$\square$
Lemma 75.6.5. Let $S$ be a scheme. Let $f : X \to Y$ be an affine morphism of algebraic spaces over $S$. For $E$ in $D_\mathit{QCoh}(\mathcal{O}_ Y)$ we have $Rf_* Lf^* E = E \otimes ^\mathbf {L}_{\mathcal{O}_ Y} f_*\mathcal{O}_ X$.
Proof.
Since $f$ is affine the map $f_*\mathcal{O}_ X \to Rf_*\mathcal{O}_ X$ is an isomorphism (Cohomology of Spaces, Lemma 69.8.2). There is a canonical map $E \otimes ^\mathbf {L} f_*\mathcal{O}_ X = E \otimes ^\mathbf {L} Rf_*\mathcal{O}_ X \to Rf_* Lf^* E$ adjoint to the map
\[ Lf^*(E \otimes ^\mathbf {L} Rf_*\mathcal{O}_ X) = Lf^*E \otimes ^\mathbf {L} Lf^*Rf_*\mathcal{O}_ X \longrightarrow Lf^* E \otimes ^\mathbf {L} \mathcal{O}_ X = Lf^* E \]
coming from $1 : Lf^*E \to Lf^*E$ and the canonical map $Lf^*Rf_*\mathcal{O}_ X \to \mathcal{O}_ X$. To check the map so constructed is an isomorphism we may work locally on $Y$. Hence we may assume $Y$ and therefore $X$ is affine. In this case the problem reduces to the case of schemes (Derived Categories of Schemes, Lemma 36.5.3) via Lemma 75.4.2 and Remark 75.6.3.
$\square$
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