36.4 Total direct image
The following lemma is the analogue of Cohomology of Schemes, Lemma 30.4.5.
Lemma 36.4.1. Let $f : X \to S$ be a morphism of schemes. Assume that $f$ is quasi-separated and quasi-compact.
The functor $Rf_*$ sends $D_\mathit{QCoh}(\mathcal{O}_ X)$ into $D_\mathit{QCoh}(\mathcal{O}_ S)$.
If $S$ is quasi-compact, there exists an integer $N = N(X, S, f)$ 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 $S$ is quasi-compact we can find $N = N(X, S, f)$ such that for every morphism of schemes $S' \to S$ the same conclusion holds for the functor $R(f')_*$ where $f' : X' \to S'$ 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. The question is local on $S$, hence we may assume $S$ is quasi-compact. Pick $N = N(X, S, f)$ as in Cohomology of Schemes, Lemma 30.4.5. Thus $R^ pf_*\mathcal{F} = 0$ for all quasi-coherent $\mathcal{O}_ X$-modules $\mathcal{F}$ and all $p \geq N$ and the same remains 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 $U \subset S$ be affine open. By Cohomology of Schemes, Lemma 30.4.6 and our choice of $N$ we have $H^ p(f^{-1}(U), \mathcal{F}) = 0$ for $p \geq N$ and any quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$. Hence we may apply Lemma 36.3.4 to the functor $\Gamma (f^{-1}(U), -)$ to see that
\[ R\Gamma (U, Rf_*E) = R\Gamma (f^{-1}(U), E) \]
has vanishing cohomology in degrees $\geq N$. Since this holds for all $U \subset S$ affine open 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 36.4.2. Let $f : X \to S$ be a quasi-separated and quasi-compact morphism of schemes. Let $\mathcal{F}^\bullet $ be a complex of quasi-coherent $\mathcal{O}_ X$-modules each of which is right acyclic for $f_*$. Then $f_*\mathcal{F}^\bullet $ represents $Rf_*\mathcal{F}^\bullet $ in $D(\mathcal{O}_ S)$.
Proof.
There is always a canonical map $f_*\mathcal{F}^\bullet \to Rf_*\mathcal{F}^\bullet $. Our task is to show that this is an isomorphism on cohomology sheaves. As the statement is invariant under shifts it suffices to show that $H^0(f_*(\mathcal{F}^\bullet )) \to R^0f_*\mathcal{F}^\bullet $ is an isomorphism. The statement is local on $S$ hence we may assume $S$ affine. By Lemma 36.4.1 we have $R^0f_*\mathcal{F}^\bullet = R^0f_*\tau _{\geq -n}\mathcal{F}^\bullet $ for all sufficiently large $n$. Thus we may assume $\mathcal{F}^\bullet $ bounded below. As each $\mathcal{F}^ n$ is right $f_*$-acyclic by assumption we see that $f_*\mathcal{F}^\bullet \to Rf_*\mathcal{F}^\bullet $ is a quasi-isomorphism by Leray's acyclicity lemma (Derived Categories, Lemma 13.16.7).
$\square$
Lemma 36.4.3. Let $X$ be a quasi-separated and quasi-compact scheme. Let $\mathcal{F}^\bullet $ be a complex of quasi-coherent $\mathcal{O}_ X$-modules each of which is right acyclic for $\Gamma (X, -)$. Then $\Gamma (X, \mathcal{F}^\bullet )$ represents $R\Gamma (X, \mathcal{F}^\bullet )$ in $D(\Gamma (X, \mathcal{O}_ X)$.
Proof.
Apply Lemma 36.4.2 to the canonical morphism $X \to \mathop{\mathrm{Spec}}(\Gamma (X, \mathcal{O}_ X))$. Some details omitted.
$\square$
Lemma 36.4.4. Let $X$ be a quasi-separated and quasi-compact scheme. For any object $K$ of $D_\mathit{QCoh}(\mathcal{O}_ X)$ the spectral sequence
\[ E_2^{i, j} = H^ i(X, H^ j(K)) \Rightarrow H^{i + j}(X, K) \]
of Cohomology, Example 20.29.3 is bounded and converges.
Proof.
By the construction of the spectral sequence via Cohomology, Lemma 20.29.1 using the filtration given by $\tau _{\leq -p}K$, we see that suffices to show that given $n \in \mathbf{Z}$ we have
\[ H^ n(X, \tau _{\leq -p}K) = 0 \text{ for } p \gg 0 \]
and
\[ H^ n(X, K) = H^ n(X, \tau _{\leq -p}K) \text{ for } p \ll 0 \]
The first follows from Lemma 36.3.4 applied with $F = \Gamma (X, -)$ and the bound in Cohomology of Schemes, Lemma 30.4.5. The second holds whenever $-p \leq n$ for any ringed space $(X, \mathcal{O}_ X)$ and any $K \in D(\mathcal{O}_ X)$.
$\square$
Lemma 36.4.5. Let $f : X \to S$ be a quasi-separated and quasi-compact morphism of schemes. Then $Rf_* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ S)$ 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 36.4.1. 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 Schemes, Lemma 30.6.1.
$\square$
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