The Stacks project

Lemma 74.6.2. 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 74.6.1. 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 68.5.2. $\square$

Comments (2)

Comment #4654 by Dylan Spence on

In the statement of the lemma, should actually be ?

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