The Stacks project

Lemma 59.96.3. Cohomology and direct sums. Let $n \geq 1$ be an integer.

  1. Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of schemes with $\text{cd}(f) < \infty $. Then the functor

    \[ Rf_* : D(X_{\acute{e}tale}, \mathbf{Z}/n\mathbf{Z}) \longrightarrow D(Y_{\acute{e}tale}, \mathbf{Z}/n\mathbf{Z}) \]

    commutes with direct sums.

  2. Let $X$ be a quasi-compact and quasi-separated scheme with $\text{cd}(X) < \infty $. Then the functor

    \[ R\Gamma (X, -) : D(X_{\acute{e}tale}, \mathbf{Z}/n\mathbf{Z}) \longrightarrow D(\mathbf{Z}/n\mathbf{Z}) \]

    commutes with direct sums.

Proof. Proof of (1). Since $\text{cd}(f) < \infty $ we see that

\[ f_* : \textit{Mod}(X_{\acute{e}tale}, \mathbf{Z}/n\mathbf{Z}) \longrightarrow \textit{Mod}(Y_{\acute{e}tale}, \mathbf{Z}/n\mathbf{Z}) \]

has finite cohomological dimension in the sense of Derived Categories, Lemma 13.32.2. Let $I$ be a set and for $i \in I$ let $E_ i$ be an object of $D(X_{\acute{e}tale}, \mathbf{Z}/n\mathbf{Z})$. Choose a K-injective complex $\mathcal{I}_ i^\bullet $ of $\mathbf{Z}/n\mathbf{Z}$-modules each of whose terms $\mathcal{I}_ i^ n$ is an injective sheaf of $\mathbf{Z}/n\mathbf{Z}$-modules representing $E_ i$. See Injectives, Theorem 19.12.6. Then $\bigoplus E_ i$ is represented by the complex $\bigoplus \mathcal{I}_ i^\bullet $ (termwise direct sum), see Injectives, Lemma 19.13.4. By Lemma 59.51.7 we have

\[ R^ qf_*(\bigoplus \mathcal{I}_ i^ n) = \bigoplus R^ qf_*(\mathcal{I}_ i^ n) = 0 \]

for $q > 0$ and any $n$. Hence we conclude by Derived Categories, Lemma 13.32.2 that we may compute $Rf_*(\bigoplus E_ i)$ by the complex

\[ f_*(\bigoplus \mathcal{I}_ i^\bullet ) = \bigoplus f_*(\mathcal{I}_ i^\bullet ) \]

(equality again by Lemma 59.51.7) which represents $\bigoplus Rf_*E_ i$ by the already used Injectives, Lemma 19.13.4.

Proof of (2). This is identical to the proof of (1) and we omit it. $\square$

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