The Stacks project

Lemma 62.10.1. Let $f : X \to Y$ be a finite type separated morphism of quasi-compact and quasi-separated schemes. Let $\Lambda $ be a ring.

  1. Let $K_ i \in D^+_{tors}(X_{\acute{e}tale}, \Lambda )$, $i \in I$ be a family of objects. Assume given $a \in \mathbf{Z}$ such that $H^ n(K_ i) = 0$ for $n < a$ and $i \in I$. Then $Rf_!(\bigoplus _ i K_ i) = \bigoplus _ i Rf_!K_ i$.

  2. If $\Lambda $ is torsion, then the functor $Rf_! : D(X_{\acute{e}tale}, \Lambda ) \to D(Y_{\acute{e}tale}, \Lambda )$ commutes with direct sums.

Proof. By construction it suffices to prove this when $f$ is an open immersion and when $f$ is a proper morphism. For any open immersion $j : U \to X$ of schemes, the functor $j_! : D(U_{\acute{e}tale}) \to D(X_{\acute{e}tale})$ is a left adjoint to pullback $j^{-1} : D(X_{\acute{e}tale}) \to D(U_{\acute{e}tale})$ and hence commutes with direct sums, see Cohomology on Sites, Lemma 21.20.8. In the proper case we have $Rf_! = Rf_*$ and we get the result from Étale Cohomology, Lemma 59.52.6 in the bounded belo case and from Étale Cohomology, Lemma 59.96.4 in the case that our coefficient ring $\Lambda $ is a torsion ring. $\square$


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