Lemma 63.10.1. Let f : X \to Y be a finite type separated morphism of quasi-compact and quasi-separated schemes. Let \Lambda be a ring.
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.
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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