Lemma 61.27.5. Let $X$ be a scheme. Let $\Lambda$ be a Noetherian ring. Let $K, L \in D_ c^-(X_{pro\text{-}\acute{e}tale}, \Lambda )$. Then $K \otimes _\Lambda ^\mathbf {L} L$ is in $D_ c^-(X_{pro\text{-}\acute{e}tale}, \Lambda )$.

Proof. Note that $H^ i(K \otimes _\Lambda ^\mathbf {L} L)$ is the same as $H^ i(\tau _{\geq i - 1}K \otimes _\Lambda ^\mathbf {L} \tau _{\geq i - 1}L)$. Thus we may assume $K$ and $L$ are bounded. In this case we can apply Lemma 61.27.4 to reduce to the case of the étale site, see Étale Cohomology, Lemma 59.76.6. $\square$

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