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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