The Stacks project

Lemma 21.53.4. Let $\mathcal{C}$ be a site. Let $\Lambda $ be a Noetherian ring. Let $K, L \in D^-(\mathcal{C}, \Lambda )$. If the cohomology sheaves of $K$ and $L$ are locally constant sheaves of $\Lambda $-modules of finite type, then the cohomology sheaves of $K \otimes _\Lambda ^\mathbf {L} L$ are locally constant sheaves of $\Lambda $-modules of finite type.

Proof. We'll prove this as an application of Lemma 21.53.1. 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. By Lemma 21.53.1 we may assume that $K$ and $L$ are represented by complexes of locally constant sheaves of $\Lambda $-modules of finite type. Then we can replace these complexes by bounded above complexes of finite free $\Lambda $-modules. In this case the result is clear. $\square$

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