The Stacks project

Lemma 58.95.5. Let $X$ be a quasi-compact and quasi-separated scheme such that $\text{cd}(X) < \infty $. Let $\Lambda $ be a torsion ring. Let $E \in D(X_{\acute{e}tale}, \Lambda )$ and $K \in D(\Lambda )$. Then

\[ R\Gamma (X, E \otimes _\Lambda ^\mathbf {L} \underline{K}) = R\Gamma (X, E) \otimes _\Lambda ^\mathbf {L} K \]

Proof. There is a canonical map from left to right by Cohomology on Sites, Section 21.48. Let $T(K)$ be the property that the statement of the lemma holds for $K \in D(\Lambda )$. We will check conditions (1), (2), and (3) of More on Algebra, Remark 15.58.13 hold for $T$ to conclude. Property (1) holds because both sides of the equality commute with direct sums, see Lemma 58.95.3. Property (2) holds because we are comparing exact functors between triangulated categories and we can use Derived Categories, Lemma 13.4.3. Property (3) says the lemma holds when $K = \Lambda [k]$ for any shift $k \in \mathbf{Z}$ and this is obvious. $\square$


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