The Stacks project

Lemma 59.96.5. 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 21.50. 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 15.59.11 hold for $T$ to conclude. Property (1) holds because both sides of the equality commute with direct sums, see Lemma 59.96.3. 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. $\square$


Comments (1)

Comment #9563 by on

The first sentence needs to state that the map from the section on coh of sites is from right to left. Pointed out by Felix Baril Boudreau.


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