Remark 61.11.8. Let $\Lambda _1 \to \Lambda _2$ be a homomorphism of torsion rings. Let $f : X \to Y$ be a separated finite type morphism of quasi-compact and quasi-separated schemes. The diagram

$\xymatrix{ D(X_{\acute{e}tale}, \Lambda _2) \ar[r]_{res} & D(X_{\acute{e}tale}, \Lambda _1) \\ D(Y_{\acute{e}tale}, \Lambda _2) \ar[r]^{res} \ar[u]^{Rf^!} & D(Y_{\acute{e}tale}, \Lambda _1) \ar[u]_{Rf^!} }$

commutes where $res$ is the “restriction” functor which turns a $\Lambda _2$-module into a $\Lambda _1$-module using the given ring map. This holds by uniquenss of adjoints, the second commutative diagram of Remark 61.10.8 and because we have

$\mathop{\mathrm{Hom}}\nolimits _{\Lambda _2}(K_1 \otimes _{\Lambda _1}^\mathbf {L} \Lambda _2, K_2) = \mathop{\mathrm{Hom}}\nolimits _{\Lambda _1}(K_1, res(K_2))$

This equality either for objects living over $X_{\acute{e}tale}$ or on $Y_{\acute{e}tale}$ is a very special case of Cohomology on Sites, Lemma 21.19.1.

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