Remark 63.11.8. 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 63.10.8 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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