Remark 63.10.8. 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} \ar[d]_{Rf_!} & D(X_{\acute{e}tale}, \Lambda _1) \ar[d]^{Rf_!} \\ D(Y_{\acute{e}tale}, \Lambda _2) \ar[r]^{res} & D(Y_{\acute{e}tale}, \Lambda _1) }
commutes where res is the “restriction” functor which turns a \Lambda _2-module into a \Lambda _1-module using the given ring map. Writing Rf_! = R\overline{f}_* \circ j_! for a factorization f = \overline{f} \circ j as in Section 63.9, we see that the result holds for j_! by inspection and for R\overline{f}_* by Cohomology on Sites, Lemma 21.20.7. On the other hand, also the diagram
\xymatrix{ D(X_{\acute{e}tale}, \Lambda _1) \ar[r]_{- \otimes _{\Lambda _1}^\mathbf {L} \Lambda _2} \ar[d]_{Rf_!} & D(X_{\acute{e}tale}, \Lambda _2) \ar[d]^{Rf_!} \\ D(Y_{\acute{e}tale}, \Lambda _1) \ar[r]^{- \otimes _{\Lambda _1}^\mathbf {L} \Lambda _2} & D(Y_{\acute{e}tale}, \Lambda _2) }
is commutative as follows from Lemma 63.10.7.
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