The Stacks project

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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