Lemma 26.7.3. Let $(X, \mathcal{O}_ X) = (\mathop{\mathrm{Spec}}(S), \mathcal{O}_{\mathop{\mathrm{Spec}}(S)})$, $(Y, \mathcal{O}_ Y) = (\mathop{\mathrm{Spec}}(R), \mathcal{O}_{\mathop{\mathrm{Spec}}(R)})$ be affine schemes. Let $\psi : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of affine schemes, corresponding to the ring map $\psi ^\sharp : R \to S$ (see Lemma 26.6.5).

We have $\psi ^* \widetilde M = \widetilde{S \otimes _ R M}$ functorially in the $R$-module $M$.

We have $\psi _* \widetilde N = \widetilde{N_ R}$ functorially in the $S$-module $N$.

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