Lemma 26.7.3 (01I9)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 26: Schemes
Section 26.7: Quasi-coherent sheaves on affines
Lemma 26.7.3 (cite)

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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 $N$ .

Proof. The first assertion follows from the identification in Lemma 26.7.1 and the result of Modules, Lemma 17.10.7. The second assertion follows from the fact that $\psi ^{-1}(D(f)) = D(\psi ^\sharp (f))$ and hence

$\psi _* \widetilde N(D(f)) = \widetilde N(D(\psi ^\sharp (f))) = N_{\psi ^\sharp (f)} = (N_ R)_ f = \widetilde{N_ R}(D(f))$

as desired. $\square$

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