The Stacks project

Lemma 30.5.1. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Assume $f$ is affine. In this case $f_*\mathcal{F} \cong Rf_*\mathcal{F}$ is a quasi-coherent sheaf, and for every base change diagram ( we have

\[ g^*f_*\mathcal{F} = f'_*(g')^*\mathcal{F}. \]

Proof. The vanishing of higher direct images is Lemma 30.2.3. The statement is local on $S$ and $S'$. Hence we may assume $X = \mathop{\mathrm{Spec}}(A)$, $S = \mathop{\mathrm{Spec}}(R)$, $S' = \mathop{\mathrm{Spec}}(R')$ and $\mathcal{F} = \widetilde{M}$ for some $A$-module $M$. We use Schemes, Lemma 26.7.3 to describe pullbacks and pushforwards of $\mathcal{F}$. Namely, $X' = \mathop{\mathrm{Spec}}(R' \otimes _ R A)$ and $\mathcal{F}'$ is the quasi-coherent sheaf associated to $(R' \otimes _ R A) \otimes _ A M$. Thus we see that the lemma boils down to the equality

\[ (R' \otimes _ R A) \otimes _ A M = R' \otimes _ R M \]

as $R'$-modules. $\square$

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