Lemma 17.10.7. Let $(X, \mathcal{O}_ X)$ be a ringed space. Set $R = \Gamma (X, \mathcal{O}_ X)$. Let $M$ be an $R$-module. Let $\mathcal{F}_ M$ be the quasi-coherent sheaf of $\mathcal{O}_ X$-modules associated to $M$. If $g : (Y, \mathcal{O}_ Y) \to (X, \mathcal{O}_ X)$ is a morphism of ringed spaces, then $g^*\mathcal{F}_ M$ is the sheaf associated to the $\Gamma (Y, \mathcal{O}_ Y)$-module $\Gamma (Y, \mathcal{O}_ Y) \otimes _ R M$.

Proof. The assertion follows from the first description of $\mathcal{F}_ M$ in Lemma 17.10.5 as $\pi ^*M$, and the following commutative diagram of ringed spaces

$\xymatrix{ (Y, \mathcal{O}_ Y) \ar[r]_-\pi \ar[d]_ g & (\{ *\} , \Gamma (Y, \mathcal{O}_ Y)) \ar[d]^{\text{induced by }g^\sharp } \\ (X, \mathcal{O}_ X) \ar[r]^-\pi & (\{ *\} , \Gamma (X, \mathcal{O}_ X)) }$

(Also use Sheaves, Lemma 6.26.3.) $\square$

Comment #2037 by Matthieu Romagny on

Typo in 1st sentence of staatement : a ringed space

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