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.) $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (2)

Comment #2037 by Matthieu Romagny on

Comment #2075 by Johan on