Exercise 110.40.2. General facts.

Show that an invertible ${\mathcal O}_ X$-module on a scheme $X$ is quasi-coherent.

Suppose $X\to Y$ is a morphism of locally ringed spaces, and ${\mathcal L}$ an invertible ${\mathcal O}_ Y$-module. Show that $f^\ast {\mathcal L}$ is an invertible ${\mathcal O}_ X$ module.

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