Lemma 17.24.5. Let $(X, \mathcal{O}_ X)$ be a ringed space.

1. If $\mathcal{L}$, $\mathcal{N}$ are invertible $\mathcal{O}_ X$-modules, then so is $\mathcal{L} \otimes _{\mathcal{O}_ X} \mathcal{N}$.

2. If $\mathcal{L}$ is an invertible $\mathcal{O}_ X$-module, then so is $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{L}, \mathcal{O}_ X)$ and the evaluation map $\mathcal{L} \otimes _{\mathcal{O}_ X} \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{L}, \mathcal{O}_ X) \to \mathcal{O}_ X$ is an isomorphism.

Proof. Part (1) is clear from the definition and part (2) follows from Lemma 17.24.2 and its proof. $\square$

Comment #9 by Pieter Belmans on

The second and third item shouldn't be plural.

Comment #10 by Pieter Belmans on

Why refrain from using the phrase "locally ringed space" by the way?

Comment #19 by Johan on

@#9: Thanks. Fixed. @#10: Because we introduce the later in the chapter on schemes. But actually, that should be changed. The section on locally ringed spaces should go in the chapter on modules on spaces, or perhaps in its own chapter.

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