Lemma 18.32.4 (040A)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 18: Modules on Sites
Section 18.32: Invertible modules
Lemma 18.32.4 (cite)

numberstags

Lemma 18.32.4. Let $(\mathcal{C}, \mathcal{O})$ be a ringed space.

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

Proof. Part (1) is clear from the definition and part (2) follows from Lemma 18.32.2 and its proof. $\square$

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