The Stacks project

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. $\square$


Comments (3)

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.


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Your email address will not be published.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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 01CT. Beware of the difference between the letter 'O' and the digit '0'.