# The Stacks Project

## Tag: 01CT

This tag has label modules-lemma-constructions-invertible and it points to

The corresponding content:

Lemma 16.21.2. Let $(X, \mathcal{O}_X)$ be a ringed space. Assume that all stalks $\mathcal{O}_{X, x}$ are local rings.
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 $\mathcal{L}^{\otimes -1} = \mathop{\mathcal{H}\!{\it om}}\nolimits_{\mathcal{O}_X}(\mathcal{L}, \mathcal{O}_X)$.
3. If $\mathcal{L}$ is an invertible $\mathcal{O}_X$-module, then the evaluation map $\mathcal{L} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes -1} \to \mathcal{O}_X$ is an isomorphism.

Proof. Omitted. $\square$

\begin{lemma}
\label{lemma-constructions-invertible}
Let $(X, \mathcal{O}_X)$ be a ringed space.
Assume that all stalks $\mathcal{O}_{X, x}$ are local rings.
\begin{enumerate}
\item If $\mathcal{L}$, $\mathcal{N}$ are invertible
$\mathcal{O}_X$-modules, then so is
$\mathcal{L} \otimes_{\mathcal{O}_X} \mathcal{N}$.
\item If $\mathcal{L}$ is an invertible
$\mathcal{O}_X$-module, then so is
$\mathcal{L}^{\otimes -1} = \SheafHom_{\mathcal{O}_X}(\mathcal{L}, \mathcal{O}_X)$.
\item If $\mathcal{L}$ is an invertible
$\mathcal{O}_X$-module, then the evaluation map
$\mathcal{L} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes -1} \to \mathcal{O}_X$ is an isomorphism.
\end{enumerate}
\end{lemma}

\begin{proof}
Omitted.
\end{proof}


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Comment #9 by Pieter Belmans on July 22, 2012 at 9:08 am UTC

The second and third item shouldn't be plural.

Comment #10 by Pieter Belmans on July 22, 2012 at 9:15 am UTC

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

Comment #19 by Johan on July 22, 2012 at 10:56 pm UTC

@#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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