The Stacks project

Lemma 17.26.1. Let $X$ be a ringed space. Let $0 \to \mathcal{E}' \to \mathcal{E} \to \mathcal{E}'' \to 0$ be a short exact sequence of finite locally free $\mathcal{O}_ X$-modules. Then there is a canonical isomorphism

\[ \det (\mathcal{E}') \otimes _{\mathcal{O}_ X}\det (\mathcal{E}'') \longrightarrow \det (\mathcal{E}) \]

of $\mathcal{O}_ X$-modules.

Proof. We can decompose $X$ into disjoint open and closed subsets such that both $\mathcal{E}'$ and $\mathcal{E}''$ have constant rank on them. Thus we reduce to the case where $\mathcal{E}'$ and $\mathcal{E}''$ have constant rank, say $r'$ and $r''$. In this situation we define

\[ \wedge ^{r'}(\mathcal{E}') \otimes _{\mathcal{O}_ X} \wedge ^{r''}(\mathcal{E}'') \longrightarrow \wedge ^{r' + r''}(\mathcal{E}) \]

as follows. Given local sections $s'_1, \ldots , s'_{r'}$ of $\mathcal{E}'$ and local sections $s''_1, \ldots , s''_{r''}$ of $\mathcal{E}''$ we map

\[ s'_1 \wedge \ldots \wedge s'_{r'} \otimes s''_1 \wedge \ldots \wedge s''_{r''} \quad \text{to}\quad s'_1 \wedge \ldots \wedge s'_{r'} \wedge \tilde s''_1 \wedge \ldots \wedge \tilde s''_{r''} \]

where $\tilde s''_ i$ is a local lift of the section $s''_ i$ to a section of $\mathcal{E}$. We omit the details. $\square$


Comments (2)

There are also:

  • 4 comment(s) on Section 17.26: Rank and determinant

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Your email address will not be published.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 0B38. Beware of the difference between the letter 'O' and the digit '0'.