The Stacks project

Remark 91.4.9. Let $(X, \mathcal{O}_ X)$ be a ringed space. We define a sequence of morphisms of first order thickenings

\[ (X'_1, \mathcal{O}_{X'_1}) \to (X'_2, \mathcal{O}_{X'_2}) \to (X'_3, \mathcal{O}_{X'_3}) \]

of $(X, \mathcal{O}_ X)$ to be a complex if the corresponding maps between the ideal sheaves $\mathcal{I}_ i$ give a complex of $\mathcal{O}_ X$-modules $\mathcal{I}_3 \to \mathcal{I}_2 \to \mathcal{I}_1$ (i.e., the composition is zero). In this case the composition $(X'_1, \mathcal{O}_{X'_1}) \to (X_3', \mathcal{O}_{X'_3})$ factors through $(X, \mathcal{O}_ X) \to (X'_3, \mathcal{O}_{X'_3})$, i.e., the first order thickening $(X'_1, \mathcal{O}_{X'_1})$ of $(X, \mathcal{O}_ X)$ is trivial and comes with a canonical trivialization $\pi : (X'_1, \mathcal{O}_{X'_1}) \to (X, \mathcal{O}_ X)$.

We say a sequence of morphisms of first order thickenings

\[ (X'_1, \mathcal{O}_{X'_1}) \to (X'_2, \mathcal{O}_{X'_2}) \to (X'_3, \mathcal{O}_{X'_3}) \]

of $(X, \mathcal{O}_ X)$ is a short exact sequence if the corresponding maps between ideal sheaves is a short exact sequence

\[ 0 \to \mathcal{I}_3 \to \mathcal{I}_2 \to \mathcal{I}_1 \to 0 \]

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

Comments (0)

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Your email address will not be published.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 08LE. Beware of the difference between the letter 'O' and the digit '0'.



View Remark 91.4.9 as pdf