The Stacks project

Lemma 29.30.9. Let $k$ be a field. Let $X$ be a scheme locally of finite type over $k$. The following are equivalent:

  1. $X$ is a local complete intersection over $k$,

  2. for every $x \in X$ there exists an affine open $U = \mathop{\mathrm{Spec}}(R) \subset X$ neighbourhood of $x$ such that $R \cong k[x_1, \ldots , x_ n]/(f_1, \ldots , f_ c)$ is a global complete intersection over $k$, and

  3. for every $x \in X$ the local ring $\mathcal{O}_{X, x}$ is a complete intersection over $k$.

Proof. The corresponding algebra results can be found in Algebra, Lemmas 10.135.8 and 10.135.9. $\square$

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Your email address will not be published.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 01UG. Beware of the difference between the letter 'O' and the digit '0'.