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. $\square$

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).