The Stacks project

Lemma 53.19.15. Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme equidimensional of dimension $1$. The following are equivalent

  1. the singularities of $X$ are at-worst-nodal, and

  2. $X$ is a local complete intersection over $k$ and the closed subscheme $Z \subset X$ cut out by the first fitting ideal of $\Omega _{X/k}$ is unramified over $k$.

Proof. We urge the reader to find their own proof of this lemma; what follows is just putting together earlier results and may hide what is really going on.

Assume (2). Since $Z \to \mathop{\mathrm{Spec}}(k)$ is quasi-finite (Morphisms, Lemma 29.35.10) we see that the residue fields of points $x \in Z$ are finite over $k$ (as well as separable) by Morphisms, Lemma 29.20.5. Hence each $x \in Z$ is a closed point of $X$ by Morphisms, Lemma 29.20.2. The local ring $\mathcal{O}_{X, x}$ is Cohen-Macaulay by Algebra, Lemma 10.135.3. Since $\dim (\mathcal{O}_{X, x}) = 1$ by dimension theory (Varieties, Section 33.20), we conclude that $\text{depth}(\mathcal{O}_{X, x})) = 1$. Thus $x$ is a node by Lemma 53.19.7. If $x \in X$, $x \not\in Z$, then $X \to \mathop{\mathrm{Spec}}(k)$ is smooth at $x$ by Divisors, Lemma 31.10.3.

Assume (1). Under this assumption $X$ is geometrically reduced at every closed point (see Varieties, Lemma 33.6.6). Hence $X \to \mathop{\mathrm{Spec}}(k)$ is smooth on a dense open by Varieties, Lemma 33.25.7. Thus $Z$ is closed and consists of closed points. By Divisors, Lemma 31.10.3 the morphism $X \setminus Z \to \mathop{\mathrm{Spec}}(k)$ is smooth. Hence $X \setminus Z$ is a local complete intersection by Morphisms, Lemma 29.34.7 and the definition of a local complete intersection in Morphisms, Definition 29.30.1. By Lemma 53.19.7 for every point $x \in Z$ the local ring $\mathcal{O}_{Z, x}$ is equal to $\kappa (x)$ and $\kappa (x)$ is separable over $k$. Thus $Z \to \mathop{\mathrm{Spec}}(k)$ is unramified (Morphisms, Lemma 29.35.11). Finally, Lemma 53.19.7 via part (3) of Lemma 53.19.3, shows that $\mathcal{O}_{X, x}$ is a complete intersection in the sense of Divided Power Algebra, Definition 23.8.5. However, Divided Power Algebra, Lemma 23.8.8 and Morphisms, Lemma 29.30.9 show that this agrees with the notion used to define a local complete intersection scheme over a field and the proof is complete. $\square$

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