Remark 53.19.8 (The quadratic extension associated to a node). Let $k$ be a field. Let $(A, \mathfrak m, \kappa )$ be a Noetherian local $k$-algebra. Assume that either $(A, \mathfrak m, \kappa )$ is as in Lemma 53.19.3, or $A$ is Nagata as in Lemma 53.19.4, or $A$ is complete and as in Lemma 53.19.6. Then $A$ defines canonically a degree $2$ separable $\kappa$-algebra $\kappa '$ as follows

1. let $q = ax^2 + bxy + cy^2$ be a nondegenerate quadric as in Lemma 53.19.3 with coordinates $x, y$ chosen such that $a \not= 0$ and set $\kappa ' = \kappa [x]/(ax^2 + bx + c)$,

2. let $A' \supset A$ be the integral closure of $A$ in its total ring of fractions and set $\kappa ' = A'/\mathfrak m A'$, or

3. let $\kappa '$ be the $\kappa$-algebra such that $\text{Proj}(\bigoplus _{n \geq 0} \mathfrak m^ n/\mathfrak m^{n + 1}) = \mathop{\mathrm{Spec}}(\kappa ')$.

The equivalence of (1) and (2) was shown in the proof of Lemma 53.19.4. We omit the equivalence of this with (3). If $X$ is a locally Noetherian $k$-scheme and $x \in X$ is a point such that $\mathcal{O}_{X, x} = A$, then (3) shows that $\mathop{\mathrm{Spec}}(\kappa ') = X^\nu \times _ X \mathop{\mathrm{Spec}}(\kappa )$ where $\nu : X^\nu \to X$ is the normalization morphism.

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