Lemma 53.19.11. Let k be a field. Let X be a 1-dimensional algebraic k-scheme. Let x \in X be a closed point. The following are equivalent
x is a split node,
x is a node and there are exactly two points x_1, x_2 of the normalization X^\nu lying over x with k = \kappa (x_1) = \kappa (x_2),
\mathcal{O}_{X, x}^\wedge \cong k[[x, y]]/(xy) as a k-algebra, and
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