Lemma 33.41.2. Let $X$ be a reduced Nagata scheme of dimension $1$. Let $\nu : X^\nu \to X$ be the normalization. Let $x \in X$ denote a closed point. Then

$\nu : X^\nu \to X$ is finite, surjective, and birational,

$\mathcal{O}_ X \subset \nu _*\mathcal{O}_{X^\nu }$ and $\nu _*\mathcal{O}_{X^\nu }/\mathcal{O}_ X$ is a direct sum of skyscraper sheaves $\mathcal{Q}_ x$ in the singular points $x$ of $X$,

$A' = (\nu _*\mathcal{O}_{X^\nu })_ x$ is the integral closure of $A = \mathcal{O}_{X, x}$ in its total ring of fractions,

$\mathcal{Q}_ x = A'/A$ has finite length equal to the $\delta $-invariant of $X$ at $x$,

$A'$ is a semi-local ring which is a finite product of Dedekind domains,

$A^\wedge $ is a reduced Noetherian complete local ring of dimension $1$,

$(A')^\wedge $ is the integral closure of $A^\wedge $ in its total ring of fractions,

$(A')^\wedge $ is a finite product of complete discrete valuation rings, and

$A'/A \cong (A')^\wedge /A^\wedge $.

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