The Stacks project

Lemma 58.20.1. Let $(A, \mathfrak m)$ be a Noetherian local ring. Set $X = \mathop{\mathrm{Spec}}(A)$ and let $U = X \setminus \{ \mathfrak m\} $. Let $\pi : Y \to X$ be a finite morphism such that $\text{depth}(\mathcal{O}_{Y, y}) \geq 2$ for all closed points $y \in Y$. Then $Y$ is the spectrum of $B = \mathcal{O}_ Y(\pi ^{-1}(U))$.

Proof. Set $V = \pi ^{-1}(U)$ and denote $\pi ' : V \to U$ the restriction of $\pi $. Consider the $\mathcal{O}_ X$-module map

\[ \pi _*\mathcal{O}_ Y \longrightarrow j_*\pi '_*\mathcal{O}_ V \]

where $j : U \to X$ is the inclusion morphism. We claim Divisors, Lemma 31.5.11 applies to this map. If so, then $B = \Gamma (Y, \mathcal{O}_ Y)$ and we see that the lemma holds. Let $x \in X$ be the closed point. It suffices to show that $\text{depth}((\pi _*\mathcal{O}_ Y)_ x) \geq 2$. Let $y_1, \ldots , y_ n \in Y$ be the points mapping to $x$. By Algebra, Lemma 10.72.11 it suffices to show that $\text{depth}(\mathcal{O}_{Y, y_ i}) \geq 2$ for $i = 1, \ldots , n$. Since this is the assumption of the lemma the proof is complete. $\square$

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