Lemma 54.16.2 (0C5K)—The Stacks project

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Part 3: Topics in Scheme Theory
Chapter 54: Resolution of Surfaces
Section 54.16: Contracting exceptional curves
Lemma 54.16.2 (cite)

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Lemma 54.16.2. Let $X$ be a Noetherian scheme. Let $E \subset X$ be an exceptional curve of the first kind. If there exists a contraction of $E$, then it is unique up to unique isomorphism.

Proof. This is immediate from the universal property of Lemma 54.16.1. $\square$

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