Lemma 29.35.17 (04HB)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.35: Unramified morphisms
Lemma 29.35.17 (cite)

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Lemma 29.35.17. Let $S$ be a scheme. Let $X$, $Y$ be schemes over $S$. Let $f, g : X \to Y$ be morphisms over $S$. Let $x \in X$. Assume that

the structure morphism $Y \to S$ is unramified,
$f(x) = g(x)$ in $Y$, say $y = f(x) = g(x)$, and
the induced maps $f^\sharp , g^\sharp : \kappa (y) \to \kappa (x)$ are equal.

Then there exists an open neighbourhood of $x$ in $X$ on which $f$ and $g$ are equal.

Proof. Consider the morphism $(f, g) : X \to Y \times _ S Y$. By assumption (1) and Lemma 29.35.13 the inverse image of $\Delta _{Y/S}(Y)$ is open in $X$. And assumptions (2) and (3) imply that $x$ is in this open subset. $\square$

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