Lemma 29.56.3 (03GU)—The Stacks project

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Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.56: Zariski's Main Theorem (algebraic version)
Lemma 29.56.3 (cite)

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Lemma 29.56.3. Let $f : Y \to X$ be a morphism of schemes. Assume

$X$ and $Y$ are affine, and
$f$ is quasi-finite.

Then there exists a diagram

\[ \xymatrix{ Y \ar[rd]_ f \ar[rr]_ j & & Z \ar[ld]^\pi \\ & X & } \]

with $Z$ affine, $\pi $ finite and $j$ an open immersion.

Proof. This is Algebra, Lemma 10.123.14 reformulated in the language of schemes. $\square$

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