Lemma 49.7.2 (0BVX)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 49: Discriminants and Differents
Section 49.7: The Kähler different
Lemma 49.7.2 (cite)

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Lemma 49.7.2. Consider a cartesian diagram of schemes

\[ \xymatrix{ Y' \ar[d]_{f'} \ar[r] & Y \ar[d]^ f \\ X' \ar[r]^ g & X } \]

with $f$ locally of finite type. Let $R \subset Y$, resp. $R' \subset Y'$ be the closed subscheme cut out by the Kähler different of $f$, resp. $f'$. Then $Y' \to Y$ induces an isomorphism $R' \to R \times _ Y Y'$.

Proof. This is true because $\Omega _{Y'/X'}$ is the pullback of $\Omega _{Y/X}$ (Morphisms, Lemma 29.32.10) and then we can apply More on Algebra, Lemma 15.8.4. $\square$

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