Lemma 31.19.4 (0635)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 31: Divisors
Section 31.19: The normal cone of an immersion
Lemma 31.19.4 (cite)

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

be a fibre product diagram in the category of schemes with $i$ , $i'$ immersions. Then the canonical map $f^*\mathcal{C}_{Z'/X', *} \to \mathcal{C}_{Z/X, *}$ of Lemma 31.19.3 is surjective. If $g$ is flat, then it is an isomorphism.

Proof. Let $R' \to R$ be a ring map, and $I' \subset R'$ an ideal. Set $I = I'R$ . Then $(I')^ n/(I')^{n + 1} \otimes _{R'} R \to I^ n/I^{n + 1}$ is surjective. If $R' \to R$ is flat, then $I^ n = (I')^ n \otimes _{R'} R$ and we see the map is an isomorphism. $\square$

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