Lemma 29.31.4 (0473)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.31: Conormal sheaf of an immersion
Lemma 29.31.4 (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 29.31.3 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 $I'/(I')^2 \otimes _{R'} R \to I/I^2$ is surjective. If $R' \to R$ is flat, then $I = I' \otimes _{R'} R$ and $I^2 = (I')^2 \otimes _{R'} R$ and we see the map is an isomorphism. $\square$

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