The Stacks project

Lemma 76.6.4. Let $S$ be a scheme. Let

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

be a cartesian square of algebraic spaces over $S$ with $i$, $i'$ immersions. Then the canonical map $f^*\mathcal{C}_{Z'/X', *} \to \mathcal{C}_{Z/X, *}$ of Lemma 76.6.3 is surjective. If $g$ is flat, then it is an isomorphism.

Proof. We may check the statement after ├ętale localizing $X'$. In this case we may assume $X' \to X$ is a morphism of schemes, hence $Z$ and $Z'$ are schemes and the result follows from the case of schemes, see Divisors, Lemma 31.19.4. $\square$


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