The Stacks project

Lemma 76.5.5. 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 fibre product diagram of algebraic spaces over $S$. Assume $i$, $i'$ immersions. Then the canonical map $f^*\mathcal{C}_{Z'/X'} \to \mathcal{C}_{Z/X}$ of Lemma 76.5.3 is surjective. If $g$ is flat, then it is an isomorphism.

Proof. Choose a commutative diagram

\[ \xymatrix{ U \ar[r] \ar[d] & X \ar[d] \\ U' \ar[r] & X' } \]

where $U$, $U'$ are schemes and the horizontal arrows are surjective and étale, see Spaces, Lemma 65.11.6. Then using Lemmas 76.5.2 and 76.5.4 we see that the question reduces to the case of a morphism of schemes. In the schemes case this is Morphisms, Lemma 29.31.4. $\square$

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