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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