Lemma 76.15.10 (060C)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 76: More on Morphisms of Spaces
Section 76.15: Universal first order thickenings
Lemma 76.15.10 (cite)

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Lemma 76.15.10. Let $S$ be a scheme. Let

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

be a fibre product diagram of algebraic spaces over $S$ with $h'$ formally unramified and $g$ flat. In this case the corresponding map $Z' \to W'$ of universal first order thickenings is flat, and $f^*\mathcal{C}_{W/Y} \to \mathcal{C}_{Z/X}$ is an isomorphism.

Proof. Flatness is preserved under base change, see Morphisms of Spaces, Lemma 67.30.4. Hence the first statement follows from the description of $W'$ in Lemma 76.15.9. It is clear that $X \times _ Y W'$ is a first order thickening. It is straightforward to check that it has the universal property because $W'$ has the universal property (by mapping properties of fibre products). See Lemma 76.5.5 for why this implies that the map of conormal sheaves is an isomorphism. $\square$

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