Lemma 37.7.7 (04F9)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 37: More on Morphisms
Section 37.7: Universal first order thickenings
Lemma 37.7.7 (cite)

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

be a fibre product diagram in the category of schemes 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, Lemma 29.25.8. Hence the first statement follows from the description of $W'$ in Lemma 37.7.6. 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 Morphisms, Lemma 29.31.4 for why this implies that the map of conormal sheaves is an isomorphism. $\square$

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