The Stacks project

Lemma 76.15.11. Taking the universal first order thickenings commutes with étale localization. More precisely, let $h : Z \to X$ be a formally unramified morphism of algebraic spaces over a base scheme $S$. Let

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

be a commutative diagram with étale vertical arrows. Let $Z'$ be the universal first order thickening of $Z$ over $X$. Then $V \to U$ is formally unramified and the universal first order thickening $V'$ of $V$ over $U$ is étale over $Z'$. In particular, $\mathcal{C}_{Z/X}|_ V = \mathcal{C}_{V/U}$.

Proof. The first statement is Lemma 76.14.2. The compatibility of universal first order thickenings is a consequence of Lemmas 76.15.2 and 76.15.3. $\square$

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