Lemma 37.7.8. Taking the universal first order thickenings commutes with taking opens. More precisely, let $h : Z \to X$ be a formally unramified morphism of schemes. Let $V \subset Z$, $U \subset X$ be opens such that $h(V) \subset U$. Let $Z'$ be the universal first order thickening of $Z$ over $X$. Then $h|_ V : V \to U$ is formally unramified and the universal first order thickening of $V$ over $U$ is the open subscheme $V' \subset Z'$ such that $V = Z \cap V'$. In particular, $\mathcal{C}_{Z/X}|_ V = \mathcal{C}_{V/U}$.

Proof. The first statement is Lemma 37.6.5. The compatibility of universal thickenings can be deduced from the proof of Lemma 37.7.1, or from Algebra, Lemma 10.149.4 or deduced from Lemma 37.7.7. $\square$

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