Lemma 93.9.8 (0DZM)—The Stacks project

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Table of contents
Part 6: Deformation Theory
Chapter 93: Deformation Problems
Section 93.9: Schemes
Lemma 93.9.8 (cite)

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Lemma 93.9.8. In Example 93.9.1 let $X$ be a scheme over $k$ Let $p \in X$ be a point. With $\mathcal{D}\! \mathit{ef}_{\mathcal{O}_{X, p}}$ as in Example 93.8.1 there is a natural functor

$\mathcal{D}\! \mathit{ef}_ X \longrightarrow \mathcal{D}\! \mathit{ef}_{\mathcal{O}_{X, p}}$

of deformation categories.

Proof. Choose an affine open $U = \mathop{\mathrm{Spec}}(P) \subset X$ containing $p$ . Then $\mathcal{O}_{X, p}$ is a localization of $P$ . We combine the functors from Lemmas 93.9.6, 93.9.7, and 93.8.7. $\square$

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