Lemma 93.9.7 (0DYX)—The Stacks project

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Table of contents
Part 6: Deformation Theory
Chapter 93: Deformation Problems
Section 93.9: Schemes
Lemma 93.9.7 (cite)

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Lemma 93.9.7. In Example 93.9.1 let $X = \mathop{\mathrm{Spec}}(P)$ be an affine scheme over $k$ . With $\mathcal{D}\! \mathit{ef}_ P$ as in Example 93.8.1 there is a natural equivalence

$\mathcal{D}\! \mathit{ef}_ X \longrightarrow \mathcal{D}\! \mathit{ef}_ P$

of deformation categories.

Proof. The functor sends $(A, Y)$ to $\Gamma (Y, \mathcal{O}_ Y)$ . This works because any deformation of $X$ is affine by More on Morphisms, Lemma 37.2.3. $\square$

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