Lemma 37.9.2 (02H5)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 37: More on Morphisms
Section 37.9: Infinitesimal deformations of maps
Lemma 37.9.2 (cite)

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Lemma 37.9.2. Let $S$ be a scheme. Let $(a, a') : (X \subset X') \to (Y \subset Y')$ be a morphism of first order thickenings over $S$ . Let

$\theta : a^*\Omega _{Y/S} \to \mathcal{C}_{X/X'}$

be an $\mathcal{O}_ X$ -linear map. Then there exists a unique morphism of pairs $(b, b') : (X \subset X') \to (Y \subset Y')$ such that (1) and (2) of Lemma 37.9.1 hold and the derivation $D$ and $\theta$ are related by Equation (37.9.1.1).

Proof. We simply set $b = a$ and we define $(b')^\sharp$ to be the map

$(a')^\sharp + D : a^{-1}\mathcal{O}_{Y'} \to \mathcal{O}_{X'}$

where $D$ is as in Equation (37.9.1.1). We omit the verification that $(b')^\sharp$ is a map of sheaves of $\mathcal{O}_ S$ -algebras and that (1) and (2) of Lemma 37.9.1 hold. Equation (37.9.1.1) holds by construction. $\square$

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