Lemma 37.9.2 (02H5)—The Stacks project

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

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. $\square$

The Stacks project

Comments (0)

Post a comment