The Stacks project

Lemma 90.15.1. Let $\mathcal{F}$ be a predeformation category. Let $\xi $ be a versal formal object of $\mathcal{F}$ such that ( holds. Then $\xi $ is a minimal versal formal object. In particular, such $\xi $ are unique up to isomorphism.

Proof. If $\xi $ is not minimal, then there exists a morphism $\xi ' \to \xi $ lying over $R' \to R$ such that $R = R'[[x_1, \ldots , x_ n]]$ with $n > 0$, see Lemma 90.14.5. Thus $d\underline{\xi }$ factors as

\[ \text{Der}_\Lambda (R, k) \to \text{Der}_\Lambda (R', k) \to T\mathcal{F} \]

and we see that ( cannot hold because $D : f \mapsto \partial /\partial x_1(f) \bmod \mathfrak m_ R$ is an element of the kernel of the first arrow which is not in the image of $\text{Der}_\Lambda (k, k) \to \text{Der}_\Lambda (R, k)$. $\square$

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