The Stacks project

Lemma 89.27.2. Let $(U, R, s, t, c)$ be a smooth prorepresentable groupoid in functors on $\mathcal{C}_\Lambda $.

  1. $(U, R, s, t, c)$ is normalized if and only if the morphism $U \to [U/R]$ induces an isomorphism on tangent spaces, and

  2. $(U, R, s, t, c)$ is minimal if and only if the kernel of $TU \to T[U/R]$ is contained in the image of $\text{Der}_\Lambda (k, k) \to TU$.

Proof. Part (1) follows immediately from the definitions. To see part (2) set $\mathcal{F} = [U/R]$. Since $\mathcal{F}$ has a presentation it is a deformation category, see Theorem 89.26.4. In particular it satisfies (RS), (S1), and (S2), see Lemma 89.16.6. Recall that minimal versal formal objects are unique up to isomorphism, see Lemma 89.14.5. By Theorem 89.15.5 a minimal versal object induces a map $\underline{\xi } : \underline{R}|_{\mathcal{C}_\Lambda } \to \mathcal{F}$ satisfying (89.15.0.2). Since $U \cong \underline{R}|_{\mathcal{C}_\Lambda }$ over $\mathcal{F}$ we see that $TU \to T\mathcal{F} = T[U/R]$ satisfies the property as stated in the lemma. $\square$


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