The Stacks project

Lemma 90.24.2. Let $(U, R, s, t, c)$ be a smooth groupoid in functors on $\mathcal{C}_\Lambda $. Assume $U$ and $R$ are deformation functors. Then:

  1. The quotient $[U/R]$ is a deformation category.

  2. The tangent space of $[U/R]$ is

    \[ T[U/R] = \mathop{\mathrm{Coker}}(ds-dt: TR \to TU). \]

  3. The space of infinitesimal automorphisms of $[U/R]$ is

    \[ \text{Inf}([U/R]) = \mathop{\mathrm{Ker}}(ds \oplus dt : TR \to TU \oplus TU). \]

Proof. Since $U$ and $R$ are deformation functors $[U/R]$ is a predeformation category. Since (RS) holds for deformation functors by definition we see that (RS) holds for [U/R] by Lemma 90.24.1. Hence $[U/R]$ is a deformation category. Statements (2) and (3) follow directly from the definitions. $\square$

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