The Stacks project

Lemma 90.28.3. Let $\mathcal{F}$ be a deformation category over $\mathcal{C}_\Lambda $ with $\dim _ k T\mathcal{F} < \infty $ and $\dim _ k \text{Inf}(\mathcal{F}) < \infty $. Let $\xi $ be a versal formal object lying over $R$. Let $\eta $ be a formal object lying over $S$. Then any two maps

\[ f, g : R \to S \]

such that $f_*\xi \cong \eta \cong g_*\xi $ are formally homotopic.

Proof. By Theorem 90.26.4 and its proof, $\mathcal{F}$ has a presentation by a smooth prorepresentable groupoid

\[ (\underline{R}, \underline{R_1}, s, t, c, e, i)|_{\mathcal{C}_\Lambda } \]

in functors on $\mathcal{C}_\lambda $ such that $\mathcal{F}$. Then the maps $s : R \to R_1$ and $t : R \to R_1$ are formally smooth ring maps and $e : R_1 \to R$ is a section. In particular, we can choose an isomorphism $R_1 = R[[t_1, \ldots , t_ r]]$ for some $r \geq 0$ such that $s$ is the embedding $R \subset R[[t_1, \ldots , t_ r]]$ and $t$ corresponds to a map $h : R \to R[[t_1, \ldots , t_ r]]$ with $h(a) \bmod (t_1, \ldots , t_ r) = a$ for all $a \in R$. The existence of the isomorphism $\alpha : f_*\xi \to g_*\xi $ means exactly that there is a map $k : R_1 \to S$ such that $f = k \circ s$ and $g = k \circ t$. This exactly means that $(r, h, k)$ is a formal homotopy between $f$ and $g$. $\square$

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