The Stacks project

Lemma 88.14.3. Let $\mathcal{F}$ be a category cofibred in groupoids over $\mathcal{C}_\Lambda $ which has (S1). Let $\xi $ be a versal formal object of $\mathcal{F}$ lying over $R$. Let $\xi ' \to \xi $ be a morphism of formal objects lying over $R' \subset R$ as constructed in Lemma 88.14.2. Then

\[ R \cong R'[[x_1, \ldots , x_ r]] \]

is a power series ring over $R'$. Moreover, $\xi '$ is a versal formal object too.

Proof. By Lemma 88.8.11 there exists a morphism $\xi \to \xi '$. By Lemma 88.14.2 the corresponding map $f : R \to R'$ induces a surjection $f|_{R'} : R' \to R'$. This is an isomorphism by Algebra, Lemma 10.31.10. Hence $I = \mathop{\mathrm{Ker}}(f)$ is an ideal of $R$ such that $R = R' \oplus I$. Let $x_1, \ldots , x_ n \in I$ be elements which form a basis for $I/\mathfrak m_ RI$. Consider the map $S = R'[[X_1, \ldots , X_ r]] \to R$ mapping $X_ i$ to $x_ i$. For every $n \geq 1$ we get a surjection of Artinian $R'$-algebras $B = S/\mathfrak m_ S^ n \to R/\mathfrak m_ R^ n = A$. Denote $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(B)$, resp. $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(A))$ the pushforward of $\xi '$ along $R' \to S \to B$, resp. $R' \to S \to A$. Note that $x$ is also the pushforward of $\xi $ along $R \to A$ as $\xi $ is the pushforward of $\xi '$ along $R' \to R$. Thus we have a solid diagram

\[ \vcenter { \xymatrix{ & y \ar[d] \\ \xi \ar[r] \ar@{..>}[ru] & x } } \quad \text{lying over}\quad \vcenter { \xymatrix{ & S/\mathfrak m_ S^ n \ar[d] \\ R \ar[r] \ar@{..>}[ru] & R/\mathfrak m_ R^ n } } \]

Because $\xi $ is versal, using Remark 88.8.10 we obtain the dotted arrows fitting into these diagrams. In particular, the maps $S/\mathfrak m_ S^ n \to R/\mathfrak m_ R^ n$ have sections $h_ n : R/\mathfrak m_ R^ n \to S/\mathfrak m_ S^ n$. It follows from Lemma 88.4.9 that $S \to R$ is an isomorphism.

As $\xi $ is a pushforward of $\xi '$ along $R' \to R$ we obtain from Remark 88.7.13 a commutative diagram

\[ \xymatrix{ \underline{R}|_{\mathcal{C}_\Lambda } \ar[rr] \ar[rd]_{\underline{\xi }} & & \underline{R'}|_{\mathcal{C}_\Lambda } \ar[ld]^{\underline{\xi '}} \\ & \mathcal{F} } \]

Since $R' \to R$ has a left inverse (namely $R \to R/I = R'$) we see that $\underline{R}|_{\mathcal{C}_\Lambda } \to \underline{R'}|_{\mathcal{C}_\Lambda }$ is essentially surjective. Hence by Lemma 88.8.7 we see that $\underline{\xi '}$ is smooth, i.e., $\xi '$ is a versal formal object. $\square$


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