The Stacks project

Lemma 90.29.6. With notation and assumptions as in Situation 90.29.1. Let $\xi $ be a versal formal object for $\mathcal{F}$ lying over $R \in \mathop{\mathrm{Ob}}\nolimits (\widehat{\mathcal{C}}_{\Lambda , k})$. Then there exist

  1. an $S \in \mathop{\mathrm{Ob}}\nolimits (\widehat{\mathcal{C}}_{\Lambda , l})$ and a local $\Lambda $-algebra homomorphism $R \to S$ which is formally smooth in the $\mathfrak m_ S$-adic topology and induces the given field extension $l/k$ on residue fieds, and

  2. a versal formal object of $\mathcal{F}_{l/k}$ lying over $S$.

Proof. Construction of $S$. Choose a surjection $R[x_1, \ldots , x_ n] \to l$ of $R$-algebras. The kernel is a maximal ideal $\mathfrak m$. Set $S$ equal to the $\mathfrak m$-adic completion of the Noetherian ring $R[x_1, \ldots , x_ n]$. Then $S$ is in $\widehat{\mathcal{C}}_{\Lambda , l}$ by Algebra, Lemma 10.97.6. The map $R \to S$ is formally smooth in the $\mathfrak m_ S$-adic topology by More on Algebra, Lemmas 15.37.2 and 15.37.4 and the fact that $R \to R[x_1, \ldots , x_ n]$ is formally smooth. (Compare with the proof Lemma 90.9.5.)

Since $\xi $ is versal, the transformation $\underline{\xi } : \underline{R}|_{\mathcal{C}_{\Lambda , k}} \to \mathcal{F}$ is smooth. By Lemma 90.29.5 the induced map

\[ (\underline{R}|_{\mathcal{C}_{\Lambda , k}})_{l/k} \longrightarrow \mathcal{F}_{l/k} \]

is smooth. Thus it suffices to construct a smooth morphism $\underline{S}|_{\mathcal{C}_{\Lambda , l}} \to (\underline{R}|_{\mathcal{C}_{\Lambda , k}})_{l/k}$. To give such a map means for every object $B$ of $\mathcal{C}_{\Lambda , l}$ a map of sets

\[ \mathop{\mathrm{Mor}}\nolimits _{\widehat{\mathcal{C}}_{\Lambda , l}}(S, B) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _{\widehat{\mathcal{C}}_{\Lambda , k}}(R, B \times _ l k) \]

functorial in $B$. Given an element $\varphi : S \to B$ on the left hand side we send it to the composition $R \to S \to B$ whose image is contained in the sub $\Lambda $-algebra $B \times _ l k$. Smoothness of the map means that given a surjection $B' \to B$ and a commutative diagram

\[ \xymatrix{ S \ar[r] & B \ar@{=}[r] & B \\ R \ar[u] \ar[r] & B' \times _ l k \ar[u] \ar[r] & B' \ar[u] } \]

we have to find a ring map $S \to B'$ fitting into the outer rectangle. The existence of this map is guaranteed as we chose $R \to S$ to be formally smooth in the $\mathfrak m_ S$-adic topology, see More on Algebra, Lemma 15.37.5. $\square$

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