The Stacks project

Remark 105.2.9 (Upgrading versal rings). In Situation 105.2.1 let $x_0 : \mathop{\mathrm{Spec}}(k) \to \mathcal{X}$ be a morphism, where $k$ is a finite type field over $S$. Let $A$ be a versal ring to $\mathcal{X}$ at $x_0$. By Artin's Axioms, Lemma 96.9.5 our versal formal object in fact comes from a morphism

\[ \mathop{\mathrm{Spec}}(A) \longrightarrow \mathcal{X} \]

over $S$. Moreover, the results above each can be upgraded to be compatible with this morphism. Here is a list:

  1. in Lemma 105.2.4 the isomorphism $A \cong A'[[t_1, \ldots , t_ r]]$ or $A' \cong A[[t_1, \ldots , t_ r]]$ may be chosen compatible with these morphisms,

  2. in Lemma 105.2.5 the homomorphism $A \to A'$ may be chosen compatible with these morphisms,

  3. in Lemma 105.2.6 the morphism $\mathop{\mathrm{Spec}}(\mathcal{O}_{U, u_0}^\wedge ) \to \mathcal{X}$ is the composition of the canonical map $\mathop{\mathrm{Spec}}(\mathcal{O}_{U, u_0}^\wedge ) \to U$ and the given map $U \to \mathcal{X}$,

  4. in Lemma 105.2.8 the isomorphism $\mathcal{O}_{U, u_0}^\wedge \cong A$ may be chosen so $\mathop{\mathrm{Spec}}(A) \to \mathcal{X}$ corresponds to the canonical map in the item above.

In each case the statement follows from the fact that our maps are compatible with versal formal elements; we note however that the implied diagrams are $2$-commutative only up to a (noncanonical) choice of a $2$-arrow. Still, this means that the implied map $A' \to A$ or $A \to A'$ in (1) is well defined up to formal homotopy, see Formal Deformation Theory, Lemma 88.28.3.


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