The Stacks project

Remark 89.8.10. Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal C_\Lambda $, and let $\xi $ be a formal object of $\mathcal{F}$. It follows from the definition of smoothness that versality of $\xi $ is equivalent to the following condition: If

\[ \xymatrix{ & y \ar[d] \\ \xi \ar[r] & x } \]

is a diagram in $\widehat{\mathcal{F}}$ such that $y \to x$ lies over a surjective map $B \to A$ of Artinian rings (we may assume it is a small extension), then there exists a morphism $\xi \to y$ such that

\[ \xymatrix{ & y \ar[d] \\ \xi \ar[r] \ar[ur] & x } \]

commutes. In particular, the condition that $\xi $ be versal does not depend on the choices of pushforwards made in the construction of $\underline{\xi } : \underline{R}|_{\mathcal{C}_\Lambda } \to \mathcal{F}$ in Remark 89.7.12.

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