Definition 97.9.1. Let $S$ be a locally Noetherian scheme. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids.

A

*formal object*$\xi = (R, \xi _ n, f_ n)$ of $\mathcal{X}$ consists of a Noetherian complete local $S$-algebra $R$, objects $\xi _ n$ of $\mathcal{X}$ lying over $\mathop{\mathrm{Spec}}(R/\mathfrak m_ R^ n)$, and morphisms $f_ n : \xi _ n \to \xi _{n + 1}$ of $\mathcal{X}$ lying over $\mathop{\mathrm{Spec}}(R/\mathfrak m^ n) \to \mathop{\mathrm{Spec}}(R/\mathfrak m^{n + 1})$ such that $R/\mathfrak m$ is a field of finite type over $S$.A

*morphism of formal objects*$a : \xi = (R, \xi _ n, f_ n) \to \eta = (T, \eta _ n, g_ n)$ is given by morphisms $a_ n : \xi _ n \to \eta _ n$ such that for every $n$ the diagram\[ \xymatrix{ \xi _ n \ar[r]_{f_ n} \ar[d]_{a_ n} & \xi _{n + 1} \ar[d]^{a_{n + 1}} \\ \eta _ n \ar[r]^{g_ n} & \eta _{n + 1} } \]is commutative. Applying the functor $p$ we obtain a compatible collection of morphisms $\mathop{\mathrm{Spec}}(R/\mathfrak m_ R^ n) \to \mathop{\mathrm{Spec}}(T/\mathfrak m_ T^ n)$ and hence a morphism $a_0 : \mathop{\mathrm{Spec}}(R) \to \mathop{\mathrm{Spec}}(T)$ over $S$. We say that $a$

*lies over*$a_0$.

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