Lemma 98.12.7. Let S be a locally Noetherian scheme. Let p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf} be a category fibred in groupoids. Let \xi = (R, \xi _ n, f_ n) be a formal object of \mathcal{X} with \xi _1 lying over \mathop{\mathrm{Spec}}(k) \to S with image s \in S. Assume
\xi is versal,
\xi is effective,
\mathcal{O}_{S, s} is a G-ring, and
p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf} is limit preserving on objects.
Then there exist a morphism of finite type U \to S, a finite type point u_0 \in U with residue field k, and an object x of \mathcal{X} over U such that x is versal at u_0 and such that x|_{\mathop{\mathrm{Spec}}(\mathcal{O}_{U, u_0}/\mathfrak m_{u_0}^ n)} \cong \xi _ n.
Proof.
Choose an object x_ R of \mathcal{X} lying over \mathop{\mathrm{Spec}}(R) whose associated formal object is \xi . Let N = 2 and apply Lemma 98.10.1. We obtain A, \mathfrak m_ A, x_ A, \ldots . Let \eta = (A^\wedge , \eta _ n, g_ n) be the formal object associated to x_ A|_{\mathop{\mathrm{Spec}}(A^\wedge )}. We have a diagram
\vcenter { \xymatrix{ & \eta \ar[d] \\ \xi \ar[r] \ar@{..>}[ru] & \xi _2 = \eta _2 } } \quad \text{lying over}\quad \vcenter { \xymatrix{ & A^\wedge \ar[d] \\ R \ar[r] \ar@{..>}[ru] & R/\mathfrak m_ R^2 = A/\mathfrak m_ A^2 } }
The versality of \xi means exactly that we can find the dotted arrows in the diagrams, because we can successively find morphisms \xi \to \eta _3, \xi \to \eta _4, and so on by Formal Deformation Theory, Remark 90.8.10. The corresponding ring map R \to A^\wedge is surjective by Formal Deformation Theory, Lemma 90.4.2. On the other hand, we have \dim _ k \mathfrak m_ R^ n/\mathfrak m_ R^{n + 1} = \dim _ k \mathfrak m_ A^ n/\mathfrak m_ A^{n + 1} for all n by construction. Hence R/\mathfrak m_ R^ n and A/\mathfrak m_ A^ n have the same (finite) length as \Lambda -modules by additivity of length and Formal Deformation Theory, Lemma 90.3.4. It follows that R/\mathfrak m_ R^ n \to A/\mathfrak m_ A^ n is an isomorphism for all n, hence R \to A^\wedge is an isomorphism. Thus \eta is isomorphic to a versal object, hence versal itself. By Lemma 98.12.3 we conclude that x_ A is versal at the point u_0 of U = \mathop{\mathrm{Spec}}(A) corresponding to \mathfrak m_ A.
\square
Comments (0)