Proof.
Let R = \mathcal{O}_{U, u}^\wedge . Let k = \kappa (u) be the residue field of R. Let \xi be the formal object of \mathcal{X} over R associated to x. Since x is not versal at u, we see that \xi is not versal, see Lemma 98.12.3. By the discussion following Definition 98.12.1 this means we can find morphisms \xi _1 \to x_ A \to x_ B of \mathcal{X} lying over closed immersions \mathop{\mathrm{Spec}}(k) \to \mathop{\mathrm{Spec}}(A) \to \mathop{\mathrm{Spec}}(B) where A, B are Artinian local rings with residue field k, an n \geq 1 and a commutative diagram
\vcenter { \xymatrix{ & x_ A \ar[ld] \\ \xi _ n & \xi _1 \ar[u] \ar[l] } } \quad \text{lying over}\quad \vcenter { \xymatrix{ & \mathop{\mathrm{Spec}}(A) \ar[ld] \\ \mathop{\mathrm{Spec}}(R/\mathfrak m^ n) & \mathop{\mathrm{Spec}}(k) \ar[u] \ar[l] } }
such that there does not exist an m \geq n and a commutative diagram
\vcenter { \xymatrix{ & & x_ B \ar[lldd] \\ & & x_ A \ar[ld] \ar[u] \\ \xi _ m & \xi _ n \ar[l] & \xi _1 \ar[u] \ar[l] } } \quad \text{lying over} \vcenter { \xymatrix{ & & \mathop{\mathrm{Spec}}(B) \ar[lldd] \\ & & \mathop{\mathrm{Spec}}(A) \ar[ld] \ar[u] \\ \mathop{\mathrm{Spec}}(R/\mathfrak m^ m) & \mathop{\mathrm{Spec}}(R/\mathfrak m^ n) \ar[l] & \mathop{\mathrm{Spec}}(k) \ar[u] \ar[l] } }
We may moreover assume that B \to A is a small extension, i.e., that the kernel I of the surjection B \to A is isomorphic to k as an A-module. This follows from Formal Deformation Theory, Remark 90.8.10. Then we simply define
T = U \amalg _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(B)
By property (RS*) we find y over T whose restriction to \mathop{\mathrm{Spec}}(B) is x_ B and whose restriction to U is x (this gives the arrow x \to y lying over U \to T). To finish the proof we verify conditions (1), (2), and (3).
By the construction of the pushout we have a commutative diagram
\xymatrix{ 0 \ar[r] & I \ar[r] & B \ar[r] & A \ar[r] & 0 \\ 0 \ar[r] & I \ar[r] \ar[u] & \Gamma (T, \mathcal{O}_ T) \ar[r] \ar[u] & \Gamma (U, \mathcal{O}_ U) \ar[r] \ar[u] & 0 }
with exact rows. This immediately proves (1) and (2). To finish the proof we will argue by contradiction. Assume we have a pair (W, \beta ) as in (3). Since \mathop{\mathrm{Spec}}(B) \to T factors through W we get the morphism
x_ B \to y|_ W \xrightarrow {\beta } x
Since B is Artinian local with residue field k = \kappa (u) we see that x_ B \to x lies over a morphism \mathop{\mathrm{Spec}}(B) \to U which factors through \mathop{\mathrm{Spec}}(\mathcal{O}_{U, u}/\mathfrak m_ u^ m) for some m \geq n. In other words, x_ B \to x factors through \xi _ m giving a map x_ B \to \xi _ m. The compatibility condition on the morphism \alpha in condition (3) translates into the condition that
\xymatrix{ x_ B \ar[d] & x_ A \ar[d] \ar[l] \\ \xi _ m & \xi _ n \ar[l] }
is commutative. This gives the contradiction we were looking for.
\square
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