The Stacks project

Lemma 96.19.1. Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$ having (RS*). Let $x$ be an object of $\mathcal{X}$ over an affine scheme $U$ of finite type over $S$. Let $u \in U$ be a finite type point such that $x$ is not versal at $u$. Then there exists a morphism $x \to y$ of $\mathcal{X}$ lying over $U \to T$ satisfying

  1. the morphism $U \to T$ is a first order thickening,

  2. we have a short exact sequence

    \[ 0 \to \kappa (u) \to \mathcal{O}_ T \to \mathcal{O}_ U \to 0 \]
  3. there does not exist a pair $(W, \alpha )$ consisting of an open neighbourhood $W \subset T$ of $u$ and a morphism $\beta : y|_ W \to x$ such that the composition

    \[ x|_{U \cap W} \xrightarrow {\text{restriction of }x \to y} y|_ W \xrightarrow {\beta } x \]

    is the canonical morphism $x|_{U \cap W} \to x$.

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 96.12.3. By the discussion following Definition 96.12.1 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 88.8.10. 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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