The Stacks project

Lemma 115.18.1. 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 $u_ n \in U$, $n \geq 1$ be pairwise distinct finite type points such that $x$ is not versal at $u_ n$ for all $n$. After replacing $u_ n$ by a subsequence, there exist morphisms

over $S$ such that

1. for each $n$ the morphism $U \to U_ n$ is a first order thickening,

2. for each $n$ we have a short exact sequence

   $$0 \to \kappa (u_ n) \to \mathcal{O}_{U_ n} \to \mathcal{O}_{U_{n - 1}} \to 0$$

   with $U_0 = U$ for $n = 1$,

3. for each $n$ there does not exist a pair $(W, \alpha )$ consisting of an open neighbourhood $W \subset U_ n$ of $u_ n$ and a morphism $\alpha : x_ n|_ W \to x$ such that the composition

   $$x|_{U \cap W} \xrightarrow {\text{restriction of }x \to x_ n} x_ n|_ W \xrightarrow {\alpha } x$$

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