The Stacks project

Lemma 113.15.1. 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

\[ x \to x_1 \to x_2 \to \ldots \quad \text{in }\mathcal{X}\text{ lying over }\quad U \to U_1 \to U_2 \to \ldots \]

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 $x|_{U \cap W} \to x$.

Proof. This lemma was originally used in the proof of a criterion for openness of versality (Artin's Axioms, Lemma 96.20.3) but it got replaced by Artin's Axioms, Lemma 96.20.1 from which it readily follows. Namely, after replacing $u_ n$, $n \geq 1$ by a subsequence we may and do assume that there are no specializations among these points, see Properties, Lemma 28.5.11. Then we can apply Artin's Axioms, Lemma 96.20.1 to finish the proof. $\square$


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