The Stacks project

Lemma 85.16.9. Let $S$ be a scheme. Let $X$ be a locally Noetherian formal algebraic space over $S$. Then $X = \mathop{\mathrm{colim}}\nolimits X_ n$ for a system $X_1 \to X_2 \to X_3 \to \ldots $ of finite order thickenings of locally Noetherian algebraic spaces over $S$ where $X_1 = X_{red}$ and $X_ n$ is the $n$th infinitesimal neighbourhood of $X_1$ in $X_ m$ for all $m \geq n$.

Proof. We only sketch the proof and omit some of the details. Set $X_1 = X_{red}$. Define $X_ n \subset X$ as the subfunctor defined by the rule: a morphism $f : T \to X$ where $T$ is a scheme factors through $X_ n$ if and only if the $n$th power of the ideal sheaf of the closed immersion $X_1 \times _ X T \to T$ is zero. Then $X_ n \subset X$ is a subsheaf as vanishing of quasi-coherent modules can be checked fppf locally. We claim that $X_ n \to X$ is representable by schemes, a closed immersion, and that $X = \mathop{\mathrm{colim}}\nolimits X_ n$ (as fppf sheaves). To check this we may work ├ętale locally on $X$. Hence we may assume $X = \text{Spf}(A)$ is a Noetherian affine formal algebraic space. Then $X_1 = \mathop{\mathrm{Spec}}(A/\mathfrak a)$ where $\mathfrak a \subset A$ is the ideal of topologically nilpotent elements of the Noetherian adic topological ring $A$. Then $X_ n = \mathop{\mathrm{Spec}}(A/\mathfrak a^ n)$ and we obtain what we want. $\square$


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