The Stacks project

Lemma 86.19.1. Let $X$ be a locally Noetherian formal algebraic space over a complete discrete valuation ring $A$. Then there exists a closed immersion $X' \to X$ of formal algebraic spaces such that $X'$ is flat over $A$ and such that any morphism $Y \to X$ of locally Noetherian formal algebraic spaces with $Y$ flat over $A$ factors through $X'$.

Proof. Let $\pi \in A$ be the uniformizer. Recall that an $A$-module is flat if and only if the $\pi $-power torsion is $0$.

First assume that $X$ is an affine formal algebraic space. Then $X = \text{Spf}(B)$ with $B$ an adic Noetherian $A$-algebra. In this case we set $X' = \text{Spf}(B')$ where $B' = B/\pi \text{-power torsion}$. It is clear that $X'$ is flat over $A$ and that $X' \to X$ is a closed immersion. Let $g : Y \to X$ be a morphism of locally Noetherian formal algebraic spaces with $Y$ flat over $A$. Choose a covering $\{ Y_ j \to Y\} $ as in Formal Spaces, Definition 85.7.1. Then $Y_ j = \text{Spf}(C_ j)$ with $C_ j$ flat over $A$. Hence the morphism $Y_ j \to X$, which correspond to a continuous $R$-algebra map $B \to C_ j$, factors through $X'$ as clearly $B \to C_ j$ kills the $\pi $-power torsion. Since $\{ Y_ j \to Y\} $ is a covering and since $X' \to X$ is a monomorphism, we conclude that $g$ factors through $X'$.

Let $X$ and $\{ X_ i \to X\} _{i \in I}$ be as in Formal Spaces, Definition 85.7.1. For each $i$ let $X'_ i \to X_ i$ be the flat part as constructed above. For $i, j \in I$ the projection $X'_ i \times _ X X_ j \to X'_ i$ is an étale (by assumption) morphism of schemes (by Formal Spaces, Lemma 85.5.11). Hence $X'_ i \times _ X X_ j$ is flat over $A$ as morphisms representable by algebraic spaces and étale are flat (Lemma 86.13.8). Thus the projection $X'_ i \times _ X X_ j \to X_ j$ factors through $X'_ j$ by the universal property. We conclude that

\[ R_{ij} = X'_ i \times _ X X_ j = X'_ i \times _ X X'_ j = X_ i \times _ X X'_ j \]

because the morphisms $X'_ i \to X_ i$ are injections of sheaves. Set $U = \coprod X'_ i$, set $R = \coprod R_{ij}$, and denote $s, t : R \to U$ the two projections. As a sheaf $R = U \times _ X U$ and $s$ and $t$ are étale. Then $(t, s) : R \to U$ defines an étale equivalence relation by our observations above. Thus $X' = U/R$ is an algebraic space by Spaces, Theorem 63.10.5. By construction the diagram

\[ \xymatrix{ \coprod X'_ i \ar[r] \ar[d] & \coprod X_ i \ar[d] \\ X' \ar[r] & X } \]

is cartesian. Since the right vertical arrow is étale surjective and the top horizontal arrow is representable and a closed immersion we conclude that $X' \to X$ is representable by Bootstrap, Lemma 78.5.2. Then we can use Spaces, Lemma 63.5.6 to conclude that $X' \to X$ is a closed immersion.

Finally, suppose that $Y \to X$ is a morphism with $Y$ a locally Noetherian formal algebraic space flat over $A$. Then each $X_ i \times _ X Y$ is étale over $Y$ and therefore flat over $A$ (see above). Then $X_ i \times _ X Y \to X_ i$ factors through $X'_ i$. Hence $Y \to X$ factors through $X'$ because $\{ X_ i \times _ X Y \to Y\} $ is an étale covering. $\square$


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