The Stacks project

Lemma 86.17.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of locally Noetherian formal algebraic spaces over $S$. The following are equivalent

  1. $f$ is rig-étale,

  2. for every commutative diagram

    \[ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } \]

    with $U$ and $V$ affine formal algebraic spaces, $U \to X$ and $V \to Y$ representable by algebraic spaces and étale, the morphism $U \to V$ corresponds to a rig-étale map in $\textit{WAdm}^{Noeth}$,

  3. there exists a covering $\{ Y_ j \to Y\} $ as in Formal Spaces, Definition 85.7.1 and for each $j$ a covering $\{ X_{ji} \to Y_ j \times _ Y X\} $ as in Formal Spaces, Definition 85.7.1 such that each $X_{ji} \to Y_ j$ corresponds to a rig-étale map in $\textit{WAdm}^{Noeth}$, and

  4. there exist a covering $\{ X_ i \to X\} $ as in Formal Spaces, Definition 85.7.1 and for each $i$ a factorization $X_ i \to Y_ i \to Y$ where $Y_ i$ is an affine formal algebraic space, $Y_ i \to Y$ is representable by algebraic spaces and étale, and $X_ i \to Y_ i$ corresponds to a rig-étale map in $\textit{WAdm}^{Noeth}$.

Proof. The equivalence of (1) and (2) is Definition 86.17.1. The equivalence of (2), (3), and (4) follows from the fact that being rig-étale is a local property of arrows of $\text{WAdm}^{Noeth}$ by Lemma 86.16.3 and an application of the variant of Formal Spaces, Lemma 85.17.3 for morphisms between locally Noetherian algebraic spaces mentioned in Formal Spaces, Remark 85.17.5. $\square$


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