The Stacks project

Lemma 86.11.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of locally adic* formal algebraic spaces over $S$. The following are equivalent

  1. 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 ├ętale, the morphism $U \to V$ corresponds to an arrow of $\textit{WAdm}^{adic*}$ which is adic and topologically of finite type,

  2. 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 an arrow of $\textit{WAdm}^{adic*}$ which is adic and topologically of finite type,

  3. 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 ├ętale, and $X_ i \to Y_ i$ corresponds to an arrow of $\textit{WAdm}^{adic*}$ which is adic and topologically of finite type, and

  4. $f$ is locally of finite type.

Proof. Immediate consequence of the equivalence of (1) and (2) in Lemma 86.11.1 and Formal Spaces, Lemma 85.25.9. $\square$


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