The Stacks project

Lemma 86.12.6. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of locally countably indexed 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 étale, the morphism $U \to V$ corresponds to a morphism of $\textit{WAdm}^{count}$ satisfying the property defined in Lemma 86.12.1,

  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 a morphism of $\textit{WAdm}^{count}$ satisfying the property defined in Lemma 86.12.1,

  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 étale, and $X_ i \to Y_ i$ corresponds to a morphism of $\textit{WAdm}^{count}$ satisfying the property defined in Lemma 86.12.1, and

  4. the morphism $f_{red} : X_{red} \to Y_{red}$ is locally of finite type.

Proof. The equivalence of (1), (2), and (3) follows from Lemma 86.12.1 and an application of Formal Spaces, Lemma 85.17.3. Let $Y_ j$ and $X_{ji}$ be as in (2). Then

  • The families $\{ Y_{j, red} \to Y_{red}\} $ and $\{ X_{ji, red} \to X_{red}\} $ are étale coverings by affine schemes. This follows from the discussion in the proof of Formal Spaces, Lemma 85.8.1 or directly from Formal Spaces, Lemma 85.8.3.

  • If $X_{ji} \to Y_ j$ corresponds to the morphism $B_ j \to A_{ji}$ of $\textit{WAdm}^{count}$, then $X_{ji, red} \to Y_{j, red}$ corresponds to the ring map $B_ j/\mathfrak b_ j \to A_{ji}/\mathfrak a_{ji}$ where $\mathfrak b_ j$ and $\mathfrak a_{ji}$ are the ideals of topologically nilpotent elements. This follows from Formal Spaces, Example 85.8.2. Hence $X_{ji, red} \to Y_{j, red}$ is locally of finite type if and only if $B_ j \to A_{ji}$ satisfies the property defined in Lemma 86.12.1.

The equivalence of (2) and (4) follows from these remarks because being locally of finite type is a property of morphisms of algebraic spaces which is étale local on source and target, see discussion in Morphisms of Spaces, Section 65.23. $\square$


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