The Stacks project

Lemma 87.29.8. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of affine formal algebraic spaces. Assume $Y$ countably indexed. The following are equivalent

  1. $f$ is locally of finite type,

  2. $f$ is of finite type,

  3. $f$ corresponds to a morphism $B \to A$ of $\textit{WAdm}^{count}$ (Section 87.21) satisfying the equivalent conditions of Lemma 87.29.6.

Proof. Since $X$ and $Y$ are affine it is clear that conditions (1) and (2) are equivalent. In cases (1) and (2) the morphism $f$ is representable by algebraic spaces by definition, hence affine by Lemma 87.19.7. Thus if (1) or (2) holds we see that $X$ is countably indexed by Lemma 87.19.9. Write $X = \text{Spf}(A)$ and $Y = \text{Spf}(B)$ for topological $S$-algebras $A$ and $B$ in $\textit{WAdm}^{count}$, see Lemma 87.10.4. By Lemma 87.9.10 we see that $f$ corresponds to a continuous map $B \to A$. Hence now the result follows from Lemma 87.29.2. $\square$

Comments (2)

Comment #1974 by Brian Conrad on

The applicability of the Lemma at the end of the 2nd sentence of the proof requires knowing that is representable and affine. So it would help the reader if it is said that under (1) and (2) the map is representable in algebraic spaces by the definitions and thus it is also affine by Tag 0AKN.

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