The Stacks project

Lemma 86.20.1. In the situation above. If $f$ is locally of finite type, then $f_{/T}$ is locally of finite type.

Proof. (Finite type morphisms of formal algebraic spaces are discussed in Formal Spaces, Section 85.20.) Namely, suppose that $Z \to X$ is a morphism from a scheme into $X$ such that $|Z|$ maps into $T$. From the cartesian square above we see that $Z \times _ X X'$ is an algebraic space representing $Z \times _{X_{/T}} X'_{/T'}$. Since $Z \times _ X X' \to Z$ is locally of finite type by Morphisms of Spaces, Lemma 65.23.3 we conclude. $\square$


Comments (3)

Comment #1965 by Brian Conrad on

In the statement of the result, "algebraic spaces" should be "formal algebraic spaces". On the 2nd line of the proof remove "". On the 3rd line of the proof, in the first fiber product the placement of and should be swapped.

Comment #1966 by Brian Conrad on

Please disregard the first sentence of my preceding comment.


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