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. $\square$

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 "$\rightarrow X$". On the 3rd line of the proof, in the first fiber product the placement of $T$ and $T'$ should be swapped.

Comment #1966 by Brian Conrad on

Please disregard the first sentence of my preceding comment.

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