Lemma 87.12.1. For an arrow $\varphi : A \to B$ in $\text{WAdm}^{count}$ consider the property $P(\varphi )=$“the induced ring homomorphism $A/\mathfrak a \to B/\mathfrak b$ is of finite type” where $\mathfrak a \subset A$ and $\mathfrak b \subset B$ are the ideals of topologically nilpotent elements. Then $P$ is a local property as defined in Formal Spaces, Situation 86.21.2.
Proof. Consider a commutative diagram
\[ \xymatrix{ B \ar[r] & (B')^\wedge \\ A \ar[r] \ar[u]^\varphi & (A')^\wedge \ar[u]_{\varphi '} } \]
as in Formal Spaces, Situation 86.21.2. Taking $\text{Spf}$ of this diagram we obtain
\[ \xymatrix{ \text{Spf}(B) \ar[d] & \text{Spf}((B')^\wedge ) \ar[l] \ar[d] \\ \text{Spf}(A) & \text{Spf}((A')^\wedge ) \ar[l] } \]
of affine formal algebraic spaces whose horizontal arrows are representable by algebraic spaces and étale by Formal Spaces, Lemma 86.19.13. Hence we obtain a commutative diagram of affine schemes
\[ \xymatrix{ \text{Spf}(B)_{red} \ar[d]^ f & \text{Spf}((B')^\wedge )_{red} \ar[l]^ g \ar[d]^{f'} \\ \text{Spf}(A)_{red} & \text{Spf}((A')^\wedge )_{red} \ar[l] } \]
whose horizontal arrows are étale by Formal Spaces, Lemma 86.12.3. By Formal Spaces, Example 86.12.2 and Lemma 86.19.14 conditions (1), (2), and (3) of Formal Spaces, Situation 86.21.2 translate into the following statements
if $f$ is locally of finite type, then $f'$ is locally of finite type,
if $f'$ is locally of finite type and $g$ is surjective, then $f$ is locally of finite type, and
if $T_ i \to S$, $i = 1, \ldots , n$ are locally of finite type, then $\coprod _{i = 1, \ldots , n} T_ i \to S$ is locally of finite type.
Properties (1) and (2) follow from the fact that being locally of finite type is local on the source and target in the étale topology, see discussion in Morphisms of Spaces, Section 66.23. Property (3) is a straightforward consequence of the definition. $\square$
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