The Stacks project

Lemma 86.12.1. For a morphism $\varphi : A \to B$ 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 85.17.2.

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 85.17.2. 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 85.15.13. 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 85.8.3. By Formal Spaces, Example 85.8.2 and Lemma 85.15.14 conditions (1), (2), and (3) of Formal Spaces, Situation 85.17.2 translate into the following statements

  1. if $f$ is locally of finite type, then $f'$ is locally of finite type,

  2. if $f'$ is locally of finite type and $g$ is surjective, then $f$ is locally of finite type, and

  3. 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 65.23. Property (3) is a straightforward consequence of the definition. $\square$


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