Lemma 88.12.1. 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 87.21.2.
88.12 Finite type on reductions
In this section we talk a little bit about morphisms $X \to Y$ of locally countably indexed formal algebraic spaces such that $X_{red} \to Y_{red}$ is locally of finite type. We will translate this into an algebraic condition. To understand this algebraic condition it pays to keep in mind the following:
If $A$ is a weakly admissible topological ring, then the set $\mathfrak a \subset A$ of topological nilpotent elements is an open, radical ideal and $\text{Spf}(A)_{red} = \mathop{\mathrm{Spec}}(A/\mathfrak a)$.
See Formal Spaces, Definition 87.4.8, Lemma 87.4.10, and Example 87.12.2.
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 87.21.2. 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 87.19.13. 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 87.12.3. By Formal Spaces, Example 87.12.2 and Lemma 87.19.14 conditions (1), (2), and (3) of Formal Spaces, Situation 87.21.2 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 67.23. Property (3) is a straightforward consequence of the definition. $\square$
Lemma 88.12.2. Consider the property $P$ on arrows of $\textit{WAdm}^{count}$ defined in Lemma 88.12.1. Then $P$ is stable under base change (Formal Spaces, Situation 87.21.6).
Proof. The statement makes sense by Lemma 88.12.1. To see that it is true assume we have morphisms $B \to A$ and $B \to C$ in $\textit{WAdm}^{count}$ such that $B/\mathfrak b \to A/\mathfrak a$ is of finite type where $\mathfrak b \subset B$ and $\mathfrak a \subset A$ are the ideals of topologically nilpotent elements. Since $A$ and $B$ are weakly admissible, the ideals $\mathfrak a$ and $\mathfrak b$ are open. Let $\mathfrak c \subset C$ be the (open) ideal of topologically nilpotent elements. Then we find a surjection $A \widehat{\otimes }_ B C \to A/\mathfrak a \otimes _{B/\mathfrak b} C/\mathfrak c$ whose kernel is a weak ideal of definition and hence consists of topologically nilpotent elements (please compare with the proof of Formal Spaces, Lemma 87.4.12). Since already $C/\mathfrak c \to A/\mathfrak a \otimes _{B/\mathfrak b} C/\mathfrak c$ is of finite type as a base change of $B/\mathfrak b \to A/\mathfrak a$ we conclude. $\square$
Lemma 88.12.3. Consider the property $P$ on arrows of $\textit{WAdm}^{count}$ defined in Lemma 88.12.1. Then $P$ is stable under composition (Formal Spaces, Situation 87.21.11).
Proof. Omitted. Hint: compositions of finite type ring maps are of finite type. $\square$
Lemma 88.12.4. Let $\varphi : A \to B$ be an arrow of $\textit{WAdm}^{count}$. If $\varphi $ is taut and topologically of finite type, then $\varphi $ satisfies the condition defined in Lemma 88.12.1.
Proof. This is an easy consequence of the definitions. $\square$
Lemma 88.12.5. Let $\varphi : A \to B$ be an arrow of $\textit{WAdm}^{Noeth}$ satisfying the condition defined in Lemma 88.12.1. Then $A \to B$ is topologically of finite type.
Proof. Let $\mathfrak b \subset B$ be the ideal of topologically nilpotent elements. Choose $b_1, \ldots , b_ r \in B$ which map to generators of $B/\mathfrak b$ over $A$. Choose generators $b_{r + 1}, \ldots , b_ s$ of the ideal $\mathfrak b$. We claim that the image of
\[ \varphi : A[x_1, \ldots , x_ s] \longrightarrow B, \quad x_ i \longmapsto b_ i \]
has dense image. Namely, if $b \in \mathfrak b^ n$ for some $n \geq 0$, then we can write $b = \sum b_ E b_{r + 1}^{e_{r + 1}} \ldots b_ s^{e_ s}$ for multiindices $E = (e_{r + 1}, \ldots , e_ s)$ with $|E| = \sum e_ i = n$ and $b_ E \in B$. Next, we can write $b_ E = f_ E(b_1, \ldots , b_ r) + b'_ E$ with $b'_ E \in \mathfrak b$ and $f_ E \in A[x_1, \ldots , x_ r]$. Combined we obtain $b \in \mathop{\mathrm{Im}}(\varphi ) + \mathfrak b^{n + 1}$. By induction we see that $B = \mathop{\mathrm{Im}}(\varphi ) + \mathfrak b^ n$ for all $n \geq 0$ which mplies what we want as $\mathfrak b$ is an ideal of definition of $B$. $\square$
Lemma 88.12.6. Let $\varphi : A \to B$ be an arrow of $\textit{WAdm}^{Noeth}$. If $\varphi $ is adic the following are equivalent
Proof. Omitted. Hint: For the proof of (1) $\Rightarrow $ (2) use Lemma 88.12.5. $\square$
Lemma 88.12.7. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of locally countably indexed formal algebraic spaces over $S$. The following are equivalent
for every commutative diagram
with $U$ and $V$ affine formal algebraic spaces, $U \to X$ and $V \to Y$ representable by algebraic spaces and étale, the morphism $U \to V$ corresponds to an arrow of $\textit{WAdm}^{count}$ satisfying the property defined in Lemma 88.12.1,
there exists a covering $\{ Y_ j \to Y\} $ as in Formal Spaces, Definition 87.11.1 and for each $j$ a covering $\{ X_{ji} \to Y_ j \times _ Y X\} $ as in Formal Spaces, Definition 87.11.1 such that each $X_{ji} \to Y_ j$ corresponds to an arrow of $\textit{WAdm}^{count}$ satisfying the property defined in Lemma 88.12.1,
there exist a covering $\{ X_ i \to X\} $ as in Formal Spaces, Definition 87.11.1 and for each $i$ a factorization $X_ i \to Y_ i \to Y$ where $Y_ i$ is an affine formal algebraic space, $Y_ i \to Y$ is representable by algebraic spaces and étale, and $X_ i \to Y_ i$ corresponds to an arrow of $\textit{WAdm}^{count}$ satisfying the property defined in Lemma 88.12.1, and
the morphism $f_{red} : X_{red} \to Y_{red}$ is locally of finite type.
\[ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } \]
Proof. The equivalence of (1), (2), and (3) follows from Lemma 88.12.1 and an application of Formal Spaces, Lemma 87.21.3. Let $Y_ j$ and $X_{ji}$ be as in (2). Then
The families $\{ Y_{j, red} \to Y_{red}\} $ and $\{ X_{ji, red} \to X_{red}\} $ are étale coverings by affine schemes. This follows from the discussion in the proof of Formal Spaces, Lemma 87.12.1 or directly from Formal Spaces, Lemma 87.12.3.
If $X_{ji} \to Y_ j$ corresponds to the morphism $B_ j \to A_{ji}$ of $\textit{WAdm}^{count}$, then $X_{ji, red} \to Y_{j, red}$ corresponds to the ring map $B_ j/\mathfrak b_ j \to A_{ji}/\mathfrak a_{ji}$ where $\mathfrak b_ j$ and $\mathfrak a_{ji}$ are the ideals of topologically nilpotent elements. This follows from Formal Spaces, Example 87.12.2. Hence $X_{ji, red} \to Y_{j, red}$ is locally of finite type if and only if $B_ j \to A_{ji}$ satisfies the property defined in Lemma 88.12.1.
The equivalence of (2) and (4) follows from these remarks because being locally of finite type is a property of morphisms of algebraic spaces which is étale local on source and target, see discussion in Morphisms of Spaces, Section 67.23. $\square$
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