## 87.12 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 86.4.8, Lemma 86.4.10, and Example 86.12.2.

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

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

Lemma 87.12.2. Consider the property $P$ on arrows of $\textit{WAdm}^{count}$ defined in Lemma 87.12.1. Then $P$ is stable under base change (Formal Spaces, Situation 86.21.6).

Proof. The statement makes sense by Lemma 87.12.1. 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 86.4.12). 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 87.12.3. Consider the property $P$ on arrows of $\textit{WAdm}^{count}$ defined in Lemma 87.12.1. Then $P$ is stable under composition (Formal Spaces, Situation 86.21.11).

Proof. Omitted. Hint: compositions of finite type ring maps are of finite type. $\square$

Lemma 87.12.4. 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 87.12.1.

Proof. This is an easy consequence of the definitions. $\square$

Lemma 87.12.5. Let $\varphi : A \to B$ be an arrow of $\textit{WAdm}^{Noeth}$ satisfying the condition defined in Lemma 87.12.1. 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 87.12.6. Let $\varphi : A \to B$ be an arrow of $\textit{WAdm}^{Noeth}$. If $\varphi$ is adic the following are equivalent

1. $\varphi$ satisfies the condition defined in Lemma 87.12.1 and

2. $\varphi$ satisfies the condition defined in Lemma 87.11.1.

Proof. Omitted. Hint: For the proof of (1) $\Rightarrow$ (2) use Lemma 87.12.5. $\square$

Lemma 87.12.7. 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

1. for every commutative diagram

$\xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y }$

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 87.12.1,

2. there exists a covering $\{ Y_ j \to Y\}$ as in Formal Spaces, Definition 86.11.1 and for each $j$ a covering $\{ X_{ji} \to Y_ j \times _ Y X\}$ as in Formal Spaces, Definition 86.11.1 such that each $X_{ji} \to Y_ j$ corresponds to an arrow of $\textit{WAdm}^{count}$ satisfying the property defined in Lemma 87.12.1,

3. there exist a covering $\{ X_ i \to X\}$ as in Formal Spaces, Definition 86.11.1 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 87.12.1, and

4. the morphism $f_{red} : X_{red} \to Y_{red}$ is locally of finite type.

Proof. The equivalence of (1), (2), and (3) follows from Lemma 87.12.1 and an application of Formal Spaces, Lemma 86.21.3. 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 86.12.1 or directly from Formal Spaces, Lemma 86.12.3.

• 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 86.12.2. 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 87.12.1.

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 66.23. $\square$

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