In this section we denote $\textit{WAdm}$ the category of weakly admissible topological rings and continuous ring homomorphisms. We define full subcategories
\[ \textit{WAdm} \supset \textit{WAdm}^{count} \supset \textit{WAdm}^{cic} \supset \textit{WAdm}^{weakly\ adic} \supset \textit{WAdm}^{adic*} \supset \textit{WAdm}^{Noeth} \]
whose objects are
$\textit{WAdm}^{count}$: those weakly admissible topological rings $A$ which have a countable fundamental system of open ideals,
$\textit{WAdm}^{cic}$: the admissible topological rings $A$ which have a countable fundamental system of open ideals,
$\textit{WAdm}^{weakly\ adic}$: the weakly adic topological rings (Section 87.7),
$\textit{WAdm}^{adic*}$: the adic topological rings which have a finitely generated ideal of definition, and
$\textit{WAdm}^{Noeth}$: the adic topological rings which are Noetherian.
Clearly, the formal spectra of these types of rings are the basic building blocks of locally countably indexed, locally countably indexed and classical, locally weakly adic, locally adic*, and locally Noetherian formal algebraic spaces.
We briefly review the relationship between morphisms of countably indexed, affine formal algebraic spaces and morphisms of $\textit{WAdm}^{count}$. Let $S$ be a scheme. Let $X$ and $Y$ be countably indexed, affine formal algebraic spaces. Write $X = \text{Spf}(A)$ and $Y = \text{Spf}(B)$ topological $S$-algebras $A$ and $B$ in $\textit{WAdm}^{count}$, see Lemma 87.10.4. By Lemma 87.9.10 there is a 1-to-1 correspondence between morphisms $f : X \to Y$ and continuous maps
\[ \varphi : B \longrightarrow A \]
of topological $S$-algebras. The relationship is given by $f \mapsto f^\sharp $ and $\varphi \mapsto \text{Spf}(\varphi )$.
Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of locally countably indexed formal algebraic spaces. Consider a commutative diagram
\[ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } \]
with $U$ and $V$ affine formal algebraic spaces and $U \to X$ and $V \to Y$ representable by algebraic spaces and étale. By Definition 87.20.7 (and hence via Lemma 87.20.6) we see that $U$ and $V$ are countably indexed affine formal algebraic spaces. By the discussion in the previous paragraph we see that $U \to V$ is isomorphic to $\text{Spf}(\varphi )$ for some continuous map
\[ \varphi : B \longrightarrow A \]
of topological $S$-algebras in $\textit{WAdm}^{count}$.
Lemma 87.21.1. Let $A \in \mathop{\mathrm{Ob}}\nolimits (\textit{WAdm})$. Let $A \to A'$ be a ring map (no topology). Let $(A')^\wedge = \mathop{\mathrm{lim}}\nolimits _{I \subset A\text{ w.i.d}} A'/IA'$ be the object of $\textit{WAdm}$ constructed in Example 87.19.11.
If $A$ is in $\textit{WAdm}^{count}$, so is $(A')^\wedge $.
If $A$ is in $\textit{WAdm}^{cic}$, so is $(A')^\wedge $.
If $A$ is in $\textit{WAdm}^{weakly\ adic}$, so is $(A')^\wedge $.
If $A$ is in $\textit{WAdm}^{adic*}$, so is $(A')^\wedge $.
If $A$ is in $\textit{WAdm}^{Noeth}$ and $A'$ is Noetherian, then $(A')^\wedge $ is in $\textit{WAdm}^{Noeth}$.
Proof.
Recall that $A \to (A')^\wedge $ is taut, see discussion in Example 87.19.11. Hence statements (1), (2), (3), and (4) follow from Lemmas 87.5.7, 87.5.9, 87.7.5, and 87.6.5. Finally, assume that $A$ is Noetherian and adic. By (4) we know that $(A')^\wedge $ is adic. By Algebra, Lemma 10.97.6 we see that $(A')^\wedge $ is Noetherian. Hence (5) holds.
$\square$
Situation 87.21.2. Let $P$ be a property of morphisms of $\textit{WAdm}^{count}$. Consider commutative diagrams
87.21.2.1
\begin{equation} \label{formal-spaces-equation-localize} \vcenter { \xymatrix{ A \ar[r] & (A')^\wedge \\ B \ar[r] \ar[u]^\varphi & (B')^\wedge \ar[u]_{\varphi '} } } \end{equation}
satisfying the following conditions
$A$ and $B$ are objects of $\textit{WAdm}^{count}$,
$A \to A'$ and $B \to B'$ are étale ring maps,
$(A')^\wedge = \mathop{\mathrm{lim}}\nolimits A'/IA'$, resp. $(B')^\wedge = \mathop{\mathrm{lim}}\nolimits B'/JB'$ where $I \subset A$, resp. $J \subset B$ runs through the weakly admissible ideals of definition of $A$, resp. $B$,
$\varphi : B \to A$ and $\varphi ' : (B')^\wedge \to (A')^\wedge $ are continuous.
By Lemma 87.21.1 the topological rings $(A')^\wedge $ and $(B')^\wedge $ are objects of $\textit{WAdm}^{count}$. We say $P$ is a local property if the following axioms hold:
for any diagram (87.21.2.1) we have $P(\varphi ) \Rightarrow P(\varphi ')$,
for any diagram (87.21.2.1) with $A \to A'$ faithfully flat we have $P(\varphi ') \Rightarrow P(\varphi )$,
if $P(B \to A_ i)$ for $i = 1, \ldots , n$, then $P(B \to \prod _{i = 1, \ldots , n} A_ i)$.
Axiom (3) makes sense as $\textit{WAdm}^{count}$ has finite products.
Lemma 87.21.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of locally countably indexed formal algebraic spaces over $S$. Let $P$ be a local property of morphisms of $\textit{WAdm}^{count}$. The following are equivalent
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 a morphism of $\textit{WAdm}^{count}$ with property $P$,
there exists a covering $\{ Y_ j \to Y\} $ as in Definition 87.11.1 and for each $j$ a covering $\{ X_{ji} \to Y_ j \times _ Y X\} $ as in Definition 87.11.1 such that each $X_{ji} \to Y_ j$ corresponds to a morphism of $\textit{WAdm}^{count}$ with property $P$, and
there exist a covering $\{ X_ i \to X\} $ as in 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 a morphism of $\textit{WAdm}^{count}$ with property $P$.
Proof.
It is clear that (1) implies (2) and that (2) implies (3). Assume $\{ X_ i \to X\} $ and $X_ i \to Y_ i \to Y$ as in (3) and let a diagram as in (1) be given. Since $Y_ i \times _ Y V$ is a formal algebraic space (Lemma 87.15.2) we may pick coverings $\{ Y_{ij} \to Y_ i \times _ Y V\} $ as in Definition 87.11.1. For each $(i, j)$ we may similarly choose coverings $\{ X_{ijk} \to Y_{ij} \times _{Y_ i} X_ i \times _ X U\} $ as in Definition 87.11.1. Since $U$ is quasi-compact we can choose $(i_1, j_1, k_1), \ldots , (i_ n, j_ n, k_ n)$ such that
\[ X_{i_1 j_1 k_1} \amalg \ldots \amalg X_{i_ n j_ n k_ n} \longrightarrow U \]
is surjective. For $s = 1, \ldots , n$ consider the commutative diagram
\[ \xymatrix{ & & & X_{i_ s j_ s k_ s} \ar[ld] \ar[d] \ar[rd] \\ X \ar[d] & X_{i_ s} \ar[l] \ar[d] & X_{i_ s} \times _ X U \ar[l] \ar[d] & Y_{i_ s j_ s} \ar[ld] \ar[rd] & X_{i_ s} \times _ X U \ar[d] \ar[r] & U \ar[d] \ar[r] & X \ar[d] \\ Y & Y_{i_ s} \ar[l] & Y_{i_ s} \times _ Y V \ar[l] & & Y_{i_ s} \times _ Y V \ar[r] & V \ar[r] & Y } \]
Let us say that $P$ holds for a morphism of countably indexed affine formal algebraic spaces if it holds for the corresponding morphism of $\textit{WAdm}^{count}$. Observe that the maps $X_{i_ s j_ s k_ s} \to X_{i_ s}$, $Y_{i_ s j_ s} \to Y_{i_ s}$ are given by completions of étale ring maps, see Lemma 87.19.13. Hence we see that $P(X_{i_ s} \to Y_{i_ s})$ implies $P(X_{i_ s j_ s k_ s} \to Y_{i_ s j_ s})$ by axiom (1). Observe that the maps $Y_{i_ s j_ s} \to V$ are given by completions of étale rings maps (same lemma as before). By axiom (2) applied to the diagram
\[ \xymatrix{ X_{i_ s j_ s k_ s} \ar@{=}[r] \ar[d] & X_{i_ s j_ s k_ s} \ar[d] \\ Y_{i_ s j_ s} \ar[r] & V } \]
(this is permissible as identities are faithfully flat ring maps) we conclude that $P(X_{i_ s j_ s k_ s} \to V)$ holds. By axiom (3) we find that $P(\coprod _{s = 1, \ldots , n} X_{i_ s j_ s k_ s} \to V)$ holds. Since the morphism $\coprod X_{i_ s j_ s k_ s} \to U$ is surjective by construction, the corresponding morphism of $\textit{WAdm}^{count}$ is the completion of a faithfully flat étale ring map, see Lemma 87.19.14. One more application of axiom (2) (with $B' = B$) implies that $P(U \to V)$ is true as desired.
$\square$
Situation 87.21.6. Let $P$ be a local property of morphisms of $\textit{WAdm}^{count}$, see Situation 87.21.2. We say $P$ is stable under base change if given $B \to A$ and $B \to C$ in $\textit{WAdm}^{count}$ we have $P(B \to A) \Rightarrow P(C \to A \widehat{\otimes }_ B C)$. This makes sense as $A \widehat{\otimes }_ B C$ is an object of $\textit{WAdm}^{count}$ by Lemma 87.4.12.
Lemma 87.21.7. Let $S$ be a scheme. Let $P$ be a local property of morphisms of $\textit{WAdm}^{count}$ which is stable under base change. Let $f : X \to Y$ and $g : Z \to Y$ be morphisms of locally countably indexed formal algebraic spaces over $S$. If $f$ satisfies the equivalent conditions of Lemma 87.21.3 then so does $\text{pr}_2 : X \times _ Y Z \to Z$.
Proof.
Choose a covering $\{ Y_ j \to Y\} $ as in Definition 87.11.1. For each $j$ choose a covering $\{ X_{ji} \to Y_ j \times _ Y X\} $ as in Definition 87.11.1. For each $j$ choose a covering $\{ Z_{jk} \to Y_ j \times _ Y Z\} $ as in Definition 87.11.1. Observe that $X_{ji} \times _{Y_ j} Z_{jk}$ is an affine formal algebraic space which is countably indexed, see Lemma 87.20.10. Then we see that
\[ \{ X_{ji} \times _{Y_ j} Z_{jk} \to X \times _ Y Z\} \]
is a covering as in Definition 87.11.1. Moreover, the morphisms $X_{ji} \times _{Y_ j} Z_{jk} \to Z$ factor through $Z_{jk}$. By assumption we know that $X_{ji} \to Y_ j$ corresponds to a morphism $B_ j \to A_{ji}$ of $\text{WAdm}^{count}$ having property $P$. The morphisms $Z_{jk} \to Y_ j$ correspond to morphisms $B_ j \to C_{jk}$ in $\text{WAdm}^{count}$. Since $X_{ji} \times _{Y_ j} Z_{jk} = \text{Spf}(A_{ji} \widehat{\otimes }_{B_ j} C_{jk})$ by Lemma 87.16.4 we see that it suffices to show that $C_{jk} \to A_{ji} \widehat{\otimes }_{B_ j} C_{jk}$ has property $P$ which is exactly what the condition that $P$ is stable under base change guarantees.
$\square$
then the equivalent conditions of Lemma 87.21.3 hold for $\text{pr}_2 : X \times _ Y Z \to Z$ and $P$.
Situation 87.21.11. Let $P$ be a local property of morphisms of $\textit{WAdm}^{count}$, see Situation 87.21.2. We say $P$ is stable under composition if given $B \to A$ and $C \to B$ in $\textit{WAdm}^{count}$ we have $P(B \to A) \wedge P(C \to B) \Rightarrow P(C \to A)$.
Lemma 87.21.12. Let $S$ be a scheme. Let $P$ be a local property of morphisms of $\textit{WAdm}^{count}$ which is stable under composition. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of locally countably indexed formal algebraic spaces over $S$. If $f$ and $g$ satisfies the equivalent conditions of Lemma 87.21.3 then so does $g \circ f : X \to Z$.
Proof.
Choose a covering $\{ Z_ k \to Z\} $ as in Definition 87.11.1. For each $k$ choose a covering $\{ Y_{kj} \to Z_ k \times _ Z Y\} $ as in Definition 87.11.1. For each $k$ and $j$ choose a covering $\{ X_{kji} \to Y_{kj} \times _ Y X\} $ as in Definition 87.11.1. If $f$ and $g$ satisfies the equivalent conditions of Lemma 87.21.3 then $X_{kji} \to Y_{jk}$ and $Y_{jk} \to Z_ k$ correspond to arrows $B_{kj} \to A_{kji}$ and $C_ k \to B_{kj}$ of $\text{WAdm}^{count}$ having property $P$. Hence the compositions do too and we conclude.
$\square$
Situation 87.21.15. Let $P$ be a local property of morphisms of $\textit{WAdm}^{count}$, see Situation 87.21.2. We say $P$ has the cancellation property if given $B \to A$ and $C \to B$ in $\textit{WAdm}^{count}$ we have $P(C \to B) \wedge P(C \to A) \Rightarrow P(B \to A)$.
Lemma 87.21.16. Let $S$ be a scheme. Let $P$ be a local property of morphisms of $\textit{WAdm}^{count}$ which has the cancellation property. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of locally countably indexed formal algebraic spaces over $S$. If $g \circ f$ and $g$ satisfies the equivalent conditions of Lemma 87.21.3 then so does $f : X \to Y$.
Proof.
Choose a covering $\{ Z_ k \to Z\} $ as in Definition 87.11.1. For each $k$ choose a covering $\{ Y_{kj} \to Z_ k \times _ Z Y\} $ as in Definition 87.11.1. For each $k$ and $j$ choose a covering $\{ X_{kji} \to Y_{kj} \times _ Y X\} $ as in Definition 87.11.1. Let $X_{kji} \to Y_{jk}$ and $Y_{jk} \to Z_ k$ correspond to arrows $B_{kj} \to A_{kji}$ and $C_ k \to B_{kj}$ of $\text{WAdm}^{count}$. If $g \circ f$ and $g$ satisfies the equivalent conditions of Lemma 87.21.3 then $C_ k \to B_{kj}$ and $C_ k \to A_{kji}$ satisfy $P$. Hence $B_{kj} \to A_{kji}$ does too and we conclude.
$\square$
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