85.17 Morphisms and continuous ring maps

In this section we denote $\textit{WAdm}$ the category of weakly admissible topological rings and continuous ring homomorphisms. We define full subcategories

$\textit{WAdm} \supset \textit{WAdm}^{count} \supset \textit{WAdm}^{adic*} \supset \textit{WAdm}^{Noeth}$

whose objects are

1. $\textit{WAdm}^{count}$: those weakly admissible topological rings $A$ which have a countable fundamental system of neighbourhoods of $0$,

2. $\textit{WAdm}^{adic*}$: the adic topological rings which have a finitely generated ideal of definition, and

3. $\textit{WAdm}^{Noeth}$: 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 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 85.6.4. By Lemma 85.5.10 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 85.16.5 (and hence via Lemma 85.16.4) 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 85.17.1. 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 85.15.11.

1. If $A$ is in $\textit{WAdm}^{count}$, so is $(A')^\wedge$.

2. If $A$ is in $\textit{WAdm}^{adic*}$, so is $(A')^\wedge$.

3. If $A$ is in $\textit{WAdm}^{Noeth}$ and $A'$ is Noetherian, then $(A')^\wedge$ is in $\textit{WAdm}^{Noeth}$.

Proof. Part (1) is clear from the construction. Assume $A$ has a finitely generated ideal of definition $I \subset A$. Then $I^ n(A')^\wedge = \mathop{\mathrm{Ker}}((A')^\wedge \to A'/I^ nA')$ by Algebra, Lemma 10.96.3. Thus $I(A')^\wedge$ is a finitely generated ideal of definition and we see that (2) holds. Finally, assume that $A$ is Noetherian and adic. By (2) we know that $(A')^\wedge$ is adic. By Algebra, Lemma 10.97.6 we see that $(A')^\wedge$ is Noetherian. Hence (3) holds. $\square$

Situation 85.17.2. Let $P$ be a property of morphisms of $\textit{WAdm}^{count}$. Consider commutative diagrams

85.17.2.1
$$\label{formal-spaces-equation-localize} \vcenter { \xymatrix{ A \ar[r] & (A')^\wedge \\ B \ar[r] \ar[u]^\varphi & (B')^\wedge \ar[u]_{\varphi '} } }$$

satisfying the following conditions

1. $A$ and $B$ are objects of $\textit{WAdm}^{count}$,

2. $A \to A'$ and $B \to B'$ are étale ring maps,

3. $(A')^\wedge = \mathop{\mathrm{lim}}\nolimits A'/IA'$, 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. $B$,

4. $\varphi : B \to A$ and $\varphi ' : (B')^\wedge \to (A')^\wedge$ are continuous.

By Lemma 85.17.1 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:

1. for any diagram (85.17.2.1) we have $P(\varphi ) \Rightarrow P(\varphi ')$,

2. for any diagram (85.17.2.1) with $A \to A'$ faithfully flat we have $P(\varphi ') \Rightarrow P(\varphi )$,

3. 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 85.17.3. 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

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 a morphism of $\textit{WAdm}^{count}$ with property $P$,

2. there exists a covering $\{ Y_ j \to Y\}$ as in Definition 85.7.1 and for each $j$ a covering $\{ X_{ji} \to Y_ j \times _ Y X\}$ as in Definition 85.7.1 such that each $X_{ji} \to Y_ j$ corresponds to a morphism of $\textit{WAdm}^{count}$ with property $P$, and

3. there exist a covering $\{ X_ i \to X\}$ as in Definition 85.7.1 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 85.11.2) we may pick coverings $\{ Y_{ij} \to Y_ i \times _ Y V\}$ as in Definition 85.7.1. 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 85.7.1. 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 85.15.13. 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 85.15.14. One more application of axiom (2) (with $B' = B$) implies that $P(U \to V)$ is true as desired. $\square$

Remark 85.17.4 (Variant for adic-star). Let $P$ be a property of morphisms of $\textit{WAdm}^{adic*}$. We say $P$ is a local property if axioms (1), (2), (3) of Situation 85.17.2 hold for morphisms of $\textit{WAdm}^{adic*}$. In exactly the same way we obtain a variant of Lemma 85.17.3 for morphisms between locally adic* formal algebraic spaces over $S$.

Remark 85.17.5 (Variant for Noetherian). Let $P$ be a property of morphisms of $\textit{WAdm}^{Noeth}$. We say $P$ is a local property if axioms (1), (2), (3), of Situation 85.17.2 hold for morphisms of $\textit{WAdm}^{Noeth}$. In exactly the same way we obtain a variant of Lemma 85.17.3 for morphisms between locally Noetherian formal algebraic spaces over $S$.

Situation 85.17.6. Let $P$ be a local property of morphisms of $\textit{WAdm}^{count}$, see Situation 85.17.2. 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 85.4.16.

Lemma 85.17.7. 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 85.17.3 then so does $\text{pr}_2 : X \times _ Y Z \to Z$.

Proof. Choose a covering $\{ Y_ j \to Y\}$ as in Definition 85.7.1. For each $j$ choose a covering $\{ X_{ji} \to Y_ j \times _ Y X\}$ as in Definition 85.7.1. For each $j$ choose a covering $\{ Z_{jk} \to Y_ j \times _ Y Z\}$ as in Definition 85.7.1. Observe that $X_{ji} \times _{Y_ j} Z_{jk}$ is an affine formal algebraic space which is countably indexed, see Lemma 85.16.8. Then we see that

$\{ X_{ji} \times _{Y_ j} Z_{jk} \to X \times _ Y Z\}$

is a covering as in Definition 85.7.1. 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 85.12.4 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. $\square$

Remark 85.17.8 (Variant for adic-star). Let $P$ be a local property of morphisms of $\textit{WAdm}^{adic*}$, see Remark 85.17.4. We say $P$ is stable under base change if given $B \to A$ and $B \to C$ in $\textit{WAdm}^{adic*}$ 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 $\textit{WAdm}^{adic*}$ by Lemma 85.4.16. In exactly the same way we obtain a variant of Lemma 85.17.7 for morphisms between locally adic* formal algebraic spaces over $S$.

Remark 85.17.9 (Variant for Noetherian). Let $P$ be a local property of morphisms of $\textit{WAdm}^{Noeth}$, see Remark 85.17.5. We say $P$ is stable under base change if given $B \to A$ and $B \to C$ in $\textit{WAdm}^{Noeth}$ the property $P(B \to A)$ implies both that $A \widehat{\otimes }_ B C$ is adic Noetherian1 and that $P(C \to A \widehat{\otimes }_ B C)$. In exactly the same way we obtain a variant of Lemma 85.17.7 for morphisms between locally Noetherian formal algebraic spaces over $S$.

Remark 85.17.10 (Another variant for Noetherian). Let $P$ and $Q$ be local properties of morphisms of $\textit{WAdm}^{Noeth}$, see Remark 85.17.5. We say $P$ is stable under base change by $Q$ if given $B \to A$ and $B \to C$ in $\textit{WAdm}^{Noeth}$ satisfying $P(B \to A)$ and $Q(B \to C)$, then $A \widehat{\otimes }_ B C$ is adic Noetherian and $P(C \to A \widehat{\otimes }_ B C)$ holds. Arguing exactly as in the proof of Lemma 85.17.7 we obtain the following statement: given morphisms $f : X \to Y$ and $g : Y \to Z$ of locally Noetherian formal algebraic spaces over $S$ such that

1. the equivalent conditions of Lemma 85.17.3 hold for $f$ and $P$,

2. the equivalent conditions of Lemma 85.17.3 hold for $g$ and $Q$,

then the equivalent conditions of Lemma 85.17.3 hold for $\text{pr}_2 : X \times _ Y Z \to Z$ and $P$.

Situation 85.17.11. Let $P$ be a local property of morphisms of $\textit{WAdm}^{count}$, see Situation 85.17.2. 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 85.17.12. 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 85.17.3 then so does $g \circ f : X \to Z$.

Proof. Choose a covering $\{ Z_ k \to Z\}$ as in Definition 85.7.1. For each $k$ choose a covering $\{ Y_{kj} \to Z_ k \times _ Z Y\}$ as in Definition 85.7.1. For each $k$ and $j$ choose a covering $\{ X_{kji} \to Y_{kj} \times _ Y X\}$ as in Definition 85.7.1. If $f$ and $g$ satisfies the equivalent conditions of Lemma 85.17.3 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. $\square$

Remark 85.17.13 (Variant for adic-star). Let $P$ be a local property of morphisms of $\textit{WAdm}^{adic*}$, see Remark 85.17.4. We say $P$ is stable under composition if given $B \to A$ and $C \to B$ in $\textit{WAdm}^{adic*}$ we have $P(B \to A) \wedge P(C \to B) \Rightarrow P(C \to A)$. In exactly the same way we obtain a variant of Lemma 85.17.12 for morphisms between locally adic* formal algebraic spaces over $S$.

Remark 85.17.14 (Variant for Noetherian). Let $P$ be a local property of morphisms of $\textit{WAdm}^{Noeth}$, see Remark 85.17.5. We say $P$ is stable under composition if given $B \to A$ and $C \to B$ in $\textit{WAdm}^{Noeth}$ we have $P(B \to A) \wedge P(C \to B) \Rightarrow P(C \to A)$. In exactly the same way we obtain a variant of Lemma 85.17.12 for morphisms between locally Noetherian formal algebraic spaces over $S$.

Situation 85.17.15. Let $P$ be a local property of morphisms of $\textit{WAdm}^{count}$, see Situation 85.17.2. 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 85.17.16. 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 85.17.3 then so does $f : X \to Y$.

Proof. Choose a covering $\{ Z_ k \to Z\}$ as in Definition 85.7.1. For each $k$ choose a covering $\{ Y_{kj} \to Z_ k \times _ Z Y\}$ as in Definition 85.7.1. For each $k$ and $j$ choose a covering $\{ X_{kji} \to Y_{kj} \times _ Y X\}$ as in Definition 85.7.1. 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 85.17.3 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. $\square$

Remark 85.17.17 (Variant for adic-star). Let $P$ be a local property of morphisms of $\textit{WAdm}^{adic*}$, see Remark 85.17.4. We say $P$ has the cancellation property if given $B \to A$ and $C \to B$ in $\textit{WAdm}^{adic*}$ we have $P(C \to A) \wedge P(C \to B) \Rightarrow P(B \to A)$. In exactly the same way we obtain a variant of Lemma 85.17.12 for morphisms between locally adic* formal algebraic spaces over $S$.

Remark 85.17.18 (Variant for Noetherian). Let $P$ be a local property of morphisms of $\textit{WAdm}^{Noeth}$, see Remark 85.17.5. We say $P$ has the cancellation property if given $B \to A$ and $C \to B$ in $\textit{WAdm}^{Noeth}$ we have $P(C \to B) \wedge P(C \to A) \Rightarrow P(C \to B)$. In exactly the same way we obtain a variant of Lemma 85.17.12 for morphisms between locally Noetherian formal algebraic spaces over $S$.

[1] See Lemma 85.4.16 for a criterion.

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