The Stacks project

86.21 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

\[ \textit{WAdm} \supset \textit{WAdm}^{count} \supset \textit{WAdm}^{cic} \supset \textit{WAdm}^{weakly\ adic} \supset \textit{WAdm}^{adic*} \supset \textit{WAdm}^{Noeth} \]

whose objects are

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

  2. $\textit{WAdm}^{cic}$: the admissible topological rings $A$ which have a countable fundamental system of open ideals,

  3. $\textit{WAdm}^{weakly\ adic}$: the weakly adic topological rings (Section 86.7),

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

  5. $\textit{WAdm}^{Noeth}$: 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 86.10.4. By Lemma 86.9.10 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 86.20.7 (and hence via Lemma 86.20.6) 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 86.21.1. 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 86.19.11.

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

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

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

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

  5. 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 86.19.11. Hence statements (1), (2), (3), and (4) follow from Lemmas 86.5.7, 86.5.9, 86.7.5, and 86.6.5. 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 86.21.2. Let $P$ be a property of morphisms of $\textit{WAdm}^{count}$. Consider commutative diagrams

86.21.2.1
\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

  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 86.21.1 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:

  1. for any diagram (86.21.2.1) we have $P(\varphi ) \Rightarrow P(\varphi ')$,

  2. for any diagram (86.21.2.1) 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 86.21.3. 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

  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 86.11.1 and for each $j$ a covering $\{ X_{ji} \to Y_ j \times _ Y X\} $ as in Definition 86.11.1 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 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 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 86.15.2) we may pick coverings $\{ Y_{ij} \to Y_ i \times _ Y V\} $ as in Definition 86.11.1. 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 86.11.1. 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 86.19.13. 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 86.19.14. One more application of axiom (2) (with $B' = B$) implies that $P(U \to V)$ is true as desired. $\square$

Remark 86.21.4 (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 86.21.2 hold for morphisms of $\textit{WAdm}^{adic*}$. In exactly the same way we obtain a variant of Lemma 86.21.3 for morphisms between locally adic* formal algebraic spaces over $S$.

Remark 86.21.5 (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 86.21.2 hold for morphisms of $\textit{WAdm}^{Noeth}$. In exactly the same way we obtain a variant of Lemma 86.21.3 for morphisms between locally Noetherian formal algebraic spaces over $S$.

Situation 86.21.6. Let $P$ be a local property of morphisms of $\textit{WAdm}^{count}$, see Situation 86.21.2. 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 86.4.12.

Lemma 86.21.7. 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 86.21.3 then so does $\text{pr}_2 : X \times _ Y Z \to Z$.

Proof. Choose a covering $\{ Y_ j \to Y\} $ as in Definition 86.11.1. For each $j$ choose a covering $\{ X_{ji} \to Y_ j \times _ Y X\} $ as in Definition 86.11.1. For each $j$ choose a covering $\{ Z_{jk} \to Y_ j \times _ Y Z\} $ as in Definition 86.11.1. Observe that $X_{ji} \times _{Y_ j} Z_{jk}$ is an affine formal algebraic space which is countably indexed, see Lemma 86.20.10. Then we see that

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

is a covering as in Definition 86.11.1. 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 86.16.4 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. $\square$

Remark 86.21.8 (Variant for adic-star). Let $P$ be a local property of morphisms of $\textit{WAdm}^{adic*}$, see Remark 86.21.4. 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 86.4.12. In exactly the same way we obtain a variant of Lemma 86.21.7 for morphisms between locally adic* formal algebraic spaces over $S$.

Remark 86.21.9 (Variant for Noetherian). Let $P$ be a local property of morphisms of $\textit{WAdm}^{Noeth}$, see Remark 86.21.5. 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 86.21.7 for morphisms between locally Noetherian formal algebraic spaces over $S$.

Remark 86.21.10 (Another variant for Noetherian). Let $P$ and $Q$ be local properties of morphisms of $\textit{WAdm}^{Noeth}$, see Remark 86.21.5. 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 86.21.7 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 86.21.3 hold for $f$ and $P$,

  2. the equivalent conditions of Lemma 86.21.3 hold for $g$ and $Q$,

then the equivalent conditions of Lemma 86.21.3 hold for $\text{pr}_2 : X \times _ Y Z \to Z$ and $P$.

Situation 86.21.11. Let $P$ be a local property of morphisms of $\textit{WAdm}^{count}$, see Situation 86.21.2. 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 86.21.12. 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 86.21.3 then so does $g \circ f : X \to Z$.

Proof. Choose a covering $\{ Z_ k \to Z\} $ as in Definition 86.11.1. For each $k$ choose a covering $\{ Y_{kj} \to Z_ k \times _ Z Y\} $ as in Definition 86.11.1. For each $k$ and $j$ choose a covering $\{ X_{kji} \to Y_{kj} \times _ Y X\} $ as in Definition 86.11.1. If $f$ and $g$ satisfies the equivalent conditions of Lemma 86.21.3 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. $\square$

Remark 86.21.13 (Variant for adic-star). Let $P$ be a local property of morphisms of $\textit{WAdm}^{adic*}$, see Remark 86.21.4. 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 86.21.12 for morphisms between locally adic* formal algebraic spaces over $S$.

Remark 86.21.14 (Variant for Noetherian). Let $P$ be a local property of morphisms of $\textit{WAdm}^{Noeth}$, see Remark 86.21.5. 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 86.21.12 for morphisms between locally Noetherian formal algebraic spaces over $S$.

Situation 86.21.15. Let $P$ be a local property of morphisms of $\textit{WAdm}^{count}$, see Situation 86.21.2. 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 86.21.16. 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 86.21.3 then so does $f : X \to Y$.

Proof. Choose a covering $\{ Z_ k \to Z\} $ as in Definition 86.11.1. For each $k$ choose a covering $\{ Y_{kj} \to Z_ k \times _ Z Y\} $ as in Definition 86.11.1. For each $k$ and $j$ choose a covering $\{ X_{kji} \to Y_{kj} \times _ Y X\} $ as in Definition 86.11.1. 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 86.21.3 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. $\square$

Remark 86.21.17 (Variant for adic-star). Let $P$ be a local property of morphisms of $\textit{WAdm}^{adic*}$, see Remark 86.21.4. 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 86.21.12 for morphisms between locally adic* formal algebraic spaces over $S$.

Remark 86.21.18 (Variant for Noetherian). Let $P$ be a local property of morphisms of $\textit{WAdm}^{Noeth}$, see Remark 86.21.5. 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 86.21.12 for morphisms between locally Noetherian formal algebraic spaces over $S$.

[1] See Lemma 86.4.12 for a criterion.

Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0ANA. Beware of the difference between the letter 'O' and the digit '0'.