The Stacks project

85.16 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 85.15.5 (and hence via Lemma 85.15.4) 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 85.16.1. 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 85.14.11.

  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 10.95.3. 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 10.96.6 we see that $(A')^\wedge $ is Noetherian. Hence (3) holds. $\square$

Situation 85.16.2. Let $P$ be a property of morphisms of $\textit{WAdm}^{count}$. Consider commutative diagrams

85.16.2.1
\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. $\varphi : A \to B$ and $\varphi ' : (A')^\wedge \to (B')^\wedge $ are continuous.

By Lemma 85.16.1 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 (85.16.2.1) we have $P(\varphi ) \Rightarrow P(\varphi ')$,

  2. for any diagram (85.16.2.1) 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 85.16.3. 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 85.10.2) 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 85.14.13. 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 85.14.14. One more application of axiom (2) (with $B' = B$) implies that $P(U \to V)$ is true as desired. $\square$

Remark 85.16.4 (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 85.16.2 hold for morphisms of $\textit{WAdm}^{adic*}$. In exactly the same way we obtain a variant of Lemma 85.16.3 for morphisms between locally adic* formal algebraic spaces over $S$.

Remark 85.16.5 (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 85.16.2 hold for morphisms of $\textit{WAdm}^{Noeth}$. In exactly the same way we obtain a variant of Lemma 85.16.3 for morphisms between locally Noetherian formal algebraic spaces over $S$.

Lemma 85.16.6. Let $B \to A$ be an arrow of $\textit{WAdm}^{count}$. The following are equivalent

  1. $B \to A$ is taut (Definition 85.4.11),

  2. for $B \supset J_1 \supset J_2 \supset J_3 \supset \ldots $ a fundamental system of weak ideals of definitions there exist a commutative diagram

    \[ \xymatrix{ A \ar[r] & \ldots \ar[r] & A_3 \ar[r] & A_2 \ar[r] & A_1 \\ B \ar[r] \ar[u] & \ldots \ar[r] & B/J_3 \ar[r] \ar[u] & B/J_2 \ar[r] \ar[u] & B/J_1 \ar[u] } \]

    such that $A_{n + 1}/J_ nA_{n + 1} = A_ n$ and $A = \mathop{\mathrm{lim}}\nolimits A_ n$ as topological ring.

Moreover, these equivalent conditions define a local property, i.e., they satisfy axioms (1), (2), (3).

Proof. The equivalence of (a) and (b) is immediate. Below we will give an algebraic proof of the axioms, but it turns out we've already proven them. Namely, using Lemma 85.14.10 (a) and (b) translate to saying the corresponding morphism of affine formal algebraic spaces is representable, and this condition is “étale local on the source and target” by Lemma 85.14.4.

Let a diagram (85.16.2.1) as in Situation 85.16.2 be given. By Example 85.14.11 the maps $A \to (A')^\wedge $ and $B \to (B')^\wedge $ satisfy (a) and (b).

Assume (a) and (b) hold for $\varphi $. Let $J \subset B$ be a weak ideal of definition. Then the closure of $JA$, resp. $J(B')^\wedge $ is a weak ideal of definition $I \subset A$, resp. $J' \subset (B')^\wedge $. Then the closure of $I(A')^\wedge $ is a weak ideal of definition $I' \subset (A')^\wedge $. A topological argument shows that $I'$ is also the closure of $J(A')^\wedge $ and of $J'(A')^\wedge $. Finally, as $J$ runs over a fundamental system of weak ideals of definition of $B$ so do the ideals $I$ and $I'$ in $A$ and $(A')^\wedge $. It follows that (a) holds for $\varphi '$. This proves (1).

Assume $A \to A'$ is faithfully flat and that (a) and (b) hold for $\varphi '$. Let $J \subset B$ be a weak ideal of definition. Using (a) and (b) for the maps $B \to (B')^\wedge \to (A')^\wedge $ we find that the closure $I'$ of $J(A')^\wedge $ is a weak ideal of definition. In particular, $I'$ is open and hence the inverse image of $I'$ in $A$ is open. Now we have (explanation below)

\begin{align*} A \cap I' & = A \cap \bigcap (J(A')^\wedge + \mathop{\mathrm{Ker}}((A')^\wedge \to A'/I_0A')) \\ & = A \cap \bigcap \mathop{\mathrm{Ker}}((A')^\wedge \to A'/JA' + I_0 A') \\ & = \bigcap (JA + I_0) \end{align*}

which is the closure of $JA$ by Lemma 85.4.2. The intersections are over weak ideals of definition $I_0 \subset A$. The first equality because a fundamental system of neighbourhoods of $0$ in $(A')^\wedge $ are the kernels of the maps $(A')^\wedge \to A'/I_0A'$. The second equality is trivial. The third equality because $A \to A'$ is faithfully flat, see Algebra, Lemma 10.81.11. Thus the closure of $JA$ is open. By Lemma 85.4.10 the closure of $JA$ is a weak ideal of definition of $A$. Finally, given a weak ideal of definition $I \subset A$ we can find $J$ such that $J(A')^\wedge $ is contained in the closure of $I(A')^\wedge $ by property (a) for $B \to (B')^\wedge $ and $\varphi '$. Thus we see that (a) holds for $\varphi $. This proves (2).

We omit the proof of (3). $\square$

Lemma 85.16.7. Let $P=$"taut" viewed as a property of morphisms of $\textit{WAdm}^{count}$. Then under the assumptions of Lemma 85.16.3 the equivalent conditions (1), (2), and (3) are also equivalent to the condition

  1. $f$ is representable by algebraic spaces.

Proof. Property $P$ is a local property by Lemma 85.16.6. By Lemma 85.14.10 condition $P$ on morphisms of $\textit{WAdm}^{count}$ corresponds to “representable by algebraic spaces” for the corresponding morphisms of countably indexed affine formal algebraic spaces. Thus the lemma follows from Lemma 85.14.4. $\square$


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