The Stacks project

85.6 Countably indexed affine formal algebraic spaces

These are the affine formal algebraic spaces as in the following lemma.

Lemma 85.6.1. Let $S$ be a scheme. Let $X$ be an affine formal algebraic space over $S$. The following are equivalent

  1. there exists a system $X_1 \to X_2 \to X_3 \to \ldots $ of thickenings of affine schemes over $S$ such that $X = \mathop{\mathrm{colim}}\nolimits X_ n$,

  2. there exists a choice $X = \mathop{\mathrm{colim}}\nolimits X_\lambda $ as in Definition 85.5.1 such that $\Lambda $ is countable.

Proof. This follows from the observation that a countable directed set has a cofinal subset isomorphic to $(\mathbf{N}, \geq )$. See proof of Algebra, Lemma 10.85.3. $\square$

Definition 85.6.2. Let $S$ be a scheme. Let $X$ be an affine formal algebraic space over $S$. We say $X$ is countably indexed if the equivalent conditions of Lemma 85.6.1 are satisfied.

In the language of [BVGD] this is expressed by saying that $X$ is an $\aleph _0$-ind scheme.

Lemma 85.6.3. Let $X$ be an affine formal algebraic space over a scheme $S$.

  1. If $X$ is Noetherian, then $X$ is adic*.

  2. If $X$ is adic*, then $X$ is adic.

  3. If $X$ is adic, then $X$ is countably indexed.

  4. If $X$ is countably indexed, then $X$ is McQuillan.

Proof. Parts (1) and (2) are immediate from the definitions.

Proof of (3). By definition there exists an adic topological ring $A$ such that $X = \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Spec}}(A/I)$ where the colimit is over the ideals of definition of $A$. As $A$ is adic, there exits an ideal $I$ such that $\{ I^ n\} $ forms a fundamental system of neighbourhoods of $0$. Then each $I^ n$ is an ideal of definition and $X = \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Spec}}(A/I^ n)$. Thus $X$ is countably indexed.

Proof of (4). Write $X = \mathop{\mathrm{colim}}\nolimits X_ n$ for some system $X_1 \to X_2 \to X_3 \to \ldots $ of thickenings of affine schemes over $S$. Then

\[ A = \mathop{\mathrm{lim}}\nolimits \Gamma (X_ n, \mathcal{O}_{X_ n}) \]

surjects onto each $\Gamma (X_ n, \mathcal{O}_{X_ n})$ because the transition maps are surjections as the morphisms $X_ n \to X_{n + 1}$ are closed immersions. $\square$

Lemma 85.6.4. Let $S$ be a scheme. Let $X$ be a presheaf on $(\mathit{Sch}/S)_{fppf}$. The following are equivalent

  1. $X$ is a countably indexed affine formal algebraic space,

  2. $X = \text{Spf}(A)$ where $A$ is a weakly admissible topological $S$-algebra which has a countable fundamental system of neighbourhoods of $0$,

  3. $X = \text{Spf}(A)$ where $A$ is a weakly admissible topological $S$-algebra which has a fundamental system $A \supset I_1 \supset I_2 \supset I_3 \supset \ldots $ of weak ideals of definition,

  4. $X = \text{Spf}(A)$ where $A$ is a complete topological $S$-algebra with a fundamental system of open neighbourhoods of $0$ given by a countable sequence $A \supset I_1 \supset I_2 \supset I_3 \supset \ldots $ of ideals such that $I_ n/I_{n + 1}$ is locally nilpotent, and

  5. $X = \text{Spf}(A)$ where $A = \mathop{\mathrm{lim}}\nolimits B/J_ n$ with the limit topology where $B \supset J_1 \supset J_2 \supset J_3 \supset \ldots $ is a sequence of ideals in an $S$-algebra $B$ with $J_ n/J_{n + 1}$ locally nilpotent.

Proof. Assume (1). By Lemma 85.6.3 we can write $X = \text{Spf}(A)$ where $A$ is a weakly admissible topological $S$-algebra. For any presentation $X = \mathop{\mathrm{colim}}\nolimits X_ n$ as in Lemma 85.6.1 part (1) we see that $A = \mathop{\mathrm{lim}}\nolimits A_ n$ with $X_ n = \mathop{\mathrm{Spec}}(A_ n)$ and $A_ n = A/I_ n$ for some weak ideal of definition $I_ n \subset A$. This follows from the final statement of Lemma 85.5.6 which moreover implies that $\{ I_ n\} $ is a fundamental system of open neighbourhoods of $0$. Thus we have a sequence

\[ A \supset I_1 \supset I_2 \supset I_3 \supset \ldots \]

of weak ideals of definition with $A = \mathop{\mathrm{lim}}\nolimits A/I_ n$. In this way we see that condition (1) implies each of the conditions (2) – (5).

Assume (5). First note that the limit topology on $A = \mathop{\mathrm{lim}}\nolimits B/J_ n$ is a linearly topologized, complete topology, see More on Algebra, Section 15.35. If $f \in A$ maps to zero in $B/J_1$, then some power maps to zero in $B/J_2$ as its image in $J_1/J_2$ is nilpotent, then a further power maps to zero in $J_2/J_3$, etc, etc. In this way we see the open ideal $\mathop{\mathrm{Ker}}(A \to B/J_1)$ is a weak ideal of definition. Thus $A$ is weakly admissible. In this way we see that (5) implies (2).

It is clear that (4) is a special case of (5) by taking $B = A$. It is clear that (3) is a special case of (2).

Assume $A$ is as in (2). Let $E_ n$ be a countable fundamental system of neighbourhoods of $0$ in $A$. Since $A$ is a weakly admissible topological ring we can find open ideals $I_ n \subset E_ n$. We can also choose a weak ideal of definition $J \subset A$. Then $J \cap I_ n$ is a fundamental system of weak ideals of definition of $A$ and we get $X = \text{Spf}(A) = \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Spec}}(A/(J \cap I_ n))$ which shows that $X$ is a countably indexed affine formal algebraic space. $\square$

Lemma 85.6.5. Let $S$ be a scheme. Let $X$ be an affine formal algebraic space. The following are equivalent

  1. $X$ is Noetherian,

  2. $X$ is adic* and for some choice of $X = \mathop{\mathrm{colim}}\nolimits X_\lambda $ as in Definition 85.5.1 the schemes $X_\lambda $ are Noetherian,

  3. $X$ is adic* and for any closed immersion $T \to X$ with $T$ a scheme, $T$ is Noetherian.

Proof. This follows from the fact that if $A$ is a ring complete with respect to a finitely generated ideal $I$, then $A$ is Noetherian if and only if $A/I$ is Noetherian, see Algebra, Lemma 10.96.5. Details omitted. $\square$


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