The Stacks project

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 15.36. 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. $\square$


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