The Stacks project

Proof. Assume (1). By Lemma 87.10.3 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 87.10.1 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 87.9.6 which moreover implies that $\{ I_ n\}$ is a fundamental system of open neighbourhoods of $0$. Thus we have a sequence

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. $\square$