Constructing a ring whose spectrum is given by order ideals of Z with generic point

Question

Put the following topology on $\mathbf{Z}_{>0} \cup \infty$: the closed sets are the initial invervals $\{1,\dots,n\}$ for all $n$. If I understood Hochster's characterization of the underlying spaces of affine schemes, this space can be realized as Spec(R) for some commutative ring $R$. My question is how to give a construction of $R$. The simpler the description, the better, but I am not interested in a reference to general results which guarantee that this can be done without giving an explicit construction.

Related question, how does one realize the subspaces $\{1,\dots,n\}$ (with the induced topology) as the spectrum of a ring? When $n=2$, take any DVR, but for $n=3$ it's already not clear to me what one should think about.

The motivation for this problem comes from joint work with Andrew Snowden on "twisted commutative algebras" (tca) where these kinds of spaces arise as some notion of spectra of tca and we were wondering if it connects to classical ring theory in any reasonable sense.

Answer 1

Consider the abelian group $\Gamma:=\mathbb{Z}^{(\mathbb{Z}_{>0})}$ of sequences $f:\mathbb{Z}_{>0}\to\mathbb{Z}$ with finite support. Order $\Gamma$ "inverse-lexicographically" by declaring a sequence $f:\mathbb{Z}_{>0}\to\mathbb{Z}$ to be $>0$ if its last nonzero term is $>0$.

We get a totally ordered abelian group $(\Gamma,\leq)$. Its convex subgroups are $$\Gamma_0=\{0\}\;\subset\; \Gamma_1=\mathbb{Z}\times\{0\}\times\cdots\subset \Gamma_2=\mathbb{Z}\times\mathbb{Z}\times\{0\}\times\cdots\;\subset\;\cdots\subset \Gamma_\infty=\Gamma. $$

Now let $(R,v)$ be any valuation ring with value group $\Gamma$. Its prime ideals are in bijection with the above subgroups, by the rule $\mathfrak{p}_n=\{x\,\vert\, v(x)>\Gamma_n\}$. Thus $\mathrm{Spec}(R)$ is the space you want. The finite variant is obtained similarly, replacing sequences by $n$-tuples of integers.

To get an explicit valuation ring $R$ with any given group $(\Gamma,+,\leq)$, fix a field $k$ and take the ring of "Hahn series" associated to $k$ and $\Gamma$: these are formal sums $\sum_{\gamma\in\Gamma_{\geq0}}a_\gamma x^{\gamma}$ with $a_\gamma$ in $k$, subject to the condition that the support $\{\gamma\,\vert\, a_\gamma\neq0\}$ is well-ordered; the ring operations and the valuation are the obvious ones.