In An elementary characterisation of Krull dimension and A short proof for the Krull dimension of a polynomial ring, Coquand, Lombardi, and Roy give an elementary characterization of Krull dimension, which inductively makes use of one of two notions of the "boundary" of a subvariety, given as follows:
Let $R$ be a commutative ring, and $x\in R$. \begin{align*} & \operatorname{upper boundary} R^{\{x\}} \mathrel{:=} R/I^{\{x\}}, && I^{\{x\}} \mathrel{:=} xR + (\sqrt{0}:x) \\ & \operatorname{lower boundary} R_{\{x\}} \mathrel{:=} S_{\{x\}}^{-1}R, && S_{\{x\}} \mathrel{:=} x^{\mathbb{N}}(1+xR) \end{align*} where $(\sqrt{0}:x)$ is the ideal quotient of the nilradical, and $x^{\mathbb{N}}(1+xR) = \{x^n(1+rx) \mathrel\vert \text{$n\in\mathbb{N}$, $r\in R$}\}$.
Upon inspection, $\mathrm{Spec}(R^{\{x\}})$ is $V(x) \cap \overline{\mathrm{Spec}R\setminus V(x)}$, and $\mathrm{Spec}(R_{\{x\}})$ is a localization (not quite open) that is disjoint from the locus $V(x)$. Also, both are trivial exactly when $x\in R^\times \cup \sqrt{0}$.
However, I do not have good intuition for these subschemes.
How to think about these boundary schemes? Do they represent anything in particular?
Do these constructions appear anywhere else in the literature? I have not been able to find anything.
Are they commutative, in that $R^{\{x\}\{y\}} = R^{\{y\}\{x\}}$ and $R_{\{x\}\{y\}} = R_{\{y\}\{x\}}$?
I suspect they are commutative, but am unable to prove it, and I have reservations stemming from the fact that permutations of a regular sequence are not necessarily regular.
- Are these very natural constructions? I.e. would it be worth studying them in more detail, in specific cases, or are they primarily instrumental in the characterization of Krull dimension?
I am willing to restrict to cases where $R$ is integral and Noetherian or has a finitely generated function field. It seems best to consider first the $R^{\{x_0\}...\{x_k\}}$, where $x_0, ..., x_k$ form a regular sequence, but I was not able to get much further with this assumption.