TITLE: How to understand the "boundary" of subscheme, as defined in "An elementary characterisation of Krull dimension"
QUESTION [5 upvotes]: 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.

REPLY [3 votes]: This is also not really an answer, but it explains an alternative version of the characterisation that I find easier to work with.  Let $P_d(R)$ be the polynomial ring over $R$ in variables $x_0,\dotsc,x_d$, and order the monomials lexicographically.  Say that $f\in P_d(R)$ is comonic if the lowest monomial has coefficient one.  Say that an $R$-algebra homomorphism $\phi\colon P_d(R)\to R$ is thin if the kernel contains a comonic polynomial.  After a little translation, the results of Coquand and Lombardi say that $R$ has dimension $\leq d$ iff every such homomorphism is thin.