Continuous closure, axes closure, and natural closure

TL;DR

This paper explores different notions of closure for ideals in algebraic rings, comparing continuous, axes, and natural closures, establishing their relationships, and providing specific characterizations and examples in polynomial rings.

Contribution

It extends axes closure to general Noetherian rings, introduces natural closure, and compares these with continuous closure, including explicit characterizations and counterexamples.

Findings

Natural closure is contained in axes closure, which contains continuous closure.

In polynomial rings, continuous closure equals axes closure in two variables.

Counterexamples show strict inclusions among closures in higher dimensions.

Abstract

Let $R$ be a reduced affine $\mathbb C$-algebra, with corresponding affine algebraic set $X$. Let $\mathcal C(X)$ be the ring of continuous (Euclidean topology) $\mathbb C$-valued functions on $X$. Brenner defined the \emph{continuous closure} $I^{\rm cont}$ of an ideal $I$ as $I\mathcal C(X) \cap R$. He also introduced an algebraic notion of \emph{axes closure} $I^{\rm ax}$ that always contains $I^{\rm cont}$, and asked whether they coincide. We extend the notion of axes closure to general Noetherian rings, defining $f \in I^{\rm ax}$ if its image is in $IS$ for every homomorphism $R \to S$, where $S$ is a one-dimensional complete seminormal local ring. We also introduce the \emph{natural closure} $I^\natural$ of $I$. One of many characterizations is $I^\natural = I + \{f \in R: \exists n >0 \text{ with } f^n \in I^{n+1}\}$. We show that $I^\natural \subseteq I^{\rm ax}$, and that when continuous closure is defined, $I^\natural \subseteq I^{\rm cont }\subseteq I^{\rm ax}$. Under mild hypotheses on the ring, we show that $I^\natural= I^{\rm ax}$ when $I$ is primary to a maximal ideal, and that if $I$ has no embedded primes, then $I = I^\natural$ if and only if $I = I^{\rm ax}$, so that $I^{\rm cont}$ agrees as well. We deduce that in the polynomial ring $\mathbb C[x_1, \ldots, x_n]$, if $f = 0$ at all points where all of the ${\partial f \over \partial x_i}$ are 0, then $f \in ( {\partial f \over \partial x_1}, \, \ldots, \, {\partial f \over \partial x_n})R$. We characterize $I^{\rm cont}$ for monomial ideals in polynomial rings over $\mathbb C$, but we show that the inequalities $I^\natural \subset I^{\rm cont}$ and $I^{\rm cont} \subset I^{\rm ax}$ can be strict for monomial ideals even in dimension 3. Thus, $I^{\rm cont}$ and $I^{\rm ax}$ need not agree, although we prove they are equal in $\mathbb C[x_1, x_2]$.