Continuous closure, axes closure, and natural closure

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 continuous closure $I^{\mathrm {cont}}$ of an ideal $I$ as $I\mathcal {C}(X) \cap R$. He also introduced an algebraic notion of axes closure $I^{\mathrm {ax}}$ that always contains $I^{\mathrm {cont}}$, and asked whether they coincide. We extend the notion of axes closure to general Noetherian rings, defining $f \in I^{\mathrm {ax}}$ 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 natural closure $I^{\natural }$ of $I$. One of many characterizations is $I^{\natural } = I + \{f \in R: \exists n >0 \mathrm {\ with\ } f^n \in I^{n+1}\}$. We show that $I^{\natural } \subseteq I^{\mathrm {ax}}$ and that when continuous closure is defined, $I^{\natural } \subseteq I^{\mathrm {cont}} \subseteq I^{\mathrm {ax}}$. Under mild hypotheses on the ring, we show that $I^{\natural } = I^{\mathrm {ax}}$ when $I$ is primary to a maximal ideal and that if $I$ has no embedded primes, then $I = I^{\natural }$ if and only if $I = I^{\mathrm {ax}}$, so that $I^{\mathrm {cont}}$ agrees as well. We deduce that in the polynomial ring $\mathbb {C} \lbrack x_1, \ldots , x_n \rbrack$, 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 $I^{\mathrm {cont}}$ for monomial ideals in polynomial rings over $\mathbb {C}$, but we show that the inequalities $I^{\natural } \subseteq I^{\mathrm {cont}}$ and $I^{\mathrm {cont}} \subseteq I^{\mathrm {ax}}$ can be strict for monomial ideals even in dimension 3. Thus, $I^{\mathrm {cont}}$ and $I^{\mathrm {ax}}$ need not agree, although we prove they are equal in $\mathbb {C}[x_1, x_2]$.