Continuous closure, axes closure, and natural closure

Details Continuous closure, axes closure, and natural closure Continuous closure, axes closure, and natural closure

2011-06-17

arxiv.org/abs/1106.3462v1

Mathematics

Commutative Algebra

43 pages

Scientific paper

Let $R$ be a reduced affine $\C$-algebra, and let $X$ be the corresponding affine algebraic set. Brenner defined the \emph{continuous closure} $I\ct$ of an ideal $I$ as the ideal of elements of $R$ that can be written as linear combinations of elements of $I$ with coefficients from the ring of $\C$-valued continuous (in the Euclidean topology) functions on $X$. He also introduced an algebraic notion of \emph{axes closure} $I\ax$ in such a ring $R$ that always contains $I\ct$, and he raised the question of whether they coincide. To attack this problem, we extend the notion of axes closure to general Noetherian rings, defining $f \in I\ax$ 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} $I\ncl$ of $I$. One characterization among many is that $I\ncl$ is the sum of $I$ and the ideal of all elements $f \in R$ such that $f^n \in I^{n+1}$ for some $n >0$. We show that $I\ncl \subseteq I\ax$, and that whenever continuous closure is defined, we have $I\ncl \subseteq I\ct \subseteq I\ax$. Under mild hypotheses on the ring, we show that $I\ncl = I\ax$ when $I$ is primary to a maximal ideal, and that if $I$ has no embedded primes, then $I = I\ncl$ if and only if $I = I\ax$, so that $I\ct$, trapped in between, agrees as well. One consequence is that if a polynomial over $\C$ vanishes whenever its partial derivatives vanish, then it is in the continuous closure of the ideal they generate. We show that for monomial ideals in polynomial rings over $\C$ that $I\ncl = I\ct$, but we show by example that the inequality $I\ct \subset I\ax$ can be strict even for monomial ideals in dimension 3. Thus, $I\ct$ and $I\ax$ do not agree in general, although we prove that they do agree in polynomial rings of dimension at most 2 over $\C$.

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