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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Continuous closure, axes closure, and natural closure

Authors: Neil Epstein and Melvin Hochster
Journal: Trans. Amer. Math. Soc. 370 (2018), 3315-3362
MSC (2010): Primary 13B22, 13F45; Secondary 13A18, 46E25, 13B40, 13A15
Published electronically: December 26, 2017
MathSciNet review: 3766851
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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 $\mathbb {C}[x_1, x_2]$.

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Additional Information

Neil Epstein
Affiliation: Department of Mathematical Sciences, George Mason University, Fairfax, Virginia 22030
MR Author ID: 768826

Melvin Hochster
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1043
MR Author ID: 86705
ORCID: 0000-0002-9158-6486

Keywords: Continuous closure, axes closure, natural closure, seminormal ring
Received by editor(s): July 2, 2015
Received by editor(s) in revised form: July 20, 2016, and July 27, 2017
Published electronically: December 26, 2017
Additional Notes: The second-named author is grateful for support from the National Science Foundation, grants DMS-0901145 and DMS-1401384.
Article copyright: © Copyright 2017 American Mathematical Society