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Continuous closure, axes closure, and natural closure

Authors: Neil Epstein and Melvin Hochster
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 13B22, 13F45; Secondary 13A18, 46E25, 13B40, 13A15
Published electronically: December 26, 2017
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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

Melvin Hochster
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1043

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

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