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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Continuous closure, axes closure, and natural closure
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by Neil Epstein and Melvin Hochster PDF
Trans. Amer. Math. Soc. 370 (2018), 3315-3362 Request permission

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
  • Email: nepstei2@gmu.edu
  • Melvin Hochster
  • Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1043
  • MR Author ID: 86705
  • ORCID: 0000-0002-9158-6486
  • Email: hochster@umich.edu
  • 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.
  • © Copyright 2017 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 370 (2018), 3315-3362
  • MSC (2010): Primary 13B22, 13F45; Secondary 13A18, 46E25, 13B40, 13A15
  • DOI: https://doi.org/10.1090/tran/7031
  • MathSciNet review: 3766851