Continuous closure, axes closure, and natural closure
HTML articles powered by AMS MathViewer
- 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 $\mathbb {C}[x_1, x_2]$.References
- A. Andreotti and E. Bombieri, Sugli omeomorfismi delle varietà algebriche, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 23 (1969), 431–450 (Italian). MR 266923
- Ian M. Aberbach, Melvin Hochster, and Craig Huneke, Localization of tight closure and modules of finite phantom projective dimension, J. Reine Angew. Math. 434 (1993), 67–114. MR 1195691, DOI 10.1515/crll.1993.434.67
- M. Artin, Algebraic approximation of structures over complete local rings, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 23–58. MR 268188, DOI 10.1007/BF02684596
- E. Bombieri, Seminormalità e singolarità ordinarie, Symposia Mathematica, Vol. XI (Convegno di Algebra Commutativa, INDAM, Roma, Novembre 1971), Academic Press, London, 1973, pp. 205–210. MR 0429874
- Holger Brenner, Continuous solutions to algebraic forcing equations, arXiv:math.AC /0608611v2, 2006.
- Paul M. Eakin Jr., The converse to a well known theorem on Noetherian rings, Math. Ann. 177 (1968), 278–282. MR 225767, DOI 10.1007/BF01350720
- David Eisenbud and Melvin Hochster, A Nullstellensatz with nilpotents and Zariski’s main lemma on holomorphic functions, J. Algebra 58 (1979), no. 1, 157–161. MR 535850, DOI 10.1016/0021-8693(79)90196-0
- Neil Epstein, Reductions and special parts of closures, J. Algebra 323 (2010), no. 8, 2209–2225. MR 2596375, DOI 10.1016/j.jalgebra.2010.02.015
- Christopher Francisco, Lee Klingler, Sean Sather-Wagstaff, and Janet C. Vassilev (eds.), Progress in commutative algebra 2, Walter de Gruyter GmbH & Co. KG, Berlin, 2012. Closures, finiteness and factorization. MR 2920512, DOI 10.1515/9783110278606
- Charles Fefferman and János Kollár, Continuous solutions of linear equations, From Fourier analysis and number theory to Radon transforms and geometry, Dev. Math., vol. 28, Springer, New York, 2013, pp. 233–282. MR 2986959, DOI 10.1007/978-1-4614-4075-8_{1}0
- Gavin J. Gibson, Seminormality and $F$-purity in local rings, Osaka J. Math. 26 (1989), no. 2, 245–251. MR 1017583
- S. Greco and C. Traverso, On seminormal schemes, Compositio Math. 40 (1980), no. 3, 325–365. MR 571055
- Terence Gaffney and Marie A. Vitulli, Weak subintegral closure of ideals, Adv. Math. 226 (2011), no. 3, 2089–2117. MR 2739774, DOI 10.1016/j.aim.2010.09.020
- Shiro Goto and Keiichi Watanabe, The structure of one-dimensional $F$-pure rings, J. Algebra 49 (1977), no. 2, 415–421. MR 453729, DOI 10.1016/0021-8693(77)90250-2
- Melvin Hochster and Craig Huneke, $F$-regularity, test elements, and smooth base change, Trans. Amer. Math. Soc. 346 (1994), no. 1, 1–62. MR 1273534, DOI 10.1090/S0002-9947-1994-1273534-X
- Melvin Hochster and Craig Huneke, Tight closure of parameter ideals and splitting in module-finite extensions, J. Algebraic Geom. 3 (1994), no. 4, 599–670. MR 1297848
- Melvin Hochster and Joel L. Roberts, Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay, Advances in Math. 13 (1974), 115–175. MR 347810, DOI 10.1016/0001-8708(74)90067-X
- Craig Huneke and Irena Swanson, Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, vol. 336, Cambridge University Press, Cambridge, 2006. MR 2266432
- Shiroh Itoh, On weak normality and symmetric algebras, J. Algebra 85 (1983), no. 1, 40–50. MR 723066, DOI 10.1016/0021-8693(83)90117-5
- János Kollár, Continuous closure of sheaves, Michigan Math. J. 61 (2012), no. 3, 475–491. MR 2975256, DOI 10.1307/mmj/1347040253
- John V. Leahy and Marie A. Vitulli, Seminormal rings and weakly normal varieties, Nagoya Math. J. 82 (1981), 27–56. MR 618807, DOI 10.1017/S0027763000019279
- Mirella Manaresi, Some properties of weakly normal varieties, Nagoya Math. J. 77 (1980), 61–74. MR 556308, DOI 10.1017/S0027763000018663
- Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
- Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0155856
- Masayoshi Nagata, A type of subrings of a noetherian ring, J. Math. Kyoto Univ. 8 (1968), 465–467. MR 236162, DOI 10.1215/kjm/1250524062
- Richard G. Swan, On seminormality, J. Algebra 67 (1980), no. 1, 210–229. MR 595029, DOI 10.1016/0021-8693(80)90318-X
- Carlo Traverso, Seminormality and Picard group, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 24 (1970), 585–595. MR 277542
- Marie A. Vitulli, Weak normality and seminormality, Commutative algebra—Noetherian and non-Noetherian perspectives, Springer, New York, 2011, pp. 441–480. MR 2762521, DOI 10.1007/978-1-4419-6990-3_{1}7
- Ken-ichi Yoshida, On birational-integral extension of rings and prime ideals of depth one, Japan. J. Math. (N.S.) 8 (1982), no. 1, 49–70. MR 722521, DOI 10.4099/math1924.8.49
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