Tight closure does not commute with localization

Pages 571-588 from Volume 171 (2010), Issue 1 by Holger Brenner, Paul Monsky

Abstract

We give an example showing that tight closure does not commute with localization.

  • [brennertightproj] Go to document H. Brenner, "Tight closure and projective bundles," J. Algebra, vol. 265, iss. 1, pp. 45-78, 2003.
    @article {brennertightproj, MRKEY = {1984899},
      AUTHOR = {Brenner, Holger},
      TITLE = {Tight closure and projective bundles},
      JOURNAL = {J. Algebra},
      FJOURNAL = {Journal of Algebra},
      VOLUME = {265},
      YEAR = {2003},
      NUMBER = {1},
      PAGES = {45--78},
      ISSN = {0021-8693},
      CODEN = {JALGA4},
      MRCLASS = {13A35 (14H60)},
      MRNUMBER = {2004h:13008},
      MRREVIEWER = {Nobuo Hara},
      DOI = {10.1016/S0021-8693(03)00222-9},
      ZBLNUMBER = {1099.13010},
      }
  • [brennerslope] Go to document H. Brenner, "Slopes of vector bundles on projective curves and applications to tight closure problems," Trans. Amer. Math. Soc., vol. 356, iss. 1, pp. 371-392, 2004.
    @article {brennerslope, MRKEY = {2020037},
      AUTHOR = {Brenner, Holger},
      TITLE = {Slopes of vector bundles on projective curves and applications to tight closure problems},
      JOURNAL = {Trans. Amer. Math. Soc.},
      FJOURNAL = {Transactions of the American Mathematical Society},
      VOLUME = {356},
      YEAR = {2004},
      NUMBER = {1},
      PAGES = {371--392},
      ISSN = {0002-9947},
      CODEN = {TAMTAM},
      MRCLASS = {13A35},
      MRNUMBER = {2004m:13017},
      MRREVIEWER = {Anurag K. Singh},
      DOI = {10.1090/S0002-9947-03-03391-9},
      ZBLNUMBER = {1041.13002},
      }
  • [brennerhilbertkunz] Go to document H. Brenner, "The rationality of the Hilbert-Kunz multiplicity in graded dimension two," Math. Ann., vol. 334, iss. 1, pp. 91-110, 2006.
    @article {brennerhilbertkunz, MRKEY = {2208950},
      AUTHOR = {Brenner, Holger},
      TITLE = {The rationality of the {H}ilbert-{K}unz multiplicity in graded dimension two},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {334},
      YEAR = {2006},
      NUMBER = {1},
      PAGES = {91--110},
      ISSN = {0025-5831},
      CODEN = {MAANA},
      MRCLASS = {13D40 (13A35 13D02)},
      MRNUMBER = {2007d:13022},
      MRREVIEWER = {Florian Enescu},
      DOI = {10.1007/s00208-005-0703-x},
      ZBLNUMBER = {1098.13017},
      }
  • [brennertightplus] Go to document H. Brenner, "Tight closure and plus closure in dimension two," Amer. J. Math., vol. 128, iss. 2, pp. 531-539, 2006.
    @article {brennertightplus, MRKEY = {2214902},
      AUTHOR = {Brenner, Holger},
      TITLE = {Tight closure and plus closure in dimension two},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {128},
      YEAR = {2006},
      NUMBER = {2},
      PAGES = {531--539},
      ISSN = {0002-9327},
      CODEN = {AJMAAN},
      MRCLASS = {13A35},
      MRNUMBER = {2006m:13010},
      MRREVIEWER = {Adela N. Vraciu},
      URL = {http://muse.jhu.edu/journals/american_journal_of_mathematics/v128/128.2brenner.pdf},
      ZBLNUMBER = {1102.13002},
      }
  • [brennerkatzmanarithmetic] Go to document H. Brenner and M. Katzman, "On the arithmetic of tight closure," J. Amer. Math. Soc., vol. 19, iss. 3, pp. 659-672, 2006.
    @article {brennerkatzmanarithmetic, MRKEY = {2220102},
      AUTHOR = {Brenner, Holger and Katzman, Mordechai},
      TITLE = {On the arithmetic of tight closure},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {19},
      YEAR = {2006},
      NUMBER = {3},
      PAGES = {659--672},
      ISSN = {0894-0347},
      MRCLASS = {13A35 (11A41)},
      MRNUMBER = {2006m:13011},
      MRREVIEWER = {Karl Schwede},
      DOI = {10.1090/S0894-0347-05-00514-X},
      ZBLNUMBER = {1096.13005},
      }
  • [brunsherzog] W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge, UK: Cambridge Univ. Press, 1998, vol. 39.
    @book{brunsherzog,
      author={Bruns, W. and Herzog, J.},
      TITLE={Cohen-Macaulay Rings},
      SERIES={Cambridge Stud. Adv. Math.},
      VOLUME={39},
      YEAR={1998},
      PUBLISHER={Cambridge Univ. Press},
      ADDRESS={Cambridge, UK},
      MRNUMBER={95h:13020},
      MRREVIEWER={Matthew Miller},
      ZBLNUMBER={0909.13005},
      }
  • [hararational] Go to document N. Hara, "A characterization of rational singularities in terms of injectivity of Frobenius maps," Amer. J. Math., vol. 120, iss. 5, pp. 981-996, 1998.
    @article {hararational, MRKEY = {1646049},
      AUTHOR = {Hara, Nobuo},
      TITLE = {A characterization of rational singularities in terms of injectivity of {F}robenius maps},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {120},
      YEAR = {1998},
      NUMBER = {5},
      PAGES = {981--996},
      ISSN = {0002-9327},
      CODEN = {AJMAAN},
      MRCLASS = {13A99 (14B05 14E05)},
      MRNUMBER = {99h:13005},
      MRREVIEWER = {Karen E. Smith},
      URL = {http://muse.jhu.edu/journals/american_journal_of_mathematics/v120/120.5hara.pdf},
      ZBLNUMBER = {0942.13006},
      }
  • [hartshorneample] Go to document R. Hartshorne, "Ample vector bundles," Inst. Hautes Études Sci. Publ. Math., iss. 29, pp. 63-94, 1966.
    @article {hartshorneample, MRKEY = {0193092},
      AUTHOR = {Hartshorne, Robin},
      TITLE = {Ample vector bundles},
      JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      NUMBER = {29},
      YEAR = {1966},
      PAGES = {63--94},
      ISSN = {0073-8301},
      MRCLASS = {14.55 (14.20)},
      MRNUMBER = {33 \#1313},
      MRREVIEWER = {M. F. Atiyah},
      URL = {http://www.numdam.org/item?id=PMIHES_1966__29__63_0},
      ZBLNUMBER = {0173.49003},
      }
  • [haramp] R. Hartshorne, Ample Subvarieties of Algebraic Varieties, New York: Springer-Verlag, 1970, vol. 156.
    @book {haramp, MRKEY = {0282977},
      AUTHOR = {Hartshorne, Robin},
      TITLE = {Ample Subvarieties of Algebraic Varieties},
      SERIES = {Lecture Notes in Math.},
      VOLUME={156},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1970},
      PAGES = {xiv+256},
      MRCLASS = {14.05},
      MRNUMBER = {44 \#211},
      MRREVIEWER = {M. Miyanishi},
      ZBLNUMBER = {0208.48901},
      }
  • [haralg] R. Hartshorne, Algebraic Geometry, New York: Springer-Verlag, 1977.
    @book {haralg, MRKEY = {0463157},
      AUTHOR = {Hartshorne, Robin},
      TITLE = {Algebraic Geometry},
      SERIES = {Graduate Texts in Math.},
      NUMBER={52},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1977},
      PAGES = {xvi+496},
      ISBN = {0-387-90244-9},
      MRCLASS = {14-01},
      MRNUMBER = {57 \#3116},
      MRREVIEWER = {Robert Speiser},
      ZBLNUMBER = {0367.14001},
      }
  • [hochstersolid] M. Hochster, "Solid closure," in Commutative Algebra: Syzygies, Multiplicities, and Birational Algebra (South Hadley, MA, 1992), Providence, RI: Amer. Math. Soc., 1994, pp. 103-172.
    @incollection {hochstersolid, MRKEY = {1266182},
      AUTHOR = {Hochster, Melvin},
      TITLE = {Solid closure},
      BOOKTITLE = {Commutative Algebra: Syzygies, Multiplicities, and Birational Algebra ({S}outh {H}adley, {\rm MA},
      1992)},
      SERIES = {Contemp. Math.},
      NUMBER = {159},
      PAGES = {103--172},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {1994},
      MRCLASS = {13B99 (13A35)},
      MRNUMBER = {95a:13011},
      MRREVIEWER = {Ian M. Aberbach},
      ZBLNUMBER = {0812.13006},
      }
  • [hochstertightsolid] M. Hochster, "Tight closure in equal characteristic, big Cohen-Macaulay algebras, and solid closure," in Commutative Algebra: Syzygies, Multiplicities, and Birational Algebra (South Hadley, MA, 1992), Providence, RI: Amer. Math. Soc., 1994, pp. 173-196.
    @incollection {hochstertightsolid, MRKEY = {1266183},
      AUTHOR = {Hochster, Melvin},
      TITLE = {Tight closure in equal characteristic, big {C}ohen-{M}acaulay algebras, and solid closure},
      BOOKTITLE = {Commutative Algebra: Syzygies, Multiplicities, and Birational Algebra ({S}outh {H}adley, {\rm MA},
      1992)},
      SERIES = {Contemp. Math.},
      NUMBER = {159},
      PAGES = {173--196},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {1994},
      MRCLASS = {13B99 (13H10)},
      MRNUMBER = {95a:13012},
      ZBLNUMBER = {0803.13004},
      }
  • [hochsterhunekebriancon] Go to document M. Hochster and C. Huneke, "Tight closure, invariant theory, and the Brian\c con-Skoda theorem," J. Amer. Math. Soc., vol. 3, iss. 1, pp. 31-116, 1990.
    @article {hochsterhunekebriancon, MRKEY = {1017784},
      AUTHOR = {Hochster, Melvin and Huneke, Craig},
      TITLE = {Tight closure, invariant theory, and the {B}rian\c con-{S}koda theorem},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {3},
      YEAR = {1990},
      NUMBER = {1},
      PAGES = {31--116},
      ISSN = {0894-0347},
      MRCLASS = {13C05 (13A15 13A50 13B99 13D02)},
      MRNUMBER = {91g:13010},
      MRREVIEWER = {Luchezar L. Avramov},
      DOI = {10.2307/1990984},
      ZBLNUMBER={0701.13002},
      }
  • [hochsterhuneketestexponent] Go to document M. Hochster and C. Huneke, "Localization and test exponents for tight closure," Michigan Math. J., vol. 48, pp. 305-329, 2000.
    @article {hochsterhuneketestexponent, MRKEY = {1786493},
      AUTHOR = {Hochster, Melvin and Huneke, Craig},
      TITLE = {Localization and test exponents for tight closure},
      JOURNAL = {Michigan Math. J.},
      FJOURNAL = {The Michigan Mathematical Journal},
      VOLUME = {48},
      YEAR = {2000},
      PAGES = {305--329},
      ISSN = {0026-2285},
      MRCLASS = {13A35 (13B30 13D40)},
      MRNUMBER = {2002a:13001},
      MRREVIEWER = {Nobuo Hara},
      DOI = {10.1307/mmj/1030132721},
      ZBLNUMBER = {0993.13003},
      }
  • [hunekeapplication] C. Huneke, Tight Closure and its Applications, Providence, RI: Amer. Math. Soc., 1996.
    @book {hunekeapplication, MRKEY = {1377268},
      AUTHOR = {Huneke, Craig},
      TITLE = {Tight Closure and its Applications},
      SERIES = {CBMS Regional Conf. Ser. Math.},
      NUMBER = {88},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS={Providence, RI},
      YEAR = {1996},
      PAGES = {x+137},
      ISBN = {0-8218-0412-X},
      MRCLASS = {13-02 (13A35 13C14 13H10)},
      MRNUMBER = {96m:13001},
      MRREVIEWER = {Ian M. Aberbach},
      ZBLNUMBER = {0930.13004},
      }
  • [hunekeparameter] C. Huneke, "Tight closure, parameter ideals, and geometry," in Six Lectures on Commutative Algebra (Bellaterra, 1996), Basel: Birkhäuser, 1998, pp. 187-239.
    @incollection {hunekeparameter, MRKEY = {1648666},
      AUTHOR = {Huneke, Craig},
      TITLE = {Tight closure, parameter ideals, and geometry},
      BOOKTITLE = {Six Lectures on Commutative Algebra ({B}ellaterra, 1996)},
      SERIES = {Progr. Math.},
      NUMBER = {166},
      PAGES = {187--239},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Basel},
      YEAR = {1998},
      MRCLASS = {13A35 (13-02)},
      MRNUMBER = {99j:13001},
      MRREVIEWER = {Karen E. Smith},
      ZBLNUMBER = {0930.13005},
      }
  • [lyubezniksmithgraded] Go to document G. Lyubeznik and K. E. Smith, "Strong and weak $F$-regularity are equivalent for graded rings," Amer. J. Math., vol. 121, iss. 6, pp. 1279-1290, 1999.
    @article {lyubezniksmithgraded, MRKEY = {1719806},
      AUTHOR = {Lyubeznik, Gennady and Smith, Karen E.},
      TITLE = {Strong and weak {$F$}-regularity are equivalent for graded rings},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {121},
      YEAR = {1999},
      NUMBER = {6},
      PAGES = {1279--1290},
      ISSN = {0002-9327},
      CODEN = {AJMAAN},
      MRCLASS = {13A35 (13A02)},
      MRNUMBER = {2000m:13006},
      MRREVIEWER = {J. K. Verma},
      URL = {http://muse.jhu.edu/journals/american_journal_of_mathematics/v121/121.6lyubeznik.pdf},
      ZBLNUMBER = {0970.13003},
      }
  • [monskyhilbertkunzpointquartic] Go to document P. Monsky, "Hilbert-Kunz functions in a family: Point-$S_4$ quartics," J. Algebra, vol. 208, iss. 1, pp. 343-358, 1998.
    @article {monskyhilbertkunzpointquartic, MRKEY = {1644019},
      AUTHOR = {Monsky, Paul},
      TITLE = {Hilbert-{K}unz functions in a family: Point-{$S\sb 4$} quartics},
      JOURNAL = {J. Algebra},
      FJOURNAL = {Journal of Algebra},
      VOLUME = {208},
      YEAR = {1998},
      NUMBER = {1},
      PAGES = {343--358},
      ISSN = {0021-8693},
      CODEN = {JALGA4},
      MRCLASS = {13A35 (13D40)},
      MRNUMBER = {99k:13005},
      MRREVIEWER = {Elsa Zatini},
      DOI = {10.1006/jabr.1998.7500},
      ZBLNUMBER={0932.13010},
      }
  • [monskyhilbertkunzzd] P. Monsky, "On the Hilbert-Kunz function of $z^D-p_4(x,y)$," J. Algebra, vol. 291, iss. 2, pp. 350-372, 2005.
    @article {monskyhilbertkunzzd, MRKEY = {2163477},
      AUTHOR = {Monsky, Paul},
      TITLE = {On the {H}ilbert-{K}unz function of {$z\sp D-p\sb 4(x,y)$}},
      JOURNAL = {J. Algebra},
      FJOURNAL = {Journal of Algebra},
      VOLUME = {291},
      YEAR = {2005},
      NUMBER = {2},
      PAGES = {350--372},
      ISSN = {0021-8693},
      CODEN = {JALGA4},
      MRCLASS = {13D40 (28A80)},
      MRNUMBER = {2006m:13019},
      ZBLNUMBER = {1100.13018},
      }
  • [monskyfailure] P. Monsky, Tight closure’s failure to localize – a self-contained exposition, 2009.
    @misc{monskyfailure,
      author = {Monsky, Paul},
      TITLE = {Tight closure's failure to localize - a self-contained exposition},
      YEAR={2009},
      ARXIV={0902.1120},
      }
  • [smithparameter] Go to document K. E. Smith, "Tight closure of parameter ideals," Invent. Math., vol. 115, iss. 1, pp. 41-60, 1994.
    @article {smithparameter, MRKEY = {1248078},
      AUTHOR = {Smith, K. E.},
      TITLE = {Tight closure of parameter ideals},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {115},
      YEAR = {1994},
      NUMBER = {1},
      PAGES = {41--60},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {13A35 (13B99 13C99 13F40)},
      MRNUMBER = {94k:13006},
      MRREVIEWER = {Ian M. Aberbach},
      DOI = {10.1007/BF01231753},
      ZBLNUMBER = {0820.13007},
      }
  • [smithbinomial] Go to document K. E. Smith, "Tight closure commutes with localization in binomial rings," Proc. Amer. Math. Soc., vol. 129, iss. 3, pp. 667-669, 2001.
    @article {smithbinomial, MRKEY = {1706969},
      AUTHOR = {Smith, Karen E.},
      TITLE = {Tight closure commutes with localization in binomial rings},
      JOURNAL = {Proc. Amer. Math. Soc.},
      FJOURNAL = {Proceedings of the American Mathematical Society},
      VOLUME = {129},
      YEAR = {2001},
      NUMBER = {3},
      PAGES = {667--669},
      ISSN = {0002-9939},
      CODEN = {PAMYAR},
      MRCLASS = {13A35},
      MRNUMBER = {2001f:13008},
      MRREVIEWER = {Irena Swanson},
      DOI = {10.1090/S0002-9939-00-05626-4},
      ZBLNUMBER = {0966.13004},
      }
  • [smithvanishingsingularities] K. E. Smith, "Vanishing, singularities and effective bounds via prime characteristic local algebra," in Algebraic Geometry—Santa Cruz 1995, Providence, RI: Amer. Math. Soc., 1997, pp. 289-325.
    @incollection {smithvanishingsingularities, MRKEY = {1492526},
      AUTHOR = {Smith, Karen E.},
      TITLE = {Vanishing, singularities and effective bounds via prime characteristic local algebra},
      BOOKTITLE = {Algebraic Geometry---{S}anta {C}ruz 1995},
      SERIES = {Proc. Sympos. Pure Math.},
      NUMBER = {62},
      PAGES = {289--325},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {1997},
      MRCLASS = {14F17 (13A35 13D45 14B05)},
      MRNUMBER = {99a:14026},
      MRREVIEWER = {Ian M. Aberbach},
      ZBLNUMBER = {0913.13004},
      }

Mathematical Subject Classification
Primary: ,
DOI
http://doi.org/10.4007/annals.2010.171.571
MR
2630050
zbMATH
05712738
Milestones
Received: 30 October 2007
Accepted: 18 May 2009
Published online: 17 March 2010

Authors

Holger Brenner

Fachbereich Mathematik und Informatik, Universität Osnabrück, Albrechtstr. 28A, 49069 Osnabrück, Germany

Paul Monsky

Mathematics Department, >MS 050, Brandeis University, Waltham, MA 02254-9110, United States