Abstract
We give an example showing that tight closure does not commute with localization.
-
[brennertightproj]
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]
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},
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ZBLNUMBER = {1041.13002},
} -
[brennerhilbertkunz]
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AUTHOR = {Brenner, Holger},
TITLE = {The rationality of the {H}ilbert-{K}unz multiplicity in graded dimension two},
JOURNAL = {Math. Ann.},
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VOLUME = {334},
YEAR = {2006},
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ISSN = {0025-5831},
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MRCLASS = {13D40 (13A35 13D02)},
MRNUMBER = {2007d:13022},
MRREVIEWER = {Florian Enescu},
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ZBLNUMBER = {1098.13017},
} -
[brennertightplus]
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},
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FJOURNAL = {American Journal of Mathematics},
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NUMBER = {2},
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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]
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AUTHOR = {Brenner, Holger and Katzman, Mordechai},
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MRNUMBER = {2006m:13011},
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ZBLNUMBER = {1096.13005},
} -
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author={Bruns, W. and Herzog, J.},
TITLE={Cohen-Macaulay Rings},
SERIES={Cambridge Stud. Adv. Math.},
VOLUME={39},
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PUBLISHER={Cambridge Univ. Press},
ADDRESS={Cambridge, UK},
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ZBLNUMBER={0909.13005},
} -
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AUTHOR = {Hara, Nobuo},
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} -
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AUTHOR = {Hartshorne, Robin},
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} -
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AUTHOR = {Hartshorne, Robin},
TITLE = {Algebraic Geometry},
SERIES = {Graduate Texts in Math.},
NUMBER={52},
PUBLISHER = {Springer-Verlag},
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MRCLASS = {14-01},
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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]
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]
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]
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]
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]
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]
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},
}