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| In [[mathematics]], in the area of [[commutative algebra]], '''tight closure''' is an operation defined on [[ideal (ring theory)|ideals]] in positive [[characteristic of a ring|characteristic]]. It was introduced by {{harvs|txt|authorlink=Melvin Hochster|first=Melvin|last= Hochster|first2=Craig |last2=Huneke|author2-link=Craig Huneke|year1=1988|year2= 1990}}.
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| Let <math>R</math> be a commutative noetherian [[ring (mathematics)|ring]] containing a [[field (mathematics)|field]] of characteristic <math>p > 0</math>. Hence <math>p</math> is a [[prime number]].
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| Let <math>I</math> be an ideal of <math>R</math>. The tight closure of <math>I</math>, denoted by <math>I^*</math>, is another ideal of <math>R</math> containing <math>I</math>. The ideal <math>I^*</math> is defined as follows.
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| :<math>z \in I^*</math> if and only if there exists a <math>c \in R</math>, where <math>c</math> is not contained in any minimal prime ideal of <math>R</math>, such that <math>c z^{p^e} \in I^{[p^e]}</math> for all <math>e \gg 0</math>. If <math>R</math> is reduced, then one can instead consider all <math>e > 0</math>.
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| Here <math>I^{[p^e]}</math> is used to denote the ideal of <math>R</math> generated by the <math>p^e</math>'th powers of elements of <math>I</math>, called the <math>e</math>th [[Frobenius endomorphism|Frobenius]] power of <math>I</math>.
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| An ideal is called tightly closed if <math>I = I^*</math>. A ring in which all ideals are tightly closed is called weakly <math>F</math>-regular (for Frobenius regular). A previous major open question in tight closure is whether the operation of tight closure commutes with [[localization of a ring|localization]], and so there is the additional notion of <math>F</math>-regular, which says that all ideals of the ring are still tightly closed in localizations of the ring.
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| {{harvtxt|Brenner|Monsky|2010}} found a counterexample to the localization property of tight closure. However, there is still an open question of whether every weakly <math>F</math>-regular ring is <math>F</math>-regular. That is, if every ideal in a ring is tightly closed, is true that every ideal in every localization of that ring also tightly closed?
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| ==References==
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| *{{Citation | last1=Brenner | first1=Holger | last2=Monsky | first2=Paul | title=Tight closure does not commute with localization | url=http://arxiv.org/abs/0710.2913 | doi=10.4007/annals.2010.171.571 | id={{MR|2630050}} | year=2010 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=171 | issue=1 | pages=571–588}}
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| *{{Citation | last1=Hochster | first1=Melvin | last2=Huneke | first2=Craig | title=Tightly closed ideals | url=http://dx.doi.org/10.1090/S0273-0979-1988-15592-9 | doi=10.1090/S0273-0979-1988-15592-9 | id={{MR|919658}} | year=1988 | journal=American Mathematical Society. Bulletin. New Series | issn=0002-9904 | volume=18 | issue=1 | pages=45–48}}
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| *{{Citation | last1=Hochster | first1=Melvin | last2=Huneke | first2=Craig | title=Tight closure, invariant theory, and the Briançon–Skoda theorem | url=http://dx.doi.org/10.2307/1990984 | doi=10.2307/1990984 | id={{MR|1017784}} | year=1990 | journal=[[Journal of the American Mathematical Society]] | issn=0894-0347 | volume=3 | issue=1 | pages=31–116}}
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| [[Category:Commutative algebra]]
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| [[Category:Ideals]]
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