In mathematics, in the area of commutative algebra, tight closure is an operation defined on ideals in positive characteristic.
It was introduced by Melvin Hochster and Craig Huneke (1988, 1990).
be a commutative noetherian ring containing a field of characteristic
is used to denote the ideal of
th Frobenius power of
An ideal is called tightly closed if
A ring in which all ideals are tightly closed is called weakly
A previous major open question in tight closure is whether the operation of tight closure commutes with localization, and so there is the additional notion of
-regular, which says that all ideals of the ring are still tightly closed in localizations of the ring.
Brenner & Monsky (2010) found a counterexample to the localization property of tight closure.
However, there is still an open question of whether every weakly
That is, if every ideal in a ring is tightly closed, is it true that every ideal in every localization of that ring is also tightly closed?
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