Tight closure

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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