Tight closure

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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).

Let R be a commutative noetherian ring containing a field of characteristic p > 0. Hence p is a prime number.

Let I be an ideal of R. The tight closure of I, denoted by I^*, is another ideal of R containing I. The ideal I^* is defined as follows.

z \in I^* if and only if there exists a c \in R, where c is not contained in any minimal prime ideal of R, such that c z^{p^e} \in I^{[p^e]} for all e \gg 0. If R is reduced, then one can instead consider all e > 0.

Here I^{[p^e]} is used to denote the ideal of R generated by the p^e'th powers of elements of I, called the eth Frobenius power of I.

An ideal is called tightly closed if I = I^*. A ring in which all ideals are tightly closed is called weakly F-regular (for Frobenius regular). 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 F-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 F-regular ring is F-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?