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 be a commutative noetherian ring containing a field of characteristic . Hence is a prime number.

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

if and only if there exists a , where is not contained in any minimal prime ideal of , such that for all . If is reduced, then one can instead consider all .

Here is used to denote the ideal of generated by the 'th powers of elements of , called the th Frobenius power of .

An ideal is called tightly closed if . A ring in which all ideals are tightly closed is called weakly -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 -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 -regular ring is -regular. 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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