- 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 Mel Hochster and Craig Huneke in the 1980s.
Let R be a commutative noetherian ring containing a field of characteristic p > 0. Hence p is a prime.
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.
- if and only if there exists a , where c is not contained in any minimal prime ideal of R, such that for all . If R is reduced, then one can instead consider all e > 0.
Here is used to denote the ideal of R generated by the pe'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.
In October 2007, Paul Monsky announced in a talk at Brandeis University that he and Brenner have found a counterexample to the localization property of tight closure. A preprint of this result is also available on the mathematics arXiv. 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 every ideal in every localization of that ring also tightly closed).
External links
- Abstract of Monsky's talk
- Preprint on the example of Monsky and Brenner that tight closure does not commute with localization.
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