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 ∈ I ∗ if and only if there exists a
c ∈ R , where
c is not contained in any minimal prime ideal of
R , such that
c z p e ∈ I [ p e ] for all
e ≫ 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 e th 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 it true that every ideal in every localization of that ring also tightly closed?
