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.

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 $$e$$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.

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 is also tightly closed?