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 Melvin Hochster and Craig Huneke (1988, 1990).

Let ${\displaystyle R}$ be a commutative noetherian ring containing a field of characteristic ${\displaystyle p>0}$. Hence ${\displaystyle p}$ is a prime number.

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

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

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

An ideal is called tightly closed if ${\displaystyle I=I^{*}}$. A ring in which all ideals are tightly closed is called weakly ${\displaystyle F}$-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 ${\displaystyle F}$-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 ${\displaystyle F}$-regular ring is ${\displaystyle F}$-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?