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 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.

$z \in I^*$ 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.

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 is true that 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.