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.

$z \in I^*$ 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.