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 Template:Harvs.

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.

Template:Harvtxt 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?

References

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