Tight closure explained

In mathematics, in the area of commutative algebra, tight closure is an operation defined on ideals in positive characteristic. It was introduced by .

Let

be a commutative noetherian ring containing a field of characteristic

p>0

. Hence

is a prime number.

Let

be an ideal of

. The tight closure of

, denoted by

I^*

, is another ideal of

containing

. The ideal

I^*

is defined as follows.

z\inI^*

if and only if there exists a

c\inR

, where

is not contained in any minimal prime ideal of

, such that

	p^e
z

\in

	[p^e]
I

for all

e\gg0

. If

is reduced, then one can instead consider all

e>0

Here

	[p^e]
I

is used to denote the ideal of

generated by the

p^e

'th powers of elements of

, called the

th Frobenius power of

An ideal is called tightly closed if

I=I^*

. A ring in which all ideals are tightly closed is called weakly

-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

-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

-regular ring is

-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