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
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
∈
I
∗
if and only if there exists a
c
∈
R
, where
c
is not contained in any minimal prime ideal of
R
, such that
c
z
p
e
∈
I
[
p
e
]
for all
e
≫
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.
Brenner & Monsky (2010) 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 also tightly closed?