Tight closure
Encyclopedia
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
be a commutative noetherian ring
containing a field
of characteristic
. Hence 
is a prime
.
Let
be an ideal of 
. The tight closure of 
, denoted by 
, is another ideal of 
containing 
. The ideal 
is defined as follows.
if and only if there exists a 
, where 
is not contained in any minimal prime ideal of 
, such that 
for all 
. If 
is reduced, then one can instead consider all 
.
Here
is used to denote the ideal of 
generated by the 
'th powers of elements of 
, called the 
th Frobenius
power of
.
An ideal is called tightly closed if
. 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.
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
-regular ring is 
-regular (that is, if every ideal in a ring is tightly closed, is every ideal in every localization of that ring also tightly closed).
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, in the area of commutative algebra
Commutative algebra
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
, tight closure is an operation defined on ideals
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
in positive characteristic. It was introduced by Mel Hochster
Melvin Hochster
Melvin Hochster is an eminent American mathematician, regarded as one of the leading commutative algebraists active today. He is currently the Jack E. McLaughlin Distinguished University Professor of Mathematics at the University of Michigan.Hochster attended Stuyvesant High School, where he was...
and Craig Huneke in the 1980s.
Let
be a commutative noetherian ringRing (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
containing a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
of characteristic
. Hence is a primePrime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
.
Let
be an ideal of . The tight closure of , denoted by , is another ideal of containing . The ideal is defined as follows. if and only if there exists a , where is not contained in any minimal prime ideal of , such that for all . If is reduced, then one can instead consider all .Here
is used to denote the ideal of generated by the 'th powers of elements of , called the th FrobeniusFrobenius endomorphism
In commutative algebra and field theory, the Frobenius endomorphism is a special endomorphism of commutative rings with prime characteristic p, an important class which includes finite fields. The endomorphism maps every element to its pth power...
power of
.An ideal is called tightly closed if
. 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 localizationLocalization of a ring
In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. Given a ring R and a subset S, one wants to construct some ring R* and ring homomorphism from R to R*, such that the image of S consists of units in R*...
, 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.In October 2007, Paul Monsky
Paul Monsky
Paul Monsky is an American mathematician and professor at Brandeis University.After earning a Bachelors degree from Swarthmore College, he received his Ph. D. in 1962 from the University of Chicago under the supervision of Walter Bailey...
announced in a talk at Brandeis University
Brandeis University
Brandeis University is an American private research university with a liberal arts focus. It is located in the southwestern corner of Waltham, Massachusetts, nine miles west of Boston. The University has an enrollment of approximately 3,200 undergraduate and 2,100 graduate students. In 2011, it...
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
-regular ring is -regular (that is, if every ideal in a ring is tightly closed, is every ideal in every localization of that ring also tightly closed).