# Tight closure

Last updated

In mathematics, in the area of commutative algebra, tight closure is an operation defined on ideals in positive characteristic. It was introduced by MelvinHochster and Craig Huneke  ( 1988 , 1990 ).

Let ${\displaystyle R}$ be a commutative noetherian ring containing a field of characteristic ${\displaystyle p>0}$. Hence ${\displaystyle p}$ is a prime number.

Let ${\displaystyle I}$ be an ideal of ${\displaystyle R}$. The tight closure of ${\displaystyle I}$, denoted by ${\displaystyle I^{*}}$, is another ideal of ${\displaystyle R}$ containing ${\displaystyle I}$. The ideal ${\displaystyle I^{*}}$ is defined as follows.

${\displaystyle z\in I^{*}}$ if and only if there exists a ${\displaystyle c\in R}$, where ${\displaystyle c}$ is not contained in any minimal prime ideal of ${\displaystyle R}$, such that ${\displaystyle cz^{p^{e}}\in I^{[p^{e}]}}$ for all ${\displaystyle e\gg 0}$. If ${\displaystyle R}$ is reduced, then one can instead consider all ${\displaystyle e>0}$.

Here ${\displaystyle I^{[p^{e}]}}$ is used to denote the ideal of ${\displaystyle R}$ generated by the ${\displaystyle p^{e}}$'th powers of elements of ${\displaystyle I}$, called the ${\displaystyle e}$th Frobenius power of ${\displaystyle I}$.

An ideal is called tightly closed if ${\displaystyle I=I^{*}}$. A ring in which all ideals are tightly closed is called weakly ${\displaystyle F}$-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 ${\displaystyle F}$-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 ${\displaystyle F}$-regular ring is ${\displaystyle F}$-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?

## Related Research Articles

In algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by , is the set of all prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. A locally ringed space of this form is called an affine scheme.

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of noncommutative rings where multiplication is not required to be commutative.

Commutative algebra is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers ; and p-adic integers.

Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. This is a glossary of some terms of the subject.

In mathematics, a Cohen–Macaulay ring is a commutative ring with some of the algebro-geometric properties of a smooth variety, such as local equidimensionality. Under mild assumptions, a local ring is Cohen–Macaulay exactly when it is a finitely generated free module over a regular local subring. Cohen–Macaulay rings play a central role in commutative algebra: they form a very broad class, and yet they are well understood in many ways.

In commutative algebra, a Gorenstein local ring is a commutative Noetherian local ring R with finite injective dimension as an R-module. There are many equivalent conditions, some of them listed below, often saying that a Gorenstein ring is self-dual in some sense.

In ring theory, a ring R is called a reduced ring if it has no non-zero nilpotent elements. Equivalently, a ring is reduced if it has no non-zero elements with square zero, that is, x2 = 0 implies x = 0. A commutative algebra over a commutative ring is called a reduced algebra if its underlying ring is reduced.

In mathematics, a sober space is a topological space X such that every irreducible closed subset of X is the closure of exactly one point of X: that is, every irreducible closed subset has a unique generic point.

In mathematics, ideal theory is the theory of ideals in commutative rings; and is the precursor name for the contemporary subject of commutative algebra. The name grew out of the central considerations, such as the Lasker–Noether theorem in algebraic geometry, and the ideal class group in algebraic number theory, of the commutative algebra of the first quarter of the twentieth century. It was used in the influential van der Waerden text on abstract algebra from around 1930.

Melvin Hochster is an American mathematician working in commutative algebra. He is currently the Jack E. McLaughlin Distinguished University Professor of Mathematics at the University of Michigan.

In commutative algebra, an element b of a commutative ring B is said to be integral overA, a subring of B, if there are n ≥ 1 and aj in A such that

In mathematics, a Prüfer domain is a type of commutative ring that generalizes Dedekind domains in a non-Noetherian context. These rings possess the nice ideal and module theoretic properties of Dedekind domains, but usually only for finitely generated modules. Prüfer domains are named after the German mathematician Heinz Prüfer.

In commutative algebra, a Krull ring or Krull domain is a commutative ring with a well behaved theory of prime factorization. They were introduced by Wolfgang Krull (1931). They are a higher-dimensional generalization of Dedekind domains, which are exactly the Krull domains of dimension at most 1.

In commutative algebra, an integrally closed domainA is an integral domain whose integral closure in its field of fractions is A itself. Spelled out, this means that if x is an element of the field of fractions of A which is a root of a monic polynomial with coefficients in A, then x is itself an element of A. Many well-studied domains are integrally closed: fields, the ring of integers Z, unique factorization domains and regular local rings are all integrally closed.

In algebra, the integral closure of an ideal I of a commutative ring R, denoted by , is the set of all elements r in R that are integral over I: there exist such that

Craig Lee Huneke is an American mathematician specializing in commutative algebra. He is a professor at the University of Virginia.

Karen Ellen Smith is an American mathematician, specializing in commutative algebra and algebraic geometry. She completed her bachelor's degree in mathematics at Princeton University before earning her PhD in mathematics at the University of Michigan in 1993. Currently she is the Keeler Professor of Mathematics at the University of Michigan. In addition to being a researcher in algebraic geometry and commutative algebra, Smith with others wrote the textbook An Invitation to Algebraic Geometry.

In algebra, the Hilbert–Kunz function of a local ring of prime characteristic p is the function

In algebra and algebraic geometry, given a commutative Noetherian ring and an ideal in it, the n-th symbolic power of is the ideal

## References

• Brenner, Holger; Monsky, Paul (2010), "Tight closure does not commute with localization", Annals of Mathematics , Second Series, 171 (1): 571–588, arXiv:, doi:10.4007/annals.2010.171.571, ISSN   0003-486X, MR   2630050
• Hochster, Melvin; Huneke, Craig (1988), "Tightly closed ideals", Bulletin of the American Mathematical Society, New Series, 18 (1): 45–48, doi:, ISSN   0002-9904, MR   0919658
• Hochster, Melvin; Huneke, Craig (1990), "Tight closure, invariant theory, and the Briançon–Skoda theorem", Journal of the American Mathematical Society , 3 (1): 31–116, doi:10.2307/1990984, ISSN   0894-0347, JSTOR   1990984, MR   1017784