Ideal reduction

Last updated August 13, 2023

The reduction theory goes back to the influential 1954 paper by Northcott and Rees, the paper that introduced the basic notions. In algebraic geometry, the theory is among the essential tools to extract detailed information about the behaviors of blow-ups.

Given ideals J ⊂ I in a ring R, the ideal J is said to be a reduction of I if there is some integer m > 0 such that $JI^{m}=I^{m+1}$ .^[1] For such ideals, immediately from the definition, the following hold:

For any k, $J^{k}I^{m}=J^{k-1}I^{m+1}=\cdots =I^{m+k}$ .
J and I have the same radical and the same set of minimal prime ideals over them^[2] (the converse is false).

If R is a Noetherian ring, then J is a reduction of I if and only if the Rees algebra R[It] is finite over R[Jt].^[3] (This is the reason for the relation to a blow up.)

A closely related notion is that of analytic spread. By definition, the fiber cone ring of a Noetherian local ring (R, ${\mathfrak {m}}$ ) along an ideal I is

{\mathcal {F}}_{I}(R)=R[It]\otimes _{R}\kappa ({\mathfrak {m}})\simeq \bigoplus _{n=0}^{\infty }I^{n}/{\mathfrak {m}}I^{n}

.

The Krull dimension of ${\mathcal {F}}_{I}(R)$ is called the analytic spread of I. Given a reduction $J\subset I$ , the minimum number of generators of J is at least the analytic spread of I.^[4] Also, a partial converse holds for infinite fields: if $R/{\mathfrak {m}}$ is infinite and if the integer $\ell$ is the analytic spread of I, then each reduction of I contains a reduction generated by $\ell$ elements.^[5]

Related Research Articles

In commutative algebra, the Krull dimension of a commutative ring R, named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally the Krull dimension can be defined for modules over possibly non-commutative rings as the deviation of the poset of submodules.

In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noetherian respectively. That is, every increasing sequence $of left ideals has a largest element; that is, there exists an n such that:$

Commutative algebra, first known as ideal theory, 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 $; and p -adic integers .$

In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their representations, or, in different language, modules, special classes of rings, as well as an array of properties that proved to be of interest both within the theory itself and for its applications, such as homological properties and polynomial identities.

In commutative algebra and algebraic geometry, localization is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing ring/module R, so that it consists of fractions $such that the denominator s belongs to a given subset S of R. If S is the set of the non-zero elements of an integral domain, then the localization is the field of fractions: this case generalizes the construction of the field of rational numbers from the ring of integers.$

In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral domain are like ideals where denominators are allowed. In contexts where fractional ideals and ordinary ring ideals are both under discussion, the latter are sometimes termed integral ideals for clarity.

In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let A be a Noetherian local ring with maximal ideal m, and suppose a₁, ..., a_n is a minimal set of generators of m. Then by Krull's principal ideal theorem n ≥ dim A, and A is defined to be regular if n = dim A.

In ring theory, a branch of mathematics, a ring 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 abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal.

In mathematics, more specifically abstract algebra and commutative algebra, Nakayama's lemma — also known as the Krull–Azumaya theorem — governs the interaction between the Jacobson radical of a ring and its finitely generated modules. Informally, the lemma immediately gives a precise sense in which finitely generated modules over a commutative ring behave like vector spaces over a field. It is an important tool in algebraic geometry, because it allows local data on algebraic varieties, in the form of modules over local rings, to be studied pointwise as vector spaces over the residue field of the ring.

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 a_j in A such that

In mathematics, the Artin–Rees lemma is a basic result about modules over a Noetherian ring, along with results such as the Hilbert basis theorem. It was proved in the 1950s in independent works by the mathematicians Emil Artin and David Rees; a special case was known to Oscar Zariski prior to their work.

In commutative algebra, a quasi-excellent ring is a Noetherian commutative ring that behaves well with respect to the operation of completion, and is called an excellent ring if it is also universally catenary. Excellent rings are one answer to the problem of finding a natural class of "well-behaved" rings containing most of the rings that occur in number theory and algebraic geometry. At one time it seemed that the class of Noetherian rings might be an answer to this problem, but Masayoshi Nagata and others found several strange counterexamples showing that in general Noetherian rings need not be well-behaved: for example, a normal Noetherian local ring need not be analytically normal.

In mathematics, specifically in algebraic geometry, a formal scheme is a type of space which includes data about its surroundings. Unlike an ordinary scheme, a formal scheme includes infinitesimal data that, in effect, points in a direction off of the scheme. For this reason, formal schemes frequently appear in topics such as deformation theory. But the concept is also used to prove a theorem such as the theorem on formal functions, which is used to deduce theorems of interest for usual schemes.

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 in 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, as shown by the following chain of class inclusions:

In commutative algebra, the Rees algebra of an ideal I in a commutative ring R is defined to be

In mathematics, an algebraic number field is an extension field $of the field of rational numbers such that the field extension has finite degree . Thus is a field that contains and has finite dimension when considered as a vector space over .$

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

In algebra, an analytically unramified ring is a local ring whose completion is reduced.

References

↑ Huneke & Swanson 2006 , Definition 1.2.1
↑ Huneke & Swanson 2006 , Lemma 8.1.10
↑ Huneke & Swanson 2006 , Theorem 8.2.1.
↑ Huneke & Swanson 2006 , Corollary 8.2.5.
↑ Huneke & Swanson 2006 , Proposition 8.3.7

Huneke, Craig; Swanson, Irena (2006), Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, vol. 336, Cambridge, UK: Cambridge University Press, ISBN 978-0-521-68860-4, MR 2266432, archived from the original on 2019-11-15, retrieved 2022-05-29

This commutative algebra-related article is a stub. You can help Wikipedia by expanding it.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] Huneke & Swanson 2006 , Definition 1.2.1

[2] Huneke & Swanson 2006 , Lemma 8.1.10

[3] Huneke & Swanson 2006 , Theorem 8.2.1.

[4] Huneke & Swanson 2006 , Corollary 8.2.5.

[5] Huneke & Swanson 2006 , Proposition 8.3.7

[1]

[2]

[3]

[4]

[5]