**Serre’s Inequality on Height**
**Definition**
Serre’s inequality on height is a fundamental result in commutative algebra and algebraic geometry that provides an upper bound on the height of the sum of two ideals in a Noetherian ring. Specifically, it states that for two ideals ( I ) and ( J ) in a Noetherian ring ( R ), the height of their sum satisfies the inequality
[
operatorname{ht}(I + J) leq operatorname{ht}(I) + operatorname{ht}(J).
]
—
## Introduction
Serre’s inequality on height is a classical and widely used inequality in the theory of commutative algebra and algebraic geometry. It relates the heights of ideals in a Noetherian ring, providing a crucial tool for understanding the structure of prime ideals and their intersections. The inequality is named after the French mathematician Jean-Pierre Serre, who made significant contributions to algebraic geometry and commutative algebra.
The concept of height of an ideal is central in algebraic geometry and commutative algebra, as it measures the codimension of the variety defined by the ideal or, equivalently, the minimal number of elements needed to generate a prime ideal containing it. Serre’s inequality gives a bound on how the height behaves under the operation of taking sums of ideals, which corresponds geometrically to the intersection of algebraic sets.
This article provides a detailed exposition of Serre’s inequality on height, including its statement, proof outline, examples, applications, and related results.
—
## Background Concepts
### Noetherian Rings
A **Noetherian ring** is a ring in which every ascending chain of ideals stabilizes, meaning that there is no infinite strictly increasing sequence of ideals. This property ensures that ideals are finitely generated and is fundamental in algebraic geometry and commutative algebra.
### Ideals and Prime Ideals
An **ideal** ( I ) in a ring ( R ) is a subset closed under addition and multiplication by elements of ( R ). A **prime ideal** ( mathfrak{p} ) is an ideal such that if the product ( ab in mathfrak{p} ), then either ( a in mathfrak{p} ) or ( b in mathfrak{p} ).
### Height of an Ideal
The **height** (or codimension) of a prime ideal ( mathfrak{p} ) in a ring ( R ), denoted ( operatorname{ht}(mathfrak{p}) ), is the supremum of the lengths ( n ) of strictly increasing chains of prime ideals
[
mathfrak{p}_0 subsetneq mathfrak{p}_1 subsetneq cdots subsetneq mathfrak{p}_n = mathfrak{p}.
]
For a general ideal ( I ), the height is defined as the minimum height among all prime ideals containing ( I ):
[
operatorname{ht}(I) = min { operatorname{ht}(mathfrak{p}) mid mathfrak{p} supseteq I, mathfrak{p} text{ prime} }.
]
Geometrically, the height corresponds to the codimension of the algebraic set defined by the ideal.
—
## Statement of Serre’s Inequality on Height
Let ( R ) be a Noetherian ring, and let ( I, J subseteq R ) be two ideals. Then Serre’s inequality states that
[
operatorname{ht}(I + J) leq operatorname{ht}(I) + operatorname{ht}(J).
]
This inequality provides an upper bound on the height of the sum of two ideals in terms of the heights of the individual ideals.
—
## Intuition and Geometric Interpretation
In algebraic geometry, ideals correspond to algebraic sets (varieties or schemes). The sum of two ideals ( I + J ) corresponds to the intersection of the algebraic sets defined by ( I ) and ( J ). The height of an ideal corresponds to the codimension of the associated variety.
Serre’s inequality thus states that the codimension of the intersection of two algebraic sets is at most the sum of their codimensions. This aligns with the intuitive geometric idea that the intersection of two subvarieties of codimensions ( a ) and ( b ) should have codimension at most ( a + b ).
—
## Proof Outline
The proof of Serre’s inequality involves several key ideas from commutative algebra, including the use of prime ideals, chains of prime ideals, and properties of Noetherian rings. While the full proof is technical, the main steps can be summarized as follows:
1. **Reduction to Prime Ideals:**
Since the height of an ideal is defined via prime ideals containing it, the problem reduces to understanding the heights of prime ideals containing ( I + J ).
2. **Chains of Prime Ideals:**
Consider a chain of prime ideals leading up to a prime ideal ( mathfrak{p} ) containing ( I + J ). The goal is to relate this chain to chains of prime ideals containing ( I ) and ( J ) separately.
3. **Use of Prime Avoidance and Localization:**
Techniques such as prime avoidance and localization at prime ideals are used to analyze the structure of the ideals and their sums.
4. **Application of Dimension Theory:**
The dimension theory of Noetherian rings, including the behavior of heights under ring homomorphisms and extensions, is employed to establish the inequality.
5. **Combining Chains:**
By carefully combining chains of prime ideals containing ( I ) and ( J ), one constructs a chain containing ( I + J ) whose length is bounded by the sum of the lengths of the original chains.
The detailed proof can be found in advanced texts on commutative algebra and algebraic geometry.
—
## Examples
### Example 1: Polynomial Ring in Two Variables
Consider the polynomial ring ( R = k[x,y] ) over a field ( k ). Let
[
I = (x), quad J = (y).
]
Then ( I + J = (x,y) ), which is the maximal ideal corresponding to the origin.
– ( operatorname{ht}(I) = 1 ) since ( (x) ) is a prime ideal of height 1.
– ( operatorname{ht}(J) = 1 ) similarly.
– ( operatorname{ht}(I + J) = operatorname{ht}((x,y)) = 2 ).
Serre’s inequality states:
[
operatorname{ht}(I + J) leq operatorname{ht}(I) + operatorname{ht}(J) implies 2 leq 1 + 1 = 2,
]
which holds with equality.
### Example 2: Ideals in a Local Ring
Let ( R ) be a Noetherian local ring with maximal ideal ( mathfrak{m} ). Suppose ( I ) and ( J ) are ideals such that ( I subseteq mathfrak{m} ), ( J subseteq mathfrak{m} ), and ( operatorname{ht}(I) = 2 ), ( operatorname{ht}(J) = 3 ). Then
[
operatorname{ht}(I + J) leq 2 + 3 = 5.
]
This provides a useful bound on the height of the sum ideal.
—
## Applications
### Algebraic Geometry
In algebraic geometry, Serre’s inequality is used to estimate the codimension of intersections of algebraic varieties. It helps in understanding how the dimension of varieties behaves under intersection and is fundamental in intersection theory.
### Commutative Algebra
The inequality is a key tool in the study of prime ideals, dimension theory, and the structure of Noetherian rings. It is used in proofs of other important results, such as the Krull height theorem and properties of regular sequences.
### Homological Algebra
Serre’s inequality also appears in homological contexts, particularly in the study of depth and dimension of modules over Noetherian rings.
—
## Related Results
### Krull’s Height Theorem
Krull’s height theorem states that if an ideal ( I ) in a Noetherian ring ( R ) is generated by ( n ) elements, then every minimal prime ideal over ( I ) has height at most ( n ). This theorem complements Serre’s inequality by providing bounds on heights in terms of generators.
### Dimension Formula
The dimension formula relates the dimensions of rings and their extensions, often used in conjunction with Serre’s inequality to analyze the behavior of heights under ring homomorphisms.
### Subadditivity of Height
Serre’s inequality is an example of a subadditivity property of height functions, reflecting the general principle that codimension behaves subadditively under intersection.
—
## Generalizations and Extensions
Serre’s inequality has been generalized and extended in various directions, including:
– **Modules and Associated Primes:** Extensions to heights of associated primes of modules.
– **Non-Noetherian Settings:** Attempts to generalize the inequality to certain non-Noetherian rings.
– **Intersection Theory:** More refined inequalities in the context of intersection multiplicities and cycles.
—
## Historical Context
Jean-Pierre Serre introduced this inequality in the mid-20th century as part of his foundational work in algebraic geometry and commutative algebra. His insights into the structure of Noetherian rings and algebraic varieties have had a lasting impact on the field.
—
## Summary
Serre’s inequality on height is a fundamental inequality in commutative algebra and algebraic geometry that bounds the height of the sum of two ideals by the sum of their heights. It provides a crucial link between algebraic and geometric properties of ideals and varieties, facilitating the study of dimension, codimension, and intersection theory.
—
## References for Further Study
While this article does not include external links, readers interested in a deeper understanding of Serre’s inequality on height may consult advanced textbooks on commutative algebra and algebraic geometry, such as:
– „Commutative Algebra” by H. Matsumura
– „Introduction to Commutative Algebra” by M.F. Atiyah and I.G. Macdonald
– „Algebraic Geometry” by R. Hartshorne
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**Meta Description:**
Serre’s inequality on height is a key result in commutative algebra that bounds the height of the sum of two ideals by the sum of their individual heights, with important applications in algebraic geometry and dimension theory.