11.2 Primary decomposition and associated primes
4 min read•july 30, 2024
Primary decomposition and associated primes are powerful tools in algebraic geometry. They help us break down complex ideals into simpler parts, giving us insights into the structure of algebraic sets and modules.
These concepts connect algebraic geometry to commutative algebra by linking geometric properties to algebraic ones. They're crucial for understanding the local behavior of varieties, their irreducible components, and singularities.
Primary Decomposition of Ideals and Modules
Definitions and Properties
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- A proper ideal Q in a commutative ring R is called primary if for any a,b ∈ R with ab ∈ Q, either a ∈ Q or b^n ∈ Q for some positive integer n
- The radical of a Q, denoted √Q, is a prime ideal
- If √Q = P, then Q is called P-primary
- Every proper ideal in a can be expressed as a finite intersection of primary ideals, called a primary decomposition
- A primary decomposition is called minimal if:
- The prime ideals √Qi are all distinct
- Qi ⊈ ⋂_{j≠i} Qj for each i
Associated Primes and Modules
- The prime ideals √Qi in a minimal primary decomposition are called the associated primes of the ideal
- The concept of primary decomposition can be extended to modules over a Noetherian ring
- Submodules take the place of ideals in this context
- Associated primes play a key role in understanding the structure of modules
- The set of associated primes of a module M is denoted Ass(M)
- For an ideal I, Ass(R/I) corresponds to the associated primes of I
Computing Primary Decompositions
Polynomial Rings and Gr�bner Bases
- To find a primary decomposition of an ideal I in a polynomial ring k[x1, ..., xn]:
- Start by finding a decomposition I = Q1 ∩ ... ∩ Qr, where the Qi are primary ideals
- The associated primes are then Pi = √Qi for each i
- The associated primes can be computed using techniques such as finding the minimal primes containing I
- Gr�bner basis techniques can be used to compute primary decompositions in more complex cases
- Buchberger's algorithm for computing Gr�bner bases is a key tool (Buchberger, 1965)
Simple Cases and Examples
- In simple cases, the primary decomposition can often be found by inspection or using the structure theorem for modules over a PID
- Example: In the ring Z, the ideal (12) has primary decomposition (12) = (2^2) ∩ (3)
- Monomial ideals in polynomial rings also have straightforward primary decompositions
- Example: In k[x,y], the ideal (x^2, xy) has primary decomposition (x^2, xy) = (x) ∩ (x^2, y)
Primary Decomposition for Algebraic Varieties
Irreducible Components and Dimension
- If I is an ideal in a polynomial ring k[x1, ..., xn], the algebraic set V(I) is the set of points in k^n where all polynomials in I vanish
- The minimal associated primes of I correspond to the irreducible components of V(I)
- The dimension of each component is given by the codimension of the corresponding prime ideal
- Example: Let I = (xy, xz) in C[x,y,z]. Then V(I) has two irreducible components:
- V(x), which has codimension 1 and dimension 2
- V(y,z), which has codimension 2 and dimension 1
Embedded Components and Singularities
- The primary decomposition of I can be used to study the local structure of V(I) at its singular points
- The embedded primes (associated primes that are not minimal) correspond to the embedded components of V(I)
- The multiplicity of a component can be determined from the length of the corresponding primary ideal in the at a point
- Example: Let I = (x^2, xy) in C[x,y]. The primary decomposition is I = (x) ∩ (x^2, y)
- V(I) has one irreducible component V(x) of dimension 1
- V(x^2, y) is an embedded component consisting of the origin, which is a singular point
Primary Decomposition vs Geometric Properties
Irreducible and Embedded Components
- The irreducible components of an algebraic set V(I) correspond to the minimal associated primes of I
- These are the primary components of I of maximum dimension
- The embedded components of V(I) correspond to the embedded primes of I
- These are the primary components of I of smaller dimension that are "embedded" in the larger components
- Example: In the previous example, V(x) is the irreducible component and V(x^2, y) is the embedded component
Local Rings and Support
- The local ring of V(I) at a point P is obtained by localizing the coordinate ring k[x1, ..., xn]/I at the maximal ideal corresponding to P
- The primary decomposition of I determines the structure of this local ring
- The support of a module M over k[x1, ..., xn]/I is the set of points in V(I) where the localization of M is non-zero
- This can be determined from the associated primes of M
- Example: Let M = (k[x,y]/(xy))_x be the localization of k[x,y]/(xy) at the prime ideal (x). Then:
- Ass(M) = {(x), (y)}
- Supp(M) = V(x) ∪ V(y), the union of the x and y axes in k^2
Invariants of Algebraic Sets
- The primary decomposition of I can be used to compute invariants of V(I) such as:
- Dimension: the maximum dimension of the irreducible components
- Degree: the sum of the degrees of the irreducible components, weighted by their multiplicities
- Arithmetic genus: a measure of the "complexity" of the singularities of V(I)
- These invariants provide key information about the geometry and topology of the algebraic set V(I)
- They can be used to classify algebraic varieties and study their moduli spaces
Key Terms to Review (16)
Associated Prime: An associated prime of a ring is a prime ideal that corresponds to the zero divisors of a module over that ring, indicating where the module fails to be free. These primes reveal important structural information about the module and are closely related to primary decomposition, as they help classify the components of the module into more manageable pieces. Understanding associated primes provides insights into the depth and regularity of modules, especially in the context of Cohen-Macaulay rings.
Cohen-Macaulay ring: A Cohen-Macaulay ring is a type of commutative ring that satisfies certain depth and dimension conditions, making it a key object of study in algebraic geometry and commutative algebra. It is characterized by having a well-behaved structure, where the depth of every ideal equals its height, ensuring that the ring has desirable properties such as finite generation of its modules. This notion plays an important role in understanding primary decomposition and associated primes.
David Hilbert: David Hilbert was a renowned German mathematician who made significant contributions to various fields of mathematics, particularly algebra, geometry, and mathematical logic. His work laid the foundations for much of modern mathematics and provided deep insights into the relationships between algebraic structures and geometric concepts.
Groebner Basis: A Groebner basis is a particular kind of generating set for an ideal in a polynomial ring, which has desirable algorithmic properties that simplify the process of solving systems of polynomial equations. It allows for effective computation in algebraic geometry and commutative algebra, particularly when working with primary decomposition and associated primes.
Height of a Prime: The height of a prime ideal in a ring is defined as the maximum length of a chain of prime ideals contained within it. This concept is crucial in understanding the structure of algebraic varieties and their associated prime ideals, particularly in the context of primary decomposition and associated primes.
Homology: Homology refers to a mathematical concept in algebraic topology that provides a way to associate a sequence of algebraic objects, typically groups, with a topological space. This association allows for the study of the topological properties of spaces by translating them into algebraic terms, offering insights into their structure and relationships. In the context of primary decomposition and associated primes, homology plays a crucial role in understanding the intersection and union of subspaces, which can be analyzed through their homological features.
Integral Closure: Integral closure refers to the set of elements in a ring that are integral over that ring, meaning they satisfy a polynomial equation with coefficients in that ring. This concept helps in understanding how certain subrings can be extended to encompass all elements that behave nicely with respect to the original ring, especially in relation to ideals and primary decomposition.
Irreducible variety: An irreducible variety is a type of algebraic variety that cannot be expressed as the union of two or more proper subvarieties. This concept is fundamental in understanding how varieties can be decomposed and analyzed in terms of their structure. An irreducible variety represents a 'building block' in algebraic geometry, serving as a critical component when discussing decomposition and the relationship between primary decomposition and associated primes.
Local Ring: A local ring is a type of ring that has a unique maximal ideal, which means it is focused around a single point or a specific 'local' aspect. This structure allows for the study of properties and behaviors of algebraic objects in a neighborhood, making it essential in various areas like algebraic geometry and commutative algebra.
Noether's Primary Decomposition Theorem: Noether's Primary Decomposition Theorem states that for any ideal in a Noetherian ring, it can be expressed as an intersection of primary ideals. This theorem plays a crucial role in understanding the structure of ideals and their associated varieties, as it provides a way to analyze the geometric properties of the underlying algebraic objects by breaking down complex ideals into simpler components.
Noetherian Ring: A Noetherian ring is a ring in which every ascending chain of ideals eventually stabilizes, meaning there are no infinitely increasing sequences of ideals. This property is crucial in algebraic geometry because it ensures that every ideal is finitely generated, which facilitates the construction and understanding of polynomial rings, coordinate rings, and local rings. Additionally, Noetherian rings help simplify the study of Krull dimension and primary decomposition by providing a framework where these concepts can be effectively analyzed.
Oscar Zariski: Oscar Zariski was a prominent mathematician known for his foundational contributions to algebraic geometry and commutative algebra. His work laid the groundwork for many modern developments in these fields, including the concept of primary decomposition, which relates closely to the study of ideals in rings and their geometric interpretations.
Primary Component: A primary component in algebraic geometry is a building block of a primary decomposition of an ideal, specifically representing the set of points that correspond to the closure of the zero set of that ideal. Each primary component captures a certain aspect of the algebraic structure, helping to decompose the ideal into simpler, manageable parts, and making it easier to analyze properties like dimension and associated primes.
Primary Ideal: A primary ideal is an ideal in a ring such that if the product of two elements belongs to the ideal, then at least one of the elements is in the ideal or some power of the other element is in the ideal. This concept connects deeply with the structure of rings and ideals, as it allows for a refined understanding of their decomposition and associated prime ideals.
Radical Ideal: A radical ideal is an ideal in a ring such that if a power of an element is in the ideal, then the element itself is also in the ideal. This concept connects deeply with the structure of coordinate rings, where radical ideals help describe the properties of affine varieties and their points. Radical ideals play a crucial role in the Zariski topology, as they relate to the closure of sets and help understand the relationship between algebraic sets and their corresponding coordinate rings.
Zariski's Lemma: Zariski's Lemma is a fundamental result in algebraic geometry that states that if a point is contained in an affine variety, then the maximal ideal of the coordinate ring corresponding to that point can be represented by a set of generators consisting of polynomials vanishing at that point. This lemma connects deeply with various aspects of algebraic varieties, such as coordinate rings, projective varieties, isomorphisms, and primary decomposition, by providing a clear relationship between geometric properties and algebraic structures.