🧮Commutative Algebra Unit 8 – Primary Decomposition & Associated Primes

Key Concepts and Definitions

• Primary ideals are proper ideals $Q$ of a commutative ring $R$ where for every $a,b \in R$, if $ab \in Q$, then either $a \in Q$ or $b^n \in Q$ for some positive integer $n$
• Associated primes of an $R$-module $M$ are prime ideals $P$ of $R$ such that $P = Ann(m)$ for some $m \in M$, where $Ann(m) = \{r \in R : rm = 0\}$
• Minimal primes are prime ideals that do not properly contain any other prime ideal
• Irreducible ideals cannot be written as the intersection of two strictly larger ideals
• Primary decomposition of an ideal $I$ is an expression of $I$ as a finite intersection of primary ideals, i.e., $I = Q_1 \cap Q_2 \cap \cdots \cap Q_n$, where each $Q_i$ is primary
  • Minimal primary decomposition has no redundant components and the radicals of the $Q_i$ are all distinct
• Support of a module $M$ is the set of prime ideals $P$ such that the localization $M_P \neq 0$
• Artinian rings satisfy the descending chain condition on ideals, meaning every descending chain of ideals stabilizes

Historical Context and Development

• Primary decomposition has its roots in the work of David Hilbert and Emmy Noether in the early 20th century
• Hilbert's Nullstellensatz, which relates ideals in polynomial rings to algebraic varieties, motivated the study of primary decomposition
• Emmy Noether's work on the decomposition of ideals in polynomial rings laid the foundation for the general theory
• The concept of associated primes was introduced by Wolfgang Krull in the 1920s
• Krull's principal ideal theorem states that in a Noetherian ring, every minimal prime over a principal ideal has height at most 1
• The theory of primary decomposition was further developed by mathematicians such as Claude Chevalley, Masayoshi Nagata, and Irving Kaplansky in the mid-20th century
• Primary decomposition has since become a fundamental tool in commutative algebra and algebraic geometry

Fundamental Theorems

• Lasker-Noether Theorem: In a Noetherian ring, every ideal admits a primary decomposition
  • The radical of each primary component is a prime ideal, called an associated prime of the original ideal
• Uniqueness Theorem: In a Noetherian ring, every ideal has a minimal primary decomposition, which is unique up to the order of the components
• Krull's Intersection Theorem: In a Noetherian local ring $(R, \mathfrak{m})$, the intersection of all powers of the maximal ideal $\mathfrak{m}$ is zero, i.e., $\bigcap_{n=1}^\infty \mathfrak{m}^n = (0)$
• Krull's Principal Ideal Theorem: If $R$ is a Noetherian ring and $a \in R$ is a non-zero divisor, then every minimal prime over $(a)$ has height at most 1
• Associativity Formula: For a finitely generated module $M$ over a Noetherian ring $R$, $Ass(M) = \bigcup_{i=0}^n Ass(Ext_R^i(M,R))$

Examples and Applications

• In the ring of integers $\mathbb{Z}$, the ideal $(12)$ has primary decomposition $(12) = (2^2) \cap (3)$, where $(2^2)$ is primary and $(3)$ is prime
• In the polynomial ring $k[x,y]$ over a field $k$, the ideal $(x^2, xy)$ has primary decomposition $(x^2, xy) = (x) \cap (x^2, y)$, where $(x)$ is prime and $(x^2, y)$ is primary
• Primary decomposition is used in algebraic geometry to study the structure of algebraic varieties
  • The associated primes of an ideal correspond to the irreducible components of the corresponding algebraic variety
• In the study of singularities, primary decomposition helps analyze the local structure of a variety near a singular point
• Primary decomposition is applied in solving systems of polynomial equations and in computational algebraic geometry
• Associated primes are used to understand the support and annihilator of a module
• Primary decomposition is a key tool in the study of local cohomology and the computation of Hilbert functions

Computational Techniques

• Gröbner bases can be used to compute primary decompositions in polynomial rings
  • Buchberger's algorithm is a key method for computing Gröbner bases
• Symbolic computation software like Macaulay2, Singular, and Sage have built-in functions for primary decomposition
• Shimoyama-Yokoyama algorithm is a method for computing primary decompositions in polynomial rings over fields
• Eisenbud-Huneke-Vasconcelos algorithm computes primary decompositions in local rings
• Gianni-Trager-Zacharias algorithm is another method for primary decomposition in polynomial rings
• Homological techniques, such as the computation of Ext and Tor functors, are used to find associated primes
• Localization and completion are often employed to reduce primary decomposition problems to simpler cases

Relationship to Other Algebraic Structures

• Primary decomposition is closely related to the theory of modules over a ring
  • Associated primes and primary submodules play a crucial role in the structure theory of modules
• In the context of group rings, primary decomposition is connected to the study of group representations and characters
• Primary decomposition is a key tool in the study of homological algebra, particularly in the computation of local cohomology and the structure of derived categories
• In algebraic geometry, primary decomposition is related to the decomposition of algebraic varieties into irreducible components
• The concept of primary decomposition has analogues in non-commutative settings, such as the theory of Artinian rings and modules
• Primary decomposition is connected to the theory of Stanley-Reisner rings and simplicial complexes in combinatorial commutative algebra

Advanced Topics and Extensions

• Symbolic powers of ideals are related to primary decomposition and have applications in algebraic geometry and commutative algebra
• The Hilbert-Kunz function, which measures the growth of the length of quotients by powers of an ideal, is related to the associated primes of the ideal
• Tight closure is a closure operation on ideals that extends the notion of integral closure and is connected to primary decomposition
• Multiplier ideals, which arise in complex analytic geometry, are related to primary decomposition and have applications in birational geometry
• Rees algebras and associated graded rings provide a way to study the asymptotic behavior of ideals and their powers
• Intersection theory in algebraic geometry uses primary decomposition to define intersection multiplicities and study the geometry of intersections
• Characteristic cycles and characteristic varieties, which encode topological information about a module, are related to associated primes and primary decomposition

Common Pitfalls and Misconceptions

• Not every ideal in a ring admits a primary decomposition; the ring must be Noetherian for the Lasker-Noether Theorem to apply
• The primary decomposition of an ideal is not always unique; uniqueness holds only for minimal primary decompositions in Noetherian rings
• Associated primes are not the same as the prime factors in a primary decomposition; they are the radicals of the primary components
• The set of associated primes of an ideal can be strictly smaller than the set of prime ideals containing the ideal
• The support of a module is not always a closed subset of the prime spectrum; it is closed in the Zariski topology only for finitely generated modules over Noetherian rings
• The primary decomposition of a module is not as well-behaved as that of an ideal; modules may have infinite primary decompositions or no primary decomposition at all
• Computing primary decompositions can be computationally expensive, especially in rings with many variables or large degree polynomials
• The concept of primary decomposition does not extend directly to non-commutative rings; analogues such as tertiary decomposition and the Artin-Wedderburn theorem are used instead