Abstract Algebra

Commutative Algebra

When Hilbert proved his basis theorem in 1888, Gordan exclaimed: "This is theology, not mathematics!"-because Hilbert proved a finite basis exists without constructing it. Today, Buchberger's algorithm builds Gröbner bases explicitly, and commutative algebra has become a computational science.

  • Computer algebra systems (Mathematica, Sage): Gröbner bases solve polynomial systems-a direct application of Noetherian-ness
  • Robotics and kinematics: the configuration space of a robot is an algebraic variety whose ideal is computed via Gröbner bases
  • Number theory: rings of algebraic integers are Noetherian; Minkowski's lattice basis theorem

Предварительные знания

  • Linear Codes over Galois Fields

Primary Ideals and Primary Decomposition

A **primary ideal** q ⊊ R is an ideal where ab ∈ q implies a ∈ q or b^n ∈ q for some n ≥ 1. Its radical √q is a prime ideal. Analogy: prime ideals are like prime numbers; primary ideals are like prime powers (p^k). **Lasker-Noether theorem:** In a Noetherian ring, every ideal I decomposes as an intersection of primary ideals: I = q₁ ∩ q₂ ∩ ... ∩ qₙ (primary decomposition). This is the ideal-theoretic analog of prime factorization for integers! **Isolated vs. embedded components:** √qᵢ = pᵢ is the associated prime. The component qᵢ is isolated if pᵢ contains no other pⱼ. Embedded components correspond geometrically to "components with multiplicity".

**Uniqueness:** The isolated components of a primary decomposition are uniquely determined (Noether's theorem). Embedded components are not unique; their choice depends on the decomposition. This algebraically encodes the notion of "multiplicity": a point of tangency has an embedded component of the zero cycle.

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Is the ideal (4) in Z primary?

Hilbert's Nullstellensatz

The **Nullstellensatz** is the fundamental theorem connecting ideals of polynomials to their geometric zeros. **Weak form:** Let k be algebraically closed, I ⊊ k[x₁,...,xₙ] a proper ideal. Then V(I) = {a ∈ kⁿ : f(a)=0 for all f∈I} ≠ ∅. **Strong form:** If f ∈ k[x₁,...,xₙ] vanishes at every point of V(I), then f^m ∈ I for some m. That is: I(V(I)) = √I (the radical of I). **Galois correspondence:** {radical ideals I ⊆ k[x₁,...,xₙ]} ←→ {affine varieties V ⊆ kⁿ} I ↦ V(I) = {zeros of I}, I(V) ← V. **Maximal ideals:** m ↔ points {a} ∈ kⁿ; m = (x₁-a₁, ..., xₙ-aₙ). "Nullstellen" (zeros) are points; the ideal of a point is maximal.

**Algebraic closure is essential:** Over ℝ the theorem fails: I = (x²+1) is a proper ideal of ℝ[x] with V(I) = ∅. Over ℂ: V(x²+1) = {i, -i} ≠ ∅. This is why algebraic geometry works over algebraically closed fields-primarily ℂ or one of its subfields.

Let k be algebraically closed, I = (x², y) ⊆ k[x,y]. What is I(V(I))?

Noetherian Rings and Localization

A ring R is **Noetherian** if any of the following equivalent conditions holds: 1. (ACC) Every ascending chain of ideals I₁ ⊆ I₂ ⊆ ... stabilizes. 2. Every ideal is finitely generated. 3. Every nonempty family of ideals has a maximal element. **Hilbert's basis theorem:** If R is Noetherian, then R[x] is Noetherian. Corollary: k[x₁,...,xₙ] is Noetherian for any field k. Every ideal in it is finitely generated! **Localization S⁻¹R:** Let S ⊆ R be a multiplicative set (closed under multiplication, 1∈S). S⁻¹R = {a/s : a∈R, s∈S} / ~, where a/s ~ b/t iff ∃u∈S: u(at-bs)=0. **Localization at a prime:** R_p = S⁻¹R, S = R \ p. This is a **local ring** with unique maximal ideal S⁻¹p. Geometrically: "functions defined in a neighborhood of the point p".

**Geometric meaning of localization:** If R = k[V] is the coordinate ring of an affine variety V, then the localization R_p at a prime p (corresponding to a point x ∈ V) is the ring of "germs of regular functions at x": fractions f/g where g(x) ≠ 0. The local ring R_p "sees" only the local geometry of V near x-this is the basis of sheaf theory and schemes.

The ring k[x₁,...,xₙ] is Noetherian (Hilbert's basis theorem). What does this mean in practice?

Key Ideas

  • Primary decomposition: I = q₁ ∩ ... ∩ qₙ-the analog of factorization for ideals
  • Nullstellensatz: I(V(I)) = √I-algebra and geometry mirror each other
  • Noetherian ring: ACC for ideals iff every ideal is finitely generated
  • Hilbert's basis theorem: R Noetherian implies R[x] Noetherian implies k[x₁,...,xₙ] Noetherian

Further Directions

Commutative algebra is the foundation of algebraic geometry. Noetherian rings, primary decomposition, and localization translate directly into the language of affine schemes. Grothendieck's theory of schemes is commutative algebra made geometric.

  • Algebraic Geometry — Affine varieties are V(I) for ideals I ⊆ k[x₁,...,xₙ]; the Nullstellensatz is the dictionary between algebra and geometry
  • Morita Theory — Local rings are the simplest locally ringed spaces; module theory over local rings is the foundation of Morita theory

Вопросы для размышления

  • Prove Hilbert's basis theorem: if R is Noetherian then R[x] is Noetherian. Use leading coefficients.
  • Find the primary decomposition of the ideal (x²y, xy²) in k[x,y]. What are the associated prime ideals?
  • Show that the localization k[x,y]_(x,y) (at the maximal ideal of the origin) is a local ring. What is its maximal ideal and residue field?

Связанные уроки

  • nt-09
Commutative Algebra