2.2 Ideals and varieties correspondence

Ideals and varieties form a bridge between algebra and geometry. They connect polynomial equations to geometric shapes, allowing us to study geometric objects using algebraic tools. This correspondence is fundamental to algebraic geometry.

The Nullstellensatz is key to understanding this relationship. It establishes a bijection between radical ideals and affine varieties, showing how algebraic and geometric objects are intimately linked in this field.

Algebraic sets and affine varieties

Definition and notation

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• An algebraic set is the set of solutions to a system of polynomial equations over a field $k$, denoted $V(S)$ where $S$ is a set of polynomials in $k[x₁, ..., xₙ]$
• An affine variety is an irreducible algebraic set, meaning it cannot be expressed as the union of two proper algebraic subsets

Zariski topology and geometric interpretation

• The Zariski topology on affine $n$-space over $k$ is defined by taking the closed sets to be the algebraic sets
• Affine varieties are the building blocks of algebraic geometry, analogous to manifolds in differential geometry (smooth manifolds, Riemannian manifolds)

Ideals and affine varieties

Correspondence between ideals and algebraic sets

• For any set $S$ of polynomials in $k[x₁, ..., xₙ]$, the set $I(S) = \{f ∈ k[x₁, ..., xₙ] : f(p) = 0$ for all $p ∈ V(S)\}$ is an ideal, called the ideal of $V(S)$
• For any ideal $I$ in $k[x₁, ..., xₙ]$, the set $V(I) = \{(a₁, ..., aₙ) ∈ k^n : f(a₁, ..., aₙ) = 0$ for all $f ∈ I\}$ is an algebraic set, called the zero set or vanishing set of $I$
• The correspondences $I ↦ V(I)$ and $V ↦ I(V)$ are inclusion-reversing: if $I₁ ⊆ I₂$ then $V(I₂) ⊆ V(I₁)$, and if $V₁ ⊆ V₂$ then $I(V₂) ⊆ I(V₁)$

Nullstellensatz and radical ideals

• The Nullstellensatz states that for any ideal $I$, $I(V(I)) = Rad(I)$, where $Rad(I)$ is the radical of $I$
  • The radical of an ideal $I$ is defined as $Rad(I) = \{f ∈ k[x₁, ..., xₙ] : f^m ∈ I$ for some $m ∈ ℕ\}$
  • An ideal $I$ is called a radical ideal if $I = Rad(I)$
• The Nullstellensatz establishes a bijective correspondence between radical ideals and affine varieties
  • Every affine variety $V$ corresponds to a unique radical ideal $I(V)$
  • Every radical ideal $I$ corresponds to a unique affine variety $V(I)$

Ideals of affine varieties

Definition and properties

• The ideal of an affine variety $V$, denoted $I(V)$, is the set of all polynomials that vanish on every point of $V$
• $I(V)$ is a radical ideal, meaning it is equal to its own radical: $I(V) = Rad(I(V))$
• The ideal $I(V)$ captures all the algebraic relations satisfied by the points of $V$

Computing generators for I(V)

• To find generators for $I(V)$, one can use elimination theory techniques such as Gröbner bases
  • A Gröbner basis is a particular generating set of an ideal with nice algorithmic properties
  • Buchberger's algorithm is a method for computing Gröbner bases
• Elimination theory deals with the problem of eliminating variables from a system of polynomial equations to obtain relations among the remaining variables
  • Resultants and discriminants are classical tools from elimination theory that can be used to compute generators for $I(V)$

Affine varieties from ideals

Definition and irreducibility

• The affine variety associated with an ideal $I$, denoted $V(I)$, is the set of all points in affine space that satisfy every polynomial in $I$
• $V(I)$ is an algebraic set, and it is irreducible (hence an affine variety) if and only if $I$ is a prime ideal
  • An ideal $I$ is called a prime ideal if whenever $fg ∈ I$ for some polynomials $f,g$, then either $f ∈ I$ or $g ∈ I$
  • Geometrically, this means that $V(I)$ cannot be decomposed as the union of two proper closed subsets

Dimension and explicit construction

• The dimension of $V(I)$ is equal to the Krull dimension of the quotient ring $k[x₁, ..., xₙ]/I$
  • The Krull dimension of a ring $R$ is the supremum of the lengths of all chains of prime ideals in $R$
  • Intuitively, it measures the "size" or "complexity" of the ring
• Constructing $V(I)$ explicitly involves solving a system of polynomial equations, which can be done using techniques from computational algebraic geometry
  • Gröbner bases can be used to transform the system into a triangular form that is easier to solve
  • Homotopy continuation methods numerically track solution paths from a simple start system to the target system
  • Resultants and eigenvalue methods can be used to reduce the problem to solving univariate polynomials