15.3 Correspondence between ideals and varieties

Radical ideals and algebraic varieties form a powerful connection in commutative algebra. This correspondence links abstract algebra to geometry, allowing us to visualize polynomial equations as shapes in space.

Understanding this relationship helps us solve equations, analyze geometric properties, and explore the interplay between algebra and geometry. It's a fundamental tool for studying polynomial systems and their solutions.

Correspondence between Ideals and Varieties

Radical ideals vs algebraic varieties

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• Bijective correspondence establishes one-to-one relationship between radical ideals and affine algebraic varieties
• Radical ideals defined as $I$ where $\sqrt{I} = I$ characterize when $f^n \in I$ implies $f \in I$ (roots)
• Affine algebraic varieties represent common zeros of polynomials in affine space $V(I) = \{x \in k^n : f(x) = 0 \text{ for all } f \in I\}$ (plane curves, surfaces)
• Hilbert's Nullstellensatz proves $I(V(J)) = \sqrt{J}$ connecting radical ideals and varieties (fundamental bridge)

Ideals and varieties computation

• Ideal of variety $I(V)$ contains all polynomials vanishing on $V$ $I(V) = \{f \in k[x_1, \ldots, x_n] : f(x) = 0 \text{ for all } x \in V\}$ (polynomial constraints)
• Variety of ideal $V(I)$ gives common zeros of polynomials in $I$ $V(I) = \{x \in k^n : f(x) = 0 \text{ for all } f \in I\}$ (solution sets)
• Computing $I(V)$ requires finding vanishing polynomials (interpolation)
• Finding $V(I)$ involves solving polynomial system $f(x) = 0$ for $f \in I$ (root finding)

Prime ideals and irreducible varieties

• Prime ideals $P$ satisfy $ab \in P$ implies $a \in P$ or $b \in P$ making $k[x_1, \ldots, x_n]/P$ integral domain
• Irreducible varieties cannot decompose into proper subvarieties yielding integral domain coordinate ring $k[V]$
• Prime ideals correspond to irreducible varieties: $V(P)$ irreducible for prime $P$, $I(V)$ prime for irreducible $V$
• Geometrically prime ideals represent indivisible varieties (elliptic curves, irreducible polynomials)

Applications of ideal-variety correspondence

• Variety intersections match ideal sums: $V(I) \cap V(J) = V(I + J)$ (intersection of curves)
• Variety unions relate to ideal products: $V(I) \cup V(J) = V(IJ)$ (union of surfaces)
• Variety containment links to radical containment: $V(I) \subseteq V(J)$ iff $\sqrt{J} \subseteq \sqrt{I}$ (subvarieties)
• Variety dimension connects to coordinate ring Krull dimension (curves, surfaces, hypersurfaces)
• Analyzing local rings of ideals reveals variety singularities and smoothness (cusps, nodes)
• Ideal primary decomposition corresponds to variety decomposition (factoring curves)
• Zariski topology defines closed sets as algebraic varieties enabling geometric study (continuity, compactness)