Introduction to Algebraic Geometry — Mathematics Knowledge System

Overview

Algebraic geometry studies geometric objects — varieties — defined as solution sets of polynomial equations. Classical algebraic geometry works over algebraically closed fields (especially $\mathbb{C}$); modern algebraic geometry (after Grothendieck) works with schemes, sheaves, and cohomology over arbitrary rings. The field unifies algebra, geometry, topology, and number theory.

Intuition

Algebraic geometry makes precise the geometric study of polynomial equations. Just as a circle $x^2 + y^2 = 1$ is a geometric object with algebraic structure, arbitrary polynomial systems define varieties. The profound insight (Hilbert, Grothendieck) is that geometry and algebra are two sides of the same coin: geometric properties translate into algebraic properties of coordinate rings, and vice versa.

Formal Definition

Definition

Let $k$ be an algebraically closed field and $\mathbb{A}^n = k^n$ affine $n$-space. An affine variety is the zero locus of a collection of polynomials. The Zariski topology on $\mathbb{A}^n$ makes closed sets exactly the affine varieties.

V(I) = \{\mathbf{x} \in \mathbb{A}^n : f(\mathbf{x}) = 0 \text{ for all } f \in I\} \quad (I \subset k[x_1,\ldots,x_n])

Affine variety

I(V) = \{f \in k[x_1,\ldots,x_n] : f(\mathbf{x}) = 0 \text{ for all } \mathbf{x} \in V\}

Ideal of a variety

\sqrt{I} = I(V(I)) \quad \text{(Hilbert's Nullstellensatz)}

Nullstellensatz

\{\text{affine varieties over }k\}^{\text{op}} \longleftrightarrow \{\text{reduced finitely generated }k\text{-algebras}\}

Algebra-geometry duality

Worked Examples

$V(x^2 - y) = \{(t, t^2) : t \in \mathbb{C}\}$, the parabola.
$V(x^2 - y) = \{(t,t^2) : t \in \mathbb{C}\}$
The coordinate ring is $k[V] = k[x,y]/(x^2-y) \cong k[t]$ via $x \mapsto t$, $y \mapsto t^2$.
$k[V] = \mathbb{C}[x,y]/(x^2-y) \cong \mathbb{C}[t]$
This is a polynomial ring (no relations), reflecting that the parabola is isomorphic (as a variety) to the affine line $\mathbb{A}^1$.
$V(x^2-y) \cong \mathbb{A}^1$

Answer: The parabola $V(x^2-y)$ has coordinate ring $\mathbb{C}[t]$ and is isomorphic to $\mathbb{A}^1$.

Practice Problems

Difficulty 7/10

Describe the Zariski topology on $\mathbb{A}^1$ over an algebraically closed field $k$ and compare it to the usual Euclidean topology on $\mathbb{R}$.

Difficulty 8/10

Define an irreducible variety and show that $V(xy) \subset \mathbb{A}^2$ is reducible.

Difficulty 9/10

Define a projective variety and explain why projective space $\mathbb{P}^n$ is a compactification of $\mathbb{A}^n$.

Common Mistakes

Common Mistake

The Zariski topology is Hausdorff.

The Zariski topology is almost never Hausdorff (except in trivial cases). Any two nonempty open sets in the Zariski topology on an irreducible variety intersect nontrivially.

Common Mistake

Every ideal $I$ satisfies $I(V(I)) = I$.

By Nullstellensatz, $I(V(I)) = \sqrt{I}$, the radical. Only radical ideals satisfy $I(V(I)) = I$. For example, $I = (x^2)$ and $I(V(x^2)) = I(V(x)) = (x) = \sqrt{(x^2)} \neq (x^2)$.

Quiz

Summary

Algebraic geometry studies varieties — zero sets of polynomials — using the Zariski topology and coordinate rings.
The Nullstellensatz establishes a bijection between radical ideals and affine varieties over algebraically closed fields.
Irreducible varieties correspond to prime ideals; the coordinate ring is an integral domain.
Projective space $\mathbb{P}^n$ compactifies $\mathbb{A}^n$ by adding a 'hyperplane at infinity'.
Modern algebraic geometry extends to schemes (Grothendieck), allowing study over arbitrary rings and in mixed characteristic.

References

WebsiteWikipedia — Algebraic geometry