---
tag: YA2NN86Y11
kind: definition
name: Affine variety
msc: [14A10]
defines: JERVQT4C12
uses: [0PF6TM8TWT, VHFMQQYSTN, SY2MWC434Z, HH6VEVM6Z0, 00MHHZ9SNE, D85VZWS9QK]
sources:
  - url: https://en.wikipedia.org/wiki/Affine_variety
---

## Definition

Let $k$ be an [[algebraically closed field|0PF6TM8TWT]] and let $n \ge 0$.
Give $k^n$ the Zariski topology, whose closed sets are the [[algebraic sets|VHFMQQYSTN]].
An *affine variety* in $k^n$ is an algebraic set $X \subseteq k^n$ that is [[irreducible|SY2MWC434Z]].
This means that $X$ is nonempty and that $X$ is not the union of two algebraic sets strictly smaller than $X$.

## Remarks

An algebraic set $X$ is irreducible if and only if its [[ideal|HH6VEVM6Z0]] $I(X)$ is a [[prime ideal|00MHHZ9SNE]].
For instance, if $X$ is irreducible and $fg \in I(X)$, then $X = (X \cap V(f)) \cup (X \cap V(g))$, and irreducibility forces $f \in I(X)$ or $g \in I(X)$.
With [[Hilbert's Nullstellensatz|D85VZWS9QK]], affine varieties in $k^n$ correspond to prime ideals of $k[x_1, \dots, x_n]$.

Example: the parabola $X = V(y - x^2) \subseteq k^2$ is an affine variety.
Its ideal is $(y - x^2)$, and $k[x, y]/(y - x^2) \cong k[x]$ is an integral domain, so the ideal is prime.

Non-example: $V(xy) \subseteq k^2$ is an algebraic set but not an affine variety.
It is the union of the two axes, each strictly smaller than $V(xy)$.

Many texts, especially those based on schemes, call any algebraic set an affine variety, or use the term for a variety isomorphic to a closed subvariety of some $k^n$.