---
tag: JERVQT4C12
kind: concept
name: Affine variety
msc: [14A10]
specializes: [VHFMQQYSTN]
sources:
  - url: https://en.wikipedia.org/wiki/Affine_variety
same_as:
  wikidata: Q3554813
  wikipedia: Affine variety
---

## Summary

An affine variety is an irreducible algebraic set in affine space $k^n$ over an algebraically closed field $k$.
Affine varieties are the local building blocks of all algebraic varieties.
An affine variety corresponds to a prime ideal of the polynomial ring, and it is determined up to isomorphism by its coordinate ring.
Some authors drop the irreducibility condition and call every algebraic set an affine variety.

## Explanations

### L1 human

An affine variety is the solution set of polynomial equations that cannot be broken into smaller solution sets.
The parabola $y = x^2$ in the plane is one: it is a single piece.
The set $xy = 0$ is not, since it splits into the two axes.
Lines, planes, smooth conics and the whole space $k^n$ are all affine varieties.