# affine space

###### Definition.

Let $K$ be a field and let $n$ be a positive integer. In algebraic geometry^{} we define affine space^{} (or affine $n$-space) to be the set

$$\{({k}_{1},\mathrm{\dots},{k}_{n}):{k}_{i}\in K\}.$$ |

Affine space is usually denoted by ${K}^{n}$ or ${\mathrm{A}}^{n}$ (or ${\mathrm{A}}^{n}\mathit{}\mathrm{(}K\mathrm{)}$ if we want to emphasize the field of definition).

In Algebraic Geometry, we consider affine space as a topological space, with the usual Zariski topology^{} (see also algebraic set^{}, affine variety^{}). The polynomials^{} in the ring $K[{x}_{1},\mathrm{\dots},{x}_{n}]$ are regarded as functions (algebraic functions^{}) on ${\mathbb{A}}^{n}(K)$. “Gluing” several copies of affine space one obtains a projective space.

###### Lemma.

If $K$ is algebraically closed^{}, affine space ${\mathrm{A}}^{n}\mathit{}\mathrm{(}K\mathrm{)}$ is an irreducible^{} algebraic variety.

## References

- 1 R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York.

Title | affine space |
---|---|

Canonical name | AffineSpace |

Date of creation | 2013-03-22 15:14:21 |

Last modified on | 2013-03-22 15:14:21 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 8 |

Author | alozano (2414) |

Entry type | Definition |

Classification | msc 14R10 |

Classification | msc 14-00 |

Related topic | ProjectiveSpace |

Related topic | AffineVariety |