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
{(k1,…,kn):ki∈K}. |
Affine space is usually denoted by Kn or An (or An(K) 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[x1,…,xn] are regarded as functions (algebraic functions
) on 𝔸n(K). “Gluing” several copies of affine space one obtains a projective space.
Lemma.
If K is algebraically closed, affine space An(K) 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 |