affine space


Let K be a field and let n be a positive integer. In algebraic geometryMathworldPlanetmathPlanetmath we define affine spaceMathworldPlanetmath (or affine n-space) to be the set


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 topologyMathworldPlanetmath (see also algebraic setMathworldPlanetmath, affine varietyMathworldPlanetmath). The polynomialsPlanetmathPlanetmath in the ring K[x1,,xn] are regarded as functions (algebraic functionsMathworldPlanetmath) on 𝔸n(K). “Gluing” several copies of affine space one obtains a projective space.


If K is algebraically closedMathworldPlanetmath, affine space An(K) is an irreduciblePlanetmathPlanetmath algebraic variety.


  • 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