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

 $\{(k_{1},\ldots,k_{n}):k_{i}\in K\}.$

Affine space is usually denoted by $K^{n}$ or $\mathbb{A}^{n}$ (or $\mathbb{A}^{n}(K)$ 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 $K[x_{1},\ldots,x_{n}]$ 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 $\mathbb{A}^{n}(K)$ is an irreducible algebraic variety.

References

• 1 R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York.
Title affine space AffineSpace 2013-03-22 15:14:21 2013-03-22 15:14:21 alozano (2414) alozano (2414) 8 alozano (2414) Definition msc 14R10 msc 14-00 ProjectiveSpace AffineVariety