PlanetMath: affine space

(more info)

Math for the people, by the people.

Main Menu

affine space

(Definition)

Definition 1 Let be a field and let be a positive integer. In algebraic geometry we define affine space (or affine -space) to be the set

$\displaystyle \{ (k_1,\ldots,k_n): k_i \in K\}.$

Affine space is usually denoted by 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 $\mathbb{A}^n(K)$ . “Glueing” several copies of affine space one obtains a projective space.

Lemma 1 If is algebraically closed, affine space $\mathbb{A}^n(K)$ is an irreducible algebraic variety.

Bibliography

Anyone with an account can edit this entry. Please help improve it!

"affine space" is owned by alozano. [ full author list (3) ]

(view preamble)

Log in to rate this entry.
(view current ratings)

AMS MSC:	14-00 (Algebraic geometry :: General reference works )
	14R10 (Algebraic geometry :: Affine geometry :: Affine spaces )

Pending Errata and Addenda

None.[ View all 1 ]

Discussion

forum policy

No messages.

Interact