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affine space
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(Definition)
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Definition 1 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 $\mathbb{A}^n(K)$ . ``Glueing'' several copies of affine space one obtains a projective space.
- 1
- R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York.
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"affine space" is owned by alozano. [ full author list (3) ]
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Cross-references: variety, algebraic, irreducible, algebraically closed, projective space, algebraic functions, functions, ring, polynomials, affine variety, algebraic set, Zariski topology, topological space, algebraic geometry, integer, positive, field
There are 11 references to this entry.
This is version 4 of affine space, born on 2005-05-05, modified 2006-02-21.
Object id is 7013, canonical name is AffineSpace3.
Accessed 6981 times total.
Classification:
| AMS MSC: | 14-00 (Algebraic geometry :: General reference works ) | | | 14R10 (Algebraic geometry :: Affine geometry :: Affine spaces ) |
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