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algebraic map (Definition)

A map $ f:X\to Y$ between quasi-affine varieties $ X\subset k^n,Y\subset k^m$ over a field $ k$ is called algebraic if there is a map $ f':k^n\to k^m$ whose component functions are polynomials, such that $ f'$ restricts to $ f$ on $ X$.

Alternatively, $ f$ is algebraic if the pullback map $ f^*:C(Y)\to C(X)$ maps the coordinate ring of $ Y$, $ k[Y]$, to the coordinate ring of $ X$, $ k[X]$.



"algebraic map" is owned by mathcam. [ full author list (2) | owner history (1) ]
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Other names:  algebraic
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Cross-references: ring, coordinate, pullback, polynomials, functions, component, field, quasi-affine varieties, map
There are 62 references to this entry.

This is version 4 of algebraic map, born on 2003-08-22, modified 2006-08-10.
Object id is 4641, canonical name is AlgebraicMap.
Accessed 8493 times total.

Classification:
AMS MSC14A10 (Algebraic geometry :: Foundations :: Varieties and morphisms)

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