regular map

A regular map ϕ:knkm between affine spacesPlanetmathPlanetmath over an algebraically closed field is merely one given by polynomials. That is, there are m polynomials F1,,Fm in n variables such that the map is given by ϕ(x1,,xn)=(F1(x),,Fm(x)) where x stands for the many components xi.

A regular map ϕ:VW between affine varietiesMathworldPlanetmath is one which is the restriction of a regular map between affine spaces. That is, if Vkn and Wkm, then there is a regular map ψ:knkm with ψ(V)W and ϕ=ψ|V. So, this is a map given by polynomials, whose image lies in the intended target.

A regular map between algebraic varieties is a locally regular map. That is ϕ:VW is regularPlanetmathPlanetmathPlanetmath if around each point x there is an affine variety Vx and around each point f(x)W there is an affine variety Wf(x) with ϕ(Vx)Wf(x) and such that the restriction VxWf(x) is a regular map of affine varieties.

Title regular map
Canonical name RegularMap
Date of creation 2013-03-22 12:04:00
Last modified on 2013-03-22 12:04:00
Owner nerdy2 (62)
Last modified by nerdy2 (62)
Numerical id 6
Author nerdy2 (62)
Entry type Definition
Classification msc 14A10
Synonym regular morphism