# regular map

A regular map $\varphi :{k}^{n}\to {k}^{m}$ between affine spaces^{} over an algebraically closed field is merely one given by polynomials. That is, there are $m$ polynomials ${F}_{1},\mathrm{\dots},{F}_{m}$ in $n$ variables such that the map is given by $\varphi ({x}_{1},\mathrm{\dots},{x}_{n})=({F}_{1}(x),\mathrm{\dots},{F}_{m}(x))$ where $x$ stands for the many components ${x}_{i}$.

A regular map $\varphi :V\to W$ between affine varieties^{} is one which is the restriction of a regular map between affine spaces. That is, if $V\subset {k}^{n}$ and $W\subset {k}^{m}$, then there is a regular map $\psi :{k}^{n}\to {k}^{m}$ with $\psi (V)\subset W$ and $\varphi ={\psi |}_{V}$. 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 $\varphi :V\to W$ is regular^{} if around each point $x$ there is an affine variety ${V}_{x}$ and around each point $f(x)\in W$ there is an affine variety ${W}_{f(x)}$ with $\varphi ({V}_{x})\subset {W}_{f(x)}$ and such that the restriction ${V}_{x}\to {W}_{f(x)}$ 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 |