# regular map

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

A regular map $\phi:V\to W$ 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 $\phi=\psi|_{V}$. 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 $\phi:V\to W$ 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 $\phi(V_{x})\subset W_{f(x)}$ and such that the restriction $V_{x}\to W_{f(x)}$ is a regular map of affine varieties.

Title regular map RegularMap 2013-03-22 12:04:00 2013-03-22 12:04:00 nerdy2 (62) nerdy2 (62) 6 nerdy2 (62) Definition msc 14A10 regular morphism