I am reading Hartshorne's algebraic geometry. According to the definition about morphism of varieties in Hartshorne's book, a morphism $\varphi:X \to Y$ should not only be continuous but also preserve all local regular functions on Y. I am naturally thinking about when a continuous map $\varphi:X \to Y$ is not a morphism of varieties?(Examples?)
At the same times, I assume that if the continuous map $\varphi:X \to Y$ is in 'the fraction form of polynomials', i.e.
$\varphi(x_1,\cdots,x_m)=(\frac{f_1(x_1,\cdots,x_m)}{g_1(x_1,\cdots,x_m)},\cdots,\frac{f_n(x_1,\cdots,x_m)}{g_n(x_1,\cdots,x_m)})$ if $X\subset \mathbb{A}^m$ and $Y\subset\mathbb{A}^n$
$\varphi(x_1,\cdots,x_m)=[\frac{f_0(x_1,\cdots,x_m)}{g_0(x_1,\cdots,x_m)},\cdots,\frac{f_n(x_1,\cdots,x_m)}{g_n(x_1,\cdots,x_m)}]$ if $X\subset \mathbb{A}^m$ and $Y\subset\mathbb{P}^n$
$\varphi[x_0,\cdots,x_m]=[\frac{f_0(x_0,\cdots,x_m)}{g_0(x_0,\cdots,x_m)},\cdots,\frac{f_n(x_1,\cdots,x_m)}{g_n(x_1,\cdots,x_m)}]$ if $X\subset \mathbb{P}^m$ and $Y\subset\mathbb{P}^n$
$\varphi[x_0,\cdots,x_m]=[\frac{f_1(x_0,\cdots,x_m)}{g_1(x_0,\cdots,x_m)},\cdots,\frac{f_n(x_1,\cdots,x_m)}{g_n(x_1,\cdots,x_m)}]$ if $X\subset \mathbb{P}^m$ and $Y\subset\mathbb{A}^n$
(obviously there are some conditions on polynomials $f_i,g_i$ especially when When $X$ and $Y$ are involved in projective varieties, but they do not matter here)
then $\varphi:X\to Y$ is a morphism of varieties, since for each locally regular function $\frac{p}{q}:U\subset Y\to \mathbb{K}$, the composition $\frac{p}{q}\circ \varphi:\varphi^{-1}(U)\to \mathbb{K}$ could also be represented by the form $\frac{p'}{q'}$ of polynomials ($q'$ is obviously nowhere zero on $\varphi^{-1}(U)$ ).
So does the second condition 'preserve all local regular functions' just mean the continuous map should be in 'the fraction form of polynomials'?
Best Answer
If $X, Y$ were both an algebraically closed field $K$, then a continuous map $K\to K$ is one that pulls back closed sets, i.e., finitely many points, to closed sets. Any injective map, for example, is continuous in this case and there are plenty of these that are not morphisms of affine varieties.
Our varieties have two structures: a topology, and what's called the "structure sheaf" which in this case is the collection of local regular functions. Maps between varieties should be ones that preserve both structures, i.e., open sets should pull back to open sets and regular functions pull back to regular functions.
It's not true that all morphisms look like fractions globally (there's an example in Mumford's Red Book pg. 21), but locally the coordinates are fractions (because projecting onto the $i$th coordinate is a regular function on $Y$).