Polynomial Maps Between Affine Varieties
Table of Contents

Recall from The Coordinate Ring of an Affine Variety page that if $K$ is a field and $V \subseteq \mathbb{A}^n(K)$ is a nonempty affine variety then the coordinate ring of $V$ is defined as:
(1)\begin{align} \quad \Gamma(V) = K[x_1, x_2, ..., x_n]/I(V) \end{align}
We noted that since $V$ is an affine variety, $I(V)$ is a prime ideal and so $\Gamma(V)$ is an integral domain.
Definition: Let $K$ be a field and let $V \subseteq \mathbb{A}^n(K)$ be a nonempty affine variety. The Set of Functions from $V$ to $K$ is denoted by $\mathcal F(V, K)$, and is a ring with the operation of addition defined for all $f, g \in \mathcal F(V, K)$ by $(f + g)(x) = f(x) + g(x)$ and the operation of multiplication defined for all $f, g \in \mathcal F(V, K)$ by $(fg)(x) = f(x)g(x)$. 
Definition: Let $K$ be a field and let $V \subseteq \mathbb{A}^n(K)$ be a nonempty affine variety. A function $f \in \mathcal F(V, K)$ is a Polynomial Function if there exists a polynomial $F \in K[x_1, x_2, ..., x_n]$ such that $f(\mathbf{p}) = F(\mathbf{p})$ for every $\mathbf{p} \in V$. The Set of Polynomial Functions from $V$ to $K$ is denoted by $\mathcal P(V, K)$. 
Definition: Let $K$ be a field and let $V \subseteq \mathbb{A}^n(K)$ and $W \subseteq \mathbb{A}^m(K)$. A Polynomial Map if a function $\varphi : V \to W$ such that there exists polynomials $T_1, T_2, ..., T_m \in K[x_1, x_2, ..., x_n]$ such that $\varphi(\mathbf{p}) = (T_1(\mathbf{p}), T_2(\mathbf{p}), ..., T_m(\mathbf{p})$ for every $\mathbf{p} \in V$. A polynomial map $\phi : V \to W$ is said to be an Isomorphism if there exists another polynomial map $\psi : W \to V$ such that $\psi \circ \phi = \mathrm{id}_V$ and $\phi \circ \psi = \mathrm{id}_W$. 
If $V \subseteq \mathbb{A}^n(K)$ and $W \subseteq \mathbb{A}^m(K)$ are affine varieties, then a natural homomorphism exists between the two.
Theorem 1: Let $K$ be a field and let $V \subseteq \mathbb{A}^n(K)$ and $W \subseteq \mathbb{A}^m(K)$ be affine varieties and let $\varphi : V \to W$ be a polynomial map. Then the function $\phi : \mathcal F(W, K) \to \mathcal F(V, K)$ defined for all functions $f \in \mathcal F(V, K)$ by $\phi(f) = f \circ \varphi$ is a ring homomorphism. 
 Proof: Let $f, g \in \mathcal F(W, K)$. Then:
\begin{align} \quad \phi(f + g) = (f + g) \circ \varphi = (f \circ \varphi) + (g \circ \varphi) = \phi(f) + \phi(g) \end{align}
(3)
\begin{align} \quad \phi(fg) = (fg) \circ \varphi = (f \circ \varphi) \cdot (g \circ \varphi) = \phi(f) \cdot \phi(g) \end{align}
 So $\phi : \mathcal F(W, K) \to \mathcal F(V, K)$ is a ring homomorphism. $\blacksquare$
The following theorem tells us that there is a onetoone correspondence between the set of polynomial maps from two nonempty varieties and the set of homomorphisms from the corresponding coordinate rings.
Theorem 2: Let $K$ be a field and let $V \subseteq \mathbb{A}^n(K)$ and $W \subseteq \mathbb{A}^m(K)$ be nonempty affine varieties. Then there is a onetoone correspondence between the set of polynomial maps $\varphi : V \to W$ and the set of ring homomorphisms $\phi : \Gamma(W) \to \Gamma(V)$. 
Corollary 3: Let $K$ be a field and let $V \subseteq \mathbb{A}^n(K)$ and $W \subseteq \mathbb{A}^m(K)$ be nonempty affine varieties. Then a polynomial map $\phi : V \to W$ is an isomorphism if and only if $\Gamma(W)$ and $\Gamma(V)$ are ring isomorphic. 