Morphism of varieties

Remarks

If $X \subseteq k^m$ and $Y \subseteq k^n$ are affine varieties, a map $\varphi \colon X \to Y$ is a morphism if and only if $\varphi = (\varphi_1, \dots, \varphi_n)$ with each $\varphi_i$ a polynomial function on $X$. A morphism $\varphi$ gives the ring homomorphism $\varphi^* \colon k[Y] \to k[X]$, $f \mapsto f \circ \varphi$, on coordinate rings. This gives a bijection between morphisms $X \to Y$ and $k$-algebra homomorphisms $k[Y] \to k[X]$.

Example: $\varphi \colon k \to V(y - x^2)$, $t \mapsto (t, t^2)$, is an isomorphism with inverse $(x, y) \mapsto x$.

Non-example: $\psi \colon k \to C = V(y^2 - x^3)$, $t \mapsto (t^2, t^3)$, is a bijective morphism but not an isomorphism. Its comorphism $k[C] \to k[t]$ has image $k[t^2, t^3]$, which does not contain $t$, so $\psi^*$ is not surjective and $\psi$ has no inverse morphism.