$\overline{h(X)}$, the closure of $h(X)$ in $Y$ is a variety whose coordinate ring is isomorphic to $A(Y)/\mathrm{rad}(I)$, where $I=\ker(h^*)$

Question

Show that if $h:X\to Y$ a variety morphism, then $\overline{h(X)}$, the closure of $h(X)$ in $Y$; it is a variety whose coordinate ring is isomorphic to $A(Y)/\mathrm{rad}(I)$, where $I=\ker(h^*)$

In this case $h^*:A(Y)\to A(X)$ represents the homomorphism induced by $h:X\to Y$.

I do not know where to start and would appreciate any suggestions that lead me to solve the problem, thank you very much.

Answer 1

Let $I$ be the ideal of the closure of $h(X)$. Try to prove directly that the kernel of $h^*$ is exactly $I$, by showing that any element in the kernel vanishes on $h(X)$.