Exercise 3, Appendix 1

Assume that $k$ is algebraically closed. Let A and B be two closed subsets of V, and $\mathfrak{a}$ (resp. $\mathfrak{b}$) the set of $f\in A_V$ which vanish on A (resp. B). Prove the equivalence of the following properties:

(i) $A\cap B =\emptyset$.

(ii) $\mathfrak{a}+\mathfrak{b}= A_V$.

(iii) There exists a polynomial function $f$ on V which is equal to 1 on A and to 0 on B.

(Use Hilbert’s theorem of zeros (Commutative Algebra, Chap. V, §3, no. 3) to prove that (i) $=\Rightarrow$ (ii).)