What does $J_1\cap J_2=\emptyset$ mean algebraically for two varieties in $\Bbb{C}^n$?

Let $J_1, J_2$ be two varieties in ${\Bbb C}^n$. Then $$J_i=V(I_i)\quad i=1,2.$$ for some $I_i\subset\Bbb{C}[x_1,\cdots, x_n]$ and $$J_1\cap J_2=V(I_1\cup I_2)$$ and $$J_1\cup J_2=V(I_1I_2).$$

An exercise in Artin's Algebra asks the following questions:

• What does $J_1\cap J_2=\emptyset$ mean algebraically?
• What does $J_1\cup J_2=\Bbb{C}^n$ mean algebraically?

I don't understand the underlying picture of these two questions. What am I supposed to answer? (Would "polynomials in $I_2$ and $I_2$ have no common divisor" count as an algebraic meaning? Or "these two statement imply that one can define the Zariski topology"?) This exercise looks rather subjective to me. Would anybody help to clarify it?

• The author probably is thinking about statements like $I_1 + I_2 = \mathbb{C}[x_1, \dots, x_n]$ and the like. – Najib Idrissi Feb 22 '14 at 16:28
• It means: If you have $J_1 \cap J_2 = \emptyset$, the ideals $I_1$ and $I_2$ must satisfy a certain property - which one? Same with $J_1 \cup J_2 = {\Bbb C}^n$. – user101036 Feb 22 '14 at 16:37

1 Answer

$$J_1\cap J_2=\emptyset \iff I_1+I_2=\Bbb{C}[x_1,\cdots, x_n]$$ $$J_1\cup J_2=\Bbb{C}^n \iff I_1=(0)\; \text {or} \; I_2=(0)$$

Edit (answer to Jack's request in the comments)
The second equivalence relies on $\mathbb C^n$ being irreducible in the Zariski topology.
This means that the union of two closed subsets $J_1,J_2\subset \mathbb C^n$ is the whole of $\mathbb C^n$ iff one of them already equals $\mathbb C^n$.
[The criterion for irreducibility of an affine variety is that its ring be a domain; here the ring is $\mathbb C[X_1,...,X_n]$ which is certainly a domain so that indeed $\mathbb C^n$ is irreducible]

Hence $J_1\cup J_2=\Bbb{C}^n \iff J_1=\Bbb{C}^n \;\text {or} \;J_2=\Bbb{C}^n$ and finally $J_i=\mathbb C^n \iff I_i=(0)$ .

• Using the weak nullstellensatz, I'm able to deduce the first one. Would you elaborate the second one? – Jack Feb 23 '14 at 20:04
• Done: I have added a few words of explanation in an Edit. – Georges Elencwajg Feb 23 '14 at 20:31
• Thanks a lot. I have to admit that I have almost zero knowledge about algebraic geometry. I used the algebraic-geometry tag since this is an exercise from a very tiny section in Artin's Algebra (2nd), which is by no means even an introductory book in algebraic geometry. I don't know what is irreducibility of ${\Bbb C}^n$ though, is the topology you use Zariski topology? – Jack Feb 23 '14 at 20:40
• Dear Jack: yes the topology is the Zariski topology. I have slightly expanded my edit so as to make it understandable practically without any knowledge of algebraic geometry. – Georges Elencwajg Feb 23 '14 at 20:58