Hints for proof that Variety $X$ of (dimension $0$) is discrete.

Question

I have come across a statement in a textbook which has been left as an exercises for the reader, and as such there is no proof.

I am looking for some HINTS to get me started on a proof, as I am do not think I am fully understanding the "of dimension zero" aspect of the statement.

If we have a variety $X$, where $\dim(X) = 0$, then $X$ is discrete.

I think I should reduce it to the affine case, but I am not sure where to go from there ...?

Note we are assuming all varieties are finite type over the field K, where K is an algebraically closed field, thus Noetherian.

Answer 1

Note that the result is true for any noetherian zero-dimensional scheme. All other properties of a variety are not needed.

It is easy to reduce to the affine case: If a topological space is covered by discrete open sets, it is itself discrete. This a very easy exercise in point set topology.

For the affine case, let $M_1, \dotsc, M_n$ be the finitely many prime (=maximal) ideals of a zero-dimensional noetherian ring $A$. Note that there are finitely many, because the ring is noetherian. For any given subset of those prime ideals (i.e. any subset of $\operatorname{Spec} A$), let $I$ be their product. Then $V(I)$ equals the given subset, i.e. it is closed.