I am not looking for someone to give me an explicit example. I want to work this out myself if possible.
Trying to learn schemes by reading The Geometry of Schemes by Eisenbud and Harris. Problem I-5 is to play with $X=\{0,1\}$ and two topologies
(1) $\tau=\{\emptyset,\{0\},\{1\},\{0,1\}\}$ (2) $\tau=\{\emptyset,\{0\},\{0,1\}\}$
The main problem is not difficult but as an aside, it is stated that both topologies can be realized as $\mbox{Spec}\,R$ for some ring $R$. For the first topology, I think $\mathbb{Z}_6$ works. I cannot find an example for the second topology. My thinking: the easiest example would be an integral domain with exactly two prime ideals, then the zero ideal would give $\{0,1\}$ as a closed set and the elements of the other ideal would give $\{1\}$ as a closed set. Since a finite integral domain is a field, the example I am looking for is infinite. Then I realized that I don't know too many examples, all the integral domains I know do not work.
Questions:
- Am I on the right track?
- If I am on the right track, is the example one that is obtainable (even for non-experts?)
Thanks.
What you are looking for is a DVR, i.e. a PID that is a local ring. Such a ring has exactly $2$ prime ideals, with one contained in the other: the $0$ ideal and a unique maximal ideal. Since a localization of a PID is again a PID, try localizing a PID (e.g. $\mathbb{Z}$ or $k[x]$, for $k$ a field) at a maximal ideal.