In the case that $X$ is a locally noetherian scheme, if $\{U_i\}$ is an affine open cover of $X$, we have $\dim X = 0\Leftrightarrow \dim U_i = 0\Leftrightarrow \dim \Gamma(U_i,\mathcal{O}_X) = 0\Leftrightarrow \Gamma(U_i,\mathcal{O}_X)$ is Artinian. So in this case, each $U_i$ has finitely many points. As a result, $\dim X = 0$ if and only if $X$ carries the discrete topology. It seems that the condition being locally noetherian plays a very important role here. Is there any example of a zero-dimensional scheme without being discrete, if we do not assume it is locally noetherian?
Asked
Active
Viewed 27 times
2
-
The linked duplicate contains good material on this. Briefly, if one considers $\operatorname{Spec} \prod^{\Bbb Z} \Bbb F_2$, one gets the Stone-Cech compactification of $\Bbb Z$. That's an infinite compact hausdorff space, which cannot be discrete. – KReiser Mar 20 '22 at 18:52