Let $K$ be a number field with ring of integers $O_K$. Now consider a finite and dominant (this condition was added later) morphism of schemes:
$$\pi:X\to \operatorname{Spec }O_K$$
Do we have some results which tell us what is the scheme $X$? For instance if $L$ is a finite extension of $K$, then one possibility is $X= \operatorname{Spec }O_L $. Then what else? Do we have some classification theorem for $X$?
Edit: I'm trying to prove that when $X$ is integral then the unique possibility is $X=\operatorname{Spec }O_L$ for a finite extension $K\subset L$. Let $L$ be the function field of $X$, then by the valuative criterion of properness I get a morphism over $\operatorname{Spec }O_K$: $$f: \operatorname{Spec }O_L\to X$$ The last step is to show that it is actually an isomorphism... Any suggestions?
Without a smoothness hypothesis, non-maximal orders give counterexamples, e.g. $\text{Spec } \mathbb{Z}[2i]$ is finite dominant over $\text{Spec } \mathbb{Z}$.