The Stacks project

Example 41.8.3. Fix an extension of number fields $L/K$ with rings of integers $\mathcal{O}_ L$ and $\mathcal{O}_ K$. The injection $K \to L$ defines a morphism $f : \mathop{\mathrm{Spec}}(\mathcal{O}_ L) \to \mathop{\mathrm{Spec}}(\mathcal{O}_ K)$. As discussed above, the points where $f$ is unramified in our sense correspond to the set of points where $f$ is unramified in the conventional sense. In the conventional sense, the locus of ramification in $\mathop{\mathrm{Spec}}(\mathcal{O}_ L)$ can be defined by vanishing set of the different; this is an ideal in $\mathcal{O}_ L$. In fact, the different is nothing but the annihilator of the module $\Omega _{\mathcal{O}_ L/\mathcal{O}_ K}$. Similarly, the discriminant is an ideal in $\mathcal{O}_ K$, namely it is the norm of the different. The vanishing set of the discriminant is precisely the set of points of $K$ which ramify in $L$. Thus, denoting by $X$ the complement of the closed subset defined by the different in $\mathop{\mathrm{Spec}}(\mathcal{O}_ L)$, we obtain a morphism $X \to \mathop{\mathrm{Spec}}(\mathcal{O}_ K)$ which is unramified. Furthermore, this morphism is also flat, as any local homomorphism of discrete valuation rings is flat, and hence this morphism is actually étale. If $L/K$ is finite Galois, then denoting by $Y$ the complement of the closed subset defined by the discriminant in $\mathop{\mathrm{Spec}}(\mathcal{O}_ K)$, we see that we get even a finite étale morphism $X \to Y$. Thus, this is an example of a finite étale covering.