Example 41.8.1. Let $k$ be a field. Unramified quasi-compact morphisms $X \to \mathop{\mathrm{Spec}}(k)$ are affine. This is true because $X$ has dimension $0$ and is Noetherian, hence is a finite discrete set, and each point gives an affine open, so $X$ is a finite disjoint union of affines hence affine. Noether normalization forces $X$ to be the spectrum of a finite $k$-algebra $A$. This algebra is a product of finite separable field extensions of $k$. Thus, an unramified quasi-compact morphism to $\mathop{\mathrm{Spec}}(k)$ corresponds to a finite number of finite separable field extensions of $k$. In particular, an unramified morphism with a connected source and a one point target is forced to be a finite separable field extension. As we will see later, $X \to \mathop{\mathrm{Spec}}(k)$ is étale if and only if it is unramified. Thus, in this case at least, we obtain a very easy description of the étale topology of a scheme. Of course, the cohomology of this topology is another story.

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