Lemma 29.35.11 (02G7)—The Stacks project

Lemma 29.35.11. Fibres of unramified morphisms.

Let $X$ be a scheme over a field $k$. The structure morphism $X \to \mathop{\mathrm{Spec}}(k)$ is unramified if and only if $X$ is a disjoint union of spectra of finite separable field extensions of $k$.
If $f : X \to S$ is an unramified morphism then for every $s \in S$ the fibre $X_ s$ is a disjoint union of spectra of finite separable field extensions of $\kappa (s)$.

Proof. Part (2) follows from part (1) and Lemma 29.35.5. Let us prove part (1). We first use Algebra, Lemma 10.151.7. This lemma implies that if $X$ is a disjoint union of spectra of finite separable field extensions of $k$ then $X \to \mathop{\mathrm{Spec}}(k)$ is unramified. Conversely, suppose that $X \to \mathop{\mathrm{Spec}}(k)$ is unramified. By Algebra, Lemma 10.151.5 for every $x \in X$ the residue field extension $\kappa (x)/k$ is finite separable. Since $X \to \mathop{\mathrm{Spec}}(k)$ is locally quasi-finite (Lemma 29.35.10) we see that all points of $X$ are isolated closed points, see Lemma 29.20.6. Thus $X$ is a discrete space, in particular the disjoint union of the spectra of its local rings. By Algebra, Lemma 10.151.5 again these local rings are fields, and we win. $\square$

Comments (3)

Comment #2245 by comment on September 27, 2016 at 19:06

I do not quite follow how one concludes in the proof that X is discrete. Does one apply that $(5)\Rightarrow (1)$ from Tag 00PJ by using the fact that the local ring at $x$ is a field (as follows from $(1)$ in Tag 00UW )?

Comment #2247 by JuanPablo on October 04, 2016 at 02:19

I didn't see immediately how one concludes that $X$ is discrete, either. I think that one can argue as follows:

$X \to \text{Spec}(k)$ is unramified so locally of finite type, so $X$ is locally Noetherian. So the affine opens of $X$ are spectra of Noetherian rings, and by Lemma 10.36.2 (tag 00FR) they're finite discrete. So points are open.

Maybe state that locally Noetherian zero dimensional squemes are discrete as a separate lemma in section 27.10? (it's similar to lemma 27.10.6 tag 0CKV).

Comment #2280 by Johan on November 03, 2016 at 18:24

Dear comment and JuanPablo, instead of your suggestions (which are fine too), I've fixed it by using that an unramified morphism is locally quasi-finite and that unramified morphisms have discrete fibres. Here is the commit.

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