# Local structure of finite unramified morphisms

It is well known that a finite étale morphism of schemes is étale locally given by a disjoint union of isomorphisms, i.e. there exists an étale cover such that the pullback is given by . Something similar is true for finite unramified morphisms:

Lemma. Let be a finite unramified¹ morphism of schemes. Then there exists an étale cover such that the pullback is given by , where are closed immersions of finite presentation.

Proof. Let be a point, let be the strict henselisation of at , and let be the base change of along . Then is unramified, so by Tag 04GL it splits as where is surjective for each and no prime of lies above . But is also finite, so by Tag 00GU the map hits the maximal ideal if . Thus, we conclude that , hence is a product of quotients of .

But is the colimit of for an étale neighbourhood inducing a separable extension . Since is of finite presentation, each of the ideals and the projections are defined over some étale neighbourhood . Then the pullback is given by a finite disjoint union of closed immersions in .

Then might not be a covering, but since was arbitrary we can do this for each point separately and take a disjoint union. Remark. The number of needed is locally bounded, but if is not quasi-compact it might be infinite. For example, we can take an infinite disjoint union of points, and such that the fibre over for has points.

Remark. In the étale case, we may actually take finite étale, by taking to be the Galois closure of , which exists in reasonable cases². For example, if is normal, we may take to be the integral closure of in the field extension corresponding to the Galois closure of . In general, if is connected it follows from Tag 0BN2 that a suitable component of the -fold fibre product of over is a Galois closure of . If the connected components of are open, apply this construction to each component.

In the unramified case, this is too much to hope for. For example, if , then we may take to be a nontrivial finite étale cover of an elliptic curve . This is finite and unramified, but does not split over any finite étale cover of since there aren’t any. In fact, it cannot split over any connected étale cover whose image contains , since that implies the image only misses finitely many points (as is ample), which is again impossible since .

¹For the purposes of this post, unramified means in the sense of Grothendieck, i.e. including the finite presentation hypothesis. In Raynaud’s work on henselisations, this was weakened to finite type. See Tag 00US for definitions.

²I’m not sure what happens in general.

# P¹ is simply connected

This is a cute proof that I ran into of the simple connectedness of . It does not use Riemann–Hurwitz or differentials, and instead relies on a purely geometric argument.

Lemma. Let be an algebraically closed field. Then is simply connected.

Proof. Let be a finite étale Galois cover with Galois group . We have to show that is an isomorphism. The diagonal is ample, so the same goes for the pullback to [Hart, Exc. III.5.7(d)]. In particular, is connected [Hart, Cor. III.7.9].

But is isomorphic to copies of because the action is an isomorphism. If is connected, this forces , so is an isomorphism. The proof actually shows that if is a smooth projective variety such that is a set-theoretic complete intersection of ample divisors, then is simply connected.

Example. For a smooth projective curve of genus , the diagonal cannot be ample, as . We already knew this by computing the self-intersection , but the argument above is more elementary.

References.

[Hart] Hartshorne, Algebraic geometry. GTM 52, Springer, 1977.