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.