Fix a nonarchimedean field of residue characteristic , and let be a normal rigid analytic space over . Suppose we’re given a closed nowhere-dense analytic subspace and a finite etale cover of . It’s natural to ask if can be extended to a finite cover , and whether some further conditions on pin such an extension down uniquely. Although the analogous problem for complex analytic spaces was solved by Stein and Grauert-Remmert in the 50s (cf. Grauert and Remmert’s article here), there isn’t very much literature on this problem in the rigid analytic context, with the notable exception of Lutkebohmert’s paper, about which more in a minute. Anyway, it turns out that at least for a base field of characteristic zero, this problem has a very satisfying answer, and the proof is a fun exercise in swinging lots of big hammers.

First, here’s the precise definition of “cover” which we’ll use.

**Definition. **Let be a normal rigid analytic space. A *cover *of is a finite surjective map from a normal rigid space , such that one of the following two equivalent conditions holds:

1. There exists a closed nowhere-dense analytic subset such that is nowhere-dense and is finite etale.

2. Each irreducible component of maps surjectively onto an irreducible component of , and contains a point such that is etale.

Equivalence of these conditions is a fun exercise left to the reader; note that the second requirement in 2. is automatic when has characteristic 0.

**Theorem. **Let be a normal rigid analytic space over a characteristic zero nonarchimedean field , and let be any closed nowhere-dense analytic subset. Then any finite etale cover of extends uniquely to a cover of .

In other words, the restriction functor from {covers of etale over } to {finite etale covers of } is an equivalence of categories.

The **uniqueness **holds without any condition on , and is an easy consequence of a powerful theorem due to Bartenwerfer. To explain this result, let be a normal rigid space and let be any closed nowhere-dense analytic subset. Then Barternwerfer proved that any bounded function on extends (uniquely) to a function on . In particular, if is a cover and is any open affinoid subset, then depends only on the restriction of to . Since the affinoids cover , this gives the desired uniqueness. More generally, this argument shows that for any closed nowhere-dense analytic subset , the restriction functor from covers of to covers of is fully faithful.

The **existence** of an extension is harder, of course. Until further notice, assume has characteristic zero. Note that by the uniqueness argument, we can always work locally on when extending a finite etale cover of . Now the key input is the following base case, due to Lutkebohmert:

**Theorem (Lutkebohmert): **If is a smooth rigid space and is a simple normal crossings divisor, then any finite etale cover of extends to a cover of .

This is more or less an immediate consequence of Lemma 3.3 in Lutkebohmert’s paper, although he doesn’t state this result so explicitly (and curiously, he never discusses the uniqueness of extensions). The main ingredient aside from this Lemma is a result of Kiehl on “tubular neighborhoods”, which says (among other things) that if is a snc divisor in a smooth rigid space, then for any point in at which components of meet, we can find some small affinoid neighborhood of in together with a smooth affinoid and an isomorphism (where denotes the r-dimensional closed ball) under which the individual components of meeting identify with the zero loci of the coordinate functions .

Granted these results, we argue as follows. Clearly we can assume that is quasicompact. We now argue by induction on the maximal number of irreducible components of passing through any individual point of . Let me sketch the induction informally. If , then is smooth, so Kiehl’s result puts us exactly in the situation covered by the case of Lemma 3.3. If , then locally on we can assume that has two smooth components and . By the previous case, any finite etale cover of extends uniquely to covers of , which then glue to a cover of . But now locally along , Kiehl’s result puts is in the situation covered by the case of Lemma 3.3, and then extends to a cover of . If , then locally on we can assume that has three smooth components . By the previous case, any finite etale cover of extends to a cover of , for each ; here we use the fact that for viewed as a strict normal crossings divisor in . Again the ‘s glue to a cover of , and again locally along Kiehl’s result puts us in the situation handled by Lemma 3.3, so extends to a cover of . Etc.

To get existence in the general case, we use some recent results of Temkin on resolution of singularities. More precisely, suppose is an affinoid rigid space, and is a closed nowhere-dense subset as before; note that is also affinoid, so we get a corresponding closed immersion of schemes . These are quasi-excellent schemes over , so according to Theorem 1.1.11 in Temkin’s paper, we can find a projective birational morphism such that is regular and is a strict normal crossings divisor, and such that is an isomorphism away from . Analytifying, we get a proper morphism of rigid spaces with smooth such that is an snc divisor etc.

Suppose now that we’re given a finite etale cover of . Pulling back along , we get a finite etale cover of , which then extends to a cover by our previous arguments. Now, since is proper, the sheaf defines a sheaf of coherent -algebras. Taking the normalization of the affinoid space associated with the global sections of this sheaf, we get a normal affinoid together with a finite map and a canonical isomorphism . The cover we seek can then be defined, finally, as the Zariski closure of in : this is just a union of irreducible components of , so it’s still normal, and it’s easy to check that satisfies condition 1. in the definition of a cover. Finally, since and are canonically isomorphic after restriction to , the uniqueness argument shows that this isomorphism extends to an isomorphism . This concludes the proof.

Combining this existence theorem with classical Zariski-Nagata purity, one gets a purity theorem for rigid spaces:

**Corollary. **Let X be a smooth rigid analytic space over a characteristic zero nonarchimedean field, and let be any closed analytic subset which is everywhere of codimension . Then finite etale covers of are equivalent to finite etale covers of .

Presumably this result has other fun corollaries. I’d be happy to know more.