## Hard to believe

In this brief post, I want to draw attention to an amazing theorem which deserves to be well-known.  Probably many readers are familiar with Nagata’s compactification theorem: if $S$ is any qcqs scheme and $f: X \to S$ is a separated morphism of finite type, then $f$ can be factored as an open embedding followed by a proper morphism. This is a by-now-classical result, and has many applications.

Less well-known, however, is the following result of Temkin (cf. Theorem 1.1.3 here):

Temkin’s Factorization Theorem. Let $f:X \to Y$ be any separated morphism of qcqs schemes. Then $f$ can be factored as an affine morphism followed by a proper morphism.

Telling other people about this theorem is an amusing experience. Invariably, their first reaction is that it simply cannot be true, and that the inclusion map $i: \mathbf{A}^2 - \{ 0,0 \} \to \mathbf{A}^2$ should give a counterexample. But then they realize (or I point out) that $i$ can be factored as $p \circ j$, where $p: X \to \mathbf{A}^2$ is the blowup of $\mathbf{A}^2$ at the origin and $j$ is the natural (affine!) open immersion of $\mathbf{A}^2 - \{ 0,0 \}$ into $X$. Then they are convinced.

Unrelated: JW pointed out to me that I am now a professional writer of appendices. Maybe this should worry me?

Advertisement

## Extending finite etale coverings

Fix a nonarchimedean field $K$ of residue characteristic $p$, and let $X$ be a normal rigid analytic space over $K$.  Suppose we’re given a closed nowhere-dense analytic subspace $Z \subset X$ and a finite etale cover $Y$ of $X \smallsetminus Z$.  It’s natural to ask if $Y$ can be extended to a finite cover $Y' \to X$, and whether some further conditions on $Y'$ 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 $K$ 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 $X$ be a normal rigid analytic space.  A cover of $X$ is a finite surjective map $\pi: Y \to X$ from a normal rigid space $Y$, such that one of the following two equivalent conditions holds:
1. There exists a closed nowhere-dense analytic subset $Z \subset X$ such that $\pi^{-1}(Z)$ is nowhere-dense and $Y \smallsetminus \pi^{-1}(Z) \to X \smallsetminus Z$ is finite etale.
2.  Each irreducible component $Y_i$ of $Y$ maps surjectively onto an irreducible component $X_i$ of $X$, and contains a point $y_i$ such that $\mathcal{O}_{X,\pi(y_i)} \to \mathcal{O}_{Y_i,y_i}$ is etale.

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

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

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

The uniqueness holds without any condition on $K$, and is an easy consequence of a powerful theorem due to Bartenwerfer.  To explain this result, let $X$ be a normal rigid space and let $Z \subset X$ be any closed nowhere-dense analytic subset. Then Barternwerfer proved that any bounded function on $X \smallsetminus Z$ extends (uniquely) to a function on $X$. In particular, if $Y \to X$ is a cover and $U \subset X$ is any open affinoid subset, then $\mathcal{O}_Y(\pi^{-1}(U)) \cong \mathcal{O}_{Y}^{+}(\pi^{-1}(U \smallsetminus U \cap Z))[1/ \varpi]$ depends only on the restriction of $Y$ to $X \smallsetminus Z$. Since the affinoids $\pi^{-1}(U)$ cover $Y$, this gives the desired uniqueness.  More generally, this argument shows that for any closed nowhere-dense analytic subset $Z \subset X$, the restriction functor from covers of $X$ to covers of $X \smallsetminus Z$ is fully faithful.

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

Theorem (Lutkebohmert): If $X$ is a smooth rigid space and $Z \subset X$ is a simple normal crossings divisor, then any finite etale cover of $X \smallsetminus Z$ extends to a cover of $X$.

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 $D \subset X$ is a snc divisor in a smooth rigid space, then for any point $x$ in $D$ at which $r$ components of $D$ meet, we can find some small affinoid neighborhood $U$ of $x$ in $X$ together with a smooth affinoid $S$ and an isomorphism $U \simeq S \times B^r$ (where $B^r = \mathrm{Sp}K \left\langle X_1, \dots, X_r \right\rangle$ denotes the r-dimensional closed ball) under which the individual components of $D$ meeting $x$ identify with the zero loci of the coordinate functions $X_i$.

Granted these results, we argue as follows.  Clearly we can assume that $X$ is quasicompact.  We now argue by induction on the maximal number $i(D)$ of irreducible components of $D$ passing through any individual point of $X$. Let me sketch the induction informally. If $i(D)=1$, then $D$ is smooth, so Kiehl’s result puts us exactly in the situation covered by the case $r=1$ of Lemma 3.3. If $i(D)=2$, then locally on $X$ we can assume that $D$ has two smooth components $D_1$ and $D_2$. By the previous case, any finite etale cover $Y$ of $X \smallsetminus D$ extends uniquely to covers $Y_i$ of $X \smallsetminus D_i$, which then glue to a cover $Y_0$ of $X \smallsetminus D_1 \cap D_2$.  But now locally along $D_1 \cap D_2$, Kiehl’s result puts is in the situation covered by the case $r=2$ of Lemma 3.3, and then $Y_0$ extends to a cover of $X$.  If $i(D)=3$, then locally on $X$ we can assume that $D$ has three smooth components $D_1, D_2, D_3$. By the previous case, any finite etale cover of $X \smallsetminus D$ extends to a cover $Y_i$ of $X \smallsetminus D_i$, for each $i \in \{1,2,3\}$; here we use the fact that $i(D \smallsetminus D_i) \leq 2$ for $D \smallsetminus D_i$ viewed as a strict normal crossings divisor in $X \smallsetminus D_i$.  Again the $Y_i$‘s glue to a cover $Y_0$ of $X \smallsetminus D_1 \cap D_2 \cap D_3$, and again locally along $D_1 \cap D_2 \cap D_3$ Kiehl’s result puts us in the situation handled by Lemma 3.3, so $Y_0$ extends to a cover of $X$.  Etc.

To get existence in the general case, we use some recent results of Temkin on resolution of singularities.  More precisely, suppose $X = \mathrm{Sp}(A)$ is an affinoid rigid space, and $Z \subset X$ is a closed nowhere-dense subset as before; note that $Z=\mathrm{Sp}(B)$ is also affinoid, so we get a corresponding closed immersion of schemes $\mathcal{Z} = \mathrm{Spec}(B) \to \mathcal{X} = \mathrm{Spec}(A)$.  These are quasi-excellent schemes over $\mathbf{Q}$, so according to Theorem 1.1.11 in Temkin’s paper, we can find a projective birational morphism $f: \mathcal{X}' \to \mathcal{X}$ such that $\mathcal{X}'$ is regular and $(\mathcal{X}' \times_{\mathcal{X}} \mathcal{Z})^{\mathrm{red}}$ is a strict normal crossings divisor, and such that $f$ is an isomorphism away from $\mathcal{Z} \cup \mathcal{X}^{\mathrm{sing}}$.  Analytifying, we get a proper morphism of rigid spaces $g: X' \to X$ with $X'$ smooth such that $g^{-1}(Z)^{\mathrm{red}}$ is an snc divisor etc.

Suppose now that we’re given a finite etale cover $Y$ of $X \smallsetminus Z$.   Pulling back along $g$, we get a finite etale cover of $X' \smallsetminus g^{-1}(Z)$, which then extends to a cover $h: Y'\to X'$ by our previous arguments. Now, since $g \circ h$ is proper, the sheaf $(g \circ h)_{\ast} \mathcal{O}_{Y'}$ defines a sheaf of coherent $\mathcal{O}_X$-algebras. Taking the normalization of the affinoid space associated with the global sections of this sheaf, we get a normal affinoid $Y''$ together with a finite map $Y'' \to X$ and a canonical isomorphism $Y''|_{(X \smallsetminus Z)^{\mathrm{sm}}} \cong Y|_{(X \smallsetminus Z)^{\mathrm{sm}}}$. The cover we seek can then be defined, finally, as the Zariski closure $Y'''$ of $Y''|_{(X \smallsetminus Z)^{\mathrm{sm}}}$ in $Y''$: this is just a union of irreducible components of $Y''$, so it’s still normal, and it’s easy to check that $Y'''$ satisfies condition 1. in the definition of a cover. Finally, since $Y'''$ and $Y$ are canonically isomorphic after restriction to $(X \smallsetminus Z)^{\mathrm{sm}}$, the uniqueness argument shows that this isomorphism extends to an isomorphism $Y'''|_{X \smallsetminus Z} \cong Y$. 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 $Z \subset X$ be any closed analytic subset which is everywhere of codimension $\geq 2$.  Then finite etale covers of $X$ are equivalent to finite etale covers of $X \smallsetminus Z$.

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