## 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.

## Diamonds for all!

Regular readers of this blog probably know that I’m obsessed with diamonds.  They can thus imagine my happiness when Peter posted an official foundational reference for diamonds a few weeks ago.

I want to use this occasion to make a remark aimed at graduate students etc. who might be wondering whether they should bother learning this stuff: in my opinion, spending time with difficult* manuscripts like the one above usually pays off in the long run. Of course, this only works if you invest a reasonable amount of time, and there’s some initial period where you’re completely befuddled, but after some months the befuddlement metamorphoses into understanding, and then you have a new set of tools in your toolkit! This shouldn’t be so surprising, though; after all, papers like this are difficult precisely because they are so rich in new ideas and tools.

Really, I’ve had this experience many times now – with the paper linked above and its precedent, with the Kedlaya-Liu “Relative p-adic Hodge theory” series, with Kato’s paper on p-adic Hodge theory and zeta functions of modular forms, etc. – and it was the same every time: for some period of months (or years) I would just read the thing for its own sake, but then at some point something in it would congeal with the rest of the swirling fragments in my head and stimulate me to an idea which never would’ve occurred to me otherwise. It’s the most fun thing in the world. Try it yourself.

*Here by “difficult” I don’t mean anything negative, but rather some combination of dense/forbidding/technical – something with a learning curve.  Of course, there are plenty of papers which are difficult for bad reasons, e.g. because they’re poorly written.  Don’t read them.

## Nowhere-vanishing sections of vector bundles

Let $X/k$ be a proper variety over some field, and let $\mathcal{E}$ be a vector bundle on $X$.  The functor of global sections of $\mathcal{E}$, i.e. the functor sending a scheme $f:S \to \mathrm{Spec}\,k$ to the set $H^0(S \times_{k} X, (f \times \mathrm{id})^{\ast} \mathcal{E})$, is (representable by) a nice affine $k$-scheme, namely the scheme $\mathcal{S}(\mathcal{E}) = \mathrm{Spec}(\mathrm{Sym}_k H^0(X,\mathcal{E})^{\vee})$. Let $\mathcal{S}(\mathcal{E})^{\times} \subset \mathcal{S}(\mathcal{E})$ denote the subfunctor corresponding to nowhere-vanishing sections $s \in H^0(S \times_k X, (f \times \mathrm{id})^{\ast} \mathcal{E})$. We’d like this subfunctor to be representable by an open subscheme. How should we prove this?

Let $p: \mathcal{S}(\mathcal{E}) \to \mathrm{Spec}\,k$ be the structure map. The identity map $\mathcal{S}(\mathcal{E}) \to \mathcal{S}(\mathcal{E})$ corresponds to a universal section $s^{\mathrm{univ}} \in H^0(\mathcal{S}(\mathcal{E}) \times_k X, (p \times \mathrm{id})^{\ast}\mathcal{E})$. Let $Z\subset |\mathcal{S}(\mathcal{E}) \times_k X|$ denote the zero locus of $s^{\mathrm{univ}}$. This is a closed subset. But now we observe that the projection $\pi: \mathcal{S}(\mathcal{E}) \times_k X \to \mathcal{S}(\mathcal{E})$ is proper, hence universally closed, and so $|\pi|(Z)$ is a closed subset of $|\mathcal{S}(\mathcal{E})|$.  One then checks directly that $\mathcal{S}(\mathcal{E})^{\times}$ is the open subscheme corresponding to the open subset $|\mathcal{S}(\mathcal{E})| \smallsetminus |\pi|(Z)$, so we win.

I guess this sort of thing is child’s play for an experienced algebraic geometer, and indeed it took Johan about 0.026 seconds to suggest that one should try to argue using the universal section.  I only cared about the above problem, though, as a toy model for the same question in the setting of a vector bundle $\mathcal{E}$ over a relative Fargues-Fontaine curve $\mathcal{X}_S$. In this situation, $\mathcal{S}(\mathcal{E})$ is a diamond over $S^\lozenge$, cf. Theorem 22.5 here, but it turns out the above argument still works after some minor changes.

## Artin vanishing is false in rigid geometry

Sorry for the lack of blogging.  It’s been a busy semester.

Let $k$ be an algebraically closed field, and let $X$ be a $d$-dimensional affine variety over $k$.  According to a famous theorem of Artin (Corollaire XIV.3.5 in SGA 4 vol. 3), the etale cohomology groups $H^i_{\mathrm{et}}(X,G)$ vanish for any $i > d$ and any torsion abelian sheaf $G$ on $X_{\mathrm{et}}$. This is a pretty useful result.

It’s natural to ask if there’s an analogous result in rigid geometry.  More precisely, fix a complete algebraically closed extension $k / \mathbf{Q}_p$ and a $d$-dimensional affinoid rigid space $X=\mathrm{Spa}(A,A^\circ)$ over $k$.  Is it true that $H^i_{\mathrm{et}}(X,G)$ vanishes for (say) any $i>d$ and any $\mathbf{Z}/n\mathbf{Z}$-sheaf $G$ on $X_{\mathrm{et}}$ for $n$ prime to $p$?

I spent some time trying to prove this before realizing that it fails quite badly.  Indeed, there are already counterexamples in the case where $X=\mathrm{Spa}(k \langle T_1,\dots,T_d \rangle,k^\circ \langle T_1, \dots, T_d \rangle)$ is the $d$-variable affinoid disk over $k$.  To make a counterexample in this case, let $Y$ be the interior of the (closed, in the adic world) subset of $X$ defined by the inequalities $|T_i| < |p|$ for all $i$; more colloquially, $Y$ is just the adic space associated to the open subdisk of (poly)radius $1/p$. Let $j: Y \to X$ be the natural inclusion.  I claim that $G = j_! \mathbf{Z}/n\mathbf{Z}$ is then a counterexample.  This follows from the fact that $H^i_{\mathrm{et}}(X,G)$ is naturally isomorphic to $H^i_{\mathrm{et},c}(Y,\mathbf{Z}/n\mathbf{Z})$, together with the nonvanishing of the latter group in degree $i = 2d$.

Note that although I formulated this in the language of adic spaces, the sheaf $G$ is overconvergent, and so this example descends to the Berkovich world thanks to the material in Chapter 8 of Huber’s book.

It does seem possible, though, that Artin vanishing might hold in the rigid world if we restrict our attention to sheaves which are Zariski-constructible.  As some (very) weak evidence in this direction, I managed to check that $H^2_{\mathrm{et}}(X,\mathbf{Z}/n \mathbf{Z})$ vanishes for any one-dimensional affinoid rigid space $X$.  (This is presumably well-known to experts.)

## Riemann-Roch sur la courbe

Let $C/\mathbf{Q}_p$ be a complete algebraically closed extension, and let $X = X_{C^\flat}$ be the Fargues-Fontaine curve associated with $C^\flat$.  If $\mathcal{E}$ is any vector bundle on $X$, the cohomology groups $H^i(X,\mathcal{E})$ vanish for all $i>1$ and are naturally Banach-Colmez Spaces for $i=0,1$.  Recall that the latter things are roughly “finite-dimensional $C$-vector spaces up to finite-dimensional $\mathbf{Q}_p$-vector spaces”. By a hard and wonderful theorem of Colmez, these Spaces form an abelian category, and they have a well-defined Dimension valued in $\mathbf{N} \times \mathbf{Z}$ which is (componentwise-) additive in short exact sequences.  The Dimension roughly records the $C$-dimension and the $\mathbf{Q}_p$-dimension, respectively.  Typical examples are $H^0(X, \mathcal{O}(1)) = B_{\mathrm{crys}}^{+,\varphi = p}$, which has Dimension $(1,1)$, and $H^1(X,\mathcal{O}(-1)) = C/\mathbf{Q}_p$, which has Dimension $(1,-1)$.

Here I want to record the following beautiful Riemann-Roch formula.

Theorem. If $\mathcal{E}$ is any vector bundle on $X$, then $\mathrm{Dim}\,H^0(X,\mathcal{E}) - \mathrm{Dim}\,H^1(X,\mathcal{E}) = (\mathrm{deg}(\mathcal{E}), \mathrm{rk}(\mathcal{E}))$.

One can prove this by induction on the rank of $\mathcal{E}$, reducing to line bundles; the latter were classified by Fargues-Fontaine, and one concludes by an explicit calculation in that case.  In particular, the proof doesn’t require the full classification of bundles.

So cool!

## w-local spaces are amazing

Let $X$ be a spectral space.  Following Bhatt-Scholze, say $X$ is w-local if the subset $X^c$ of closed points is closed and if every connected component of $X$ has a unique closed point.  This implies that the natural composite map $X^c \to X \to \pi_0(X)$ is a homeomorphism (cf. Lemma 2.1.4 of BS).

For the purposes of this post, a w-local adic space is a qcqs analytic adic space whose underlying spectral topological space is w-local.  These are very clean sorts of spaces: in particular, each connected component of such a space is of the form $\mathrm{Spa}(K,K^+)$, where $K$ is a nonarchimedean field and $K^+$ is an open and bounded valuation subring of $K$, and therefore has a unique closed point and a unique generic point.

I’ve been slowly internalizing the philosophy that w-local affinoid perfectoid spaces have a lot of amazing properties.  Here I want to record an example of this sort of thing.

Given a perfectoid space $\mathcal{X}$ together with a subset $T \subseteq |\mathcal{X}|$, let’s say $T$ is perfectoid (resp. affinoid perfectoid) if there is a pair $(\mathcal{T},f)$ where $\mathcal{T}$ is a perfectoid space (resp. affinoid perfectoid space) and $f: \mathcal{T} \to \mathcal{X}$ is a map of adic spaces identifying $|\mathcal{T}|$ homeomorphically with $T$ and which is universal for maps of perfectoid spaces $\mathcal{Y} \to \mathcal{X}$ which factor through $T$ on topological spaces. Note that if the pair $(\mathcal{T},f)$ exists, it’s unique up to unique isomorphism.

Theorem. Let $\mathcal{X}$ be a w-local affinoid perfectoid space. Then any subset $T$ of $X = |\mathcal{X}|$ which is closed and generalizing, or which is quasicompact open, is affinoid perfectoid.

Proof when $T$ is closed and generalizing. The key point here is that the map $\gamma: X \to \pi_0(X)$ defines a bijection between closed generalizing subsets of $X$ and closed subsets of the (profinite) space $pi_0(X)$, by taking preimages of the latter or images of the former. To check that this is true, note that if $T$ is closed and generalizing, then its intersection with a connected component $X'$ of $X$ being nonempty implies (since $T$ is generalizing) that $T \cap X'$ contains the unique rank one point of $X'$. But then $T \cap X'$ contains all specializations of that point (since $T \cap X'$ is closed in $X'$), so $T \cap X' = X'$, so any given connected component of $X$ is either disjoint from $T$ or contained entirely in $T$.  This implies that $T$ can be read off from which closed points of $X$ it contains.  Finally, one easily checks that $\gamma(T)$ is closed in $\pi_0(X)$, since $\pi_0(X)$ is profinite and $\gamma(T)$ is quasicompact.  Therefore $T = \gamma^{-1}(\gamma(T))$.

Returning to the matter at hand, write $\gamma(T)$ as a cofiltered intersection of qc opens $V_i \subset \pi_0(X)$, $i \in I$. But qc opens in $\pi_0(X)$ are the same as open-closed subsets, so each $V_i$ pulls back to an open-closed subset $U_i \subset X$, and its easy to check that any such $U_i$ comes from a unique rational subset $\mathcal{U}_i \subset \mathcal{X}$.  Then $\mathcal{T} := \lim_{\leftarrow i \in I} \mathcal{U}_i$ is the perfectoid space we seek.

Proof when $T$ is quasicompact open.

First we prove the result when $X$ is connected, i.e. when $\mathcal{X} = \mathrm{Spa}(K,K^+)$ as above.  We claim that in fact $T$ is a rational subset of $X$. When $T$ is empty, this is true in many stupid ways, so we can assume $T$ is nonempty. Since $T$ is a qc open, we can find finitely many nonempty rational subsets $\mathcal{W}_i = \mathrm{Spa}(K,K^{+}_{(i)}) \subset \mathcal{X}$ such that $T = \cup_i |\mathcal{W}_i|$.  But the $\mathcal{W}_i$‘s are totally ordered, since any finite set of open bounded valuation subrings of $K$ is totally ordered by inclusion (in the opposite direction), so $T = |\mathcal{W}|$ where $\mathcal{W}$ is the largest $\mathcal{W}_i$.

Now we turn to the general case. For each point $x \in \pi_0(X)$, we’ve proved that $T \cap \gamma^{-1}(x)$ is a rational subset (possibly empty) of the fiber $\gamma^{-1}(x)$.  Since $\gamma^{-1}(x) = \lim_{\substack{\leftarrow}{V_x \subset \pi_0(X) \mathrm{qc\,open}, x\in V_x}} \gamma^{-1}(V_x)$ and each $\gamma^{-1}(V_x)$ is the topological space of a rational subset $\mathcal{U}_x$ of $\mathcal{X}$, it’s now easy to check* that for every $x$ and for some small $V_x$ as above, there is a rational subset $\mathcal{T}_x \subset \mathcal{U}_x$ such that $|\mathcal{T}_x| = U_x \cap T$. Choose such a $V_x$ for each point in $\pi_0(X)$.  Since $\pi_0(X) = \cup_x V_x$, we can choose finitely many $x$‘s $\{x_i\}_{i\in I}$ such that the $V_{x_i}$‘s give a covering of $\pi_0(X)$.  Since each of these subsets is open-closed in $\pi_0(X)$, we can refine this covering to a covering of $\pi_0(X)$ by finitely many pairwise-disjoint open-closed subsets $W_j, j \in J$ where $W_j \subseteq V_{x_{i(j)}}$ for all $j$ and for some (choice of) $i(j) \in I$. Then $\gamma^{-1}(W_j)$ again comes from a rational subset $\mathcal{S}_j$ of $\mathcal{X}$, so the intersection $|\mathcal{T}_{x_{i(j)}}| \cap \gamma^{-1}(W_j)$ comes from the rational subset $\mathcal{T}_j := \mathcal{T}_{x_{i(j)}} \times_{\mathcal{U}_{x_{i(j)}}} \mathcal{S}_j$ of $X$, and since $|\mathcal{T}_j| = T \cap \gamma^{-1}(W_j)$ by design, we (finally) have that $\mathcal{T} = \coprod_{j} \mathcal{T}_j \subset \coprod_{j} S_j = \mathcal{X}$ is affinoid perfectoid. Whew! $\square$

*Here we’re using the “standard” facts that if $X_i$ is a cofiltered inverse system of affinoid perfectoid spaces with limit $X$, then $|X| = \lim_{\leftarrow i} |X_i|$, and any rational subset $W \subset X$ is the preimage of some rational subset $W_i \subset X_i$, and moreover if we have two such pairs $(i,W_i)$ and $(j,W_j)$ with the $W_{\bullet}$‘s both pulling back to $W$ then they pull back to the same rational subset of $X_k$ for some large $k \geq i,j$.

Let $T$ be a subset of a spectral space $X$; according to the incredible Lemma recorded in Tag 0A31 in the Stacks Project, the following are equivalent:

• $T$ is generalizing and pro-constructible;
• $T$ is generalizing and quasicompact;
• $T$ is an intersection of quasicompact open subsets of $X$.

Moreover, if $T$ has one of these equivalent properties, $T$ is spectral. (Johan tells me this lemma is “basically due to Gabber”.) Combining this result with the Theorem above, and using the fact that the category of affinoid perfectoid spaces has all small limits, we get the following disgustingly general statement.

Theorem. Let $\mathcal{X}$ be a w-local affinoid perfectoid space. Then any generalizing quasicompact subset $T \subset |\mathcal{X}|$ is affinoid perfectoid.

By an easy gluing argument, this implies even more generally (!) that if $T \subset |\mathcal{X}|$ is a subset such that every point $t\in T$ has a qc open neighborhood $U_t$ in $|\mathcal{X}|$ such that $T \cap U_t$ is quasicompact and generalizing, then $T$ is perfectoid (not necessarily affinoid perfectoid).  This condition* holds, for example, if $T$ is locally closed and generalizing; in that situation, I’d managed to prove that $T$ is perfectoid back in May (by a somewhat clumsy argument, cf. Section 2.7 of this thing if you’re curious) after Peter told me it was so.  But the argument here gives a lot more.

*Johan’s opinion of this condition: “I have no words for how nasty this is.”

## What does an inadmissible locus look like?

Let $H/ \overline{\mathbf{F}_p}$ be some p-divisible group of dimension d and height h, and let $\mathcal{M}$ be the rigid generic fiber (over $\mathrm{Spa}\,\breve{\mathbf{Q}}_p$) of the associated Rapoport-Zink space. This comes with its Grothendieck-Messing period map $\pi: \mathcal{M} \to \mathrm{Gr}(d,h)$, where $\mathrm{Gr}(d,h)$ is the rigid analytic Grassmannian paramatrizing rank d quotients of the (covariant) rational Dieudonne module $M(H) /\breve{\mathbf{Q}}_p$. Note that $\mathrm{Gr}(d,h)$ is a very nice space: it’s a smooth connected homogeneous rigid analytic variety, of dimension d(h-d).

The morphism $\pi$ is etale and partially proper (i.e. without boundary in Berkovich’s sense), and so the image of $\pi$ is an open and partially proper subspace* of the Grassmannian, which is usually known as the admissible locus. Let’s denote this locus by $\mathrm{Gr}(d,h)^a$. The structure of the admissible locus is understood in very few cases, and getting a handle on it more generally is a famous and difficult problem first raised by Grothendieck (cf. the Remarques on p. 435 of his 1970 ICM article). About all we know so far is the following:

• When d=1 (so $\mathrm{Gr}(d,h) = \mathbf{P}^{h-1}$) and $H$ is connected, we’re in the much-studied Lubin-Tate situation. Here, Gross and Hopkins famously proved that $\pi$ is surjective, not just on classical rigid points but on all adic points, so $\mathrm{Gr}(d,h)^a = \mathrm{Gr}(d,h)$ is the whole space. This case (along with the “dual” case where h>2,d=h-1) turns out to be the only case where $\mathrm{Gr}(d,h)^a = \mathrm{Gr}(d,h)$, cf. Rapoport’s appendix to Scholze’s paper on the Lubin-Tate tower.
• When $H \simeq \mathbf{G}_m^{d} \oplus (\mathbf{Q}_p/\mathbf{Z}_p)^{h-d}$, i.e. when $H$ has no bi-infinitesimal component, it turns out that $\mathrm{Gr}(d,h)^a = \mathbf{A}^{d(h-d)}$ is isomorphic to rigid analytic affine space of the appropriate dimension, and can be identified with the open Bruhat cell inside $\mathrm{Gr}(d,h)$. This goes back to Dwork, who proved it when d=1,h=2. (I don’t know a citation for the general result, but presumably for arbitrary d,h this is morally due to Serre-Tate/Katz?)
• In general there’s also the so-called weakly admissible locus $\mathrm{Gr}(d,h)^{wa} \subset \mathrm{Gr}(d,h)$, which contains the admissible locus and is defined in some fairly explicit way. It’s also characterized as the maximal admissible open subset of the Grassmannian with the same classical points as the admissible locus. In the classical rigid language, the map $\mathrm{Gr}(d,h)^a \to \mathrm{Gr}(d,h)^{wa}$ is etale and bijective; this is the terminology used e.g. in Rapoport-Zink’s book.
• In general, the admissible and weakly admissible loci are very different.  For example, when $H$ is isoclinic and (d,h)=1 (i.e. when $M(H)$ is irreducible as a $\varphi$-module), $\mathrm{Gr}(d,h)^a$ contains every classical point, and $\mathrm{Gr}(d,h)^{wa} = \mathrm{Gr}(d,h)$, so the weakly admissible locus tells you zilch about the admissible locus in this situation (and they really are different for any $1 < d < h-1$).

That’s about it for general results.

To go further, let’s switch our perspective a little. Since $\mathrm{Gr}(d,h)^a$ is an open and partially proper subspace of $\mathrm{Gr}(d,h)$, the subset $|\mathrm{Gr}(d,h)^a| \subseteq |\mathrm{Gr}(d,h)|$ is open and specializing, so its complement is closed and generalizing.  Now, according to a very general theorem of Scholze, namely Theorem 2.42 here (for future readers, in case the numbering there changes: it’s the main theorem in the section entitled “The miracle theorems”), if $\mathcal{D}$ is any diamond and $E \subset |\mathcal{D}|$ is any locally closed generalizing subset, there is a functorially associated subdiamond $\mathcal{E} \subset \mathcal{D}$ with $|\mathcal{E}| = E$ inside $|\mathcal{D}|$. More colloquially, one can “diamondize” any locally closed generalizing subset of $|\mathcal{D}|$, just as any locally closed subspace of $|X|$ for a scheme $X$ comes from a unique (reduced) subscheme of $X$.

Definition. The inadmissible/nonadmissible locus $\mathrm{Gr}(d,h)^{na}$ is the subdiamond of $\mathrm{Gr}(d,h)^{\lozenge}$ obtained by diamondizing the topological complement of the admissible locus, i.e. by diamondizing the closed generalizing subset $|\mathrm{Gr}(d,h)^a|^c \subset |\mathrm{Gr}(d,h)| \cong |\mathrm{Gr}(d,h)^{\lozenge}|$.

It turns out that one can actually get a handle on $\mathrm{Gr}(d,h)^{na}$ in a bunch of cases!  This grew out of some conversations with Jared Weinstein – back in April, Jared raised the question of understanding the inadmissible locus in a certain particular period domain for $\mathrm{GL}_2$ with non-minuscule Hodge numbers, and we managed to describe it completely in that case (see link below). Last night, though, I realized we hadn’t worked out any interesting examples in the minuscule (i.e. p-divisible group) setting! Here I want to record two such examples, hot off my blackboard, one simple and one delightfully bizarre.

Example 1. Take h=4, d=2 and $H$ isoclinic. Then $|\mathrm{Gr}(d,h)^a|^c$ is a single classical point, corresponding to the unique filtration on $M(H)$ with Hodge numbers $0,0,1,1$ which is not weakly admissible. So $\mathrm{Gr}(d,h)^a = \mathrm{Gr}(d,h)^{wa}$ in this case.

Example 2. Take h=5, d=2 and $H$ isoclinic\$.  Now things are much stranger.  Are you ready?
Theorem. In this case, the locus $\mathrm{Gr}^{na}$ is naturally isomorphic to the diamond $(X \smallsetminus 0)^{\lozenge} / \underline{D^\times}$, where $X$ is an open perfectoid unit disk in one variable over $\breve{\mathbf{Q}}_p$ and $D=D_{1/3}$ is the division algebra over $\mathbf{Q}_p$ with invariant 1/3, acting freely on $X \smallsetminus 0$ in a certain natural way. Precisely, the disk $X$ arises as the universal cover of the connected p-divisible group of dimension 1 and height 15, and its natural $D$-action comes from the natural $D_{1/15}$-action on $X$ via the map $D_{1/3} \to D_{1/3} \otimes D_{-2/5} \simeq D_{-1/15} \simeq D_{1/15}^{op}$.

This explicit description is actually equivariant for the $D_{2/5}$-actions on $X$ and $Gr$. As far as diamonds go, $(X \smallsetminus 0)^{\lozenge}/\underline{D^{\times}}$ is pretty high-carat: it’s spatial (roughly, its qcqs with lots of qcqs open subdiamonds), and its structure morphism to $\mathrm{Spd}\,\breve{\mathbf{Q}}_p$ is separated, smooth, quasicompact, and partially proper in the appropriate senses. Smoothness, in particular, is meant in the sense of Definition 6.1 here (cf. also the discussion in Section 4.3 here). So even though this beast doesn’t have any points over any finite extension of $\breve{\mathbf{Q}}_p$, it’s still morally a diamondly version of a smooth projective curve!

The example Jared and I had originally worked out is recorded in section 5.5 here. The reader may wish to try adapting our argument from that situation to the cases mentioned above – this is a great exercise in actually using the classification of vector bundles on the Fargues-Fontaine curve in a hands-on calculation.

Anyway, here’s a picture of $(X \smallsetminus 0)^{\lozenge} / \underline{D^{\times}}$, with some other inadmissible loci in the background:

*All rigid spaces here and throughout the post are viewed as adic spaces: in the classical language, $\mathrm{Gr}(d,h)^a$ does not generally correspond to an admissible open subset of $\mathrm{Gr}(d,h)$, so one would be forced to say that there exists a rigid space $\mathrm{Gr}(d,h)^a$ together with an etale monomorphism $\mathrm{Gr}(d,h)^a \to \mathrm{Gr}(d,h)$. But in the adic world it really is a subspace.