The Newton stratification is true

Let G be a connected reductive group over \mathbf{Q}_p, and let \mu be a G-valued (geometric) conjugacy class of minuscule cocharacters, with reflex field E. In their Annals paper, Caraiani and Scholze defined a very interesting stratification of the flag variety \mathcal{F}\ell_{G,\mu} (regarded as an adic space over E) into strata \mathcal{F}\ell_{G,\mu}^{b}, where b runs over the Kottwitz set B(G,\mu^{-1}). Let me roughly recall how this goes: any (geometric) point x \to \mathcal{F}\ell_{G,\mu} determines a canonical modification \mathcal{E}_x \to \mathcal{E}_{triv} of the trivial G-bundle on the Fargues-Fontaine curve, meromorphic at \infty and with “mermorphy \mu” in the usual sense. On the other hand, Fargues proved that G-bundles on the curve are classified up to isomorphism by B(G), and then Caraiani-Scholze and Rapoport proved that \mu-meromorphic modifications of the trivial bundle are exactly classified by the subset B(G,\mu^{-1}) (CS proved that only these elements can occur; R proved that all of these elements occur). The Newton stratification just records which element of this set parametrizes the bundle \mathcal{E}_x.

The individual strata are pretty weird. For example, if G=GL_n and \mu=(1,0,\dots,0), then \mathcal{F}\ell_{G,\mu} \simeq \mathbf{P}^{n-1} and the open stratum is just the usual Drinfeld space \Omega^{n-1}, but the other strata are of the form \Omega^{n-i-1} \times^{P_{n-i,i}(\mathbf{Q}_p)} GL_n(\mathbf{Q}_p), where P_{n-i,i} is the evident parabolic in GL_n and the action on \Omega^{n-i-1} is via the natural map P_{n-i,i}(\mathbf{Q}_p) \twoheadrightarrow GL_{n-i}(\mathbf{Q}_p). Qualitatively, this says that they’re unions of profinitely many copies of lower-dimensional Drinfeld spaces. In particular, the non-open strata are not rigid analytic spaces. There are also examples of strata which don’t have any classical rigid analytic points. However, the \mathcal{F}\ell_{G,\mu}^{b}‘s are always perfectly well-defined from the topological or diamond point of view.

Anyway, I’m getting to the following thing, which settles a question left open by Caraiani-Scholze.

Theorem. Topologically, the Newton stratification of \mathcal{F}\ell_{G,\mu} is a true stratification: the closure of any stratum is a union of strata.

The idea is as follows. After base-changing from E to the completed maximal unramified extension E' (which is a harmless move), there is a canonical map \zeta: \mathcal{F}\ell_{G,\mu,E'} \to \mathrm{Bun}_{G} sending x to the isomorphism class of \mathcal{E}_x. Here \mathrm{Bun}_{G} denotes the stack of G-bundles on the Fargues-Fontaine curve, regarded as a stack on the category of perfectoid spaces over \overline{\mathbf{F}_p}. This stack is stratified by locally closed substacks \mathrm{Bun}_{G}^{b} defined in the obvious way, and by construction the Newton stratification is just the pullback of this stratification along \zeta. Now, by Fargues’s theorem we get an identification |\mathrm{Bun}_{G}| = B(G), so it is completely trivial to see that the stratification of \mathrm{Bun}_{G} is a true stratification (at the level of topological spaces). We then conclude by the following observation:

Proposition. The map \zeta is universally open.

The idea is to observe that \zeta factors as a composition of two maps \mathcal{F}\ell_{G,\mu,E'} \to [\mathcal{F}\ell_{G,\mu,E'}/\underline{G(\mathbf{Q}_p)}] \to \mathrm{Bun}_{G}. Here the first map is a \underline{G(\mathbf{Q}_p)}-torsor by construction, so it’s universally open by e.g. Lemma 10.13 here. More subtly, the second map is also universally open. Why? Because it is cohomologically smooth in the sense of Definition 23.8 here; universal openness then follows by Proposition 23.11 in the same document.

For the cohomological smoothness claim, take any affinoid perfectoid space with a map T \to \mathrm{Bun}_{G}, corresponding to some bundle \mathcal{F} / \mathcal{X}_T. After some thought, one works out the fiber product X = T \times_{\mathrm{Bun}_{G}} [\mathcal{F}\ell_{G,\mu,E'}/\underline{G(\mathbf{Q}_p)}] “explicitly”: it parametrizes untilts of T over E' together with isomorphism classes of \mu^{-1}-meromorphic modifications \mathcal{E}\to \mathcal{F} supported along the section T^{\sharp} \to \mathcal{X}_T induced by our preferred untilt, with the property that \mathcal{E} is trivial at every geometric point of T. Without the final condition, we get a larger functor X' which etale-locally on T is isomorphic to T \times_{\mathrm{Spd}(\overline{\mathbf{F}_p})} \mathcal{F}\ell_{G,\mu^{-1},E'}^{\lozenge}. (To get the latter description, note that etale-locally on T we can trivialize \mathcal{F} on the formal completion of the curve along T^{\sharp}, and then use Beauville-Laszlo to interpret the remaining data as a suitably restricted modification of the trivial G-torsor on \mathrm{Spec} \mathbf{B}_{dR}^{+}(\mathcal{O}(T^{\sharp})). This is a Schubert cell in a Grassmannian. Then use Caraiani-Scholze’s results on the Bialynicki-Birula map.) Anyway anyway, after a little more fiddling around the point is basically that the projection X' \to T is cohomologically smooth because it’s the base change of a smooth map of rigid spaces. By Kedlaya-Liu plus epsilon, the natural map X \to X' is an open immersion, so X \to T is cohomologically smooth. Since T was arbitrary, this is enough.

 

 

 

 

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:

diamond

 

 

*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.