I wonder what this wild mess –
Let be a complete algebraically closed extension, and let be the Fargues-Fontaine curve associated with . If is any vector bundle on , the cohomology groups vanish for all and are naturally Banach-Colmez Spaces for . Recall that the latter things are roughly “finite-dimensional -vector spaces up to finite-dimensional -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 which is (componentwise-) additive in short exact sequences. The Dimension roughly records the -dimension and the -dimension, respectively. Typical examples are , which has Dimension , and , which has Dimension .
Here I want to record the following beautiful Riemann-Roch formula.
Theorem. If is any vector bundle on , then .
One can prove this by induction on the rank of , 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.
If you missed the awesome p-adic Langlands conference at Indiana University this past May, videos of all the talks are now available here.
Some talks I really liked: Bergdall, Cais, Hellmann, Ludwig.
Weirdest talk: [redacted]
Let be a spectral space. Following Bhatt-Scholze, say is w-local if the subset of closed points is closed and if every connected component of has a unique closed point. This implies that the natural composite map 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 , where is a nonarchimedean field and is an open and bounded valuation subring of , 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 together with a subset , let’s say is perfectoid (resp. affinoid perfectoid) if there is a pair where is a perfectoid space (resp. affinoid perfectoid space) and is a map of adic spaces identifying homeomorphically with and which is universal for maps of perfectoid spaces which factor through on topological spaces. Note that if the pair exists, it’s unique up to unique isomorphism.
Theorem. Let be a w-local affinoid perfectoid space. Then any subset of which is closed and generalizing, or which is quasicompact open, is affinoid perfectoid.
Proof when is closed and generalizing. The key point here is that the map defines a bijection between closed generalizing subsets of and closed subsets of the (profinite) space , by taking preimages of the latter or images of the former. To check that this is true, note that if is closed and generalizing, then its intersection with a connected component of being nonempty implies (since is generalizing) that contains the unique rank one point of . But then contains all specializations of that point (since is closed in ), so , so any given connected component of is either disjoint from or contained entirely in . This implies that can be read off from which closed points of it contains. Finally, one easily checks that is closed in , since is profinite and is quasicompact. Therefore .
Returning to the matter at hand, write as a cofiltered intersection of qc opens , . But qc opens in are the same as open-closed subsets, so each pulls back to an open-closed subset , and its easy to check that any such comes from a unique rational subset . Then is the perfectoid space we seek.
Proof when is quasicompact open.
First we prove the result when is connected, i.e. when as above. We claim that in fact is a rational subset of . When is empty, this is true in many stupid ways, so we can assume is nonempty. Since is a qc open, we can find finitely many nonempty rational subsets such that . But the ‘s are totally ordered, since any finite set of open bounded valuation subrings of is totally ordered by inclusion (in the opposite direction), so where is the largest .
Now we turn to the general case. For each point , we’ve proved that is a rational subset (possibly empty) of the fiber . Since and each is the topological space of a rational subset of , it’s now easy to check* that for every and for some small as above, there is a rational subset such that . Choose such a for each point in . Since , we can choose finitely many ‘s such that the ‘s give a covering of . Since each of these subsets is open-closed in , we can refine this covering to a covering of by finitely many pairwise-disjoint open-closed subsets where for all and for some (choice of) . Then again comes from a rational subset of , so the intersection comes from the rational subset of , and since by design, we (finally) have that is affinoid perfectoid. Whew!
*Here we’re using the “standard” facts that if is a cofiltered inverse system of affinoid perfectoid spaces with limit , then , and any rational subset is the preimage of some rational subset , and moreover if we have two such pairs and with the ‘s both pulling back to then they pull back to the same rational subset of for some large .
Let be a subset of a spectral space ; according to the incredible Lemma recorded in Tag 0A31 in the Stacks Project, the following are equivalent:
Moreover, if has one of these equivalent properties, 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 be a w-local affinoid perfectoid space. Then any generalizing quasicompact subset is affinoid perfectoid.
By an easy gluing argument, this implies even more generally (!) that if is a subset such that every point has a qc open neighborhood in such that is quasicompact and generalizing, then is perfectoid (not necessarily affinoid perfectoid). This condition* holds, for example, if is locally closed and generalizing; in that situation, I’d managed to prove that 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.”
Let be some p-divisible group of dimension d and height h, and let be the rigid generic fiber (over ) of the associated Rapoport-Zink space. This comes with its Grothendieck-Messing period map , where is the rigid analytic Grassmannian paramatrizing rank d quotients of the (covariant) rational Dieudonne module . Note that is a very nice space: it’s a smooth connected homogeneous rigid analytic variety, of dimension d(h-d).
The morphism is etale and partially proper (i.e. without boundary in Berkovich’s sense), and so the image of is an open and partially proper subspace* of the Grassmannian, which is usually known as the admissible locus. Let’s denote this locus by . 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:
That’s about it for general results.
To go further, let’s switch our perspective a little. Since is an open and partially proper subspace of , the subset 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 is any diamond and is any locally closed generalizing subset, there is a functorially associated subdiamond with inside . More colloquially, one can “diamondize” any locally closed generalizing subset of , just as any locally closed subspace of for a scheme comes from a unique (reduced) subscheme of .
Definition. The inadmissible/nonadmissible locus is the subdiamond of obtained by diamondizing the topological complement of the admissible locus, i.e. by diamondizing the closed generalizing subset .
It turns out that one can actually get a handle on 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 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 isoclinic. Then is a single classical point, corresponding to the unique filtration on with Hodge numbers which is not weakly admissible. So in this case.
Example 2. Take h=5, d=2 and isoclinic$. Now things are much stranger. Are you ready?
Theorem. In this case, the locus is naturally isomorphic to the diamond , where is an open perfectoid unit disk in one variable over and is the division algebra over with invariant 1/3, acting freely on in a certain natural way. Precisely, the disk arises as the universal cover of the connected p-divisible group of dimension 1 and height 15, and its natural -action comes from the natural -action on via the map .
This explicit description is actually equivariant for the -actions on and . As far as diamonds go, is pretty high-carat: it’s spatial (roughly, its qcqs with lots of qcqs open subdiamonds), and its structure morphism to 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 , 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 , 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, does not generally correspond to an admissible open subset of , so one would be forced to say that there exists a rigid space together with an etale monomorphism . But in the adic world it really is a subspace.
The 2017 Arizona Winter School on perfectoid spaces is officially happening! (I’ll be there, as Kiran’s project assistant.)