This post is an outgrowth of my (ongoing) attempt to understand Peter Scholze’s remarkable course at Berkeley. The first couple of paragraphs below are a rather comically compressed summary of some portions of Jared Weinstein’s excellent notes (though of course all mistakes are my own).
Let denote the category of perfectoid spaces in characteristic , equipped with its qpf topology (qpf = quasiprofinite, a slight loosening of the pro-etale topology). If is any perfectoid space in characteristic , defines a qpf sheaf on , and the association defines a fully faithful functor . A diamond is a sheaf on which admits a (relatively representable) qpf surjection from a representable sheaf. If is a perfectoid Tate-Huber pair in characteristic , we write for the sheaf regarded as a diamond. (Note: If is a diamond, I will write or for the sections of over some . Note also that if is another diamond, one can make sense of the sections of over : taking any qpf surjection , is the equalizer of . So diamonds can be regarded as certain sheaves on the category of diamonds…)
Here is one way to make diamonds. If is an adic space over , define a presheaf on by where the right-hand side denotes isomorphism classes of situations with a morphism to from a perfectoid space over and an isomorphism. So is “morphisms to from untilts of ”. This turns out to be a diamond: if is a proetale perfectoid covering of , then admits a proetale surjection from the representable sheaf . If is an affinoid adic space over , let denote the associated diamond . In particular we set . An element of is simply a pair where is a perfectoid space over and is an isomorphism. The association defines a functor from adic spaces over to diamonds; since this is indeed a functor, the structure map induces a morphism of diamonds .
However, there are plenty of diamonds which don’t arise in this manner. For example, given a reduced rational , consider the sheafification of the functor which sends a perfectoid Tate-Huber pair in characteristic to the set , where is any pseudouniformizer. This is independent of and (hence the notation), and is representable by a diamond , in the sense that . When , is not of the form . Note also that does not come with any canonical map to . For an example which does come with a map to , but still isn’t of the form , we have the diamond . This represents the sheafification of the functor which takes a map ( as above) to , where denotes the untilt of defined by the map to .
An aside: Note that if is an affinoid perfectoid space over , there is a natural morphism of diamonds given by sending to . This is even a homeomorphism on the underlying topological spaces – nevertheless, the map is something like a closed immersion, despite the fact that is a single point. This is one sense in which the fiber product of diamonds is occuring over a deeper base.
What good is this formalism? Here is one likely application. Let be a perfect field of characteristic , , a -module over . Let and . Let be a filtration type, i.e. a function such that . Define , and let denote the multiset of integers appearing in where is included in with multiplicity . Note that either of or determines the other uniquely. Set . We assume that .
For a complete discretely valued extension, we say a filtration of by -subvectorspaces is of type if
for all . By Colmez-Fontaine, there is an equivalence of categories between weakly admissible filtrations of of type , and pairs where is a crystalline representation of with Hodge-Tate multiset and is a -linear isomorphism .
Suppose now that is any perfectoid -algebra. We have the usual rings and . A -lattice in is a free -submodule such that
We say is of type if for some basis of we have , where is a generator of and denote the elements of in any order.
Let us connect with the more familiar notion of a filtration on .
Proposition. Suppose is a discretely valued extension with algebraic closure . Then the data of a filtration of of type is equivalent to the data of a –stable -lattice of type .
Idea of proof. Given a filtration of by subspaces , set
Conversely, given , set . See Fargues-Fontaine for details.
Let denote the sheafification of the functor on diamonds over which assigns to a connected affinoid perfectoid diamond the set of -lattices of type such that for some sufficiently large (any larger than all elements of should be fine), the map
is surjective. Here is the untilt of determined by the map . By Colmez’s theory of Finite Dimensional Banach Spaces, the source (resp. target) of is a Banach Space of dimension (resp. ) and dimensions are additive in short exact sequences of Banach Spaces, so (by our assumption ) the kernel of is a -dimensional -vector space. (Really I don’t need to assume the test diamonds are connected, but then you get some kind of sheaf of -vector spaces instead.)
“Theorem.” The functor is representable by a diamond over , in the sense that
The diamond is a subdiamond of a “ partial affine Grassmannian over .”
These ‘s should be examples of the “very general p-adic period domains” mentioned in Peter’s course description. I put quotes around the word “theorem” here because I haven’t actually proved it (though I have some idea how one might), and because if it is true then surely Peter has proven it already – indeed, should admit an etale morphism from the moduli space of “rank local shtukas with one paw over ”. The partial affine Grassmannian in question is the functor which sends to the set of all -lattices such that .
Theorem. If is a complete discretely valued extension with maximal absolutely unramified subfield , there is an identification of sets
where the right-hand side denotes pairs consisting of a crystalline representation of with Hodge-Tate multiset together with a -linear isomorphism .
Proof. Given , note that induces isomorphisms and (by Colmez-Fontaine) which we also denote by . Set , so we get an element of . Now we examine the sheaf sequence associated with the covering , i.e.
This is a necessary step: is not a perfectoid diamond, and is a stupid thing. Now its not hard to see that is the diamond associated with , and that evaluated on this diamond is (and likewise for ). Pulling back under , goes to the “constant” lattice , in the sense that “evaluating” at any we recover the lattice . On the other hand, pulling back under brings to the lattice whose evaluation at any is , i.e. the pullback of under the natural -action on . But is Galois-stable by construction, so we really do get an element of the equalizer!
The other direction is now clear as well: given , our analysis shows that that lives in the equalizer of the ‘s iff is preserved by the Galois action on , but this is exactly the condition that (a priori, nothing more that a -vector space) is -stable and thus a -representation. One easily checks that has all the claimed properties.
Proposition. If , then is representable by (the diamond associated with) an open subspace of a flag variety over .
Idea of proof. Notation as above, any agrees with , so its enough to specify the -direct summand , and such things are specified by a point , or equivalently a morphism of diamonds over .
In general one still has a rigid analytic flag variety over together with some weakly admissible locus , and there is an identification of sets for any as above. But it seems unlikely that is describable in terms of in general, or even that the two spaces are directly related.