Diamonds in the rough

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 {\mathrm{Perf}} denote the category of perfectoid spaces in characteristic {p}, equipped with its qpf topology (qpf = quasiprofinite, a slight loosening of the pro-etale topology). If {X} is any perfectoid space in characteristic {p}, {h_{X}=\mathrm{Hom}_{\mathrm{Perf}}(-,X)} defines a qpf sheaf on {\mathrm{Perf}}, and the association {X\rightsquigarrow h_{X}} defines a fully faithful functor {\mathrm{Perf}\rightarrow\mathrm{Sh}_{\mathrm{qpf}}(\mathrm{Perf})}. A diamond is a sheaf {\mathcal{D}} on {\mathrm{Perf}} which admits a (relatively representable) qpf surjection {h_{X}\rightarrow\mathcal{D}} from a representable sheaf. If {(R,R^{+})} is a perfectoid Tate-Huber pair in characteristic {p}, we write {\mathrm{Spd}(R,R^{+})} for the sheaf {h_{\mathrm{Spa}(R,R^{+})}} regarded as a diamond. (Note: If {\mathcal{D}} is a diamond, I will write {\mathrm{Hom}(X,\mathcal{D})} or {\mathcal{D}(X)} for the sections of {\mathcal{D}} over some {X\in\mathrm{Perf}}. Note also that if {\mathcal{E}} is another diamond, one can make sense of the sections of {\mathcal{D}} over {\mathcal{E}}: taking any qpf surjection {h_{X}\rightarrow\mathcal{E}}, {\mathcal{D}(\mathcal{E})} is the equalizer of {\mathcal{D}(X)\rightrightarrows\mathcal{D}(X\times_{\mathcal{E}}X)}. So diamonds can be regarded as certain sheaves on the category of diamonds…)

Here is one way to make diamonds. If {Y} is an adic space over {\mathbf{Q}_{p}}, define a presheaf {Y^{\lozenge}} on {\mathrm{Perf}} by {Y^{\lozenge}(X)=\{(X^{\sharp}\rightarrow Y,\iota)/\sim\}} where the right-hand side denotes isomorphism classes of situations with {X^{\sharp}\rightarrow Y} a morphism to {Y} from a perfectoid space {X^{\sharp}} over {\mathbf{Q}_{p}} and {\iota:X^{\sharp\flat}\overset{\sim}{\rightarrow}X} an isomorphism. So {Y^{\lozenge}(X)} is “morphisms to {Y} from untilts of {X}”. This turns out to be a diamond: if {\tilde{Y}\rightarrow Y} is a proetale perfectoid covering of {Y}, then {Y^{\lozenge}} admits a proetale surjection from the representable sheaf {h_{\tilde{Y}^{\flat}}}. If {Y=\mathrm{Spa}(S,S^{+})} is an affinoid adic space over {\mathbf{Q}_{p}}, let {\mathrm{Spd}(S,S^{+})} denote the associated diamond {Y^{\lozenge}}. In particular we set {\mathrm{Spd}\mathbf{Q}_{p}=\mathrm{Spd}(\mathbf{Q}_{p},\mathbf{Z}_{p})}. An element of {(\mathrm{Spd}\mathbf{Q}_{p})(X)} is simply a pair {(X^{\sharp},\iota)} where {X^{\sharp}} is a perfectoid space over {\mathbf{Q}_{p}} and {\iota:X^{\sharp\flat}\overset{\sim}{\rightarrow}X} is an isomorphism. The association {Y\rightsquigarrow Y^{\lozenge}} defines a functor from adic spaces over {\mathbf{Q}_{p}} to diamonds; since this is indeed a functor, the structure map {Y\rightarrow\mathrm{Spa}(\mathbf{Q}_{p},\mathbf{Z}_{p})} induces a morphism of diamonds {Y^{\lozenge}\rightarrow\mathrm{Spd}\mathbf{Q}_{p}}.

However, there are plenty of diamonds which don’t arise in this manner. For example, given a reduced rational {a/h\in\mathbf{Q}_{\geq0}}, consider the sheafification of the functor which sends a perfectoid Tate-Huber pair {(R,R^{+})} in characteristic {p} to the set {\mathbf{B}_{a/h}(R)=\mathbf{B}_{\mathrm{crys}}^{+}(R^{+}/\varpi)^{\varphi^{h}=p^{a}}}, where {\varpi} is any pseudouniformizer. This is independent of {\varpi} and {R^{+}} (hence the notation), and is representable by a diamond {\mathbb{B}_{a/h}}, in the sense that {\mathbf{B}_{a/h}(R)=\mathrm{Hom}(\mathrm{Spa}(R,R^{+}),\mathbb{B}_{a/h})}. When {a/h>1}, {\mathbb{B}_{a/h}} is not of the form {Y^{\lozenge}}. Note also that {\mathbb{B}_{a/h}} does not come with any canonical map to {\mathrm{Spd}\mathbf{Q}_{p}}. For an example which does come with a map to {\mathrm{Spd}\mathbf{Q}_{p}}, but still isn’t of the form {(-)^{\lozenge}}, we have the diamond {\mathbb{B}_{\mathrm{dR}}^{+}/\mathrm{Fil}^{i}}. This represents the sheafification of the functor which takes a map {\mathrm{Spd}(R,R^{+})\rightarrow\mathrm{Spd}\mathbf{Q}_{p}} ({(R,R^{+})} as above) to {\mathbf{B}_{\mathrm{dR}}^{+}(R^{\sharp})/\mathrm{Fil}^{i}}, where {(R^{\sharp},R^{\sharp+})} denotes the untilt of {(R,R^{+})} defined by the map to {\mathrm{Spd}\mathbf{Q}_{p}}.

An aside: Note that if {\mathrm{Spa}(S,S^{+})} is an affinoid perfectoid space over {\mathbf{Q}_{p}}, there is a natural morphism of diamonds {\mathrm{Spd}(S,S^{+})\rightarrow\mathrm{Spd}(S^{\flat},S^{\flat+})} given by sending {(f:X^{\sharp}\rightarrow\mathrm{Spa}(S,S^{+}),\iota)\in\mathrm{Spd}(S,S^{+})(X)} to {(f^{\flat}\circ\iota^{-1}:X\rightarrow\mathrm{Spa}(S^{\flat},S^{\flat+}))\in\mathrm{Spd}(S^{\flat},S^{\flat+})(X)}. This is even a homeomorphism on the underlying topological spaces – nevertheless, the map {\mathrm{Spd}(S,S^{+})\rightarrow\mathrm{Spd}(S^{\flat},S^{\flat+})\times\mathrm{Spd}\mathbf{Q}_{p}} is something like a closed immersion, despite the fact that {|\mathrm{Spd}\mathbf{Q}_{p}|} 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 {k} be a perfect field of characteristic {p}, {K_{0}=W(k)[\frac{1}{p}]}, {D} a {\varphi}-module over {K_{0}}. Let {d=\mathrm{dim}_{K_{0}}D} and {t_{N}(D)=v_{p}(\det\varphi|D)}. Let {\mathbf{h}} be a filtration type, i.e. a function {\mathbf{h}:\mathbf{Z}\rightarrow\mathbf{Z}_{\geq0}} such that {\sum_{i\in\mathbf{Z}}\mathbf{h}(i)=d}. Define {\mathrm{supp}\mathbf{h}=\left\{ i\in\mathbf{Z}\mid\mathbf{h}(i)\neq0\right\} }, and let {\mathrm{HT}(\mathbf{h})} denote the multiset of integers appearing in {\mathrm{supp}\mathbf{h}} where {i} is included in {\mathrm{HT}(\mathbf{h})} with multiplicity {\mathbf{h}(i)}. Note that either of {\mathbf{h}} or {\mathrm{HT}(\mathbf{h})} determines the other uniquely. Set {t_{H}(\mathbf{h})=\sum_{i}i\mathbf{h}(i)=\sum_{j\in\mathrm{HT}(\mathbf{h})}j}. We assume that {t_{H}(\mathbf{h})=t_{N}(D)}.

For {L/K_{0}} a complete discretely valued extension, we say a filtration of {D_{L}:=D\otimes_{K_{0}}L} by {L}-subvectorspaces {\cdots\subset\mathrm{Fil}^{i+1}D_{L}\subset\mathrm{Fil}^{i}D_{L}\subset\cdots} is of type {\mathbf{h}} if

\displaystyle \begin{array}{rcl} \mathrm{dim}_{L}\mathrm{Fil}^{i}D_{L} & = & \sum_{j\geq i}\mathbf{h}(j)\\ & = & |\mathrm{HT}(\mathbf{h})\cap\mathbf{Z}_{\geq i}|\end{array}

for all {i\in\mathbf{Z}}. By Colmez-Fontaine, there is an equivalence of categories between weakly admissible filtrations of {D_{L}} of type {\mathbf{h}}, and pairs {(V,\gamma)} where {V} is a crystalline representation of {G_{L}} with Hodge-Tate multiset {\mathrm{HT}(\mathbf{h})} and {\gamma} is a {\varphi}-linear isomorphism {\gamma:\mathbf{D}_{\mathrm{crys}}(V)\overset{\sim}{\rightarrow}D\otimes_{K_{0}}L_{0}}.

Suppose now that {S} is any perfectoid {K_{0}}-algebra. We have the usual rings {\mathbf{B}_{\mathrm{dR}}^{+}(S)} and {\mathbf{B}_{\mathrm{dR}}(S)}. A {\mathbf{B}_{\mathrm{dR}}^{+}(S)}-lattice in {D\otimes_{K_{0}}\mathbf{B}_{\mathrm{dR}}(S)} is a free {\mathbf{B}_{\mathrm{dR}}^{+}(S)}-submodule {\mathbf{M}\subset D\otimes_{K_{0}}\mathbf{B}_{\mathrm{dR}}(S)} such that

\displaystyle \mathbf{M}\otimes_{\mathbf{B}_{\mathrm{dR}}^{+}(S)}\mathbf{B}_{\mathrm{dR}}(S)=D\otimes_{K_{0}}\mathbf{B}_{\mathrm{dR}}(S).

We say {\mathbf{M}} is of type {\mathbf{h}} if for some basis {e_{1},\dots,e_{d}} of {D\otimes_{K_{0}}\mathbf{B}_{\mathrm{dR}}^{+}(S)} we have {\mathbf{M}=\oplus_{i=1}^{d}\xi^{-w_{i}}\mathbf{B}_{\mathrm{dR}}^{+}(S)\cdot e_{i}}, where {\xi} is a generator of {\ker\left(\mathbf{B}_{\mathrm{dR}}^{+}(S)\overset{\theta}{\rightarrow}S\right)} and {w_{1},\dots,w_{d}} denote the elements of {\mathrm{HT}(\mathbf{h})} in any order.

Let us connect {\mathbf{M}} with the more familiar notion of a filtration on {D_{L}}.

Proposition. Suppose {L/K_{0}} is a discretely valued extension with algebraic closure {\overline{L}}. Then the data of a filtration of {D_{L}} of type {\mathbf{h}} is equivalent to the data of a {G_{L}}stable {\mathbf{B}_{\mathrm{dR}}^{+}(\widehat{\overline{L}})}-lattice {\mathbf{M}\subset D\otimes_{K_{0}}\mathbf{B}_{\mathrm{dR}}(\widehat{\overline{L}})} of type {\mathbf{h}}.

Idea of proof. Given a filtration of {D_{L}} by subspaces {\cdots\subset\mathrm{Fil}^{i}D_{L}\subset\mathrm{Fil}^{i-1}D_{L}\subset\cdots}, set

\displaystyle \begin{array}{rcl} \mathbf{M} & = & \mathrm{Fil}^{0}\left(D_{L}\otimes_{L}\mathbf{B}_{\mathrm{dR}}(\widehat{\overline{L}})\right)\\ & \subset & D_{L}\otimes_{L}\mathbf{B}_{\mathrm{dR}}(\widehat{\overline{L}})\\ & = & D\otimes_{K_{0}}\mathbf{B}_{\mathrm{dR}}(\widehat{\overline{L}}).\end{array}

Conversely, given {\mathbf{M}}, set {\mathrm{Fil}^{i}D_{L}=\left(t^{i}\mathbf{M}\right)^{G_{L}}}. See Fargues-Fontaine for details. {\square}

Let {\mathcal{M}_{D,\mathbf{h}}} denote the sheafification of the functor on diamonds over {\mathrm{Spd}K_{0}} which assigns to a connected affinoid perfectoid diamond {\mathrm{Spd}(R,R^{+})\rightarrow\mathrm{Spd}K_{0}} the set of {\mathbf{B}_{\mathrm{dR}}^{+}(R^{\sharp})}-lattices {\mathbf{M}\subset D\otimes_{K_{0}}\mathbf{B}_{\mathrm{dR}}(R^{\sharp})} of type {\mathbf{h}} such that for some sufficiently large {n} (any {n} larger than all elements of {\mathrm{supp}\mathbf{h}} should be fine), the map

\displaystyle \left(D\otimes_{K_{0}}t^{-n}\mathbf{B}_{\mathrm{crys}}^{+}(R^{\sharp})\right)^{\varphi=1}\overset{\alpha_{\mathbf{M}}}{\rightarrow}\left(D\otimes_{K_{0}}t^{-n}\mathbf{B}_{\mathrm{dR}}^{+}(R^{\sharp})\right)/\mathbf{M}

is surjective. Here {R^{\sharp}} is the untilt of {R} determined by the map {\mathrm{Spd}(R,R^{+})\rightarrow\mathrm{Spd}K_{0}}. By Colmez’s theory of Finite Dimensional Banach Spaces, the source (resp. target) of {\alpha_{\mathbf{M}}} is a Banach Space of dimension {(nd-t_{N}(D),d)} (resp. {(nd-t_{H}(\mathbf{h}),0)}) and dimensions are additive in short exact sequences of Banach Spaces, so (by our assumption {t_{N}=t_{H}}) the kernel of {\alpha_{\mathbf{M}}} is a {d}-dimensional {\mathbf{Q}_{p}}-vector space. (Really I don’t need to assume the test diamonds are connected, but then you get some kind of sheaf of {\mathbf{Q}_{p}}-vector spaces instead.)

“Theorem.” The functor {\mathcal{M}_{D,\mathbf{h}}} is representable by a diamond over {\mathrm{Spd}K_{0}}, in the sense that

\displaystyle \mathcal{M}_{D,\mathbf{h}}(\mathcal{D})=\mathrm{Hom}_{\mathrm{Dia_{/\mathrm{Spd}K_{0}}}}(\mathcal{D},\mathcal{M}_{D,\mathbf{h}}).

The diamond {\mathcal{M}_{D,\mathbf{h}}} is a subdiamond of a “ partial affine Grassmannian over {\mathrm{Spd}K_{0}}.”

These {\mathcal{M}_{D,\mathbf{h}}}‘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, {\mathcal{M}_{D,\mathbf{h}}} should admit an etale morphism from the moduli space of “rank {d} local shtukas with one paw over {\mathrm{Spd}K_{0}}”. The partial affine Grassmannian in question is the functor which sends {\mathrm{Spd}(R,R^{+})\rightarrow\mathrm{Spd}K_{0}} to the set of all {\mathbf{B}_{\mathrm{dR}}^{+}(R^{\sharp})}-lattices {\mathbf{M}\subset D\otimes_{K_{0}}\mathbf{B}_{\mathrm{dR}}(R^{\sharp})} such that {D\otimes_{K_{0}}t^{n}\mathbf{B}_{\mathrm{dR}}^{+}(R^{\sharp})\subseteq\mathbf{M}\subseteq D\otimes_{K_{0}}t^{-n}\mathbf{B}_{\mathrm{dR}}^{+}(R^{\sharp})}.

Theorem. If {L/K_{0}} is a complete discretely valued extension with maximal absolutely unramified subfield {L_{0}}, there is an identification of sets

\displaystyle \mathcal{M}_{D,\mathbf{h}}(\mathrm{Spd}L)=\left\{ (V,\gamma)\right\}

where the right-hand side denotes pairs consisting of a crystalline representation {V} of {G_{L}} with Hodge-Tate multiset {\mathbf{h}} together with a {\varphi}-linear isomorphism {\gamma:\mathbf{D}_{\mathrm{crys}}(V)\overset{\sim}{\rightarrow}D\otimes_{K_{0}}L_{0}}.

Proof. Given {(V,\gamma)}, note that {\gamma} induces isomorphisms {\mathbf{D}_{\mathrm{dR}}(V)\overset{\sim}{\rightarrow}D\otimes_{K_{0}}L} and (by Colmez-Fontaine) {V\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{dR}}(\widehat{\overline{L}})\overset{\sim}{\rightarrow}D\otimes_{K_{0}}\mathbf{B}_{\mathrm{dR}}(\widehat{\overline{L}})} which we also denote by {\gamma}. Set {\mathbf{M}=\gamma\left(V\otimes_{\mathbf{Q}_{p}}\mathbf{B}_{\mathrm{dR}}^{+}(\widehat{\overline{L}})\right)}, so we get an element of {\mathcal{M}_{D,\mathbf{h}}(\mathrm{Spd}\widehat{\overline{L}})}. Now we examine the sheaf sequence associated with the covering {\mathrm{Spd}\widehat{\overline{L}}\rightarrow\mathrm{Spd}L}, i.e.

\displaystyle \mathcal{M}_{D,\mathbf{h}}(\mathrm{Spd}L)\rightarrow\mathcal{M}_{D,\mathbf{h}}(\mathrm{Spd}\widehat{\overline{L}})\overset{\mathrm{pr}_{1}^{\ast}}{\underset{\mathrm{pr}_{2}^{\ast}}{\rightrightarrows}}\mathcal{M}_{D,\mathbf{h}}(\mathrm{Spd}\widehat{\overline{L}}\times_{\mathrm{Spd}L}\mathrm{Spd}\widehat{\overline{L}}).

This is a necessary step: {\mathrm{Spd}L} is not a perfectoid diamond, and {\mathbf{B}_{\mathrm{dR}}(L)} is a stupid thing. Now its not hard to see that {\mathrm{Spd}\widehat{\overline{L}}\times_{\mathrm{Spd}L}\mathrm{Spd}\widehat{\overline{L}}} is the diamond associated with {\mathrm{Spa}\widehat{\overline{L}}\times G_{L}}, and that {\mathbf{B}_{\mathrm{dR}}} evaluated on this diamond is {\mathrm{Hom}_{\mathrm{cts}}(G_{L},\mathbf{B}_{\mathrm{dR}}(\widehat{\overline{L}}))} (and likewise for {\mathbf{B}_{\mathrm{dR}}^{+}}). Pulling back under {\mathrm{pr}_{1}^{\ast}}, {\mathbf{M}} goes to the “constant” lattice {\mathbf{M}\subset D\otimes_{K_{0}}\mathrm{Hom}_{\mathrm{cts}}(G_{L},\mathbf{B}_{\mathrm{dR}}(\widehat{\overline{L}}))}, in the sense that “evaluating” at any {g\in G_{L}} we recover the lattice {\mathbf{M}}. On the other hand, pulling back under {\mathrm{pr}_{2}^{\ast}} brings {\mathbf{M}} to the lattice {\mathbf{M}'\subset D\otimes_{K_{0}}\mathrm{Hom}_{\mathrm{cts}}(G_{L},\mathbf{B}_{\mathrm{dR}}(\widehat{\overline{L}}))} whose evaluation at any {g} is {g^{\ast}(\mathbf{M})\subset D\otimes_{K_{0}}\mathbf{B}_{\mathrm{dR}}(\widehat{\overline{L}})}, i.e. the pullback of {\mathbf{M}} under the natural {G_{L}}-action on {D\otimes_{K_{0}}\mathbf{B}_{\mathrm{dR}}(\widehat{\overline{L}})}. But {\mathbf{M}} is Galois-stable by construction, so we really do get an element of the equalizer!

The other direction is now clear as well: given {\mathbf{M}\in\mathcal{M}_{D,\mathbf{h}}(\mathrm{Spd}\widehat{\overline{L}})}, our analysis shows that that {\mathbf{M}} lives in the equalizer of the {\mathrm{pr}_{i}^{\ast}}‘s iff {\mathbf{M}} is preserved by the Galois action on {D\otimes_{K_{0}}\mathbf{B}_{\mathrm{dR}}(\widehat{\overline{L}})}, but this is exactly the condition that {V_{\mathbf{M}}:=\ker\alpha_{\mathbf{M}}} (a priori, nothing more that a {\mathbf{Q}_{p}}-vector space) is {G_{L}}-stable and thus a {G_{L}}-representation. One easily checks that {V_{\mathbf{M}}} has all the claimed properties. {\square}

Proposition. If {\mathrm{supp}\mathbf{h}\subset\{0,-1\}}, then {\mathcal{M}_{D,\mathbf{h}}} is representable by (the diamond associated with) an open subspace of a flag variety {\mathcal{F}l_{D,\mathbf{h}}} over {K_{0}}.

Idea of proof. Notation as above, any {\mathbf{M}\in\mathcal{M}_{D,\mathbf{h}}(\mathrm{Spd}(R,R^{+}))} agrees with {\theta^{-1}(\theta(\mathbf{M}))}, so its enough to specify the {R^{\sharp}}-direct summand {\theta(\mathbf{M})\subset D\otimes_{K_{0}}R^{\sharp}}, and such things are specified by a point {\mathrm{Spa}(R^{\sharp},R^{\sharp+})\rightarrow\mathcal{F}l_{D,\mathbf{h}}}, or equivalently a morphism of diamonds {\mathrm{Spd}(R,R^{+})\rightarrow\mathcal{F}l_{D,\mathbf{h}}^{\lozenge}} over {\mathrm{Spd}K_{0}}. {\square}

In general one still has a rigid analytic flag variety {\mathcal{F}l_{D,\mathbf{h}}} over {K_{0}} together with some weakly admissible locus {\mathcal{F}l_{D,\mathbf{h}}^{\mathrm{wa}}}, and there is an identification of sets {\mathcal{F}l_{D,\mathbf{h}}^{\mathrm{wa}}(L)=\mathcal{M}_{D,\mathbf{h}}(\mathrm{Spd}L)} for any {L} as above. But it seems unlikely that {\mathcal{M}_{D,\mathbf{h}}} is describable in terms of {(\mathcal{F}l_{D,\mathbf{h}}^{\mathrm{wa}})^{\lozenge}} in general, or even that the two spaces are directly related.

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