## 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.