The smooth locus in infinite-level Rapoport-Zink spaces

TL;DR

This paper proves that in infinite-level Rapoport-Zink spaces, the dense open subset of points with no extra endomorphisms is cohomologically smooth, revealing a deep geometric structure related to supersingular elliptic curves.

Contribution

It establishes the existence of a dense cohomologically smooth locus in infinite-level Rapoport-Zink spaces, characterized by the absence of extra endomorphisms, extending understanding of their geometric properties.

Findings

The cohomologically smooth locus is dense and open in the infinite-level Rapoport-Zink space.

This locus corresponds to $p$-divisible groups without extra endomorphisms.

In the modular curve case, it matches supersingular elliptic curves with no extra endomorphisms.

Abstract

Rapoport-Zink spaces are deformation spaces for $p$-divisible groups with additional structure. At infinite level, they become preperfectoid spaces. Let $\mathscr{M}_{\infty}$ be an infinite-level Rapoport-Zink space of EL type, and let $\mathscr{M}_{\infty}^\circ$ be one geometrically connected component of it. We show that $\mathscr{M}_{\infty}^{\circ}$ contains a dense open subset which is cohomologically smooth in the sense of Scholze. This is the locus of $p$-divisible groups which do not have any extra endomorphisms. As a corollary, we find that the cohomologically smooth locus in the infinite-level modular curve $X(p^\infty)^{\circ}$ is exactly the locus of elliptic curves $E$ with supersingular reduction, such that the formal group of $E$ has no extra endomorphisms.

Equations

$$0\to G_{\eta }^{\mathrm{ad}}[p^{\infty }](R,R^{+})\to G_{\eta }^{\mathrm{ad}}(R,R^{+})\to \mathrm{Lie}G[1/p].$$

$$0\to VG(R,R^{+})\to {\tilde{G}}_{\eta }^{\mathrm{ad}}(R,R^{+})\to \mathrm{Lie}G[1/p].$$

$${\tilde{H}}_{\eta }(R,R^{+})\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$0\to {\mathbf{Q}}_p^n\stackrel{(s_1,\dots ,s_n)}{\to }{\tilde{H}}_{\eta }^{\mathrm{ad}}(C,{\mathcal{O}}_C)\to W{\otimes }_{R}C\to 0$$

$$s:V\to {\tilde{H}}_{\eta }^{\mathrm{ad}}(R,R^{+}),$$

$$V{\otimes }_{{\mathbf{Q}}_p}R\stackrel{{\mathrm{qlog}}_H\circ s}{⟶}M(H){\otimes }_{W(k)}R\to W\to 0.$$

$$0\to V\to \tilde{H}({\mathcal{O}}_C)\to W{\otimes }_{R}C\to 0$$

$${\mathcal{E}}_S(H)={\mathcal{E}}_S(N){\otimes }_{{\mathcal{O}}_{X_S}}{\mathcal{O}}_{X_S}(1).$$

$${\tilde{H}}_{\eta }^{\mathrm{ad}}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\mathrm{Hom}}_{R^{\circ }/p}({\mathbf{Q}}_p/{\mathbf{Z}}_p,H)[1/p]\cong {\mathrm{Hom}}_{A_{\mathrm{cris}}(R^{\circ }),\varphi }(M({\mathbf{Q}}_p/{\mathbf{Z}}_p),M(H))[1/p],$$

$$\tilde{H}(R^{\circ })\cong (B_{\mathrm{cris}}^{+}(R^{\circ }){\otimes }_{W(k)[1/p]}N{)}^{\varphi =p}.$$

$$0\to {\mathcal{O}}_{X_S}^n\stackrel{s}{\to }{\mathcal{E}}_S(H)\to i_{\infty \ast }W\to 0,$$

$$0\to G_{\eta }^{\mathrm{ad}}[p^{\infty }]\to G_{\eta }^{\mathrm{ad}}\to \mathrm{Lie}G[1/p]\to 0.$$

$$0\to VG\to {\tilde{G}}_{\eta }^{\mathrm{ad}}\to \mathrm{Lie}G[1/p]\to 0.$$

$$0\to V{\otimes }_{{\mathbf{Q}}_p}{\mathcal{O}}_{X_S}\stackrel{s}{\to }{\mathcal{E}}_S(H)\to i_{\infty \ast }W\to 0,$$

$${\mathrm{End}}^{\circ }H{\otimes }_{{\mathbf{Q}}_p}W(k)[1/p]\to {\mathrm{End}}_{W(k)[1/p]}M(H)[1/p]$$

$${\mathrm{End}}_B^{\circ }(H){\otimes }_{{\mathbf{Q}}_p}W(k)[1/p]\to {\mathrm{End}}_{B{\otimes }_{{\mathbf{Q}}_p}W(k)[1/p]}M(H)[1/p]$$

$$D^{\prime }{\otimes }_{{\mathbf{Q}}_p}{\mathcal{O}}_{X_C}\to {\mathrm{End}}_{(B{\otimes }_{{\mathbf{Q}}_p}{\mathcal{O}}_{X_C})}{\mathcal{E}}_C(H)$$

$$D$$

$$D^{\prime }$$

$$A_{x}\cong\left\{(\alpha,\alpha^{\prime})\in D\times D^{\prime}\biggm{|}s\circ\alpha=\alpha^{\prime}\circ s\right\}.$$

$$R\mathrm{Hom}(R{\Gamma }_c(Y^{\prime },\Lambda ),\Lambda )\cong R\Gamma (Y^{\prime },\Lambda )[n].$$

$$Z\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$T_0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$X{\times }_k{T}_0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\mathrm{Hom}}_{{\mathcal{O}}_{X{\times }_k{T}_0}}(s_0^{\ast }{\Omega }_{Z/X},{\pi }_0^{\ast }I)\cong s_0^{\ast }{\mathrm{Tan}}_{Z/X}\otimes {\pi }_0^{\ast }I.$$

$$H^1(X{\times }_k{T}_0,s_0^{\ast }{\mathrm{Tan}}_{Z/X}\otimes {\pi }_0^{\ast }I).$$

$$H^p(T_0,R^q{\pi }_{0\ast }(s_0^{\ast }{\mathrm{Tan}}_{Z/X}\otimes {\pi }_0^{\ast }I)).$$

$$H^1(X{\times }_k{T}_0,s_0^{\ast }{\mathrm{Tan}}_{Z/X}\otimes {\pi }_0^{\ast }I)\cong H^0(T_0,R^1{\pi }_{0\ast }(s_0^{\ast }{\mathrm{Tan}}_{Z/X}\otimes {\pi }_0^{\ast }I)).$$

$$R{\pi }_{0\ast }(s_0^{\ast }{\mathrm{Tan}}_{Z/X}\otimes {\pi }_0^{\ast }I)\cong R{\pi }_{0\ast }{s}_0^{\ast }{\mathrm{Tan}}_{Z/X}{\otimes }^{\mathbf{L}}I.$$

$${\pi }_{\ast }{s}^{\ast }{\mathrm{Tan}}_{Z/X}\cong {\mathrm{Tan}}_{{\mathcal{M}}_Z/\mathrm{Spec}k}.$$

Full text

1 Main theorem

Let $p$ be a prime number. Rapoport-Zink spaces [RZ96] are deformation spaces of $p$-divisible groups equipped with some extra structure. This article concerns the geometry of Rapoport-Zink spaces of EL type (endomorphisms + level structure). In particular we consider the infinite-level spaces ${\mathscr{M}}_{{\mathcal{D}},\infty}$, which are preperfectoid spaces [SW13]. An example is the space ${\mathscr{M}}_{H,\infty}$, where $H/\overline{{\mathbf{F}}}_{p}$ is a $p$-divisible group of height $n$. The points of ${\mathscr{M}}_{H,\infty}$ over a nonarchimedean field $K$ containing $W(\overline{{\mathbf{F}}}_{p})$ are in correspondence with isogeny classes of $p$-divisible groups $G/\mathcal{O}_{K}$ equipped with a quasi-isogeny $G\otimes_{\mathcal{O}_{K}}\mathcal{O}_{K}/p\to H\otimes_{\overline{{\mathbf{F}}}_{p}}\mathcal{O}_{K}/p$ and an isomorphism $\mathbf{Q}_{p}^{n}\cong VG$ (where $VG$ is the rational Tate module).

The infinite-level space ${\mathscr{M}}_{{\mathcal{D}},\infty}$ appears as the limit of finite-level spaces, each of which is a smooth rigid-analytic space. We would like to investigate the question of smoothness for the space ${\mathscr{M}}_{{\mathcal{D}},\infty}$ itself, which is quite a different matter. We need the notion of cohomological smoothness [Sch17], which makes sense for general morphisms of analytic adic spaces, and which is reviewed in Section 4. Roughly speaking, an adic space is cohomologically smooth over $C$ (where $C/\mathbf{Q}_{p}$ is complete and algebraically closed) if it satisfies local Verdier duality. In particular, if $U$ is a quasi-compact adic space which is cohomologically smooth over $\operatorname{Spa}(C,\mathcal{O}_{C})$, then the cohomology group $H^{i}(U,{\mathbf{F}}_{\ell})$ is finite for all $i$ and all primes $\ell\neq p$.

Our main theorem shows that each connected component of the geometric fiber of ${\mathscr{M}}_{{\mathcal{D}},\infty}$ has a dense open subset which is cohomologically smooth.

Theorem 1.0.1**.**

Let ${\mathcal{D}}$ be a basic EL datum (cf. Section 2). Let $C$ be a complete algebraically closed extension of the field of scalars of ${\mathscr{M}}_{{\mathcal{D}},\infty}$, and let ${\mathscr{M}}_{{\mathcal{D}},\infty}^{\circ}$ be a connected component of the base change ${\mathscr{M}}_{{\mathcal{D}},\infty,C}$. Let ${\mathscr{M}}_{{\mathcal{D}},\infty}^{\circ,\operatorname{non-sp}}\subset{\mathscr{M}}_{{\mathcal{D}},\infty}^{\circ}$ be the non-special locus (cf. Section 3.5), corresponding to $p$-divisible groups without extra endomorphisms. Then ${\mathscr{M}}_{{\mathcal{D}},\infty}^{\circ,\operatorname{non-sp}}$ is cohomologically smooth over $C$.

We remark that outside of trivial cases, $\pi_{0}({\mathscr{M}}_{{\mathcal{D}},\infty,C})$ has no isolated points, which implies that no open subset of ${\mathscr{M}}_{{\mathcal{D}},\infty,C}$ can be cohomologically smooth. (Indeed, the $H^{0}$ of any quasi-compact open fails to be finitely generated.) Therefore it really is necessary to work with individual connected components of the geometric fiber of ${\mathscr{M}}_{{\mathcal{D}},\infty}$.

Theorem 1.0.1 is an application of the perfectoid version of the Jacobian criterion for smoothness, due to Fargues–Scholze [FS]; cf. Theorem 4.2.1. The latter theorem involves the Fargues-Fontaine curve $X_{C}$ (reviewed in Section 3). It asserts that a functor ${\mathscr{M}}$ on perfectoid spaces over $\operatorname{Spa}(C,\mathcal{O}_{C})$ is cohomologically smooth, when ${\mathscr{M}}$ can be interpreted as global sections of a smooth morphism $Z\to X_{C}$, subject to a certain condition on the tangent bundle $\operatorname{Tan}_{Z/X_{C}}$.

In our application to Rapoport-Zink spaces, we construct a smooth morphism $Z\to X_{C}$, whose moduli space of global sections is isomorphic to ${\mathscr{M}}_{{\mathcal{D}},\infty}^{\circ}$ (Lemma 5.2.1). Next, we show that a geometric point $x\in{\mathscr{M}}_{{\mathcal{D}},\infty}^{\circ}(C)$ lies in ${\mathscr{M}}_{{\mathcal{D}},\infty}^{\circ,{\rm non-sp}}(C)$ if and only if the corresponding section $s\colon X_{C}\rightarrow Z$ satisfies the condition that all slopes of the vector bundle $s^{\ast}\operatorname{Tan}_{Z/X_{C}}$ on $X_{C}$ are positive (Theorem 5.5.1). This is exactly the condition on $\operatorname{Tan}_{Z/X_{C}}$ required by Theorem 4.2.1, so we can conclude that ${\mathscr{M}}_{{\mathcal{D}},\infty}^{\circ}$ is cohomologically smooth.

The geometry of Rapoport-Zink spaces is related to the geometry of Shimura varieties. As an example, consider the tower of classical modular curves $X(p^{\infty})$, considered as rigid spaces over $C$. There is a perfectoid space $X(p^{\infty})$ over $C$ for which $X(p^{\infty})\sim\varprojlim_{n}X(p^{n})$, and a Hodge-Tate period map $\pi_{HT}\colon X(p^{\infty})\to{\mathbf{P}}^{1}_{C}$ [Sch15], which is $\operatorname{GL}_{2}(\mathbf{Q}_{p})$-equivariant. Let $X(p^{\infty})^{\circ}\subset X(p^{\infty})$ be a connected component.

Corollary 1.0.2**.**

The following are equivalent for a $C$-point $x$ of $X(p^{\infty})^{\circ}$.

The point $x$ corresponds to an elliptic curve $E$, such that the $p$-divisible group $E[p^{\infty}]$ has $\operatorname{End}E[p^{\infty}]=\mathbf{Z}_{p}$. 2. 2.

The stabilizer of $\pi_{HT}(x)$ in $\operatorname{PGL}_{2}(\mathbf{Q}_{p})$ is trivial. 3. 3.

There is a neighborhood of $x$ in $X(p^{\infty})^{\circ}$ which is cohomologically smooth over $C$.

2 Review of Rapoport-Zink spaces at infinite level

2.1 The infinite-level Rapoport-Zink space ${\mathscr{M}}_{H,\infty}$

Let $k$ be a perfect field of characteristic $p$, and let $H$ be a $p$-divisible group of heignt $n$ and dimension $d$ over $k$. We review here the definition of the infinite-level Rapoport-Zink space associated with $H$.

First there is the formal scheme ${\mathscr{M}}_{H}$ over $\operatorname{Spf}W(k)$ parametrizing deformations of $H$ up to isogeny, as in [RZ96]. For a $W(k)$-algebra $R$ in which $p$ is nilpotent, ${\mathscr{M}}_{H}(R)$ is the set of isomorphism classes of pairs $(G,\rho)$, where $G/R$ is a $p$-divisible group and $\rho\colon H\otimes_{k}R/p\to G\otimes_{R}R/p$ is a quasi-isogeny.

The formal scheme ${\mathscr{M}}_{H}$ locally admits a finitely generated ideal of definition. Therefore it makes sense to pass to its adic space ${\mathscr{M}}_{H}^{\mathop{\rm ad}}$, which has generic fiber $({\mathscr{M}}_{H}^{\mathop{\rm ad}})_{\eta}$, a rigid-analytic space over $\operatorname{Spa}(W(k)[1/p],W(k))$. Then $({\mathscr{M}}_{H}^{\mathop{\rm ad}})_{\eta}$ has the following moduli interpretation: it is the sheafification of the functor assigning to a complete affinoid $(W(k)[1/p],W(k))$-algebra $(R,R^{+})$ the set of pairs $(G,\rho)$, where $G$ is a $p$-divisible group defined over an open and bounded subring $R_{0}\subset R^{+}$, and $\rho\colon H\otimes_{k}R_{0}/p\to G\otimes_{R_{0}}R_{0}/p$ is a quasi-isogeny. There is an action of $\operatorname{Aut}H$ on ${\mathscr{M}}_{H}^{\mathop{\rm ad}}$ obtained by composition with $\rho$.

Given such a pair $(G,\rho)$, Grothendieck-Messing theory produces a surjection $M(H)\otimes_{W(k)}R\to\operatorname{Lie}G[1/p]$ of locally free $R$-modules, where $M(H)$ is the covariant Dieudonné module. There is a Grothendieck-Messing period map $\pi_{GM}\colon({\mathscr{M}}_{H}^{\mathop{\rm ad}})_{\eta}\to{\mathcal{F}}\ell$, where ${\mathcal{F}}\ell$ is the rigid-analytic space parametrizing rank $d$ locally free quotients of $M(H)[1/p]$. The morphism $\pi_{GM}$ is equivariant for the action of $\operatorname{Aut}H$. It has open image ${\mathcal{F}}\ell^{a}$ (the admissible locus).

We obtain a tower of rigid-analytic spaces over $({\mathscr{M}}_{H}^{\mathop{\rm ad}})_{\eta}$ by adding level structures. For a complete affinoid $(W(k)[1/p],W(k))$-algebra $(R,R^{+})$, and an element of $({\mathscr{M}}_{H}^{\mathop{\rm ad}})_{\eta}(R,R^{+})$ represented locally on $\operatorname{Spa}(R,R^{+})$ by a pair $(G,\rho)$ as above, we have the Tate module $TG=\varprojlim_{m}G[p^{m}]$, considered as an adic space over $\operatorname{Spa}(R,R^{+})$ with the structure of a $\mathbf{Z}_{p}$-module [SW13, (3.3)]. Finite-level spaces ${\mathscr{M}}_{H,m}$ are obtained by trivializing the $G[p^{m}]$; these are finite étale covers of $({\mathscr{M}}_{H}^{\mathop{\rm ad}})_{\eta}$. The infinite-level space is obtained by trivializing all of $TG$ at once, as in the following definition.

Definition 2.1.1** ([SW13, Definition 6.3.3]).**

Let ${\mathscr{M}}_{H,\infty}$ be the functor which sends a complete affinoid $(W(k)[1/p],W(k))$-algebra $(R,R^{+})$ to the set of triples $(G,\rho,\alpha)$, where $(G,\rho)$ is an element of $({\mathscr{M}}_{H})^{\mathop{\rm ad}}_{\eta}(R,R^{+})$, and $\alpha\colon\mathbf{Z}_{p}^{n}\to TG$ is a $\mathbf{Z}_{p}$-linear map which is an isomorphism pointwise on $\operatorname{Spa}(R,R^{+})$.

There is an equivalent definition in terms of isogeny classes of triples $(G,\rho,\alpha)$, where this time $\alpha\colon\mathbf{Q}_{p}^{n}\to VG$ is a trivialization of the rational Tate module. Using this definition, it becomes clear that ${\mathscr{M}}_{H,\infty}$ admits an action of the product $\operatorname{GL}_{n}(\mathbf{Q}_{p})\times\operatorname{Aut}^{0}H$, where $\operatorname{Aut}^{0}$ means automorphisms in the isogeny category. Then the period map $\pi_{GM}\colon{\mathscr{M}}_{H,\infty}\to{\mathcal{F}}\ell$ is equivariant for $\operatorname{GL}_{n}(\mathbf{Q}_{p})\times\operatorname{Aut}^{0}H$, where $\operatorname{GL}_{n}(\mathbf{Q}_{p})$ acts trivially on ${\mathcal{F}}\ell$.

We remark that ${\mathscr{M}}_{H,\infty}\sim\varprojlim_{m}{\mathscr{M}}_{H,m}$ in the sense of [SW13, Definition 2.4.1].

One of the main theorems of [SW13] is the following.

Theorem 2.1.2**.**

The adic space ${\mathscr{M}}_{H,\infty}$ is a preperfectoid space.

This means that for any perfectoid field $K$ containing $W(k)$, the base change ${\mathscr{M}}_{H,\infty}\times_{\operatorname{Spa}(W(k)[1/p],W(k))}\operatorname{Spa}(K,\mathcal{O}_{K})$ becomes perfectoid after $p$-adically completing.

We sketch here the proof of Theorem 2.1.2. Consider the “universal cover” $\tilde{H}=\varprojlim_{p}H$ as a sheaf of $\mathbf{Q}_{p}$-vector spaces on the category of $k$-algebras. This has a canonical lift to the category of $W(k)$-algebras [SW13, Proposition 3.1.3(ii)], which we continue to call $\tilde{H}$. The adic generic fiber $\tilde{H}^{\mathop{\rm ad}}_{\eta}$ is a preperfectoid space, as can be checked “by hand”: it is a product of the $d$-dimensional preperfectoid open ball $(\operatorname{Spa}W(k)\llbracket T_{1}^{1/p^{\infty}},\dots,T_{d}^{1/p^{\infty}}\rrbracket)_{\eta}$ by the constant adic space $VH^{\text{\rm\'{e}t}}$, where $H^{\text{\rm\'{e}t}}$ is the étale part of $H$. Given a triple $(G,\rho,\alpha)$ representing an element of ${\mathscr{M}}_{H,\infty}(R,R^{+})$, the quasi-isogeny $\rho$ induces an isomorphism $\tilde{H}^{\mathop{\rm ad}}_{\eta}\times_{\operatorname{Spa}(W(k)[1/p],W(k))}\operatorname{Spa}(R,R^{+})\to\tilde{G}^{\mathop{\rm ad}}_{\eta}$; composing this with $\alpha$ gives a morphism $\mathbf{Q}_{p}^{n}\to\tilde{H}^{\mathop{\rm ad}}_{\eta}(R,R^{+})$. We have therefore described a morphism ${\mathscr{M}}_{H,\infty}\to(\tilde{H}^{\mathop{\rm ad}}_{\eta})^{n}$.

Theorem 2.1.2 follows from the fact that the morphism ${\mathscr{M}}_{H,\infty}\to(\tilde{H}^{\mathop{\rm ad}})_{\eta}^{n}$ presents ${\mathscr{M}}_{H,\infty}$ as an open subset of a Zariski closed subset of $(\tilde{H}^{\mathop{\rm ad}})_{\eta}^{n}$. We conclude this subsection by spelling out how this is done. We have a quasi-logarithm map $\operatorname{qlog}_{H}\colon\tilde{H}^{\mathop{\rm ad}}_{\eta}\to M(H)[1/p]\otimes_{W(k)[1/p]}{\mathbf{G}}_{a}$ [SW13, Definition 3.2.3], a $\mathbf{Q}_{p}$-linear morphism of adic spaces over $\operatorname{Spa}(W(k)[1/p],W(k))$.

Now suppose $(G,\rho)$ is a deformation of $H$ to $(R,R^{+})$. The logarithm map on $G$ fits into an exact sequence of $\mathbf{Z}_{p}$-modules:

[TABLE]

After taking projective limits along multiplication-by-$p$, this turns into an exact sequence of $\mathbf{Q}_{p}$-vector spaces,

[TABLE]

On the other hand, we have a commutative diagram

[TABLE]

The lower horizontal map $M(H)\otimes_{W(k)}R\to\operatorname{Lie}G[1/p]$ is the quotient by the $R$-submodule of $M(H)\otimes_{W(k)}R$ generated by the image of $VG(R,R^{+})\to\tilde{G}_{\eta}^{\mathop{\rm ad}}(R,R^{+})\cong\tilde{H}_{\eta}^{\mathop{\rm ad}}(R,R^{+})\to M(H)\otimes_{W(k)}R$.

Thus if we have a triple $(G,\rho,\alpha)$ representing an element of ${\mathscr{M}}_{H,\infty}(R,R^{+})$, then we have a map $\mathbf{Q}_{p}^{n}\to\tilde{H}_{\eta}^{\mathop{\rm ad}}(R,R^{+})$, such that the cokernel of $\mathbf{Q}_{p}^{n}\to\tilde{H}_{\eta}^{\mathop{\rm ad}}(R,R^{+})\to M(H)\otimes_{W(k)}R$ is a projective $R$-module of rank $d$, namely $\operatorname{Lie}G[1/p]$. This condition on the cokernel allows us to formulate an alternate description of ${\mathscr{M}}_{H,\infty}$ which is independent of deformations.

Proposition 2.1.3**.**

The adic space ${\mathscr{M}}_{H,\infty}$ is isomorphic to the functor which assigns to a complete affinoid $(W(k)[1/p],W(k))$-algebra $(R,R^{+})$ the set of $n$-tuples $(s_{1},\dots,s_{n})\in\tilde{H}^{\mathop{\rm ad}}_{\eta}(R,R^{+})^{n}$ such that the following conditions are satisfied:

The quotient of $M(H)\otimes_{W(k)}R$ by the span of the $\operatorname{qlog}(s_{i})$ is a projective $R$-module $W$ of rank $d$. 2. 2.

For all geometric points $\operatorname{Spa}(C,\mathcal{O}_{C})\to\operatorname{Spa}(R,R^{+})$, the sequence

[TABLE]

is exact.

2.2 Infinite-level Rapoport-Zink spaces of EL type

This article treats the more general class of Rapoport-Zink spaces of EL type. We review these here.

Definition 2.2.1**.**

Let $k$ be an algebraically closed field of characteristic $p$. A rational EL datum is a quadruple ${\mathcal{D}}=(B,V,H,\mu)$, where

• •

$B$ is a semisimple $\mathbf{Q}_{p}$-algebra,

• •

$V$ is a finite $B$-module,

• •

$H$ is an object of the isogeny category of $p$-divisible groups over $k$, equipped with an action $B\to\operatorname{End}H$,

• •

$\mu$ is a conjugacy class of $\overline{\mathbf{Q}}_{p}$-rational cocharacters $\mathbf{G}_{m}\to{\mathbf{G}}$, where ${\mathbf{G}}/\mathbf{Q}_{p}$ is the algebraic group $\operatorname{GL}_{B}(V)$.

These are subject to the conditions:

• •

If $M(H)$ is the (rational) Dieudonné module of $H$, then there exists an isomorphism $M(H)\cong V\otimes_{\mathbf{Q}_{p}}W(k)[1/p]$ of $B\otimes_{\mathbf{Q}_{p}}W(k)[1/p]$-modules. In particular $\dim V=\operatorname{ht}H$.

• •

In the weight decomposition of $V\otimes_{\mathbf{Q}_{p}}\overline{\mathbf{Q}}_{p}\cong\bigoplus_{i\in\mathbf{Z}}V_{i}$ determined by $\mu$, only weights 0 and 1 appear, and $\dim V_{0}=\dim H$.

The reflex field $E$ of ${\mathcal{D}}$ is the field of definition of the conjugacy class $\mu$. We remark that the weight filtration (but not necessarily the weight decomposition) of $V\otimes_{\mathbf{Q}_{p}}\overline{\mathbf{Q}}_{p}$ may be descended to $E$, and so we will be viewing $V_{0}$ and $V_{1}$ as $B\otimes_{\mathbf{Q}_{p}}E$-modules.

The infinite-level Rapoport-Zink space ${\mathscr{M}}_{{\mathcal{D}},\infty}$ is defined in [SW13] in terms of moduli of deformations of the $p$-divisible group $H$ along with its $B$-action. It admits an alternate description along the lines of Proposition 2.1.3.

Proposition 2.2.2** ([SW13, Theorem 6.5.4]).**

Let ${\mathcal{D}}=(B,V,H,\mu)$ be a rational EL datum. Let $\breve{E}=E\cdot W(k)$. Then ${\mathscr{M}}_{{\mathcal{D}},\infty}$ is isomorphic to the functor which inputs a complete affinoid $(\breve{E},\mathcal{O}_{\breve{E}})$-algebra $(R,R^{+})$ and outputs the set of $B$-linear maps

[TABLE]

subject to the following conditions.

• •

Let $W$ be the quotient

[TABLE]

Then $W$ is a finite projective $R$-module, which locally on $R$ is isomorphic to $V_{0}\otimes_{E}R$ as a $B\otimes_{\mathbf{Q}_{p}}R$-module.

• •

For any geometric point $x=\operatorname{Spa}(C,\mathcal{O}_{C})\to\operatorname{Spa}(R,R^{+})$, the sequence of $B$-modules

[TABLE]

is exact.

If ${\mathcal{D}}=(\mathbf{Q}_{p},\mathbf{Q}_{p}^{n},H,\mu)$, where $H$ has height $n$ and dimension $d$ and $\mu(t)=(t^{\oplus d},1^{\oplus(n-d)})$, then $E=\mathbf{Q}_{p}$ and ${\mathscr{M}}_{{\mathcal{D}},\infty}={\mathscr{M}}_{H,\infty}$.

In general, we call $\breve{E}$ the field of scalars of ${\mathscr{M}}_{{\mathcal{D}},\infty}$, and for a complete algebraically closed extension $C$ of $\breve{E}$, we write ${\mathscr{M}}_{{\mathcal{D}},\infty,C}={\mathscr{M}}_{{\mathcal{D}},\infty}\times_{\operatorname{Spa}(\breve{E},\mathcal{O}_{\breve{E}})}\operatorname{Spa}(C,\mathcal{O}_{C})$ for the corresponding geometric fiber of ${\mathscr{M}}_{{\mathcal{D}},\infty}$.

The space ${\mathscr{M}}_{{\mathcal{D}},\infty}$ admits an action by the product group ${\mathbf{G}}(\mathbf{Q}_{p})\times J(\mathbf{Q}_{p})$, where $J/\mathbf{Q}_{p}$ is the algebraic group $\operatorname{Aut}_{B}^{\circ}(H)$. A pair $(\alpha,\alpha^{\prime})\in{\mathbf{G}}(\mathbf{Q}_{p})\times J(\mathbf{Q}_{p})$ sends $s$ to $\alpha^{\prime}\circ s\circ\alpha^{-1}$.

There is once again a Grothendieck-Messing period map $\pi_{GM}\colon{\mathscr{M}}_{{\mathcal{D}},\infty}\to{\mathcal{F}}\ell_{\mu}$ onto the rigid-analytic variety whose $(R,R^{+})$-points parametrize $B\otimes_{\mathbf{Q}_{p}}R$-module quotients of $M(H)\otimes_{W(k)}R$ which are projective over $R$, and which are of type $\mu$ in the sense that they are (locally on $R$) isomorphic to $V_{0}\otimes_{E}R$. The morphism $\pi_{GM}$ sends an $(R,R^{+})$-point of ${\mathscr{M}}_{{\mathcal{D}},\infty}$ to the quotient $W$ of $M(H)\otimes_{W(k)}R$ as above. It is equivariant for the action of ${\mathbf{G}}(\mathbf{Q}_{p})\times J(\mathbf{Q}_{p})$, where ${\mathbf{G}}(\mathbf{Q}_{p})$ acts trivially on ${\mathcal{F}}\ell_{\mu}$. In terms of deformations of the $p$-divisible group $H$, the period map $\pi_{GM}$ sends a deformation $G$ to $\operatorname{Lie}G$.

There is also a Hodge-Tate period map $\pi_{HT}\colon{\mathscr{M}}_{{\mathcal{D}},\infty}\to{\mathcal{F}}\ell_{\mu}^{\prime}$, where ${\mathcal{F}}\ell_{\mu}^{\prime}(R,R^{+})$ parametrizes $B\otimes_{\mathbf{Q}_{p}}R$-module quotients of $V\otimes_{\mathbf{Q}_{p}}R$ which are projective over $R$, and which are (locally on $R$) isomorphic to $V_{1}\otimes_{E}R$. The morphism $\pi_{HT}$ sends an $(R,R^{+})$-point of ${\mathscr{M}}_{{\mathcal{D}},\infty}$ to the image of $V\otimes_{\mathbf{Q}_{p}}R\to M(H)\otimes_{W(k)}R$. It is equivariant for the action of ${\mathbf{G}}(\mathbf{Q}_{p})\times J(\mathbf{Q}_{p})$, where this time $J(\mathbf{Q}_{p})$ acts trivially on ${\mathcal{F}}\ell_{\mu}^{\prime}(R,R^{+})$. In terms of deformations of the $p$-divisible group $H$, the period map $\pi_{HT}$ sends a deformation $G$ to $(\operatorname{Lie}G^{\vee})^{\vee}$.

3 The Fargues-Fontaine curve

3.1 Review of the curve

We briefly review here some constructions and results from [FF]. First we review the absolute curve, and then we cover the version of the curve which works in families.

Fix a perfectoid field $F$ of characteristic $p$, with $F^{\circ}\subset F$ its ring of integral elements. Let $\varpi\in F^{\circ}$ be a pseudo-uniformizer for $F$, and let $k$ be the residue field of $F$. Let $W(F^{\circ})$ be the ring of Witt vectors, which we equip with the $(p,[\varpi])$-adic topology. Let ${\mathcal{Y}}_{F}=\operatorname{Spa}(W(F^{\circ}),W(F^{\circ}))\backslash\left\{\left\lvert p[\varpi]\right\rvert=0\right\}$. Then ${\mathcal{Y}}_{F}$ is an analytic adic space over $\mathbf{Q}_{p}$. The Frobenius automorphism of $F$ induces an automorphism $\phi$ of ${\mathcal{Y}}_{F}$. Let $B_{F}=H^{0}({\mathcal{Y}}_{F},\mathcal{O}_{{\mathcal{Y}}_{F}})$, a $\mathbf{Q}_{p}$-algebra endowed with an action of $\phi$. Let $P_{F}$ be the graded ring $P_{F}=\bigoplus_{n\geq 0}B_{F}^{\phi=p^{n}}$. Finally, the Fargues-Fontaine curve is $X_{F}=\operatorname{Proj}P_{F}$. It is shown in [FF] that $X_{F}$ is the union of spectra of Dedekind rings, which justifies the use of the word “curve” to describe $X_{F}$. Note however that there is no “structure morphism” $X_{F}\to\operatorname{Spec}F$.

If $x\in X_{F}$ is a closed point, then the residue field of $x$ is a perfectoid field $F_{x}$ containing $\mathbf{Q}_{p}$ which comes equipped with an inclusion $i\colon F\hookrightarrow F_{x}^{\flat}$, which presents $F_{x}^{\flat}$ as a finite extension of $F$. Such a pair $(F_{x},i)$ is called an untilt of $F$. Then $x\mapsto(F_{x},i)$ is a bijection between closed points of $X_{F}$ and isomorphism classes of untilts of $F$, modulo the action of Frobenius on $i$. Thus if $F=E^{\flat}$ is the tilt of a given perfectoid field $E/\mathbf{Q}_{p}$, then $X_{E^{\flat}}$ has a canonical closed point $\infty$, corresponding to the untilt $E$ of $E^{\flat}$.

An important result in [FF] is the classification of vector bundles on $X_{F}$. (By a vector bundle on $X_{F}$ we are referring to a locally free $\mathcal{O}_{X_{F}}$-module ${\mathcal{E}}$ of finite rank. We will use the notation $V({\mathcal{E}})$ to mean the corresponding geometric vector bundle over $X_{F}$, whose sections correspond to sections of ${\mathcal{E}}$.) Recall that an isocrystal over $k$ is a finite-dimensional vector space $N$ over $W(k)[1/p]$ together with a Frobenius semi-linear automorphism $\phi$ of $N$. Given $N$, we have the graded $P_{F}$-module $\bigoplus_{n\geq 0}(N\otimes_{W(k)[1/p]}B_{F})^{\phi=p^{n}}$, which corresponds to a vector bundle ${\mathcal{E}}_{F}(N)$ on $X_{F}$. Then the Harder-Narasimhan slopes of ${\mathcal{E}}_{F}(N)$ are negative to those of $N$. If $F$ is algebraically closed, then every vector bundle on $X_{F}$ is isomorphic to ${\mathcal{E}}_{F}(N)$ for some $N$.

It is straightforward to “relativize” the above constructions. If $S=\operatorname{Spa}(R,R^{+})$ is an affinoid perfectoid space over $k$, one can construct the adic space ${\mathcal{Y}}_{S}$, the ring $B_{S}$, the scheme $X_{S}$, and the vector bundles ${\mathcal{E}}_{S}(N)$ as above. Frobenius-equivalences classes of untilts of $S$ correspond to effective Cartier divisors of $X_{S}$ of degree 1.

In our applications, we will start with an affinoid perfectoid space $S$ over $\mathbf{Q}_{p}$. We will write $X_{S}=X_{S^{\flat}}$, and we will use $\infty$ to refer to the canonical Cartier divisor of $X_{S}$ corresponding to the untilt $S$ of $S^{\flat}$. Thus if $N$ is an isocrystal over $k$, and $S=\operatorname{Spa}(R,R^{+})$ is an affinoid perfectoid space over $W(k)[1/p]$, then the fiber of ${\mathcal{E}}_{S}(N)$ over $\infty$ is $N\otimes_{W(k)[1/p]}R$.

Let $S=\operatorname{Spec}(R,R^{+})$ be as above and let $\infty$ be the corresponding Cartier divisor. We denote the completion of the ring of functions on $\mathcal{Y}_{S}$ along $\infty$ by $B_{\operatorname{dR}}^{+}(R)$. It comes equipped with a surjective homomorphism $\theta\colon B_{\operatorname{dR}}^{+}(R)\rightarrow R$, whose kernel is a principal ideal $\ker(\theta)=(\xi)$.

3.2 Relation to $p$-divisible groups

Here we recall the relationships between $p$-divisible groups and global sections of vector bundles on the Fargues-Fontaine curve. Let us fix a perfect field $k$ of characteristic $p$, and write $\operatorname{Perf}_{W(k)[1/p]}$ for the category of perfectoid spaces over $W(k)[1/p]$. Given a $p$-divisible group $H$ over $k$ with covariant isocrystal $N$, if $H$ has slopes $s_{1},\dots,s_{k}\in\mathbb{Q}$, then $N$ has the slopes $1-s_{1},\dots,1-s_{k}$. For an object $S$ in $\operatorname{Perf}_{W(k)[1/p]}$ we define the vector bundle ${\mathcal{E}}_{S}(H)$ on $X_{S}$ by

[TABLE]

Under this normalization, the Harder-Narasimhan slopes of ${\mathcal{E}}_{S}(H)$ are (pointwise on $S$) the same as the slopes of $H$.

Let us write $H^{0}({\mathcal{E}}(H))$ for the sheafification of the functor on $\operatorname{Perf}_{W(k)[1/p]}$, which sends $S$ to $H^{0}(X_{S},{\mathcal{E}}_{S}(H))$.

Proposition 3.2.1**.**

Let $H$ be a $p$-divisible group over a perfect field $k$ of characteristic $p$, with isocrystal $N$. There is an isomorphism $\tilde{H}^{\mathop{\rm ad}}_{\eta}\cong H^{0}({\mathcal{E}}(H))$ of sheaves on $\operatorname{Perf}_{W(k)[1/p]}$ making the diagram commute:

[TABLE]

where the morphism $H^{0}({\mathcal{E}}(H))\to N\otimes_{W(k)[1/p]}{\mathbf{G}}_{a}$ sends a global section of ${\mathcal{E}}(H)$ to its fiber at $\infty$.

Proof.

Let $S=\operatorname{Spa}(R,R^{+})$ be an affinoid perfectoid space over $W(k)[1/p]$. Then $\tilde{H}^{\mathop{\rm ad}}_{\eta}(R,R^{+})\cong\tilde{H}(R^{\circ})\cong\tilde{H}(R^{\circ}/p)$. Observe that $\tilde{H}(R^{\circ}/p)=\operatorname{Hom}_{R^{\circ}/p}(\mathbf{Q}_{p}/\mathbf{Z}_{p},H)[1/p]$, where the Hom is taken in the category of $p$-divisible groups over $R^{\circ}/p$. Recall the crystalline Dieudonné functor $G\mapsto M(G)$ from $p$-divisible groups to Dieudonné crystals [Mes72]. Since the base ring $R^{\circ}/p$ is semiperfect, the latter category is equivalent to the category of finite projective modules over Fontaine’s period ring $A_{\rm{cris}}(R^{\circ}/p)=A_{\rm{cris}}(R^{\circ})$, equipped with Frobenius and Verschiebung.

Now we apply [SW13, Theorem A]: since $R^{\circ}/p$ is f-semiperfect, the crystalline Dieudonné functor is fully faithful up to isogeny. Thus

[TABLE]

where the latter Hom is in the category of modules over $A_{\rm{cris}}(R^{\circ})$ equipped with Frobenius. Recall that $B_{\rm{cris}}^{+}(R^{\circ})=A_{\rm{cris}}(R^{\circ})[1/p]$. Since $H$ arises via base change from $k$, we have $M(H)[1/p]=B_{\rm{cris}}^{+}(R^{\circ})\otimes_{W(k)[1/p]}N$. For its part, $M(\mathbf{Q}_{p}/\mathbf{Z}_{p})[1/p]=B_{\rm{cris}}^{+}(R^{\circ})e$, for a basis element $e$ on which Frobenius acts as $p$. Therefore

[TABLE]

On the Fargues-Fontaine curve side, we have by definition $H^{0}(X_{S},{\mathcal{E}}_{S}(H))=(B_{S}\otimes_{W(k)[1/p]}N)^{\phi=p}$. The isomorphism between $(B_{S}\otimes_{W(k)[1/p]}N)^{\phi=p}$ and $(B_{\rm{cris}}^{+}(R^{\circ})\otimes_{W(k)[1/p]}N)^{\phi=p}$ is discussed in [LB18, Remarque 6.6].

The commutativity of the diagram in the proposition is [SW13, Proposition 5.1.6(ii)], at least in the case that $S$ is a geometric point, but this suffices to prove the general case. ∎

With Proposition 3.2.1 we can reinterpret the infinite-level Rapoport Zink spaces as moduli spaces of modifications of vector bundles on the Fargues-Fontaine curve. First we do this for ${\mathscr{M}}_{H,\infty}$. In the following, we consider ${\mathscr{M}}_{H,\infty}$ as a sheaf on the category of perfectoid spaces over $W(k)[1/p]$.

Proposition 3.2.2**.**

Let $H$ be a $p$-divisible group of height $n$ and dimension $d$ over a perfect field $k$. Let $N$ be the associated isocrystal over $k$. Then ${\mathscr{M}}_{H,\infty}$ is isomorphic to the functor which inputs an affinoid perfectoid space $S=\operatorname{Spa}(R,R^{+})$ over $W(k)[1/p]$ and outputs the set of exact sequences

[TABLE]

where $i_{\infty}\colon\operatorname{Spec}R\to X_{S}$ is the inclusion, and $W$ is a projective $\mathcal{O}_{S}$-module quotient of $N\otimes_{W(k)[1/p]}\mathcal{O}_{S}$ of rank $d$.

Proof.

We briefly describe this isomorphism on the level of points over $S=\operatorname{Spa}(R,R^{+})$. Suppose that we are given a point of ${\mathscr{M}}_{H,\infty}(S)$, corresponding to a $p$-divisible group $G$ over $R^{\circ}$, together with a quasi-isogeny $\iota\colon H\otimes_{k}R^{\circ}/p\to G\otimes_{R^{\circ}}R^{\circ}/p$ and an isomorphism $\alpha\colon\mathbf{Q}_{p}^{n}\to VG$ of sheaves of $\mathbf{Q}_{p}$-vector spaces on $S$. The logarithm map on $G$ fits into an exact sequence of sheaves of $\mathbf{Z}_{p}$-modules on $S$,

[TABLE]

After taking projective limits along multiplication-by-$p$, this turns into an exact sequence of sheaves of $\mathbf{Q}_{p}$-vector spaces on $S$,

[TABLE]

The quasi-isogeny induces an isomorphism $\tilde{H}_{\eta}^{\mathop{\rm ad}}\times_{\operatorname{Spa}W(k)[1/p]}S\cong\tilde{G}_{\eta}^{\mathop{\rm ad}}$; composing this with the level structure gives an injective map $\mathbf{Q}_{p}^{n}\to\tilde{H}^{\mathop{\rm ad}}_{\eta}(S)$, whose cokernel $W$ is isomorphic to the projective $R$-module $\operatorname{Lie}G$ of rank $d$. In light of Theorem 3.2.1, the map $\mathbf{Q}_{p}^{n}\to\tilde{H}^{\mathop{\rm ad}}_{\eta}(S)$ corresponds to an $\mathcal{O}_{X_{S}}$-linear map $s\colon\mathcal{O}_{X_{S}}^{n}\to{\mathcal{E}}_{S}(H)$, which fits into the exact sequence in (3.2.1). ∎

Similarly, we have a description of ${\mathscr{M}}_{{\mathcal{D}},\infty}$ in terms of modifications.

Proposition 3.2.3**.**

Let ${\mathcal{D}}=(B,V,H,\mu)$ be a rational EL datum. Then ${\mathscr{M}}_{{\mathcal{D}},\infty}$ is isomorphic to the functor which inputs an affinoid perfectoid space $S$ over $\breve{E}$ and outputs the set of exact sequences of $B\otimes_{\mathbf{Q}_{p}}\mathcal{O}_{X_{S}}$-modules

[TABLE]

where $W$ is a finite projective $\mathcal{O}_{S}$-module, which is locally isomorphic to $V_{0}\otimes_{\mathbf{Q}_{p}}\mathcal{O}_{S}$ as a $B\otimes_{\mathbf{Q}_{p}}\mathcal{O}_{S}$-module (using notation from Definition 2.2.1).

3.3 The determinant morphism, and connected components

If we are given a rational EL datum ${\mathcal{D}}$, there is a determinant morphism $\det\colon{\mathscr{M}}_{{\mathcal{D}},\infty}\to{\mathscr{M}}_{\det{\mathcal{D}},\infty}$, which we review below. For an algebraically closed perfectoid field $C$ containing $W(k)[1/p]$, the base change ${\mathscr{M}}_{\det{\mathcal{D}},\infty,C}$ is a locally profinite set of copies of $\operatorname{Spa}C$. For a point $\tau\in{\mathscr{M}}_{\det{\mathcal{D}},\infty}(C)$, let ${\mathscr{M}}_{{\mathcal{D}},\infty}^{\tau}$ be the fiber of ${\mathscr{M}}_{{\mathcal{D}},\infty}\to{\mathscr{M}}_{\det{\mathcal{D}},\infty}$ over $\tau$. We will prove in Section 5 that each ${\mathscr{M}}_{{\mathcal{D}},\infty}^{\tau,\operatorname{non-sp}}$ is cohomologically smooth if ${\mathcal{D}}$ is basic. This implies that $\pi_{0}({\mathscr{M}}_{{\mathcal{D}},\infty}^{\tau,\operatorname{non-sp}})$ is discrete, so that cohomogical smoothness of ${\mathscr{M}}_{{\mathcal{D}},\infty}^{\tau,\operatorname{non-sp}}$ is inherited by each of its connected components. This is Theorem 1.0.1. In certain cases (for example Lubin-Tate space) it is known that ${\mathscr{M}}_{{\mathcal{D}},\infty}^{\tau}$ is already connected [Che14].

We first review the determinant morphism for the space ${\mathscr{M}}_{H,\infty}$, where $H$ is a $p$-divisible group of height $n$ and dimension $d$ over a perfect field $k$ of characteristic $p$. Let $\breve{E}=W(k)[1/p]$. For a perfectoid space $S$ over $\breve{E}$, we have the vector bundle ${\mathcal{E}}_{S}(H)$ and its determinant $\det{\mathcal{E}}_{S}(H)$, a line bundle of degree $d$. (This does not correspond to a $p$-divisible group “$\det H$” unless $d\leq 1$.) We define ${\mathscr{M}}_{\det H,\infty}(S)$ to be the functor which inputs a perfectoid space $S=\operatorname{Spa}(R,R^{+})$ over $\breve{E}$ and outputs the set of morphisms $s\colon\mathcal{O}_{X_{S}}\to\det{\mathcal{E}}_{S}(H)$, such that the cokernel of $s$ is a projective $B_{\operatorname{dR}}^{+}(R)/(\xi)^{d}$-module of rank 1, where $(\xi)$ is the kernel of $B_{\operatorname{dR}}^{+}(R)\to R$. Then for an algebraically closed perfectoid field $C/\breve{E}$, the set ${\mathscr{M}}_{\det H,\infty}(C)$ is a $\mathbf{Q}_{p}^{\times}$-torsor. The morphism $\det\colon{\mathscr{M}}_{H,\infty}\to{\mathscr{M}}_{\det H,\infty}$ is simply $s\mapsto\det s$.

For the general case, let ${\mathcal{D}}=(B,V,H,\mu)$ be a rational EL datum. Let $F=Z(B)$ be the center of $B$. Then $F$ is a semisimple commutative $\mathbf{Q}_{p}$-algebra, and $V$ is free as an $F$-module. We put ${\mathbf{G}}=\operatorname{Aut}_{B}(V)$ (as an algebraic group), and then ${\mathbf{G}}^{\text{ab}}={\mathbf{G}}/{\mathbf{G}}^{\text{der}}=\operatorname{Aut}_{F}(\det_{F}V)\cong\mathop{\rm Res}_{F/\mathbf{Q}_{p}}{\mathbf{G}}_{m}$. Let $\mu^{\text{ab}}$ be the composition of $\mu$ with ${\mathbf{G}}\to{\mathbf{G}}^{\text{ab}}$. Let ${\mathscr{M}}_{\det{\mathcal{D}},\infty}$ be the functor which inputs a perfectoid space $S=\operatorname{Spa}(R,R^{+})$ over $\breve{E}$ and outputs the set of $F$-linear morphisms $s\colon\det_{F}V\otimes_{\mathbf{Q}_{p}}\mathcal{O}_{X_{S}}\to\det_{F}{\mathcal{E}}_{S}(H)$, such that the cokernel of $s$ is of the form $i_{\infty\ast}W$, where $W$ is a finite projective $\mathcal{O}_{S}$-module, locally isomorphic to $\det_{F}V_{0}\otimes_{\mathbf{Q}_{p}}F$ as a $F\otimes_{\mathbf{Q}_{p}}\mathcal{O}_{S}$-module.

3.4 Basic Rapoport-Zink spaces

The main theorem of this article concerns basic Rapoport-Zink spaces, so we recall some facts about these here.

Let $H$ be a $p$-divisible group over a perfect field $k$ of characteristic $p$. The space ${\mathcal{M}}_{H,\infty}$ is said to be basic when the $p$-divisible group $H$ (or rather, its Dieudonné module $M(H)$) is isoclinic. This is equivalent to saying that the natural map

[TABLE]

is an isomorphism, where on the right the endomorphisms are not required to commute with Frobenius.

More generally we have a notion of basicness for a rational EL datum $(B,H,V,\mu)$, referring to the following equivalent conditions:

• •

The ${\mathbf{G}}$-isocrystal (${\mathbf{G}}=\operatorname{Aut}_{B}V$) associated to $H$ is basic in the sense of Kottwitz [Kot85].

• •

The natural map

[TABLE]

is an isomorphism.

• •

Considered as an algebraic group over $\mathbf{Q}_{p}$, the automorphism group $J=\operatorname{Aut}^{\circ}_{B}H$ is an inner form of ${\mathbf{G}}$.

• •

Let $D^{\prime}=\operatorname{End}^{\circ}_{B}H$. For any algebraically closed perfectoid field $C$ containing $W(k)$, the map

[TABLE]

is an isomorphism.

In brief, the duality theorem from [SW13] says the following. Given a basic EL datum ${\mathcal{D}}$, there is a dual datum $\check{{\mathcal{D}}}$, for which the roles of the groups ${\mathbf{G}}$ and $J$ are reversed. There is a ${\mathbf{G}}(\mathbf{Q}_{p})\times J(\mathbf{Q}_{p})$-equivariant isomorphism ${\mathscr{M}}_{{\mathcal{D}},\infty}\cong{\mathscr{M}}_{\check{{\mathcal{D}}},\infty}$ which exchanges the roles of $\pi_{GM}$ and $\pi_{HT}$.

3.5 The special locus

Let ${\mathcal{D}}=(B,V,H,\mu)$ be a basic rational EL datum relative to a perfect field $k$ of characteristic $p$, with reflex field $E$. Let $F$ be the center of $B$. Define $F$-algebras $D$ and $D^{\prime}$ by

[TABLE]

Finally, let ${\mathbf{G}}=\operatorname{Aut}_{B}V$ and $J=\operatorname{Aut}_{B}H$, considered as algebraic groups over $\mathbf{Q}_{p}$. Then ${\mathbf{G}}$ and $J$ both contain $\mathop{\rm Res}_{F/\mathbf{Q}_{p}}{\mathbf{G}}_{m}$.

Let $C$ be an algebraically closed perfectoid field containing $\breve{E}$, and let $x\in{\mathscr{M}}_{{\mathcal{D}},\infty}(C)$. Then $x$ corresponds to a $p$-divisible group $G$ over $\mathcal{O}_{C}$ with endomorphisms by $B$, and also it corresponds to a $B\otimes_{\mathbf{Q}_{p}}\mathcal{O}_{X_{C}}$-linear map $s\colon V\otimes_{\mathbf{Q}_{p}}\mathcal{O}_{X}\to{\mathcal{E}}_{C}(N)$ as in Proposition 3.2.3. Define $A_{x}=\operatorname{End}_{B}G$ (endomorphisms in the isogeny category). Then $A_{x}$ is a semisimple $F$-algebra. In light of Proposition 3.2.3, an element of $A_{x}$ is a pair $(\alpha,\alpha^{\prime})$, where $\alpha\in\operatorname{End}_{B\otimes_{\mathbf{Q}_{p}}\mathcal{O}_{X_{C}}}V\otimes\mathcal{O}_{X_{C}}=\operatorname{End}_{B}V=D$ and $\alpha^{\prime}\in\operatorname{End}_{B\otimes_{\mathbf{Q}_{p}}\mathcal{O}_{X_{C}}}{\mathscr{E}}_{C}(H)=D^{\prime}$ (the last equality is due to basicness), such that $s\circ\alpha=\alpha^{\prime}\circ s$. Thus:

[TABLE]

Lemma 3.5.1**.**

The following are equivalent:

The $F$-algebra $A_{x}$ strictly contains $F$. 2. 2.

The stabilizer of $\pi_{GM}(x)\in\mathcal{F}\ell_{\mu}(C)$ in $J(\mathbf{Q}_{p})$ strictly contains $F^{\times}$. 3. 3.

The stabilizer of $\pi_{HT}(x)\in\mathcal{F}\ell_{\mu}^{\prime}(C)$ in ${\mathbf{G}}(\mathbf{Q}_{p})$ strictly contains $F^{\times}$.

Proof.

As in Proposition 3.2.3, let $s\colon V\otimes_{\mathbf{Q}_{p}}\mathcal{O}_{X_{S}}\stackrel{{\scriptstyle s}}{{\to}}{\mathcal{E}}_{S}(H)$ be the modification corresponding to $x$.

Note that the condition (1) is equivalent to the existence of an invertible element $(\alpha,\alpha^{\prime})\in A_{x}$ not contained in (the diagonally embedded) $F$. Also note that if one of $\alpha,\alpha^{\prime}$ lies in $F$, then so does the other, in which case they are equal.

Suppose $(\alpha,\alpha^{\prime})\in A_{x}$ is invertible. The point $\pi_{GM}(x)\in{\mathcal{F}}\ell_{\mu}$ corresponds to the cokernel of the fiber of $s$ at $\infty$. Since $\alpha^{\prime}\circ s=s\circ\alpha$, the cokernels of $\alpha^{\prime}\circ s$ and $s$ are the same, which means exactly that $\alpha^{\prime}\in J(\mathbf{Q}_{p})$ stabilizes $\pi_{GM}(x)$. Thus (1) implies (2). Conversely, if there exists $\alpha^{\prime}\in J(\mathbf{Q}_{p})\backslash F^{\times}$ which stabilizes $\pi_{GM}(x)$, it means that the $B\otimes_{\mathbf{Q}_{p}}\mathcal{O}_{X_{C}}$-linear maps $s$ and $\alpha^{\prime}\circ s$ have the same cokernel, and therefore there exists $\alpha\in\operatorname{End}_{B\otimes_{\mathbf{Q}_{p}}\mathcal{O}_{X_{C}}}V\otimes_{\mathbf{Q}_{p}}\mathcal{O}_{X_{C}}=D$ such that $s\circ\alpha=\alpha^{\prime}\circ s$, and then $(\alpha,\alpha^{\prime})\in A_{x}\backslash F^{\times}$. This shows that (2) implies (1).

The equivalence between (1) and (3) is proved similarly. ∎

Definition 3.5.2**.**

The special locus in ${\mathscr{M}}_{{\mathcal{D}},\infty}$ is the subset ${\mathscr{M}}_{{\mathcal{D}},\infty}^{\operatorname{sp}}$ defined by the condition $A_{x}\neq F$. The non-special locus ${\mathscr{M}}_{{\mathcal{D}},\infty}^{\operatorname{non-sp}}$ is the complement of the special locus.

The special locus is built out of “smaller” Rapoport-Zink spaces, in the following sense. Let $A$ be a semisimple $F$-algebra, equipped with two $F$-embeddings $A\to D$ and $A\to D^{\prime}$, so that $A\otimes_{F}B$ acts on $V$ and $H$. Also assume that a cocharacter in the conjugacy class $\mu$ factors through a cocharacter $\mu_{0}\colon{\mathbf{G}}_{m}\to\operatorname{Aut}_{A\otimes_{F}B}V$. Let ${\mathcal{D}}_{0}=(A\otimes_{F}B,V,H,\mu_{0})$. Then there is an evident morphism ${\mathscr{M}}_{{\mathcal{D}}_{0},\infty}\to{\mathscr{M}}_{{\mathcal{D}},\infty}$. The special locus ${\mathscr{M}}_{{\mathcal{D}},\infty}^{\operatorname{sp}}$ is the union of the images of all the ${\mathscr{M}}_{{\mathcal{D}}_{0},\infty}$, as $A$ ranges through all semisimple $F$-subalgebras of $D\times D^{\prime}$ strictly containing $F$.

4 Cohomological smoothness

Let $\operatorname{Perf}$ be the category of perfectoid spaces in characteristic $p$, with its pro-étale topology [Sch17, Definition 8.1]. For a prime $\ell\neq p$, there is a notion of $\ell$-cohomological smoothness [Sch17, Definition 23.8]. We only need the notion for morphisms $f\colon Y^{\prime}\to Y$ between sheaves on $\operatorname{Perf}$ which are separated and representable in locally spatial diamonds. If such an $f$ is $\ell$-cohomologically smooth, and $\Lambda$ is an $\ell$-power torsion ring, then the relative dualizing complex $Rf^{!}\Lambda$ is an invertible object in $D_{\text{\rm\'{e}t}}(Y^{\prime},\Lambda)$ (thus, it is v-locally isomorphic to $\Lambda[n]$ for some $n\in{\mathbf{Z}}$), and the natural transformation $Rf^{!}\Lambda\otimes f^{*}\to Rf^{!}$ of functors $D_{\text{\rm\'{e}t}}(Y,\Lambda)\to D_{\text{\rm\'{e}t}}(Y^{\prime},\Lambda)$ is an equivalence [Sch17, Proposition 23.12]. In particular, if $f$ is projection onto a point, and $Rf^{!}\Lambda\cong\Lambda[n]$, one derives a statement of Poincaré duality for $Y^{\prime}$:

[TABLE]

We will say that $f$ is cohomologically smooth if it is $\ell$-cohomologically smooth for all $\ell\neq p$. As an example, if $f\colon Y^{\prime}\to Y$ is a separated smooth morphism of rigid-analytic spaces over $\mathbf{Q}_{p}$, then the associated morphism of diamonds $f^{\diamond}\colon(Y^{\prime})^{\diamond}\to Y^{\diamond}$ is cohomologically smooth [Sch17, Proposition 24.3]. There are other examples where $f$ does not arise from a finite-type map of adic spaces. For instance, if $\tilde{B}_{C}=\operatorname{Spa}C\left<T^{1/p^{\infty}}\right>$ is the perfectoid closed ball over an algebraically closed perfectoid field $C$, then $\tilde{B}_{C}$ is cohomologically smooth over $C$.

If $Y$ is a perfectoid space over an algebraically closed perfectoid field $C$, it seems quite difficult to detect whether $Y$ is cohomologically smooth over $C$. We will review in Section 4.2 a “Jacobian criterion” from [FS] which applies to certain kinds of $Y$. But first we give a classical analogue of this criterion in the context of schemes.

4.1 The Jacobian criterion: classical setting

Proposition 4.1.1**.**

Let $X$ be a smooth projective curve over an algebraically closed field $k$. Let $Z\to X$ be a smooth morphism. Define ${\mathscr{M}}_{Z}$ to be the functor which inputs a $k$-scheme $T$ and outputs the set of sections of $Z\to X$ over $X_{T}$, that is, the set of morphisms $s$ making

[TABLE]

commute, subject to the condition that, fiberwise on $T$, the vector bundle $s^{*}\operatorname{Tan}_{Z/X}$ has vanishing $H^{1}$. Then ${\mathscr{M}}_{Z}\to\operatorname{Spec}k$ is formally smooth.

Here $\operatorname{Tan}_{Z/X}$ is the tangent bundle, equal to the $\mathcal{O}_{Z}$-linear dual of the sheaf of differentials $\Omega_{Z/X}$, which is locally free of finite rank. Let $\pi\colon X\times_{k}T\to T$ be the projection. For $t\in T$, let $X_{t}$ be the fiber of $\pi$ over $t$, and let $s_{t}\colon X_{t}\to Z$ be the restriction of $s$ to $X_{t}$. By proper base change, the fiber of $R^{1}\pi_{*}s^{*}\operatorname{Tan}_{Z/X}$ at $t\in T$ is $H^{1}(X_{t},s_{t}^{*}\operatorname{Tan}_{Z/X})$. The condition about the vanishing of $H^{1}$ in the proposition is equivalent to $H^{1}(X_{t},s_{t}^{*}\operatorname{Tan}_{Z/X})=0$ for each $t\in T$. By Nakayama’s lemma, this condition is equivalent to $R^{1}\pi_{*}s^{*}\operatorname{Tan}_{Z/X}=0$.

Proof.

Suppose we are given a commutative diagram

[TABLE]

where $T_{0}\to T$ is a first-order thickening of affine schemes; thus $T_{0}$ is the vanishing locus of a square-zero ideal sheaf $I\subset\mathcal{O}_{T}$. Note that $I$ becomes an $\mathcal{O}_{T_{0}}$-module.

The morphism $T_{0}\to{\mathscr{M}}_{Z}$ in (4.1.1) corresponds to a section of $Z\to X$ over $T_{0}$. Thus there is a solid diagram

[TABLE]

We claim that there exists a dotted arrow making the diagram commute. Since $Z\to X$ is smooth, it is formally smooth, and therefore this arrow exists Zariski-locally on $X$. Let $\pi\colon X\times_{k}T\to T$ and $\pi_{0}\colon X\times_{k}T_{0}\to T_{0}$ be the projections. Then $X\times_{k}T_{0}$ is the vanishing locus of the ideal sheaf $\pi^{*}I\subset\mathcal{O}_{X\times_{k}T}$. Note that sheaves of sets on $X\times_{k}T$ are equivalent to sheaves of sets on $X\times_{k}T_{0}$; under this equivalence, $\pi^{*}I$ and $\pi_{0}^{*}I$ correspond. By [Sta14, Remark 36.9.6], the set of such morphisms form a (Zariski) sheaf of sets on $X\times_{k}T$, which when viewed as a sheaf on $X\times_{k}T_{0}$ is a torsor for

[TABLE]

This torsor corresponds to class in

[TABLE]

This $H^{1}$ is the limit of a spectral sequence with terms

[TABLE]

But since $T_{0}$ is affine and $R^{q}\pi_{0*}(s_{0}^{*}\operatorname{Tan}_{Z/X}\otimes\pi_{0}^{*}I)$ is quasi-coherent, the above terms vanish for all $p>0$, and therefore

[TABLE]

Since $s_{0}^{*}\operatorname{Tan}_{Z/X}$ is locally free, we have $s_{0}^{*}\operatorname{Tan}_{Z/X}\otimes\pi_{0}^{*}I\cong s_{0}^{*}\operatorname{Tan}_{Z/X}\otimes^{{\mathbf{L}}}\pi_{0*}I$, and we may apply the projection formula [Sta14, Lemma 35.21.1] to obtain

[TABLE]

Now we apply the hypothesis about vanishing of $H^{1}$, which implies that $R\pi_{0*}s_{0}^{*}\operatorname{Tan}_{Z/X}$ is quasi-isomorphic to the locally free sheaf $\pi_{0*}s_{0}^{*}\operatorname{Tan}_{Z/X}$ in degree 0. Therefore the complex displayed above has $H^{1}=0$.

Thus our torsor is trivial, and so a morphism $s\colon X\times_{k}T\to Z$ exists filling in (4.1.2). The final thing to check is that $s$ corresponds to a morphism $T\to{\mathscr{M}}_{Z}$, i.e., that it satisfies the fiberwise $H^{1}=0$ condition. But this is automatic, since $T_{0}$ and $T$ have the same schematic points. ∎

In the setup of Proposition 4.1.1, let $s\colon X\times_{k}{\mathscr{M}}_{Z}\to Z$ be the universal section. That is, the pullback of $s$ along a morphism $T\to{\mathscr{M}}_{Z}$ is the section $X\times_{k}T\to Z$ to which this morphism corresponds. Let $\pi\colon X\times_{k}{\mathscr{M}}_{Z}\to{\mathscr{M}}_{Z}$ be the projection. By Proposition 4.1.1 ${\mathscr{M}}_{Z}\to\operatorname{Spec}k$ is formally smooth. There is an isomorphism

[TABLE]

Indeed, the proof of Proposition 4.1.1 shows that $\pi_{*}s^{*}\operatorname{Tan}_{Z/X}$ has the same universal property with respect to first order deformations as $\operatorname{Tan}_{{\mathscr{M}}_{Z}/\operatorname{Spec}k}$.

The following example is of similar spirit as our main application of the perfectoid Jacobian criterion below.

Example 4.1.2**.**

Let $X={\mathbf{P}}^{1}$ over the algebraically closed field $k$. For $d\in{\mathbf{Z}}$, let $V_{d}=\underline{\smash{\mathrm{Spec}}}_{X}{\rm\operatorname{Sym}}_{{\mathcal{O}}_{X}}({\mathcal{O}}(-d))$ be the geometric vector bundle over $X$ whose global sections are $\Gamma(X,{\mathcal{O}}(d))$. Fix integers $n,d,\delta>0$ and let $P$ be a homogeneous polynomial over $k$ of degree $\delta$ in $n$ variables. Then $P$ defines a morphism $P\colon\prod_{i=1}^{n}V_{d}\rightarrow V_{d\delta}$, by sending sections $(s_{i})_{i=1}^{n}$ of $V_{d}$ to the section $P(s_{1},\dots,s_{n})$ of $V_{d\delta}$. Fix a global section $f\colon X\rightarrow V_{d\delta}$ to the projection morphism and consider the pull-back of $P$ along $f$:

[TABLE]

Moreover, let $Z$ be the smooth locus of $P^{-1}(f)$ over $X$. It is an open subset. The derivatives $\frac{\partial P}{\partial x_{i}}$ of $P$ are homogeneous polynomials of degree $\delta-1$ in $n$ variables, hence can be regarded as functions $\prod_{i=1}^{n}V_{d}\rightarrow V_{d(\delta-1)}$. A point $y\in P^{-1}(f)$ lies in $Z$ if and only if $\frac{\partial P}{\partial x_{i}}(y)$, $i=1,\dots,n$ are not all zero. We wish to apply Proposition 4.1.1 to $Z/X$. Let ${\mathscr{M}}^{\prime}_{Z}$ denote the space of global sections of $Z$ over $X$, that is for a $k$-scheme $T$, ${\mathscr{M}}_{Z}^{\prime}(T)$ is the set of morphisms $s\colon X\times_{k}T\rightarrow Z$ as in the proposition (without any further conditions). A $k$-point $g\in{\mathscr{M}}^{\prime}_{Z}(k)$ corresponds to a section $g\colon X\rightarrow\prod_{i=1}^{n}V_{d}$, satisfying $P\circ g=f$. In general, for a (geometric) vector bundle $V$ on $X$ with corresponding locally free ${\mathcal{O}}_{X}$-module ${\mathscr{E}}$, the pull-back of the tangent space $\mathrm{Tan}_{V/X}$ along a section $s\colon X\rightarrow V$ is canonically isomorphic to ${\mathscr{E}}$. Hence in our situation (using that $Z\subseteq P^{-1}(f)$ is open) the tangent space $g^{\ast}\mathrm{Tan}_{Z/X}$ can be computed from the short exact sequence,

[TABLE]

where $D_{g}P$ is the derivative of $P$ at $g$. It is the ${\mathcal{O}}_{X}$-linear map given by $(t_{i})_{i=1}^{n}\mapsto\sum_{i=1}^{n}\frac{\partial P}{\partial x_{i}}(g)t_{i}$ (note that $\frac{\partial P}{\partial x_{i}}(g)$ are global sections of ${\mathcal{O}}(d(\delta-1))$). Note that $D_{g}P$ is surjective: by Nakayama, it suffices to check this fiberwise, where it is true by the condition defining $Z$.

The space ${\mathscr{M}}_{Z}$ is the subfunctor of ${\mathscr{M}}^{\prime}_{Z}$ consisting of all $g$ such that (fiberwise) $g^{\ast}\mathrm{Tan}_{Z/X}=\mathrm{ker}(D_{g}P)$ has vanishing $H^{1}$. Writing $\mathrm{ker}(D_{g}P)=\bigoplus_{i=1}^{r}{\mathcal{O}}(m_{i})$ ($m_{i}\in{\mathbf{Z}}$), this is equivalent to $m_{i}\geq-1$. By the Proposition 4.1.1 we conclude that ${\mathscr{M}}_{Z}$ is formally smooth over $k$.

Consider now a numerical example. Let $n=3$, $d=1$ and $\delta=4$ and let $g\in{\mathscr{M}}^{\prime}_{Z}(k)$. Then $D_{g}P\in\mathrm{Hom}_{{\mathcal{O}}_{X}}({\mathcal{O}}(1)^{\oplus 3},{\mathcal{O}}(4))=\Gamma(X,{\mathcal{O}}(3)^{\oplus 3})$, a $12$-dimensional $k$-vector space, and moreover, $D_{g}P$ lies in the open subspace of surjective maps. We have the short exact sequence of ${\mathcal{O}}_{X}$-modules

[TABLE]

This shows that $g^{\ast}\mathrm{Tan}_{Z/X}$ has rank $2$ and degree $-1$. Moreover, being a subbundle of ${\mathcal{O}}(1)^{\oplus 3}$ it only can have slopes $\leq 1$. There are only two options, either $g^{\ast}\mathrm{Tan}_{Z/X}\cong{\mathcal{O}}(-1)\oplus{\mathcal{O}}$ or $g^{\ast}\mathrm{Tan}_{Z/X}\cong{\mathcal{O}}(-2)\oplus{\mathcal{O}}(1)$. The point $g$ lies in ${\mathscr{M}}_{Z}$ if and only if the first option occurs for $g$. Which option occurs can be seen from the long exact cohomology sequence associated to (4.1.3):

[TABLE]

It is clear that $\Gamma(X,g^{\ast}\mathrm{Tan}_{Z/X})$ is $1$-dimesional if and only if $g^{\ast}\mathrm{Tan}_{Z/X}\cong{\mathcal{O}}(-1)\oplus{\mathcal{O}}$ and $2$-dimensional otherwise. The first option is generic, i.e., ${\mathscr{M}}_{Z}$ is an open subscheme of ${\mathscr{M}}^{\prime}_{Z}$.

4.2 The Jacobian criterion: perfectoid setting

We present here the perfectoid version of Proposition 4.1.1.

Theorem 4.2.1** (Fargues-Scholze [FS]).**

Let $S=\operatorname{Spa}(R,R^{+})$ be an affinoid perfectoid space in characteristic $p$. Let $Z\to X_{S}$ be a smooth morphism of schemes. Let ${\mathscr{M}}_{Z}^{>0}$ be the functor which inputs a perfectoid space $T\to S$ and outputs the set of sections of $Z\to X_{S}$ over $T$, that is, the set of morphisms $s$ making

[TABLE]

commute, subject to the condition that, fiberwise on $T$, all Harder-Narasimhan slopes of the vector bundle $s^{*}\operatorname{Tan}_{Z/X_{S}}$ are positive. Then ${\mathscr{M}}_{Z}^{>0}\to S$ is a cohomologically smooth morphism of locally spatial diamonds.

Example 4.2.2**.**

Let $S=\eta=\operatorname{Spa}(C,\mathcal{O}_{C})$, where $C$ is an algebraically closed perfectoid field of characteristic [math], and let $Z={\mathbf{V}}({\mathcal{E}}_{S}(H))\to X_{S}$ be the geometric vector bundle attached to ${\mathcal{E}}_{S}(H)$, where $H$ is a $p$-divisible group over the residue field of $C$. Then ${\mathscr{M}}_{Z}=H^{0}({\mathcal{E}}_{S}(H))$ is isomorphic to $\tilde{H}_{\eta}^{\mathop{\rm ad}}$ by Proposition 3.2.1. Let $s\colon X_{{\mathscr{M}}_{Z}}\to Z$ be the universal morphism; then $s^{*}\operatorname{Tan}_{Z/X_{S}}$ is the constant Banach-Colmez space associated to $H$ (i.e., the pull-back of ${\mathcal{E}}_{S}(H)$ along $X_{{\mathscr{M}}_{Z}}\rightarrow X_{S}$). This has vanishing $H^{1}$ if and only if $H$ has no étale part. This is true if and only if ${\mathscr{M}}_{Z}^{>0}$ is isomorphic to a perfectoid open ball. The perfectoid open ball is cohomologically smooth, in accord with Theorem 4.2.1. In contrast, if the étale quotient $H^{\text{\rm\'{e}t}}$ has height $d>0$, then $\pi_{0}(\tilde{H}_{\eta}^{\mathop{\rm ad}})\cong\mathbf{Q}_{p}^{d}$ implies that $\tilde{H}_{\eta}^{\mathop{\rm ad}}$ is not cohomologically smooth.

In the setup of Theorem 4.2.1, suppose that $x=\operatorname{Spa}(C,\mathcal{O}_{C})\to S$ is a geometric point, and that $x\to{\mathscr{M}}_{Z}^{>0}$ is an $S$-morphism, corresponding to a section $s\colon X_{C}\to Z$. Then $s^{*}\operatorname{Tan}_{Z/X_{S}}$ is a vector bundle on $X_{C}$. In light of the discussion in the previous section, we are tempted to interpret $H^{0}(X_{C},s^{*}\operatorname{Tan}_{Z/X_{S}})$ as the “tangent space of ${\mathscr{M}}_{Z}^{>0}\to S$ at $x$”. At points $x$ where $s^{*}\operatorname{Tan}_{Z/X_{S}}$ has only positive Harder-Narasimhan slopes, this tangent space is a perfectoid open ball.

5 Proof of the main theorem

5.1 Dilatations and modifications

As preparation for the proof of Theorem 1.0.1, we review the notion of a dilatation of a scheme at a locally closed subscheme [BLR90, §3.2].

Throughout this subsection, we fix some data. Let $X$ be a curve, meaning that $X$ is a scheme which is locally the spectrum of a Dedekind ring. Let $\infty\in X$ be a closed point with residue field $C$. Let $i_{\infty}\colon\operatorname{Spec}C\to X$ be the embedding, and let $\xi\in\mathcal{O}_{X,\infty}$ be a local uniformizer at $\infty$.

Proposition 5.1.1**.**

Let $V\to X$ be a morphism of finite type, and let $Y\subset V_{\infty}$ be a locally closed subscheme of the fiber of $V$ at $\infty$.

There exists a morphism of $X$-schemes $V^{\prime}\to V$ which is universal for the following property: $V^{\prime}\to X$ is flat at $\infty$, and $V_{\infty}^{\prime}\to V_{\infty}$ factors through $Y\subset V_{\infty}$.

The $X$-scheme $V^{\prime}$ is the dilatation of $V$ at $Y$. We review here its construction.

First suppose that $Y\subset V_{\infty}$ is closed. Let ${\mathscr{I}}\subset\mathcal{O}_{V}$ be the ideal sheaf which cuts out $Y$. Let $B\to V$ be the blow-up of $V$ along $Y$. Then ${\mathscr{I}}\cdot\mathcal{O}_{B}$ is a locally principal ideal sheaf. The dilatation $V^{\prime}$ of $V$ at $Y$ is the open subscheme of $B$ obtained by imposing the condition that the ideal $({\mathscr{I}}\cdot\mathcal{O}_{B})_{x}\subset\mathcal{O}_{B,x}$ is generated by $\xi$ at all $x\in B$ lying over $\infty$.

We give here an explicit local description of the dilatation $V^{\prime}$. Let $\operatorname{Spec}A$ be an affine neighborhood of $\infty$, such that $\xi\in A$, and let $\operatorname{Spec}R\subset V$ be an open subset lying over $\operatorname{Spec}A$. Let $I=(f_{1},\dots,f_{n})$ be the restriction of ${\mathscr{I}}$ to $\operatorname{Spec}R$, so that $I$ cuts out $Y\cap\operatorname{Spec}A$. Then the restriction of $V^{\prime}\to V$ to $\operatorname{Spec}R$ is $\operatorname{Spec}R^{\prime}$, where

[TABLE]

Now suppose $Y\subset V_{\infty}$ is only locally closed, so that $Y$ is open in its closure $\overline{Y}$. Then the dilatation of $V$ at $Y$ is the dilatation of $V\backslash(\overline{Y}\backslash Y)$ at $Y$.

Note that a dilatation $V^{\prime}\to V$ is an isomorphism away from $\infty$, and that it is affine.

Example 5.1.2**.**

Let

[TABLE]

be an exact sequence of $\mathcal{O}_{X}$-modules, where ${\mathcal{E}}$ (and thus ${\mathcal{E}}^{\prime}$) is locally free, and $W$ is a $C$-vector space. (This is an elementary modification of the vector bundle ${\mathcal{E}}$.) Let $K=\ker({\mathcal{E}}_{\infty}\to W)$.

Let ${\mathbf{V}}({\mathcal{E}})\to X$ be the geometric vector bundle corresponding to ${\mathcal{E}}$. Similarly, we have ${\mathbf{V}}({\mathcal{E}}^{\prime})\to X$, and an $X$-morphism ${\mathbf{V}}({\mathcal{E}}^{\prime})\to{\mathbf{V}}({\mathcal{E}})$. Let ${\mathbf{V}}(K)\subset{\mathbf{V}}({\mathcal{E}})_{\infty}$ be the affine space associated to $K\subset{\mathcal{E}}_{\infty}$. We claim that ${\mathbf{V}}({\mathcal{E}}^{\prime})$ is isomorphic to the dilatation ${\mathbf{V}}({\mathcal{E}})^{\prime}$ of ${\mathbf{V}}({\mathcal{E}})$ at ${\mathbf{V}}(K)$. Indeed, by the universal property of dilatations, there is a morphism ${\mathbf{V}}({\mathcal{E}}^{\prime})\to{\mathbf{V}}({\mathcal{E}})^{\prime}$, which is an isomorphism away from $\infty$.

To see that ${\mathbf{V}}({\mathcal{E}}^{\prime})\to{\mathbf{V}}({\mathcal{E}})^{\prime}$ is an isomorphism, it suffices to work over $\mathcal{O}_{X,\infty}$. Over this base, we may give a basis $f_{1},\dots,f_{n}$ of global sections of ${\mathcal{E}}$, with $f_{1},\dots,f_{k}$ lifting a basis for $K\subset{\mathcal{E}}_{\infty}$. Then the localization of ${\mathbf{V}}({\mathcal{E}})^{\prime}\to{\mathbf{V}}({\mathcal{E}})$ at $\infty$ is isomorphic to

[TABLE]

This agrees with the localization of ${\mathbf{V}}({\mathcal{E}}^{\prime})\to{\mathbf{V}}({\mathcal{E}})$ at $\infty$.

Lemma 5.1.3**.**

Let $V\to X$ be a smooth morphism, let $Y\subset V_{\infty}$ be a smooth locally closed subscheme, and let $\pi\colon V^{\prime}\to V$ be the dilatation of $V$ at $Y$. Then $V^{\prime}\to X$ is smooth, and $\operatorname{Tan}_{V^{\prime}/X}$ lies in an exact sequence of $\mathcal{O}_{V^{\prime}}$-modules

[TABLE]

where $N_{Y/V_{\infty}}$ is the normal bundle of $Y\subset V_{\infty}$, and $j\colon Y\to V$ is the inclusion.

Finally, let $T\to X$ be a morphism which is flat at $\infty$, and let $s\colon T\to V$ be a morphism of $X$-schemes, such that $s_{\infty}$ factors through $Y$. By the universal property of dilatations, $s$ factors through a morphism $s^{\prime}\colon T\to V^{\prime}$. Then we have an exact sequence of $\mathcal{O}_{V}$-modules

[TABLE]

Proof.

One reduces to the case that $Y$ is closed in $V_{\infty}$. The smoothness of $V^{\prime}\to X$ is [BLR90, §3.2, Proposition 3]. We turn to the exact sequence (5.1.1). The morphism $\operatorname{Tan}_{V^{\prime}/X}\to\pi^{*}\operatorname{Tan}_{V/X}$ comes from functoriality of the tangent bundle. To construct the morphism $\pi^{*}\operatorname{Tan}_{V/X}\to\pi^{*}j_{*}N_{Y/V_{\infty}}$, we consider the diagram

[TABLE]

in which the outer rectangle is cartesian. For its part, the normal bundle $N_{Y/V_{\infty}}$ sits in an exact sequence of $\mathcal{O}_{Y}$-modules

[TABLE]

The composite

[TABLE]

induces by adjunction a morphism

[TABLE]

where the last step is justified because $j$ is a closed immersion.

We check that (5.1.1) is exact using our explicit description of $V^{\prime}$. The sequence is clearly exact away from the preimage of $Y$ in $V^{\prime}$, since on the complement of this locus, the morphism $\pi$ is an isomorphism, and $\pi^{*}j_{*}=0$. Therefore we let $y\in Y$ and check exactness after localization at $y$. Let ${\mathcal{I}}\subset\mathcal{O}_{V}$ be the ideal sheaf which cuts out $Y$, and let $I\subset\mathcal{O}_{V,y}$ be the localization of ${\mathcal{I}}$ at $y$. Then $\mathcal{O}_{V_{\infty},y}=\mathcal{O}_{V,y}/\xi$. Since $Y\subset V_{\infty}$ are both smooth at $y$, we can find a system of local coordinates $\overline{f}_{1},\dots,\overline{f}_{n}\in\mathcal{O}_{V_{\infty},y}$ (meaning that the differentials $d\overline{f}_{i}$ form a basis for $\Omega^{1}_{V_{\infty}/C,y}$), such that $\overline{f}_{k+1},\dots,\overline{f}_{n}$ generate $I/\xi$. If $\partial/\partial\overline{f}_{i}$ are the dual basis, then the stalk of $N_{Y/V_{\infty}}$ at $y$ is the free $\mathcal{O}_{Y,y}$-module with basis $\partial/\partial\overline{f}_{k+1},\dots,\partial/\partial\overline{f}_{n}$.

Choose lifts $f_{i}\in\mathcal{O}_{V,y}$ of the $\overline{f}_{i}$. Then $I$ is generated by $\xi,f_{k},\dots,f_{n}$. The localization of $V^{\prime}\to V$ over $y$ is $\operatorname{Spec}\mathcal{O}_{V^{\prime},y}$, where $\mathcal{O}_{V^{\prime},y}=\mathcal{O}_{V,y}[g_{k+1},\dots,g_{n}]/(\xi\text{-torsion})$, where $\xi g_{i}=f_{i}$ for $i=k+1,\dots,n$. Then the stalk of $\operatorname{Tan}_{V^{\prime}/X}$ at $y$ is the free $\mathcal{O}_{V^{\prime},y}$-module with basis $\partial/\partial f_{1},\dots,\partial/\partial f_{k},\partial/\partial g_{k+1},\dots,\partial/\partial g_{n}$, whereas the stalk of $\pi^{*}\operatorname{Tan}_{V/X}$ at $y$ is the free $\mathcal{O}_{V^{\prime},y}$-module with basis $\partial/\partial f_{1},\dots,\partial/\partial f_{n}$. The quotient between these stalks is evidently the free module over $\mathcal{O}_{V^{\prime},y}/\xi$ with basis $\partial/\partial f_{k+1},\dots,\partial/\partial f_{n}$, and this agrees with the stalk of $\pi^{*}j_{*}N_{Y/V_{\infty}}$.

Given a morphism of $X$-schemes $s\colon T\to V$ as in the lemma, we apply $(s^{\prime})^{*}$ to (5.1.1); this is exact because $s^{\prime}$ is flat. The term on the right is $s^{*}j_{*}N_{Y/V_{\infty}}\cong i_{T_{\infty}*}s_{\infty}^{*}N_{Y/V_{\infty}}$ (once again, this is valid because $j$ is a closed immersion). ∎

5.2 The space ${\mathscr{M}}_{H,\infty}$ as global sections of a scheme over $X_{C}$

We will prove Theorem 1.0.1 for the Rapoport-Zink spaces of the form ${\mathscr{M}}_{H,\infty}$ before proceeding to the general case. Let $H$ be a $p$-divisible group of height $n$ and dimension $d$ over a perfect field $k$. In this context, $\breve{E}=W(k)[1/p]$. Let ${\mathcal{E}}={\mathcal{E}}_{C}(H)$. Throughout, we will be interpreting ${\mathscr{M}}_{H,\infty}$ as a functor on $\operatorname{Perf}_{\breve{E}}$ as in Proposition 3.2.2.

We have a determinant morphism $\det\colon{\mathscr{M}}_{H,\infty}\to{\mathscr{M}}_{\det H,\infty}$. Let $\tau\in{\mathscr{M}}_{\det H,\infty}(C)$ be a geometric point of ${\mathscr{M}}_{\det H,\infty}$. This point corresponds to a section $\tau$ of ${\mathbf{V}}(\det{\mathcal{E}})\to X_{C}$, which we also call $\tau$. Let ${\mathscr{M}}_{H,\infty}^{\tau}$ be the fiber of $\det$ over $\tau$.

Our first order of business is to express ${\mathscr{M}}_{H,\infty}^{\tau}$ as the space of global sections of a smooth morphism $Z\to X_{C}$, defined as follows. We have the geometric vector bundle ${\mathbf{V}}({\mathcal{E}}^{n})\to X$, whose global sections parametrize morphisms $s\colon\mathcal{O}_{X_{C}}^{n}\to{\mathcal{E}}$. Let $U_{n-d}$ be the locally closed subscheme of the fiber of ${\mathbf{V}}({\mathcal{E}}^{n})$ over $\infty$, which parametrizes all morphisms of rank $n-d$. We consider the dilatation ${\mathbf{V}}({\mathcal{E}}^{n})^{\operatorname{rk}_{\infty}=n-d}\rightarrow{\mathbf{V}}({\mathcal{E}}^{n})$ of ${\mathbf{V}}({\mathcal{E}}^{n})$ along $U_{n-d}$. For any flat $X_{C}$-scheme $T$, ${\mathbf{V}}({\mathcal{E}}^{n})^{\operatorname{rk}_{\infty}=n-d}(T)$ is the set of all $s\colon{\mathcal{O}}_{T}^{n}\rightarrow{\mathcal{E}}_{T}$ such that ${\rm cok}(s)\otimes C$ is projective ${\mathcal{O}}_{T}\otimes C$-module of rank $d$. Define $Z$ as the Cartesian product:

[TABLE]

Lemma 5.2.1**.**

Let ${\mathscr{M}}_{Z}$ be the functor which inputs a perfectoid space $T/C$ and outputs the set of sections of $Z\to X_{C}$ over $X_{T}$. Then ${\mathscr{M}}_{Z}$ is isomorphic to ${\mathscr{M}}_{H,\infty}^{\tau}$.

Proof.

Let $T=\operatorname{Spa}(R,R^{+})$ be an affinoid perfectoid space over $C$. The morphism $X_{T}\to X_{C}$ is flat. (This can be checked locally: $B_{\operatorname{dR}}^{+}(R)$ is torsion-free over the discrete valuation ring $B_{\operatorname{dR}}^{+}(C)$, and so it is flat.) By the description in (5.2.1), an $X_{T}$-point of ${\mathscr{M}}_{Z}$ corresponds to a morphism $\sigma\colon\mathcal{O}_{X_{T}}^{n}\to{\mathcal{E}}_{T}(H)$ which has the properties:

1. (1)

The cokernel of $\sigma_{\infty}$ is a projective $R$-module quotient of ${\mathcal{E}}_{T}(H)_{\infty}$ of rank $d$. 2. (2)

The determinant of $\sigma$ equals $\tau$.

On the other hand, by Proposition 3.2.2, ${\mathscr{M}}_{H,\infty}(T)$ is the set of morphisms $\sigma\colon\mathcal{O}_{X_{T}}^{n}\to{\mathcal{E}}_{T}(H)$ satisfying

1. ($1^{\prime}$)

The cokernel of $\sigma$ is $i_{\infty*}W$, for a projective $R$-module quotient $W$ of ${\mathcal{E}}_{T}(H)_{\infty}$ of rank $d$. 2. ($2$)

The determinant of $\sigma$ equals $\tau$.

We claim the two sets of conditions are equivalent for a morphism $\sigma\colon\mathcal{O}_{X_{T}}^{n}\to{\mathcal{E}}_{T}(H)$. Clearly ($1^{\prime}$) implies (1), so that ($1^{\prime}$) and ($2$) together imply (1) and (2) together. Conversely, suppose (1) and (2) hold. Since $\tau$ represents a point of ${\mathscr{M}}_{\det H,\infty}$, it is an isomorphism outside of $\infty$, and therefore so is $\sigma$. This means that $\operatorname{cok}\sigma$ is supported at $\infty$. Thus $\operatorname{cok}\sigma$ is a $B_{\operatorname{dR}}^{+}(R)$-module. For degree reasons, the length of $(\operatorname{cok}\sigma)\otimes_{B_{\operatorname{dR}}^{+}(R)}B_{\operatorname{dR}}^{+}(C^{\prime})$ has length $d$ for every geometric point $\operatorname{Spa}(C^{\prime},(C^{\prime})^{+})\to T$. Whereas condition (1) says that $(\operatorname{cok}\sigma)\otimes_{B_{\operatorname{dR}}^{+}(R)}R$ is a projective $R$-module of rank $d$. This shows that $(\operatorname{cok}\sigma)$ is already a projective $R$-module of rank $d$, which is condition ($1^{\prime}$). ∎

Lemma 5.2.2**.**

The morphism $Z\to X_{C}$ is smooth.

Proof.

Let $\infty^{\prime}\in X_{C}$ be a closed point, with residue field $C^{\prime}$. It suffices to show that the stalk of $Z$ at $\infty^{\prime}$ is smooth over $\operatorname{Spec}B_{\operatorname{dR}}^{+}(C^{\prime})$.

If $\infty^{\prime}\neq\infty$, then this stalk is isomorphic to the variety $({\mathbf{A}}^{n^{2}})^{\det=\tau}$ consisting of $n\times n$ matrices with fixed determinant $\tau$. Since $\tau$ is invertible in $B_{\operatorname{dR}}^{+}(C^{\prime})$, this variety is smooth.

Now suppose $\infty^{\prime}=\infty$. Let $\xi$ be a generator for the kernel of $B_{\operatorname{dR}}^{+}(C)\to C$. Then the stalk of $Z$ at $\infty$ is isomorphic to the flat $B_{\operatorname{dR}}^{+}(C)$-scheme $Y$, whose $T$-points for a flat $B_{\operatorname{dR}}^{+}(C)$-scheme $T$ are $n\times n$ matrices with coefficients in $\Gamma(T,\mathcal{O}_{T})$, which are rank $n-d$ modulo $\xi$, and which have fixed determinant $\tau$ (which must equal $u\xi^{d}$ for a unit $u\in B_{\operatorname{dR}}^{+}(C)$). Consider the open subset $Y_{0}\subset Y$ consisting of matrices $M$ where the first $(n-d)$ columns have rank $(n-d)$. Then the final $d$ columns of $M$ are congruent modulo $\xi$ to a linear combination of the first $(n-d)$ columns. After row reduction operations only depending on those first $(n-d)$ columns, $M$ becomes

[TABLE]

with $\det Q=w$ for a unit $w\in B_{\operatorname{dR}}^{+}(C)$ which only depends on the first $(n-d)$ columns of $M$. We therefore have a fibration $Y_{0}\to{\mathbf{A}}^{n(n-d)}$, namely projection onto the first $(n-d)$ columns, whose fibers are ${\mathbf{A}}^{d(n-d)}\times({\mathbf{A}}^{d^{2}})^{\det=w}$, which is smooth. Therefore $Y_{0}$ is smooth. The variety $Y$ is covered by opens isomorphic to $Y_{0}$, and so it is smooth. ∎

We intend to apply Theorem 4.2.1 to the morphism $Z\to X$, and so we need some preparations regarding the relative tangent space of ${\mathbf{V}}({\mathcal{E}}^{n})^{\operatorname{rk}_{\infty}=n-d}\to X_{C}$.

5.3 A linear algebra lemma

Let $f\colon V^{\prime}\to V$ be a rank $r$ linear map between $n$-dimensional vector spaces over a field $C$. Thus there is an exact sequence

[TABLE]

with $\dim W=\dim W^{\prime}=n-r$.

Consider the minor map $\Lambda\colon\operatorname{Hom}(V^{\prime},V)\to\operatorname{Hom}(\bigwedge^{r+1}V^{\prime},\bigwedge^{r+1}V)$ given by $\sigma\mapsto\bigwedge^{r+1}\sigma$. This is a polynomial map, whose derivative at $f$ is a linear map

[TABLE]

Explicitly, this map is

[TABLE]

Lemma 5.3.1**.**

Let

[TABLE]

be the kernel of the map $\sigma\mapsto q\circ(\sigma|_{W^{\prime}})$. Then $\ker D_{f}\Lambda=K$.

Proof.

Suppose $\sigma\in K$. Since $f$ has rank $r$, the exterior power $\bigwedge^{r+1}V^{\prime}$ is spanned over $C$ by elements of the form $v_{1}\wedge\cdots\wedge v_{r+1}$, where $v_{r+1}\in\ker f=W^{\prime}$. Since $f(v_{r+1})=0$, the sum in (5.3) reduces to

[TABLE]

Since $\sigma\in K$ and $v_{r+1}\in W^{\prime}$ we have $\sigma(v_{r+1})\in\ker q=f(V^{\prime})$, which means that $D_{f}\Lambda(\sigma)(v_{1},\dots,v_{r+1})\in\bigwedge^{r+1}f(V^{\prime})=0$. Thus $\sigma\in\ker D_{f}\Lambda$.

Now suppose $\sigma\in\ker D_{f}\Lambda$. Let $w\in W^{\prime}$. We wish to show that $\sigma(w)\in f(V^{\prime})$. Let $v_{1},\dots,v_{r}\in V^{\prime}$ be vectors for which $f(v_{1}),\cdots,f(v_{r})$ is a basis for $f(V^{\prime})$. Since $\sigma\in\ker D_{f}\Lambda$, we have $D_{f}\Lambda(\sigma)(v_{1}\wedge\cdots\wedge v_{r}\wedge w)=0$. On the other hand,

[TABLE]

because all other terms in the sum in (5.3) are 0, owing to $f(w)=0$. Since the wedge product above is 0, and the $f(v_{i})$ are a basis for $f(V^{\prime})$, we must have $\sigma(w)\in f(V^{\prime})$. Thus $\sigma\in K$. ∎

We interpret Lemma 5.3.1 as the calculation of a certain normal bundle. Let $Y={\mathbf{V}}(\operatorname{Hom}(V^{\prime},V))$ be the affine space over $C$ representing morphisms $V^{\prime}\to V$ over a $C$-scheme, and let $j\colon Y^{\operatorname{rk}=r}\to Y$ be the locally closed subscheme representing morphisms which are everywhere of rank $r$. Thus, $Y^{\operatorname{rk}=r}$ is an open subset of the fiber over 0 of (the geometric version of) the minor map $\Lambda$. It is well known that $Y^{\operatorname{rk}=r}/C$ is smooth of codimension $(n-r)^{2}$ in $Y/C$, and so the normal bundle $N_{Y^{\operatorname{rk}=r}/Y}$ is locally free of this rank.

We have a universal morphism of $\mathcal{O}_{Y^{\operatorname{rk}=r}}$-modules $\sigma\colon\mathcal{O}_{Y^{\operatorname{rk}=r}}\otimes_{C}V^{\prime}\to\mathcal{O}_{Y^{\operatorname{rk}=r}}\otimes_{C}V$. Let ${\mathcal{W}}^{\prime}=\ker\sigma$ and ${\mathcal{W}}=\operatorname{cok}\sigma$, so that ${\mathcal{W}}^{\prime}$ and ${\mathcal{W}}$ are locally free $\mathcal{O}_{Y^{\operatorname{rk}=r}}$-modules of rank $n-r$. We also have the $\mathcal{O}_{Y^{\operatorname{rk}=r}}$-linear morphism $D\Lambda\colon\mathcal{O}_{Y^{\operatorname{rk}=r}}\otimes_{C}\operatorname{Hom}(V^{\prime},V)\rightarrow\mathcal{O}_{Y^{\operatorname{rk}=r}}\otimes_{C}\operatorname{Hom}(\Lambda^{r+1}V^{\prime},\Lambda^{r+1}V)$, whose kernel is precisely $\operatorname{Tan}_{Y^{\operatorname{rk}=r}/C}$. The geometric interpretation of Lemma 5.3.1 is a commutative diagram with short exact rows:

[TABLE]

5.4 Moduli of morphisms of vector bundles with fixed rank at $\infty$

We return to the setup of §5.1. We have a curve $X$ and a closed point $\infty\in X$, with inclusion map $i_{\infty}$ and residue field $C$.

Let ${\mathcal{E}}$ and ${\mathcal{E}}^{\prime}$ be rank $n$ vector bundles over $X$, with fibers $V={\mathcal{E}}_{\infty}$ and $V^{\prime}={\mathcal{E}}^{\prime}_{\infty}$. We have the geometric vector bundle ${\mathbf{V}}(\operatorname{\mathscr{H}\text{\kern-3.0pt{\large om}}\,}({\mathcal{E}}^{\prime},{\mathcal{E}}))\to X$. If $f\colon T\to X$ is a morphism, then $T$-points of ${\mathbf{V}}(\operatorname{\mathscr{H}\text{\kern-3.0pt{\large om}}\,}({\mathcal{E}}^{\prime},{\mathcal{E}}))$ classify $\mathcal{O}_{T}$-linear maps $f^{*}{\mathcal{E}}^{\prime}\to f^{*}{\mathcal{E}}$.

Let ${\mathbf{V}}(\operatorname{\mathscr{H}\text{\kern-3.0pt{\large om}}\,}({\mathcal{E}}^{\prime},{\mathcal{E}}))^{\operatorname{rk}_{\infty}=r}$ be the dilatation of ${\mathbf{V}}(\operatorname{\mathscr{H}\text{\kern-3.0pt{\large om}}\,}({\mathcal{E}}^{\prime},{\mathcal{E}}))$ at the locally closed subscheme ${\mathbf{V}}(\operatorname{Hom}(V^{\prime},V))^{\operatorname{rk}=r}$ of the fiber ${\mathbf{V}}(\operatorname{\mathscr{H}\text{\kern-3.0pt{\large om}}\,}({\mathcal{E}}^{\prime},{\mathcal{E}}))_{\infty}={\mathbf{V}}(\operatorname{Hom}(V^{\prime},V))$. This has the following property, for a flat morphism $f\colon T\to X$: the $X$-morphisms $s\colon T\to{\mathbf{V}}(\operatorname{\mathscr{H}\text{\kern-3.0pt{\large om}}\,}({\mathcal{E}}^{\prime},{\mathcal{E}}))^{\operatorname{rk}_{\infty}=r}$ parametrize those $\mathcal{O}_{T}$-linear maps $\sigma\colon f^{*}{\mathcal{E}}^{\prime}\to f^{*}{\mathcal{E}}$, for which the fiber $\sigma_{\infty}\colon f_{\infty}^{*}V^{\prime}\to f_{\infty}^{*}V$ has rank $r$ everywhere on $T_{\infty}$.

Given a morphism $s$ as above, corresponding to a morphism $\sigma\colon f^{*}{\mathcal{E}}^{\prime}\to f^{*}{\mathcal{E}}$, we let ${\mathcal{W}}^{\prime}$ and ${\mathcal{W}}$ denote the kernel and cokernel of $\sigma_{\infty}$. Then ${\mathcal{W}}^{\prime}$ and ${\mathcal{W}}$ are locally free $\mathcal{O}_{T_{\infty}}$-modules of rank $r$. Let $i_{T_{\infty}}\colon T_{\infty}\to T$ denote the pullback of $i_{\infty}$ through $f$.

We intend to use Lemma 5.1.3 to compute $s^{*}\operatorname{Tan}_{{\mathbf{V}}(\operatorname{\mathscr{H}\text{\kern-3.0pt{\large om}}\,}({\mathcal{E}}^{\prime},{\mathcal{E}}))^{\operatorname{rk}_{\infty}=r}/X}$. The tangent bundle $\operatorname{Tan}_{{\mathbf{V}}(\operatorname{\mathscr{H}\text{\kern-3.0pt{\large om}}\,}({\mathcal{E}}^{\prime},{\mathcal{E}}))/X}$ is isomorphic to the pullback $f^{*}\operatorname{\mathscr{H}\text{\kern-3.0pt{\large om}}\,}({\mathcal{E}}^{\prime},{\mathcal{E}})$. Also, we have identified the normal bundle $N_{{\mathbf{V}}(\operatorname{Hom}(V^{\prime},V))^{\operatorname{rk}=r}/{\mathbf{V}}(\operatorname{Hom}(V^{\prime},V)}$ in (5.3.1). So when we apply the lemma to this situation, we obtain an exact sequence of $\mathcal{O}_{T}$-modules

[TABLE]

where the third arrow is adjoint to the map

[TABLE]

which sends $\sigma\in\operatorname{\mathscr{H}\text{\kern-3.0pt{\large om}}\,}(f_{\infty}^{*}V^{\prime},f_{\infty}^{*}V)$ to the composite

[TABLE]

The short exact sequence in (5.4.1) identifies the $\mathcal{O}_{T}$-module $s^{*}\mathrm{Tan}_{{\mathbf{V}}(\operatorname{\mathscr{H}\text{\kern-3.0pt{\large om}}\,}({\mathcal{E}}^{\prime},{\mathcal{E}}))^{\operatorname{rk}_{\infty}=r}/X}$ as a modification of $f^{*}\operatorname{\mathscr{H}\text{\kern-3.0pt{\large om}}\,}({\mathcal{E}}^{\prime},{\mathcal{E}})$ at the divisor $T_{\infty}$. We can say a little more in the case that $\sigma$ itself is a modification. Let us assume that $\sigma$ fits into an exact sequence

[TABLE]

Dualizing gives another exact sequence

[TABLE]

Then

[TABLE]

The kernel of $\alpha\otimes\alpha^{\prime}$ can be computed in terms of $\ker\alpha=f^{\ast}{\mathcal{E}}^{\prime}$ and $\ker\alpha^{\prime}=f^{\ast}({\mathcal{E}}^{\vee})$, see Lemma 5.4.1 below. It sits in a diagram

[TABLE]

Lemma 5.4.1**.**

Let ${\mathcal{A}}$ be an abelian $\otimes$-category. Let

[TABLE]

[TABLE]

be two exact sequences in ${\mathcal{A}}$, with $A,A^{\prime},K,K^{\prime}$ projective. The homology of the complex

[TABLE]

is given by $H_{2}=0$, $H_{1}\cong\operatorname{Tor}_{1}(B,B^{\prime})$, and $H_{0}\cong B\otimes B^{\prime}$. Thus, $K^{\prime\prime}=\ker(f\otimes f^{\prime}\colon A\otimes A^{\prime}\to B\otimes B^{\prime})$ appears in a diagram

[TABLE]

where both sequences are exact.

Proof.

Let $C_{\bullet}$ be the complex $K\to A$, and let $C^{\prime}_{\bullet}$ be the complex $K^{\prime}\to A^{\prime}$. Since $C^{\prime}_{\bullet}$ is a projective resolution of $B^{\prime}$, we have a Tor spectral sequence [Sta14, Tag 061Z]

[TABLE]

We have $E^{2}_{0,0}=B\otimes B^{\prime}$ and $E^{2}_{0,1}=\operatorname{Tor}_{1}(B,B^{\prime})$, and $E^{2}_{i,j}=0$ for all other $(i,j)$. Therefore $H_{0}(C_{\bullet}\otimes C^{\prime}_{\bullet})\cong B\otimes B^{\prime}$ and $H_{1}(C_{\bullet}\otimes C^{\prime}_{\bullet})\cong\operatorname{Tor}_{1}(B,B^{\prime})$, which is the lemma. ∎

5.5 A tangent space calculation

We return to the setup of §5.2. Thus we have fixed a $p$-divisible group $H$ over a perfect field $k$, and an algebraically closed perfectoid field $C$ containing $W(k)[1/p]$. But now we specialize to the case that $H$ is isoclinic. Therefore $D=\operatorname{End}H$ (up to isogeny) is a central simple $\mathbf{Q}_{p}$-algebra. Let ${\mathcal{E}}={\mathcal{E}}_{C}(H)$; we have $\operatorname{\mathscr{H}\text{\kern-3.0pt{\large om}}\,}({\mathcal{E}},{\mathcal{E}})\cong D\otimes_{\mathbf{Q}_{p}}\mathcal{O}_{X_{C}}$.

Recall the scheme $Z\to X_{C}$, defined as a fiber product in (5.2.1). Let $s\colon X_{C}\to Z$ be a section. This corresponds to a morphism $\sigma\colon\mathcal{O}_{X_{C}}^{n}\to{\mathcal{E}}$. Let $W^{\prime}$ and $W$ be the cokernel of $\sigma_{\infty}$; these are $C$-vector spaces.

We are interested in the vector bundle $s^{*}\mathrm{Tan}_{Z/X_{C}}$. This is the kernel of the derivative of the determinant map:

[TABLE]

We apply (5.4.2) to give a description of $s^{*}\mathrm{Tan}_{{\mathbf{V}}({\mathcal{E}}^{n})^{\operatorname{rk}_{\infty}=n-d}/X_{C}}$. We get a diagram of $\mathcal{O}_{X_{C}}$-modules

[TABLE]

On the other hand, the horizontal exact sequence fits into a diagram

[TABLE]

The arrow labeled ${\rm tr}$ is induced from the $\mathbf{Q}_{p}$-linear map $M_{n}(\mathbf{Q}_{p})\times D\to\mathbf{Q}_{p}$ carrying $(\alpha^{\prime},\alpha)$ to ${\rm tr}(\alpha^{\prime})-{\rm tr}(\alpha)$ (reduced trace on $D$). The commutativity of the lower right square boils down to the identity, valid for sections $s_{1},\dots,s_{n}\in H^{0}(X_{C},{\mathcal{E}})$ and $\alpha\in D$:

[TABLE]

(There is a similar identity for $\alpha^{\prime}\in M_{n}(\mathbf{Q}_{p})$.) Because the arrow labeled $\tau$ is injective, we can combine (5.5.1) and (5.5.2) to arrive at a description of $s^{*}\operatorname{Tan}_{Z/X_{C}}$:

[TABLE]

We pass to duals to obtain

[TABLE]

The dotted arrow is induced from the map $(M_{n}(\mathbf{Q}_{p})\times D)\otimes_{\mathbf{Q}_{p}}\mathcal{O}_{X_{C}}\to{\mathcal{E}}^{n}$ sending $(\alpha^{\prime},\alpha)\otimes 1$ to $\alpha\circ\sigma-\sigma\circ\alpha^{\prime}$.

Theorem 5.5.1**.**

If $s$ is a section to $Z\rightarrow X_{C}$ corresponding, under the isomorphism of Lemma 5.2.1, to a point $x\in{\mathcal{M}}_{H,\infty}^{\tau}(C)$, then the following are equivalent:

The vector bundle $s^{*}\mathrm{Tan}_{Z/X_{C}}$ has a Harder-Narasimhan slope which is $\leq 0$. 2. 2.

The point $x$ lies in the special locus ${\mathscr{M}}_{H,\infty}^{\tau,\operatorname{sp}}$.

Proof.

Let $\sigma\colon\mathcal{O}_{X_{C}}^{n}\to{\mathcal{E}}$ denote the homomorphism corresponding to $x$. Condition (1) is true if and only if $H^{0}(X_{C},s^{*}\mathrm{Tan}_{Z/X_{C}}^{\vee})\neq 0$. We now take $H^{0}$ of (5.5.4), noting that $H^{0}(X_{C},{\mathcal{F}}^{\vee})\to H^{0}(X_{C},{\mathcal{E}}^{n})$ is injective. We find that

[TABLE]

This is nonzero exactly when $x$ lies in the special locus. ∎

Combining Theorem 5.5.1 with the criterion for cohomological smoothness in Theorem 4.2.1 proves Theorem 1.0.1 for the space ${\mathscr{M}}_{H,\infty}$.

Naturally we wonder whether it is possible to give a complete discription of $s^{*}\operatorname{Tan}_{Z/X_{C}}$, as this is the “tangent space” of ${\mathscr{M}}_{H,\infty}^{\tau}$ at the point $x$. Note that $s^{*}\operatorname{Tan}_{Z/X_{C}}$ can only have nonnegative slopes, since it is a quotient of a trivial bundle. Therefore Theorem 5.5.1 says that 0 appears as a slope of $s^{*}\operatorname{Tan}_{Z/X_{C}}$ if and only if $s$ corresponds to a special point of ${\mathscr{M}}_{H,\infty}^{\tau}$.

Example 5.5.2**.**

Consider the case that $H$ has dimension 1 and height $n$, so that ${\mathscr{M}}_{H,\infty}$ is an infinite-level Lubin-Tate space. Suppose that $x\in{\mathscr{M}}_{H,\infty}(C)$ corresponds to a section $s\colon X_{C}\to Z$. Then $s^{*}\operatorname{Tan}_{Z/X_{C}}$ is a vector bundle of rank $n^{2}-1$ and degree $n-1$, with slopes lying in $[0,1/n]$; this already limits the possibilities for the slopes to a finite list.

If $n=2$ there are only two possibilities for the slopes appearing in $s^{*}\operatorname{Tan}_{Z/X_{C}}$: $\left\{1/3\right\}$ and $\left\{0,1/2\right\}$. These correspond exactly to the nonspecial and special loci, respectively.

If $n=3$, there are a priori five possibilities for the slopes appearing in $s^{*}\operatorname{Tan}_{Z/X_{C}}$: $\left\{1/4,1/4\right\}$, $\left\{1/3,1/5\right\}$, $\left\{1/3,1/3,0,0\right\}$, $\left\{2/7,0\right\}$, and $\left\{1/3,1/4,0\right\}$. But in fact the final two cases cannot occur: if 0 appears as a slope, then $x$ lies in the special locus, so that $A_{x}\neq\mathbf{Q}_{p}$. But as $A_{x}$ is isomorphic to a subalgebra of $\operatorname{End}^{\circ}H$, the division algebra of invariant 1/3, it must be the case that $\dim_{\mathbf{Q}_{p}}A_{x}=3$, which forces 0 to appear as a slope with multiplicity $\dim_{\mathbf{Q}_{p}}A_{x}/\mathbf{Q}_{p}=2$. On the nonspecial locus, we suspect that the generic (semistable) case $\left\{1/4,1/4\right\}$ always occurs, as otherwise there would be some unexpected stratification of ${\mathscr{M}}_{H,\infty}^{\circ,\operatorname{non-sp}}$. But currently we do not know how to rule out the case $\left\{1/3,1/5\right\}$.

5.6 The general case

Let ${\mathcal{D}}=(B,V,H,\mu)$ be a rational EL datum over $k$, with reflex field $E$. Let $F$ be the center of $B$. As in Section 3.5, let $D=\operatorname{End}_{B}V$ and $D^{\prime}=\operatorname{End}_{B}H$, so that $D$ and $D^{\prime}$ are both $F$-algebras.

Let $C$ be a perfectoid field containing $\breve{E}$, and let $\tau\in{\mathscr{M}}_{\det{\mathcal{D}},\infty}(C)$. Let ${\mathscr{M}}_{{\mathcal{D}},\infty}^{\tau}$ be the fiber of the determinant map over $\tau$. We will sketch the proof that ${\mathscr{M}}_{{\mathcal{D}},\infty}^{\tau}\to\operatorname{Spa}C$ is cohomologically smooth. It is along the same lines as the proof for ${\mathscr{M}}_{H,\infty}$, but with some extra linear algebra added.

The space ${\mathscr{M}}_{{\mathcal{D}},\infty}^{\tau}$ may be expressed as the space of global sections of a smooth morphism $Z\to X_{C}$, defined as follows. We have the geometric vector bundle ${\mathbf{V}}(\operatorname{\mathscr{H}\text{\kern-3.0pt{\large om}}\,}_{B}(V\otimes_{\mathbf{Q}_{p}}\mathcal{O}_{X},{\mathcal{E}}_{C}(H)))$. In its fiber over $\infty$, we have the locally closed subscheme whose $R$-points for a $C$-algebra $R$ are morphisms, whose cokernel is as a $B\otimes_{{\mathbf{Q}}_{p}}R$-module isomorphic to $V_{0}\otimes_{\breve{E}}R$, where $V_{0}$ is the weight 0 subspace of $V\otimes_{\mathbf{Q}_{p}}\breve{E}$ determined by $\mu$. We then have the dilatation ${\mathbf{V}}(\operatorname{\mathscr{H}\text{\kern-3.0pt{\large om}}\,}_{B}(V\otimes_{\mathbf{Q}_{p}}\mathcal{O}_{X_{C}},{\mathcal{E}}_{C}(H)))^{\mu}$ of ${\mathbf{V}}(\operatorname{\mathscr{H}\text{\kern-3.0pt{\large om}}\,}_{B}(V\otimes_{\mathbf{Q}_{p}}\mathcal{O}_{X},{\mathcal{E}}_{C}(H)))$ at this locally closed subscheme. Its points over $S=\operatorname{Spa}(R,R^{+})$ parametrize $B$-linear morphisms $s\colon V\otimes_{\mathbf{Q}_{p}}\mathcal{O}_{X_{S}}\to{\mathcal{E}}_{S}(H)$, such that (locally on $S$) the cokernel of the fiber $s_{\infty}$ is isomorphic as a $(B\otimes_{\mathbf{Q}_{p}}R)$-module to $V_{0}\otimes_{\breve{E}}R$. Finally, the morphism $Z\to X_{C}$ is defined by the cartesian diagram

[TABLE]

Let $x\in{\mathscr{M}}_{{\mathcal{D}},\infty}(C)$ correspond to a $B$-linear morphism $s\colon V\otimes_{\mathbf{Q}_{p}}\mathcal{O}_{X_{C}}\to{\mathcal{E}}_{C}(H)$ and a section of $Z\to X_{C}$ which we also call $s$. Define $B\otimes_{\mathbf{Q}_{p}}C$-modules $W^{\prime}$ and $W$ by

[TABLE]

The analogue of (5.5.4) is a diagram which computes the dual of $s^{*}\operatorname{Tan}_{Z/X_{C}}$:

[TABLE]

This time, the dotted arrow is induced from the map $(D^{\prime}\times D)\otimes_{\mathbf{Q}_{p}}\mathcal{O}_{X_{C}}\to\operatorname{\mathscr{H}\text{\kern-3.0pt{\large om}}\,}(V\otimes_{\mathbf{Q}_{p}}\mathcal{O}_{X_{C}},{\mathcal{E}}_{C}(H))$ sending $(\alpha^{\prime},\alpha)\otimes 1$ to $\alpha\circ s-s\circ\alpha^{\prime}$. Taking $H^{0}$ in (5.6.1) shows that $H^{0}(X_{C},s^{*}\operatorname{Tan}_{Z/X_{C}}^{\vee})=A_{x}/F$, and this is nonzero exactly when $x$ lies in the special locus.

5.7 Proof of Corollary 1.0.2

We conclude with a discussion of the infinite-level modular curve $X(p^{\infty})$. Recall from [Sch15] the following facts about the Hodge-Tate period map $\pi_{HT}\colon X(p^{\infty})\to{\mathbf{P}}^{1}$. The ordinary locus in $X(p^{\infty})$ is sent to ${\mathbf{P}}^{1}(\mathbf{Q}_{p})$. The supersingular locus is isomorphic to finitely many copies of ${\mathscr{M}}_{H,\infty,C}$, where $H$ is a connected $p$-divisible group of height 2 and dimension 1 over the residue field of $C$; the restriction of $\pi_{HT}$ to this locus agrees with the $\pi_{HT}$ we had already defined on each ${\mathscr{M}}_{H,\infty,C}$.

We claim that the following are equivalent for a $C$-point $x$ of $X(p^{\infty})^{\circ}$:

The point $x$ corresponds to an elliptic curve $E/\mathcal{O}_{C}$, such that the $p$-divisible group $E[p^{\infty}]$ has $\operatorname{End}E[p^{\infty}]=\mathbf{Z}_{p}$. 2. 2.

The stabilizer of $\pi_{HT}(x)$ in $\operatorname{PGL}_{2}(\mathbf{Q}_{p})$ is trivial. 3. 3.

There is a neighborhood of $x$ in $X(p^{\infty})^{\circ}$ which is cohomologically smooth over $C$.

First we discuss the equivalence of (1) and (2). If $E$ is ordinary, then $E[p^{\infty}]\cong\mathbf{Q}_{p}/\mathbf{Z}_{p}\times\mu_{p^{\infty}}$ certainly has endomorphism ring larger than $\mathbf{Z}_{p}$, so that (1) is false. Meanwhile, the stabilizer of $\pi_{HT}(x)$ in $\operatorname{PGL}_{2}(\mathbf{Q}_{p})$ is a Borel subgroup, so that (2) is false as well. The equivalence between (1) and (2) in the supersingular case is a special case of the equivalence discussed in Section 3.5.

Theorem 1.0.1 tells us that ${\mathscr{M}}_{H,\infty}^{\circ,\operatorname{non-sp}}$ is cohomologically smooth, which implies that shows that (2) implies (3). We therefore are left with showing that if (2) is false for a point $x\in X(p^{\infty})^{\circ}$, then no neighborhood of $x$ is cohomologically smooth.

First suppose that $x$ lies in the ordinary locus. This locus is fibered over ${\mathbf{P}}^{1}(\mathbf{Q}_{p})$. Suppose $U$ is a sufficiently small neighborhood of $x$. Then $U$ is contained in the ordinary locus, and so $\pi_{0}(U)$ is nondiscrete. This implies that $H^{0}(U,{\mathbf{F}}_{\ell})$ is infinite, and so $U$ cannot be cohomologically smooth.

Now suppose that $x$ lies in the supersingular locus, and that $\pi_{HT}(x)$ has nontrivial stabilizer in $\operatorname{PGL}_{2}(\mathbf{Q}_{p})$. We can identify $x$ with a point in ${\mathscr{M}}_{H,\infty}^{\circ,\operatorname{sp}}(C)$. We intend to show that every neighborhood of $x$ in ${\mathscr{M}}_{H,\infty}^{\circ}$ fails to be cohomologically smooth.

Not knowing a direct method, we appeal to the calculations in [Wei16], which constructed semistable formal models for each ${\mathscr{M}}_{H,m}^{\circ}$. The main result we need is Theorem 5.1.2, which uses the term “CM points” for what we have called special points. There exists a decreasing basis of neighborhoods $Z_{x,0}\supset Z_{x,1}\supset\cdots$ of $x$ in ${\mathscr{M}}_{H,\infty}^{\circ}$. For each affinoid $Z=\operatorname{Spa}(R,R^{+})$, let $\overline{Z}=\operatorname{Spec}R^{+}\otimes_{\mathcal{O}_{C}}\kappa$, where $\kappa$ is the residue field of $C$. For each $m\geq 0$, there exists a nonconstant morphism $\overline{Z}_{x,m}\to C_{x,m}$, where $C_{x,m}$ is an explicit nonsingular affine curve over $\kappa$. This morphism is equivariant for the action of the stabilizer of $Z_{x,m}$ in $\operatorname{SL}_{2}(\mathbf{Q}_{p})$. For infinitely many $m$, the completion $C_{x,m}^{\rm{cl}}$ of $C_{x,m}$ is a projective curve with positive genus.

Let $U\subset{\mathscr{M}}_{H,\infty}^{\circ}$ be an affinoid neighborhood of $x$. Then there exists $N\geq 0$ such that $Z_{x,m}\subset U$ for all $m\geq N$. Let $K\subset\operatorname{SL}_{2}(\mathbf{Q}_{p})$ be a compact open subgroup which stabilizes $U$, so that $U/K$ is an affinoid subset of the rigid-analytic curve ${\mathscr{M}}_{H,\infty}^{\circ}/K$. For each $m\geq N$, let $K_{m}\subset K$ be the stabilizer of $Z_{x,m}$, so that $K_{m}$ acts on $C_{x,m}$.

There exists an integral model of $U/K$ whose special fiber contains as a component the completion of each $\overline{Z}_{x,m}/K_{m}$ which has positive genus. Since there is a nonconstant morphism $\overline{Z}_{x,m}/K_{m}\to C_{x,m}/K_{m}$, we must have

[TABLE]

Now we take a limit as $K$ shrinks. Since $U\sim\varprojlim U/K$, we have $H^{1}(U,{\mathbf{F}}_{\ell})\cong\varinjlim H^{1}(U/K,{\mathbf{F}}_{\ell})$. Also, for each $m$, the action of $K_{m}$ on $C_{x,m}$ is trivial for all sufficiently small $K$. Therefore

[TABLE]

This shows that $U$ is not cohomologically smooth.

Acknowledgements. The authors want to thank Peter Scholze for his help and his interest in their work. Also they thank Andreas Mihatsch for pointing out a mistake in a previous version of the manuscript. The first named author was supported by Peter Scholze’s Leibniz Preis. The second author was supported by NSF Grant No. DMS-1440140 while in residence at the Mathematical Sciences Research Institute in Berkeley, California.