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 height $n$ 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 [Reference Rapoport and ZinkRZ96]. 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 ${{\mathscr M}}_H^{\operatorname{ad}}$, which has generic fiber $({{\mathscr M}}_H^{\operatorname{ad}})_\eta$, a rigid-analytic space over $\operatorname {Spa}(W(k)[1/p],W(k))$. Then $({{\mathscr M}}_H^{\operatorname{ad}})_{\eta }$ 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 ${{\mathscr M}}_H^{\operatorname{ad}}$ 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 $\pi _{GM}\colon ({{\mathscr M}}_H^{\operatorname{ad}})_{\eta }\to {{\mathcal {F}}}\ell$, 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 $({{\mathscr M}}_H^{\operatorname{ad}})_{\eta }$ by adding level structures. For a complete affinoid $(W(k)[1/p],W(k))$-algebra $(R,R^{+})$, and an element of $({{\mathscr M}}_H^{\operatorname{ad}})_{\eta }(R,R^{+})$ 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 [Reference Scholze and WeinsteinSW13, (3.3)]. Finite-level spaces ${{\mathscr M}}_{H,m}$ are obtained by trivializing the $G[p^{m}]$; these are finite étale covers of $({{\mathscr M}}_H^{\operatorname{ad}})_{\eta }$. The infinite-level space is obtained by trivializing all of $TG$ at once, as in the following definition.

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 [Reference Scholze and WeinsteinSW13, Definition 2.4.1].

One of the main theorems of [Reference Scholze and WeinsteinSW13] is the following.

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 [Reference Scholze and WeinsteinSW13, Proposition 3.1.3(ii)], which we continue to call $\tilde {H}$. The adic generic fiber $\tilde {H}^{\operatorname{ad}}_{\eta }$ is a preperfectoid space, as can be checked ‘by hand’: it is a product of the $d$-dimensional preperfectoid open ball $(\operatorname {Spa} W(k) [\kern-1pt[ {T_1^{1/p^{\infty }},\ldots ,T_d^{1/p^{\infty }}} ]\kern-1pt])_{\eta }$ by the constant adic space $VH^{\mathop e\limits^{{\prime}}t}$, where $H^{\mathop e\limits{{\prime}}t}$ 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 $\tilde {H}^{\operatorname{ad}}_{\eta }\times _{\operatorname {Spa}(W(k)[1/p],W(k))} \operatorname {Spa}(R,R^{+})\to \tilde {G}^{\operatorname{ad}}_{\eta }$; composing this with $\alpha$ gives a morphism $\mathbf {Q}_p^{n}\to \tilde {H}^{\operatorname{ad}}_{\eta }(R,R^{+})$. We have therefore described a morphism ${{\mathscr M}}_{H,\infty } \to (\tilde {H}^{\operatorname{ad}}_{\eta })^{n}$.

Theorem 2.1.2 follows from the fact that the morphism ${{\mathscr M}}_{H,\infty }\to (\tilde {H}^{\operatorname{ad}})_{\eta }^{n}$ presents ${{\mathscr M}}_{H,\infty }$ as an open subset of a Zariski closed subset of $(\tilde {H}^{\operatorname{ad}})_{\eta }^{n}$. We conclude this subsection by spelling out how this is done. We have a quasi-logarithm map $\operatorname {qlog}_H\colon \tilde {H}^{\operatorname{ad}}_{\eta } \to M(H)[1/p] \otimes _{W(k)[1/p]} {{\mathbf {G}}}_a$ [Reference Scholze and WeinsteinSW13, 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:

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

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

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

On the other hand, we have a commutative diagram.

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 $VG(R,R^{+}) \to \tilde {G}_{\eta }^{\operatorname{ad}}(R,R^{+})\cong \tilde {H}_{\eta }^{\operatorname{ad}}(R,R^{+})\to M(H)\otimes _{W(k)} R$.

Now suppose we have a point of ${{\mathscr M}}_{H,\infty }(R,R^{+})$ represented by a triple $(G,\rho ,\alpha )$. Then we have a $\mathbf {Q}_p$-linear map $\mathbf {Q}_p^{n}\to \tilde {H}_{\eta }^{\operatorname{ad}}(R,R^{+})\to M(H)\otimes _{W(k)} R$. The cokernel of its $R$-extension $R^{n}\to M(H)\otimes _{W(k)} R$ 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.

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.

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 [Reference Scholze and WeinsteinSW13] 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

\[ s\colon V\to \tilde{H}_\eta^{\operatorname{ad}}(R,R^{+}), \]

subject to the following conditions.

1. – Let $W$ be the quotient

   \[ V\otimes_{\mathbf{Q}_p} R\stackrel{\operatorname{qlog}_H\circ s}{\longrightarrow} M(H)\otimes_{W(k)} R \to W \to 0. \]
   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.

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

   \[ 0\to V \to \tilde{H}(\mathcal{O}_C) \to W\otimes_R C \to 0 \]
   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 $(\alpha ,\alpha ')\in {{\mathbf {G}}}(\mathbf {Q}_p)\times J(\mathbf {Q}_p)$ sends $s$ to $\alpha '\circ s\circ \alpha ^{-1}$.

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 $\pi _{HT}\colon {{\mathscr M}}_{{{\mathcal {D}}},\infty }\to {{\mathcal {F}}}\ell _{\mu }'$, where ${{\mathcal {F}}}\ell _\mu '(R,R^{+})$ 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 ${{\mathcal {F}}}\ell _\mu '(R,R^{+})$. In terms of deformations of the $p$-divisible group $H$, the period map $\pi _{HT}$ sends a deformation $G$ to $(\operatorname {Lie} G^{\vee })^{\vee }$.

3.1 Review of the curve

We briefly review here some constructions and results from [Reference Fargues and FontaineFF18]. 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 ${{\mathcal {Y}}}_F=\operatorname {Spa}(W(F^{\circ }),W(F^{\circ }))\backslash \{ {\left \lvert {p[\varpi ]} \right \rvert =0} \}$. 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 [Reference Fargues and FontaineFF18] 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 [Reference Fargues and FontaineFF18] 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 semilinear 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 $S = \operatorname {Spa}(R,R^{+})$ 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 $s_1,\ldots ,s_k \in \mathbb {Q}$, then $N$ has the slopes $1-s_1, \ldots ,1-s_k$. For an object $S$ in $\operatorname {Perf}_{W(k)[1/p]}$ we define the vector bundle ${{\mathcal {E}}}_S(H)$ on $X_S$ by

\[ {{\mathcal{E}}}_S(H) = {{\mathcal{E}}}_S(N) \otimes_{\mathcal{O}_{X_S}} \mathcal{O}_{X_S}(1). \]

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 $\tilde {H}^{\operatorname{ad}}_{\eta }\cong H^{0}({{\mathcal {E}}}(H))$ of sheaves on $\operatorname {Perf}_{W(k)[1/p]}$ making the following diagram commute,

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 $\tilde {H}^{\operatorname{ad}}_{\eta }(R,R^{+})\cong \tilde {H}(R^{\circ })\cong \tilde {H}(R^{\circ }/p)$. 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 [Reference MessingMes72]. 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 $A_\textrm {cris}(R^{\circ }/p)=A_\textrm {cris}(R^{\circ })$, equipped with Frobenius and Verschiebung.

Now we apply [Reference Scholze and WeinsteinSW13, Theorem A]: since $R^{\circ }/p$ is f-semiperfect, the crystalline Dieudonné functor is fully faithful up to isogeny. Thus

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

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

\[ \tilde{H}(R^{\circ})\cong ( B_\textrm{cris}^{+}(R^{\circ}) \otimes_{W(k)[1/p]} N )^{\phi=p}. \]

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 $(B_\textrm {cris}^{+}(R^{\circ }) \otimes _{W(k)[1/p]} N )^{\phi =p}$ is discussed in [Reference Le BrasLB18, Remarque 6.6].

The commutativity of the diagram in the proposition is [Reference Scholze and WeinsteinSW13, 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

(3.2.1)\begin{equation} 0 \to \mathcal{O}_{X_S}^{n} \stackrel{s}{\to} {{\mathcal{E}}}_S(H) \to i_{\infty *} W \to 0, \end{equation}

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 $S$,

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

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

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

The quasi-isogeny induces an isomorphism $\tilde {H}_{\eta }^{\operatorname{ad}} \times _{\operatorname {Spa} W(k)[1/p]} S \cong \tilde {G}_{\eta }^{\operatorname{ad}}$; composing this with the level structure gives an injective map $\mathbf {Q}_p^{n}\to \tilde {H}^{\operatorname{ad}}_{\eta }(S)$, 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 $\mathbf {Q}_p^{n}\to \tilde {H}^{\operatorname{ad}}_{\eta }(S)$ 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

\[ 0 \to V\otimes_{\mathbf{Q}_p} \mathcal{O}_{X_S} \stackrel{s}{\to} {{\mathcal{E}}}_S(H) \to i_{\infty*} W \to 0, \]

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 § 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 [Reference ChenChe14].

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 $S=\operatorname {Spa}(R,R^{+})$ 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 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 $B_{\operatorname {dR}}^{+}(R)\to R$. The morphism $\det \colon {{\mathscr M}}_{H,\infty }\to {{\mathscr M}}_{\det H,\infty }$ is simply $s\mapsto \det s$.

Regarding the structure of ${{\mathscr M}}_{\det H,\infty }$: we claim that for an algebraically closed perfectoid field $C/\breve {E}$, the set ${{\mathscr M}}_{\det H,\infty }(C)$ is a $\mathbf {Q}_p^{\times }$-torsor. Indeed, since the vector bundle ${{\mathcal {E}}}_C(H)$ has degree $d$, so does the line bundle $\det {{\mathcal {E}}}_C(H)$, so that $\det {{\mathcal {E}}}_C(H)\cong \mathcal {O}_{X_C}(d)$. A $C$-point of ${{\mathscr M}}_{\det H,\infty }$ is therefore a global section of $\mathcal {O}_{X_C}(d)$ with a zero of order $d$ at $\infty$. In other words, it is a nonzero element of $\operatorname {Fil}^{0}B_C^{\phi =p^{d}}\cong \mathbf {Q}_p(d)$.

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; i.e., it is a product of fields. The idea is now to construct the determinant datum $\det {{\mathcal {D}}}=(F,\det _F V,\det _F H,\det _F\circ \mu )$, noting once again that there may not be a $p$-divisible group ‘$\det _F H$’. The determinant $\det _F V$ is a free $F$-module of rank 1. For a perfectoid space $S=\operatorname {Spa}(R,R^{+})$ over $\breve {E}$, we have the $F\otimes _{\mathbf {Q}_p}{{\mathcal {O}}}_{X_S}$-module ${{\mathcal {E}}}_S(H)$ and its determinant $\det _F {{\mathcal {E}}}_S(H)$; the latter is a locally free $F\otimes _{\mathbf {Q}_p}{{\mathcal {O}}}_{X_S}$-module of rank 1. Let $d$ be the degree of $\det _F{{\mathcal {E}}}_S(H)$, considered as a function on $\operatorname {Spec} F$. We define ${{\mathscr M}}_{\det {{\mathcal {D}}},\infty }(S)$ to be 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 (locally on $\operatorname {Spec} F$) a projective $B_{\operatorname {dR}}^{+}(R)/(\xi )^{d}$-module of rank 1. (We remark here that the $\det _F$ in $\det _F\circ \mu$ means the morphism from ${{\mathbf {G}}}=\operatorname {Aut}_B(V)$ to ${{\mathbf {G}}}^{\text {ab}}=\operatorname {Aut}_F(\det _F V)=\mathop { \rm Res}_{F/\mathbf {Q}_p}{{\mathbf {G}}}_m$. If $\det _F \mu$ is a minuscule cocharacter, meaning that it is a vector of only 0s and 1s in the character group $X_*({{\mathbf {G}}}^{\text {ab}})\cong \mathbf {Z}^{[F:\mathbf {Q}_p]})$, then $\det {{\mathcal {D}}}$ is an honest rational EL datum.) The morphism ${{\mathscr M}}_{{{\mathcal {D}}},\infty }\to {{\mathscr M}}_{\det {{\mathcal {D}}},\infty }$ sends a $B\otimes _{\mathbf {Q}_p}\mathcal {O}_{X_S}$-linear map $s\colon V\otimes _{\mathbf {Q}_p} \mathcal {O}_{X_S} \to {{\mathcal {E}}}_S(H)$ to the $F\otimes _{\mathbf {Q}_p}\mathcal {O}_{X_S}$-linear map $\det s\colon \det _F V\otimes _{\mathbf {Q}_p}\mathcal {O}_{X_S} \to \det _F {{\mathcal {E}}}_S(H)$.

An argument similar to the above shows that for an algebraically closed perfectoid field $C/\breve {E}$, the set ${{\mathscr M}}_{\det {{\mathcal {D}}},\infty }(C)$ is an $F^{\times }$-torsor, equal to the set of $F$-bases for $F(d)$. Here the Tate twist is interpreted (locally on $\operatorname {Spec} F$) as the $d$th tensor power of the rational Tate module of the Lubin–Tate module for $F$.

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

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

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.

1. – The ${{\mathbf {G}}}$-isocrystal (${{\mathbf {G}}}=\operatorname {Aut}_B V$) associated to $H$ is basic in the sense of Kottwitz [Reference KottwitzKot85].

2. – The natural map

   \[ \operatorname{End}^{\circ}_B(H) \otimes_{\mathbf{Q}_p} W(k)[1/p] \to \operatorname{End}_{B\otimes_{\mathbf{Q}_p} W(k)[1/p]} M(H)[1/p] \]
   is an isomorphism.

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

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

   \[ D'\otimes_{\mathbf{Q}_p} \mathcal{O}_{X_C} \to \operatorname{\mathscr{E}{\mathcal{nd}}\,}_{(B\otimes_{\mathbf{Q}_p} \mathcal{O}_{X_C})} {{\mathcal{E}}}_C(H) \]
   is an isomorphism.

In brief, the duality theorem from [Reference Scholze and WeinsteinSW13] 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 $D'$ by

\begin{gather*} D= \operatorname{End}_{B} V,\\ D'= \operatorname{End}_{B} H. \end{gather*}

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 $(\alpha ,\alpha ')$, 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 $\alpha '\in \operatorname {End}_{B\otimes _{\mathbf {Q}_p} \mathcal {O}_{X_C}} {{\mathscr E}}_C(H)=D'$ (the last equality is due to basicness), such that $s\circ \alpha =\alpha '\circ s$. Thus we have

\[ A_x\cong \{ {(\alpha,\alpha')\in D\times D' \mid s\circ\alpha = \alpha'\circ s} \}. \]

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 $A\to D'$, 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 $D\times D'$ strictly containing $F$.

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

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,

(4.1.1)

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.

(4.1.2)

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

\[ \operatorname{\mathscr{H}{\mathcal{om}}\,}_{\mathcal{O}_{X\times_k T_0}}(s_{0}^{*}\Omega_{Z/X}, \pi_0^{*}I)\cong s_0^{*}\operatorname{Tan}_{Z/X} \otimes \ \pi_0^{*}I. \]

This torsor corresponds to class in

\[ H^{1}(X\times_k T_0, s_0^{*}\operatorname{Tan}_{Z/X}\otimes \ \pi_0^{*}I). \]

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

\[ H^{p}(T_0, R^{q}\pi_{0*}(s_0^{*}\operatorname{Tan}_{Z/X}\otimes \ \pi_0^{*}I)). \]

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

\[ H^{1}(X\times_k T_0, s_0^{*}\operatorname{Tan}_{Z/X}\otimes \ \pi_0^{*}I) \cong H^{0}(T_0,R^{1}\pi_{0*}(s_0^{*}\operatorname{Tan}_{Z/X}\otimes \ \pi_0^{*}I)). \]

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

\[ R\pi_{0*}(s_0^{*}\operatorname{Tan}_{Z/X}\otimes \ \pi_0^{*}I)\cong R\pi_{0*}s_0^{*}\operatorname{Tan}_{Z/X} \otimes^{{{\mathbf{L}}}} \, I. \]

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 pull-back 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

\[ \pi_*s^{*}\operatorname{Tan}_{Z/X} \cong \operatorname{Tan}_{{{\mathscr M}}_Z/\operatorname{Spec} k}. \]

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

\[ V_d = \underline {\smash {\mathrm {Spec}}}_X{\rm {\rm Sym}}_{{\mathcal {O}}_X}({\mathcal {O}}(-d)) \]

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 $P(s_1,\ldots ,s_n)$ 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 $f$.

Moreover, let $Z$ be the smooth locus of $P^{-1}(f)$ over $X$. It is an open subset. The derivatives ${\partial P}/{\partial x_i}$ 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 $({\partial P}/{\partial x_i})(y)$, $i=1,\ldots ,n$ are not all zero. We wish to apply Proposition 4.1.1 to $Z/X$. Let ${{\mathscr M}}'_Z$ denote the space of global sections of $Z$ over $X$, that is for a $k$-scheme $T$, ${{\mathscr M}}_Z'(T)$ is the set of morphisms $s \colon X \times _k T \rightarrow Z$ as in the proposition (without any further conditions). A $k$-point $g \in {{\mathscr M}}'_Z(k)$ 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,

\[ 0 \rightarrow g^{\ast} \mathrm{Tan}_{Z/X} \rightarrow \bigoplus_{i=1}^{n} {{\mathcal{O}}}(d) \stackrel{D_g P}{\longrightarrow} {{\mathcal{O}}}(d\delta) \rightarrow 0, \]

where $D_g P$ is the derivative of $P$ at $g$. It is the ${{\mathcal {O}}}_X$-linear map given by $(t_i)_{i=1}^{n} \mapsto \sum _{i = 1}^{n} ({\partial P}/{\partial x_i})(g)t_i$ (note that $({\partial P}/{\partial x_i})(g)$ 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 ${{\mathscr M}}'_Z$ 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 $g \in {{\mathscr M}}'_Z(k)$. 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

(4.1.3)\begin{equation} 0 \rightarrow g^{\ast} \mathrm{Tan}_{Z/X} \rightarrow {{\mathcal{O}}}(1)^{\oplus 3} \stackrel{D_g P}{\longrightarrow} {{\mathcal{O}}}(4) \rightarrow 0 . \end{equation}

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 less than or equal to $1$. 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):

\[ 0 \rightarrow \Gamma(X, g^{\ast} \mathrm{Tan}_{Z/X}) \rightarrow \underbrace{\Gamma(X, {{\mathcal{O}}}(1))^{\oplus 3}}_{\text{6-dim'l}} \stackrel{\Gamma(D_g P)}{\longrightarrow} \underbrace{\Gamma(X, {{\mathcal{O}}}(4))}_{\text{5-dim'l}} \rightarrow {\textrm{H}}^{1}(X, g^{\ast} \mathrm{Tan}_{Z/X}) \rightarrow 0 . \]

It is clear that $\Gamma (X, g^{\ast } \mathrm {Tan}_{Z/X})$ is $1$-dimensional 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 ${{\mathscr M}}'_Z$.

4.2 The Jacobian criterion: perfectoid setting

We present here the perfectoid version of Proposition 4.1.1

Theorem 4.2.1 (Fargues and 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

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.

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.

5.1 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 [Reference Bosch, Lütkebohmert and RaynaudBLR90, § 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 $\infty$.

The $X$-scheme $V'$ 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 $V'$ 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 $V'$. 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 $I=(f_1,\ldots ,f_n)$ be the restriction of ${{\mathscr I}}$ to $\operatorname {Spec} R$, so that $I$ cuts out $Y\cap \operatorname {Spec} A$. Then the restriction of $V'\to V$ to $\operatorname {Spec} R$ is $\operatorname {Spec} R'$, where

\[ R'= R\biggl[\frac{f_1}{\xi},\ldots,\frac{f_n}{\xi}\biggr]\biggr /(\xi\text{-torsion}). \]

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 $V'\to V$ is an isomorphism away from $\infty$, and that it is affine.

Example 5.1.2 Let

\[ 0 \to {{\mathcal{E}}}' \to {{\mathcal{E}}} \to i_{\infty *} W \to 0 \]

be an exact sequence of $\mathcal {O}_X$-modules, where ${{\mathcal {E}}}$ (and thus ${{\mathcal {E}}}'$) 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 ${{\mathbf {V}}}({{\mathcal {E}}}')\to X$, and an $X$-morphism ${{\mathbf {V}}}({{\mathcal {E}}}')\to {{\mathbf {V}}}({{\mathcal {E}}})$. Let ${{\mathbf {V}}}(K)\subset {{\mathbf {V}}}({{\mathcal {E}}})_\infty$ be the affine space associated to $K\subset {{\mathcal {E}}}_\infty$. We claim that ${{\mathbf {V}}}({{\mathcal {E}}}')$ is isomorphic to the dilatation ${{\mathbf {V}}}({{\mathcal {E}}})'$ of ${{\mathbf {V}}}({{\mathcal {E}}})$ at ${{\mathbf {V}}}(K)$. Indeed, by the universal property of dilatations, there is a morphism ${{\mathbf {V}}}({{\mathcal {E}}}')\to {{\mathbf {V}}}({{\mathcal {E}}})'$, which is an isomorphism away from $\infty$.

To see that ${{\mathbf {V}}}({{\mathcal {E}}}')\to {{\mathbf {V}}}({{\mathcal {E}}})'$ is an isomorphism, it suffices to work over $\mathcal {O}_{X,\infty }$. Over this base, we may give a basis $f_1,\ldots ,f_n$ of global sections of ${{\mathcal {E}}}$, with $f_1,\ldots ,f_k$ lifting a basis for $K\subset {{\mathcal {E}}}_\infty$. Then the localization of ${{\mathbf {V}}}({{\mathcal {E}}})'\to {{\mathbf {V}}}({{\mathcal {E}}})$ at $\infty$ is isomorphic to

\[ \operatorname{Spec} \mathcal{O}_{X,\infty}\biggl[\frac{f_1}{\xi},\ldots,\frac{f_k}{\xi},f_{k+1},\ldots,f_n\biggr] \to \operatorname{Spec} \mathcal{O}_{X,\infty}[f_1,\ldots,f_n]. \]

This agrees with the localization of ${{\mathbf {V}}}({{\mathcal {E}}}')\to {{\mathbf {V}}}({{\mathcal {E}}})$ 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 $\pi \colon V'\to V$ be the dilatation of $V$ at $Y$. Then $V'\to X$ is smooth, and $\operatorname {Tan}_{V'/X}$ lies in an exact sequence of $\mathcal {O}_{V'}$-modules

(5.1.1)\begin{equation} 0 \to \operatorname{Tan}_{V'/X}\to \pi^{*}\operatorname{Tan}_{V/X} \to \pi^{*}j_*N_{Y/V_\infty}\to 0, \end{equation}

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 $s'\colon T\to V'$. Then we have an exact sequence of $\mathcal {O}_V$-modules

(5.1.2)\begin{equation} 0 \to (s')^{*}\operatorname{Tan}_{V'/X} \to s^{*}\operatorname{Tan}_{V/X} \to i_{T_\infty*}s_\infty^{*}N_{Y/V_\infty} \to 0. \end{equation}

Proof. One reduces to the case that $Y$ is closed in $V_\infty$. The smoothness of $V'\to X$ is [Reference Bosch, Lütkebohmert and RaynaudBLR90, § 3.2, Proposition 3]. We turn to the exact sequence (5.1.1). The morphism $\operatorname {Tan}_{V'/X}\to \pi ^{*}\operatorname {Tan}_{V/X}$ 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

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

\[ 0 \to \operatorname{Tan}_{Y/C}\to i_Y^{*}\operatorname{Tan}_{V_\infty/C} \to N_{Y/V_\infty} \to 0. \]

The composite

\begin{align*} i_{V'}^{*}\pi^{*}\operatorname{Tan}_{V/X} & = \pi_\infty^{*} i_V^{*} \operatorname{Tan}_{V/X}\\ &\cong \pi_\infty^{*} \operatorname{Tan}_{V_\infty/C} \\ &=(\pi_\infty')^{*} i_Y^{*} \operatorname{Tan}_{V/C} \\ &\to (\pi_\infty')^{*} N_{Y/V_\infty} \end{align*}

induces by adjunction a morphism

\[ \pi^{*}\operatorname{Tan}_{V/X}\to i_{V'*}(\pi_\infty')^{*} N_{Y/V_\infty} \cong \pi^{*}j_* N_{Y/V_\infty}, \]

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 $V'$. The sequence is clearly exact away from the preimage of $Y$ in $V'$, 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 $\overline {f}_1,\ldots ,\overline {f}_n\in \mathcal {O}_{V_\infty ,y}$ (meaning that the differentials $d\overline {f}_i$ form a basis for $\Omega ^{1}_{V_{\infty }/C,y}$), such that $\overline {f}_{k+1},\ldots ,\overline {f}_n$ 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 $\partial /\partial \overline {f}_{k+1},\ldots ,\partial /\partial \overline {f}_n$.

Choose lifts $f_i\in \mathcal {O}_{V,y}$ of the $\overline {f}_i$. Then $I$ is generated by $\xi ,f_k,\ldots ,f_n$. The localization of $V'\to V$ over $y$ is $\operatorname {Spec} \mathcal {O}_{V',y}$, where $\mathcal {O}_{V',y}=\mathcal {O}_{V,y}[g_{k+1},\ldots ,g_n]/(\xi \text {-torsion})$, where $\xi g_i=f_i$ for $i=k+1,\ldots ,n$. Then the stalk of $\operatorname {Tan}_{V'/X}$ at $y$ is the free $\mathcal {O}_{V',y}$-module with basis $\partial /\partial f_1,\ldots ,\partial /\partial f_k,\partial /\partial g_{k+1},\ldots ,\partial /\partial g_n$, whereas the stalk of $\pi ^{*}\operatorname {Tan}_{V/X}$ at $y$ is the free $\mathcal {O}_{V',y}$-module with basis $\partial /\partial f_1,\ldots ,\partial / \partial f_n$. The quotient between these stalks is evidently the free module over $\mathcal {O}_{V',y}/\xi$ with basis $\partial /\partial f_{k+1},\ldots ,\partial /\partial f_n$, 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 $(s')^{*}$ to (5.1.1); this is exact because $s'$ 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 $\textrm {cok}(s) \otimes C$ is projective ${{\mathcal {O}}}_T \otimes C$-module of rank $d$. Define $Z$ as the following cartesian product.

(5.2.1)

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

If $\infty '\neq \infty$, 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 $B_{\operatorname {dR}}^{+}(C')$, this variety is smooth.

Now suppose $\infty '=\infty$. 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

\[ \left( \begin{array}{c|c} I_{n-d} & P \\\hline 0 & \xi Q \end{array} \right), \]

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

\[ 0 \to W'\to V' \stackrel{f}{\to} V \stackrel{q}{\to} W \to 0, \]

with $\dim W=\dim W'=n-r$.

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

\[ D_f\Lambda\colon \operatorname{Hom}(V',V) \to \operatorname{Hom} \biggl (\bigwedge^{r+1}V',\bigwedge^{r+1} V \biggr ). \]

Explicitly, this map is

(5.3.1)\begin{equation} D_f\Lambda(\sigma)(v_1\wedge\cdots \wedge v_{r+1}) = \sum_{i=1}^{r+1} f(v_1)\wedge f(v_2)\wedge \cdots \wedge \sigma(v_i) \wedge \cdots \wedge f(v_{r+1}). \end{equation}

Lemma 5.3.1 Let

\[ K=\ker(\operatorname{Hom}(V',V) \to \operatorname{Hom}(W',W)) \]

be the kernel of the map $\sigma \mapsto q\circ (\sigma \vert _{W'})$. Then $\ker D_f\Lambda = K$.

Proof. Suppose $\sigma \in K$. Since $f$ has rank $r$, the exterior power $\bigwedge ^{r+1} V'$ is spanned over $C$ by elements of the form $v_1\wedge \cdots \wedge v_{r+1}$, where $v_{r+1}\in \ker f = W'$. Since $f(v_{r+1})=0$, the sum in (5.3.1) reduces to

\[ D_f\Lambda(\sigma)(v_1\wedge\cdots\wedge v_{r+1}) = f(v_1)\wedge \cdots \wedge f(v_r) \wedge \sigma(v_{r+1}). \]

Since $\sigma \in K$ and $v_{r+1}\in W'$ we have $\sigma (v_{r+1})\in \ker q=f(V')$, which means that

\[ D_f\Lambda (\sigma ) (v_1,\ldots ,v_{r+1})\in \bigwedge ^{r+1}f(V')=0. \]

Thus $\sigma \in \ker D_f\Lambda$.

Now suppose $\sigma \in \ker D_f\Lambda$. Let $w\in W'$. We wish to show that $\sigma (w)\in f(V')$. Let $v_1,\ldots ,v_r\in V'$ be vectors for which $f(v_1),\ldots ,f(v_r)$ is a basis for $f(V')$. 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,

\[ D_f\Lambda(\sigma)(v_1\wedge \cdots\wedge v_r\wedge w) = f(v_1)\wedge\cdots \wedge f(v_r) \wedge \sigma(w), \]

because all other terms in the sum in (5.3.1) are 0, owing to $f(w)=0$. Since the wedge product above is 0, and the $f(v_i)$ are a basis for $f(V')$, we must have $\sigma (w)\in f(V')$. Thus $\sigma \in K$.

We interpret Lemma 5.3.1 as the calculation of a certain normal bundle. Let $Y={{\mathbf {V}}}(\operatorname {Hom}(V',V))$ be the affine space over $C$ representing morphisms $V'\to V$ 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 $\sigma \colon \mathcal {O}_{Y^{\operatorname {rk} = r}}\otimes _C V'\to \mathcal {O}_{Y^{\operatorname {rk} = r}}\otimes _C V$. Let ${{\mathcal {W}}}'=\ker \sigma$ and ${{\mathcal {W}}}=\operatorname {cok} \sigma$, so that ${{\mathcal {W}}}'$ 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 $D\Lambda \colon \mathcal {O}_{Y^{\operatorname {rk}=r}} \otimes _C \operatorname {Hom}(V',V) \rightarrow \mathcal {O}_{Y^{\operatorname {rk}=r}} \otimes _C \operatorname {Hom}(\Lambda ^{r+1}V',\Lambda ^{r+1}V)$, 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.

(5.3.2)

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 ${{\mathcal {E}}}'$ be rank $n$ vector bundles over $X$, with fibers $V={{\mathcal {E}}}_\infty$ and $V'={{\mathcal {E}}}'_\infty$. We have the geometric vector bundle ${{\mathbf {V}}}(\operatorname {\mathscr {H} {\mathcal{om}}\,}({{\mathcal {E}}}',{{\mathcal {E}}}))\to X$. If $f\colon T\to X$ is a morphism, then $T$-points of ${{\mathbf {V}}}(\operatorname {\mathscr {H}{\mathcal{om}}\,}({{\mathcal {E}}}',{{\mathcal {E}}}))$ classify $\mathcal {O}_T$-linear maps $f^{*}{{\mathcal {E}}}'\to f^{*}{{\mathcal {E}}}$.

Let ${{\mathbf {V}}}(\operatorname {\mathscr {H}{\mathcal{om}}\,}({{\mathcal {E}}}',{{\mathcal {E}}}))^{\operatorname {rk}_\infty = r}$ be the dilatation of ${{\mathbf {V}}}(\operatorname {\mathscr {H}{\mathcal{om}}\,}({{\mathcal {E}}}',{{\mathcal {E}}}))$ at the locally closed subscheme ${{\mathbf {V}}}(\operatorname {Hom}(V',V))^{\operatorname {rk} =r}$ of the fiber ${{\mathbf {V}}}(\operatorname {\mathscr {H}{\mathcal{om}}\,}({{\mathcal {E}}}',{{\mathcal {E}}}))_\infty ={{\mathbf {V}}}(\operatorname {Hom}(V',V))$. This has the following property, for a flat morphism $f\colon T\to X$: the $X$-morphisms $s\colon T\to {{\mathbf {V}}}(\operatorname {\mathscr {H}{\mathcal{om}}\,}({{\mathcal {E}}}',{{\mathcal {E}}}))^{\operatorname {rk}_\infty = r}$ parametrize those $\mathcal {O}_T$-linear maps $\sigma \colon f^{*}{{\mathcal {E}}}'\to f^{*}{{\mathcal {E}}}$, for which the fiber $\sigma _\infty \colon f_\infty ^{*}V'\to f_\infty ^{*}V$ has rank $r$ everywhere on $T_\infty$.

Given a morphism $s$ as above, corresponding to a morphism $\sigma \colon f^{*}{{\mathcal {E}}}'\to f^{*}{{\mathcal {E}}}$, we let ${{\mathcal {W}}}'$ and ${{\mathcal {W}}}$ denote the kernel and cokernel of $\sigma _\infty$. Then ${{\mathcal {W}}}'$ 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 pull-back of $i_\infty$ through $f$.