2.1 ${\mathcal O}_E$ -prisms

Let A be an ${\mathcal O}_E$ -algebra. A $\delta _E$ -structure on A is a map $\delta _E \colon A \to A$ of sets with the following properties:

• • $\delta _E(xy)=x^q\delta _E(y)+y^q\delta _E(x)+\pi \delta _E(x)\delta _E(y)$ .

• • $\delta _E(x+y)=\delta _E(x)+\delta _E(y)+(x^q+y^q-(x+y)^q)/\pi $ .

• • For an element $x \in {\mathcal O}_E$ , we have $\delta _E(x)=(x-x^q)/\pi $ .

A $\delta _E$ -ring is an ${\mathcal O}_E$ -algebra A equipped with a $\delta _E$ -structure. We define

$$\begin{align*}\phi_A \colon A \to A, \quad x \mapsto x^q+\pi\delta_E(x). \end{align*}$$

Then $\phi _A$ is a homomorphism of ${\mathcal O}_E$ -algebras and is a lift of the q-th power Frobenius $A/\pi \to A/\pi $ , $x \mapsto x^q$ . The homomorphism $\phi _A$ is called the Frobenius of the $\delta _E$ -ring A. When there is no ambiguity, we omit the subscript and simply write $\phi =\phi _A$ .

An ${\mathcal O}_E$ -prism is a pair $(A, I)$ of a $\delta _E$ -ring A and a Cartier divisor $I \subset A$ such that A is derived $(\pi , I)$ -adically complete and $\pi \in I + \phi (I)A$ . We say that $(A, I)$ is bounded if $A/I$ has bounded $p^\infty $ -torsion (i.e., $(A/I)[p^n]=(A/I)[p^\infty ]$ for some integer $n \geq 1$ ). In this case, A is $(\pi , I)$ -adically complete ([Reference ItoIto23, Remark 2.3.2]). If ${\mathcal O}_E={\mathbb Z}_p$ , then bounded ${\mathcal O}_E$ -prisms are the same as bounded prisms in the sense of [Reference Bhatt and ScholzeBS22]. We say that $(A, I)$ is orientable if I is principal. An ${\mathcal O}_E$ -prism $(A, I)$ with a homomorphism ${\mathcal O} \to A$ of $\delta _E$ -rings is called an ${\mathcal O}_E$ -prism over ${\mathcal O}$ . We refer to [Reference MarksMar23] and [Reference ItoIto23] for details.

We give some examples of ${\mathcal O}_E$ -prisms, which play a central role in this paper. We set

$$\begin{align*}\mathfrak{S}_{\mathcal O}:={\mathcal O}[[t_1, \dotsc, t_n]] \end{align*}$$

for $n \geq 0$ , which admits a unique $\delta _E$ -structure such that the Frobenius $\phi \colon \mathfrak {S}_{\mathcal O} \to \mathfrak {S}_{\mathcal O}$ is given by $\phi (t_i)=t^q_i$ ( $1 \leq i \leq n$ ). For every integer $m \geq 1$ , the quotient

$$\begin{align*}\mathfrak{S}_{{\mathcal O}, m}:={\mathcal O}[[t_1, \dotsc, t_n]]/(t_1, \dotsc, t_n)^m \end{align*}$$

admits a unique $\delta _E$ -structure that is compatible with the one on $\mathfrak {S}_{\mathcal O}$ .

Our next example is related to (integral) perfectoid rings. We refer to [Reference Bhatt, Morrow and ScholzeBMS18, Section 3] and [Reference Česnavičius and ScholzeČS24, Section 2] for the definition and basic properties of perfectoid rings.

Let S be a perfectoid ring over ${\mathcal O}$ (i.e., S is a perfectoid ring with a ring homomorphism ${\mathcal O} \to S$ ). Let $ S^\flat :=\varprojlim _{x \mapsto x^p} S/p $ be the tilt of S, which is a perfect k-algebra. Then

$$\begin{align*}W_{{\mathcal O}_E}(S^\flat):=W(S^\flat) \otimes_{W({\mathbb F}_q)} {\mathcal O}_E \end{align*}$$

is naturally an ${\mathcal O}$ -algebra. Let $\phi \colon W_{{\mathcal O}_E}(S^\flat ) \to W_{{\mathcal O}_E}(S^\flat )$ be the base change of the q-th power Frobenius of $W(S^\flat )$ . Since $W_{{\mathcal O}_E}(S^\flat )$ is $\pi $ -torsion free, we obtain the corresponding $\delta _E$ -structure on $W_{{\mathcal O}_E}(S^\flat )$ . Moreover, for any element $x \in S^\flat $ , the quotient $W_{{\mathcal O}_E}(S^\flat )/[x]$ admits a unique $\delta _E$ -structure that is compatible with the one on $W_{{\mathcal O}_E}(S^\flat )$ . Here, $[-]$ denotes the Teichmüller lift.

We note the following fact.

Let $ \theta \colon W(S^\flat ) \to S $ be the unique ring homomorphism whose reduction modulo p is the projection map $S^\flat \to S/p$ , $(x_0, x_1, \dotsc ) \mapsto x_0$ . Let $\theta _{{\mathcal O}_E} \colon W_{{\mathcal O}_E}(S^\flat ) \to S$ be the homomorphism induced from $\theta $ . We write $I_S:= {\mathrm {Ker}} \theta _{{\mathcal O}_E}$ for the kernel of $\theta _{{\mathcal O}_E}$ .

For the purposes of this paper, it will be convenient to introduce the following (slightly nonstandard) definition:

2.2 Prismatic sites

We say that a map $(A, I) \to (A', I')$ of bounded ${\mathcal O}_E$ -prisms is a flat map (resp. a faithfully flat map) if $A \to A'$ is $(\pi ,I)$ -completely flat (resp. $(\pi ,I)$ -completely faithfully flat) in the sense of [Reference Bhatt and ScholzeBS22, Notation 1.2].

Let R be a $\pi $ -adically complete ${\mathcal O}_E$ -algebra. As in [Reference ItoIto23, Definition 2.5.3], let

be the category of bounded ${\mathcal O}_E$ -prisms $(A, I)$ together with a homomorphism $R \to A/I$ of ${\mathcal O}_E$ -algebras. We endow the opposite category with the flat topology – that is, the topology generated by the faithfully flat maps. It follows from [Reference ItoIto23, Remark 2.5.4] that is a site. If ${\mathcal O}_E={\mathbb Z}_p$ , then is the same as the category introduced in [Reference Bhatt and ScholzeBS22, Remark 4.7].

Remark 2.2.1. The functors

form sheaves with respect to the flat topology. Here, $\mathrm {Set}$ is the category of sets.

Let $(A, I)$ be a bounded ${\mathcal O}_E$ -prism. We write

$$\begin{align*}(A, I)_{\mathrm{\acute{e}t}} \end{align*}$$

for the category of $(\pi , I)$ -completely étale A-algebras (in the sense of [Reference Bhatt and ScholzeBS22, Notation 1.2]). We endow $ (A, I)^{ {\mathrm {op}} }_{\mathrm {\acute {e}t}} $ with the $(\pi , I)$ -completely étale topology – that is, the topology generated by the $(\pi , I)$ -completely étale coverings $B \to B'$ (i.e., $B \to B'$ is $(\pi , I)$ -completely étale and $(\pi , I)$ -completely faithfully flat). Every $B \in (A, I)_{\mathrm {\acute {e}t}}$ admits a unique $\delta _E$ -structure compatible with that on A, and the pair $(B, IB)$ is a bounded ${\mathcal O}_E$ -prism by [Reference ItoIto23, Lemma 2.5.10].

We recall the following fact (see [Reference ItoIto23, Example 2.5.11] and the references therein):

2.3 Display groups

Let A be an ${\mathcal O}$ -algebra and $I \subset A$ an ideal generated by a nonzerodivisor $d \in A$ . We assume that A is I-adically complete. We set $A[1/I]:=A[1/d]$ . As in [Reference ItoIto23, Section 4], the display group $G_\mu (A, I)$ is defined by

$$\begin{align*}G_\mu(A, I):=\{ \, g \in G(A) \, \vert \, \mu(d)g\mu(d)^{-1} \, \, \text{lies in} \, \, G(A) \subset G(A[1/I]) \, \}. \end{align*}$$

We shall recall a structural result about $G_\mu (A, I)$ .

Let $P_\mu , U^{-}_{\mu } \subset G_{\mathcal O}$ be the closed subgroup schemes defined by, for every ${\mathcal O}$ -algebra R,

$$ \begin{align*} P_\mu(R)&=\{ \, g \in G(R) \, \vert \, \lim_{t \to 0} \mu(t)g\mu(t)^{-1} \, \text{exists} \, \},\\ U^{-}_{\mu}(R)&=\{ \, g \in G(R) \, \vert \, \lim_{t \to 0} \mu(t)^{-1}g\mu(t)=1 \, \}. \end{align*} $$

The group schemes $P_\mu $ and $U^{-}_{\mu }$ are smooth over ${\mathcal O}$ .

2.4 G- $\mu $ -displays

Let $(A, I)$ be a bounded ${\mathcal O}_E$ -prism over ${\mathcal O}$ . We recall the definition of G- $\mu $ -displays over $(A, I)$ .

We assume for a moment that $(A, I)$ is orientable. The results in Section 2.3 apply to $(A, I)$ . In particular, we have the display group $G_\mu (A, I)$ . For each generator $d \in I$ , we define the following homomorphism:

$$\begin{align*}\sigma_{\mu, d} \colon G_\mu(A, I) \to G(A), \quad g \mapsto \phi(\mu(d)g\mu(d)^{-1}). \end{align*}$$

Let $G(A)_d$ be the set $G(A)$ together with the following action of $G_\mu (A, I)$ :

$$\begin{align*}G(A) \times G_\mu(A, I) \to G(A), \quad (X, g) \mapsto g^{-1}X\sigma_{\mu, d}(g). \end{align*}$$

For another generator $d' \in I$ , we have $d=ud'$ for a unique $u \in A^\times $ . The map $G(A)_d \to G(A)_{d'}$ defined by $X \mapsto X\phi (\mu (u))$ is $G_\mu (A, I)$ -equivariant. Thus, we can define

$$\begin{align*}G(A)_I := {\varprojlim}_{d} G(A)_d, \end{align*}$$

where d runs over the set of generators $d \in I$ . The set $G(A)_I$ carries a natural action of $G_\mu (A, I)$ . The projection map $G(A)_I \to G(A)_d$ is a $G_\mu (A, I)$ -equivariant bijection. For an element $X \in G(A)_I$ , let

(2.1) $$ \begin{align} X_d \in G(A)_d \end{align} $$

denote the image of X.

Let $G_{\mu , A, I}$ and be the sheaves on $(A, I)^{ {\mathrm {op}} }_{\mathrm {\acute {e}t}}$ defined by

respectively. The sheaf is equipped with a natural action of the group sheaf $G_{\mu , A, I}$ . In fact, we can extend these definitions to (not necessarily orientable) bounded ${\mathcal O}_E$ -prisms $(A, I)$ over ${\mathcal O}$ ; see [Reference ItoIto23, Section 4.3] for details.

Definition 2.4.1 (G- $\mu $ -display).

Let $(A, I)$ be a bounded ${\mathcal O}_E$ -prism over ${\mathcal O}$ . A G- $\mu $ -display over $(A, I)$ is a pair

$$\begin{align*}(\mathcal{Q}, \alpha_{\mathcal{Q}}), \end{align*}$$

where $\mathcal {Q}$ is a $G_{\mu , A, I}$ -torsor and is a $G_{\mu , A, I}$ -equivariant map of sheaves on $(A, I)^{ {\mathrm {op}} }_{\mathrm {\acute {e}t}}$ . When there is no possibility of confusion, we write $\mathcal {Q}$ instead of $(\mathcal {Q}, \alpha _{\mathcal {Q}})$ . We say that $(\mathcal {Q}, \alpha _{\mathcal {Q}})$ is banal if $\mathcal {Q}$ is a trivial $G_{\mu , A, I}$ -torsor.

Isomorphisms between G- $\mu $ -displays over $(A, I)$ are defined in the obvious way. We write

for the groupoid of G- $\mu $ -displays over $(A, I)$ and the groupoid of banal G- $\mu $ -displays over $(A, I)$ , respectively.

Remark 2.4.2. Assume that $(A, I)$ is orientable. Let $ [G(A)_I/G_\mu (A, I)] $ be the groupoid whose objects are the elements $X \in G(A)_I$ and whose morphisms are defined by

$$\begin{align*}{\mathrm{Hom}}(X, X')=\{\, g \in G_\mu(A, I) \, \vert \, X'\cdot g=X \, \}. \end{align*}$$

Here, $(-)\cdot g$ denotes the action of $g \in G_\mu (A, I)$ . To each $X \in G(A)_I$ , we attach a banal G- $\mu $ -display

$$\begin{align*}\mathcal{Q}_X:=(G_{\mu, A, I}, \alpha_X) \end{align*}$$

over $(A, I)$ , where is given by $1 \mapsto X$ . This construction gives an equivalence of groupoids.

For a map $f \colon (A, I) \to (A', I')$ of bounded ${\mathcal O}_E$ -prisms over ${\mathcal O}$ , we have a base change functor

More precisely, we have a fibered category

over .

In the following, we recall more concrete descriptions of G- $\mu $ -displays. Let $(A, I)$ be a bounded ${\mathcal O}_E$ -prism over ${\mathcal O}$ .

Definition 2.4.4. A G-Breuil–Kisin module over $(A, I)$ is a G-torsor $\mathcal {P}$ over $ \mathrm {Spec} A$ (with respect to the étale topology) with an isomorphism $ F_{\mathcal {P}} \colon (\phi ^*\mathcal {P})[1/I] \overset {\sim }{\to } \mathcal {P}[1/I] $ of G-torsors over $ \mathrm {Spec} A[1/I]$ . We call $F_{\mathcal {P}}$ the Frobenius of $\mathcal {P}$ . We say that $\mathcal {P}$ is of type $\mu $ if $(\pi , I)$ -completely étale locally on A, there exists some trivialization $\mathcal {P} \simeq G_A$ under which the isomorphism $F_{\mathcal {P}}$ is given by $g \mapsto Yg$ for an element Y in the double coset

$$\begin{align*}G(A)\mu(d)G(A) \subset G(A[1/I]), \end{align*}$$

where $d \in I$ is a generator.

Example 2.4.6. Assume that $(A, I)$ is orientable. Let $d \in I$ be a generator. For an element $X \in G(A)_I$ , the trivial G-torsor $G_A$ with the isomorphism

$$\begin{align*}(\phi^*G_A)[1/I]=G_A[1/I] \overset{\sim}{\to} G_A[1/I], \quad g \mapsto (\mu(d)X_d)g \end{align*}$$

is a G-Breuil–Kisin module of type $\mu $ over $(A, I)$ , and this is isomorphic to the one $(\mathcal {Q}_X)_{\mathrm {BK}}$ associated with .

Example 2.4.7 (Minuscule Breuil–Kisin module).

We assume that $G= {\mathrm {GL}} _N$ . Let $\mu \colon {\mathbb G}_m \to {\mathrm {GL}} _{N}$ be the cocharacter defined by

$$\begin{align*}t \mapsto {\mathrm{diag}}{(\underbrace{t, \dotsc, t}_s, \underbrace{1, \dotsc, 1}_{N-s})}, \end{align*}$$

which is 1-bounded. A minuscule Breuil–Kisin module over $(A, I)$ is a finite projective A-module M together with an A-linear homomorphism

$$\begin{align*}F_M \colon \phi^*M=A \otimes_{\phi, A} M \to M \end{align*}$$

whose cokernel is killed by I. We set $ {\mathrm {Fil}} ^1(\phi ^*M):= \{ \, x \in \phi ^*M \, \vert \, F_M(x) \in IM \, \} $ and let $ P^1 \subset (\phi ^*M)/I(\phi ^*M) $ be the image of $ {\mathrm {Fil}} ^1(\phi ^*M)$ . By [Reference ItoIto23, Proposition 3.1.6], we see that $P^1$ is a direct summand of $(\phi ^*M)/I(\phi ^*M)$ . We say that M is of type $\mu $ if the rank of M (resp. $P^1$ ) is constant and equal to N (resp. s). Let $ \mathrm {BK}_\mu (A, I)^{\simeq } $ be the groupoid of minuscule Breuil–Kisin modules over $(A, I)$ of type $\mu $ . In [Reference ItoIto23, Example 5.3.3, Corollary 5.3.11], we constructed a natural equivalence of groupoids:

We set $ {\mathrm {Fil}} ^1_\mu := A^s \oplus I^{N-s} \subset A^N$ . Then the underlying $G_{\mu , A, I}$ -torsor of $\mathcal {Q}(M)$ is the functor

$$\begin{align*}\underline{\mathrm{Isom}}_{{\mathrm{Fil}}}(A^N, \phi^*M) \colon (A, I)_{\mathrm{\acute{e}t}} \to \mathrm{Set} \end{align*}$$

defined by sending $B \in (A, I)_{\mathrm {\acute {e}t}}$ to the set of isomorphisms $B^N \overset {\sim }{\to } (\phi ^*M) \otimes _A B$ under which $ {\mathrm {Fil}} ^1_\mu $ agrees with $ {\mathrm {Fil}} ^1(\phi ^*M)$ .

2.5 Hodge filtrations and G- $\phi $ -modules

Let $(A, I)$ be an orientable and bounded ${\mathcal O}_E$ -prism over ${\mathcal O}$ . Here, we recall the definitions of the Hodge filtration $P(\mathcal {Q})_{A/I}$ and the underlying G- $\phi $ -module $\mathcal {Q}_\phi $ of a G- $\mu $ -display $\mathcal {Q}$ over $(A, I)$ introduced in [Reference ItoIto23].

Let , , and be the sheaves on $(A, I)^{ {\mathrm {op}} }_{\mathrm {\acute {e}t}}$ defined by

respectively. Let be the inclusion. The composition of $\tau $ with the projection is denoted by $\overline {\tau }$ , which factors through a homomorphism by Proposition 2.3.3. We have a commutative diagram

Remark 2.5.1. For a G-torsor $\mathcal {P}$ over $ \mathrm {Spec} A$ (with respect to the étale topology), the sheaf on $(A, I)^{ {\mathrm {op}} }_{\mathrm {\acute {e}t}}$ defined by $B \mapsto \mathcal {P}(B)$ is a -torsor. By [Reference ItoIto23, Proposition 4.3.1], this construction induces an equivalence of categories

We will make no distinction between a G-torsor over $ \mathrm {Spec} A$ and the corresponding -torsor. Similarly, the category of G-torsors over $ \mathrm {Spec} A/I$ (resp. $P_\mu $ -torsors over $ \mathrm {Spec} A/I$ ) is equivalent to that of -torsors (resp. -torsors).

Let . We write

$$\begin{align*}\mathcal{Q}_{A}:=\mathcal{Q}^{\tau} \quad (\text{resp.}\ \mathcal{Q}_{A/I}:=\mathcal{Q}^{\overline{\tau}},\, \text{resp.}\ P(\mathcal{Q})_{A/I}:=\mathcal{Q}^{\overline{\tau}_P}) \end{align*}$$

for the pushout of $\mathcal {Q}$ along $\tau $ (resp. $\overline {\tau }$ , resp. $\overline {\tau }_P$ ).

We recall the following useful fact:

A G- $\phi $ -module over $(A, I)$ is a G-torsor $\mathcal {P}$ over $ \mathrm {Spec} A$ with an isomorphism

$$\begin{align*}\phi_{\mathcal{P}} \colon (\phi^*\mathcal{P}) \times_{\mathrm{Spec} A} \mathrm{Spec} A[1/\phi(I)] \overset{\sim}{\to} \mathcal{P} \times_{\mathrm{Spec} A} \mathrm{Spec} A[1/\phi(I)] \end{align*}$$

of G-torsors over $ \mathrm {Spec} A[1/\phi (I)]$ . Here, we set $A[1/\phi (I)]:=A[1/\phi (d)]$ for a generator $d \in I$ . We call $\phi _{\mathcal {P}}$ the Frobenius of $\mathcal {P}$ .

In [Reference ItoIto23, Section 5.5], we constructed a functor

If $\mathcal {Q}=\mathcal {Q}_X$ is the banal G- $\mu $ -display over $(A, I)$ associated with an element $X \in G(A)_I$ (see Remark 2.4.2), then the isomorphism $\phi _{\mathcal {Q}_A}$ is defined as follows. We set

(2.2) $$ \begin{align} X_\phi := X_d\phi(\mu(d)) \in G(A[1/\phi(I)]), \end{align} $$

which is independent of the choice of a generator $d \in I$ . (See (2.1) for the notation $X_d$ .) We have $\mathcal {Q}_A=G_A$ , and hence, $\phi ^*(\mathcal {Q}_A)=G_A$ . The isomorphism $\phi _{\mathcal {Q}_A}$ is then given by $ g \mapsto X_\phi g. $

Example 2.5.5. We retain the notation of Example 2.4.7. Let $\mathcal {Q}:=\mathcal {Q}(M)$ be the $ {\mathrm {GL}} _N$ - $\mu $ -display over $(A, I)$ associated with a minuscule Breuil–Kisin module M over $(A, I)$ of type $\mu $ . The underlying $ {\mathrm {GL}} _N$ - $\phi $ -module $\mathcal {Q}_{\phi }$ is the one naturally associated with the pair

$$\begin{align*}(\phi^*M, \phi^*(F_M)). \end{align*}$$

For the Hodge filtration $P(\mathcal {Q})_{A/I}$ , we have

$$\begin{align*}P(\mathcal{Q})_{A/I} \simeq \underline{\mathrm{Isom}}_{{\mathrm{Fil}}}((A/I)^N, (\phi^*M)/I(\phi^*M)), \end{align*}$$

where the right-hand side is the scheme of isomorphisms $f \colon (A/I)^N \overset {\sim }{\to } (\phi ^*M)/I(\phi ^*M)$ such that $f((A/I)^s \oplus 0)=P^1$ .

2.6 Prismatic G- $\mu $ -displays over complete regular local rings

In this subsection, we recall the main result of [Reference ItoIto23].

Definition 2.6.1. Let R be a $\pi $ -adically complete ${\mathcal O}$ -algebra. We define the following groupoid:

A prismatic G- $\mu $ -display over R is an object of .

For a homomorphism $h \colon R \to R'$ of $\pi $ -adically complete ${\mathcal O}$ -algebras, we have a base change functor

Remark 2.6.2. Giving a prismatic G- $\mu $ -display $\mathfrak {Q}$ over R is equivalent to giving a section of the fibered category over . We thus have the associated G- $\mu $ -display $\mathfrak {Q}_{(A, I)}$ over $(A, I)$ for each and a natural isomorphism

$$\begin{align*}\gamma_f \colon f^*(\mathfrak{Q}_{(A, I)}) \overset{\sim}{\to} \mathfrak{Q}_{(A', I')} \end{align*}$$

for each morphism $f \colon (A, I) \to (A', I')$ in . We call $\mathfrak {Q}_{(A, I)}$ the value of $\mathfrak {Q}$ at . Recall that an object of is given by a bounded ${\mathcal O}_E$ -prism $(A, I)$ over ${\mathcal O}$ together with a homomorphism $g \colon R \to A/I$ over ${\mathcal O}$ . We also write $ \mathfrak {Q}_{g}:=\mathfrak {Q}_{(A, I)} $ in order to emphasize that it depends on the homomorphism g. For a homomorphism $h \colon R \to R'$ and an object , we have

$$\begin{align*}(h^*\mathfrak{Q})_g=\mathfrak{Q}_{g \circ h}. \end{align*}$$

Example 2.6.3. The functor

is an equivalence since is an initial object ([Reference ItoIto23, Example 2.5.8]).

Let $\mathcal {C}_{\mathcal O}$ be the category of complete regular local rings R with a local homomorphism ${\mathcal O} \to R$ which induces an isomorphism on the residue fields. The morphisms in $\mathcal {C}_{\mathcal O}$ are the local homomorphisms over ${\mathcal O}$ .

An important feature of the category $\mathcal {C}_{\mathcal O}$ is the following:

Let $R \in \mathcal {C}_{\mathcal O}$ . We choose an ${\mathcal O}_E$ -prism $(\mathfrak {S}_{\mathcal O}, (\mathcal {E}))$ of Breuil–Kisin type with an isomorphism $ R \simeq \mathfrak {S}_{\mathcal O}/\mathcal {E} $ over ${\mathcal O}$ . The following result is a key ingredient in our deformation theory:

Theorem 2.6.5. The functor

is an equivalence if the cocharacter $\mu $ is 1-bounded.

3.1 Deformations

Let $f \colon (A', I') \to (A, I)$ be a map of orientable and bounded ${\mathcal O}_E$ -prisms over ${\mathcal O}$ such that $f \colon A' \to A$ is surjective.

We can define a deformation of a G- $\mu $ -display in the usual way:

Definition 3.1.1. Let $\mathcal {Q}$ be a G- $\mu $ -display over $(A, I)$ .

1. (1) A deformation of $\mathcal {Q}$ over $(A', I')$ is a pair $ (\mathscr {Q}, h) $ consisting of a G- $\mu $ -display $\mathscr {Q}$ over $(A', I')$ and an isomorphism $ h \colon f^*\mathscr {Q} \overset {\sim }{\to } \mathcal {Q} $ of G- $\mu $ -displays over $(A, I)$ . If there is no ambiguity, we simply write $\mathscr {Q}$ instead of $(\mathscr {Q}, h)$ .

2. (2) An isomorphism $ g \colon (\mathscr {Q}, h) \overset {\sim }{\to } (\mathscr {R}, i) $ of deformations of $\mathcal {Q}$ over $(A', I')$ is an isomorphism $g \colon \mathscr {Q} \overset {\sim }{\to } \mathscr {R}$ of G- $\mu $ -display over $(A', I')$ such that $i \circ (f^*g)=h$ .

The set of isomorphism classes of deformations of $\mathcal {Q}$ over $(A', I')$ is denoted by

$$\begin{align*}{\mathrm{Def}}(\mathcal{Q})_{(A', I')}. \end{align*}$$

The groupoid of deformations of $\mathcal {Q}$ over $(A', I')$ is denoted by $ \mathbf {Def}(\mathcal {Q})_{(A', I')}, $ using boldface letters.

We collect some preliminary results on deformations which will be used in the sequel. Let $J \subset A'$ (resp. $K \subset A'/I'$ ) be the kernel of $A' \to A$ (resp. $A'/I' \to A/I$ ).

We first note the following fact:

The next lemma is an analogue of [Reference LauLau21, Lemma 7.1.4 (b)].

We have the following useful description of deformations for banal G- $\mu $ -displays.

Proposition 3.1.4. Assume that $A'$ is J-adically complete and $A'/I'$ is K-adically complete. Let $X \in G(A)_I$ be an element and $\mathcal {Q}_X$ the banal G- $\mu $ -display over $(A, I)$ associated with X. Let

$$\begin{align*}\mathbf{Def}(X)_{(A', I')} \end{align*}$$

be the groupoid whose objects are the elements $Y \in G(A')_{I'}$ such that $f(Y)=X$ , and whose morphisms are given by

$$\begin{align*}{\mathrm{Hom}}_{\mathbf{Def}(X)_{(A', I')}}(Y, Z)=\{\, g \in G_\mu(A', I') \, \vert \, Z \cdot g =Y \, \, \mathrm{and} \, \, f(g)=1 \, \}. \end{align*}$$

For each $Y \in \mathbf {Def}(X)_{(A', I')}$ , we have a natural isomorphism $f^*(\mathcal {Q}_Y) \overset {\sim }{\to } \mathcal {Q}_X$ given by $1 \in G_{\mu }(A, I)$ . The construction $Y \mapsto (\mathcal {Q}_Y, 1)$ gives an equivalence

$$\begin{align*}\mathbf{Def}(X)_{(A', I')} \overset{\sim}{\to} \mathbf{Def}(\mathcal{Q}_X)_{(A', I')}. \end{align*}$$

For later use, we also record the following result.

Proof. (1) We claim that

(3.1) $$ \begin{align} G(A)_I \overset{\sim}{\to} {\varprojlim}_{m} G(A/J^m)_I \quad \text{and} \quad G_\mu(A, I)\overset{\sim}{\to}{\varprojlim}_{m} G_\mu(A/J^m, I). \end{align} $$

Indeed, the first equality is clear since G is affine. The second equality follows from Proposition 2.3.3 (2).

In order to prove (1), it suffices to show that the natural functor

$$\begin{align*}[G(A)_I/G_\mu(A, I)] \to \mathcal{C}:={2-\varprojlim}_{m} [G(A/J^m)_I/G_\mu(A/J^m, I)] \end{align*}$$

is an equivalence (cf. Remark 2.4.2). The fully faithfulness follows from (3.1). We shall prove that the functor is essentially surjective. An object of $\mathcal {C}$ can be identified with a family of objects $\{ X_m \}_{m \geq 1}$ , where $X_m \in G(A/J^m)_I$ , together with a family of isomorphisms $\{ g_m \}_{m \geq 1}$ , where $g_m \in G_\mu (A/J^m, I)$ is such that the equality $X_m\cdot g_m=X_{m+1}$ holds in $G(A/J^m)_I$ . Applying Proposition 3.1.4 to the natural map $p_m \colon (A/J^{m+1}, I) \to (A/J^{m}, I)$ repeatedly, we can find an element $(X^{\prime }_m)_{m \geq 1} \in \varprojlim _{m} G(A/J^m)_I$ such that the corresponding object of $\mathcal {C}$ is isomorphic to the object $\{ X_m \}_{m \geq 1}$ . Let $X \in G(A)_I$ be the element corresponding to $(X^{\prime }_m)_{m \geq 1}$ . Then the image of X under the above functor is isomorphic to the object $\{ X_m \}_{m \geq 1}$ .

(2) An object of can be identified with a family $\{ \mathcal {Q}_m \}_{m \geq 1}$ , where $\mathcal {Q}_m$ is a G- $\mu $ -display over $(A/J^m, I)$ , together with isomorphisms $p^*_m(\mathcal {Q}_{m+1}) \overset {\sim }{\to } \mathcal {Q}_{m}$ ( $m \geq 1$ ). Our assumption and Lemma 3.1.2 imply that there is a finite étale covering $A \to B$ such that for any $m \geq 1$ , the base change of $\mathcal {Q}_m$ to $(B/J^mB, I(B/J^mB))$ is banal. We note that for a finite étale covering $A \to B$ , the ideal $JB$ satisfies the above conditions (a), (b), (c). Using these observations and finite étale descent for G- $\mu $ -displays, we see that (2) follows from (1).

Example 3.1.6. Let $(\mathfrak {S}_{\mathcal O}, (\mathcal {E}))$ be an ${\mathcal O}_E$ -prism of Breuil–Kisin type as in Example 2.1.1, where $\mathfrak {S}_{\mathcal O}={\mathcal O}[[t_1, \dotsc , t_n]]$ . Let $\mathcal {Q}$ be a G- $\mu $ -display over $(\mathfrak {S}_{\mathcal O}, (\mathcal {E}))$ . By Proposition 2.5.3, there is a finite extension $\widetilde {k}$ of k such that the base change of $\mathcal {Q}$ to $(\mathfrak {S}_{\widetilde {{\mathcal O}}}, (\mathcal {E}))$ is banal, where $\widetilde {{\mathcal O}}:= W(\widetilde {k}) \otimes _{W({\mathbb F}_q)} {\mathcal O}_E$ and $\mathfrak {S}_{\widetilde {{\mathcal O}}}:=\mathfrak {S}_{\mathcal O} \otimes _{\mathcal O} \widetilde {{\mathcal O}} = \widetilde {{\mathcal O}}[[t_1, \dotsc , t_n]]$ . Therefore, by Proposition 3.1.5, we have

where $\mathfrak {S}_{{\mathcal O}, m}={\mathcal O}[[t_1, \dotsc , t_n]]/(t_1, \dotsc , t_n)^m$ .

3.2 Special nilpotent thickenings

Let $f \colon (A', I') \to (A, I)$ be a map of orientable and bounded ${\mathcal O}_E$ -prisms over ${\mathcal O}$ such that $f \colon A' \to A$ is surjective. Let $J \subset A'$ be the kernel of f. We assume that $J^n=0$ for some $n \geq 1$ . In this case, we say that f is a nilpotent thickening. The functor

$$\begin{align*}(A', I')_{\mathrm{\acute{e}t}} \to (A, I)_{\mathrm{\acute{e}t}}, \quad B' \mapsto B, \end{align*}$$

where B is the $(\pi , I)$ -adic completion of $B' \otimes _{A'} A$ , is an equivalence (see [Reference ItoIto23, Lemma 2.5.9] for example).

Proof. We have the following exact sequence for every $m \geq 1$ :

$$\begin{align*}0 \to (J+(\pi, I')^m)/(\pi, I')^m \to A'/(\pi, I')^m \to A/(\pi, I)^m \to 0. \end{align*}$$

Since $A'/(\pi , I')^m \to B'/(\pi , I')^m$ is flat, the induced sequence

$$\begin{align*}0 \to (JB'+(\pi, I')^m)/(\pi, I')^m \to B'/(\pi, I')^m \to B/(\pi, I)^m \to 0 \end{align*}$$

is also exact. By taking the limit over m, we see that $B' \to B$ is surjective, and

$$\begin{align*}J_B \simeq {\varprojlim}_m (JB'+(\pi, I')^m)/(\pi, I')^m. \end{align*}$$

Using this isomorphism, we see that $J^n_B=0$ , and that $\phi _{B'}(J_B) = 0$ if $\phi _{A'}(J) =0$ .

Example 3.2.6. Let the notation be as in Example 3.2.4 and Example 3.2.5. In this paper, we say that a nilpotent thickening $f \colon (A', I') \to (A, I)$ is of Breuil–Kisin type (resp. of perfectoid type) if it is of the form

$$ \begin{align*} (\mathfrak{S}_{{\mathcal O}, m+1}, (\mathcal{E})) &\to (\mathfrak{S}_{{\mathcal O}, m}, (\mathcal{E})) \\ (\text{resp.\ } (W_{{\mathcal O}_E}(S^\flat)/[a^\flat]^{m+1}, I_S) &\to (W_{{\mathcal O}_E}(S^\flat)/[a^\flat]^{m}, I_S)). \end{align*} $$

In either case, f is a special nilpotent thickening.

Deformations of G- $\phi $ -modules are defined in the usual way. If $\mathcal {Q}$ is a G- $\mu $ -display over $(A, I)$ and $\mathscr {Q}$ is a deformation of $\mathcal {Q}$ over $(A', I')$ , then $\mathscr {Q}_\phi $ is naturally a deformation of $\mathcal {Q}_\phi $ over $(A', I')$ . Here, $\mathscr {Q}_\phi $ and $\mathcal {Q}_\phi $ are the underlying G- $\phi $ -modules; see Definition 2.5.4. The following proposition is the key ingredient in the construction of period maps introduced in Section 3.3 below.

Proof. (1) The problem is $(\pi , I)$ -completely étale local on A by $(\pi , I)$ -completely étale descent for G- $\phi $ -modules (which follows from the same argument as in [Reference ItoIto23, Remark 5.1.3]) and the uniqueness assertion. Thus, we may assume that $\mathscr {Q}$ , $\mathscr {R}$ and $\mathcal {Q}$ are banal. We can identify $\mathcal {Q}$ with $\mathcal {Q}_X$ for some $X \in G(A)_I$ . By Proposition 3.1.4, we may assume that $(\mathscr {Q}, h)$ and $(\mathscr {R}, i)$ are of the form $(\mathcal {Q}_Y, 1)$ and $(\mathcal {Q}_Z, 1)$ , respectively, for some $Y, Z \in \mathbf {Def}(X)_{(A', I')}$ . It suffices to prove that there exists a unique element $\psi \in G(A')$ such that $f(\psi )=1$ in $G(A)$ and

(3.2) $$ \begin{align} \psi^{-1}Z_\phi \phi(\psi) =Y_\phi \quad \text{in} \quad G(A'[1/\phi(I')]). \end{align} $$

Here, $Y_\phi , Z_\phi \in G(A'[1/\phi (I')])$ are the elements defined in (2.2).

We fix a generator $d \in I'$ . We shall show that the element

$$\begin{align*}\psi:=Z_d(Y_d)^{-1} \in G(A') \end{align*}$$

satisfies $f(\psi )=1$ and (3.2). Indeed, since $X=f(Y)=f(Z)$ , we have $f(\psi )=1$ . This implies that $f(\mu (d)\psi \mu (d)^{-1})=1$ . By our assumption that $\phi _{A'}(J)=0$ for the kernel $J \subset A'$ of $f \colon A' \to A$ , it then follows that $\phi (\mu (d)\psi \mu (d)^{-1})=1$ , whence (3.2) holds.

It remains to show the uniqueness of $\psi $ . Let $\psi \in G(A')$ be an element satisfying $f(\psi )=1$ and (3.2). As explained above, the equality $f(\psi )=1$ implies $\phi (\mu (d)\psi \mu (d)^{-1})=1$ . Then it follows from (3.2) that $\psi =Z_d(Y_d)^{-1}$ in $G(A'[1/\phi (I')])$ . Since the natural homomorphism $G(A') \to G(A'[1/\phi (I')])$ is injective by the condition (2) in Definition 3.2.2, we conclude that $\psi =Z_d(Y_d)^{-1}$ in $G(A')$ .

(2) We note that the functor $\mathcal {Q} \mapsto \mathcal {Q}_{\phi }$ from to the groupoid of G- $\phi $ -modules over $(A, I)$ is faithful (since the functor from the groupoid of $G_{\mu , A, I}$ -torsors to that of -torsors obtained by pushout along the inclusion is faithful). Then (2) follows from the uniqueness assertion in (1).

Remark 3.2.8. Let the notation be as in Proposition 3.1.4. Let $Y, Z \in \mathbf {Def}(X)_{(A', I')}$ . For the deformations $(\mathcal {Q}_Y, 1)$ and $(\mathcal {Q}_Z, 1)$ of the G- $\mu $ -display $\mathcal {Q}_X$ , the proof of Proposition 3.2.7 shows that for any generator $d \in I'$ ,

$$\begin{align*}(\mathcal{Q}_Y, 1) \simeq (\mathcal{Q}_Z, 1) \quad \text{if and only if} \quad Z_d(Y_d)^{-1} \in G_{\mu}(A', I'). \end{align*}$$

If this is the case, the unique isomorphism $(\mathcal {Q}_Y, 1) \overset {\sim }{\to } (\mathcal {Q}_Z, 1)$ is given by $Z_d(Y_d)^{-1}$ .

Corollary 3.2.9. Assume that $f \colon (A', I') \to (A, I)$ is a special nilpotent thickening. Let $\mathcal {Q}$ be a G- $\mu $ -display over $(A, I)$ . For each $B' \in (A', I')_{\mathrm {\acute {e}t}}$ with corresponding $B \in (A, I)_{\mathrm {\acute {e}t}}$ , we denote the base change of $\mathcal {Q}$ to $(B, IB)$ by $\mathcal {Q}_{(B, IB)}$ . Then the functor

$$\begin{align*}\underline{{\mathrm{Def}}}(\mathcal{Q}) \colon (A', I')_{\mathrm{\acute{e}t}} \to \mathrm{Set}, \quad B' \mapsto \underline{{\mathrm{Def}}}(\mathcal{Q})(B')={\mathrm{Def}}(\mathcal{Q}_{(B, IB)})_{(B', I'B')} \end{align*}$$

forms a sheaf with respect to the $(\pi , I')$ -completely étale topology.

3.3 Period maps

Let $f \colon (A', I') \to (A, I)$ be a special nilpotent thickening of orientable and bounded ${\mathcal O}_E$ -prisms over ${\mathcal O}$ . Let $\mathcal {Q}$ be a G- $\mu $ -display over $(A, I)$ . Recall the Hodge filtration $P(\mathcal {Q})_{A/I} \to \mathcal {Q}_{A/I}$ from Section 2.5.

Definition 3.3.1. Let $\mathscr {Q}$ be a G-torsor over $ \mathrm {Spec} A'/I'$ with an isomorphism

$$\begin{align*}f^*\mathscr{Q}:=\mathscr{Q} \times_{\mathrm{Spec} A'/I'} \mathrm{Spec} A/I \overset{\sim}{\to} \mathcal{Q}_{A/I} \end{align*}$$

of G-torsors over $ \mathrm {Spec} A/I$ . A lift of the Hodge filtration $P(\mathcal {Q})_{A/I} \to \mathcal {Q}_{A/I}$ in $\mathscr {Q}$ is a pair $(\mathscr {P}, \iota )$ of a $P_\mu $ -torsor $\mathscr {P}$ over $ \mathrm {Spec} A'/I'$ and a $P_\mu $ -equivariant morphism $\iota \colon \mathscr {P} \to \mathscr {Q}$ such that the isomorphism $f^*\mathscr {Q} \overset {\sim }{\to } \mathcal {Q}_{A/I}$ restricts to an isomorphism $f^*\mathscr {P} \overset {\sim }{\to } P(\mathcal {Q})_{A/I}$ .

There is an obvious notion of isomorphism of lifts of the Hodge filtration. We note that there is at most one isomorphism between two lifts. Let

$$\begin{align*}{\mathrm{Lift}}(P(\mathcal{Q})_{A/I}, \mathscr{Q}) \end{align*}$$

be the set of isomorphism classes of lifts of the Hodge filtration $P(\mathcal {Q})_{A/I} \to \mathcal {Q}_{A/I}$ in $\mathscr {Q}$ .

Example 3.3.2. For a deformation $\mathscr {Q}$ of $\mathcal {Q}$ over $(A', I')$ , the pair

$$\begin{align*}(P(\mathscr{Q})_{A'/I'}, P(\mathscr{Q})_{A'/I'} \to \mathscr{Q}_{A'/I'}) \end{align*}$$

is a lift of the Hodge filtration $P(\mathcal {Q})_{A/I}$ in $\mathscr {Q}_{A'/I'}$ .

Definition 3.3.3 (Period map).

Let $\mathscr {Q}$ be a deformation of $\mathcal {Q}$ over $(A', I')$ . We define a map of sets

$$\begin{align*}{\mathrm{Per}}_{\mathscr{Q}} \colon {\mathrm{Def}}(\mathcal{Q})_{(A', I')} \to {\mathrm{Lift}}(P(\mathcal{Q})_{A/I}, \mathscr{Q}_{A'/I'}) \end{align*}$$

as follows. Let $\mathscr {R} \in {\mathrm {Def}} (\mathcal {Q})_{(A', I')}$ be a deformation of $\mathcal {Q}$ over $(A', I')$ . By Proposition 3.2.7, there exists a unique isomorphism $ \psi \colon \mathscr {R}_{\phi } \overset {\sim }{\to } \mathscr {Q}_{\phi } $ of deformations of the underlying G- $\phi $ -module $\mathcal {Q}_{\phi }$ . In particular, $\psi $ induces an isomorphism $\mathscr {R}_{A'/I'} \overset {\sim }{\to } \mathscr {Q}_{A'/I'}$ of G-torsors over $ \mathrm {Spec} A'/I'$ . We define the map $ {\mathrm {Per}} _{\mathscr {Q}}$ by sending $\mathscr {R}$ to the lift

$$\begin{align*}(P(\mathscr{R})_{A'/I'}, \, P(\mathscr{R})_{A'/I'} \to \mathscr{R}_{A'/I'} \overset{\sim}{\to} \mathscr{Q}_{A'/I'}) \in {\mathrm{Lift}}(P(\mathcal{Q})_{A/I}, \mathscr{Q}_{A'/I'}). \end{align*}$$

We call $ {\mathrm {Per}} _{\mathscr {Q}}$ the period map associated with the deformation $\mathscr {Q}$ .

The main result of this section is the following theorem, which can be seen as an analogue of the Grothendieck–Messing deformation theory.

Theorem 3.3.4. Let $f \colon (A', I') \to (A, I)$ be a special nilpotent thickening of orientable and bounded ${\mathcal O}_E$ -prisms over ${\mathcal O}$ . Let $\mathcal {Q}$ be a G- $\mu $ -display over $(A, I)$ , and we fix a deformation $\mathscr {Q}$ of $\mathcal {Q}$ over $(A', I')$ . If the cocharacter $\mu $ is 1-bounded, then the period map

$$\begin{align*}{\mathrm{Per}}_{\mathscr{Q}} \colon {\mathrm{Def}}(\mathcal{Q})_{(A', I')} \to {\mathrm{Lift}}(P(\mathcal{Q})_{A/I}, \mathscr{Q}_{A'/I'}) \end{align*}$$

is bijective.

Proof. By Corollary 3.2.9, we may work $(\pi , I')$ -completely étale locally on $A'$ . We may thus assume that $\mathcal {Q}=\mathcal {Q}_X$ for some $X \in G(A)_I$ . We shall prove that $ {\mathrm {Per}} _{\mathscr {Q}}$ is injective. Let $\mathscr {R}$ and $\mathscr {S}$ be two deformations of $\mathcal {Q}$ over $(A', I')$ such that $P(\mathscr {R})_{A'/I'} = P(\mathscr {S})_{A'/I'}$ in $ {\mathrm {Lift}} (P(\mathcal {Q})_{A/I}, \mathscr {Q}_{A'/I'})$ . By Proposition 3.1.4, we may further assume that

$$\begin{align*}\mathscr{Q}=(\mathcal{Q}_{X'}, 1), \quad \mathscr{R}=(\mathcal{Q}_Y, 1), \quad \text{and} \quad \mathscr{S}=(\mathcal{Q}_Z, 1) \end{align*}$$

for some $X', Y, Z \in \mathbf {Def}(X)_{(A', I')}$ . Then $\mathscr {Q}_{A'/I'}$ is naturally identified with $G_{A'/I'}$ , and the isomorphism $f^*(\mathscr {Q}_{A'/I'}) \overset {\sim }{\to } \mathcal {Q}_{A/I}$ agrees with the canonical isomorphism $f^*G_{A'/I'} \overset {\sim }{\to } G_{A/I}$ . We fix a generator $d \in I'$ . The period map $ {\mathrm {Per}} _{\mathscr {Q}}$ sends $(\mathcal {Q}_Y, 1)$ and $(\mathcal {Q}_Z, 1)$ to

$$\begin{align*}((P_{\mu})_{A'/I'}, \iota_{X^{\prime}_d(Y_d)^{-1}}) \quad \text{and} \quad ((P_{\mu})_{A'/I'}, \iota_{X^{\prime}_d(Z_d)^{-1}}), \end{align*}$$

respectively, where $\iota _{X^{\prime }_d(Y_d)^{-1}} \colon (P_{\mu })_{A'/I'} \hookrightarrow G_{A'/I'}$ is defined by $g \mapsto X^{\prime }_d(Y_d)^{-1}g$ , and similarly for $\iota _{X^{\prime }_d(Z_d)^{-1}}$ . (See (2.1) for the notation $X_d$ .) Since the two lifts are isomorphic, the image of $Z_d(Y_d)^{-1}$ in $G(A'/I')$ belongs to $P_\mu (A'/I')$ . This implies that $Z_d(Y_d)^{-1} \in G_\mu (A', I')$ by Proposition 2.3.3 since $\mu $ is 1-bounded. Then $Z_d(Y_d)^{-1}$ gives $(\mathcal {Q}_Y, 1) \overset {\sim }{\to } (\mathcal {Q}_Z, 1)$ by Remark 3.2.8.

We next prove that $ {\mathrm {Per}} _{\mathscr {Q}}$ is surjective. We may assume that $\mathscr {Q}=(\mathcal {Q}_{X'}, 1)$ as above. Let $(\mathscr {P}, \iota ) \in {\mathrm {Lift}} (P(\mathcal {Q})_{A/I}, \mathscr {Q}_{A'/I'})$ . By the same argument as in the proof of Lemma 3.1.2, we see that $\mathscr {P}$ is trivial as a $P_\mu $ -torsor. Then, with the notation as above, this lift can be identified with $ ((P_{\mu })_{A'/I'}, \iota _{x}) $ for some $x \in G(A'/I')$ whose image in $G(A/I)$ is $1$ . By Lemma 3.3.5 below, after replacing x by $x t$ for some $t \in P_\mu (A'/I')$ , we can find an element $\widetilde {x} \in G(A')$ such that $f(\widetilde {x})=1$ in $G(A)$ and the image of $\widetilde {x}$ in $G(A'/I')$ coincides with x. We then define $Y \in \mathbf {Def}(X)_{(A', I')}$ to be the unique object such that $Y_d=\widetilde {x}^{-1} X^{\prime }_d$ . By construction, the period map $ {\mathrm {Per}} _{\mathscr {Q}}$ sends $(\mathcal {Q}_Y, 1)$ to $((P_{\mu })_{A'/I'}, \iota _{x})$ . This concludes the proof of the surjectivity of $ {\mathrm {Per}} _{\mathscr {Q}}$ , and hence that of Theorem 3.3.4.

The following lemma is used in the proof of Theorem 3.3.4.

3.4 Normalized period maps

Let $f \colon (A', I') \to (A, I)$ be a special nilpotent thickening of orientable and bounded ${\mathcal O}_E$ -prisms over ${\mathcal O}$ . In this subsection, we assume that f satisfies the following assumption: