2.1 Base ring

In this subsection, we introduce our base ring $R$.

In this paper, we use the crystalline period rings developed in [Reference BrinonBri08]. For this, we consider the following class of $p$-adic rings that contains connected small $\mathcal {O}_K$-algebras.

Let $R$ be a base ring as defined in Set-up 2.3. In the rest of this subsection, we introduce basic objects attached to $R$ that we use throughout this article.

Let $\varphi \colon R_0\rightarrow R_0$ denote the lift of the Frobenius on $R_0/pR_0$ with $\varphi (T_i)=T_i^p$; this uniquely determines $\varphi$. Let $\widehat {\Omega }_{R_0}$ denote the module of continuous Kähler differentials $\varprojlim _n\Omega _{(R_0/p^nR_0)/(W/p^nW)}$. By [Reference BrinonBri08, Proposition 2.0.2], we have $\widehat {\Omega }_{R_0} = \bigoplus _{i=1}^d R_0 \cdot d\log {T_i}$.

Let $\overline {R}$ denote the union of finite $R$-subalgebras $R'$ of a fixed algebraic closure of $\operatorname {Frac}{R}$ such that $R'[p^{-1}]$ is étale over $R[p^{-1}]$. Set

\[ \mathcal{G}_{R}:= \operatorname{Gal}(\overline{R}[p^{-1}]/R[p^{-1}]). \]

Let $\mathrm {Rep}_{\mathbf {Q}_p}(\mathcal {G}_R)$ denote the category of finite-dimensional $\mathbf {Q}_p$-vector spaces with continuous $\mathcal {G}_R$-action. We call its objects $\mathbf {Q}_p$-representations of $\mathcal {G}_R$ for short. Similarly, let $\mathrm {Rep}_{\mathbf {Z}_p}(\mathcal {G}_R)$ denote the category of finite $\mathbf {Z}_p$-modules equipped with continuous $\mathcal {G}_R$-action and let $\mathrm {Rep}_{\mathbf {Z}_p}^{\mathrm {pr}}(\mathcal {G}_R)$ denote the full subcategory consisting of finite free objects.

Let $\overline {R}^\wedge$ be the $p$-adic completion of $\overline {R}$ and let $\overline {R}^{\flat }$ be its tilt $\varprojlim _\varphi \overline {R}/p\overline {R}$. Set $\mathbf {A}_{\mathrm {inf}}(\overline {R}) := W(\overline {R}^{\flat })$. The first projection $\overline {R}^{\flat }\rightarrow \overline {R}/p\overline {R}$ lifts uniquely to a surjective $W$-algebra homomorphism $\theta \colon \mathbf {A}_{\mathrm {inf}}(\overline {R})\rightarrow \overline {R}^\wedge$.

We recall a result about the Frobenius on $\mathfrak {S}$.

Notation 2.8 Note that $(p)$ (respectively, $(\pi )$) is a prime ideal of $R_0$ (respectively, $R$). We let $\mathcal {O}_{L_0}$ (respectively, $\mathcal {O}_L$) denote the $p$-adic completion of the localization $(R_0)_{(p)}$ (respectively, $R_{(\pi )}$). Then $\mathcal {O}_{L_0}$ and $\mathcal {O}_L$ are CDVRs with the same residue field $\operatorname {Frac} (R_0/(p))=\operatorname {Frac} (R/(\pi ))$. Set $L_0:= \mathcal {O}_{L_0}[p^{-1}]$ and $L:= \mathcal {O}_L[p^{-1}]$. The Frobenius $\varphi$ on $R_0$ extends to $\varphi \colon \mathcal {O}_{L_0}\rightarrow \mathcal {O}_{L_0}$. Note that $\mathcal {O}_L$ is also a base ring. When we work on $\mathcal {O}_L$ for a fixed base ring $R$, we simply write $R=\mathcal {O}_L$ by abuse of notation.

Define $\mathcal {O}_{K_{0,g}}$ to be the $p$-adic completion of $\varinjlim _{\varphi }\mathcal {O}_{L_0}$ and let $\mathcal {O}_{K_g}:= \mathcal {O}_{K_{0,g}}\otimes _{W}\mathcal {O}_K$. Set $K_{0,g}:= \mathcal {O}_{K_{0,g}}[p^{-1}]$ and $K_g:= \mathcal {O}_{K_g}[p^{-1}]$. Note that there is a unique $\varphi$-compatible isomorphism $\mathcal {O}_{K_{0,g}}\cong W(k_g)$ that reduces to the identity modulo $p$, where $k_g$ denotes $\varinjlim _{\varphi }\operatorname {Frac}(R_0/(p))$. Hence $\mathcal {O}_{K_{0,g}}$ and $\mathcal {O}_{K_g}$ are CDVRs with the same perfect residue field $k_g$. Note that the structure map $R_0\rightarrow \mathcal {O}_{K_{0,g}}$ factors through $\mathcal {O}_{L_0}\rightarrow \mathcal {O}_{K_{0,g}}=W(k_g)$.

We often deduce our statements over $\mathfrak {S}$ from those over $\mathcal {O}_{\mathcal {E}}$ and $\mathcal {O}_L$ (or $\mathcal {O}_{K_g}$) by taking certain intersections of modules (e.g. Construction 4.19 and the proof of Theorem 3.29(i)). For proofs, we need the following localization method (cf. [Reference BrinonBri08, § 3.3]): fix an algebraic closure $\overline {K_g}$ and let $\mathcal {O}_{\overline {K_g}}$ denote its ring of integers. Let $\mathcal {P}$ be the set of minimal prime ideals of $\overline {R}$ containing $p$. For each $\mathfrak {p}\in \mathcal {P}$, fix a continuous ring homomorphism $(\overline {R}_{\mathfrak {p}})^{\wedge } \rightarrow (\mathcal {O}_{\overline {K_g}})^{\wedge }$ extending $R_{(\pi )}\rightarrow \mathcal {O}_{\overline {K_g}}$, where $(\cdots )^{\wedge }$ denotes the $p$-adic completion. Taking the product over the $\mathfrak {p}$'s induces injective maps

\[ \overline{R}^{\wedge} \rightarrow \prod_{\mathfrak{p} \in \mathcal{P}} (\overline{R}_{\mathfrak{p}})^{\wedge} \rightarrow \prod_{\mathfrak{p} \in \mathcal{P}} (\mathcal{O}_{\overline{K_g}})^{\wedge} \quad\text{and}\quad \overline{R}^{\flat} \rightarrow \prod_{\mathfrak{p} \in \mathcal{P}} \mathcal{O}_{\overline{K_g}}^{\flat}. \]

In § 3, we consider the absolute prismatic site on a $p$-adic formal scheme. In the affine case, we usually make the following additional assumption.

2.2 Crystalline representations

Let $R$ be a base ring. In this subsection, we review the crystalline period ring $\mathbf {OB}_{\mathrm {cris}}(\overline {R})$ and the notion of crystalline representations of the Galois group $\mathcal {G}_R$ of $R[p^{-1}]$ developed in [Reference BrinonBri08, Chapitre 6].

Recall the surjective $W$-algebra homomorphism $\theta \colon \mathbf {A}_{\mathrm {inf}}(\overline {R}):= W(\overline {R}^{\flat })\rightarrow \overline {R}^\wedge$. Define $\mathbf {A}_{\mathrm {cris}}(\overline {R})$ to be the $p$-adic completion of the divided power envelope of $\mathbf {A}_{\mathrm {inf}}(\overline {R})$ with respect to $\operatorname {Ker}\theta$. Choose a non-trivial compatible system of $p$-power roots of unity: $\varepsilon _n\in \overline {R}$ with $\varepsilon _0=1$, $\varepsilon _1\neq 1$, and $\varepsilon _n=\varepsilon _{n+1}^p$. Set $\varepsilon = (\varepsilon _n)_n\in \overline {R}^{\flat }$ and $t:= \log [\varepsilon ]\in \mathbf {A}_{\mathrm {cris}}(\overline {R})$. Define $\mathbf {B}_{\mathrm {cris}}(\overline {R}):= \mathbf {A}_{\mathrm {cris}}(\overline {R})[p^{-1},t^{-1}]$.

Extend the map $\theta$ to $\theta _{R_0}\colon R_0\otimes _W \mathbf {A}_{\mathrm {inf}}(\overline {R})\rightarrow \overline {R}^{\wedge }$. Define $\mathbf {OA}_{\mathrm {cris}}(\overline {R})$ to be the $p$-adic completion of the divided power envelope of $R_0\otimes _W \mathbf {A}_{\mathrm {inf}}(\overline {R})$ with respect to $\operatorname {Ker} \theta _{R_0}$. Define $\mathbf {OB}_{\mathrm {cris}}(\overline {R}):= \mathbf {OA}_{\mathrm {cris}}(\overline {R})[p^{-1},t^{-1}]$.

The crystalline period ring $\mathbf {OB}_{\mathrm {cris}}(\overline {R})$ has a natural $\mathcal {G}_R$-action and a Frobenius endomorphism $\varphi$ extending those on $R_0\otimes _W \mathbf {A}_{\mathrm {inf}}(\overline {R})$, and there is a natural $\mathbf {B}_{\mathrm {cris}}(\overline {R})$-linear integrable connection $\nabla \colon \mathbf {OB}_{\mathrm {cris}}(\overline {R})\rightarrow \mathbf {OB}_{\mathrm {cris}}(\overline {R})\otimes _{R_0}\widehat {\Omega }_{R_0}$. Moreover, $R\otimes _{R_0}\mathbf {OB}_{\mathrm {cris}}(\overline {R})$ is equipped with a filtration by $R[p^{-1}]$-modules, which is compatible with the natural PD-filtration on $\mathbf {A}_{\mathrm {cris}}(\overline {R})$. See [Reference BrinonBri08, Chapitre 6] for the detail of these structures.

The following result on the crystalline period ring is used later.

Lemma 2.11 [Reference BrinonBri08, Proposition 6.1.5]

Choose a compatible system $(T_{i,n})$ of $p$-power roots of $T_i$ in $\overline {R}$ with $T_{i,0}=T_i$, and let $T_i^\flat \in \overline {R}^\flat$ denote the corresponding element. The map $X_i \mapsto T_i\otimes 1-1\otimes [T_i^\flat ]$ induces an $\mathbf {A}_{\mathrm {cris}}(\overline {R})$-linear isomorphism

\[ \mathbf{A}_{\mathrm{cris}}(\overline{R})\{X_1, \ldots, X_d\} \cong \mathbf{OA}_{\mathrm{cris}}(\overline{R}), \]

where the former ring denotes the $p$-adically completed divided power polynomial with variables $X_i$ and coefficients in $\mathbf {A}_{\mathrm {cris}}(\overline {R})$.

Let us recall the definition of crystalline representations.

Definition 2.12 For $V\in \mathrm {Rep}_{\mathbf {Q}_p}(\mathcal {G}_R)$, set

\[ D_{\mathrm{cris}}(V):= (\mathbf{OB}_{\mathrm{cris}}(\overline{R})\otimes_{\mathbf{Q}_p}V)^{\mathcal{G}_R}\quad \text{and}\quad D_{\mathrm{cris}}^\vee(V):= \mathrm{Hom}_{\mathcal{G}_R}(V,\mathbf{OB}_{\mathrm{cris}}(\overline{R})). \]

Then $D_{\mathrm {cris}}(V)$ is a finite projective $R_0[p^{-1}]$-module of rank at most $\dim _{\mathbf {Q}_p}V$ equipped with a natural $\varphi$ and $\nabla$ structure induced from $\mathbf {OB}_{\mathrm {cris}}(\overline {R})$, and $R\otimes _{R_0}D_{\mathrm {cris}}(V)$ has a filtration induced from $R\otimes _{R_0}\mathbf {OB}_{\mathrm {cris}}(\overline {R})$. The natural map

\[ \alpha_{\mathrm{cris}}(V)\colon \mathbf{OB}_{\mathrm{cris}}(\overline{R})\otimes_{R_0[p^{-1}]}D_{\mathrm{cris}}(V) \rightarrow \mathbf{OB}_{\mathrm{cris}}(\overline{R})\otimes_{\mathbf{Q}_p}V \]

is injective by [Reference BrinonBri08, Proposition 8.2.6]. We say that $V$ is $R_0$-crystalline if $\alpha _{\mathrm {cris}}(V)$ is an isomorphism. By [Reference BrinonBri08, Proposition 8.3.5], this notion depends only on $R$, not on $R_0$. Hence, we simply say that $V$ is crystalline from now on.

By [Reference BrinonBri08, Théorème 8.4.2], $V$ is crystalline if and only if $V^{\vee }$ is crystalline. Note also that $D_{\mathrm {cris}}^\vee (V)=D_{\mathrm {cris}}(V^\vee )=\mathrm {Hom}_{R_0[p^{-1}]}(D_{\mathrm {cris}}(V),R_0[p^{-1}])$. We mainly use $D_{\mathrm {cris}}^\vee (V)$ in this paper.

A finite free $\mathbf {Z}_p$-representation $T\in \mathrm {Rep}_{\mathbf {Z}_p}^{\mathrm {pr}}(\mathcal {G}_R)$ is called crystalline if the associated $\mathbf {Q}_p$-representation $T\otimes _{\mathbf {Z}_p}\mathbf {Q}_p$ is crystalline.

Finally, let us explain the functoriality. Let $R'=R_0'\otimes _{W}\mathcal {O}_K$ be another base ring and assume that there exists a $\varphi$-equivariant ring homomorphism $g\colon R_0\rightarrow R_0'$ that extends to $g\colon \overline {R}\rightarrow \overline {R'}$. By the change of paths for étale fundamental groups, $g$ induces a continuous group homomorphism $\mathcal {G}_{R'}\rightarrow \mathcal {G}_R$ and, thus, a natural $\otimes$-functor $\mathrm {Rep}_{\mathbf {Q}_p}(\mathcal {G}_R)\rightarrow \mathrm {Rep}_{\mathbf {Q}_p}(\mathcal {G}_{R'})$. The map $g$ also induces a ring homomorphism $\mathbf {OB}_{\mathrm {cris}}(\overline {R})\rightarrow \mathbf {OB}_{\mathrm {cris}}(\overline {R'})$, and the latter is compatible with Frobenii and Galois actions.

Lemma 2.13 With the notation as previously, for $V\in \mathrm {Rep}_{\mathbf {Q}_p}(\mathcal {G}_R)$, the map $R_0'[p^{-1}]\otimes _{R_0[p^{-1}]}\mathbf {OB}_\mathrm {cris}(\overline {R})\otimes _{\mathbf {Q}_p}V\rightarrow \mathbf {OB}_\mathrm {cris}(\overline {R'})\otimes _{\mathbf {Q}_p}V$ induces a $\varphi$-equivariant morphism of $R_0'[p^{-1}]$-modules

(2.1)\begin{equation} R_0'[p^{-1}]\otimes_{R_0[p^{-1}]}D_{\mathrm{cris}}(V)\rightarrow D_{\mathrm{cris}}(V|_{\mathcal{G}_{R'}}). \end{equation}

Moreover, if $V$ is crystalline, then $V|_{\mathcal {G}_{R'}}$ is crystalline and the above map is an isomorphism.

Proof. The first assertion is obvious. Now assume that $V$ is crystalline. Consider the composite of $\mathbf {OB}_\mathrm {cris}(\overline {R'})$-linear maps

Observe that $\alpha$ is the base change of $\alpha _\mathrm {cris}(V)$ along the map $\mathbf {OB}_\mathrm {cris}(\overline {R})\rightarrow \mathbf {OB}_\mathrm {cris}(\overline {R'})$. Since $V$ is crystalline, $\alpha$ is an isomorphism. Moreover, the second map $\alpha _\mathrm {cris}(V|_{\mathcal {G}_{R'}}\!)$ in $\alpha$ is injective. Hence, $\alpha _\mathrm {cris}(V|_{\mathcal {G}_{R'}})$ is an isomorphism and, thus, $V|_{\mathcal {G}_{R'}}$ is crystalline. We also see that the first map in $\alpha$ is an isomorphism. Since the map $R_0'[p^{-1}]\rightarrow \mathbf {OB}_\mathrm {cris}(\overline {R'})$ is faithfully flat by [Reference BrinonBri08, Théorème 6.3.8], the morphism (2.1) is an isomorphism.

2.3 Étale $\varphi$-modules

The classical theory of étale $\varphi$-modules and Galois representations is generalized to our relative setting in [Reference KimKim15]. We briefly review some necessary facts discussed in [Reference KimKim15] and [Reference Liu and MoonLM20]. Recall Notation 2.6: $\mathfrak {S}=\mathfrak {S}_R := R_0[\![u]\!]$ equipped with the Frobenius given by $\varphi (u) = u^p$; $\mathcal {O}_{\mathcal {E}}$ is the $p$-adic completion of $\mathfrak {S}[u^{-1}]$, equipped with the Frobenius $\varphi$ extending that on $\mathfrak {S}$.

We use étale $(\varphi, \mathcal {O}_{\mathcal {E}})$-modules to study certain Galois representations as follows. We refer the reader to [Reference ScholzeSch12] for definitions and facts on perfectoid algebras. Recall that $\pi$ denotes a uniformizer in $\mathcal {O}_K$. For integers $n \geq 0$, compatibly choose $\pi _n \in \overline {K}$ such that $\pi _0 = \pi$ and $\pi _{n+1}^p = \pi _n$, and let $K_{\infty }$ be the $p$-adic completion of $\bigcup _{n \geq 0} K(\pi _n)$. Then $K_{\infty }$ is a perfectoid field, and $(\overline {R}^\wedge [p^{-1}], \overline {R}^\wedge )$ is a perfectoid affinoid $K_{\infty }$-algebra. Let $K_{\infty }^\flat$ denote the tilt of $K_{\infty }$, and set $\pi ^\flat := (\pi _n) \in K_{\infty }^\flat$.

Let $E_{R_\infty }^+ = \mathfrak {S}/p\mathfrak {S}$, and let $\tilde {E}_{R_\infty }^+$ be the $u$-adic completion of $\varinjlim _{\varphi }E_{R_\infty }^+$. Let $E_{R_\infty } = E_{R_\infty }^+[u^{-1}]$ and $\tilde {E}_{R_\infty } = \tilde {E}_{R_\infty }^+[u^{-1}]$. By [Reference ScholzeSch12, Proposition 5.9], $(\tilde {E}_{R_\infty }, \tilde {E}_{R_\infty }^+)$ is a perfectoid affinoid $K_{\infty }^\flat$-algebra, and we have a natural injective map $(\tilde {E}_{R_\infty }, \tilde {E}_{R_\infty }^+) \hookrightarrow (\overline {R}^\flat [(\pi ^\flat )^{-1}], \overline {R}^\flat )$ given by $u \mapsto \pi ^\flat$.

Consider

(2.2)\begin{equation} \tilde{R}_{\infty} := W(\tilde{E}_{R_\infty}^+)\otimes_{W(K_{\infty}^{\flat \circ}), \theta} \mathcal{O}_{K_{\infty}}. \end{equation}

By [Reference ScholzeSch12, Remark 5.19], $(\tilde {R}_\infty [p^{-1}], \tilde {R}_\infty )$ is a perfectoid affinoid $K_{\infty }$-algebra whose tilt is $(\tilde {E}_{R_\infty }, \tilde {E}_{R_\infty }^+)$. Furthermore, we have a natural injective map $(\tilde {R}_\infty [p^{-1}], \tilde {R}_\infty ) \hookrightarrow (\overline {R}^\wedge [p^{-1}], \overline {R}^\wedge )$ whose tilt is $(\tilde {E}_{R_\infty }, \tilde {E}_{R_\infty }^+) \hookrightarrow (\overline {R}^\flat [(\pi ^{\flat })^{-1}], \overline {R}^\flat )$. If we write $\mathcal {G}_{\tilde {R}_\infty }$ for $\pi _1^{{\unicode{x00E9}}{\text t}}(\operatorname {Spec}\tilde {R}_\infty [p^{-1}])$, we then have a continuous map of Galois groups $\mathcal {G}_{\tilde {R}_\infty }\! \rightarrow \mathcal {G}_R$, which is a closed embedding by [Reference Gabber and RameroGR03, Proposition 5.4.54]. By [Reference ScholzeSch12, Theorem 7.12], we can canonically identify $\overline {R}^\flat [(\pi ^{\flat })^{-1}]$ with the $\pi ^\flat$-adic completion of the affine ring of a universal pro-étale covering of $\operatorname {Spec}\tilde {E}_{R_\infty }$. Let $\mathcal {G}_{\tilde {E}_{R_\infty }}$ be the Galois group corresponding to the universal pro-étale covering. Then we have a canonical isomorphism $\mathcal {G}_{\tilde {E}_{R_\infty }} \!\cong \mathcal {G}_{\tilde {R}_\infty }$.

There exists a unique $W(k)$-linear map $R_0 \rightarrow W(\overline {R}^{\flat })$ which maps $T_i$ to $[T_i^\flat ]$ and is compatible with Frobenii (see Lemma 2.11 for the definition of $[T_i^\flat ]$). This induces a $\varphi$-equivariant embedding $\mathfrak {S} \rightarrow W(\overline {R}^{\flat })$ given by $u \mapsto [\pi ^\flat ]$, which further extends to an embedding $\mathcal {O}_{\mathcal {E}} \rightarrow W(\overline {R}^\flat [(\pi ^{\flat })^{-1}])$. Let $\mathcal {O}_\mathcal {E}^{\mathrm {ur}}$ be the union of finite étale $\mathcal {O}_\mathcal {E}$-subalgebras of $W(\overline {R}^\flat [(\pi ^{\flat })^{-1}])$, and let $\widehat {\mathcal {O}}_{\mathcal {E}}^{\mathrm {ur}}$ be its $p$-adic completion. We also define $\widehat {\mathfrak {S}}^{\mathrm {ur}} := \widehat {\mathcal {O}}_{\mathcal {E}}^{\mathrm {ur}} \cap W(\overline {R}^\flat ) \subset W(\overline {R}^\flat [\pi ^\flat ]^{-1})$. We note that the definitions of these rings in [Reference KimKim15, p. 8201] are incorrect but that the results concerning these rings in [Reference KimKim15] hold with the correct definitions: since $(\mathcal {O}_{\mathcal {E}}, (p))$ is a henselian pair, $\mathcal {O}_\mathcal {E}^{\mathrm {ur}}/(p)=\widehat {\mathcal {O}}_{\mathcal {E}}^{\mathrm {ur}}/(p)$ is the union of finite étale $E_{R_\infty }$-subalgebras of $\overline {R}^\flat [(\pi ^{\flat })^{-1}]$. In particular, we have $\mathrm {Aut}_{\mathcal {O}_\mathcal {E}}(\mathcal {O}_\mathcal {E}^{\mathrm {ur}}) \cong \mathcal {G}_{E_{R_\infty }} \!:= \pi _1^{{\unicode{x00E9}}{\text t}}(\operatorname {Spec}E_{R_\infty })$. By [Reference Gabber and RameroGR03, Proposition 5.4.54] and [Reference ScholzeSch12, Lemma 7.5], we have $\mathcal {G}_{E_{R_\infty }} \cong \mathcal {G}_{\tilde {E}_{R_\infty }} \!\cong \mathcal {G}_{\tilde {R}_\infty }$. This induces $\mathcal {G}_{\tilde {R}_\infty }$-action on $\widehat {\mathcal {O}}_{\mathcal {E}}^{\mathrm {ur}}$. The following is proved in [Reference KimKim15].

Let $\mathrm {Rep}_{\mathbf {Z}_p}(\mathcal {G}_{\tilde {R}_\infty }\!)$ denote the category of finite $\mathbf {Z}_p$-modules equipped with continuous $\mathcal {G}_{\tilde {R}_\infty }$-action, and let $\mathrm {Rep}_{\mathbf {Z}_p}^{\mathrm {pr}}(\mathcal {G}_{\tilde {R}_\infty }\!)$ and $\mathrm {Rep}_{\mathbf {Z}_p}^{\mathrm {tor}}(\mathcal {G}_{\tilde {R}_\infty }\!)$, respectively, denote the full subcategories of free and torsion objects. For $\mathcal {M} \in \mathrm {Mod}_{\mathcal {O}_\mathcal {E}}$ and $T \in \mathrm {Rep}_{\mathbf {Z}_p}(\mathcal {G}_{\tilde {R}_\infty })$, define

\[ T(\mathcal{M}) := (\widehat{\mathcal{O}}_{\mathcal{E}}^{\mathrm{ur}}\otimes_{\mathcal{O}_\mathcal{E}}\mathcal{M})^{\varphi = 1} \quad\text{and}\quad \mathcal{M}(T) := (\widehat{\mathcal{O}}_{\mathcal{E}}^{\mathrm{ur}}\otimes_{\mathbf{Z}_p}T)^{\mathcal{G}_{\tilde{R}_\infty}}. \]

For a torsion étale $\varphi$-module $\mathcal {M} \in \mathrm {Mod}_{\mathcal {O}_{\mathcal {E}}}^{\mathrm {tor}}$, we define its length to be the length of $(\mathcal {O}_{\mathcal {E}})_{(p)}\otimes _{\mathcal {O}_{\mathcal {E}}}\mathcal {M}$ as an $(\mathcal {O}_{\mathcal {E}})_{(p)}$-module. The following equivalence is proved in [Reference KimKim15] (see also [Reference Liu and MoonLM20, Proposition 2.5]).

For $\mathcal {M} \in \mathrm {Mod}_{\mathcal {O}_\mathcal {E}}^{\mathrm {pr}}$ and $T \in \mathrm {Rep}_{\mathbf {Z}_p}^{\mathrm {pr}}(\mathcal {G}_{\tilde {R}_\infty })$, we can consider the contravariant functors

\[ T^{\vee}(\mathcal{M}) := \mathrm{Hom}_{\mathcal{O}_\mathcal{E}, \varphi}(\mathcal{M}, \widehat{\mathcal{O}}_{\mathcal{E}}^{\mathrm{ur}}) \quad\text{and}\quad \mathcal{M}^{\vee}(T) := \mathrm{Hom}_{\mathcal{G}_{\tilde{R}_\infty}}(T, \widehat{\mathcal{O}}_{\mathcal{E}}^{\mathrm{ur}}). \]

We have natural isomorphisms

\[ T^{\vee}(\mathcal{M}) \cong T(\mathcal{M}^{\vee}) \quad\text{and}\quad \mathcal{M}^{\vee}(T) \cong \mathcal{M}(T^{\vee}), \]

and these contravariant functors give equivalences of categories between $\mathrm {Mod}_{\mathcal {O}_{\mathcal {E}}}^{\mathrm {pr}}$ and $\mathrm {Rep}_{\mathbf {Z}_p}^{\mathrm {pr}}(\mathcal {G}_{\tilde {R}_\infty })$ by Proposition 2.16.

We now explain certain functoriality of above constructions. Let $R_0'$ be another base ring over $W(k)\langle T_1^{\pm 1}, \ldots, T_d^{\pm 1}\rangle$ as in § 2.1 equipped with Frobenius, and suppose a $\varphi$-equivariant map $R_0 \to R_0'$ of $W(k)\langle T_1^{\pm 1}, \ldots, T_d^{\pm 1}\rangle$-algebras is given. Consider the induced $\mathcal {O}_K$-linear extension $R = R_0\otimes _{W(k)}\mathcal {O}_K \rightarrow R' := R_0'\otimes _{W(k)}\mathcal {O}_K$. By fixing an algebraic closure of $\mathrm {Frac}(R')$, we have a map $\overline {R} \rightarrow \overline {R'}$, and this induces $\tilde {R}_{\infty } \rightarrow \tilde {R'}_{\infty }$ by the constructions given in (2.2). Hence, the corresponding map of Galois groups $\mathcal {G}_{R'} \rightarrow \mathcal {G}_R$ restricts to $\mathcal {G}_{\tilde {R'}_\infty } \rightarrow \mathcal {G}_{\tilde {R}_\infty }$. Let $\mathfrak {S}_{R'} = R_0'[\![u]\!]$ and let $\mathcal {O}_{\mathcal {E}, R'}$ be the $p$-adic completion of $\mathfrak {S}_{R'}[u^{-1}]$. Let $\mathcal {M}_{R'}(\cdot )$ be the functor for the base ring $R'$ constructed similarly as previously. If $T \in \mathrm {Rep}_{\mathbf {Z}_p}^{\mathrm {pr}}(\mathcal {G}_{\tilde {R}_\infty })$, then $T$ can be also considered as a $\mathcal {G}_{\tilde {R'}_\infty }$-representation via the map $\mathcal {G}_{\tilde {R'}_\infty } \rightarrow \mathcal {G}_{\tilde {R}_\infty }$. We claim that there is a natural isomorphism $\mathcal {O}_{\mathcal {E}, R'}\otimes _{\mathcal {O}_\mathcal {E}}\mathcal {M}(T) \xrightarrow {\cong } \mathcal {M}_{R'}(T)$ of étale $(\varphi, \mathcal {O}_{\mathcal {E}, R'})$-modules. Indeed, the $W(k)\langle T_1^{\pm 1}, \ldots, T_d^{\pm 1}\rangle$-algebra homomorphism $\mathcal {O}_\mathcal {E}\rightarrow \mathcal {O}_{\mathcal {E}, R'}$ extends to a map $\widehat {\mathcal {O}}_{\mathcal {E}}^{\mathrm {ur}}\rightarrow \widehat {\mathcal {O}}_{\mathcal {E},R'}^{\mathrm {ur}}$, which defines the desired map $\mathcal {O}_{\mathcal {E}, R'}\otimes _{\mathcal {O}_\mathcal {E}}\mathcal {M}(T) \rightarrow \mathcal {M}_{R'}(T)$. To see that this is an isomorphism, observe

\begin{align*} \widehat{\mathcal{O}}_{\mathcal{E}, R'}^{\mathrm{ur}} \otimes_{ \mathcal{O}_{\mathcal{E} , R'}} (\mathcal{O}_{\mathcal{E}, R'}\otimes_{\mathcal{O}_{\mathcal{E}}}\mathcal{M}(T)) &= \widehat{\mathcal{O}}_{\mathcal{E}, R'}^{\mathrm{ur}} \otimes_{ \widehat{\mathcal{O}}^\mathrm{ur} _{\mathcal{E}}} (\widehat{\mathcal{O}}^\mathrm{ur}_{\mathcal{E}}\otimes_{\mathcal{O}_{\mathcal{E}}}\mathcal{M}(T))\\ & \xrightarrow{\cong} \widehat{\mathcal{O}}_{\mathcal{E}, R'}^{\mathrm{ur}} \otimes_{ \widehat{\mathcal{O}}^\mathrm{ur} _{\mathcal{E}}} (\widehat{\mathcal{O}}^\mathrm{ur}_{\mathcal{E}}\otimes_{\mathbf{Z}_p} T)= \widehat{\mathcal{O}}_{\mathcal{E}, R'}^{\mathrm{ur}}\otimes_{\mathbf{Z}_p} T. \end{align*}

Hence, we conclude that $\mathcal {O}_{\mathcal {E}, R'}\otimes _{\mathcal {O}_\mathcal {E}}\mathcal {M}(T) \rightarrow \mathcal {M}_{R'}(T)$ is an isomorphism since it is so after the base change along the faithfully flat map $\mathcal {O}_{\mathcal {E}, R'}\rightarrow \widehat {\mathcal {O}}_{\mathcal {E}, R'}^{\mathrm {ur}}$.

We use this functoriality for the maps of base rings $R \rightarrow \mathcal {O}_L$ and $R \rightarrow \mathcal {O}_{K_g}$ as in Notation 2.8 in later sections. For $\mathcal {O}_L$ and $\mathcal {O}_{K_g}$, the relevant rings will be denoted by $\widehat {\mathcal {O}}_{\mathcal {E}, L}^{\mathrm {ur}}$, $\widehat {\mathfrak {S}}_L^{\mathrm {ur}}$, $\widehat {\mathcal {O}}_{\mathcal {E}, g}^{\mathrm {ur}}$, and $\widehat {\mathfrak {S}}_g^{\mathrm {ur}}$.

3.1 The absolute prismatic site

We first recall the definition of the absolute prismatic site from [Reference Bhatt and ScholzeBS22] and [Reference Bhatt and ScholzeBS23]. Let ${\mathfrak X}$ be a smooth $p$-adic formal scheme over $\mathcal {O}_K$ (or a CDVR of mixed characteristic $(0,p)$ such as $\mathcal {O}_L$ in Notation 2.8).

Let us explain the functoriality of the prismatic topoi. Let $f\colon {\mathfrak Y}\rightarrow {\mathfrak X}$ be a morphism of smooth $p$-adic formal schemes over $\mathcal {O}_K$. Then $f$ induces a cocontinuous functor

\[ {\mathfrak Y}_{{\mathbb{\Delta}}} \rightarrow {\mathfrak X}_{{\mathbb{\Delta}}}, \quad ((A,I),\iota\colon \operatorname{Spf} A/I\rightarrow {\mathfrak Y})\mapsto ((A,I),f\circ\iota\colon \operatorname{Spf} A/I\rightarrow {\mathfrak X}). \]

Hence, we have a morphism of topoi

\[ f_{{\mathbb{\Delta}}}=(f_{{\mathbb{\Delta}}}^{-1},f_{{{\mathbb{\Delta}}},\ast})\colon \mathrm{Shv}({\mathfrak Y}_{{\mathbb{\Delta}}})\rightarrow \mathrm{Shv}({\mathfrak X}_{{\mathbb{\Delta}}}). \]

Observe that if $f$ is an open immersion, then $\mathrm {Shv}({\mathfrak Y}_{ {\mathbb {\Delta }}})$ is an open subtopos of $\mathrm {Shv}({\mathfrak X}_{ {\mathbb {\Delta }}})$ by $f_{ {\mathbb {\Delta }}}$.

Let $R$ be small over $\mathcal {O}_K$ or $R=\mathcal {O}_L$ (Assumption 2.9). We now explain several objects of $R_{ {\mathbb {\Delta }}}$ that we use later.

Example 3.4 The Breuil–Kisin prism and its self-products

Consider the pair $(\mathfrak {S},(E))$ where $\mathfrak {S} = R_0[\![u]\!]$ and $E = E(u)$ is the Eisenstein polynomial for $\pi \in \mathcal {O}_K$ over $W$ as before. Equip $\mathfrak {S}$ with the $\delta$-structure defined by extending the fixed Frobenius $\varphi$ on $R_0$ to $\mathfrak {S}$ via $\varphi (u)=u^p$. Then $(\mathfrak {S},(E)) \in R_{ {\mathbb {\Delta }}}$, where the structure map $R \rightarrow \mathfrak {S}/(E)$ is given by the natural isomorphism $R \cong \mathfrak {S}/(E)$. We call $({\mathfrak S},(E))$ the Breuil–Kisin prism attached to $\pi$ and $R_0$.

The self-product of $(\mathfrak {S},(E))$ exists in $R_{{ {\mathbb {\Delta }}}}$ as follows. Consider the $p$-adically complete tensor-product $\mathfrak {S}\widehat {\otimes }_{\mathbf {Z}_p} \mathfrak {S}$ equipped with the induced $\otimes$-product Frobenius. We have a projection $d\colon \mathfrak {S}\widehat {\otimes }_{\mathbf {Z}_p} \mathfrak {S} \rightarrow R$ given by the composite of the multiplication $\mathfrak {S}\widehat {\otimes }_{\mathbf {Z}_p} \mathfrak {S} \rightarrow \mathfrak {S}$ and the natural projection $\mathfrak {S} \rightarrow \mathfrak {S}/(E)\cong R$. Let $J$ be the kernel of $d$, and let

\[ \mathfrak{S}^{(1)} := (\mathfrak{S}\widehat{\otimes}_{\mathbf{Z}_p} \mathfrak{S})\bigg\{\frac{J}{E}\bigg\}_\delta^{\wedge}. \]

Here $\mathfrak {S}\widehat {\otimes }_{\mathbf {Z}_p} \mathfrak {S}$ is regarded as an $\mathfrak {S}$-algebra via $a \mapsto a\otimes 1$, and $\{\cdot \}_\delta ^{\wedge }$ means adjoining elements in the category of derived $(p, E)$-complete simplicial $\delta$-$\mathfrak {S}$-algebras. Note that the $E$ in $\{{J}/{E}\}_\delta ^{\wedge }$ denotes $E\otimes 1$ but using $1\otimes E$ instead also gives the same $\mathfrak {S}^{(1)}$ (see [Reference Bhatt and ScholzeBS23, Construction 7.13]). By [Reference Bhatt and ScholzeBS21, Corollary 3.14], $(\mathfrak {S}^{(1)}, (E))$ is a $(p,E)$-completely flat prism over $(\mathfrak {S},(E))$. Furthermore, $(\mathfrak {S}^{(1)},(E))$ is bounded by [Reference Bhatt and ScholzeBS22, Lemma 3.7 (2)], so $(\mathfrak {S}^{(1)},(E)) \in R_{{ {\mathbb {\Delta }}}}$. Let $(B, I) \in R_{{ {\mathbb {\Delta }}}}$. If we are given maps $f_1, f_2\colon (\mathfrak {S}, (E)) \rightarrow (B, I)$ such that two maps $R \cong \mathfrak {S}/E \rightarrow B/I$ induced by $f_1$ and $f_2$ agree, then we have a natural induced map $f_1\otimes f_2\colon \mathfrak {S}\widehat {\otimes }_{\mathbf {Z}_p} \mathfrak {S} \rightarrow B$ of $\delta$-rings, and $(f_1\otimes f_2)(J) \subset I$. Thus, by the universal property of prismatic envelope ([Reference Bhatt and ScholzeBS21, Corollary 3.14]), we obtain a map $(\mathfrak {S}^{(1)},(E)) \rightarrow (B, I)$ in $R_{{ {\mathbb {\Delta }}}}$ uniquely determined by $f_1$, $f_2$. Thus, $(\mathfrak {S}^{(1)},(E))$ is the self-product of $(\mathfrak {S},(E))$ in $R_{{ {\mathbb {\Delta }}}}$. Similarly, the self-triple-product $(\mathfrak {S}^{(2)},(E))$ of $(\mathfrak {S},(E))$ exists in $R_{{ {\mathbb {\Delta }}}}$. Write $p_1$, $p_2$ (respectively, $q_1$, $q_2$, $q_3$) for the maps from $(\mathfrak {S},(E))$ to $(\mathfrak {S}^{(1)},(E))$ (respectively, to $(\mathfrak {S}^{(2)},(E))$).

A little more explicit description is given in [Reference Du and LiuDL23, § 4.1] as follows. Recall that $R_0$ is a $W\langle T_1^{\pm 1},\ldots,T_d^{\pm 1}\rangle$-algebra. Let $B^{\widehat {\otimes }[1]}$ denote the completion of ${\mathfrak S}\otimes _{\mathbf {Z}_p}{\mathfrak S}$ with respect to the ideal $\operatorname {Ker}({\mathfrak S}\otimes _{\mathbf {Z}_p}{\mathfrak S}\rightarrow {\mathfrak S})$. We have two natural maps $p_1,p_2\colon {\mathfrak S}\rightarrow B^{\widehat {\otimes }[1]}$ and regard $B^{\widehat {\otimes }[1]}$ as an ${\mathfrak S}$-algebra via $p_1$. If we set $s_j:= p_2(T_j)$ and $y:= p_2(u)$, then

\[ {\mathfrak S}^{\widehat{\otimes}[1]} := \mathfrak{S} [\![y-u, s_1 - T_1, \ldots , s_d - T_d]\!] \]

can be naturally considered as an $\mathfrak {S}$-subalgebra of $B^{\widehat {\otimes }[1]}$. Similarly, let $B^{\widehat {\otimes }[2]}$ denote the completion of ${\mathfrak S}\otimes _{\mathbf {Z}_p}{\mathfrak S}\otimes _{\mathbf {Z}_p}{\mathfrak S}$ with respect to the ideal $\operatorname {Ker}({\mathfrak S}\otimes _{\mathbf {Z}_p}{\mathfrak S}\otimes _{\mathbf {Z}_p}{\mathfrak S}\rightarrow {\mathfrak S})$. We have maps $q_1,q_2,q_3\colon {\mathfrak S}\rightarrow B^{\widehat {\otimes }[2]}$. Via $q_1$, we can naturally consider

\[ {\mathfrak S}^{\widehat{\otimes}[2]} := \mathfrak{S} [\![y-u, w-u,\{s_j - T_j,r_j-T_j\}_{j=1,\ldots,d}]\!] \]

as an ${\mathfrak S}$-subalgebra of $B^{\widehat {\otimes }[2]}$, where $s_j:= q_2(T_i), r_j:= q_3(T_j), y:= q_2(u)$ and $w:= q_3(u)$. Let

\begin{align*} J^{(1)}&= (E, y-u , \{s_j- T_j \}_{j = 1, \ldots, d} )\subset \mathfrak{S}^{\widehat\otimes [1]} \end{align*}

and

\begin{align*} J^{(2)}&= (E, y-u , w-u, \{s_j-T_j, r_j - T_j\}_{j = 1, \ldots, d}) \subset \mathfrak{S}^{\widehat\otimes [2]}. \end{align*}

For $i=1,2$, ${\mathfrak S}^{\widehat {\otimes }[i]}$ is naturally a $(p,E)$-completed $\delta$-${\mathfrak S}$-algebra and $\mathfrak {S}^{(i)}\cong \mathfrak {S}^{\widehat \otimes [i]}\{{J ^{(i)}}/{E}\}_\delta ^{\wedge }$.

Corollary 3.6 We have that $\mathfrak {S}^{(1)}$ is $p$-torsion free and $E$-torsion free. Furthermore,

\[ \mathfrak{S}^{(1)}[p^{-1}] \cap \mathfrak{S}^{(1)}[E^{-1}] = \mathfrak{S}^{(1)}, \]

and $\mathfrak {S}^{(1)}[E^{-1}]$ is $p$-adically separated.

Proof. Since $\mathfrak {S}$ is torsion free and $p_1\colon \mathfrak {S} \rightarrow \mathfrak {S}^{(1)}$ is classically flat, $\mathfrak {S}^{(1)}$ is $p$-torsion free and $E$-torsion free. We deduce by Lemma 3.1(i) and $\mathfrak {S}[p^{-1}] \cap \mathfrak {S}[E^{-1}] = \mathfrak {S}$ that

\[ \mathfrak{S}^{(1)}[p^{-1}] \cap \mathfrak{S}^{(1)}[E^{-1}] = (\mathfrak{S}^{(1)}\otimes_{p_1, \mathfrak{S}} \mathfrak{S}[p^{-1}]) \cap (\mathfrak{S}^{(1)}\otimes_{p_1, \mathfrak{S}}\mathfrak{S}[E^{-1}]) = \mathfrak{S}^{(1)}. \]

Since $\mathfrak {S}^{(1)}$ is $p$-adically complete, this also implies that $\mathfrak {S}^{(1)}[E^{-1}]$ is $p$-adically separated.

Example 3.8 The $\mathbf {OA}_\mathrm {cris}$-prism and its Frobenius twists

Consider the surjective map $\theta _{R_0}\colon \mathbf {OA}_{\mathrm {cris}}(\overline {R}) \rightarrow \overline {R}^\wedge$. The map

\[ \varphi\colon \mathbf{OA}_{\mathrm{cris}}(\overline{R})/(p) \rightarrow \mathbf{OA}_{\mathrm{cris}}(\overline{R})/(p) \]

factors through

\[ \mathbf{OA}_{\mathrm{cris}}(\overline{R})/(p) \rightarrow \mathbf{OA}_{\mathrm{cris}}(\overline{R})/((p)+\mathrm{ker}(\theta_{R_0})) \cong \overline{R}/(p) \xrightarrow{h} \mathbf{OA}_{\mathrm{cris}}(\overline{R})/(p). \]

The pair $(\mathbf {OA}_{\mathrm {cris}}(\overline {R}), (p))$ defines a prism in $R_{ {\mathbb {\Delta }}}$, where the structure map $R \rightarrow \mathbf {OA}_{\mathrm {cris}}(\overline {R})/(p)$ is given by the composite of $R\rightarrow \overline {R}/(p)$ and $h$ (defined in the above factorization). Consider the composite $\mathbf {A}_{\mathrm {inf}}(\overline {R}) \xrightarrow {\varphi } \mathbf {A}_{\mathrm {inf}}(\overline {R}) \rightarrow \mathbf {OA}_{\mathrm {cris}}(\overline {R})$, where the second map is the natural inclusion. This induces a map of prisms $(\mathbf {A}_{\mathrm {inf}}(\overline {R}), (\xi )) \xrightarrow {\varphi } (\mathbf {OA}_{\mathrm {cris}}(\overline {R}), (p))$ over $R$, which is compatible with Frobenii and $\mathcal {G}_R$-actions.

On the other hand, consider the prism $(R_0, (p))$ in $R_{{ {\mathbb {\Delta }}}}$, where the structure map $R \rightarrow R_0/(p)$ is given by the natural projection $R \rightarrow R/(\pi ) \cong R_0/(p)$. For any integer $j \geq 1$, write $(\phi _j \mathbf {OA}_{\mathrm {cris}}(\overline {R}), (p))$ for the prism in $R_{{ {\mathbb {\Delta }}}}$ whose underlying $\delta$-pair is $(\mathbf {OA}_{\mathrm {cris}}(\overline {R}),(p))$ and the structure map $R \rightarrow \mathbf {OA}_{\mathrm {cris}}(\overline {R})/(p)$ is given the structure map of $(\mathbf {OA}_{\mathrm {cris}}(\overline {R}),(p))$ composed with $\varphi ^j\colon \mathbf {OA}_{\mathrm {cris}}(\overline {R})\rightarrow \mathbf {OA}_{\mathrm {cris}}(\overline {R})$. For a sufficiently large $j$, there exists a map of prisms $(R_0, (p)) \rightarrow (\phi _j\mathbf {OA}_{\mathrm {cris}}(\overline {R}), (p))$ given by Dwork's trick. Indeed, let $e = [K: K_0]$ be the ramification index, and choose an integer $l$ such that $p^l \geq e$. Consider $\varphi ^{l+1}\colon \mathbf {OA}_{\mathrm {cris}}(\overline {R}) \rightarrow \mathbf {OA}_{\mathrm {cris}}(\overline {R})$. Taking modulo the ideal $(p)\subset \mathbf {OA}_{\mathrm {cris}}(\overline {R})$, this induces a map $\overline {R}/(p) \rightarrow \mathbf {OA}_{\mathrm {cris}}(\overline {R})/(p)$ as previously, which further factors through $\overline {R}/(\pi )$. Thus, the ring map $R_0 \xrightarrow {\varphi ^{l+1}} \mathbf {OA}_{\mathrm {cris}}(\overline {R})$ induces a map $(R_0, (p)) \rightarrow (\phi _{l}\mathbf {OA}_{\mathrm {cris}}(\overline {R}), (p))$ of prisms over $R$.

For $i = 1, 2$, let $S^{(i)} := D_{\mathfrak {S}^{\widehat {\otimes }[i]}} (J^{(i)})^{\wedge }$ be the $p$-adically completed PD-envelope of $\mathfrak {S}^{\widehat {\otimes }[i]}$ with respect to the ideal $J^{(i)}$. We set

\[ z_0 := \frac{y-u}{E}\quad\text{and}\quad z_j := \frac{s_j-T_j}{E} \quad(j = 1, \ldots, d). \]

Let $A_{\mathrm {max}}^{(1)}$ be the $p$-adic completion of the $\mathfrak {S}$-subalgebra of $(\mathfrak {S}[p^{-1}])[z_0, z_1, \ldots, z_d]$ generated by ${E}/{p}$ and $\{\gamma _n(z_j)\}_{n \geq 1, 0\leq j\leq d}$. By [Reference Du and LiuDL23, § 2.2],Footnote ¹ we have a ring endomorphism $\varphi \colon A_{\mathrm {max}}^{(1)} \rightarrow A_{\mathrm {max}}^{(1)}$ extending $\varphi \colon \mathfrak {S} \rightarrow \mathfrak {S}$ and satisfying

\[ \varphi(z_0) = \frac{y^p-u^p}{\varphi(E)} \quad\text{and}\quad \varphi(z_j) = \frac{s_j^p-T_j^p}{\varphi(E)} \quad(j = 1, \ldots, d). \]

Since $\varphi \colon \mathfrak {S} \rightarrow \mathfrak {S}$ is injective, $\varphi \colon A_{\mathrm {max}}^{(1)} \rightarrow A_{\mathrm {max}}^{(1)}$ is injective. By [Reference Du and LiuDL23, § 2.2], we have a natural ring map $\mathfrak {S}^{(1)} \rightarrow A_{\mathrm {max}}^{(1)}$ which is injective and compatible with $\varphi$. In fact, $A_{\mathrm {max}}^{(1)}$ is isomorphic to $\mathfrak {S}^{(1)}\langle {E}/{p} \rangle$, the $p$-adic completion of $\mathfrak {S}^{(1)}[ {E}/{p} ]$ by [Reference Du and LiuDL23, Remark 2.2.11].

Let $S_1 := \mathfrak {S}^{\widehat {\otimes }[1]} [\gamma _n(E), \gamma _n(y-u), \{\gamma _n (s_j-T_j)\}_{n \geq 1, j = 0, \ldots, d}] \subset \mathfrak {S}^{\widehat {\otimes }[1]} [p^{-1}]$. Note that $S_1 \subset A_{\mathrm {max}}^{(1)}$ since $\gamma _n(y-u) =\gamma _n(z_0)E^n \in A_{\mathrm {max}}^{(1)}$ and similarly for $\gamma _n(s_i-T_i)$. Since $E$, $y-u$, and $\{s_j-T_j\}_{j = 1, \ldots, d}$ form a regular sequence in $\mathfrak {S}^{\widehat {\otimes }[1]}$, $S_1$ is the PD-envelope of $\mathfrak {S}^{\widehat {\otimes }[1]}$ for $J^{(1)}$ by [Reference Bhatt and ScholzeBS22, Corollary 2.39]. Then $S^{(1)}$ is the $p$-adic completion of $S_1$. As a subring of $(R_0[p^{-1}])[\![u, y-u, s_1-T_1, \ldots, s_d-T_d]\!]$, we have

\begin{align*} S^{(1)} &= \bigg\{ \sum a_{i_0, \ldots, i_{d+1}} \gamma_{i_0}(E) \gamma_{i_1}(y-u)\gamma_{i_2}(s_1-T_1)\cdots \gamma_{i_{d+1}} (s_d-T_d) \bigg| \\ & \quad\quad\ a_{i_0, \ldots , i_{d+1}} \in \mathfrak{S}^{\widehat{\otimes}[1]}, a_{i_0, \ldots , i_{d+1}} \rightarrow 0 \ (\text{as } i_0+\cdots+i_{d+1}\to\infty) \bigg\}, \end{align*}

where the sum goes over the multi-index $(i_0, \ldots, i_{d+1})$ of non-negative integers and $a_{i_0, \ldots, i_{d+1}} \rightarrow 0$ means in the $p$-adic topology. Note that $S^{(1)}$ is a $\delta$-ring by [Reference Bhatt and ScholzeBS22, Corollary 2.39]. We similarly construct a $\delta$-ring $S^{(2)}$.

Proof. We prove the statement for $\varphi \colon \mathfrak {S}^{(1)} \rightarrow S^{(1)}$, and the proof for $\varphi \colon \mathfrak {S}^{(2)} \rightarrow S^{(2)}$ is analogous. It suffices to show $\varphi (\delta ^n(z_j)) \in S^{(1)}$ for $j = 0, \ldots, d$. Since $S^{(1)}$ is a $\delta$-ring, we have

\[ \varphi(z_0) = c^{-1}\frac{\varphi(y-u)}{p} = c^{-1}\bigg(\frac{(y-u)^p}{p}+\delta(y-u)\bigg) \in S^{(1)}, \]

and similarly $\varphi (z_j) \in S^{(1)}$ for $j = 1, \ldots, d$. Again since $S^{(1)}$ is a $\delta$-ring, we have $\varphi (\delta ^n(z_j)) = \delta ^n(\varphi (z_j)) \in S^{(1)}$ for any $n \geq 0$.

3.2 Completed prismatic $F$-crystals in the small affine case

In this subsection, we introduce completed prismatic crystals and completed prismatic $F$-crystals on the absolute prismatic site.

We first introduce the notion of finitely generated completed prismatic crystals. Let ${\mathfrak X}$ be a smooth $p$-adic formal scheme over $\mathcal {O}_K$ (or a CDVR of mixed characteristic $(0,p)$).

Remark 3.12 Let $\mathcal {F}$ be a finitely generated completed prismatic crystal on ${\mathfrak X}$ and let $(A,I)\rightarrow (B,IB)$ be a map of bounded prisms over ${\mathfrak X}$. Since $\mathcal {F}_A$ is a finitely generated $A$-module and $B$ is classically $(p,IB)$-complete, the natural map

(3.1)\begin{equation} B\otimes_A\mathcal{F}_A\rightarrow B\widehat{\otimes}_A\mathcal{F}_A\xrightarrow{\cong}\mathcal{F}_B \end{equation}

is surjective. Since $(p,IB)$ is a finitely generated ideal of $B$, the map (3.1) induces an isomorphism $B/(p,IB)^n\otimes _A\mathcal {F}_A\xrightarrow {\cong }\mathcal {F}_B/(p,IB)^n\mathcal {F}_B$ by [Reference YekutieliYek18, Theorem 1.2(2)]. Moreover, the map (3.1) is an isomorphism if $A$ and $B$ are both noetherian or if $A$ is noetherian and the map $A\rightarrow B$ is classically flat. The latter case follows from [Sta22, Tag 0912]. It is also an isomorphism if $\mathcal {F}_A$ is a finite projective $A$-module.

Proof. Let $\mathcal {F}$ be a finitely generated completed crystal of $\mathcal {O}_{ {\mathbb {\Delta }}}$-modules on ${\mathfrak X}_{ {\mathbb {\Delta }}}$. Let $(A,I)\rightarrow (B,IB)$ be a $(p,I)$-completely faithfully flat map of bounded prisms over ${\mathfrak X}$. Set $B'=(B\otimes _AB)^\wedge _{(p,I)}$. By Lemma 3.3, $B'$ is $(p,I)$-completely faithfully flat over $A$ and $(B',IB')\in {\mathfrak X}_{ {\mathbb {\Delta }}}$ is the self-fiber product of $(B,IB)$ over $(A,I)$. Let $p_1,p_2\colon (B,IB)\rightarrow (B',IB')$ be the two maps of bounded prisms over ${\mathfrak X}$. Since $B/(p,I)^nB$ is classically faithfully flat over $A/(p,I)^n$ and since $B/(p,I)^nB\otimes _{A/(p,I)^n}B/(p,I)^nB\cong B'/(p,I)^nB'$, we have an exact sequence

\[ 0\rightarrow \mathcal{F}_{n,A}\rightarrow B/(p,I)^nB\otimes_{A/(p,I)^n}\mathcal{F}_{n,A} \xrightarrow{p_1\otimes 1-p_2\otimes 1} B'/(p,I)^nB'\otimes_{A/(p,I)^n}\mathcal{F}_{n,A}. \]

On the other hand, since $\mathcal {F}$ is a completed prismatic crystal, we have the isomorphisms $B\widehat {\otimes }_A\mathcal {F}_A\cong \mathcal {F}_B$ and $B'\widehat {\otimes }_A\mathcal {F}_A\cong \mathcal {F}_{B'}$. It follows that the above exact sequence is identified with

\[ 0\rightarrow \mathcal{F}_{n,A}\rightarrow \mathcal{F}_{n,B}\xrightarrow{p_1^\ast-p_2^\ast} \mathcal{F}_{n,B'}. \]

This implies that $\mathcal {F}_n$ is a sheaf on ${\mathfrak X}_{ {\mathbb {\Delta }}}$, representing the quotient sheaf $\mathcal {F}/(p,\mathcal {I}_{ {\mathbb {\Delta }}})^n\mathcal {F}$. Since $\mathcal {F}_A$ is a finitely generated classically $(p,I)$-complete $A$-module, $\mathcal {F}_n$ is a finitely generated crystal of $\mathcal {O}_{{ {\mathbb {\Delta }}},n}$-modules. Moreover, we have isomorphisms of $\mathcal {O}_{ {\mathbb {\Delta }}}$-modules $\mathcal {O}_{{ {\mathbb {\Delta }}},n}\otimes _{\mathcal {O}_{{ {\mathbb {\Delta }}},n+1}}\mathcal {F}_{n+1}\xrightarrow {\cong }\mathcal {F}_n$ and $\mathcal {F}\xrightarrow {\cong } \varprojlim _n \mathcal {F}_n$. Finally, since the absolute prismatic topos is replete and $\mathcal {F}_{n+1}\rightarrow \mathcal {F}_n$ is surjective for every $n$, we obtain $\varprojlim _n\mathcal {F}_n\cong \operatorname {Rlim}\mathcal {F}_n$ by [Reference Bhatt and ScholzeBS15, Proposition 3.1.10].

Conversely, let $(\mathcal {F}_n)_n$ be an inverse system of sheaves of $\mathcal {O}_{ {\mathbb {\Delta }}}$-modules satisfying the properties as in the lemma. An argument similar to the previous paragraph shows that the association $(A,I)\mapsto A/(p,I)^n\otimes _{A/(p,I)^{n+1}}\mathcal {F}_{n+1,A}$ represents the sheaf $\mathcal {O}_{{ {\mathbb {\Delta }}},n}\otimes _{\mathcal {O}_{{ {\mathbb {\Delta }}},n+1}}\mathcal {F}_{n+1}$. Set $\mathcal {F}:= \varprojlim _n\mathcal {F}_n$ and take any $(A,I)\in {\mathfrak X}_{ {\mathbb {\Delta }}}$. Then we have $\mathcal {F}(A,I)=(\varprojlim _n\mathcal {F}_n)(A,I)=\varprojlim _n \mathcal {F}_{n,A}$ and $A/(p,I)^n\otimes _{A/(p,I)^{n+1}}\mathcal {F}_{n+1,A}\cong \mathcal {F}_{n,A}$. It follows from [Reference YekutieliYek18, Theorem 2.8] that $\mathcal {F}(A,I)$ is a finitely generated and classically $(p,I)$-complete $A$-module with $\mathcal {F}(A,I)/(p,I)^n\mathcal {F}(A,I)\cong \mathcal {F}_{A,n}$. Moreover, for a morphism $(A,I)\rightarrow (B,IB)$ of bounded prisms over ${\mathfrak X}$, we have $B\otimes _A\mathcal {F}_{n,A}\cong \mathcal {F}_{n,B}$. It follows that the canonical map $B\widehat {\otimes }_A\mathcal {F}(A,I)\rightarrow \mathcal {F}(B,IB)$ is an isomorphism. Hence, $\mathcal {F}$ is a finitely generated completed crystal of $\mathcal {O}_{ {\mathbb {\Delta }}}$-modules. Now the remaining assertions follow easily.

To define completed prismatic $F$-crystals in the affine case, let us first introduce the following terminologies.

We now introduce the notion of completed prismatic $F$-crystals on $R$, which will be our main object of study. We make Assumption 2.9: $R$ is small over $\mathcal {O}_K$ or $R=\mathcal {O}_L$.

Definition 3.16 A completed $F$-crystal of $\mathcal {O}_{ {\mathbb {\Delta }}}$-modules on $R_{ {\mathbb {\Delta }}}$ is a pair $(\mathcal {F}, \varphi _{\mathcal {F}})$, where $\mathcal {F}$ is a finitely generated completed crystal of $\mathcal {O}_{{ {\mathbb {\Delta }}}}$-modules on $R_{ {\mathbb {\Delta }}}$ and

\[ \varphi_{\mathcal{F}}\colon \mathcal{F} \rightarrow \mathcal{F} \]

is a $\varphi$-semilinear morphism of $\mathcal {O}_{{ {\mathbb {\Delta }}}}$-modules such that:

1. (i) $\mathcal {F}_{\mathfrak {S}} := \mathcal {F}(\mathfrak {S}, E)$ is projective away from $(p, E)$ and saturated;

2. (ii) the pair $(\mathcal {F}_{\mathfrak S},\varphi _{\mathcal {F}_{\mathfrak S}})$ has finite $E$-height.

We also call such an object a completed prismatic $F$-crystal on $R$. The morphisms between completed $F$-crystals of $\mathcal {O}_{ {\mathbb {\Delta }}}$-modules are $\mathcal {O}_{{ {\mathbb {\Delta }}}}$-module maps compatible with Frobenii.

We write $\mathrm {CR}^{\wedge,\varphi }(R_{{ {\mathbb {\Delta }}}})$ for the category of completed $F$-crystals of $\mathcal {O}_{ {\mathbb {\Delta }}}$-modules on $R_{ {\mathbb {\Delta }}}$. Let $\mathrm {Vect}^{\varphi }_{\mathrm {eff}}(R_{{ {\mathbb {\Delta }}}})$ denote the full subcategory of $\mathrm {CR}^{\wedge,\varphi }(R_{{ {\mathbb {\Delta }}}})$ consisting of objects $(\mathcal {F},\varphi _{\mathcal {F}})$ where $\mathcal {F}$ is a locally free $\mathcal {O}_{ {\mathbb {\Delta }}}$-module. For a fixed non-negative integer $r$, we let $\mathrm {CR}^{\wedge,\varphi }_{[0,r]}(R_{{ {\mathbb {\Delta }}}})$ and $\mathrm {Vect}^{\varphi }_{[0,r]}(R_{{ {\mathbb {\Delta }}}})$ denote the full subcategories consisting of objects for which $(\mathcal {F}_{\mathfrak S},\varphi _{\mathcal {F}_{\mathfrak S}}\!)$ has $E$-height $\leq r$.