# A new method for overconvergence of $(\varphi,\Gamma)$-modules

Heng Du  Yau Mathematical Sciences Center, Tsinghua University, Beijing
100084, China<EMAIL_ADDRESS>and  Tong Liu  Department of
Mathematics, Purdue University, 150 N. University Street, West Lafayette,
Indiana 47907, USA<EMAIL_ADDRESS>
###### Abstract.

We show all Laurent $F$-crystals over $p$-adic fields are overconvergent.

###### Contents

  1. 1 Introduction
  2. 2 Overconvergent Laurent $F$-crystals
  3. 3 Overconvergence of Laurent $F$-crystals
  4. 4 Overconvergence of $(\varphi,\Gamma)$-modules associated with $\Phi$-iterate tower.
  5. A Arithmetic Kummer tower is not decompleting

##  1\. Introduction

One aim in the study of $p$-adic Hodge theory is to relate good categories of
$p$-adic Galois representations of $\mathrm{Gal}_{\mathbb Q_{p}}$ with certain
categories of semilinear algebraic objects over some big rings. Since it is
usually easier to work with those semilinear algebraic objects when studying
Galois representations in general coefficients, and when constructing or
deforming Galois representations. When we want to consider all $p$-adic Galois
representations of $\mathrm{Gal}_{\mathbb Q_{p}}$ over $\mathbb
Z_{p}$-lattices, then the semilinear algebra category one should use is the
category of étale $(\varphi,\Gamma)$-modules over $\mathbf{A_{\mathbb
Q_{p}}}$. Here the latter category is a certain linear algebra category over
the ring

$\mathbf{A}_{\mathbb Q_{p}}\coloneqq\\{f=\sum_{i\in\mathbb Z}a_{i}X^{i}\mid
a_{i}\in\mathbb Z_{p}\text{ and }\lim_{i\to-\infty}a_{i}=0\\}$

equipped with a semilinear action $\varphi$ and $\mathbb Z_{p}^{\times}$. When
$K$ is a $p$-adic field, here in this paper we will refer to the case that $K$
being a finite extension of $\mathbb Q_{p}$, the above equivalence between
$p$-adic Galois representations and étale $(\varphi,\Gamma)$-modules are
generalized using the field of norms construction of Fontaine-
Wintenberger[FW79]. It turns out that étale $(\varphi,\Gamma)$-modules over
$\mathbf{A}_{\mathbb Q_{p}}$ always descent to so-called overconvergent étale
$(\varphi,\Gamma)$-modules and this was first established by Cherbonnier-
Colmez in [CC98]. In the case when $K=\mathbb Q_{p}$, their result implies
every étale $(\varphi,\Gamma)$-modules over $\mathbf{A}_{\mathbb Q_{p}}$
descent to the Robba ring $\mathbf{A}_{\mathbb Q_{p}}^{\dagger}$, the subring
consisting of functions $f$ that converges on some annulus $(r,1)$ with $r$
sufficiently closed to $1$.

There are two paralleled directions of generalizing the theory of étale
$(\varphi,\Gamma)$-modules and the corresponding overconvergent theories, to a
family version or to a relative version. To be more precise, the family
version usually refers to Galois representations in general coefficients, cf.
[Dee01], [BC08], and recently [Por22]. On the other hand, the relative version
refers to replacing Galois groups with étale fundamental groups of some
geometric objects over the base $p$-adic field, cf. [AB08] and [KL19]. Other
than the works we mentioned, people also ask similar questions to
$(\varphi,\tau)$-modules developed in [Car13], and the overconvergence of
$(\varphi,\tau)$-modules were obtained in [GL20a] and [GP21]. Moreover, there
is the theory about overconvergence for Lubin-Tate tower which is considered
by Berger in [Ber16].

In [KL19], Kedlaya-Liu tried to give an axiomatic approach to the theory of
overconvergence of $(\varphi,\Gamma)$-modules, and their idea is that
overconvergence is closely related to a property of a perfectoid finite étale
tower which they called decompleteness. Using such an idea, they show the
decompleteness of the cyclotomic tower, which recovers the result of
Cherbonnier-Colmez. Their formalism also succeeds in dealing with relative
situations that show relative toric towers are decompleting, and this result
plays an important role in the construction of the Riemann-Hilbert functor in
[LZ17]. Kedlaya-Liu expects their formalism to apply to the Kummer tower as
well, which will reprove the overconvergence of $(\varphi,\tau)$-modules, cf.
Conjecture 6.2.6. in $loc.$ $cit$. However, we will show that the
overconvergent period rings defined by Kedlaya-Liu in Kummer’s case are too
small and very few $(\varphi,\tau)$-modules are overconvergent in their sense,
cf. the Appendix A.

This paper gives a sheaf theoretic approach to the overconvergence of
$(\varphi,\Gamma)$-modules. We will use the theory of prisms developed by
Bhatt-Scholze in [BS22] and [BS21]. Let $K$ be a finite extension of $\mathbb
Q_{p}$, and let $X$ be a smooth formal scheme over $\mathcal{O}_{K}$. Consider
the absolute prismatic site $X_{{\mathbbl{\Delta}}}$ over $X$, which equipped
with the structure sheaf $\mathcal{O}_{{\mathbbl{\Delta}}}$ and ideal sheaf
$\mathcal{I}_{{\mathbbl{\Delta}}}$, also define
$\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}=\mathcal{O}_{{\mathbbl{\Delta}}}[1/\mathcal{I}_{{\mathbbl{\Delta}}}]^{\wedge}_{p}$.
In [BS21], they define the category of Laurent $F$-crystals consisting of
crystals in finite locally free
$\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}$-modules together with a
Frobenius semilinear endomorphism which is étale, i.e., the linearization of
the Frobenius semilinear endomorphism is an isomorphism, then they can show

######  Theorem 1.1 ([BS21, Corollary 3.8], [Wu21]).

The category of Laurent $F$-crystals over $X$ is equivalent to the category of
$\mathbb Z_{p}$-local systems on the generic fiber of $X$.

In this paper, we will construct a sheaf
$\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger}$ over
$X_{{{\mathbbl{\Delta}}}^{\circ}}$ which is the subcategory of
$X_{{{\mathbbl{\Delta}}}}$ consisting transversal prisms over $X$. Then we
define the category of overconvergent Laurent $F$-crystals to be crystals in
finite locally free
$\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger}$-modules together
with a Frobenius semilinear endomorphism which is étale in §2.

######  Theorem 1.2 (Theorem 3.5).

Let $K$ be a finite extension of $\mathbb Q_{p}$ with ring of integers
$\mathcal{O}_{K}$, and let $X=\mathop{\rm Spf}\nolimits(\mathcal{O}_{K})$, we
have all Laurent $F$-crystals over $X$ are overconvergent.

For the exact meaning of the above theorem, we will define a functor

(1)
$D_{X}:\mathrm{Vect}(X_{{{\mathbbl{\Delta}}}^{\circ}},\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger})^{\varphi=1}\to\mathrm{Vect}(X_{{{\mathbbl{\Delta}}}},\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}})^{\varphi=1}.$

in §2, where
$\mathrm{Vect}(X_{{{\mathbbl{\Delta}}}^{\circ}},\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger})^{\varphi=1}$
and
$\mathrm{Vect}(X_{{{\mathbbl{\Delta}}}},\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}})^{\varphi=1}$
are the category of overconvergent Laurent $F$-crystals and Laurent
$F$-crystals respectively. And Theorem 1.2 states that $D_{X}$ is an
equivalence of categories when $X$ is the formal spectrum of the ring of
integers of a $p$-adic field. So in particular, by the result of Bhatt-
Scholze,
$\mathrm{Vect}(X_{{{\mathbbl{\Delta}}}^{\circ}},\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger})^{\varphi=1}$
is also equivalent to the category of $G_{K}$-stable lattice in $p$-adic
representations of $G_{K}$.

We expect the above theorem to give a uniform approach to all overconvergent
theories over $p$-adic fields. In particular, it recovers all the main results
in [CC98], [GL20a] and [GP21] in the case of $p$-adic fields. Here we want to
mention the above results can also be treated uniformly using the framework of
Berger-Colmez on locally analytic vectors [BC16] and [Ber16], by showing there
are “enough” locally analytic vectors in certain perfect overconvergent period
rings. When the tower is not a $p$-adic Lie extension, Berger-Colmez’s method
doesn’t apply, and there is no general framework for proving overconvergent
results. Meanwhile, Theorem 1.2 can produce many results in the non-Lie
extension direction. Theorem 1.2 In §4, we will formulate and prove the
“decompleting” of any $\Phi$-iterate tower over $K$. Moreover, we can show for
a fixed $\mathbb Z_{p}$-Galois representation $T$, the convergent radii are
uniformly bounded for all choices of $\Phi$-iterate towers.

Why using the prismatic site. Prisms are considered as the deperfection of
perfectoid rings. We know the overconvergent theory is relatively easy over
perfect period rings. Actually, the underlying étale $\varphi$-modules already
give an equivalence of categories in many cases, cf. [KL19]. And we can show
(1) fits into the following diagram:

${\mathrm{Vect}(X_{{{\mathbbl{\Delta}}}^{\circ}},\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger})^{\varphi=1}}$${\mathrm{Vect}(X_{{{\mathbbl{\Delta}}}},\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}})^{\varphi=1}}$${\mathrm{Vect}(X_{{{\mathbbl{\Delta}}}^{\circ}}^{\mathop{\rm
perf}\nolimits},\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger})^{\varphi=1}}$${\mathrm{Vect}(X_{{\mathbbl{\Delta}}}^{\mathop{\rm
perf}\nolimits},\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}})^{\varphi=1},}$$\scriptstyle{D_{X}}$$\scriptstyle{D_{X}^{\mathop{\rm
perf}\nolimits}}$

where the vertical arrows are induced by the restrictions to perfect sites.
And $D_{X}^{\mathop{\rm perf}\nolimits}$ is the corresponding functor over
perfect sites. We will see $D_{X}^{\mathop{\rm perf}\nolimits}$ recovers the
overconvergent theory over perfect period rings in §3.1. And our theory is
indeed a deperfection of this result.

Idea of proof. We will define the overconvergent period sheaves
$\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger,n}$ and
$\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger}$ over
$X_{{{\mathbbl{\Delta}}}^{\circ}}$ in §2, to understand sections of these
sheaves, we developed a “Noetherian approximation” method in §2.1. Then we can
define the category of overconvergent Laurent $F$-crystals and we can show
that overconvergent Laurent $F$-crystals can be understood using
overconvergent étale $\varphi$-modules with “overconvergent” descent
isomorphisms in §2. Given a $\mathbb Z_{p}$-Galois representation, we will get
an overconvergent étale $\varphi$-module from the classical overconvergent
results, then the question is how to find the overconvergent descent
isomorphism. We will use the idea in [DL21]. Briefly speaking, our slogan is
that the descent isomorphism should come from the $\tau$-action, where $\tau$
is a specific Galois group element that comes from the $(\varphi,\tau)$-module
theory of Caruso in [Car13]. To apply our strategy, we will also use some
computations made in [GL20a], where they can estimate the overconvergent
radius of $\tau$ using (loose) crystalline lifting tricks together with the
theory of $(\varphi,\hat{G})$-modules. However, in [GL20a] they can only get
the overconvergent $\tau$-action in some perfect overconvergent period rings,
the main obstruction of to “deperfect” the $\tau$-action is because in
$(\varphi,\hat{G})$-modules, one can only see the $\tau$-action after a
Frobenius twist. But we will see if we replace the theory of
$(\varphi,\hat{G})$-module with the theory of prismatic
$(\varphi,\hat{G})$-modules developed in [DL21], this problem disappears and
$\tau$-action provides the descent isomorphism.

Notations and conventions. In this paper, $K$ is $p$-adic field, i.e., a
finite extension of $\mathbb Q_{p}$. We let $\mathcal{O}_{K}$ be the ring of
integers of $K$ which contains a fixed uniformizer $\varpi$, and $k$ the
residue field of $K$. We will write $W=W(k)$, $K_{0}=W[1/p]$. We fix an
algebraic closure $\overline{K}$ over $K$ and write $C$ as the $p$-adic
completion of $\overline{K}$. The ring of integers in $C$ will be denoted by
$\mathcal{O}_{C}$. Let $C^{\flat}$ (resp. $\mathcal{O}_{C}^{\flat}$) to be the
tilt of $C$ (resp. $\mathcal{O}_{C}$). Define
$A_{\mathrm{inf}}=A_{\mathrm{inf}}(\mathcal{O}_{C})$. We will fix a compatible
system $\\{\varpi_{n}\\}_{n\geq 0}$ of $p^{n}$-th roots with
$\varpi_{0}=\varpi$ as well as a compatible system $\\{\zeta_{n}\\}_{n\geq 0}$
of $p^{n}$-th roots of $1$. Write
$\underline{\varpi}^{\flat}\coloneqq(\varpi_{n})_{n\geq 0}$ and
$\underline{\zeta}^{\flat}\coloneqq(\zeta_{n})_{n\geq 0}$ as elements in
$\mathcal{O}_{C}^{\flat}$. Once the above is given, we can define a map
$W[\\![u]\\!]\to A_{\mathrm{inf}}$ given by
$u\to[\underline{\varpi}^{\flat}]$.

Let $K_{\infty}=\bigcup_{n=1}^{\infty}K(\varpi_{n})$. Then we have a closed
subgroup $G_{\infty}\coloneqq{\rm Gal}(\overline{K}/K_{\infty})$ of
$G_{K}\coloneqq{\rm Gal}(\overline{K}/K)$. Let
$L=\cup_{n=1}^{\infty}K_{\infty}(\zeta_{p^{n}})$ and
$K_{1^{\infty}}=\bigcup_{n=1}^{\infty}K(\zeta_{p^{n}})$, then
$H_{K}\coloneqq\mathop{\rm Gal}\nolimits(L/K_{\infty})$ is a closed subgroup
of $\hat{G}\coloneqq\mathop{\rm Gal}\nolimits(L/K)$. Let $\tau$ be a
topological generator of $\mathop{\rm Gal}\nolimits(L/K_{1^{\infty}})$. We
will also use $\tau$ to denote a lifting of $\tau$ in $G_{K}$ when it is
acting on an element fixed by $\mathop{\rm Gal}\nolimits(\overline{K}/L)$.

In this paper, for a prism $(A,I)$, if $I$ is generated by $d$, then we will
write $(A,d)$ instead of $(A,(d))$ to simplify our symbols.

Acknowledgments. It is our pleasure to thank Hansheng Diao, Hui Gao, Yu Min,
Yong Suk Moon, Koji Shimizu, Shanwen Wang, and Yupeng Wang for their valuable
discussions and interest during the preparation of this paper. The appendix of
this paper is based on an email correspondence between the first-named author
and Kiran Kedlaya and Ruochuan Liu, H.D. wants to thank them for the
discussions on [KL19].

##  2\. Overconvergent Laurent $F$-crystals

Let $X$ be a bounded $p$-adic formal scheme, and let
$X_{{{\mathbbl{\Delta}}}}$ be the absolute prismatic site over $X$. We will
denote $\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}$ to be the sheaf
$\mathcal{O}_{{\mathbbl{\Delta}}}[\mathcal{I}_{{\mathbbl{\Delta}}}^{-1}]^{\wedge}_{p}$,
and we use
$\mathrm{Vect}(X_{{{\mathbbl{\Delta}}}},\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}})^{\varphi=1}$
to denote the category of Laurent $F$-crystals over $X$.

Let $X_{{{\mathbbl{\Delta}}}^{\circ}}$ be the subcategory of
$X_{{{\mathbbl{\Delta}}}}$ consisting only transversal prisms over $X$. Recall
that a prism is called transversal if $A/I$ is $p$-torsion free. We recall
some basic properties of transversal prisms.

######  Proposition 2.1.

  1. (1)

If $f\colon(A,I)\to(B,J)$ is a flat morphism of prisms. If $(A,I)$ is
transversal, then so is $(B,J)$. The converse holds if $f$ is a covering;

  2. (2)

If $(A,d)$ is a transversal prism, then for all $s\neq r>0$ in $\mathbb N$,
the sequences $(p,\varphi^{s}(d))$ and $(\varphi^{r}(d),\varphi^{s}(d))$ are
regular.

###### Proof.

(1) is [BL22, Remark 2.4.4] and (2) is [ALB20, Lemma 3.3] ∎

######  Lemma 2.2.

$X_{{{\mathbbl{\Delta}}}^{\circ}}$ is a site.

###### Proof.

We need to check that the pushout of a cover along an arbitrary map is again
inside $X_{{{\mathbbl{\Delta}}}}^{\circ}$. For this, we have the pushout of a
cover along an arbitrary map is again a covering in
$X_{{{\mathbbl{\Delta}}}}$, by Proposition 2.1, we have the pushout is
transversal. ∎

######  Lemma 2.3.

Let $X$ be a $p$-torsion free quasi-syntomic $p$-adic formal scheme. Then the
following categories are equivalent:

$\mathrm{Vect}(X_{{{\mathbbl{\Delta}}}},\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}})\simeq\mathrm{Vect}(X_{{{\mathbbl{\Delta}}}^{\circ}},\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}})\simeq\mathrm{Vect}(X_{qsyn},{{\mathbbl{\Delta}}}_{\bullet}[\frac{1}{I}]^{\wedge}_{p}).$

###### Proof.

The equivalence of
$\mathrm{Vect}(X_{{{\mathbbl{\Delta}}}},\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}})$
and
$\mathrm{Vect}(X_{qsyn},{{\mathbbl{\Delta}}}_{\bullet}[\frac{1}{I}]^{\wedge}_{p})$
is established in [BS21, Proposition 2.14]. Here, one reduces to the case
$X=\mathop{\rm Spf}\nolimits(R)$ with $R$ quasi-syntomic, and the equivalence
is given by the natural functor

${{\mathbbl{\Delta}}}(-)\colon R_{qrsp}\to R_{{{\mathbbl{\Delta}}}}$

that maps $S$ to ${{\mathbbl{\Delta}}}_{S}$.

Let $S$ be a quasiregular semiperfectoid covering $R$. In particular, we have
$S$ is $p$-torsion free. This will imply that ${{\mathbbl{\Delta}}}_{S}$ is
transversal, cf. [BS22, Proposition 7.10]. Let $S^{\bullet}$ be the Čech nerve
of this map, then we have all three categories equivalent to

$\lim\mathrm{Vect}({{\mathbbl{\Delta}}}_{S^{\bullet}})$

here we use the fact that ${{\mathbbl{\Delta}}}_{S^{i}}$ are transversal for
all $i$ by Lemma 2.2. ∎

For any transversal prism $(A,I)$ and $n\in\mathbb N$ greater than $1$, let
$I_{n}\coloneqq\prod_{i=1}^{n}\varphi^{i}(I)$.

######  Lemma 2.4 ([AB20, Lemma 3.3]).

$I_{n}$ defines a Cartier divisor on $\mathop{\rm Spec}\nolimits(A)$.

######  Remark 2.5.

Note that our definition of $I_{n}$ differs from a Frobenius twist with the
one considered in [AB20].

Define

$\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger,n}(A,I)=A\langle\frac{p}{I_{n}}\rangle[\frac{1}{I_{n}}],$

where $A\langle\frac{p}{I_{n}}\rangle$ is the $(p,I_{n})$-adic completion of
$A[\frac{p}{I_{n}}]$.

######  Remark 2.6.

We have $\varphi(I)$ coincides with $I^{p}$ after modulo $p$, so for
$n\in\mathbb N$, $(p,I_{n})=(p,I^{m})$ for some $m$, and

$(p,I)^{m}\subset(p,I^{m})=(p,\prod_{i=1}^{n}\varphi^{i}(I))\subset(p,I),$

that is $(p,I)$, $(p,\varphi(I))$ and $(p,I_{n})$ defines the same topology on
$A$.

Over $A[\frac{p}{I_{n}}]$, we have $I_{n}$ divides $p$, so the
$(p,I_{n})$-adic topology is the same as the $I_{n}$-adic topology.

######  Lemma 2.7.

$\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger,n}$ defines a sheaf
on $X_{{{\mathbbl{\Delta}}}^{\circ}}$.

###### Proof.

Let $(A,I)\to(B,IB)$ be a covering in $X_{{{\mathbbl{\Delta}}}^{\circ}}$, we
need to show

$\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger,n}((A,I))\to\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger,n}((B,IB))\quad\begin{subarray}{c}\rightarrow\\\\[0.50003pt]
\rightarrow\\\\[-0.50003pt]
\end{subarray}\quad\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger,n}((B\hat{\otimes}_{A}B,IB\hat{\otimes}_{A}B))$

is exact. By the definition of
$\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger,n}$, it is enough to
show the following lemma. ∎

######  Lemma 2.8.

We have $(A,I)\mapsto A\langle\frac{p}{I_{n}}\rangle$ defines a sheaf on
$X_{{{\mathbbl{\Delta}}}^{\circ}}$.

And we will deduce Lemma 2.8 from the following two lemmas.

######  Lemma 2.9.

Let $(A,I)\to(B,IB)$ be a covering of transversal prisms, then
$A\langle\frac{p}{I_{n}}\rangle\to B\langle\frac{p}{I_{n}}\rangle$ is
$I_{n}$-complete faithfully flat.

###### Proof.

First, by [BS22, Lemma 3.6], $\varphi(I)$ is principle generated by a
distinguished element $\xi$. And we have $I_{n}$ is generated by
$\tilde{\xi}_{n}\coloneqq\prod_{i=0}^{n-1}\varphi^{i}(\xi)$.

By discussions in [BS22, page 11] or [BMS19, §4.1], we have if an ideal $J$ of
$A$ induces the same topology as $(p,I)$, then $A\to B$ is also $J$-complete
flat. In particular, we have $A\to B$ is $(p,\tilde{\xi}_{n})$-complete flat
by Remark 2.6. Moreover it is also $(p,\tilde{\xi}_{n})$-complete faithfully
flat since $A/(p,\tilde{\xi}_{n})$ and $A/(p,I)$ have the same spectrum.

For $m_{1},m_{2}\in\mathbb N$, consider $B_{m_{1},m_{2}}\coloneqq
B\otimes_{A}^{\mathbb{L}}A/(p^{m_{1}},\xi^{m_{2}})$. Then
$B_{m_{1},m_{2}}=(B\otimes_{A}^{\mathbb{L}}A/\xi^{m_{2}})\otimes_{A/\xi^{m_{2}}}^{\mathbb{L}}A/(p^{m_{1}},\xi^{m_{2}})$.
By [BMS19, Lemma 4.7] or the proof of (2) in [BS22, Lemma 3.7], we have
$B\otimes_{A}^{\mathbb{L}}A/\xi^{m_{2}}$ is discrete and equal to
$B/\xi^{m_{2}}$. Now we can repeat this argument, using the fact that
$B/\xi^{m_{2}}$ is $p$-complete flat over $A/\xi^{m_{2}}$, so
$B_{m_{1},m_{2}}$ is discrete and equal to $B/(p^{m_{1}},\xi^{m_{2}})$.

Note that by [Sta20, Tag 00M8], one also have $A[X]\to B[X]$ is
$(p,\xi)$-complete faithfully flat, and
$(p,\tilde{\xi}_{n})=(p,\xi^{m})=(\tilde{\xi}_{n},\tilde{\xi}_{n}X-p)$ in
$A[X]$ for some $m\in\mathbb N$. So same argument shows rCl B[X] ⊗_A[X]^L
A[X]/(~ξ_nX-p) & ≃ B[X]/(~ξ_nX-p) and rCl B[X] ⊗_A[X]^L A[X]/(~ξ_n^k,~ξ_nX-p)
& ≃ B[X]/(~ξ_n^k,~ξ_nX-p)
≃ B[p~ξn]/(p^k,~ξ_n^k)=B[p~ξn]/(p^k,ξ^km). When $k=1$, this implies
$B\langle\frac{p}{\tilde{\xi}_{n}}\rangle/(\tilde{\xi}_{n})\simeq
B/(p,\tilde{\xi}_{n})[X]$. So we have
$A\langle\frac{p}{\tilde{\xi}_{n}}\rangle/(\tilde{\xi}_{n})\to
B\langle\frac{p}{\tilde{\xi}_{n}}\rangle/(\tilde{\xi}_{n})$ is faithfully flat
since $A\to B$ is $(p,\tilde{\xi}_{n})$-complete faithfully flat.

It remains to show
$\mathrm{Tor}^{A\langle\frac{p}{\tilde{\xi}_{n}}\rangle}_{i}(A\langle\frac{p}{\tilde{\xi}_{n}}\rangle/(\tilde{\xi}_{n}),B\langle\frac{p}{\tilde{\xi}_{n}}\rangle)=0$
for $i>0$. It is enough to show this for $i=1$, we claim

$\mathrm{Tor}^{A\langle\frac{p}{\tilde{\xi}_{n}}\rangle}_{1}(A\langle\frac{p}{\tilde{\xi}_{n}}\rangle/(\tilde{\xi}_{n}),B\langle\frac{p}{\tilde{\xi}_{n}}\rangle)=B\langle\frac{p}{\tilde{\xi}_{n}}\rangle[\tilde{\xi}_{n}]$

vanishes. This is because when $B$ is transversal, $B\subset
B[\frac{p}{\tilde{\xi}_{n}}]\subset B[\frac{1}{\tilde{\xi}_{n}}]$ are
$\tilde{\xi}_{n}$-torsion free, and the claim comes from the following fact:
if $R$ is $d$-torsion free, let $\hat{x}=(x_{n})_{n}\in\varprojlim_{n}R/d^{n}$
be an element in the completion such that $d\hat{x}=0$, then $\hat{x}=0$. Let
$y_{n}\in R$ be any lifting of $x_{n}$, then $dy_{n}\in d^{n}R$ from the
definition. Since $R$ is $d$-torsion free, we have $y_{n}\in d^{n-1}R$ which
implies $x_{n-1}=0$. But $n$ is arbitrary, so $\hat{x}=0$. ∎

######  Lemma 2.10.

Let $(A,I)\to(B,IB)$ be a covering of transversal prisms, and let
$\tilde{\xi}_{n}$ be as in the proof of Lemma 2.9, then

$(B\hat{\otimes}_{A}B)\langle\frac{p}{\tilde{\xi}_{n}}\rangle\simeq
B\langle\frac{p}{\tilde{\xi}_{n}}\rangle\hat{\otimes}_{A\langle\frac{p}{\tilde{\xi}_{n}}\rangle}B\langle\frac{p}{\tilde{\xi}_{n}}\rangle$

where $-\hat{\otimes}-$ is the $(p,\tilde{\xi}_{n})$-complete tensor product.

###### Proof.

With all the notations used in the proof of Lemma 2.9, we have $A\to B$ is
$(p,\xi)$-complete faithfully flat, so does $A\to B\hat{\otimes}_{A}B$. So if
one replace $B$ with $B\hat{\otimes}_{A}B$ in Eq. (2) then we will get rCl
(B^⊗_A B)[X] ⊗_A[X] A[X]/(~ξ_n^k,~ξ_nX-p) & ≃ (B^⊗_A B)[p~ξn]/(p^k,ξ^km)
≃ (B⊗_A B)[X]/(p^k,ξ^km,~ξ_nX-p) where the last isomorphism is by definition.
On the other hand, we have

$B\langle\frac{p}{\tilde{\xi}_{n}}\rangle\hat{\otimes}_{A\langle\frac{p}{\tilde{\xi}_{n}}\rangle}B\langle\frac{p}{\tilde{\xi}_{n}}\rangle\simeq\varprojlim_{k}\big{(}(B\langle\frac{p}{\tilde{\xi}_{n}}\rangle{\otimes}_{A\langle\frac{p}{\tilde{\xi}_{n}}\rangle}B\langle\frac{p}{\tilde{\xi}_{n}}\rangle)\otimes_{A\langle\frac{p}{\tilde{\xi}_{n}}\rangle}A[X]/(p^{k},\xi^{km},\tilde{\xi}_{n}X-p)\big{)}.$

Using commuting with base extension property of tensor product, we have

$\big{(}B\langle\frac{p}{\tilde{\xi}_{n}}\rangle{\otimes}_{A\langle\frac{p}{\tilde{\xi}_{n}}\rangle}B\langle\frac{p}{\tilde{\xi}_{n}}\rangle\big{)}\otimes_{A\langle\frac{p}{\tilde{\xi}_{n}}\rangle}\big{(}A[X]/(p^{k},\xi^{km},\tilde{\xi}_{n}X-p)\big{)}$

is isomorphic to

$B[X]/(p^{k},\xi^{km},\tilde{\xi}_{n}X-p)\otimes_{A[X]/(p^{k},\xi^{km},\tilde{\xi}_{n}X-p)}B[X]/(p^{k},\xi^{km},\tilde{\xi}_{n}X-p)$

here we also use Eq. 2. Moreover, this is also isomorphic to Eq. 2. ∎

###### Proof of Lemma 2.8.

We have $A\langle\frac{p}{I_{n}}\rangle\to B\langle\frac{p}{I_{n}}\rangle$ is
$I_{n}$-complete faithfully flat by Lemma 2.9. So Lemma 2.8 follows from Lemma
2.10 and $I_{n}$-complete faithfully flat descent. ∎

######  Definition 2.11.

A crystal in finite free
$\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger,n}$-modules is a
sheaf $\mathcal{M}_{{{\mathbbl{\Delta}}}}$ of locally finite free
$\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger,n}$-modules such that
for all $(A,I)\to(B,IB)$ of prisms over $X$, we have the canonical map

$\mathcal{M}_{{{\mathbbl{\Delta}}}}((A,I))\widehat{\otimes}_{A}B\to\mathcal{M}_{{{\mathbbl{\Delta}}}}((B,IB))$

is an isomorphism. We will denote this category by
$\mathrm{Vect}(X_{{{\mathbbl{\Delta}}}^{\circ}},\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger,n})$.

######  Remark 2.12.

Using [BL22, Proposition 2.4.1], one could extend a crystal in
$\mathrm{Vect}(X_{{{\mathbbl{\Delta}}}^{\circ}},\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger,n})$
to a crystal in
$\mathrm{Vect}(X_{{{\mathbbl{\Delta}}}},\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger,n})$.
But we will actually only work with transversal (even oriented transversal)
prisms over $X$ in this paper.

######  Corollary 2.13.

Let $(A,I)\to(B,IB)$ be a covering of transversal prisms, then we have rCl
Vect(O_E,Δ^†,n(A,I)) ∼→ ←lim(Vect(O_E,Δ^†,n(B,I)) →→ Vect(O_E,Δ^†,n(B^⊗_A B))
→→→ ⋯) here $\mathrm{Vect}(R)$ is the category of finite projective modules of
$R$.

###### Proof.

This also follows from Lemma 2.9 and Lemma 2.10, then apply [Mat20, Theorem
5.8] or (3) in [BS21, Theorem 2.2]. ∎

We have for $n>0$, there is a natural map
$\iota_{n}\colon\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger,{n}}\to\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger,{n+1}}$,
and we define

$\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger}\coloneqq\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}}\displaylimits_{\iota_{n},n\geq
1}\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger,{n}}$

as sheaves over $X_{{{\mathbbl{\Delta}}}^{\circ}}$. On the other hand, we have
$\varphi_{{\mathbbl{\Delta}}}$ on $\mathcal{O}_{{\mathbbl{\Delta}}}$ defines a
map

$\varphi_{n}\colon\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger,{n}}\to\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger,{n+1}}$

which induces a map
$\varphi_{{\mathbbl{\Delta}}}\colon\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger}\to\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger}$.

######  Definition 2.14.

An overconvergent Laurent $F$-crystal over $X$ is a crystal
$\mathcal{M}_{{\mathbbl{\Delta}}}$ in finite free
$\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger}$-modules together
with an isomorphism

$\varphi_{\mathcal{M}_{{\mathbbl{\Delta}}}}\colon\varphi_{{\mathbbl{\Delta}}}^{\ast}\mathcal{M}_{{\mathbbl{\Delta}}}\simeq\mathcal{M}_{{\mathbbl{\Delta}}}.$

And we let
$\mathrm{Vect}(X_{{{\mathbbl{\Delta}}}^{\circ}},\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger})^{\varphi=1}$
to be category of overconvergent Laurent $F$-crystals over $X$.

Let’s recall the following general facts about sections on colimits of
sheaves.

######  Lemma 2.15 ([Sta20, Tag 0738]).

Let $\mathcal{C}$ be a site. Let $I\to\mathop{\rm Shv}\nolimits(\mathcal{C})$,
$i\mapsto\mathcal{F}_{i}$ be a filtered diagram of sheaves of sets. Let $U$ be
a quasi-compact object of $\mathcal{C}$ in the sense of [Sta20, Tag 090H].
Consider the canonical map

$\Psi\colon\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}}\displaylimits_{i}\mathcal{F}_{i}(U)\to(\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}}\displaylimits_{i}\mathcal{F}_{i})(U),$

where
$(\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}}\displaylimits_{i}\mathcal{F}_{i})$
is taken inside $\mathop{\rm Shv}\nolimits(\mathcal{C})$, then $\Psi$ is an
isomorphism if all transition maps over sections over $U$ are injective.

######  Remark 2.16.

We will see that all objects in $X_{{{\mathbbl{\Delta}}}^{\circ}}$ are quasi-
compact by definition. Moreover, all transition maps $\iota_{n}$ on sections
are injective by Corollary 2.21, so Lemma 2.15 implies taking colimits
commutes with taking sections for our definition of
$\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger,n}$.

###  2.1. Noetherian approximation via Breuil–Kisin prism

Objects in $X_{{{\mathbbl{\Delta}}}^{\circ}}$ can be very random, which makes
it very hard to understand sections of
$\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger,n}$ over them. The
strategy we are going to explain in this subsection is that we first
understand sections over prisms can be “approximated” by Noetherian ones, or
more precisely, the Breuil-Kisin prism, then descend properties from there to
general transversal prisms.

In this subsection, we only work on the case that $X=\mathop{\rm
Spf}\nolimits(\mathcal{O}_{K})$ with $K$ a $p$-adic field, but all results in
this subsection generalize to smooth affine formal schemes $\mathop{\rm
Spf}\nolimits(R)$ that are small, where we can also define the relative
Breuil–Kisin modules as in [DLMS22] using the framing map. Recall the
Breuil–Kisin prism $(\mathfrak{S},E)$ inside
$(\mathcal{O}_{K})_{{\mathbbl{\Delta}}}$ is defined as
$\mathfrak{S}=W[\\![u]\\!]$ and $E$ is an Eisenstein polynomial over $W$ that
is the minimal polynomial of a fixed uniformizer $\varpi\in\mathcal{O}_{K}$.
Let’s recall some basic properties of the Breuil–Kisin prisms.

######  Proposition 2.17.

$(\mathfrak{S},E)$ covers the final object $\ast$ in $\mathop{\rm
Shv}\nolimits(X_{{\mathbbl{\Delta}}})$ and $\mathop{\rm
Shv}\nolimits(X_{{{\mathbbl{\Delta}}}^{\circ}})$.

###### Proof.

This can be show using [BS22, Lemma 4.1], or more concretely one can show the
product of $(\mathfrak{S},E)$ with $(A,I)$ in $\mathop{\rm
Shv}\nolimits(X_{{\mathbbl{\Delta}}})$ can be computed as in Example 2.6 of
$loc.$ $cit.$ that we have for any prism $(A,I)\in X_{{\mathbbl{\Delta}}}$, we
have the product of $(A,I)$ and $(\mathfrak{S},E)$ is

$A[u]\\{\frac{u-v}{I}\\}^{\wedge}_{(p,I)}$

for some $v$ in $A$ that lifts the image of $\varpi$ under the structure map
$\mathcal{O}_{K}\to A/I$. ∎

######  Corollary 2.18.

Let $(A,I)\in X_{{{\mathbbl{\Delta}}}^{\circ}}$, and let $(B,EB)$ be the
product of $(A,I)$ and $(\mathfrak{S},E)$ in $X_{{\mathbbl{\Delta}}}$, then
the natural map $\mathfrak{S}\to B$ is classically fully faithful.

###### Proof.

First, recall that we have $IB=EB$ by the irreducibility of distinguished
elements. We have $(B,IB)$ covers $(A,I)$ from the last proposition, so by
Lemma 2.2, we have $(B,IB)$ is also transversal, i.e., $B/EB$ is $p$-torsion
free, which will imply that $\mathfrak{S}/E\to B/E$ is flat. Moreover, one can
also check that $(B,EB)$ is also a covering of $(\mathfrak{S},E)$ by the
$E$-torsionfreeness of $B$. Now we have $\mathfrak{S}\to B$ is also
classically faithfully flat since $\mathfrak{S}$ is Noetherian and using
[Sta20, Tag 0912]. ∎

######  Remark 2.19.

One should compare the above corollary with [BL22, Proposition 2.4.9].

######  Lemma 2.20.

Over the Breuil–Kisin prism

$\iota_{n,\mathfrak{S}}\colon\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger,n}((\mathfrak{S},E))\to\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger,n+1}((\mathfrak{S},E))$

is injective for all $n$.

Assume the above lemma, we will have

######  Corollary 2.21.

For any $(A,I)\in X_{{{\mathbbl{\Delta}}}^{\circ}}$, $\iota_{n,A}$ is
injective for all $n$. In particular by Lemma 2.15, we will have

$\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger}((A,I))=\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}}\displaylimits_{n}\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger,n}((A,I)).$

###### Proof.

Let $(B,IB)$ be as in Corollary 2.18, we have $\mathfrak{S}\to B$ is
faithfully flat, we first show that $\iota_{n,B}$ are injective for all $n$.
We can rewrite elements in $B\langle\frac{p}{I_{n}}\rangle$ inside
$B\langle\frac{p}{u^{N}}\rangle$ for some $N$, cf. Lemma 2.23. So it is enough
to show $B\langle\frac{p}{u^{N}}\rangle\to B\langle\frac{p}{u^{N+1}}\rangle$
is injective for all $N\in\mathbb N$. But this map is injective since it
modulo $u$ is injective by Lemma 2.20 and the flatness of $\mathfrak{S}\to B$.
Moreover, $B\langle\frac{p}{u^{N}}\rangle$ is $u$-separated, it is also
$u$-torsion free using a similar argument as in the last part of the proof of
Lemma 2.9.

To descent this to sections over $(A,I)$, let’s write
$S_{n}=B\langle\frac{p}{I_{n}}\rangle$ and
$R_{n}=A\langle\frac{p}{I_{n}}\rangle$, and for any ring $R$ over $A$, we
write $\overline{R}$ as the reduction of $R$ by $I_{n}$, then by the proof of
Lemma 2.9, we have
$\overline{S}_{n}=\overline{R}_{n}\otimes_{\overline{A}}\overline{B}$. Let $K$
be the kernel of the map $\overline{R}_{n}\to\overline{R}_{n+1}$, using the
fact that $\overline{A}\to\overline{B}$ is $p$-complete faithfully flat, we
have $K\equiv 0\mod p$. On the other hand, we have $\overline{A}$ is
classically $p$-completed, so in particular, $p$-adic separated, so $K=0$.
Repeat the last argument and using the fact that $R_{n}$ is classically
$(p,I)$-adically complete, we have $R_{n}\to R_{n+1}$ is also injective. ∎

######  Remark 2.22.

As a result, we have Corollary 2.13 still holds when replacing
$\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger,n}$ with
$\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger}$ since by Corollary
2.21, any finite projective module over
$\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger}(A,I)$ is defined
over $\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger,n}(A,I)$ for
some $n$. Moreover, again by Corollary 2.13, this “convergent radius” is
universally defined over all sections in $X_{{{\mathbbl{\Delta}}}^{\circ}}$.

######  Lemma 2.23.

For a nonzero prism $(A,I)\in X_{{{\mathbbl{\Delta}}}^{\circ}}$, assume that
an oriented prism $(A,I)$ is a covering of a prism which is Noetherian, then
there are $n_{0}<n_{1}$ depend on $n$, such that $n_{0}$ goes to $\infty$ as
$n$ goes to $\infty$. When $n>1$, $n_{0},n_{1}$ satisfy

$A\langle\frac{p}{{I}^{n_{0}}}\rangle[\frac{1}{I}]\subset\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger,n}((A,I))\subset
A\langle\frac{p}{{I}^{n_{1}}}\rangle[\frac{1}{I}].$

###### Proof.

We may assume $I=(d)$. Then by assumption, we have there is ($(p,I)$-complete)
faithfully flat map $A_{0}\to A$ with $A_{0}$ being Noetherian. Then we have
$A[p/d]$ is $d$-adically separated since $A_{0}[p/I]$ is. So in particular, we
can view $A\langle p/d\rangle$ as the closure of $A[p/d]$ for the $d$-adic
topology, and elements inside $A\langle p/d\rangle$ can be represented by
Cauchy series $\sum_{i\geq 0}a_{i}(p/d)^{i}$ with $a_{i}$ goes to $0$
$d$-adically. Here the last part is from the fact that if $\\{x_{n}\\}_{n\geq
0}\subset A[p/d]$ converges $d$-adically to some element that is not in
$A[p/d]$, we can always assume that $x_{n}$ is unbounded for the $d$-adic
topology inside $A[1/d]$.

We will prove a slightly general formula that will be used in our later
content. Assume we can find some $x\in(p,d)$ such that

$\varphi^{n}(d)=x^{p^{n}}+pu_{n}$

with $u_{n}$ being invertible for all $n\geq 1$. When $n=1$, the above
equation implies that $(x,p)$ and $(d,p)$ define the same topology. For the
statement of this lemma, one can check $x=d$ fulfill this requirement.

To prove this result, we have

$\frac{p}{\prod_{i=1}^{n}(x^{p^{i}}+pu_{i})}=\frac{p}{\prod_{i=1}^{n}x^{p^{i}}}\cdot\frac{1}{\prod_{i=1}^{n}(1+u_{i}\frac{p}{x^{p^{i}}})},$

And if we let $n_{1}=\sum_{i=1}^{n}p^{i}+n$, we will have

$A\langle\frac{p}{\prod_{i=1}^{n}(x^{p^{i}}+pu_{i})}\rangle\subset
A\langle\frac{p}{{x}^{n_{1}}}\rangle.$

On the other hand, if we let $n_{0}=\sum_{i=1}^{n}p^{i}$, we can show

$\frac{p}{x^{n_{0}}}=\frac{p}{\prod_{i=1}^{n}(x^{p^{i}}+pu_{i}-pu_{i})}=\frac{p}{\prod_{i=1}^{n}(x^{p^{i}}+pu_{i})}\frac{1}{\prod_{i=1}^{n}(1-u_{i}\frac{p}{x^{p^{i}}+pu_{i}})}.$

For each $i$ between $1$ and $n$, we have

$\frac{1}{(1-u_{i}\frac{p}{x^{p^{i}}+pu_{i}})}=\sum_{m=0}^{\infty}u_{i}^{m}\big{(}\frac{p}{x^{p^{i}}+pu_{i}}\big{)}^{m}=\sum_{m=0}^{\infty}u_{i}^{m}\big{(}\frac{p\prod_{0\leq
k\leq n,k\neq
i}(x^{p^{k}}+pu_{k})}{\prod_{k=0}^{n}(x^{p^{k}}+pu_{k})}\big{)}^{m}$

which is inside $A\langle\frac{p}{\prod_{i=1}^{n}(x^{p^{i}}+pu_{i})}\rangle$.
So this implies

$A\langle\frac{p}{{x}^{n_{0}}}\rangle[\frac{1}{x}]\subset
A\langle\frac{p}{\prod_{i=1}^{n}(x^{p^{i}}+pu_{i})}\rangle[\frac{1}{\prod_{i=1}^{n}(x^{p^{i}}+pu_{i})}].$

∎

###### Proof of Lemma 2.20.

In the proof of Lemma 2.23, if we let $(A,I)=(\mathfrak{S},E)$, and we have

$\varphi^{n}(E)=x^{p^{n}}+pu_{n}$

with $x=u^{e}$. So we have for all $n\geq 1$, we can regards
$\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger,n}((\mathfrak{S},E))$
and
$\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger,n+1}((\mathfrak{S},E))$
as subset of $\mathfrak{S}\langle\frac{p}{u^{N}}\rangle[\frac{1}{u}]$ for some
$N\gg 0$. Then it is not hard to see
$\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger,n}((\mathfrak{S},E))\subset\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger,n+1}((\mathfrak{S},E))$.
∎

With the notions in the proof of Lemma 2.23, if we write

$\mathcal{O}_{\mathcal{E}}\coloneqq\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}((\mathfrak{S},E))=\left\\{\,\sum_{i\in\mathbb
Z}\,a_{i}u^{i}\,\mid\,a_{i}\in W,\,\lim_{i\to-\infty}a_{i}=0\,\right\\},$

then we will have
$\mathcal{O}_{\mathcal{E}}^{\dagger}\coloneqq\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger,n}((\mathfrak{S},E))\subset\mathcal{O}_{\mathcal{E}}$.
We have the following generalizations of the above inclusion. For $k\in\mathbb
N$, let $(\mathfrak{S}^{k},E)$ be the $k+1$-th self-product of
$(\mathfrak{S},E)$ in $X_{{\mathbbl{\Delta}}}$, then we can have the natural
projections $p_{j}:\mathfrak{S}\to\mathfrak{S}^{k}$ are all classically
faithfully flat, cf., [DL21, Proposition 2.2.7] and [DLMS22, Lemma 3.5]. Let
$x\in\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger}(\mathfrak{S}^{k},E)$,
then by Lemma 2.23, there is $N\gg 0$ and we can write $E^{N}x=x^{+}$ with
$x^{+}=\sum\limits_{i=0}^{\infty}a_{i}(\frac{p}{E^{n}})^{i}$ for some
$n\in\mathbb N$ with $\\{a_{i}\\}$ converges $(p,E)$-adically to $0$ in
$\mathfrak{S}^{k}$. Since
$E^{ns}(\frac{p}{E^{n}})^{i}=p^{s}(\frac{p}{E^{n}})^{i-s}$, we may rewrite the
terms of $x^{+}$ so that $a_{i}\to 0$ $p$-adically in $\mathfrak{S}^{1}$.
Rewriting $x$ in this way induces a map
$c_{k}\colon\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger}(\mathfrak{S}^{k},E)\to\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}(\mathfrak{S}^{k},E)$.

One could check, as we mentioned at the beginning of this subsection, that if
$R$ is $p$-complete small $\mathcal{O}_{K}$-algebra in the sense of [DLMS22,
Definition 2.1], we could also define the (relative) Breuil–Kisin prism
$(\mathfrak{S},E)$ as in Example 3.4 of $loc.$ $cit.$, and we have the same
results carried out for the transversal prismatic site over $X=\mathop{\rm
Spf}\nolimits(R)$ and $(\mathfrak{S},E)$. In particular, the maps $c_{k}$ are
well-defined.

Now an object in
$\mathrm{Vect}(X_{{{\mathbbl{\Delta}}}},\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}})^{\varphi=1}$
(resp.
$\mathrm{Vect}(X_{{{\mathbbl{\Delta}}}^{\circ}},\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger})^{\varphi=1}$)
is equivalent to a pair $(\mathcal{M},f)$ with $\mathcal{M}$ an étale
$\varphi$-module over $\mathcal{O}_{\mathcal{E}}$ (resp.
$\mathcal{O}_{\mathcal{E}}^{\dagger}$) together with an $\varphi$-equivariant
descent isomorphism over
$\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}((\mathfrak{S}^{1},E))$ (resp.
$\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger,n}((\mathfrak{S}^{1},E))$).
Then base change along $c_{k}$ defines a functor:

(2)
$D_{X}:\mathrm{Vect}(X_{{{\mathbbl{\Delta}}}^{\circ}},\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger})^{\varphi=1}\to\mathrm{Vect}(X_{{{\mathbbl{\Delta}}}},\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}})^{\varphi=1}.$

And we can restrict this functor over the perfect site, and we get

(3) $D_{X}^{\mathop{\rm
perf}\nolimits}:\mathrm{Vect}(X_{{{\mathbbl{\Delta}}}^{\circ}}^{\mathop{\rm
perf}\nolimits},\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger})^{\varphi=1}\to\mathrm{Vect}(X_{{\mathbbl{\Delta}}}^{\mathop{\rm
perf}\nolimits},\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}})^{\varphi=1}.$

We made the following conjecture:

######  Conjecture 2.24.

If $R$ is $p$-adically complete small $\mathcal{O}_{K}$-algebra, and let
$X=\mathop{\rm Spf}\nolimits(R)$, then $D_{X}$ (resp. $D_{X}^{\mathop{\rm
perf}\nolimits}$) defines an equivalence of categories.

######  Remark 2.25.

One could define sheaves
$\mathcal{O}_{\mathcal{E},qrsp}\coloneqq{{\mathbbl{\Delta}}}_{\bullet}[1/I]^{\wedge}_{p}$
and
$\mathcal{O}_{\mathcal{E},qrsp}^{\dagger}\coloneqq\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
<EMAIL_ADDRESS>over $X_{qrsp}$ and $X_{qrsp}^{\circ}$ respectively similar as in [BS21, §6].
Here $X_{qrsp}$ is defined in Definition 2.9 of $loc.$ $cit.$ and
$X_{qrsp^{\circ}}$ is the full subcategory of $X_{qrsp}$ consisting of quasi-
regular semiperfectoid rings over $X$ that are $p$-torsion free. One could
also define
$\mathrm{Vect}(X_{qrsp^{\circ}},\mathcal{O}_{\mathcal{E},qrsp}^{\dagger})^{\varphi=1}$
similar to Definition 2.14, and the comparison result in [BS21, Proposition
2.1] gives an equivalence

$\mathrm{Vect}(X_{qrsp^{\circ}},\mathcal{O}_{\mathcal{E},qrsp}^{\dagger})^{\varphi=1}\simeq\mathrm{Vect}(X_{{{\mathbbl{\Delta}}}^{\circ}},\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger})^{\varphi=1}.$

In particular Conjecture 2.24 has a version over the quasi-syntomic sites. We
do not pursue this direction here.

######  Remark 2.26.

From Lemma 2.23, when $X=\mathop{\rm Spf}\nolimits(\mathcal{O}_{K})$, we have
$\mathcal{O}_{\mathcal{E}}^{\dagger}$ defined as above is equal to the
overconvergent period ring used in the theory of $(\varphi,\tau)$-modules.
Beyond the Kummer tower case, one could compare
$\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger,n}((A,d))$ with the
period rings used in the work of Kedlaya-Liu in [KL19, §5] or [Ked13] defined
using $\theta$-maps.

##  3\. Overconvergence of Laurent $F$-crystals

In this section, we will assume $X$ to be $\mathop{\rm
Spf}\nolimits(\mathcal{O}_{K})$. But some results, especially those in §3.1
carries over more general bases.

###  3.1. Perfect overconvergent theory

We consider the transversal perfect prismatic site
$X_{{{\mathbbl{\Delta}}}^{\circ}}^{\mathop{\rm perf}\nolimits}$ over $X$,
i.e., objects in $X_{{{\mathbbl{\Delta}}}^{\circ}}^{\mathop{\rm
perf}\nolimits}$ are perfect prisms $(A,I)$ over $\mathcal{O}_{K}$ such that
$A/I$ is $\mathcal{O}_{K}$-flat. Let $(A_{\mathrm{inf}},\xi)$ be the Fontaine
prism over $\mathcal{O}_{K}$.

######  Lemma 3.1.

The sheaf represented by $(A_{\mathrm{inf}},\xi)$ covers the final object in
$X_{{{\mathbbl{\Delta}}}^{\circ}}^{\mathop{\rm perf}\nolimits}$.

###### Proof.

It is a well-known fact that the sheaf represented by $(A_{\mathrm{inf}},\xi)$
covers the final object in $X_{{{\mathbbl{\Delta}}}}^{\mathop{\rm
perf}\nolimits}$, which can be directly deduced from [BS22, Proposition 7.10
and Proposition 7.11]. Then the result follows from Proposition 2.1. ∎

It is known that the category of perfect prisms over $X$ is equivalent to the
category of integral perfectoid rings over $\mathcal{O}_{K}$, so it admits
finite non-empty coproducts. We will denote
$((A_{\mathrm{inf}}^{\bullet})_{\mathop{\rm perf}\nolimits},\xi)$ be the Čech
nerve of $(A_{\mathrm{inf}},\xi)$ in $X_{{\mathbbl{\Delta}}}$. One can
identify $A_{\mathrm{inf}}^{i}$ with ${{\mathbbl{\Delta}}}_{R^{i}}$ where
$R^{i}$ is the $i$-th complete self-tensor product of $\mathcal{O}_{C}$ over
$\mathcal{O}_{K}$, and $(A_{\mathrm{inf}}^{i})_{\mathop{\rm perf}\nolimits}$
with $({{\mathbbl{\Delta}}}_{R^{i}})_{\mathop{\rm perf}\nolimits}$, then one
can apply [BS22, Proposition 7.10 and Proposition 7.11] to see
$((A_{\mathrm{inf}}^{i})_{\mathop{\rm perf}\nolimits},\xi)$ are transversal
for $i>0$. For any topological ring, let $C^{0}(G_{K}^{i},R)$ be the ring of
continuous functions from $G_{K}^{i}$ to $R$. Let
$\tilde{\xi}_{n}=\prod_{i=1}^{n}\varphi^{i}(\xi)$.

######  Lemma 3.2.

  1. (1)

$(A_{\mathrm{inf}}^{i})_{\mathop{\rm perf}\nolimits}$ is almost isomorphic to
$C^{0}(G_{K}^{i},A_{\mathrm{inf}})$ for $i>0$;

  2. (2)

$(A_{\mathrm{inf}}^{i})_{\mathop{\rm perf}\nolimits}[1/\xi]^{\wedge}_{p}$ is
isomorphic to $C^{0}(G_{K}^{i},W(C^{\flat}))$;

  3. (3)

$(A_{\mathrm{inf}}^{i})_{\mathop{\rm
perf}\nolimits}\langle\frac{p}{\tilde{\xi}_{n}}\rangle[\frac{1}{\tilde{\xi}_{n}}]$
is isomorphic to
$C^{0}(G_{K}^{i},A_{\mathrm{inf}}\langle\frac{p}{\tilde{\xi}_{n}}\rangle[\frac{1}{\tilde{\xi}_{n}}])$.

###### Proof.

We will have
$C^{0}(G_{K}^{i},A_{\mathrm{inf}}\langle\frac{p}{\tilde{\xi}_{n}}\rangle[\frac{1}{\tilde{\xi}_{n}}])\simeq
C^{0}(G_{K}^{i},A_{\mathrm{inf}})\langle\frac{p}{\tilde{\xi}_{n}}\rangle[\frac{1}{\tilde{\xi}_{n}}]$
by the compactness of $G_{K}$, and one can check $(1)$ implies $(2)$ and
$(3)$. We have $(1)$ follows from the theory of diamond and [Sch17, Example
3.22]. For the arguments using diamonds, one can refer to [Wu21, Lemma 5.3] or
[MW21, Lemma 2.13]. ∎

We will also denote $A_{\mathrm{inf}}$ by $\tilde{\mathbf{A}}^{+}$, and we let
$\tilde{\mathbf{A}}=W(\overline{K}^{\flat})$. For $r>0$ let

$\tilde{\mathbf{A}}^{\dagger,r}=\\{\sum_{n\geq 0}[a_{n}]p^{n}|a_{n}\in
C^{\flat}\,,\,a_{n}(p^{n/r})\to 0\\},$

and $\tilde{\mathbf{A}}^{\dagger}=\cup_{r>0}\tilde{\mathbf{A}}^{\dagger,r}$.
We have the following result.

######  Theorem 3.3.

Base extension of étale $(\varphi,\Gamma)$-modules from
$\tilde{\mathbf{A}}^{\dagger}$ to $\tilde{\mathbf{A}}$ is an equivalence of
categories, and both categories are equivalent to the category of continuous
representations of $G_{K}$ on finite free $\mathbb Z_{p}$-modules.

###### Proof.

In the proof of Lemma 2.23, if we let $d=[\underline{p}^{\flat}]-p$ and
$x=[\underline{p}^{\flat}]$, then we will have for each $n\in\mathbb N$, there
is $r,s>0$ such that

$\tilde{\mathbf{A}}^{\dagger,r}\subset\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger,n}((A_{\mathrm{inf}},\xi))\subset\tilde{\mathbf{A}}^{\dagger,s}.$

And the rest of this theorem becomes a well-known fact. Actually, we have the
corresponding categories of étale $\varphi$-modules are both equivalent to the
category of finite free $\mathbb Z_{p}$-modules. For statement about étale
$\varphi$-modules, one can refer to [KL15, Theorem 8.5.3]. On the other hand,
the statement about étale $(\varphi,\Gamma)$-modules is due to Lemma 3.2 and
[Ked15, Theorem 2.4.5]. ∎

######  Corollary 3.4.

We have (3) is an equivalence of categories.

###  3.2. The deperfection

In this section, we will show

######  Theorem 3.5.

If $K$ is a finite extension of $\mathbb Q_{p}$, and let $X=\mathop{\rm
Spf}\nolimits(\mathcal{O}_{K})$, we have (2) is an equivalence of categories.

###### Proof of fully faithfulness in Theorem 3.5.

It reduces to prove the functors

(4)
$\mathrm{Vect}(\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger}(\mathfrak{S}^{i},E))^{\varphi=1}\to\mathrm{Vect}(\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}(\mathfrak{S}^{i},E))^{\varphi=1}$

induced by base change along $c_{k}$ defined in §2.1 is fully faithful when
$k=0$ and faithful when $k=1,2$, where $\mathrm{Vect}(R)^{\varphi=1}$ denotes
the category of étale $\varphi$-modules over $R$. The fully faithfulness of
(4) when $k=0$ will follow from the fully faithfulness of the classical theory
of étale $\varphi$-modules, for instance, [KL19, Proposition 5.4.8]. When
$k=1,2$, it is enough to prove that $c_{k}$ are injective.

Let $x$ and $x^{+}$ be as in the discussions when defining $c_{k}$. To show
the injectivity of $c_{k}$ we may assume $x=x^{+}$ which is equal to
$\sum\limits_{i=0}^{\infty}a_{i}(\frac{p}{E^{n}})^{i}$ for some $n\in\mathbb
N$ with $\\{a_{i}\\}$ converges $p$-adically to $0$ in $\mathfrak{S}^{k}$.
Suppose that $c_{k}(x)=0$ in
$\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}(\mathfrak{S}^{k},E).$ For
$m\in N$, set

$x_{m}=\sum\limits_{i=0}^{m}a_{i}(\frac{p}{E^{n}})^{i}.$

It is enough to show for any $l\in\mathbb N$, $x_{m}$ are in
$p^{l}\mathfrak{S}^{k}[\frac{p}{E^{n}}]$ for $m$ sufficient large. Since
$a_{i}$ converges to $0$ $p$-adically in $\mathfrak{S}^{k}$ and $x=0$ in
$\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}(\mathfrak{S}^{k},E)$, we can
choose $m$ sufficiently large so that $x_{m}\in
p^{m+l}\mathfrak{S}^{k}[\frac{1}{E}]\subset
p^{m+l}\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}(\mathfrak{S}^{k},E)$. So
$y_{m}:=E^{mn}x_{m}\in\mathfrak{S}^{k}\cap
p^{m+l}\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}(\mathfrak{S}^{k},E)$.
Note that $\mathfrak{S}^{k}/p\mathfrak{S}^{k}$ injects to
$\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}(\mathfrak{S}^{k},E)/p\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}(\mathfrak{S}^{k},E)=\mathfrak{S}^{k}/p\mathfrak{S}^{k}[\frac{1}{u}]$
and $\mathfrak{S}^{k}$ is $p$-adically separated, both because
$\mathfrak{S}^{k}$ is faithfully flat over $\mathfrak{S}$ along any of the
projection maps. Therefore, $y_{m}$ is in $\mathfrak{S}^{k}\cap
p^{m+l}\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}(\mathfrak{S}^{k},E)=p^{m+l}\mathfrak{S}^{k}$.
Say $y_{m}=p^{m+l}z$ with $z\in\mathfrak{S}^{k}$. Then
$x_{m}=p^{l}(\frac{p}{E^{n}})^{m}\in p^{l}\mathfrak{S}^{k}[\frac{p}{E^{n}}]$
as required. ∎

We will use the same method introduced in [GL20a] to show the essential
surjectivity, except we will replace the theory of $(\varphi,\hat{G})$-modules
developed in [Liu10] with the theory of prismatic $(\varphi,\hat{G})$-modules
developed in [DL21]. Let $T$ be a $\mathbb Z_{p}$-representation of $G_{K}$
and let $(\mathcal{M},\varphi_{\mathcal{M}},\hat{G})$ be the
$(\varphi,\tau)$-module associated with $T$ in the sense of [DL21, Theorem
4.2.3]. Let $T_{n}\coloneqq T/p^{n}T$, then we have $T_{n}$ correspondence to
$\mathcal{M}_{n}=\mathcal{M}/p^{n}\mathcal{M}$ which is from the
correspondences studied in [Car13].

We will summarize the result in [GL20a, §6] in the form we need. In the
following, $M_{d}(R)$ denote the set if $d\times d$-matrices with entries in
$R$.

######  Theorem 3.6 ([GL20b]).

There is a basis $\\{e_{j}^{(n)}\\}_{j=1}^{d}$ of $\mathcal{\mathcal{M}}_{n}$,
such that $\\{\lim_{n}e_{j}\\}_{j=1}^{d}$ forms a basis of
$M=\varprojlim\mathcal{M}_{n}$. For each $n$, there is a
$\mathfrak{S}$-sublattice $\mathfrak{M}_{(n)}$ of $\mathcal{M}_{n}$, i.e,
$\mathcal{M}_{n}=\mathfrak{M}_{(n)}[1/u]$. Moreover, $\mathfrak{M}_{(n)}$ is
generated over $\\{\mathfrak{e}_{(n),j}^{(i)}\mid j\leq
n\\}\subset\mathcal{M}_{n}$ as $\mathfrak{S}$-module, satisfying that

  1. (1)

$(e_{j}^{(n)})=(\mathfrak{e}^{(n)}_{(n),j})Y_{n}^{-1}$ with
$Y_{n},Y_{n}^{-1}\in M_{d}(\mathfrak{S})$;

  2. (2)

$\mathfrak{e}^{(i)}_{(n),j}=\mathfrak{e}^{(n)}_{(n),j}Y_{i,n}$ for $Y_{i,n}\in
M_{d}(\mathfrak{S}[p/u^{2h}])$ for some $h>0$;

  3. (3)

for each $n>0$, $\mathfrak{M}_{(n)}$ is a quotient of two finite free Kisin
module $\mathfrak{M}$ (may different for each $n$) attached to a crystalline
representation of $G_{K}$ in the sense of [GL20a, Definition 4.1.1].

######  Corollary 3.7.

There is a constant $n=n(T,K)\in\mathbb N$, and a basis $\\{e_{j}\\}$ of
$\mathcal{M}$ such that $\varphi_{M}$ (resp. $\tau$) acts on $\\{e_{j}\\}$ by
a matrix inside
$M_{d}(\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger,n}((\mathfrak{S},E)))$
(resp
$M_{d}(\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger,n}((\mathfrak{S}^{1},E)))$).

###### Proof.

This is from [GL20a, §6] and replace the theory of $(\varphi,\hat{G})$-modules
with the theory of prismatic $(\varphi,\hat{G})$-modules attached with
crystalline representations. By [DL21, Corollary 3.3.4], we have $\tau$ acts
on $\mathfrak{e}^{(n)}_{(n),j}$ by a matrix in $M_{d}(\mathfrak{S}^{1})$. Here
the main difference between classical $(\varphi,\hat{G})$-modules with
prismatic $(\varphi,\hat{G})$-modules is that they are differed by a Frobenius
twist for the definition of $\hat{G}$-action, and this enables us to have a
better estimation of the $\tau$-action. ∎

###### Proof of Theorem 3.5.

It remains to prove the essential surjectivity of $D_{X}$. For any Laurent
$F$-crystals, we attach it a prismatic étale $(\varphi,\tau)$-module as in
[DL21, §4.2]. Let’s briefly recall our setup, we fix a topological generator
$\tau$ of $\mathop{\rm Gal}\nolimits(L/K_{1^{\infty}})$, we can define a map
$\mathfrak{S}^{1}\to A_{\mathrm{inf}}$ induced by the two maps
$\mathfrak{S}\hookrightarrow A_{\mathrm{inf}}$ and
$\mathfrak{S}\hookrightarrow
A_{\mathrm{inf}}\xrightarrow[]{\tau}A_{\mathrm{inf}}$. This map is injective
by Corollary 2.4.5 of $loc.$ $cit.$, and induces natural maps
$\mathfrak{S}^{1}[1/E]^{\wedge}_{p}\to W(C^{\flat})$ and
$\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger}((\mathfrak{S}^{1},E))\to\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger}((A_{\mathrm{inf}},\xi))$.
We have a Laurent $F$-crystal is equivalent to an étale $\varphi$-module
$\mathcal{M}$ over $\mathcal{O}_{\mathcal{E}}$ together with a descent
isomorphism over $\mathfrak{S}^{1}[1/E]^{\wedge}_{p}$. Then in §4.2 of $loc.$
$cit.$ we show that if we base change the descent isomorphism along
$\mathfrak{S}^{1}[1/E]^{\wedge}_{p}\to W(C^{\flat})$, then it is exactly the
$\tau$-action in the definition of $(\varphi,\tau)$-modules. By Corollary 3.7,
we have the matrix defines the $\tau$-action actually has coefficients inside
$\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger,n}((\mathfrak{S}^{1},E))$
for some $n$. The linearization of this $\tau$-action defines a descent
isomorphism of an overconvergent Laurent $F$-crystal. ∎

##  4\. Overconvergence of $(\varphi,\Gamma)$-modules associated with
$\Phi$-iterate tower.

Let $K$ be as in §2.1, and we fix a polynomial $\Phi$ in $W[X]$ such that
$\Phi(X)\equiv X^{p}\mod p$ and $\Phi(0)=0$. We fix a uniformizer
$\varpi\in\mathcal{O}_{K}$ and also fix a system
$\vec{\nu}=\\{\nu_{i}\\}_{i\geq 0}$ in $\overline{K}$ with $\nu_{0}=\varpi$
and $\Phi(\nu_{n+1})=\nu_{n}$. Let $K_{n}=K(\nu_{n})$, then define
$K_{\infty}=K_{\infty,\vec{\nu}}=\cup_{n}K_{n}$ and $G_{\infty}=\mathop{\rm
Gal}\nolimits(\overline{K}/K_{\infty})$. We define
$\mathfrak{S}_{\Phi}=W[\\![u]\\!]$ and define the $\delta$-structure of
$\mathfrak{S}_{\Phi}$ determined by $\varphi(u)=\Phi(u)$. Let $E(u)$ be the
minimal polynomial of $\varpi$ over $K_{0}$, we have $(\mathfrak{S}_{\Phi},E)$
is a transversal prism over $\mathcal{O}_{K}$. We define
$\iota\colon\mathfrak{S}_{\Phi}\to A_{\mathrm{inf}}$ by
$u\mapsto\\{[\underline{\nu}_{n}^{\flat}]\\}_{\Phi}$, where
$\\{\cdot\\}_{\Phi}$ is the unique set-theoretic section
$\mathcal{O}_{C}^{\flat}\to A_{\mathrm{inf}}$ to the reduction modulo $p$ and
satisfying $\varphi(\\{x\\}_{\Phi})=\Phi(\\{x\\}_{\Phi})$ for all
$x\in\mathcal{O}_{C}^{\flat}$, cf., [Col08, Lemme 9.3] and also [CL16, Lemma
2.1.1]. We will let
$\mathcal{O}_{\mathcal{E},\Phi}=\mathfrak{S}_{\Phi}[1/E]^{\wedge}_{p}$. Then
$\iota$ defines a map of prisms
$(\mathfrak{S}_{\Phi},E)\to(A_{\mathrm{inf}},\xi)$ and also
$\mathcal{O}_{\mathcal{E},\Phi}\to W(C^{\flat})$. One has
$\mathfrak{S}_{\Phi}\subset A_{\mathrm{inf}}^{G_{\infty}}$.

######  Definition 4.1.

For $\ast\in\\{\emptyset,\dagger\\}$, we define the category
$\mathrm{Mod}^{\text{\'{e}t},\ast}_{\Phi}(\varphi,\Gamma)$ of étale
$(\varphi,\Gamma)$-modules associated with the $\Phi$-iterate tower
$\vec{\nu}$ consists of triples
$(\mathcal{M}^{\ast},\varphi_{\mathcal{M}^{\ast}},\widehat{\mathcal{M}}^{\ast})$
where $(\mathcal{M}^{\ast},\varphi_{\mathcal{M}^{\ast}})$ is an étale
$\varphi$-module over
$\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\ast}((\mathfrak{S}_{\Phi},E))$,
and
$\widehat{\mathcal{M}}^{\ast}\coloneqq\mathcal{M}^{\ast}\otimes_{\iota}\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\ast}((A_{\mathrm{inf}},\xi))$
is equipped with a semilinear $G_{K}$-action such that regarding
$\mathcal{M}^{\ast}$ as an
$\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\ast}((\mathfrak{S}_{\Phi},E))$-submodule
in $\widehat{\mathcal{M}}^{\ast}$, we have
$\mathcal{M}^{\ast}\subset(\widehat{\mathcal{M}}^{\ast})^{G_{\infty}}$.

Given
$(\mathcal{M},\varphi_{\mathcal{M}},\widehat{\mathcal{M}})\in\mathrm{Mod}^{\text{\'{e}t}}_{\Phi}(\varphi,\Gamma)$
associated with the $\Phi$-iterate tower $\vec{\nu}$ as above, we define
$T=\widehat{\mathcal{M}}^{\varphi=1}$. Given $T\in\mathrm{Rep}_{\mathbb
Z_{p}}(G_{K})$, a $\mathbb Z_{p}$-Galois representation of $G_{K}$, and for
$\ast\in\\{\emptyset,\dagger\\}$, we define
$(\mathcal{M}^{\ast}(T|_{G_{\infty}}),\varphi_{\mathcal{M}^{\ast}(T|_{G_{\infty}})},\widehat{\mathcal{M}}^{\ast}(T))$
with $\widehat{\mathcal{M}}^{\ast}=T\otimes_{\mathbb
Z_{p}}\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\ast}((A_{\mathrm{inf}},\xi))$
and
$(\mathcal{M}^{\ast}(T|_{G_{\infty}}),\varphi_{\mathcal{M}^{\ast}(T|_{G_{\infty}})})$
the étale $\varphi$-module over associated with $T|_{G_{\infty}}$ as in
[GL20a, §2.5]. One can show $T$ induces an equivalence of
$\mathrm{Mod}^{\text{\'{e}t}}_{\Phi}(\varphi,\Gamma)$ and
$\mathrm{Rep}_{\mathbb Z_{p}}(G_{K})$ using [CL16, Theorem 3.2.2].

######  Theorem 4.2.

Bese change induces an equivalence of categories between
$\mathrm{Mod}^{\text{\'{e}t},\dagger}_{\Phi}(\varphi,\Gamma)$ and
$\mathrm{Mod}^{\text{\'{e}t}}_{\Phi}(\varphi,\Gamma)$.

###### Proof.

We will show all objects in
$\mathrm{Mod}^{\text{\'{e}t}}_{\Phi}(\varphi,\Gamma)$ are overconvergent in
the sense that if $(\mathcal{M},\varphi_{\mathcal{M}},\widehat{\mathcal{M}})$
is associated with $T\in\mathrm{Rep}_{\mathbb Z_{p}}(G_{K})$, then
$(\mathcal{M},\varphi_{\mathcal{M}},\widehat{\mathcal{M}})$ comes from the
base change from
$(\mathcal{M}^{\dagger}(T|_{G_{\infty}}),\varphi_{\mathcal{M}^{\dagger}(T|_{G_{\infty}})},\widehat{\mathcal{M}}^{\dagger}(T))$.
This follows from Theorem 3.5 by evaluation an overconvergent Laurent
$F$-crystal to the diagram $(\mathfrak{S}_{\Phi},E)\to(A_{\mathrm{inf}},\xi)$
and using
$\mathcal{M}^{\ast}\subset(\widehat{\mathcal{M}}^{\ast})^{G_{\infty}}$. ∎

######  Corollary 4.3.

There is a uniform bound for the overconvergence radii of overconvergent étale
$(\varphi,\Gamma)$-modules associated with the same Galois representation for
different $\Phi$-iterate towers.

###### Proof.

This comes from the proof of Theorem 4.2 and the fact that for an
overconvergent Laurent $F$-crystal, since it is finitely generated, it is
defined over $\mathcal{O}_{\mathcal{E},{{\mathbbl{\Delta}}}}^{\dagger,n}$ for
some $n$. ∎

##  Appendix A Arithmetic Kummer tower is not decompleting

We will show arithmetic Kummer tower over $\mathbb Q_{p}$ is not decompleting
in the sense of [KL19, Definition 5.6.1] by showing if it is decompleting,
then all $p$-adic representations of $G_{\mathbb Q_{p}}$ are potentially
unramified. In other words [KL19, Conjecture 6.2.6] does not hold.

Let $K=\mathbb Q_{p}$ and $K_{n}=K(\varpi_{n})$, where $\varpi_{0}=p$ and
$\\{\varpi_{n}\\}$ satisfies $\varpi_{n}=\varpi_{n+1}^{p}$. We define
$K_{n}^{\prime}$ be the normalizations of $K_{n}$ in $\overline{K}$. Let
$L=\cup K_{n}$ and $L^{\prime}=\cup K_{n}^{\prime}$. Then
$\\{(K_{n},\mathcal{O}_{K_{n}})\\}$ and
$\\{(K_{n}^{\prime},\mathcal{O}_{K_{n}^{\prime}})\\}$ form perfectoid finite
étale towers over $K$ in the sense of [KL19, §5.1], so we can construct the
period rings of type $\mathbf{A}$ for this two towers (where we let $E=\mathbb
Z_{p}$ as in your notions), $\widetilde{\mathbf{A}}^{*}$ for the $\\{K_{n}\\}$
tower and $\widetilde{\mathbf{A}}^{\prime*}$ for the $\\{K^{\prime}_{n}\\}$
tower, and also define their overconvergent version $\mathbf{A}^{*}$ and
$\mathbf{A}^{\prime*}$, with $*\in\\{r>0,\dagger\\}$ as been defined in
§5.2.1. of $loc.$ $cit$.

Now let’s assume that Kummer towers are decompleting, i.e., [KL19, Conjecture
6.2.6] holds, then we should have Theorem 5.7.3 of $loc.$ $cit.$ holds so that
the category of ($\varphi,\Gamma$)-modules over $\mathbf{A}^{\dagger}$ and the
category of ($\varphi,\Gamma$)-modules over $\widetilde{\mathbf{A}}^{\dagger}$
are equivalent via the base change functor. If we use the interpretations in
Example 5.5.7 of $loc.$ $cit$., ($\varphi,\Gamma$)-modules over
$\mathbf{A}^{\dagger}$ is the same as a finite projective $\varphi$-module $M$
over $\mathbf{A}^{\dagger}$ together with a semilinear $\mathop{\rm
Gal}\nolimits(L^{\prime}/K)$ action on
$M\otimes_{\mathbf{A}^{\dagger}}\mathbf{A}^{\prime\dagger}$ commutes with
$\varphi$ and whose restriction to $\mathop{\rm Gal}\nolimits(L^{\prime}/L)$
fixes $M$. Let’s assume $M$ is actually defined over $\mathbf{A}^{r}$, and we
choose $n\gg 0$ such that $\theta\circ\varphi^{-n}$ is defined, and let $T$ be
the $\mathbb Z_{p}$-Galois representation corresponds to $M$, then we will
have $T\otimes_{\mathbb Z_{p}}\mathbb
C_{p}=M\otimes_{\theta\circ\varphi^{-n}}\mathbb C_{p}$. Moreover, we have by
the definition of $\mathbf{A}^{r}$, the action of $\mathop{\rm
Gal}\nolimits_{K}$ on $M\otimes_{\theta\circ\varphi^{-n}}\mathbb C_{p}$
actually factor through the an action of $\mathop{\rm
Gal}\nolimits(L^{\prime}/K)$ on
$M\otimes_{\theta\circ\varphi^{-n}}K^{\prime}_{n}$, and there is a basis of
$M\otimes_{\theta\circ\varphi^{-n}}K^{\prime}_{n}$ fixed by $\mathop{\rm
Gal}\nolimits(L^{\prime}/L)$.

Now we claim that under the notions as above, we can show that there is an
open subgroup of $\mathop{\rm Gal}\nolimits(L^{\prime}/K)$ fixes a basis of
$M\otimes_{\theta\circ\varphi^{-n}}K^{\prime}_{n}$, and this will imply that
$T$ is $\mathbb C_{p}$-admissible, thus potentially unramified by a theorem of
Sen [Sen80, §3.2]. To show this claim, we have $\mathop{\rm
Gal}\nolimits(L^{\prime}/K^{\prime}_{n})$ acts linearly on
$M\otimes_{\theta\circ\varphi^{-n}}K^{\prime}_{n}$ and $H=\mathop{\rm
Gal}\nolimits(L^{\prime}/K^{\prime}_{n})\cap\mathop{\rm
Gal}\nolimits(L^{\prime}/L)$ in the kernel of this action, it remains to show
that the normalization of $H$ in $\mathop{\rm
Gal}\nolimits(L^{\prime}/K^{\prime}_{n})$ is $\mathop{\rm
Gal}\nolimits(L^{\prime}/K^{\prime}_{n})$. Then this is actually from Kummer
theory since $K^{\prime}_{n}$ contains both $p^{n}$-th roots of unit and a
$p^{n}$-th root of $p$. The above can also be proven by a direct computation
by identifying $\mathop{\rm Gal}\nolimits(L^{\prime}/K^{\prime}_{n})$ with
$\mathbb Z_{p}\rtimes\mathbb Z_{p}$ when $n\gg 0$.

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