# A prismatic approach to $(\varphi,\hat{G})$-modules and $F$-crystals

Heng Du  Department of Mathematics, Purdue University<EMAIL_ADDRESS>and
Tong Liu  Department of Mathematics, Purdue University
<EMAIL_ADDRESS>
###### Abstract.

We give a new construction of $(\varphi,\hat{G})$-modules using the theory of
prisms developed by Bhatt and Scholze. As an application, we give a new proof
about the equivalence between the category of prismatic $F$-crystals in finite
locally free $\mathcal{O}_{{\mathbbl{\Delta}}}$-modules over
$(\mathcal{O}_{K})_{{\mathbbl{\Delta}}}$ and the category of lattices in
crystalline representations of $G_{K}$, where $K$ is a complete discretely
valued field of mixed characteristic with perfect residue field. We also
generalize this result to semi-stable representations using the absolute
logarithmic prismatic site defined by Koshikawa.

###### Contents

  1. 1 Introduction
  2. 2 Ring Structures on certain prismatic envelope
    1. 2.1 Construction of $A^{(2)}$
    2. 2.2 The ring $A^{(2)}_{\max}$
    3. 2.3 The ring $A^{(2)}_{\mathop{\rm st}\nolimits}$
    4. 2.4 Embedding $A^{(2)}$ and $A^{(2)}_{\mathop{\rm st}\nolimits}$ to $A_{\mathrm{inf}}$
  3. 3 Application to semi-stable Galois representations
    1. 3.1 Kisin module attached to semi-stable representation
    2. 3.2 Descent of the $G_{K}$-action
    3. 3.3 Relation to $(\varphi,\hat{G})$-modules
  4. 4 Crystalline representations and prismatic $F$-crystals
    1. 4.1 Prismatic $F$-crystals in finite projective modules
    2. 4.2 $(\varphi,\tau)$-modules and prismatic $F$-crystals
    3. 4.3 Proofs of Proposition 3.2.2 and Theorem 4.1.10
  5. 5 Logarithmic prismatic $F$-crystals and semi-stable representations
  6. 6 Some discussions on base rings

##  1\. Introduction

Let $K$ be a complete discretely valued field of mixed characteristic with
perfect residue field $k$. Fix a separable closure of $\overline{K}$ of $K$
and let $G_{K}$ be the absolute Galois group of $K$. The study of stable
lattices in crystalline representations of $G_{K}$ plays an important role in
number theory. For example, in many modularity lifting results, one wants to
understand liftings of mod $p$ representations of the Galois group of a number
field $F$ to Galois representations over $\mathbb Z_{p}$-lattices with nice
properties when restricted to the Galois groups of $F_{v}$ for all places $v$
of $F$. And a reasonable property at places over $p$ is that the
representation of the Galois group of the local field is crystalline. There
are various theories about characterizing $G_{K}$-stable lattices in
crystalline representations, for example, theory of strongly divisible
lattices of Breuil(cf. [Bre02]), Wach modules(cf. [Wac96] and [Ber04]), Kisin
modules(cf. [Kis06]), Kisin-Ren’s theory(cf. [KR09]) and the theory of
$(\varphi,\widehat{G})$-modules(cf. [Liu10]). The theories above state that
one can describe lattices in crystalline representations using certain linear
algebraic data over certain commutative rings $A$.

In a recent work of Bhatt-Scholze[BS21], they give a different
characterization of the category of lattices in crystalline representations.
To explain their result, let $\mathcal{O}_{K}$ be the ring of integers in $K$,
and they consider the absolute prismatic site
$(\mathcal{O}_{K})_{{{\mathbbl{\Delta}}}}$, which is defined as the opposite
category of all bounded prisms over $\mathcal{O}_{K}$ and equipped with the
faithfully flat topology. Let $\mathcal{O}_{{\mathbbl{\Delta}}}$ be the
structure sheaf over $(\mathcal{O}_{K})_{{{\mathbbl{\Delta}}}}$, and
$\mathcal{I}_{{{\mathbbl{\Delta}}}}\subset\mathcal{O}_{{\mathbbl{\Delta}}}$ be
the ideal sheaf of the Hodge-Tate divisor, then
$\mathcal{O}_{{\mathbbl{\Delta}}}$ carries a $\varphi$-action coming from the
$\delta$-structures. A prismatic $F$-crystal in finite locally free
$\mathcal{O}_{{\mathbbl{\Delta}}}$-modules over
$(\mathcal{O}_{K})_{{{\mathbbl{\Delta}}}}$ is defined as a crystal
$\mathfrak{M}_{{\mathbbl{\Delta}}}$ over
$(\mathcal{O}_{K})_{{{\mathbbl{\Delta}}}}$ in finite locally free
$\mathcal{O}_{{\mathbbl{\Delta}}}$-modules together with an isomorphism
$(\varphi^{\ast}\mathfrak{M}_{{\mathbbl{\Delta}}})[1/\mathcal{I}_{{{\mathbbl{\Delta}}}}]\simeq\mathfrak{M}_{{\mathbbl{\Delta}}}[1/\mathcal{I}_{{{\mathbbl{\Delta}}}}]$.
The main result in [BS21] is the following:

######  Theorem 1.0.1.

([BS21, Theorem 1.2] and Theorem 4.1.10) There is an equivalence of the
category of prismatic $F$-crystals in finite locally free
$\mathcal{O}_{{\mathbbl{\Delta}}}$-modules over
$(\mathcal{O}_{K})_{{{\mathbbl{\Delta}}}}$ and the category of Galois stable
lattices in crystalline representations of $G_{K}$.

To relate the result of Bhatt-Scholze with previous works of characterizing
lattices in crystalline representations using linear algebraic data, one
should first realize the base rings $A$ used in those theories as certain
prisms $(A,I)$ over $\mathcal{O}_{K}$. Then one should expect that evaluating
the prismatic $F$-crystals on $(A,I)$ should recover the corresponding theory.
For example, in the theory of Kisin [Kis06], he uses the base ring
$A=\mathfrak{S}:=W(k)[\\![u]\\!]$ with $\delta(u)=0$, and if one fixes a
uniformizer $\varpi$ of $\mathcal{O}_{K}$ which is a zero of an Eisenstein
polynomial $E\in W(k)[u]$, then it is well-known that $(A,(E))$ is the so-
called Breuil-Kisin prism which is inside
$(\mathcal{O}_{K})_{{{\mathbbl{\Delta}}}}$. And Kisin was able to attach any
lattice $T$ in a crystalline representation of $G_{K}$ a finite free
$A$-module together with an isomorphism
$(\varphi^{\ast}\mathfrak{M})[1/E]\simeq\mathfrak{M}[1/E]$. Now, if
$\mathfrak{M}_{{{\mathbbl{\Delta}}}}$ is the prismatic $F$-crystal attaching
to $T$ under Theorem 1.0.1, then Bhatt-Scholze show that the evaluation of
$\mathfrak{M}_{{\mathbbl{\Delta}}}$ on $(A,(E))$ recovers Kisin’s theory (cf.
Theorem 1.3 of $loc.cit.$).

The first question answered in this paper is whether and how one can recover
the theory of $(\varphi,\hat{G})$-modules from the prismatic $F$-crystals
characterization of Bhatt-Scholze. The category of
$(\varphi,\hat{G})$-modules, roughly speaking, consisting of pairs
$((\mathfrak{M},\varphi_{\mathfrak{M}}),\hat{G})$, where
$(\mathfrak{M},\varphi_{\mathfrak{M}})$ is a Kisin module, and $\hat{G}$ is a
$G_{K}$-action on
$\mathfrak{M}\otimes_{\mathfrak{S},\varphi}\widehat{\mathcal{R}}$ that
commutes with $\varphi_{\mathfrak{M}}$ and satisfying some additional
properties. Here $\widehat{\mathcal{R}}$ is a subring of $A_{\mathrm{inf}}$
that is stable under $\varphi$ and $G_{K}$, where
$A_{\mathrm{inf}}=W(\mathcal{O}_{\overline{K}}^{\flat})$ introduced by
Fontaine, and there is a surjection
$\theta:A_{\mathrm{inf}}:=W(\mathcal{O}_{\overline{K}}^{\flat})\to\widehat{\mathcal{O}_{\overline{K}}}$.
However, the period ring $\widehat{\mathcal{R}}$ introduced by Liu is not
known to be $p$-adically complete or not, and it is even harder to determine
whether it can be shown up as a prism. So in order to relate the theory of
$(\varphi,\hat{G})$-modules with the category of prismatic $F$-crystals of
Bhatt-Scholze, we develop a theory of prismatic $(\varphi,\hat{G})$-modules,
in which theory the ring $\widehat{\mathcal{R}}$ is replaced by
$A^{(2)}_{\mathop{\rm st}\nolimits}$, a subring of $A_{\mathrm{inf}}$
constructed as certain prismatic envelope in §2.3.

The first result of this paper is about the theory of prismatic
$(\varphi,\hat{G})$-modules. We can show similar to the classical
$(\varphi,\hat{G})$-module theory, there is an equivalence between the
category of prismatic $(\varphi,\hat{G})$-modules and lattices in semi-stable
representations of $G_{K}$. Moreover, $(A^{(2)}_{\mathop{\rm
st}\nolimits},(E))$ is indeed a prism in
$(\mathcal{O}_{K})_{{{\mathbbl{\Delta}}}}$, it admits a map
$(A,(E))\to(A^{(2)}_{\mathop{\rm st}\nolimits},(E))$ of prisms, and carries an
action of $G_{K}$. For a $G_{K}$-stable lattice $T$ in a crystalline
representation, if $\mathfrak{M}_{{{\mathbbl{\Delta}}}}$ is the prismatic
$F$-crystal attaches to $T$, then evaluating
$\mathfrak{M}_{{{\mathbbl{\Delta}}}}$ on the morphism
$(A,(E))\to(A^{(2)}_{\mathop{\rm st}\nolimits},(E))$ recovers the prismatic
$(\varphi,\hat{G})$-module attaches to $T$. We can also show the map
$A^{(2)}_{\mathop{\rm st}\nolimits}\to
A_{\mathrm{inf}}\xrightarrow{\varphi}A_{\mathrm{inf}}$ factor through
$\widehat{\mathcal{R}}$, so the theory of prismatic
$(\varphi,\hat{G})$-modules recovers the classical theory. The ring
$A^{(2)}_{\mathop{\rm st}\nolimits}$ is simpler than $\widehat{\mathcal{R}}$
in many ways, although it is still very complicated and non-noetherian, it is
more explicitly described and is $p$-adic complete. In particular, our new
theory can be used to fix the gap [Liu07] indicated by [Gao21, Appendix B].

The second attempt made in this paper is to provide a new approach to the
equivalence between the category of prismatic $F$-crystals and the category of
lattices in crystalline representation established by Bhatt and Scholze as in
Theorem 1.0.1. That is, using the known equivalence between lattices in semi-
stable representations and prismatic $(\varphi,\hat{G})$-modules, we will
establish a functor from the category of prismatic $(\varphi,\hat{G})$-modules
that correspond to crystalline representations to prismatic $F$-crystals, and
show this functor is an equivalence.

To be more precise, let $T$ be a $G_{K}$-stable lattice in a crystalline
representation with positive Hodge-Tate weights, let $(A,E)$ be the Breuil-
Kisin prism, and let $(A^{(2)},(E))$ (resp. $(A^{(3)},(E))$) be the self-
product (self-triple-product) of $(A,(E))$ in
$(\mathcal{O}_{K})_{{\mathbbl{\Delta}}}$. Then evaluating prismatic
$F$-crystals on the diagram
$(A,(E))\xrightarrow{i_{1}}(A^{(2)},(E))\xleftarrow{i_{2}}(A,(E))$ induces an
equivalence of the category of prismatic $F$-crystals and Kisin modules with
descent data, that is pairs $((\mathfrak{M},\varphi_{\mathfrak{M}}),f)$ where
$(\mathfrak{M},\varphi_{\mathfrak{M}})$ is a Kisin module and

$f:\mathfrak{M}\otimes_{\mathfrak{S},i_{1}}A^{(2)}\simeq\mathfrak{M}\otimes_{\mathfrak{S},i_{2}}A^{(2)}$

is an isomorphism of $A^{(2)}$-modules that is compatible with $\varphi$ and
satisfies cocycle condition over $A^{(3)}$. Using this, to establish an
equivalence between prismatic $(\varphi,\hat{G})$-modules that correspond to
crystalline representations and prismatic $F$-crystals, it remains to find
certain correspondence between the $\hat{G}$-action and the descent
isomorphism $f$. We will show the descent isomorphism can be obtained by
taking the $G_{K}$-action of the $(\varphi,\widehat{G})$-module at a specific
element. To be more precise, fix a Kummer tower
$K_{\infty}=\bigcup_{n=1}^{\infty}K(\varpi_{n})$ used in the theory of Kisin,
where $\\{\varpi_{n}\\}_{n}$ is a compatible system of $p^{n}$-th roots of
$\varpi_{0}=\varpi$, and let $L$ be the normalization of $K_{\infty}$ inside
$\overline{K}$. Choose $\tau\in\hat{G}:=\mathop{\rm Gal}\nolimits(L/K)$
satisfying $\tau(\varpi_{n})=\zeta_{p^{n}}\varpi_{n}$ such that
$\\{\zeta_{p^{n}}\\}$ is a compatible system of primitive $p^{n}$-th roots of
$1$, then our slogan is that the descent isomorphism corresponds to the
$\tilde{\tau}$-action on the Kisin module $\mathfrak{M}$ inside
$T^{\vee}\otimes A_{\mathrm{inf}}$ where $\tilde{\tau}\in G_{K}$ is any
lifting of $\tau$ under the quotient map $G_{K}\to\widehat{G}$.

To sketch our idea, first we have the maps $u\mapsto[{\varpi}^{\flat}]$ and
$u\mapsto[{\tau}({\varpi}^{\flat})]$ defines two morphisms of $(A,(E))$ to
$(A_{\mathrm{inf}},\mathop{\rm Ker}\nolimits\theta)$. By the universal
property of $(A^{(2)},(E))$, these two maps induce a morphism
$(A^{(2)},(E))\to(A_{\mathrm{inf}},\mathop{\rm Ker}\nolimits\theta)$. We can
show this map is injective, and the embedding factors through
$A^{(2)}_{\mathop{\rm st}\nolimits}$, which is the base ring used in our
prismatic $(\varphi,\hat{G})$-module theory. That is, we have a chain of
subrings $A\subset A^{(2)}\subset A^{(2)}_{\mathop{\rm st}\nolimits}$ of
$A_{\mathrm{inf}}$, such that $\tilde{\tau}(A)$ is also contained in
$A^{(2)}$. We can show a prismatic $(\varphi,\hat{G})$-module corresponds to a
crystalline representation if and only if the coefficients of the
$\tilde{\tau}$-action on $\mathfrak{M}$ in $T^{\vee}\otimes A_{\mathrm{inf}}$
lie inside $A^{(2)}$. And once this is proved, the $\tilde{\tau}$-action will
induce an isomorphism:

$f_{\tau}:\mathfrak{M}\otimes_{\mathfrak{S},\tau}A^{(2)}\simeq\mathfrak{M}\otimes_{\mathfrak{S}}A^{(2)}.$

We will see $f_{\tau}$ gives the descent isomorphism. As a result, we give a
new proof for Theorem 1.0.1.

An advantage of our approach is that our new method can be easily generalized
to the semi-stable representations cases. It turns out that the prism
$(A^{(2)}_{\mathop{\rm st}\nolimits},(E))$ is isomorphic to the self-coproduct
of $(A,(E))$ in the category of logarithmic prisms over $\mathcal{O}_{K}$
defined by Koshikawa[Kos21]. Using the equivalence between prismatic
$(\varphi,\hat{G})$-modules and lattices in semi-stable representations of
$G_{K}$. we will show in §5 the following generalization of Theorem 1.0.1 for
semi-stable representations.

######  Theorem 1.0.2.

(Theorem 5.0.18) There is an equivalence of the category of prismatic
$F$-crystals in finite locally free $\mathcal{O}_{{\mathbbl{\Delta}}}$-modules
over $(\mathcal{O}_{K})_{{{\mathbbl{\Delta}}}_{\log}}$ and the category of
Galois stable lattices in semi-stable representations of $G_{K}$.

Another interesting and natural question one can ask is whether Theorem 1.0.1
and Theorem 1.0.2 can accommodate more general base rings. Motivated by our
strategy, it seems to us that the answer should be affirmative if a suitable
theory of $(\varphi,\hat{G})$-module can accommodate more general base rings,
for example, if the base ring $R$ is a complete DVR with _imperfect_ residue
field that admits a finite $p$-basis. We are working on such direction and
hopefully will report our progress in the future. So part of our paper, for
example, § 2 do allow specific general base rings.

### Acknowledgments

It is our pleasure to thank Hui Gao, Wansu Kim, Teruhisa Koshikawa, Zeyu Liu,
Yong Suk Moon, Peter Scholze, Koji Shimizu, Yupeng Wang, Zhiyou Wu and Min Yu
for comments and conversations during the preparation of this paper.

##  2\. Ring Structures on certain prismatic envelope

Recall that $K$ is a completed discrete valuation field in mix characteristic
$(0,p)$ with ring of integers of $\mathcal{O}_{K}$ and prefect residue field
$k$. Write $W=W(k)$. Let $\varpi\in\mathcal{O}_{K}$ be a uniformizer and
$E=E(u)\in W[u]$ be the Eisenstein polynomial of $\varpi$. Let $\mathbb C_{p}$
be the $p$-adic completion of $\overline{K}$, and $\mathcal{O}_{\mathbb
C_{p}}$ be the ring of integers. Let $R_{0}$ be a $W(k)$-algebra which admits
Frobenius lift $\varphi:R_{0}\to R_{0}$. Set
$R:=R_{0}\otimes_{W(k)}\mathcal{O}_{K}$. We make the following assumptions for
$R_{0}$ and $R$:

  1. (1)

Both $R_{0}$ and $R$ are $p$-adically complete integral domains, and
$R_{0}/pR_{0}=R/\varpi R$ is an integral domain;

  2. (2)

Let $\breve{R}_{0}=W\langle t_{1},\dots,t_{m}\rangle$. $R_{0}$ is a
$\breve{R}_{0}$-_formally étale_ algebra with $p$-adic topology;

  3. (3)

$\breve{R}_{0}$ admits a Frobenius lift such that $\breve{R_{0}}\to R_{0}$
defined in (2) is $\varphi$-equivalent.

  4. (4)

The $k$-algebra $R_{0}/pR_{0}$ has finite $p$-basis in the sense of [dJ95,
Definition 1.1.1].

Our main example is $R_{0}=\breve{R}_{0}=W(k).$ We will not use the finite
$p$-basis assumption until §4. The following are other examples of $R_{0}$:

######  Example 2.0.1.

  1. (1)

$R_{0}=W(k)\langle t_{1}^{\pm 1},\dots,t_{m}^{\pm 1}\rangle$ with
$\varphi(t_{j})=t^{p}_{j}$

  2. (2)

$R_{0}=W(k)[\\![t]\\!]$ with $\varphi(t)=t^{p}$ or $(1+t)^{p}-1$.

  3. (3)

$R_{0}$ is an unramified complete DVR with imperfect field $\kappa$ with
finite $p$-basis. See §6 for more discussions.

We reserve $\gamma_{i}(\cdot)$ to denote $i$-th divided power.

###  2.1. Construction of $A^{(2)}$

Let $A=\mathfrak{S}=R_{0}[\\![u]\\!]$ and extend $\varphi:A\to A$ by
$\varphi(u)=u^{p}$. It is well-known that $(A,E)$ is a prism and we can define
a surjection $\theta:A\to R$ via $u\mapsto\varpi$. We have $\mathop{\rm
Ker}\nolimits\theta=(E(u))$. Let $\breve{A}:=\breve{R}_{0}[\\![u]\\!]$ and
define $\varphi$ and
$\breve{\theta}:\breve{A}\to\breve{R}:=\mathcal{O}_{K}\otimes_{W}\breve{R}_{0}$
similarly. We set

$A^{\widehat{\otimes}2}:=A[\\![y-x,s_{1}-t_{1},\dots,s_{m}-t_{m}]\\!],\
A^{\widehat{\otimes}3}:=A[\\![y-x,w-x,\\{s_{i}-t_{i},r_{i}-t_{i}\\}_{j=1,\dots,m}]\\!].$

Note that $A^{\widehat{\otimes}2}$ (resp. $A^{\widehat{\otimes}3}$) is
$\breve{A}\otimes_{\mathbb Z_{p}}\breve{A}$(resp. $\breve{A}\otimes_{\mathbb
Z_{p}}\breve{A}\otimes_{\mathbb Z_{p}}\breve{A}$)-algebra by $u\otimes
1\mapsto x$, $1\otimes u\mapsto y$ and $1\otimes t_{i}\mapsto s_{i}$ (resp.
$1\otimes 1\otimes u\mapsto w$ and $1\otimes 1\otimes t_{i}\mapsto r_{i}$). So
in this way, we can extend Frobenius $\varphi$ of $A$, which is compatible
with that on $\breve{A}$ to $A^{\widehat{\otimes}2}$ and
$A^{\widehat{\otimes}3}$. Set
$J^{(2)}=(E,y-x,\\{s_{i}-t_{i}\\}_{i=1,\dots,m})\subset
A^{\widehat{\otimes}2}$ and
$J^{(3)}=(E,y-x,w-x,\\{s_{i}-t_{i},r_{i}-t_{i}\\}_{i=1,\dots,m})\subset
A^{\widehat{\otimes}3}.$ Clearly, we have
$A^{\widehat{\otimes}i}/J^{(i)}\simeq R$ for $i=2,3$. And we have
$A^{\widehat{\otimes}2}/(p,E)$ (resp. $A^{\widehat{\otimes}3}/(p,E)$) is a
formal power series ring over the variables
$\bar{y}-\bar{x},\\{\bar{s}_{i}-\bar{t}_{i}\\}_{i=1,\dots,m}$ (resp.
$\bar{y}-\bar{x},\bar{w}-\bar{x},\\{\bar{s}_{i}-\bar{t}_{i},\bar{r}_{i}-\bar{t}_{i}\\}_{i=1,\dots,m}$),
so $(A,(E))\to(A^{\widehat{\otimes}i},J^{(i)})$ satisfies the requirements of
in [BS22, Prop. 3.13], and we can construct the prismatic envelope with
respect to this map, which will be denoted by $A^{(i)}$. More precisely,
$A^{(i)}\simeq
A^{\widehat{\otimes}i}\left\\{\frac{J^{(i)}}{E}\right\\}_{\delta}^{\wedge}$,
here $\\{\cdot\\}_{\delta}^{\wedge}$ means freely adjoining elements in the
category of $(p,E(u))$-completed $\delta$-$A$-algebras. We will see $A^{(i)}$,
$i=2,3$ are the self product and triple product of $A$ in category
$X_{{\mathbbl{\Delta}}}$ in §4.1.

###  2.2. The ring $A^{(2)}_{\max}$

Now we set $t_{0}=x$, $s_{0}=y$ and

$z_{j}=\frac{s_{i}-t_{i}}{E}\textnormal{ and
}z_{0}=z=\frac{y-x}{E}=\frac{s_{0}-t_{0}}{E}.$

Note that $A^{(i)}$ are $A$-algebras via $u\mapsto x$.

######  Definition 2.2.1.

Let ${O}_{\mathrm{max}}$ be the $p$-adic completion of the $A$-subalgebra of
$A[\frac{1}{p}]$ generated by $p^{-1}E$. And let $A_{\max}^{(2)}$ be the
$p$-adic completion of the $A$-subalgebra of
$A[z_{j},\frac{1}{p};j=0,\dots,m]$ generated by $p^{-1}E$ and
$\\{\gamma_{i}(z_{j})\\}_{i\geq 1,j=0,\dots,m}$.

We first note that $A^{(2)}_{\max}$ is an $A^{\widehat{\otimes}2}$-algebra via
$(s_{j}-t_{j})=Ez_{j},j=0,\dots,m$. Write $\iota:A^{\widehat{\otimes}2}\to
A^{(2)}_{\max}$ for the structure map. By construction, it is easy to see that
$A^{(2)}_{\max}\subset R_{0}[\frac{1}{p}][\\![E,z_{j},j=0,\dots,m]\\!]$. In
particular, $A^{(2)}_{\max}$ is a domain and any element $b\in A^{(2)}_{\max}$
can be _uniquely_ written as
$\sum\limits_{i_{0}=0}^{\infty}\cdots\sum\limits_{i_{m}=0}^{\infty}b_{i_{1},\dots,i_{m}}\prod\limits_{j=0}^{m}\gamma_{i_{j}}(z_{j})$
with $b_{i_{0},\dots,i_{m}}\in{O}_{\mathrm{max}}$ and
$b_{i_{0},\dots,i_{m}}\to 0$ $p$-adically when $i_{0}+\cdots+i_{m}\to\infty$.
Our next aim is to define $\varphi$ on $A^{(2)}_{\max}$. For this, we need a
little preparation.

######  Lemma 2.2.2.

$c:=\frac{\varphi(E)}{p}\in{O}_{\mathrm{max}}$ and
$c^{-1}\in{O}_{\mathrm{max}}$.

###### Proof.

We have $A$ is a $\delta$-ring, and $E$ is a distinguished element, so in
particular

$\varphi(E)/p=c_{0}+E^{p}/p$

where $c_{0}=\delta(E)\in A^{\times}$. So
$c=\varphi(E)/p\in{O}_{\mathrm{max}}$, and
$c^{-1}=c_{0}^{-1}\sum\limits_{i=0}^{\infty}\frac{(-c_{0}^{-1}E^{p})^{i}}{p^{i}}\in{O}_{\mathrm{max}}.$
∎

Now we define $\varphi(z)=\varphi(z_{0})=\frac{y^{p}-x^{p}}{\varphi(E)}$ and
$\varphi(z_{j})=\frac{\varphi(s_{j})-\varphi(t_{j})}{\varphi(E)}$. Since
+rCl+x* φ(z)=yp-x pφ(E)=c^-1yp-xpp=c^-1(x+ Ez)p-xpp &=
c^-1∑_i=1^px^p-i(Ez)^i(pi)/p
= c^-1∑_i=1^pa_iz^i, where $a_{i}\in
W(k)[\\![x]\\!][\frac{E^{p}}{p}]\subset{O}_{\mathrm{max}}\subset
A^{(2)}_{\max}$ and $c$ is a unit in ${O}_{\mathrm{max}}$, we have
$\varphi(z)\in A^{(2)}_{\max}$. Then

$\gamma_{n}(\varphi(z))=\frac{\varphi(z)^{n}}{n!}=\frac{z^{n}}{n!}(c^{-1}\sum_{i=1}^{p}a_{i}z^{i-1})^{n}$

is in $A^{(2)}_{\max}.$ The argument for $\varphi(z_{j})$ for $j>1$ need a
little more details. Note that $\varphi(t_{j})=t_{j}^{p}+p\delta(t_{j})$ with
$\delta(t_{j})\in\breve{R}_{0}$ by our assumptions. It is clear that
$\delta(s_{j})-\delta(t_{j})=(s_{j}-t_{j})\lambda_{j}$ with $\lambda_{j}\in
A^{\widehat{\otimes}2}$. Using that $(s_{j}-t_{j})=Ez_{j}$, so

(1) $\varphi(z_{j})=c^{-1}(\frac{s^{p}_{j}-t^{p}_{j}}{p}+Ez_{j}\lambda_{j})$

The same argument as that for $\varphi(z_{0})$ also shows that
$\gamma_{n}(z_{j})\in A^{(2)}_{\max}$, for $j=1,\dots,m$.

Since any element $b\in A^{(2)}_{\max}$ can be uniquely written as
$\sum\limits_{i_{0}=0}^{\infty}\cdots\sum\limits_{i_{m}=0}^{\infty}b_{i_{1},\dots,i_{m}}\prod\limits_{j=0}^{m}\gamma_{i_{j}}(z_{j})$
with $b_{i_{0},\dots,i_{m}}\in{O}_{\mathrm{max}}$ and
$b_{i_{0},\dots,i_{m}}\to 0$ $p$-adically when $i_{0}+\cdots+i_{m}\to\infty$,
this allows to extend Frobenius map $\varphi$ on $A$ to a _ring_ map
$\varphi:A^{(2)}_{\max}\to A^{(2)}_{\max}$ by sending $u\mapsto u^{p}$,
$z\mapsto\frac{y^{p}-x^{p}}{\varphi(E)}$,
$\varphi(z_{j})=\frac{\varphi(s_{j})-\varphi(t_{j})}{\varphi(E)}$, and
$\gamma_{i}(z_{j})\mapsto\gamma_{i}(\varphi(z_{j}))$ as the above.

######  Remark 2.2.3.

The ring map $\varphi:A^{(2)}_{\max}\to A^{(2)}_{\max}$ is _not_ a Frobenius
lift of $A^{(2)}_{\max}/p$ because $\varphi(E/p)-(E/p)^{p}\not\in
pA^{(2)}_{\max}$. In particular, $A^{(2)}_{\max}$ is not a $\delta$-ring.

Recall that $A^{(2)}_{\max}$ is an $A^{\widehat{\otimes}2}$-algebra via map
$\iota:A^{\widehat{\otimes}2}\to A^{(2)}_{\max}$. The above construction of
Frobenius $\varphi$ on $A^{(2)}_{\max}$ is obviously compatible with $\iota$.

Our next goal is to show that $\iota$ induces a map $A^{(2)}\to
A^{(2)}_{\max}$ so that $A^{(2)}$ is a subring of $A^{(2)}_{\max}$ which is
compatible with $\varphi$-structures and filtration. We need a little
preparation. Write $\mathfrak{z}_{n}=\delta^{n}(z)$ with
$\delta_{0}(z)=z=\mathfrak{z}_{0}$, and $A_{0}=W(k)[\\![u]\\!]$.

######  Lemma 2.2.4.

$\delta^{n}(Ez)=b_{n}\mathfrak{z}_{n}+\sum_{i=0}^{p}a^{(n)}_{i}\mathfrak{z}_{n-1}^{i}.$

where $a^{(n)}_{i}\in A_{0}[\mathfrak{z}_{0},\dots,\mathfrak{z}_{n-2}]$ so
that $a^{(n)}_{p}\in A_{0}^{\times}$ and for $0\leq i\leq p-1$ each monomials
of $a^{(n)}_{i}$ contains a factor $\mathfrak{z}_{j}^{p}$ for some $0\leq
j\leq n-2$. Furthermore, $b_{n+1}=p\delta(b_{n})+b^{p}_{n}$ and
$b_{1}=p\delta(E)+E^{p}$.

###### Proof.

Given $f\in A_{0}[x_{1},\dots,x_{m}]$, if each monomials of $f$ contains
$x_{j}^{l}$ for some $j$ and $l\geq p$ then we call $f$ _good_. For example,
$f=x_{1}^{p}x_{2}+2x_{1}x_{2}^{p+3}.$ So we need to show that $a^{(n)}_{i}\in
A_{0}[\mathfrak{z}_{0},\dots,\mathfrak{z}_{n-2}]$ is good. Before making
induction on $n$, we discuss some properties of good polynomial. It is clear
that the set of good polynomials is closed under addition and multiplications.
Note that

(2)
$\delta(\mathfrak{z}_{l}^{i})=\frac{1}{p}(\varphi(\mathfrak{z}_{l}^{i})-\mathfrak{z}_{l}^{pi})=\frac{1}{p}\big{(}(p\mathfrak{z}_{l+1}+\mathfrak{z}_{l}^{p})^{i}-\mathfrak{z}_{l}^{pi}\big{)}=\sum\limits_{j=1}^{i}\binom{i}{j}(p^{j-1}\mathfrak{z}_{l}^{p(i-j)})\mathfrak{z}_{l+1}^{j}.$

In particular, given an $f\in A_{0}[\mathfrak{z}_{0},\dots,\mathfrak{z}_{m}]$,
$\delta(\mathfrak{z}_{m}^{p}f)=f^{p}\delta(\mathfrak{z}_{m}^{p})+\mathfrak{z}_{m}^{p^{2}}\delta(f)+p\delta(\mathfrak{z}_{m}^{p})\delta(f)$
is a good polynomial in $A[\mathfrak{z}_{0},\dots,\mathfrak{z}_{m+1}]$. Using
the fact that $\delta(a+b)=\delta(a)+\delta(b)+F(a,b)$ where
$F(X,Y)=\frac{1}{p}(X^{p}+Y^{p}-(X+Y)^{p})=-\sum\limits_{i=1}^{p-1}\binom{p}{i}/pX^{i}Y^{p-i}$,
together with the above argument of $\delta(\mathfrak{z}_{l}^{p}f)$, it is not
hard to show that if $g\in A_{0}[\mathfrak{z}_{0},\dots,\mathfrak{z}_{m}]$ is
good then $\delta(g)\in
A_{0}[\mathfrak{z}_{0},\dots,\mathfrak{z}_{m},\mathfrak{z}_{m+1}]$ is also
good.

Now we make induction on $n$. When $n=1$, we have

$\delta(Ez)=E^{p}\mathfrak{z}_{1}+z^{p}\delta(E)+p\delta(E)\mathfrak{z}_{1}=(p\delta(E)+E^{p})\mathfrak{z}_{1}+\delta(E)z^{p}.$

Then $b_{1}=p\delta(E)+E^{p}$, $a^{(1)}_{p}=\delta(E)\in A_{0}^{\times}$ and
$a^{(1)}_{i}=0$ for $1\leq i\leq p-1$ are required. Now assume the formula is
correct for $n$, then

$\delta^{n+1}(Ez)=\delta(b_{n}\mathfrak{z}_{n}+\sum_{i=0}^{p}a^{(n)}_{i}\mathfrak{z}_{n-1}^{i})=\delta(b_{n}\mathfrak{z}_{n})+\delta(\sum_{i=0}^{p}a^{(n)}_{i}\mathfrak{z}_{n-1}^{i}))+F(b_{n}\mathfrak{z}_{n},\sum_{i=0}^{p}a^{(n)}_{i}\mathfrak{z}_{n-1}^{i})),$

Clearly,
$F(b_{n}\mathfrak{z}_{n},\sum\limits_{i=0}^{p}a^{(n)}_{i}\mathfrak{z}_{n-1}^{i}))=\sum\limits_{j=1}^{p-1}\tilde{a}^{(n)}_{j}\mathfrak{z}_{n}^{j}$
with $\tilde{a}^{(n)}_{j}$ being good. An easy induction shows that
$\delta(\sum\limits_{i=0}^{p}a^{(n)}_{i}\mathfrak{z}_{n-1}^{i})=\sum\limits_{i=0}^{p}\delta(a^{(n)}_{i}\mathfrak{z}_{n-1}^{i})+f$
with $f\in A_{0}[\mathfrak{z}_{0},\dots,\mathfrak{z}_{n-1}]$ being good. Since
$\delta(a^{(n)}_{i}\mathfrak{z}_{n-1}^{i})=(a^{(n)}_{i})^{p}\delta(\mathfrak{z}_{n-1}^{i})+(\mathfrak{z}_{n-1}^{pi})\delta(a_{i}^{(n)})+p\delta(\mathfrak{z}_{n-1}^{i})\delta(a_{i}^{(n)})$,
by using formula of $\delta(\mathfrak{z}_{n-1}^{i})$ in (2) and that
$a^{(n)}_{i}$ is good implies that $\delta(a_{i}^{(n)})$ is also good, we
conclude that for $0\leq i\leq p-1$,

$\sum\limits_{i=0}^{p-1}\delta(a^{(n)}_{i}\mathfrak{z}_{n-1}^{i})=\sum_{i=0}^{p-1}\alpha_{i}\mathfrak{z}_{n}^{i}$

with $\alpha_{i}\in A_{0}[\mathfrak{z}_{0},\dots,\mathfrak{z}_{n-1}]$ being
good polynomials. Using that $a_{p}^{(n)}\in A_{0}^{\times}$, we compute that
$\delta(a_{p}^{(n)}\mathfrak{z}^{p}_{n-1})=\sum\limits_{i=0}^{p}\beta_{i}\mathfrak{z}_{n}^{i}$
with $\beta_{p}\in pA_{0}$ and $\beta_{j}\in
A_{0}[\mathfrak{z}_{0},\dots,\mathfrak{z}_{n-1}]$ being good for $1\leq j\leq
p-1$. Now we only need to analyze $\delta(b_{n}\mathfrak{z}_{n})$, which is
$\delta(b_{n})\mathfrak{z}_{n}^{p}+b_{n}^{p}\mathfrak{z}_{n+1}+p\delta(b_{n})\mathfrak{z}_{n+1}$.
So $b_{n+1}=p\delta(b_{n})+b_{n}^{p}$ and
$a_{p}^{(n+1)}=\delta(b_{n})+\beta_{p}$. Since $\delta(b_{n})\in
A_{0}^{\times}$, we see that $a_{p}^{(n+1)}=\delta(b_{n})+\beta_{p}\in
A_{0}^{\times}$ as required. ∎

Let
$\widetilde{A}^{(2)}:=A^{\widehat{\otimes}2}[z_{j}]_{\delta}=A^{\widehat{\otimes}2}[\delta^{n}(z_{j}),n\geq
0,j=0,\dots,m]$ and natural map
$\alpha:\widetilde{A}^{(2)}\to\widetilde{A}^{(2)}[\frac{1}{p}]$ (we do not
know $\alpha$ is injective at this moment).

######  Lemma 2.2.5.

For $i\geq 0$ and $j=0,1,\ldots,d$, there exists
$f_{ij}(X)\in\widetilde{A}^{(2)}[X]$ such that, as elements of
$\widetilde{A}^{(2)}[\frac{1}{p}]$ via
$\alpha:\widetilde{A}^{(2)}\to\widetilde{A}^{(2)}[\frac{1}{p}]$,

$\gamma_{i}(z_{j})=f_{ij}\Bigl{(}\frac{E}{p}\Bigr{)}.$

###### Proof.

Write $z=z_{j}$ for simplicity, and let $\tilde{\gamma}(z)=\frac{z^{p}}{p}$
and
$\tilde{\gamma}^{n}=\underbrace{\tilde{\gamma}\circ\tilde{\gamma}\cdots\circ\tilde{\gamma}}_{n}$.
It suffices to show that for each $n\geq 1$, we have
$\tilde{\gamma}^{n}(z)=f_{n}(\frac{E}{p})$ inside
$\widetilde{A}^{(2)}[\frac{1}{p}]$ for some
$f_{n}(X)\in\widetilde{A}^{(2)}[X]$. For an element $x\in
A[\delta^{i}(z)]_{i\geq 0}$, we say that $x$ has _$\delta$ -order $\leq n$_ if
$x\in\sum_{0\leq j\leq n}A[\\{\delta^{i}(z)\\}_{0\leq i\leq n}]\delta^{j}(z)$,
namely, if $x$ can be written as a sum of monomials such that each term is
divisible by $\delta^{j}(z)$ for some $0\leq j\leq n$.

We claim that the following two equations hold for each $n\geq 1$:

  1. (1)

We have

(3)
$\delta^{n}(z)=\nu_{n}\tilde{\gamma}^{n}(z)+P_{n}\Bigl{(}\frac{E}{p}\Bigr{)}+\frac{E^{p}}{p}d_{n}\delta^{n}(z)$

for some $\nu_{n}\in A^{\times}$, $d_{n}\in A$, and
$P_{n}(X)\in(A[\delta^{i}(z)]_{i\geq 0})[X]$ such that each coefficient of
$P_{n}(X)$ has $\delta$-order $\leq n-1$.

  2. (2)

We have

(4)
$\tilde{\gamma}(\delta^{n-1}(z))=\mu_{n-1}\tilde{\gamma}^{n}(z)+Q_{n-1}\Bigl{(}\frac{E}{p}\Bigr{)}$

for some $\mu_{n-1}\in A^{\times}$ and $Q_{n-1}(X)\in(A[\delta^{i}(z)]_{i\geq
0})[X]$ such that each coefficient of $Q_{n-1}(X)$ has $\delta$-order $\leq
n-1$.

We prove claims (1) and (2) by induction. For $n=1$, since

$\delta(Ez)=z^{p}\delta(E)+(p\delta(E)+E^{p})\delta(z)$

and $\delta(E)\in\mathfrak{S}^{\times}$, we have

$\delta(z)=-\tilde{\gamma}(z)+\delta(E)^{-1}\frac{\delta(Ez)}{p}-\delta(E)^{-1}\frac{E^{p}}{p}\delta(z).$

By easy induction, we also have $\delta^{i}(Ez)\in(Ez)A$ for each $i\geq 1$.
So claim (1) holds. Claim (2) holds for $n=1$ trivially with $Q_{0}(X)=0$.

Suppose that claims (1) and (2) hold for $1\leq n\leq m$. We will verify
claims (1) and (2) for $n=m+1$. We first consider claim (2). Since each
coefficient of $P_{m}(X)$ has $\delta$-order $\leq m-1$,
$\frac{E^{p}}{p}=p^{p-1}\bigl{(}\frac{E}{p}\bigr{)}^{p}$, and Equations (3)
and (4) hold for $1\leq n\leq m$, applying $\tilde{\gamma}(\cdot)$ to Equation
(3) for $n=m$ yields

$\tilde{\gamma}(\delta^{m}(z))=\nu_{m}^{p}\tilde{\gamma}^{m+1}(z)+Q_{m}\Bigl{(}\frac{E}{p}\Bigr{)}$

for some $Q_{m}(X)\in(\mathfrak{S}[\delta^{i}(z)]_{i\geq 0})[X]$ such that
each coefficient of $Q_{m}(X)$ has $\delta$-order $\leq m$. This proves the
claim (2) for $n=m+1$.

We now consider claim (1) for $n=m+1$. By the above Lemma for $n=m+1$ and that
$b_{n}=p\alpha_{n}+\beta_{n}E^{p}$ for some $\alpha_{n}\in A^{\times}$ and
$\beta_{n}\in A$ (via an easy induction on $n$), we have

$\alpha_{m+1}\delta^{m+1}(z)=\frac{\delta^{m+1}(Ez)}{p}-\beta_{m+1}\frac{E^{p}}{p}\delta^{m+1}(z)-a_{p}^{(m+1)}\tilde{\gamma}(\delta^{m}(z))-\frac{1}{p}\sum_{j=0}^{p-1}a_{j}^{(m+1)}(\delta^{m}(z))^{j}.$

As noted above, we have $\delta^{m+1}(Ez)\in(Ez)A$. Furthermore, by the
condition on $a_{j}^{(m+1)}$, the last term
$\frac{1}{p}\sum_{j=0}^{p-1}a_{j}^{(m+1)}(\delta^{m}(z))^{j}$ is a linear
combination of terms involving
$\tilde{\gamma}(\delta^{l}(z))=\frac{1}{p}(\delta^{l}(z))^{p}$ for some $0\leq
l\leq m-1$. Thus, by applying Equations (3) and (4) for $1\leq n\leq m$, we
see that claim (1) also holds for $n=m+1$ with
$\nu_{m+1}=-\alpha_{m+1}^{-1}a_{p}^{(m+1)}\mu_{m}$ and
$d_{m+1}=-\alpha_{m+1}^{-1}\beta_{m+1}$. This completes the induction and
prove the lemma . ∎

######  Remark 2.2.6.

In the above proof, by equation (4), we even have for each $i,j\geq 0$,
$\gamma_{i}(\delta^{j}(z))=f(\frac{E}{p})$ for some
$f\in\widetilde{A}^{(2)}[X]$.

An easy induction by (3) implies that $\alpha(\delta^{n}(z))\in
A^{\widehat{\otimes}2}[\\{\gamma_{i}(z_{j})\\}_{i\geq
0,j=1,\dots,m},\frac{E}{p}]\subset A^{(2)}_{\max}$, which satisfies equations
in Lemma 2.2.4 by replacing $\mathfrak{z}_{n}$ by $\alpha(\delta^{n}(z))$
inside $A^{(2)}_{\max}$. It is clear that $\iota$ is still Frobenius
compatible (because both $A^{\widehat{\otimes}2}$ and $A^{(2)}_{\max}$ are
domains). Since $E=p\frac{E}{p}$, $\iota$ is a continuous for $(p,E)$-topology
on $\widetilde{A}^{(2)}$ and $p$-topology on $A^{(2)}_{\max}$. Finally, we
construct a ring map $\iota:A^{(2)}\to A^{(2)}_{\max}$ so that $\iota$ is
compatible with Frobenius.

Our next goal is to show that $\iota$ is injective. Define $\mathop{\rm
Fil}\nolimits^{i}A^{(2)}_{\max}[\frac{1}{p}]:=E^{i}A^{(2)}_{\max}[\frac{1}{p}]$.
For any subring $B\subset A^{(2)}_{\max}[\frac{1}{p}]$, set

$\mathop{\rm Fil}\nolimits^{i}B:=B\cap\mathop{\rm
Fil}\nolimits^{i}A^{(2)}_{\max}[\frac{1}{p}]=B\cap
E^{i}A^{(2)}_{\max}[\frac{1}{p}].$

Let $D_{z}$ be the $p$-adic completion of $R[\gamma_{i}(z_{j}),i\geq
0;j=0,\dots,m]$.

######  Proposition 2.2.7.

  1. (1)

$\widetilde{A}^{(2)}/E=R[\gamma_{i}(z_{j}),i\geq 0;j=0,\dots,m]$.

  2. (2)

$A^{(2)}/E\simeq D_{z}$.

  3. (3)

$\iota$ is injective.

  4. (4)

$\mathop{\rm Fil}\nolimits^{1}A^{(2)}=EA^{(2)}$.

  5. (5)

$A^{(i)}$ are flat over $A$ for $i=2,3$.

###### Proof.

(1) By definition,
$\widetilde{A}^{(2)}=A^{\widehat{\otimes}2}[z^{(n)}_{j},n\geq
0;j=0,\dots,m]/J$ where $\mod J$ is equivalent the following relations (note
that $z_{0}=z$):
$Ez=(x-y),Ez_{j}=s_{j}-t_{j},\delta(z^{(n)}_{j})=z^{(n+1)}_{j},\delta^{n}(Ez)=\delta^{n}(y-x),\delta^{n}(Ez_{j})=\delta^{n}(s_{j}-t_{j}).$
Since $\delta(x-y)=\frac{(x^{p}-y^{p})-(x-y)^{p}}{p}$ and
$\delta(s_{j}-t_{j})=\frac{\varphi(s_{j}-t_{j})-(s_{j}-t_{j})^{p}}{p}$, it is
easy to prove by induction that $\delta^{n}(x-y)$ and
$\delta^{n}(s_{j}-t_{j})$ always contains a factor $(x-y)$, $s_{j}-t_{j}$ and
hence $\delta^{n}(x-y),\delta(s_{j}-t_{j})\equiv 0\mod E$. Therefore
$\delta^{n}(Ez_{j})\equiv 0\mod E$. By Lemma 2.2.4, we see that

$p\mu_{n}z^{(n)}_{j}=-\sum_{i=0}^{p}\overline{a^{(n)}_{i}}(z^{(n-1)}_{j})^{i}\mod
E\text{ and }pz^{(1)}_{j}=z_{j}^{p}\mod E$

where $\overline{a^{(n)}_{i}}=a^{(n)}_{i}\mod E$ and
$\mu_{n}=\frac{\delta(b_{n})}{p}\mod E\in\mathcal{O}_{K}^{\times}$. Using that
$a_{p}^{(n)}\in A_{0}^{\times}$, and $a_{i}^{(n)},1\leq i\leq p-1$ are good in
the sense that they contains factor of $(z^{(l)}_{j})^{p}$ for some
$l=0,\dots,n-2$, we easily see by induction that
$\widetilde{A}^{(2)}/E=R[\widetilde{\gamma}^{n}(z_{j}),n\geq 0;j=0,\dots,m]$.
But it is well-known that $R[\widetilde{\gamma}^{n}(z_{j}),n\geq
0;j=0,\dots,m]=R[\gamma_{n}(z_{j}),n\geq 0;j=0,\dots,m].$

Now we show that the natural map $\iota:\widetilde{A}^{(2)}\to
A^{(2)}_{\max}[\frac{1}{p}]$ induced by $\alpha(\delta^{n}(z_{j}))$ is
injective. Note that $\widetilde{A}^{(2)}$ is the direct limit of
$\widetilde{A}^{(2)}_{n}:=A^{\hat{\otimes}2}[\\{\delta^{i}(z_{j})\\}_{i=1,\dots,n,j=0,\dots,m}]$.
A similar argument similar as above show that $\widetilde{A}^{(2)}_{n}/E$
injects to $A^{(2)}_{\max}[\frac{1}{p}]/E=D_{z}[\frac{1}{p}]$. Since
$\widetilde{A}^{(2)}_{n}$ is $E$-separate and $A^{(2)}_{\max}$ is a domain,
this implies that $\widetilde{A}^{(2)}_{n}$ injects to
$A^{(2)}_{\max}[\frac{1}{p}]$. So $\widetilde{A}^{(2)}$ injects to
$A^{(2)}_{\max}$ via $\iota$.

(2) Since $A^{(2)}$ is $(p,E)$-completion of $\widetilde{A}^{(2)}$ 111Indeed,
$A^{(2)}$ is _derived_ $(p,E)$-completion. Since $\widetilde{A}^{(2)}/E$ is
$\mathbb Z_{p}$-flat, then derived completion coincides with the classical
completion, which is used here., we have a natural map from
$\bar{\iota}:A^{(2)}/E\to D_{z}$. The surjectivity of $\bar{\iota}$ is
straightforward as $A^{(2)}$ is also $p$-complete. To see injectivity, given
an sequence $f_{n}$ so that $f_{n+1}-f_{n}\in(p,E)^{n}\widetilde{A}^{(2)}$ and
$f_{n}=Eg_{n}$ for all $n$, we have to show that $g_{n}$ is a convergent
sequence in $A^{(2)}$. Since
$E(g_{n+1}-g_{n})=\sum_{i=0}^{n}p^{i}E^{n-i}h_{i}$ with
$h_{i}\in\widetilde{A}^{(2)}$. Then $E|p^{n}h_{n}$. Since
$\widetilde{A}^{(2)}/E$ has no $p$-torsion, we have $E|h_{n}$ and write
$h_{n}=Eh^{\prime}_{n}$. Since $\widetilde{A}^{(2)}$ is a domain as it is
inside the fraction field of $A^{\widehat{\otimes}2}$, we see that
$g_{n+1}-g_{n}=p^{n}h^{\prime}_{n}+\sum\limits_{i=0}^{n-1}p^{i}E^{n-i-1}h_{i}$.
Hence $g_{n}$ converges in $A^{(2)}$ as required.

(3) It is clear that $A^{(2)}_{\max}[\frac{1}{p}]/E\simeq D_{z}[\frac{1}{p}]$.
So the map $\iota\mod E(u)$ induces an injection $D_{z}\hookrightarrow
D_{z}[\frac{1}{p}]$. So for any $x\in\mathop{\rm Ker}\nolimits(\iota)$, we see
that $x=Ea$ for some $a\in A^{(2)}$. As $A^{(2)}_{\max}$ is $E$-torsion free
and $A^{(2)}$ is $E$-complete, we see that $x=0$ as required.

(4) By the definition of $\mathop{\rm Fil}\nolimits^{1}A^{(2)}$, we see that
$EA^{(2)}\subset\mathop{\rm Fil}\nolimits^{1}A^{(2)}$ and $A^{(2)}/\mathop{\rm
Fil}\nolimits^{1}A^{(2)}$ injects to
$A^{(2)}_{\max}[\frac{1}{p}]/E=D_{z}[\frac{1}{p}]$. But we have seen that
$A^{(2)}/E=D_{z}$ injects to $D_{z}$. Then $\mathop{\rm
Fil}\nolimits^{1}A^{(2)}=EA^{(2)}$.

(5) Both $A^{(2)}$ and $A^{(3)}$ are obtained by the construction of [BS22,
Proposition 3.13], which implies that they are $(p,E)$-complete flat over $A$.
Since $A$ is Noetherian, by [Sta20, Tag 0912], we have both $A^{(2)}$ and
$A^{(3)}$ are $A$-flat. ∎

######  Corollary 2.2.8.

  1. (1)

$\mathop{\rm Fil}\nolimits^{i}A^{(2)}=E^{i}A^{(2)}.$

  2. (2)

$A^{(i)}$ are bounded prisms for $i=2,3$.

###### Proof.

These follow that $A^{(2)}/EA^{(2)}\simeq D_{z}$ which is $\mathbb
Z_{p}$-flat. For (2), we have $A^{(2)}$ and $A^{(3)}$ are $(p,E)$-complete
flat over $A$, so boundedness follows from (2) in [BS22, Lemma 3.7]. ∎

######  Lemma 2.2.9.

$A^{(2)}$ is a closed subset inside $A^{(2)}_{\max}$.

###### Proof.

We need to show the following statement: Given $x\in\widetilde{A}^{(2)}$, if
$x=p^{n}y$ with $y\in A^{(2)}_{\max}$ then
$x=\sum\limits_{i=0}^{n}p^{n-i}E^{i}x_{i}$ with $x_{i}\in\widetilde{A}^{(2)}.$
Indeed, since $A^{(2)}/E\simeq A^{(2)}_{\max}/{\mathop{\rm
Fil}\nolimits^{1}}$, there exists $x_{0},w_{1}\in\widetilde{A}^{(2)}$ so that
$x=p^{n}x_{0}+Ew_{1}$. Then $Ew_{1}\in p^{n}A^{(2)}_{\max}$. Write
$Ew_{1}=p^{n}\sum\limits_{i=0}^{\infty}\sum\limits_{j=0}^{m}f_{ij}\gamma_{i}(z_{j})$,
we see that $f_{ij}=\sum_{l\geq 1}a_{ijl}\frac{E^{l}}{p^{l}}\in\mathop{\rm
Fil}\nolimits^{1}{O}_{\mathrm{max}}$. So it is easy to see that
$p^{n}E^{-1}f_{ij}\in p^{n-1}{O}_{\mathrm{max}}$ and then $w_{1}=p^{n-1}x_{1}$
with $x_{1}\in A^{(2)}_{\max}$. Then we may repeat the above argument to
$w_{1}$, and finally $x=\sum\limits_{i=0}^{n}p^{n-i}E^{i}x_{i}$ with
$x_{i}\in\widetilde{A}^{(2)}$ as required. ∎

Now we realize $A^{(2)}$ as a subring of $A^{(2)}_{\max}$ via $\iota$. We need
to introduce some auxiliary rings. By the description of elements in
$A^{(2)}_{\max}$, we define ${\widetilde{S}}_{0}$ be the subring of
$A^{(2)}_{\max}$ as follow

$\widetilde{S}:=A^{(2)}[\\![\frac{E^{p}}{p}]\\!]:=\\{\sum_{i\geq
0}a_{i}(\frac{E^{p}}{p})^{i}\mid a_{i}\in A^{(2)}\\}.$

And when $p=2$, we define $\widehat{S}:=A^{(2)}[\\![\frac{E^{4}}{2}]\\!]$
simiarly. We will have $\widehat{S}\subset\widetilde{S}\subset
A^{(2)}_{\max}$. Viewing $\widetilde{S}$ and $\widehat{S}$ as subrings of
$A^{(2)}_{\max}$, we give them the filtration induced from $A^{(2)}_{\max}$.
The following lemma is crucial for later applications and we thank Yong Suk
Moon for many useful comments to improve many details in the proof.

######  Lemma 2.2.10.

Fix $h\in\mathbb N$, then we have

  1. (1)

We have $\varphi(A^{(2)}_{\max})\subset\widetilde{S}\subset A^{(2)}_{\max}$,
and when $p=2$, we have
$\varphi(\widetilde{S})\subset\widehat{S}\subset\widetilde{S}$;

  2. (2)

$x\in\mathop{\rm Fil}\nolimits^{h}\widetilde{S}$ if and only if $x$ can be
written as

$x=\sum\limits_{i\geq h}a_{i}\frac{E^{i}}{p^{\lfloor\frac{i}{p}\rfloor}}$

with $a_{i}\in A^{(2)}$.

  3. (3)

when $p>2$, there is a $h_{0}>h$ such that $\varphi(\mathop{\rm
Fil}\nolimits^{m}\widetilde{S})\subset A^{(2)}+E^{h}\mathop{\rm
Fil}\nolimits^{m+1}\widetilde{S}$ for all $m>h_{0}$;

  4. (4)

when $p=2$, then $x\in\mathop{\rm Fil}\nolimits^{h}\widehat{S}$ if and only if
$x$ can be written as

$x=\sum\limits_{i\geq h}a_{i}\frac{E^{i}}{2^{\lfloor\frac{i}{4}\rfloor}}$

with $a_{i}\in A^{(2)}$;

  5. (5)

when $p=2$, there is a $h_{0}>h$ such that $\varphi(\mathop{\rm
Fil}\nolimits^{m}\widehat{S})\subset A^{(2)}+E^{h}\mathop{\rm
Fil}\nolimits^{m+1}\widehat{S}$ for all $m>h_{0}$.

###### Proof.

For $(1)$, any $a\in A^{(2)}_{\max}$, we can write

$a=\sum_{i_{0}=0}^{\infty}\cdots\sum_{i_{m}=0}^{\infty}\sum_{l=0}^{\infty}a_{i_{0},\dots,i_{m},l}\left(\frac{E}{p}\right)^{l}\prod_{j=0}^{m}\gamma_{i_{j}}(z_{j})$

where $a_{i_{0},\dots,i_{m},l}\in A$ and $a_{i_{0},\dots,i_{m},l}\to 0$
$p$-adically when $\sum_{j}i_{j}+l\to\infty$. Thanks for Lemma 2.2.5, we see
that
$b_{i_{0},\dots,i_{m},l}:=\varphi\left(\left(\frac{E}{p}\right)^{l}\prod_{j=0}^{m}\gamma_{i_{j}}(z_{j})\right)\in\widetilde{S}$.
So $\varphi(a)=\sum a_{i_{0},\dots,i_{m},l}b_{i_{0},\dots,i_{m},l}$ converges
in $\widetilde{S}$.

For the claim in $(1)$ for $p=2$, we have
$\varphi(\frac{E^{2}}{2})=(E^{2}+2b^{\prime})^{2}/2=\frac{E^{4}}{2}+2b$ for
some $b,b^{\prime}\in A$. And for $a=\sum_{i\geq
0}a_{i}(\frac{E^{p}}{p})^{i}\in\widetilde{S}$, we have

$\varphi(a)=\sum_{i\geq
0}\varphi(a_{i})(\frac{\varphi(E^{2})}{2})^{i}=\sum_{i\geq
0}\varphi(a_{i})\sum_{j=0}^{i}c_{ij}(2b)^{i-j}(\frac{E^{4}}{2})^{j}=\sum_{j\geq
0}\left(\sum_{i=j}^{\infty}\varphi(a_{i})c_{ij}(2b)^{i-j}\right)(\frac{E^{4}}{2})^{j}$

for some $c_{ij}\in\mathbb Z$. So we have $\varphi(a)\in\widehat{S}$.

For $(2)$, the if part is trivial. For the other direction, any
$x\in\mathop{\rm Fil}\nolimits^{h}\widetilde{S}$, we have

$x=\sum\limits_{i\geq 0}a_{i}\frac{E^{i}}{p^{\lfloor\frac{i}{p}\rfloor}}$

as element in $\widetilde{S}$. And if we also have $x\in\mathop{\rm
Fil}\nolimits^{h}A^{(2)}_{\max}[\frac{1}{p}]=E^{h}A^{(2)}_{\max}[\frac{1}{p}]$,
this implies for $\tilde{a}_{0}=\sum\limits_{0\leq i\leq
h}a_{i}\frac{E^{i}}{p^{\lfloor\frac{i}{p}\rfloor}}$ is in $\mathop{\rm
Fil}\nolimits^{h}A^{(2)}[\frac{1}{p}]$. This implies
$p^{\lfloor\frac{h}{p}\rfloor}\tilde{a}_{0}\in\mathop{\rm
Fil}\nolimits^{h}A^{(2)}=E^{h}A^{(2)}$. That is
$\tilde{a}_{0}={p^{-\lfloor\frac{h}{p}\rfloor}}{E^{h}}b$ for some $b\in
A^{(2)}$. So $x$ is of the given form. The proof for $(4)$ is similar.

For $(3)$, we have by $(2)$, $x\in\mathop{\rm Fil}\nolimits^{m}\widetilde{S}$,
$x$ can be written as

$x=\sum\limits_{i\geq m}a_{i}\frac{E^{i}}{p^{\lfloor\frac{i}{p}\rfloor}}.$

And use the fact $\varphi(E)=E^{p}+pb$ for some $b\in A^{(2)}$, we have

$\varphi(x)=\sum\limits_{i\geq
m}\varphi(a_{i})\sum_{j=0}^{i}\frac{c_{ij}E^{p(i-j)}p^{j}}{p^{\lfloor\frac{i}{p}\rfloor}}=\sum_{i\geq
m}\sum_{j\geq\lfloor\frac{i}{p}\rfloor}^{i}\frac{b_{ij}E^{p(i-j)}p^{j}}{p^{\lfloor\frac{i}{p}\rfloor}}+\sum_{i\geq
m}\sum_{0\leq
j<\lfloor\frac{i}{p}\rfloor}E^{h}\frac{b_{ij}E^{p(i-j)-h}p^{j}}{p^{\lfloor\frac{i}{p}\rfloor}}$

with $b_{ij}\in A^{(2)}$.

In particular, we have $\sum_{i\geq
m}\sum_{j\geq\lfloor\frac{i}{p}\rfloor}^{i}\frac{b_{ij}E^{p(i-j)}p^{j}}{p^{\lfloor\frac{i}{p}\rfloor}}$
is inside $A^{(2)}$. To prove $(3)$, it is amount to find $h_{0}$ such that
whenever $m>h_{0}$, $i\geq m$ and $0\leq j<\lfloor\frac{i}{p}\rfloor$, we have

$\sum_{i\geq m}\sum_{0\leq
j<\lfloor\frac{i}{p}\rfloor}\frac{b_{ij}E^{p(i-j)-h}p^{j}}{p^{\lfloor\frac{i}{p}\rfloor}}\in\mathop{\rm
Fil}\nolimits^{m+1}\widetilde{S}.$

The claim follows if we can find $h_{0}>h$ such that
$\frac{E^{p(i-j)-h}p^{j}}{p^{\lfloor\frac{i}{p}\rfloor}}\in\widetilde{S}$ and
$p(i-j)-h\geq m+1$ for all $m>h_{0}$, $i\geq m$ and $0\leq
j<\lfloor\frac{i}{p}\rfloor$. That is
$\lfloor\frac{p(i-j)-h}{p}\rfloor+j\geq\lfloor\frac{i}{p}\rfloor$ and
$p(i-j)-h\geq m+1$ for all $i,j,m$ in this range. And solve this we have it is
enough to choose $h_{0}>\max\\{h,\frac{p(h+1)+1}{p(p-2)}\\}$, which is valid
for $p>2$.

Statement in $(5)$ is similar to $(3)$. Any $x\in\mathop{\rm
Fil}\nolimits^{m}\widehat{S}$, $x$ can be written as

$x=\sum\limits_{i\geq m}a_{i}\frac{E^{i}}{2^{\lfloor\frac{i}{4}\rfloor}}.$

We have $\varphi(E)=E^{2}+2b$ for some $b\in A^{(2)}$, so

$\varphi(x)=\sum\limits_{i\geq
m}\varphi(a_{i})\sum_{j=0}^{i}\frac{c_{ij}E^{2(i-j)}2^{j}}{2^{\lfloor\frac{i}{4}\rfloor}}=\sum_{i\geq
m}\sum_{j\geq\lfloor\frac{i}{4}\rfloor}^{i}\frac{b_{ij}E^{2(i-j)}2^{j}}{2^{\lfloor\frac{i}{4}\rfloor}}+\sum_{i\geq
m}\sum_{0\leq
j<\lfloor\frac{i}{4}\rfloor}E^{h}\frac{b_{ij}E^{2(i-j)-h}2^{j}}{2^{\lfloor\frac{i}{4}\rfloor}}.$

Similar to the argument in $(3)$, it is amount to find $h_{0}$ such that
whenever $m>h_{0}$, $i\geq m$ and $0\leq j<\lfloor\frac{i}{4}\rfloor$, we have
$\lfloor(i-j)-\frac{h}{2}\rfloor+j\geq\lfloor\frac{i}{4}\rfloor$ and
$2(i-j)-h\geq m+1$. It is enough to choose $h_{0}>2(h+2)$. ∎

If $A$ is a ring then we denote by ${\rm M}_{d}(A)$ the set of $d\times
d$-matrices with entries in $A$.

######  Proposition 2.2.11.

Let $Y\in{\rm M}_{d}(A^{(2)}_{\max})$ so that $E^{h}Y=B\varphi(Y)C$ with $B$
and $C$ in ${\rm M}_{d}(A^{(2)})$ then $Y$ is in ${\rm
M}_{d}(A^{(2)}[\frac{1}{p}])$.

###### Proof.

First, we claim that there is a constant $s$ only depends on $h$, such that
the entries of $p^{s}Y$ is in $\widetilde{S}$. By $(1)$ of Lemma 2.2.10,
entries of $E^{h}Y$ are in $\widetilde{S}$. So for each entry $a$ of $Y$, we
can write $E^{h}a=\sum\limits_{i=0}^{\infty}a_{i}\frac{E^{pi}}{p^{i}}$ with
$a_{i}\in A^{(2)}$. It is clear that
$E^{h}p^{h}a=a^{\prime}+E^{h}\sum\limits_{i\geq h}a_{j}\frac{E^{pi-h}}{p^{i}}$
so that $a^{\prime}\in A^{(2)}$. Therefore, $a^{\prime}\in\mathop{\rm
Fil}\nolimits^{h}A^{(2)}=E^{h}A^{(2)}$ by Corollary 2.2.8. So write
$a^{\prime}=E^{h}b$, we have $p^{h}a=b^{\prime}+\sum\limits_{i\geq
h}a_{j}\frac{E^{pi-h}}{p^{i}}$. In particular, we see that
$p^{2h}a\in\widetilde{S}$, this proves our claim. When $p=2$, then we may
repeat the above argument and we can assume $p^{s}Y$ is in ${\rm
M}_{d}(\widehat{S})$.

Let $R=\widetilde{S}$ when $p>2$ and $R=\widehat{S}$ when $p=2$, then we may
assume $Y$ is inside ${\rm M}_{d}(R)$. Then we claim there is another constant
$r$ only depends on $h$, such that for each entry $a$ of $Y$, there is a
sequence $\\{b_{i}\\}_{i\geq 1}$ in $A^{(2)}$ such that we have
$a-\sum\limits_{i=0}^{m}b_{i}E^{i}\in\mathop{\rm Fil}\nolimits^{m+1}R$. Note
that once this is known, we will have $\sum\limits_{i=0}^{m}b_{i}E^{i}$
converges to an element $b$ in $A^{(2)}$, and $a-b=0$ since it is in
$\mathop{\rm Fil}\nolimits^{m}R$ for all $m\in\mathbb N$.

So it remains to show our claim. When $p>2$, let $h_{0}$ be the integer in
$(3)$ of Lemma 2.2.10, then it is easy to show there is a constant $r$ only
depends on $h_{0}$ (so only on $h$) and sequence $\\{b_{i}\\}_{i=1}^{h_{0}}$
such that for each entry $a$ of $Y^{\prime}:=p^{r}Y$, we have

$a-\sum\limits_{i=0}^{h_{0}}b_{i}E^{i}\in\mathop{\rm
Fil}\nolimits^{h_{0}+1}R.$

Now we show our claim by induction, assume for each entry $a$ in $Y^{\prime}$,
there is a sequence $\\{b_{i}\\}_{i=1}^{m}$ such that,

$a-\sum\limits_{i=0}^{m}b_{i}E^{i}\in\mathop{\rm Fil}\nolimits^{m+1}R.$

for some $m\geq h_{0}$. So we can write $Y^{\prime}$ as

$\sum_{i=0}^{m}Y_{i}E^{i}+Z_{m+1},$

with $Y_{i}\in{\rm M}_{d}(A^{(2)})$ and $Z_{m+1}\in{\rm M}_{d}(\mathop{\rm
Fil}\nolimits^{m+1}R)$. Writing $X_{m}=\sum_{i=0}^{m}Y_{i}E^{i}$, then
$E^{h}Y^{\prime}=B\varphi(Y^{\prime})C$ implies

$E^{h}Z_{m+1}=B\varphi(X_{m})C-E^{h}X_{m}+B\varphi(Z_{m+1})C.$

By $(3)$ in Lemma 2.2.10, we have $B\varphi(Z_{m+1})C=A_{m+1}+E^{h}B_{m+1}$,
with $A_{m+1}\in{\rm M}_{d}(A^{(2)})$ and $B_{m+1}\in{\rm M}_{d}(\mathop{\rm
Fil}\nolimits^{m+2}R)$. One can check
$B\varphi(X_{m})C-E^{h}X_{m}+A_{m+1}\in{\rm M}_{d}(\mathop{\rm
Fil}\nolimits^{h+m+1}A^{(2)})$, so
$B\varphi(X_{m})C-E^{h}X_{m}+A_{m+1}=E^{h+m+1}Y_{m+1}$ with $Y_{m+1}\in{\rm
M}_{d}(A^{(2)})$. And we have $Y-\sum_{i=0}^{m+1}Y_{i}E^{i}=B_{m+1}\in{\rm
M}_{d}(\mathop{\rm Fil}\nolimits^{m+2}R)$ as required.

At last when $p=2$. We know we can assume $Y$ is inside ${\rm
M}_{d}(\widehat{S})$. Then repeat the above arguments by replacing $(3)$ in
Lemma 2.2.10 with $(5)$, we can also prove our claim. ∎

###  2.3. The ring $A^{(2)}_{\mathop{\rm st}\nolimits}$

We assume that $R=\mathcal{O}_{K}$ in the following two subsections. For our
later use for semi-stable representations, we construct $A^{(2)}_{\mathop{\rm
st}\nolimits}$ as the following: Define $\varphi$ on
$W(k)[\\![x,\mathfrak{y}]\\!]$ by $\varphi(x)=x^{p}$ and
$\varphi(\mathfrak{y})=(1+\mathfrak{y})^{p}-1$ and set
$w=\frac{\mathfrak{y}}{E}$. Set $A^{(2)}_{\mathop{\rm
st}\nolimits}:=W(k)[\\![x,\mathfrak{y}]\\!]\\{w\\}_{\delta}^{\wedge}$ where
$\wedge$ means $(p,E)$-completion. Similarly, we define $A^{(3)}_{\mathop{\rm
st}\nolimits}=W(k)[\\![x,\mathfrak{y},\mathfrak{z}]\\!]\\{\frac{\mathfrak{y}}{E},\frac{\mathfrak{z}}{E}\\}^{\wedge}_{\delta}$,
with the $\delta$-structure on $W(k)[\\![x,\mathfrak{y},\mathfrak{z}]\\!]$
given by $\delta(x)=0$, $\varphi(\mathfrak{y})=(\mathfrak{y}+1)^{p}-1$ and
$\varphi(\mathfrak{z})=(\mathfrak{z}+1)^{p}-1$. Define $A^{(2)}_{\mathop{\rm
st}\nolimits,\max}$ to be the $p$-adic completion of
$W(k)[\\![x,\mathfrak{y}]\\!][w,\frac{E}{p},\gamma_{i}(w),i\geq 0].$ It is
clear that for any $f\in A^{(2)}_{\mathop{\rm st}\nolimits,\max}$ can be
written uniquely $a=\sum\limits_{i=0}^{\infty}f_{i}\gamma_{i}(w)$ with
$f_{i}\in{O}_{\mathrm{max}}$ and $f_{i}\to 0$ $p$-adically. For any subring
$B\subset A^{(2)}_{\mathop{\rm st}\nolimits,\max}[\frac{1}{p}]$, we set
$\mathop{\rm Fil}\nolimits^{i}B:=B\cap E^{i}A^{(2)}_{\mathop{\rm
st}\nolimits,\max}[\frac{1}{p}]$ and $D_{w}$ the $p$-adic completion of
$\mathcal{O}_{K}[\gamma_{i}(w),i\geq 0]$.

It turns out that $A^{(2)}$ and $A^{(2)}_{\mathop{\rm st}\nolimits}$ share
almost the same properties by replacing $z$ with $w$. So we summarize all
these properties in the following:

######  Proposition 2.3.1.

  1. (1)

One can extend Froebnius from $A$ to $A^{(2)}_{\mathop{\rm
st}\nolimits,\max}$.

  2. (2)

There exists an embedding $\iota:A^{(2)}_{\mathop{\rm
st}\nolimits}\hookrightarrow A^{(2)}_{\mathop{\rm st}\nolimits,\max}$ so that
$\iota$ commutes with Frobenius.

  3. (3)

$A^{(2)}_{\mathop{\rm st}\nolimits}\cap E^{i}A^{(2)}_{\mathop{\rm
st}\nolimits,\max}[\frac{1}{p}]=EA^{(2)}_{\mathop{\rm st}\nolimits}$.

  4. (4)

$A^{(2)}_{\mathop{\rm st}\nolimits}/E\simeq D_{w}=A^{(2)}_{\mathop{\rm
st}\nolimits,\max}/\mathop{\rm Fil}\nolimits^{1}A^{(2)}_{\mathop{\rm
st}\nolimits,\max}.$

  5. (5)

$A^{(2)}_{\mathop{\rm st}\nolimits}$ is closed in $A^{(2)}_{\mathop{\rm
st}\nolimits,\max}$.

  6. (6)

$A^{(2)}_{\mathop{\rm st}\nolimits}$ and $A^{(3)}_{\mathop{\rm st}\nolimits}$
are flat over $A$, and in particular they are bounded.

  7. (7)

Proposition 2.2.11 holds by replacing $A^{(2)}_{\max}$ and $A^{(2)}$ by
$A^{(2)}_{\mathop{\rm st}\nolimits}$ and $A^{(2)}_{\mathop{\rm
st}\nolimits,\max}$ respectively.

###### Proof.

All previous proof applies by noting the following difference

$\varphi(w)=\varphi(\frac{\mathfrak{y}}{E})=c^{-1}\frac{1}{p}\sum_{i=1}^{p}\binom{p}{i}\mathfrak{y}^{i}=c^{-1}\sum_{i=1}^{p-1}\mathfrak{y}^{i}\binom{p}{i}/p+c^{-1}\frac{E^{p}w^{p}}{p}.$

Also
$\delta(\mathfrak{y})=\sum\limits_{i=1}^{p-1}\mathfrak{y}^{i}\binom{p}{i}/p$
always contains $\mathfrak{y}$-factor and this is a key input for the analogy
of Lemma 2.2.5.

For the boundedness of $A^{(3)}_{\mathop{\rm st}\nolimits}$, we have

$W(k)[\\![x,\mathfrak{y},\mathfrak{z}]\\!]/(p,E)\simeq(\mathcal{O}_{K}/p)[\\![\bar{\mathfrak{y}},\bar{\mathfrak{z}}]\\!]$

so $\\{\mathfrak{y},\mathfrak{z}\\}$ form a $(p,E)$-complete regular sequence,
and by [BS22, Proposition 3.13], $A^{(3)}_{\mathop{\rm st}\nolimits}$ is also
$A$-flat, and this implies $A^{(3)}_{\mathop{\rm st}\nolimits}$ is bounded by
(2) in Lemma 3.7 of $loc.cit.$. ∎

Note that $A^{\widehat{\otimes}2}=W(k)[\\![x,y]\\!]\subset
W(k)[\\![x,\mathfrak{y}]\\!]$ via $y=x(\mathfrak{y}+1)$ or equivalently
$\mathfrak{y}=\frac{y}{x}-1$. It is clear that this inclusion is a map of
$\delta$-rings. By the universal property of prismatic envelope to construct
$A^{(2)}$, the inclusion induces a map of prisms $\alpha:A^{(2)}\to
A^{(2)}_{\mathop{\rm st}\nolimits}$. Since $z=xw$, we easily see that
$A^{(2)}_{\max}\subset A^{(2)}_{\mathop{\rm st}\nolimits,\max}$. So
$A^{(2)}\subset A^{(2)}_{\mathop{\rm st}\nolimits}$ via $\alpha$. We will see
that $A^{(2)}$ (resp. $A^{(2)}_{\mathop{\rm st}\nolimits}$) is the self
product of $A$ in category $X_{{{\mathbbl{\Delta}}}}$ (resp.
$(X,M_{X})_{{{\mathbbl{\Delta}}}_{\text{log}}}$) in §4.1 and §5. Then the
existence of $\alpha:A^{(2)}\to A^{(2)}_{\mathop{\rm st}\nolimits}$ can be
explained by the universal property of self product. See §5 for details.

To simplify our notation, let $B^{(2)}_{\mathop{\rm st}\nolimits}$ (resp.
$B^{(3)}_{\mathop{\rm st}\nolimits}$, $B^{(2)}$, $B^{(3)}$) be the $p$-adic
completion of ${A^{(2)}_{\mathop{\rm st}\nolimits}}[\frac{1}{E}]$ (resp.
$A^{(3)}_{\mathop{\rm st}\nolimits}[\frac{1}{E}]$, $A^{(2)}[\frac{1}{E}]$,
$A^{(3)}[\frac{1}{E}]$).

######  Lemma 2.3.2.

  1. (1)

$A^{(i)}_{\mathop{\rm st}\nolimits}\subset B^{(i)}_{\mathop{\rm
st}\nolimits}\subset B^{(i)}_{\mathop{\rm st}\nolimits}[\frac{1}{p}]$ and
$A^{(i)}\subset B^{(i)}\subset B^{(i)}[\frac{1}{p}]$ for $i=2,3$.

  2. (2)

$B^{(2)}_{\mathop{\rm st}\nolimits}\cap{A^{(2)}_{\mathop{\rm
st}\nolimits}}[\frac{1}{p}]=A^{(2)}_{\mathop{\rm st}\nolimits}$ and
$B^{(2)}\cap{A^{(2)}}[\frac{1}{p}]=A^{(2)}$.

###### Proof.

Here we only prove the case $A^{(2)}$ while the proofs for
$A^{(2)}_{\mathop{\rm st}\nolimits}$, $A^{(3)}$ and $A^{(3)}_{\mathop{\rm
st}\nolimits}$ are almost the same.

By Proposition 2.2.7, $A^{(2)}$ is a subring of $A^{(2)}_{\max}\subset
K_{0}[\\![x,z]\\!]$. So $A^{(2)}$ and hence $A^{(2)}[\frac{1}{E}]$ is an
integral domain. Then $B^{(2)}$ has no $p$-torsion: Assume that $x\in B^{(2)}$
so that $px=0$. Suppose that $x_{n}\in A^{(2)}[\frac{1}{E}]$ so that $x\equiv
x_{n}\mod p^{n}$. Then $px_{n}\equiv 0\mod p^{n}A^{(2)}[\frac{1}{E}]$. Since
$A^{(2)}[\frac{1}{E}]$ is domain, $x_{n}\equiv 0\mod p^{n-1}$. Hence $x=0$. As
$B^{(2)}$ has no $p$-torsion, we see that $B^{(2)}\subset
B^{(2)}[\frac{1}{p}]$. To see the natural map $A^{(2)}\to B^{(2)}$ is
injective, it suffices to show that $A^{(2)}/pA^{(2)}$ injects to
$A^{(2)}/pA^{(2)}[\frac{1}{u}]=B^{(2)}/pB^{(2)}$. Clearly, this is equivalent
to that $A^{(2)}/pA^{(2)}$ has no $u$-torsion. Note that $A^{(2)}$ is obtained
by taking prismatic envelope of $A^{\widehat{\otimes}2}=W(k)[\\![x,z]\\!]$ for
the ideal $I=(z)$. As mentioned before, we can apply [BS22, Prop. 3.13] to our
situation. So $A^{(2)}$ is flat over $A$ and hence $A^{(2)}/pA^{(2)}$ has no
$u$-torsion as desired.

Now we can regard $B^{(2)}$ and $A^{(2)}[\frac{1}{p}]$ as subrings of
$B^{(2)}[\frac{1}{p}]$. In particular, $B^{(2)}\cap A^{(2)}[\frac{1}{p}]$
makes sense and contains $A^{(2)}$. For any $x\in B^{(2)}\cap
A^{(2)}[\frac{1}{p}]$, if $x\not\in A^{(2)}$ but $px\in A^{(2)}$. Then the
image of $y=px$ inside $A^{(2)}/pA^{(2)}$ is nonzero but the image of $y$ in
$B^{(2)}/pB^{(2)}$ is zero. This contradicts to that $A^{(2)}/pA^{(2)}$
injects to $B^{(2)}/pB^{(2)}$. So such $x$ can not exist and we have
$B^{(2)}\cap{A^{(2)}}[\frac{1}{p}]=A^{(2)}$ as required. ∎

By [BS22, Lem. 3.9], any prism $(B,J)$ admits its perfection
$(B,J)_{\mathop{\rm perf}\nolimits}=(B_{\mathop{\rm
perf}\nolimits},JB_{\mathop{\rm perf}\nolimits})$.

######  Remark 2.3.3.

In [BS22], the underlying $\delta$-ring of $(B,J)_{\mathop{\rm
perf}\nolimits}$ is denoted by $(B_{\infty},JB_{\infty})$, and $B_{\mathop{\rm
perf}\nolimits}$ is defined as the direct perfection of $B$ in the category of
$\delta$-rings. In this paper, we write $B_{\mathop{\rm perf}\nolimits}$ as
the $(p,J)$-adic completion of $\mathrm{colim}_{\varphi}B$, which also
coincides with the derived $(p,I)$-completion of $\mathrm{colim}_{\varphi}B$
(cf. Lemma 3.9 of $loc.cit.$).

######  Lemma 2.3.4.

We have $(A^{(2)})_{\mathop{\rm perf}\nolimits}$ and $(A^{(2)}_{\mathop{\rm
st}\nolimits})_{\mathop{\rm perf}\nolimits}$ are $A$-flat.

###### Proof.

We have seen that $A^{(2)}$ is $A$-flat via $i_{1}$. And it is easy to see
$\varphi$ on $A$ is flat. Since $i_{1}$ is a $\delta$-map, so we have
$\varphi^{n}\circ i_{1}=i_{1}\circ\varphi^{n}$ which is flat. So
$\mathrm{colim}_{\varphi}A^{(2)}$ is flat over $A$. In particular, we will
have $A_{\mathop{\rm perf}\nolimits}$ is $(p,E)$-complete flat over $A$. Now
since $A$ is Noetherian, by [Sta20, Tag 0912], we have $(A^{(2)})_{\mathop{\rm
perf}\nolimits}$ is $A$-flat. The proof for $(A^{(2)}_{\mathop{\rm
st}\nolimits})_{\mathop{\rm perf}\nolimits}$ is the same. ∎

###  2.4. Embedding $A^{(2)}$ and $A^{(2)}_{\mathop{\rm st}\nolimits}$ to
$A_{\mathrm{inf}}$

Let $A_{\mathrm{inf}}=W(\mathcal{O}_{\mathbb C_{p}}^{\flat})$, then there is a
surjection $\theta:A_{\mathrm{inf}}\to\mathcal{O}_{\mathbb C_{p}}$ and
$\mathop{\rm Ker}\nolimits\theta=(E)$. And let $B_{\mathrm{dR}}^{+}$ be the
$\mathop{\rm ker}\nolimits\theta$-adic completion of
$A_{\mathrm{inf}}[\frac{1}{p}]$.

######  Definition 2.4.1.

Let $\mathbb A_{\max}$ be the $p$-adic completion of the
$A_{\mathrm{inf}}$-subalgebra of $B_{\mathrm{dR}}^{+}$ generated by $E/p$.

It can be easily seen that $\varphi(E/p):=\varphi(E)/p\in
A_{\mathrm{cris}}\subset\mathbb A_{\max}$ is well-defined and it extends the
Frobenius structure on $A_{\mathrm{inf}}$ to an endomorphism on ${\mathbb
A}_{\mathrm{max}}$.

Let $\\{\varpi_{n}\\}_{n\geq 0}$ be a compatible system of $p^{n}$-th roots of
$\varpi_{0}=\varpi$ and $\\{\zeta_{n}\\}_{n\geq 0}$ be a compatible system of
$p^{n}$-th roots of 1. Write $\varpi^{\flat}:=\\{\varpi_{n}\\}_{n\geq
0},\zeta^{\flat}:=\\{\zeta_{n}\\}_{n\geq 0}\in\mathcal{O}_{\mathbb
C_{p}}^{\flat}$ and let $u=[\varpi^{\flat}]$, $\epsilon=[\zeta^{\flat}]$,
$v=\epsilon u$ and $\mu=\epsilon-1$ be elements inside $A_{\mathrm{inf}}$. We
can regard $W(k)[\\![x,y]\\!]$ as a subring of $A_{\mathrm{inf}}$ via
$x\mapsto u$ and $y\mapsto v$. Consider $z^{\prime}=\frac{u-v}{E}\in
A_{\mathrm{inf}}[\frac{1}{E}]$. Since $u-v=u(\epsilon-1)$ is clearly inside
$\mathop{\rm Ker}\nolimits(\theta)$ and $\mathop{\rm
Ker}\nolimits(\theta)=EA_{\mathrm{inf}}$, we conclude that $z^{\prime}\in
A_{\mathrm{inf}}$. Hence we have a natural map (of $\delta$-rings)
$\iota_{A}:\widetilde{A}^{(2)}\to A_{\mathrm{inf}}$ via $z\mapsto z^{\prime}$,
which naturally extends to $\iota_{A}:A^{(2)}\to A_{\mathrm{inf}}$ because
$(p,E)$-topology of $A^{(2)}$ matches with the weak topology of
$A_{\mathrm{inf}}$. Similarly, we have map of $\delta$-rings
$\iota_{\mathop{\rm st}\nolimits}:A^{(2)}_{\mathop{\rm st}\nolimits}\to
A_{\inf}$ via $x\mapsto u$ and $\mathfrak{y}\mapsto\epsilon-1$ and
$w\mapsto\frac{\epsilon-1}{E}$.

######  Remark 2.4.2.

Once we know that $A^{(2)}$ is self-product of $A$ inside
$X_{{\mathbbl{\Delta}}}$ with $X=\mathop{\rm Spf}\nolimits(\mathcal{O}_{K})$
as explained in §4.1. The map $\iota_{A}$ can be constructed as the following:
First we fix an embedding $A\to A_{\mathrm{inf}}$ by sending $x\mapsto
u=[\varpi^{\flat}]$. Then $A\to A_{\mathrm{inf}}$ by $x\to v=\epsilon u$ is
another map of prisms. By universal property of $A^{(2)}$, these two maps
extends to a map $\iota_{A}:A^{(2)}\to A_{\mathrm{inf}}$. Clearly, the map
$\iota_{A}:A^{(2)}\to A_{\mathrm{inf}}$ depends on choice of
${\varpi}^{\flat}=(\varpi_{n})_{n\geq 0}$ and
${\zeta}^{\flat}=(\zeta_{n})_{n\geq 0}$. Also $\iota_{A}$ is a special case of
$\iota^{(2)}_{\gamma}$ defined by (14) in §4.3. Indeed if
$\gamma([w^{\flat}])=[\zeta^{\flat}][w^{\flat}]$ then
$\iota_{A}=\iota^{(2)}_{\gamma}$. Similarly comment also applies to
$\iota_{\mathop{\rm st}\nolimits}$.

######  Proposition 2.4.3.

There is a unique embedding ${A_{\max}^{(2)}}$${\mathbb A_{\max}}$ such that

${W(k)[\\![x,y]\\!]}$${A_{\mathrm{inf}}}$${A_{\max}^{(2)}}$${\mathbb
A_{\max}}$${B_{\mathrm{dR}}^{+}}$

commutes. Furthermore, $\mathop{\rm Fil}\nolimits^{i}B_{\mathrm{dR}}^{+}\cap
A^{(2)}_{\max}=\mathop{\rm Fil}\nolimits^{i}A^{(2)}_{\max}$. The same result
holds when $A^{(2)}_{\max}$ is replaced by $A^{(2)}_{\mathop{\rm
st}\nolimits,\max}$.

###### Proof.

In the following, we only treat the case of $A^{(2)}_{\mathop{\rm
st}\nolimits,\max}$ while the proof of $A^{(2)}_{\max}$ is the same by noting
that $z=uw$ in $A_{\inf}$.

The uniqueness is clear. To show the existence of the embedding, it is enough
to show $\gamma_{i}(w)\in{\mathbb A}_{\mathrm{max}}$ for all $i\geq 1$.

It is a well-known fact that ${\mathbb A}_{\mathrm{max}}$ is isomorphic to the
$p$-adic completion of $A_{\mathrm{inf}}[\frac{u^{e}}{p}]$, and ${\mathbb
A}_{\mathrm{max}}[1/p]$ is a Banach $\mathbb Q_{p}$-algebra, which is the
completion of $A_{\mathrm{inf}}[1/p]$ under the norm
$\lvert\cdot\rvert_{p^{-1}}$ such that

$\lvert x\rvert_{p^{-1}}=\sup_{n}\\{p^{-n}\lvert
x_{n}\rvert_{\mathcal{O}_{C}^{\flat}}\\}$

where $x=\sum_{n\gg 0}[x_{n}]p^{n}\in A_{\mathrm{inf}}[1/p]$. And we have for
$x\in{\mathbb A}_{\mathrm{max}}[1/p]$, $x\in{\mathbb A}_{\mathrm{max}}$ if and
only if $\lvert x\rvert_{p^{-1}}\leq 1$. Moreover $\lvert\cdot\rvert_{p^{-1}}$
is multiplicative. So now it is enough to show for $x=\gamma_{i}(w)$
considered as an element inside ${\mathbb A}_{\mathrm{max}}[1/p]$, we have
$\lvert x^{p-1}\rvert_{p^{-1}}\leq 1$. To show this, we have by [BMS18,
Proposition 3.17], $\xi:=\mu/\varphi^{-1}(\mu)$ is a generator of $\mathop{\rm
Ker}\nolimits\theta$ with $\mu=\epsilon-1$. In particular,
$w=\mu/E=a\varphi^{-1}(\mu)\in A_{\mathrm{inf}}$ with $a\in
A_{\mathrm{inf}}^{\times}$. And we can check
$\overline{w}^{p-1}=c\overline{u}^{e}$ inside
$\mathcal{O}_{C}^{\flat}=A_{\mathrm{inf}}/pA_{\mathrm{inf}}$, with $c$ a unit.
So $w^{p-1}=au^{e}+bp$ with $a,b\in A_{\mathrm{inf}}$, and

$x^{p-1}=\frac{(au^{e}+bp)^{i}}{(i!)^{p-1}}.$

Using the fact $v_{p}(i!)<\frac{i}{p-1}$, one can show each term in the
binomial expansion on the right hand side of the equation has
$\lvert\cdot\rvert_{p^{-1}}$-norm less or equal to $1$, so in particular,
$\lvert x^{p-1}\rvert_{p^{-1}}\leq 1$.

To prove that $\mathop{\rm Fil}\nolimits^{i}B_{\mathrm{dR}}^{+}\cap
A^{(2)}_{\mathop{\rm st}\nolimits,\max}=\mathop{\rm
Fil}\nolimits^{i}A^{(2)}_{\mathop{\rm st}\nolimits,\max}$, it suffices to show
that $EB_{\mathrm{dR}}^{+}\cap A^{(2)}_{\mathop{\rm
st}\nolimits,\max}[\frac{1}{p}]=EA^{(2)}_{\mathop{\rm
st}\nolimits,\max}[\frac{1}{p}]$. By Proposition 2.2.7, we reduces to prove
that the map

$\theta:D_{w}=A^{(2)}_{\mathop{\rm st}\nolimits,\max}[\frac{1}{p}]/E\to
B_{\mathrm{dR}}^{+}/E=\mathbb C_{p}$

is injective. Let $f(w)=\sum_{i\geq 0}a_{i}\gamma_{i}(w)\in\mathop{\rm
Ker}\nolimits\theta$ with $a_{i}\in\mathcal{O}_{K}$ limits to $0$
$p$-adically. Then $f(w_{0})=0$ with
$w_{0}:=\theta(w)=\theta(\frac{\epsilon-1}{E})\in\mathbb C_{p}$. Note
$v_{p}(w_{0})\geq\frac{1}{p-1}$ because it is well-known
$\frac{\epsilon-1}{\varphi^{-1}(\epsilon)-1}$ is another generator of kernel
$\theta:A_{\inf}\to\mathcal{O}_{\mathbb C_{p}}$ and then
$v_{p}(w_{0})=v_{p}(\theta(\varphi^{-1}(\epsilon)-1))=\frac{1}{p-1}$. Since we
are aiming to show that $f=0$, without loss of generality, we can assume that
$K$ contains $p_{1}=\sqrt[p-1]{p}$. Note that $v_{p}(i!)\leq\frac{1}{p-1}$, we
conclude that $\frac{w_{0}}{p_{1}}$ is a root of $f(p_{1}w)$ which is in
$\mathcal{O}_{K}\langle w\rangle$. By Weierstrass preparation theorem, $w_{0}$
is algebraic over $K$ unless $f=0$. By Lemma below,
$w_{0}:=\theta(w)\in\mathbb C_{p}$ is transcendental over $K$ and hence $f=0$.
∎

######  Lemma 2.4.4.

$w_{0}=\theta(\frac{\epsilon-1}{E})$ is transcendental over $K$.

###### Proof.

If $w_{0}$ is contained in an algebraic extension $L$ over $K$, we define
$L_{0,\infty}=\bigcup_{n}L(\varpi_{n})$. For $g\in G_{L_{0,\infty}}$, we will
have

$\theta(g(\frac{\epsilon-1}{E}))=g(w_{0})=w_{0}=\theta(\frac{\epsilon-1}{E}).$

Since $G_{L_{0,\infty}}$ fix $E$,
$\theta(\frac{g(\epsilon-1)-(\epsilon-1)}{E})=0$. This implies
$g(\epsilon-1)-(\epsilon-1)\in\mathop{\rm
Fil}\nolimits^{2}B_{\mathrm{dR}}^{+}$. Recall for $t=\log\epsilon$,
$t-(\epsilon-1)\in\mathop{\rm Fil}\nolimits^{2}B_{\mathrm{dR}}^{+}$, so we
have $g(t)-t\in\mathop{\rm Fil}\nolimits^{2}B_{\mathrm{dR}}^{+}$. But this
can’t be true. Since $L_{0,\infty}$ can only contain finitely many $p^{n}$-th
roots of $1$, for $g\in G_{L_{0,\infty}}$, $g(t)=c(g)t$ satisfying
$c(g)\in\mathbb Q_{p}$ and $c(g)\neq 1$. This implies
$g(t)-t=(c(g)-1)t\in\mathop{\rm
Fil}\nolimits^{1}B_{\mathrm{dR}}^{+}\setminus\mathop{\rm
Fil}\nolimits^{2}B_{\mathrm{dR}}^{+}$. ∎

######  Corollary 2.4.5.

The natural maps $\iota_{A}:A^{(2)}\to A_{\inf}$ and $\iota_{\mathop{\rm
st}\nolimits}:A^{(2)}_{\mathop{\rm st}\nolimits}\to A_{\inf}$ are injective.

To summarize, we have the following commutative diagram of rings inside
$B_{\mathrm{dR}}^{+}$:

${A^{(2)}}$${A^{(2)}_{\mathop{\rm
st}\nolimits}}$${A_{\mathrm{inf}}}$${A^{(2)}_{\max}}$${A^{(2)}_{\mathop{\rm
st}\nolimits,\max}}$${{\mathbb A}_{\mathrm{max}}.}$

##  3\. Application to semi-stable Galois representations

In this section, we assume that $R=\mathcal{O}_{K}$. We explain how to use the
period ring $A^{(2)}$ and $A^{(2)}_{\mathop{\rm st}\nolimits}$ to understand
lattices in crystalline and semi-stable representations. Roughly speaking, we
are going to use $A^{(2)}$ and $A^{(2)}_{\mathop{\rm st}\nolimits}$ to replace
$\widehat{\mathcal{R}}$ in the theory of $(\varphi,\hat{G})$-modules developed
in [Liu10].

Let $K_{\infty}=\bigcup_{n=1}^{\infty}K(\varpi_{n})$, $G_{\infty}:={\rm
Gal}(\overline{K}/K_{\infty})$ and $G_{K}:={\rm Gal}(\overline{K}/K)$. Recall
that $A=\mathfrak{S}=W(k)[\\![u]\\!]$. Let $S$ be the $p$-adic completion of
$W(k)[\\![u,\frac{E^{i}}{i!},i\geq 1]\\!]$, which is the PD envelope of
$W(k)[u]$ for the ideal $(E)$. It is clear that $S\subset{O}_{\mathrm{max}}$.
We define $\varphi$ and $\mathop{\rm Fil}\nolimits^{i}$ on $S$ induced that
from those on ${O}_{\mathrm{max}}$, in particular, $\mathop{\rm
Fil}\nolimits^{i}S=S\cap E^{i}{O}_{\mathrm{max}}[\frac{1}{p}]$. Note that $A$
embeds to $A_{\mathrm{inf}}$ via $u\mapsto[\varpi^{\flat}]$ is not stable
under $G_{K}$-action but only on $G_{\infty}$-action. For any $g\in G_{K}$,
define ${\underline{\varepsilon}}(g)=\frac{g(u)}{u}$. It is clear that
${\underline{\varepsilon}}(g)=\epsilon^{a(g)}$ with $a(g)\in\mathbb Z_{p}$. We
define _two_ differential operators $N_{S}$ and $\nabla_{S}$ on $S$ by
$N_{S}(f)=\frac{df}{du}u$ and $\nabla_{S}(f)=\frac{df}{du}$. We need
$\nabla_{S}$ to treat crystalline representations.

###  3.1. Kisin module attached to semi-stable representation

Fix $h\geq 0$, a _Kisin module of height $h$_ is a finite free $A$-module
$\mathfrak{M}$ with a semi-linear endomorphism
$\varphi_{\mathfrak{M}}:\mathfrak{M}\to\mathfrak{M}$ so that $\mathop{\rm
coker}\nolimits(1\otimes\varphi_{\mathfrak{M}})$ is killed by $E^{h}$, where
$1\otimes\varphi_{\mathfrak{M}}:\mathfrak{M}^{*}:=A\otimes_{\varphi,A}\mathfrak{M}\to\mathfrak{M}$
is linearization of $\varphi_{\mathfrak{M}}$. Note here we are using classical
setting of Kisin modules used in [Liu10] but it is good enough for this paper.
The following summarizes the results on Kisin modules attached to
$G_{K}$-stable $\mathbb Z_{p}$-lattices in semi-stable representations. The
details and proofs of these facts can be found in [Liu10].

Let $T$ be a $G_{K}$-stable $\mathbb Z_{p}$-lattice inside a semi-stable
representation $V$ of $G_{K}$ with Hodge-Tate weights in $\\{0,\dots,h\\}$.
Let $D:=D^{*}_{\mathop{\rm st}\nolimits}(V)=\mathop{\rm Hom}\nolimits_{\mathbb
Q_{p},G_{K}}(V,B_{\mathop{\rm st}\nolimits})$ be the filtered
$(\varphi,N)$-module attached to $V$ and $D_{K}:=K\otimes_{K_{0}}D$. Then
there exists a unique Kisin module $\mathfrak{M}:=\mathfrak{M}(T)$ of height
$h$ attached to $T$ so that

  1. (1)

$\mathop{\rm Hom}\nolimits_{\varphi,A}(\mathfrak{M},A_{\mathrm{inf}})\simeq
T|_{G_{\infty}}$.

  2. (2)

There exists an $S$-linear isomorphism

$\iota_{S}:S[\frac{1}{p}]\otimes_{\varphi,A}\mathfrak{M}\simeq
D\otimes_{W(k)}S$

so that $\iota_{S}$ is compatible with $\varphi$ on the both sides.

  3. (3)

$\iota_{S}$ also induces an isomorphism $\mathop{\rm
Fil}\nolimits^{h}(S[\frac{1}{p}]\otimes_{\varphi,A}\mathfrak{M})\simeq\mathop{\rm
Fil}\nolimits^{h}(D\otimes_{W(k)}S)$. The filtration on the both sides are
defined as following:

$\mathop{\rm
Fil}\nolimits^{h}(S[\frac{1}{p}]\otimes_{\varphi,A}\mathfrak{M}):=\left\\{x\in
S[\frac{1}{p}]\otimes_{\varphi,A}\mathfrak{M}|(1\otimes\varphi_{\mathfrak{M}}(x))\in\mathop{\rm
Fil}\nolimits^{h}S[\frac{1}{p}]\otimes_{A}\mathfrak{M}\right\\}.$

To define filtration on ${\mathcal{D}}:=S\otimes_{W(k)}D$, we first extend the
monodromy operator $N_{\mathcal{D}}$ (resp. $\nabla_{\mathcal{D}}$) on $D$ to
${\mathcal{D}}$ by $N_{{\mathcal{D}}}=1\otimes N_{D}+N_{S}\otimes 1$ (resp.
$\nabla_{\mathcal{D}}=1\otimes N_{D}+\nabla_{S}\otimes 1$). Then we define
$\mathop{\rm Fil}\nolimits^{i}{\mathcal{D}}$ by induction: set $\mathop{\rm
Fil}\nolimits^{0}{\mathcal{D}}={\mathcal{D}}$ and

$\mathop{\rm
Fil}\nolimits^{i}{\mathcal{D}}:=\\{x\in{\mathcal{D}}|N_{{\mathcal{D}}}(x)\in\mathop{\rm
Fil}\nolimits^{i-1}{\mathcal{D}},f_{\varpi}(x)\in\mathop{\rm
Fil}\nolimits^{i}D_{K}\\}$

where $f_{\varpi}:{\mathcal{D}}\to D_{K}$ is induced by $S\to\mathcal{O}_{K}$
via $u\mapsto\varpi.$

######  Remark 3.1.1 (Griffith transversality).

From the construction of $\mathop{\rm Fil}\nolimits^{i}{\mathcal{D}}$, we see
that $N_{{\mathcal{D}}}(\mathop{\rm
Fil}\nolimits^{i}{\mathcal{D}})\subset\mathop{\rm
Fil}\nolimits^{i-1}{\mathcal{D}}$. This property is called Griffith
transversality.

We only use $\nabla_{\mathcal{D}}$ when $N_{D}=0$, that is, when $V$ is
crystalline. In this case, it is clear that
$N_{\mathcal{D}}=u\nabla_{{\mathcal{D}}}$. So it is clear that
$\nabla_{\mathcal{D}}(\mathop{\rm
Fil}\nolimits^{i}{\mathcal{D}})\subset\mathop{\rm
Fil}\nolimits^{i-1}{\mathcal{D}}$.

For ease of notations, we will write $N=N_{\mathcal{D}}$ and
$\nabla=\nabla_{\mathcal{D}}$ in the following. Let $T^{\vee}:=\mathop{\rm
Hom}\nolimits_{\mathbb Z_{p}}(T,\mathbb Z_{p})$ and
$V^{\vee}:=T^{\vee}\otimes_{\mathbb Z_{p}}\mathbb Q_{p}$ denote the dual
representations. Then there exists an $A_{\inf}$-linear injection

(5) $\iota_{\mathfrak{M}}:A_{\inf}\otimes_{A}\mathfrak{M}\to
T^{\vee}\otimes_{\mathbb Z_{p}}A_{\mathrm{inf}},$

which is compatible with $G_{\infty}$-actions ($G_{\infty}$ acts on
$\mathfrak{M}$ trivially) and $\varphi$ on both sides. Applying
$S\otimes_{\varphi,A}$ and using
$\iota_{S}:=S\otimes_{\varphi,A}\iota_{\mathfrak{M}}$, we obtain the following
commutative diagram

$\textstyle{A_{\mathrm{cris}}[\frac{1}{p}]\otimes_{\varphi,A}\mathfrak{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wr}$$\scriptstyle{A_{\mathrm{cris}}\otimes_{S}\iota_{S}}$$\scriptstyle{S\otimes_{\varphi,A}\iota_{\mathfrak{M}}}$$\textstyle{V^{\vee}\otimes_{\mathbb
Z_{p}}A_{\mathrm{cris}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{\mathrm{cris}}\otimes_{W(k)}D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{V^{\vee}\otimes_{\mathbb
Z_{p}}A_{\mathrm{cris}}}$

where the second row $\alpha$ is built by the classical comparison

$B_{\mathop{\rm st}\nolimits}\otimes_{K_{0}}D^{*}_{\mathop{\rm
st}\nolimits}(V)\simeq V^{\vee}\otimes_{\mathbb Q_{p}}B_{\mathop{\rm
st}\nolimits},$

and $\alpha$ is $G_{K}$-stable on the both sides. The left side of $\alpha$ is
defined by

$\forall x\in D,\forall g\in
G_{K},g(x)=\sum_{i=0}^{\infty}N^{i}(x)\gamma_{i}(\log({\underline{\varepsilon}}(g)))$

Therefore, if we regard $\mathfrak{M}^{*}:=A\otimes_{\varphi,A}\mathfrak{M}$
as an $A$-submodule of $V^{\vee}\otimes_{\mathbb Z_{p}}A_{\mathrm{cris}}$ via
injection $\iota^{*}:=S\otimes_{\varphi,A}\iota_{A}$, one can show that:

(6) $\forall g\in
G_{K},x\in\mathfrak{M}^{*},g(x)=\sum_{i=0}^{\infty}N_{\mathcal{D}}^{i}(x)\gamma_{i}(\log({\underline{\varepsilon}}(g))).$

When $V$ is crystalline, or equivalently, $N_{D}=0$, we have ([LL21, §8.1])

(7) $\forall g\in
G_{K},x\in\mathfrak{M}^{*},g(x)=\sum_{i=0}^{\infty}\nabla_{\mathcal{D}}^{i}(x)\gamma_{i}(u{\underline{\varepsilon}}(g)).$

###  3.2. Descent of the $G_{K}$-action

Let us first discuss the $G_{K}$-action on $\mathfrak{M}\subset
T^{\vee}\otimes_{\mathbb Z_{p}}A_{\mathrm{inf}}$ via $\iota_{\mathfrak{M}}$ in
(5) in more details. We select an $A$-basis $e_{1},\dots,e_{d}$ of
$\mathfrak{M}$ so that
$\varphi(e_{1},\dots,e_{d})=(e_{1},\dots,e_{d})\mathfrak{A}$ with
$\mathfrak{A}\in{\rm M}_{d}(A)$. Then there exists a matrix $B\in{\rm
M}_{d}(A)$ so that $\mathfrak{A}B=B\mathfrak{A}=E^{h}I_{d}$. For any $g\in
G_{K},$ let $X_{g}$ be the matrix so that

$g(e_{1},\dots,e_{d})=(e_{1},\dots,e_{d})X_{g}.$

In this section, we are interested in where the entries of $X_{g}$ locates.

######  Theorem 3.2.1.

The entries of $X_{g}$ are in $A^{(2)}_{\mathop{\rm st}\nolimits}$. If $V$ is
crystalline and $g(u)-u=Ez$ then $X_{g}\in{\rm M}_{d}(A^{(2)}).$

First, it is well-known that $W(\mathbb
C_{p}^{\flat})\otimes_{A_{\mathrm{inf}}}\iota_{\mathfrak{M}}$ is an
isomorphism. So $X_{g}\in{\rm M}_{d}(W(\mathbb C_{p}^{\flat}))$. Since
$G_{K}$-actions and $\varphi$-commutes, we have

$\mathfrak{A}\varphi(X_{g})=X_{g}g(\mathfrak{A}).$

Define

$\mathop{\rm
Fil}\nolimits^{h}\mathfrak{M}^{*}:=\\{x\in\mathfrak{M}^{*}|(1\otimes\varphi_{\mathfrak{M}})(x)\in
E^{h}\mathfrak{M}\\}.$

Since $\mathfrak{M}$ has $E$-height $h$, it is easy to show that $\mathop{\rm
Fil}\nolimits^{h}\mathfrak{M}^{*}$ is a finite free $A$-module and
$\mathop{\rm Fil}\nolimits^{h}{\mathcal{D}}$ is generated by $\mathop{\rm
Fil}\nolimits^{h}\mathfrak{M}^{*}$.

To be more precise, let $\\{e^{*}_{i}:=1\otimes e_{i},i=1,\dots,d\\}$ be an
$A$-basis of $\mathfrak{M}^{*}$. It is easy to check that
$(\alpha_{1},\dots,\alpha_{d})=(e_{1}^{*},\dots,e_{d}^{*})B$ is an $A$-basis
of $\mathop{\rm Fil}\nolimits^{h}\mathfrak{M}^{*}$, and it is also an
$S[\frac{1}{p}]$-basis of $\mathop{\rm Fil}\nolimits^{h}{\mathcal{D}}$. So for
any $g\in G_{K}$, we have
$g(\alpha_{j})=\sum\limits_{i=0}^{\infty}N^{i}(\alpha_{j})\gamma_{i}(\log({\underline{\varepsilon}}(g)))$.
By Griffith transversality in Remark 3.1.1: $N(\mathop{\rm
Fil}\nolimits^{i}{\mathcal{D}})\subset\mathop{\rm
Fil}\nolimits^{i-1}{\mathcal{D}},$ we have,

(8)
$g(\alpha_{j})=\sum_{i=0}^{h}N^{i}(\alpha_{j})E^{i}\gamma_{i}(\frac{\log({\underline{\varepsilon}}(g))}{E})+\sum_{i>h}^{\infty}N^{i}(\alpha_{j})\gamma_{i}(E)(\frac{\log({\underline{\varepsilon}}(g))}{E})^{i}.$

Since $N^{i}(\alpha_{j})E^{i}\in\mathop{\rm Fil}\nolimits^{h}{\mathcal{D}}$,
$\gamma_{i}(E)$ in ${O}_{\mathrm{max}}$ and $w^{n}\to 0$ inside
$A^{(2)}_{\mathop{\rm st}\nolimits,\max}$, we see that
$g(\alpha_{1},\dots,\alpha_{d})=(\alpha_{1},\dots,\alpha_{d})Y_{g}$ with
$Y_{g}\in A^{(2)}_{\mathop{\rm st}\nolimits,\max}[\frac{1}{p}].$

In the case that $V$ is crystalline, using (7), we have

$g(\alpha_{j})=\sum_{i=0}^{h}\nabla^{i}(\alpha_{j})E^{i}\gamma_{i}(\frac{u{\underline{\varepsilon}}(g)}{E})+\sum_{i>h}^{\infty}\nabla^{i}(\alpha_{j})\gamma_{i}(E)(\frac{u{\underline{\varepsilon}}(g)}{E})^{i}$

_If $g$ is chosen so that $g(u)-u=Ez$_ then, a similar argument can shows that
$g(\alpha_{1},\dots,\alpha_{d})=(\alpha_{1},\dots,\alpha_{d})Y^{\nabla}_{g}$
with $Y^{\nabla}_{g}\in A^{(2)}_{\max}[\frac{1}{p}].$

Now $g(e_{1}^{*},\dots,e_{d}^{*})=(e_{1}^{*},\dots,e_{d}^{*})\varphi(X_{g})$.
Using similar arguments, we see that $\varphi(X_{g})$’s entry are in
$A^{(2)}_{\mathop{\rm st}\nolimits,\max}[\frac{1}{p}]$ and
$A^{(2)}_{\max}[\frac{1}{p}]$ respectively. Since
$(\alpha_{1},\dots,\alpha_{d})=(e_{1}^{*},\dots,e_{d}^{*})B$, we conclude that

$\varphi(X_{g})g(B)=BY_{g}.$

Using the formula that $\mathfrak{A}\varphi(X_{g})=X_{g}g(\mathfrak{A})$ and
$\mathfrak{A}B=B\mathfrak{A}=E^{h}I_{d}$, we conclude that
$Y_{g}=(\frac{g(E)}{E})^{h}X_{g}$. Write $r=\frac{g(E)}{E}$. We claim that $r$
is a unit in $A^{(2)}_{\mathop{\rm st}\nolimits}$. Indeed,
$\frac{g(E)}{E}=\frac{E(u\epsilon^{a(g)})}{E(u)}=\sum\limits_{i=0}^{e}E^{(i)}(u)\frac{u^{i}(\epsilon^{a(g)}-1)^{i}}{Ei!}$
is again inside $A_{\mathop{\rm st}\nolimits}^{(2)}$, where $E^{(i)}$ means
the $i$-th derivative of $E$. And it is easy to show $g(E)$ is also a
distinguished element $A_{\mathop{\rm st}\nolimits}^{(2)}$, so by [BS22, Lemma
2.24], $r$ is a unit. Similarly, when $g(u)-u=Ez$, we will have
$r=\frac{g(E)}{E}\in(A^{(2)})^{\times}$. Hence

(9) $E^{h}X_{g}=r^{-h}\mathfrak{A}\varphi(X_{g})g(B).$

Now we can apply Proposition 2.2.11 and Proposition 2.3.1 (5) to the above
formula, we conclude that for $g\in G_{K}$ (resp. $g\in G_{K}$ such that
$g(u)-u=Ez$ and $V$ is crystalline), we have $X_{g}$ has entries in
$A^{(2)}_{\mathop{\rm st}\nolimits}[\frac{1}{p}]$ (resp.
$A^{(2)}[\frac{1}{p}]$).

To complete the proof of Theorem 3.2.1, it suffices to show that entries
$X_{g}$ are in $A^{(2)}_{\mathop{\rm st}\nolimits}$ (resp. $A^{(2)}$).
Unfortunately, the proof to remove “$\frac{1}{p}$” is much harder, which needs
§4.2 and §4.3. For the remaining of this subsection, we only show that the
proof of Theorem 3.2.1 can be reduced to the case that $g=\tilde{\tau}$ for a
special selected $\tilde{\tau}\in G_{K}$.

Let $L=\bigcup\limits_{n=1}^{\infty}K_{\infty}(\zeta_{p^{n}})$,
$K_{1^{\infty}}:=\bigcup_{n=1}^{\infty}K(\zeta_{p^{n}})$,
$\hat{G}:=\mathop{\rm Gal}\nolimits(L/K)$ and $H_{K}:=\mathop{\rm
Gal}\nolimits(L/K_{\infty})$. If $p>2$ then it is known that
$\hat{G}\simeq\mathop{\rm Gal}\nolimits(L/K_{1^{\infty}})\rtimes H_{K}$ with
$\mathop{\rm Gal}\nolimits(L/K_{1^{\infty}})\simeq\mathbb Z_{p}$. Let $\tau$
be a topological generator of $\mathop{\rm Gal}\nolimits(L/K_{1^{\infty}})$.
We have $\tau(u)=\epsilon^{a}u$ with $a\in\mathbb Z_{p}^{\times}$. Without
loss of generality, we may assume that $\tau(u)=\epsilon u$. If $p=2$ then we
can still select $\tau\in\hat{G}$ so that $\tau(u)=\epsilon u$ and
$\tau,H_{K}$ topologically generate $\hat{G}$. Pick $\tilde{\tau}\in G_{K}$ a
lift of $\tau$. Clearly, we have $\tilde{\tau}(u)-u=Ez$.

######  Proposition 3.2.2.

For $g=\tilde{\tau},$ the entries of $X_{g}$ are in $A^{(2)}_{\mathop{\rm
st}\nolimits}$, and if further $V$ is crystalline, then $X_{g}\in{\rm
M}_{d}(A^{(2)}).$

######  Lemma 3.2.3.

Proposition 3.2.2 is equivalent to Theorem 3.2.1.

###### Proof.

Since $\hat{G}$ is topologically generated by $\tau$ and $H_{K}$. So $G_{K}$
is topologically generated by $G_{\infty}$ and $\tilde{\tau}$. And we have
$\tau(u)-u=(\epsilon-1)u=Ez$. Now if $X_{\tilde{\tau}}$ has coefficient in
$A^{(2)}_{\mathop{\rm st}\nolimits}$ and $X_{g}=I_{d}$ for all $g\in
G_{\infty}$ then to show that $X_{g}\in A^{(2)}_{\mathop{\rm st}\nolimits}$
for all $g\in G_{K}$, it suffices to show that $X_{\tilde{\tau}^{p^{n}}}$
converges to $I_{d}$ inside ${\rm M}_{d}(A^{(2)}_{\mathop{\rm st}\nolimits})$.
Since $A^{(2)}_{\mathop{\rm st}\nolimits}$ is closed in $A^{(2)}_{\mathop{\rm
st}\nolimits,\max}$ by Proposition 2.3.1 (5), it suffices to show that
$X_{\tilde{\tau}^{p^{n}}}$ converges inside $A^{(2)}_{\mathop{\rm
st}\nolimits,\max}$. Since $X_{g}=(\frac{E}{g(E)})^{r}Y_{g}$ and $Y_{g}$ is
defined by (8), we easily check that $X_{\tilde{\tau}^{p^{n}}}$ converges to
$I_{d}$ in $A^{(2)}_{\mathop{\rm st}\nolimits,\max}$ by using that
${\underline{\varepsilon}}(\tilde{\tau}^{p^{n}})$ converges to $0$ in
$(p,\epsilon-1)$-topology. The proof for the crystalline case is similar by
replacing $A^{(2)}_{\mathop{\rm st}\nolimits}$ with $A^{(2)}$. ∎

So it remains to prove Proposition 3.2.2 to complete the proof of Theorem
3.2.1. We will prove Proposition 3.2.2 in §4.3. Briefly speaking, for
$g=\tilde{\tau}$, we have shown that the linearization of the $g$-action
defines a $\varphi$-equivariant isomorphism:

$f_{g}:\mathfrak{M}\otimes_{A,\iota_{g}}A_{\mathop{\rm
st}\nolimits}^{(2)}[\frac{1}{p}]\simeq\mathfrak{M}\otimes_{A}A_{\mathop{\rm
st}\nolimits}^{(2)}[\frac{1}{p}]$

of $A_{\mathop{\rm st}\nolimits}^{(2)}[\frac{1}{p}]$-modules, and since
$g(u)-u=Ez$ and $V$ is crystalline, $f_{g}$ defines a $\varphi$-equivariant
isomorphism:

$f_{g}:\mathfrak{M}\otimes_{A,\iota_{g}}A^{(2)}[\frac{1}{p}]\simeq\mathfrak{M}\otimes_{A}A^{(2)}[\frac{1}{p}]$

of $A^{(2)}[\frac{1}{p}]$-modules. Here $\iota_{g}:A\to A^{(2)}_{\mathop{\rm
st}\nolimits}$ (resp. $\iota_{g}:A\to A^{(2)})$) is defined by $u\to g(u)$. On
the other hand, by [Wu21, Theorem 5.6], we will see the $g$-action on
$T^{\vee}\otimes W(\mathbb C_{p}^{\flat})$ also descent to a
$\varphi$-equivariant morphism $c$ of $B^{(2)}$-modules, and recall that
$B^{(2)}$ the is $p$-adic completion of $A^{(2)}[\frac{1}{E}]$. Then by
comparing $c$ and $f_{g}$ using the technique developed in §4.2, we will
deduce Proposition 3.2.2 from Lemma 2.3.2.

######  Remark 3.2.4.

Our original strategy to prove Theorem 3.2.1 is to show $A^{(2)}_{\mathop{\rm
st}\nolimits}[\frac{1}{p}]\cap W({\mathbb C^{\flat}_{p}})=A^{(2)}_{\mathop{\rm
st}\nolimits}$ (resp. $A^{(2)}[\frac{1}{p}]\cap W(\mathcal{O}^{\flat}_{\mathbb
C_{p}})=A^{(2)}$). This is equivalent to that $A^{(2)}/p,A^{(2)}_{\mathop{\rm
st}\nolimits}/p$ injects in $\mathbb C_{p}^{\flat}$. Unfortunately, it does
not work out though we can show
$\widetilde{A}^{(2)}/p,\widetilde{A^{(2)}_{\mathop{\rm st}\nolimits}}/p$
injects in $\mathbb C_{p}^{\flat}$.

###  3.3. Relation to $(\varphi,\hat{G})$-modules

In this subsection, we show that the base ring $\widehat{{\mathcal{R}}}$ for
the theory of $(\varphi,\hat{G})$-modules can be replaced by
$A^{(2)}_{\mathop{\rm st}\nolimits}$. To this end, this builds a new theory of
$(\varphi,\hat{G})$-modules with new base ring $A^{(2)}_{\mathop{\rm
st}\nolimits}$. Since the idea of this new theory is almost the same as that
of the old one, We will use _classical_ to indicate we are using the theory
over $\widehat{\mathcal{R}}$. For example, when we say classical
$(\varphi,\hat{G})$-module, it means a $(\varphi,\hat{G})$-module over
$\widehat{\mathcal{R}}$. Recall
$L=\bigcup\limits_{n=1}^{\infty}K_{\infty}(\zeta_{p^{n}})$,
$\hat{G}:=\mathop{\rm Gal}\nolimits(L/K)$ and $H_{K}:=\mathop{\rm
Gal}\nolimits(L/K_{\infty})$. Let $\mathfrak{m}$ be the maximal ideal of
$\mathcal{O}_{\mathbb C_{p}}^{\flat}$ and set $I_{+}=W(\mathfrak{m})$ so that
$A_{\mathrm{inf}}/I_{+}=W(\bar{k})$. For any subring $B\subset
A_{\mathrm{inf}}$ set $I_{+}B=B\cap I_{+}$. Let $t=\log\epsilon$,
$t^{(i)}=t^{r(i)}\gamma_{\tilde{q}(i)}(\frac{t^{p-1}}{p})$ where
$i=(p-1)\tilde{q}(i)+r(i)$ with $0\leq r(i)<p-1.$ Recall that
$\widehat{\mathcal{R}}:=A_{\mathrm{inf}}\cap{\mathcal{R}}_{K_{0}}$ where

${\mathcal{R}}_{K_{0}}:=\left\\{\sum_{i=0}^{\infty}f_{i}t^{(i)},f_{i}\in
S[\frac{1}{p}],f_{i}\to 0\ p{\text{-adically}}\right\\}.$

######  Lemma 3.3.1.

  1. (1)

As a subring of $A_{\mathrm{inf}}$, $A^{(2)}_{\mathop{\rm st}\nolimits}$ is
stable under $G_{K}$-action and the $G_{K}$-action factors through $\hat{G}$.

  2. (2)

$A^{(2)}_{\mathop{\rm st}\nolimits}/I_{+}A^{(2)}_{\mathop{\rm
st}\nolimits}=W(k)$.

  3. (3)

$I_{+}A^{(2)}\subset uA^{(2)}_{\mathop{\rm st}\nolimits}$.

  4. (4)

$\varphi(A^{(2)}_{\mathop{\rm st}\nolimits})\subset\widehat{\mathcal{R}}$.

###### Proof.

(1) It is clear that the $G_{K}$-action is stable on
$W(k)[\\![u,\epsilon-1]\\!]$. Since $A^{(2)}_{\mathop{\rm st}\nolimits}$ is
$(p,E)$-completion of $W(k)[\\![u,\epsilon-1]\\!][\delta^{i}(w),i\geq 0]$, to
show that $A^{(2)}_{\mathop{\rm st}\nolimits}$ is $G_{K}$-stable, it suffices
to show that $g(w)\in A^{(2)}_{\mathop{\rm st}\nolimits}$ (because $g$ and
$\delta$ commutes, if $g(x)\in A^{(2)}_{\mathop{\rm st}\nolimits}$ then so is
$g(\delta(x))$). Now $Ew=\epsilon-1$, we have
$g(E)g(w)=g(\epsilon-1)=\epsilon^{a(g)}-1$. Then
$g(w)=\frac{E}{g(E)}\frac{\epsilon^{a(g)}-1}{E}$. By [BS22, Lemma 2.24],
$E/g(E)$ is a unit in $A^{(2)}_{\mathop{\rm st}\nolimits}$, then $g(w)\in
A^{(2)}_{\mathop{\rm st}\nolimits}$.

(2) It is clear that both $u,\epsilon-1$ are in $I_{+}$. Hence $w\in I_{+}$
because $Ew=\epsilon-1\in I_{+}$ and $E\equiv p\mod I_{+}$. For any
$x=\sum\limits_{i=0}^{\infty}p^{i}[x_{i}]\in A_{\mathrm{inf}}$, $x\in I_{+}$
if and only of $x_{i}\in\mathfrak{m}$. Then it is easy to check that
$\delta(I_{+})\subset I_{+}$, and consequently all $\delta^{i}(w)\in I_{+}$.
So $I_{+}A^{(2)}_{\mathop{\rm st}\nolimits}$ is topologically generated by
$u,y=\epsilon-1,\delta^{i}(w),i\geq 0$ and hence $A^{(2)}_{\mathop{\rm
st}\nolimits}/I_{+}A^{(2)}_{\mathop{\rm st}\nolimits}=W(k)$ as required.

(3) $I_{+}A^{(2)}$ is topologically generated by $u,v=\epsilon
u,\\{\delta^{i}(z)\\},i\geq 0$. And (3) follows from $z=uw$ and
$\delta^{n}(z)=u^{p^{n}}\delta^{n}(w)$.

(4) Since $A^{(2)}_{\mathop{\rm st}\nolimits}\subset A^{(2)}_{\mathop{\rm
st}\nolimits,\max}$, it suffices to show that $\varphi(A^{(2)}_{\mathop{\rm
st}\nolimits,\max})\subset{\mathcal{R}}_{K_{0}}$. Since
$\varphi({O}_{\mathrm{max}})\subset A[\\![\frac{E^{p}}{p}]\\!]\subset S$, it
suffices to show that $\varphi(\gamma_{n}(w))\in{\mathcal{R}}_{K_{0}}$. Note
that $\varphi(E)=p\nu$ with $\nu\in A[\\![\frac{E^{p}}{p}]\\!]^{\times}$ and
$\gamma_{i}(\epsilon-1)\in{\mathcal{R}}_{K_{0}}$. And we have

$\varphi(w)=\varphi(\frac{(\epsilon-1)}{E})=\nu^{-1}(\epsilon-1)\sum_{i=1}^{p}\left(\binom{p}{i}/p\right)(\epsilon-1)^{i-1}$

which is a polynomial with coefficients in $\mathbb Z$ and in variables
$\nu^{-1}$ and $\gamma_{i}(\epsilon-1)$’s. In particular
$\varphi(\gamma_{n}(w))\in{\mathcal{R}}_{K_{0}}$ by basic properties of
divided powers. ∎

######  Definition 3.3.2.

A (finite free) $(\varphi,\hat{G})$-module of height $h$ is a (finite free)
Kisin module $(\mathfrak{M},\varphi_{\mathfrak{M}})$ of height $h$ together
with an $A^{(2)}_{\mathop{\rm st}\nolimits}$-semi-linear $\hat{G}$-action on
$\widehat{\mathfrak{M}}:=A^{(2)}_{\mathop{\rm
st}\nolimits}\otimes_{A}\mathfrak{M}$ so that

  1. (1)

The actions of $\varphi$ and $\hat{G}$ on $\widehat{\mathfrak{M}}$ commutes;

  2. (2)

$\mathfrak{M}\subset\widehat{\mathfrak{M}}^{H_{K}}$;

  3. (3)

$\hat{G}$-acts on $\widehat{\mathfrak{M}}/I^{+}A^{(2)}_{\mathop{\rm
st}\nolimits}$ trivially.

The category of $(\varphi,\hat{G})$-modules consists of the above objects and
morphism of two $(\varphi,\hat{G})$-modules is morphism of Kisn modules that
commutes with actions of $\hat{G}$. Given a $(\varphi,\hat{G})$-modules
$\widehat{\mathfrak{M}}:=(\mathfrak{M},\varphi,\hat{G})$, we define a $\mathbb
Z_{p}$-representation of $G_{K}$,

$\widehat{T}(\widehat{\mathfrak{M}}):=\mathop{\rm
Hom}\nolimits_{A^{(2)}_{\mathop{\rm
st}\nolimits},\varphi}(A^{(2)}_{\mathop{\rm
st}\nolimits}\otimes_{A}\mathfrak{M},A_{\mathrm{inf}}).$

Since $\varphi(A^{(2)}_{\mathop{\rm
st}\nolimits})\subset\widehat{\mathcal{R}}$, given a
$(\varphi,\hat{G})$-module
$\widehat{\mathfrak{M}}:=(\mathfrak{M},\varphi,\hat{G})$-defined as the above,
$(\mathfrak{M},\varphi)$ together $\hat{G}$-action on
$\widehat{\mathcal{R}}\otimes_{\varphi,A}\mathfrak{M}$ is a _classical_
$(\varphi,\hat{G})$-modules $\widehat{\mathfrak{M}}_{c}$. It is easy to check
that
$\widehat{T}(\widehat{\mathfrak{M}})=\widehat{T}(\widehat{\mathfrak{M}}_{c}):=\mathop{\rm
Hom}\nolimits_{\widehat{\mathcal{R}},\varphi}(\widehat{\mathcal{R}}\otimes_{\varphi,A}\mathfrak{M},A_{\mathrm{inf}}).$

######  Theorem 3.3.3.

The functor $\widehat{T}$ induces an anti-equivalence between the category of
$(\varphi,\hat{G})$-modules of height $h$ and the category of $G_{K}$-stable
$\mathbb Z_{p}$-lattices in semi-stable representations with Hodge-Tate
weights in $[0,\dots,h]$.

###### Proof.

Given an $(\varphi,\hat{G})$-module
$\widehat{\mathfrak{M}}=(\mathfrak{M},\varphi,\hat{G})$,
$\widehat{\mathfrak{M}}_{c}$ is a classical $(\varphi,\hat{G})$-module. So
$\widehat{T}(\widehat{\mathfrak{M}})=\widehat{T}(\widehat{\mathfrak{M}}_{c})$
is a lattice inside semi-stable representation with Hodge-Tate weights in
$[0,\dots,h]$. Conversely, given a lattice in semi-stable representation $T$
with Hodge-Tate weights in $[0,\dots,h]$, following the proof for the
existence of classical $(\varphi,\hat{G})$-module $\widehat{\mathfrak{M}}$ so
that $\widehat{T}(\widehat{\mathfrak{M}})=T$, it suffices to show that for any
$g\in G_{K}$, $g(\mathfrak{M})\subset A^{(2)}_{\mathop{\rm
st}\nolimits}\otimes_{A}\mathfrak{M}$, here $\mathfrak{M}$ and
$A^{(2)}_{\mathop{\rm st}\nolimits}\otimes_{A}\mathfrak{M}$ are regarded as
submodules of $T^{\vee}\otimes_{\mathbb Z_{p}}A_{\mathrm{inf}}$ via
$\iota_{\mathfrak{M}}$ in (5) and uses the $G_{K}$-action on
$T^{\vee}\otimes_{\mathbb Z_{p}}A_{\mathrm{inf}}$. This follows Theorem 3.2.1.
∎

Now let us discuss when $\widehat{T}(\widehat{\mathfrak{M}})$ becomes a
crystalline representation. Recall that $\tau$ is a selected topological
generator of $\mathop{\rm Gal}\nolimits(L/K_{1^{\infty}})$, and we have
$\tau(u)=\epsilon u$ and $\tau,H_{K}$ topologically generate $\hat{G}$.

######  Corollary 3.3.4.

Select $\tau\in\hat{G}$ as the above. Then
$\widehat{T}(\widehat{\mathfrak{M}})$ is crystalline if and only if
$\tau(\mathfrak{M})\subset A^{(2)}\otimes_{A}\mathfrak{M}$.

###### Proof.

Clearly for the selected $\tau$, we have $\tau(u)-u=Ez$. If
$T:=\widehat{T}(\widehat{\mathfrak{M}})$ is crystalline then Theorem 3.2.1
proves that $\tau(\mathfrak{M})\subset A^{(2)}\otimes_{A}\mathfrak{M}$.
Conversely, Suppose $\tau(\mathfrak{M})\subset
A^{(2)}\otimes_{A}\mathfrak{M}$. Then we see that $(\tau-1)\mathfrak{M}\subset
uA_{\mathrm{inf}}\otimes_{A}\mathfrak{M}$ by Lemma 3.3.1 (3). And we have this
is enough to show that $\widehat{T}(\widehat{\mathfrak{M}})$ is crystalline.
For example, We will have
$\mathfrak{M}\otimes_{A}(A_{\mathrm{inf}}[\frac{1}{p}]/\mathfrak{p})$ has a
$G_{K}$-fixed basis given by a basis of $\mathfrak{M}$, where the ideal
$\mathfrak{p}$ is defined as $\mathfrak{p}:=\cup_{n\in\mathbb
N}\varphi^{-n}(u)A_{\mathrm{inf}}[\frac{1}{p}]\subset
A_{\mathrm{inf}}[\frac{1}{p}]$. Then one can prove by the same method in
[Oze18, Thm. 3.8] or directly use [Du21, Theorem 4.2.1] that $T$ is
crystalline. ∎

######  Remark 3.3.5.

Though $A^{(2)}_{\mathop{\rm st}\nolimits}$ is still complicated, for example,
it is not noetherian, $A^{(2)}_{\mathop{\rm st}\nolimits}$ is still better
than old $\widehat{\mathcal{R}}$: at least it has explicit topological
generators. Furthermore, $A^{(2)}_{\mathop{\rm st}\nolimits}$ is $p$-adic
complete. This can help to close the gap in [Liu07] mentioned in [Gao21,
Appendix B]. Indeed, as indicated by Remark B.0.5 _loc.cit._ , if
$\widehat{\mathcal{R}}$ can be shown to be $p$-adic complete then the gap in
[Liu07] can be closed. So by replacing $\widehat{\mathcal{R}}$ by
$A^{(2)}_{\mathop{\rm st}\nolimits}$, we close the gap of [Liu07] ([Gao21]
provides another similar way to close the gap).

##  4\. Crystalline representations and prismatic $F$-crystals

In this section, we reprove the theorem of Bhatt and Scholze on the
equivalence of prismatic $F$-crystal and lattices in crystalline
representations of $G_{K}$ and complete the proof of Theorem 3.2.1. We start
to discuss some general facts on the absolute prismatic site (which allows
general base rings).

###  4.1. Prismatic $F$-crystals in finite projective modules

Let $R=R_{0}\otimes_{W}\mathcal{O}_{K}=R_{0}[u]/E$ as in the beginning of §2
and $X=\mathop{\rm Spf}\nolimits(R)$ with the $p$-adic topology.

######  Definition 4.1.1.

The (absolute) prismatic site $X_{{\mathbbl{\Delta}}}$ of $X$ is the opposite
of the category of bounded prisms $(A,I)$ that are $(p,I)$-completed together
with a map $R\to A/I$, and a morphism of prisms $(A,I)\to(B,J)$ is a covering
map if and only if $A\to B$ is $(p,I)$-completely faithfully flat.

Define the following functors:

$\mathcal{O}_{{\mathbbl{\Delta}}}:(A,I)\mapsto A,$

and for all $h\in\mathbb N$, let

$\mathcal{I}_{{\mathbbl{\Delta}}}^{h}:(A,I)\mapsto I^{h}.$

It is known in [BS22] that these are sheaves on $X_{{\mathbbl{\Delta}}}$. We
will also use
$\mathcal{O}_{{\mathbbl{\Delta}}}[1/\mathcal{I}_{{{\mathbbl{\Delta}}}}]^{\wedge}_{p}$
to denote functor assign $(A,I)$ to the $p$-adic completion of $A$ with $I$
inverted.

Now we verify $A^{(2)}$(resp. $A^{(3)})$ constructed in §2.1 is indeed self
(resp. triple) product of $A$ in $X_{{\mathbbl{\Delta}}}$. We mainly discuss
the situation of $A^{(2)}$ while the proof of $A^{(3)}$ is almost the same.
Recall that $\breve{A}=\breve{R}_{0}[\\![u]\\!]=W\langle
t_{1},\dots,t_{m}\rangle[\\![u]\\!]$.

First we want to make a remark on the existence of nonempty self-coproduct in
the category of prisms over $R$. We thank Peter Scholze for answering our
question on Mathoverflow. And we just repeat his answer here. Let
$(A_{i},I_{i})$ for $i=1,2$ that are prisms over $R$, let
$A_{0}=A_{1}\hat{\otimes}_{\mathbb Z_{p}}A_{2}$ where the completion is taken
for the $(p,I_{1},I_{2})$-adic topology. Let $J$ be the kernel of the map:

$A_{0}\to A_{1}/I_{1}\otimes_{R}A_{2}/I_{2}.$

Let $(A,I)$ be the prismatic envelope of $(A_{1},I_{1})\to(A_{0},J)$, one can
check this is the initial object in the category of prisms over $R$ that
admits maps from $(A_{i},I_{i})$ such that the two $R\to A_{i}/I_{i}\to A/I$
agree. Also we want to note that in general, we don’t know if the boundedness
of $(A_{1},I_{1})$ and $(A_{2},I_{2})$ will imply the boundedness of their
coproduct. But we have seen $A^{(2)}$ and $A^{(3)}$ are indeed bounded by
Corollary 2.2.8.

To start, note that there exists a $W$-linear map $\breve{i}_{2}:\breve{A}\to
A^{\widehat{\otimes}2}$ induced by $u\mapsto y$ and $t_{i}\mapsto s_{i}$. We
claim that $\breve{i}_{2}$ uniquely extends to $i_{2}:A\to
A^{\widehat{\otimes}2}$ which is compatible with $\delta$-structures. Indeed,
consider the following commutative diagram

$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{i}_{2}}$$\scriptstyle{i_{2,n}}$$\textstyle{A^{\widehat{\otimes}2}/(p,J^{(2)})}$$\textstyle{{\breve{A}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\breve{i}_{2,n}}$$\textstyle{A^{\widehat{\otimes}2}/(p,J^{(2)})^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Here $\breve{i}_{2,n}=\breve{i}_{2}\mod(p,J^{(2)})^{n}$ and $\overline{i}_{2}$
is induced by $A\to A/(p,E)\simeq A^{\widehat{\otimes}2}/(p,J^{(2)})$. Since
$\breve{i}_{2}(u)=y=x+(y-x)$ and
$\breve{i}_{2}(t_{i})=s_{i}=t_{i}+(s_{i}-t_{i})$, we see that the above
(outer) diagram commutes. Since $A$ is formally étale over $\breve{A}$ by
$(p,u)$-adic topology, we conclude that there exists a unique map
$i_{2,n}:A\to A^{\widehat{\otimes}2}/(p,J^{(2)})^{n}$ so that the above
diagram commutes. Since $A^{\widehat{\otimes}2}$ is $(p,J^{(2)})$-complete,
there uniquely exists $i_{2}:A\to A^{\widehat{\otimes}2}$ which extends
$\breve{i}_{2}$. To see $i_{2}$ is compatible with $\delta$-structures. it
suffices to show that $\varphi\circ i_{2}=i_{2}\circ\varphi$. But both of
$\varphi\circ i_{2}$ and $i_{2}\circ\varphi$ extend
$\breve{A}\overset{\varphi}{\to}\breve{A}\to A^{\widehat{\otimes}2}$. Again by
formally étaleness of $A$ over $\breve{A}$, we see that $\varphi\circ
i_{2}=i_{2}\circ\varphi$. Hence we obtain a map $1\otimes
i_{2}:A\otimes_{\mathbb Z_{p}}A\to A^{\widehat{\otimes}2}$. Define
$\theta^{\otimes 2}:A\otimes_{\mathbb Z_{p}}A\to R$ via $\theta^{\otimes
2}(a\otimes b)=\theta(a)\theta(b)$. By the construction of $i_{2}$, we have
the following commutative diagram

$\textstyle{A\otimes_{\mathbb
Z_{p}}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\otimes
i_{2}}$$\scriptstyle{\theta^{\otimes
2}}$$\textstyle{A^{\widehat{\otimes}2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{A^{\widehat{\otimes}2}/J^{(2)}}$

Let $\widehat{A^{\otimes 2}}$ be the $(p,\mathop{\rm
ker}\nolimits(\theta^{\otimes 2}))$-completion of $A^{\otimes
2}:=A\otimes_{\mathbb Z_{p}}A$. Hence $1\otimes i_{2}$ induces a map
$\hat{i}_{2}$ from the $\widehat{A^{\otimes 2}}$ to $A^{\widehat{\otimes}2}$
because $A^{\widehat{\otimes}2}$ is clearly $(p,J^{(2)})$-complete. To treat
$A^{\widehat{\otimes}3}$, we construct $i_{3}:A\to A^{\widehat{\otimes}3}$ by
extending $\breve{i}_{3}:A\to A^{\widehat{\otimes}3}$ by sending $u\mapsto w$
and $t_{j}\mapsto r_{j}$. The same method shows that $i_{3}$ is compatible
with $\delta$-structure and we obtain a map $1\otimes i_{2}\otimes
i_{3}:A^{\otimes 3}\to A^{\widehat{\otimes}3}$ with $A^{\otimes
3}:A\otimes_{\mathbb Z_{p}}A\otimes_{\mathbb Z_{p}}A$. Similarly, we obtain a
natural map $\hat{i}_{3}:\widehat{A^{\otimes 3}}\to A^{\widehat{\otimes}3}$.

######  Lemma 4.1.2.

For $s=2,3$, $\hat{i}_{s}:\widehat{A^{\otimes s}}\to A^{\widehat{\otimes}s}$
are isomorphisms.

###### Proof.

We need to construct an inverse of $\hat{i}_{s}$. We only show for
$\hat{i}_{2}$ and the proof for $\hat{i}_{3}$ is the same. Let
$g:A^{\widehat{\otimes}2}\to\widehat{A^{\otimes 2}}$ be the $A$-linear map by
sending $y-x\mapsto 1\otimes u-u\otimes 1$ and $s_{j}-t_{j}\mapsto 1\otimes
t_{j}-t_{j}\otimes 1$. Clearly $g$ is well-defined because $1\otimes
u-u\otimes 1$ and $1\otimes t_{j}-t_{j}\otimes 1$ are in $\mathop{\rm
Ker}\nolimits(\theta^{\otimes 2})$. Since $i_{2}(u)=y$ and
$i_{2}(t_{j})=s_{j}$, $\hat{i}_{2}\circ g$ is identity on
$A^{\widehat{\otimes}2}$. Now it suffices to show that $h:=g\circ\hat{i}_{2}$
is identity. Write $K=(p,\mathop{\rm Ker}\nolimits(\theta^{\otimes 2}))$. Note
that we have map $A\otimes_{\mathbb Z_{p}}\breve{A}\to\widehat{A^{\otimes
2}}\overset{h}{\to}\widehat{A^{\otimes 2}}$ induced by $h$ which we still call
it $\breve{h}$. Now we have the following commutative diagram

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
18.41666pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{A\otimes_{\mathbb
Z_{p}}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
33.82573pt\raise 5.43056pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{\mod K}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 75.09732pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
44.64268pt\raise-14.67471pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.82222pt\hbox{$\scriptstyle{\mod K^{n}}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
44.7555pt\raise-26.35803pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.0pt\hbox{$\scriptstyle{h_{n}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
78.00015pt\raise-29.955pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
75.09732pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{(A\otimes_{\mathbb
Z_{p}}A)/K}$}}}}}}}{\hbox{\kern-17.16666pt\raise-40.99387pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{A\otimes_{\mathbb
Z_{p}}{\breve{A}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-6.22444pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
28.65993pt\raise-34.52164pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-3.47223pt\hbox{$\scriptstyle{\breve{h}\mod K^{n}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
73.41666pt\raise-40.99387pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
73.41666pt\raise-40.99387pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{(A\otimes_{\mathbb
Z_{p}}A)/K^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
106.72922pt\raise-6.22444pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces,$

where $h_{n}$ is induced by $h\mod K^{n}$. We see that both $h_{n}$ and $\mod
K^{n}$ on the dashed arrow can make the diagram commute. Then by the formal
étaleness of $A$ over $\breve{A}$, we conclude that $h_{n}=\mod K^{n}$ and $h$
is the identity map. ∎

######  Proposition 4.1.3.

$A^{(2)}$ and $A^{(3)}$ is self-product and triple product of $A$ in
$X_{{\mathbbl{\Delta}}}$.

###### Proof.

In the following, we only treat the case of $A^{(2)}$ while the proof for
$A^{(3)}$ is the same. We need to prove that for any $B=(B,J)$ in
$X_{{\mathbbl{\Delta}}}$,

$\mathop{\rm
Hom}\nolimits_{X_{{\mathbbl{\Delta}}}^{\mathrm{opp}}}(A^{(2)},B)=\mathop{\rm
Hom}\nolimits_{X_{{\mathbbl{\Delta}}}^{\mathrm{opp}}}(A,B)\times\mathop{\rm
Hom}\nolimits_{X_{{\mathbbl{\Delta}}}^{\mathrm{opp}}}(A,B).$

By the above lemma, we have natural maps $A\otimes_{\mathbb
Z_{p}}A\to\widehat{A^{\otimes 2}}\simeq A^{\widehat{\otimes}2}$. Combined with
natural map $A^{\widehat{\otimes}2}\to A^{(2)}$ as $A^{(2)}$ is the prismatic
envelope of $A^{\widehat{\otimes}2}$ for the ideal $J^{(2)}$, we have map
$\alpha:A\otimes_{\mathbb Z_{p}}A\to A^{(2)}$ which is compatible with
$\delta$-structures. Then $\alpha$ induces map

$\beta:\mathop{\rm
Hom}\nolimits_{X_{{\mathbbl{\Delta}}}^{\mathrm{opp}}}(A^{(2)},B)\to\mathop{\rm
Hom}\nolimits_{X_{{\mathbbl{\Delta}}}^{\mathrm{opp}}}(A,B)\times\mathop{\rm
Hom}\nolimits_{X_{{\mathbbl{\Delta}}}^{\mathrm{opp}}}(A,B).$

To prove the surjectivity of $\beta$, given $f_{i}\in\mathop{\rm
Hom}\nolimits_{X_{{\mathbbl{\Delta}}}}(A,B)$ for $i=1,2$, we obtain a map
$f_{1}\otimes f_{2}:A\otimes_{\mathbb Z_{p}}A\to B$. It is clear that
$(f_{1}\otimes f_{2})(\mathop{\rm Ker}\nolimits(\theta^{\otimes 2}))\subset
J$. Since $B$ is $(p,J)$-derived complete, $f\otimes f_{2}$ extends to a map
$f_{1}\widehat{\otimes}f_{2}:\widehat{A^{\otimes 2}}\simeq
A^{\widehat{\otimes}2}\to B$ which is compatible with $\delta$-structures,
Hence $f_{1}\widehat{\otimes}f_{2}$ is a morphism of $\delta$-algebra.
Finally, by the universal properties of prismatic envelope,
$f_{1}\widehat{\otimes}f_{2}$ extends to a map of prisms
$f_{1}\widehat{\otimes}_{{{\mathbbl{\Delta}}}}f_{2}:A^{(2)}\to B$ as required.

Finally, we need to show that $\beta$ is injective. It suffices to show that
$A$-algebra structure map $i_{1}:A\to A^{(2)}$ and
$i^{\prime}_{2}:A\overset{i_{2}}{\to}A^{\widehat{\otimes}2}\to A^{(2)}$ both
are injective. Since all rings here are $(p,E)$-complete integral domains, it
suffices to check that $i_{1},i_{2}^{\prime}\mod(p,E)$ are injective. By
Proposition 2.2.7, we see that $i_{1}\mod(p,E)$ is $R/pR\to
R/pR[\\{\gamma_{i}(z_{j})\\}]$, so it is injective. By the construction
$i^{\prime}_{2}$ and $i_{2}$, we see that $i^{\prime}_{2}\mod(p,E)$ is the
same as $A/(p,E)\to A^{\widehat{\otimes}2}/(p,J^{(2)})\to A^{(2)}/(p,E)$,
which is same as $R/pR\to R/pR[\\{\gamma_{i}(z_{j})\\}]$. So it is injective.
∎

######  Remark 4.1.4.

When $R=\mathcal{O}_{K}$ is a complete DVR with perfect residue field $k$, we
know a priori, the self-product $A^{(2)}$ of $(A,(E))$ in
$X_{{\mathbbl{\Delta}}}$ can be constructed as the prismatic envelope of
$(A,(E))\to(B,I)$, where $B$ is the $(p,E(u),E(v))$-adic completion of
$W(k)[\\![u]\\!]\otimes_{\mathbb Z_{p}}W(k)[\\![v]\\!]$ and $I$ is the kernel
of the map:

$B\to A/(E)\otimes_{R}A/(E)=R.$

On the other hand, $W(k)$ is formally étale over $\mathbb Z_{p}$ for the
$p$-adic topology, so for all $(C,J)\in X_{{\mathbbl{\Delta}}}$, the map
$W(k)\to R\to C/J$ lifts uniquely to a map $W(k)\to C$. In particular, for all
$(C,J)\in X_{{\mathbbl{\Delta}}}$, $C$ has a natural $W(k)$-algebra structure.
So when we construct the self-product, we can also consider $A^{(2)}$ as the
prismatic envelope of $(A,(E))\to(C,J)$, where $C$ is the $(p,E(u),E(v))$-adic
completion of $A\otimes_{W(k)}A$ and $J$ is the kernel of the map:

$C\to A/(E)\otimes_{R}A/(E)=R.$

We have $C\simeq W(k)[\\![u,v]\\!]$, $J=(E(u),u-v)$ and
$A^{(2)}=W(k)[\\![u,v]\\!]\\{\frac{u-v}{E}\\}^{\wedge}_{\delta}$.

######  Definition 4.1.5.

  1. (1)

A prismatic crystal over $X_{{\mathbbl{\Delta}}}$ in finite locally free
$\mathcal{O}_{{\mathbbl{\Delta}}}$-modules (resp.
$\mathcal{O}_{{\mathbbl{\Delta}}}[1/I]^{\wedge}_{p}$-modules) is a finite
locally free $\mathcal{O}_{{\mathbbl{\Delta}}}$-module (resp.
$\mathcal{O}_{{\mathbbl{\Delta}}}[1/I]^{\wedge}_{p}$-module)
$\mathfrak{M}_{{{\mathbbl{\Delta}}}}$ such that for all morphisms
$f:(A,I)\to(B,J)$ of prisms, it induces an isomorphism:

$f^{\ast}\mathfrak{M}_{{{\mathbbl{\Delta}}},A}:=\mathfrak{M}_{{{\mathbbl{\Delta}}}}((A,I))\otimes_{A}B\simeq\mathfrak{M}_{{{\mathbbl{\Delta}}},B}:=\mathfrak{M}_{{{\mathbbl{\Delta}}}}((B,J))$
$(resp.\quad
f^{\ast}\mathfrak{M}_{{{\mathbbl{\Delta}}},A}:=\mathfrak{M}_{{{\mathbbl{\Delta}}}}((A,I))\otimes_{A[1/I]^{\wedge}_{p}}B[1/I]^{\wedge}_{p}\simeq\mathfrak{M}_{{{\mathbbl{\Delta}}},B}:=\mathfrak{M}_{{{\mathbbl{\Delta}}}}((B,J))).$

  2. (2)

A prismatic $F$-crystal over $X_{{\mathbbl{\Delta}}}$ of height $h$ in finite
locally free $\mathcal{O}_{{\mathbbl{\Delta}}}$-modules is a prismatic crystal
$\mathfrak{M}_{{{\mathbbl{\Delta}}}}$ in finite locally free
$\mathcal{O}_{{\mathbbl{\Delta}}}$-modules together with a
$\varphi_{\mathcal{O}_{{\mathbbl{\Delta}}}}$-semilinear endomorphism
$\varphi_{\mathfrak{M}_{{{\mathbbl{\Delta}}}}}$ of the
$\mathcal{O}_{{\mathbbl{\Delta}}}$-module
$\mathfrak{M}_{{{\mathbbl{\Delta}}}}:\mathfrak{M}_{{{\mathbbl{\Delta}}}}\to\mathfrak{M}_{{{\mathbbl{\Delta}}}}$
such that the cokernel of the linearization
$\varphi^{\ast}\mathfrak{M}_{{{\mathbbl{\Delta}}}}\to\mathfrak{M}_{{{\mathbbl{\Delta}}}}$
is killed by $\mathcal{I}^{h}$.

######  Proposition 4.1.6.

If the sheaf represented by $(B,I)$ in $\mathop{\rm
Shv}\nolimits(X_{{\mathbbl{\Delta}}})$ covers the final object $\ast$ in
$\mathop{\rm Shv}\nolimits(X_{{\mathbbl{\Delta}}})$, i.e., for any $(C,J)$ in
$X_{{\mathbbl{\Delta}}}$, there is a $(P,J)$ lies over $(B,I)$ and covers
$(C,J)$. Also assume that the self-coproduct $B^{(2)}$ and self-triple-
coproduct $B^{(3)}$ of $(B,I)$ are inside $X_{{{\mathbbl{\Delta}}}}$, i.e.,
they are bounded. Then a prismatic crystal
$\mathfrak{M}_{{{\mathbbl{\Delta}}}}$ over $X$ in finite locally free
$\mathcal{O}_{{\mathbbl{\Delta}}}$-modules is the same as a finite projective
module $\mathfrak{M}$ over $B$ together with a descent data
$\psi:\mathfrak{M}\otimes_{i_{1},B}B^{(2)}\simeq\mathfrak{M}\otimes_{i_{2},B}B^{(2)}$
satisfies the cocycle condition. Here $i_{j}:B\to B^{(2)}$ $(j=1,2)$ are the
two natural maps.

###### Proof.

First let $\mathfrak{M}$ be a prismatic crystal in finite projective modules.
Define $\mathfrak{M}=\mathfrak{M}_{{{\mathbbl{\Delta}}}}((B,I))$, and the
descent data comes from the crystal property:

$\psi:\mathfrak{M}\otimes_{i_{1},B}B^{(2)}\simeq\mathfrak{M}_{{{\mathbbl{\Delta}}}}((B^{(2)},I))\simeq\mathfrak{M}\otimes_{i_{2},B}B^{(2)}.$

Now given $(\mathfrak{M},\psi)$, then for any $(C,J)$ in
$X_{{\mathbbl{\Delta}}}$, we need to construct a finite projective module over
$C$. We choose the $(P,J)$ as in the assumption, let
$\mathfrak{M}_{P}=\mathfrak{M}\otimes_{B}P$, and consider the following
diagram:

${C}$${P}$${P^{(2)}_{C}}$${B}$${B^{(2)}}$${B}$${P}$${C}$$\scriptstyle{f_{1}}$$\scriptstyle{i_{1}}$$\scriptstyle{f}$$\scriptstyle{i_{2}}$$\scriptstyle{f_{2}}$

Here $(P^{(2)}_{C},J)$ is the self-coproduct of $(P,J)$ in the category of
prisms over $(C,J)$, and the existence of $(P^{(2)}_{C},J)$ is from [BS22,
Corollary 3.12], where they also show that $P^{(2)}_{C}$ is the derived
$(p,J)$-completion of $P\otimes L_{C}P$ and $(P^{(2)}_{C},J)$ is bounded. As a
bounded prism over $(C,J)$, $(P^{(2)}_{C},J)$ is naturally inside
$X_{{\mathbbl{\Delta}}}$, so $f$ exists by the universal property of
$B^{(2)}$. So if we take the base change of $\psi$ along $f$, we get

$f^{\ast}\psi:(\mathfrak{M}\otimes_{i_{1},B}B^{(2)})\otimes_{B^{(2)},f}P^{(2)}_{C}\simeq(\mathfrak{M}\otimes_{i_{2},B}B^{(2)})\otimes_{B^{(2)},f}P^{(2)}_{C}$

which is the same as an isomorphism:

$\psi_{C}:\mathfrak{M}_{P}\otimes_{P,f_{1}}P^{(2)}_{C}\simeq\mathfrak{M}_{P}\otimes_{P,f_{2}}P^{(2)}_{C}.$

Similar arguments will show $\psi_{C}$ satisfies the cocycle condition. And
$\mathfrak{M}_{P}$ descents to a finite projective module over $C$ by [AB21,
Proposition A.12]. ∎

######  Remark 4.1.7.

We want to note that the structures of finite nonempty coproducts in the
category of bounded prisms over a prism $(A,I)$ is much simpler compared with
the structure of finite nonempty products in the category
$(R/A)_{{{\mathbbl{\Delta}}}}$ (cf. [Bha18, Lecture V, Corollary 5.2]).

######  Lemma 4.1.8.

The prism $(A,(E))$ defined in §2.1 covers the final object $\ast$ in
$\mathop{\rm Shv}\nolimits(X_{{\mathbbl{\Delta}}})$ in the sense of
Proposition 4.1.6. And $A^{(2)}$ and $A^{(3)}$ are bounded.

###### Proof.

The proof is similar to [AB21, Lemma 5.2.8], we need to show for $R$ defined
as in §2.1, there exists a quasi-syntomic perfectoid cover of $R$. We will
construct this perfectoid cover similar to [Kim14, §7.1].

First recall we have $R=\mathcal{O}_{K}\otimes_{W}R_{0}$, and we fix a
compatible system $\\{\varpi_{n}\\}_{n\geq 0}$ of $p^{n}$-th roots of a
uniformizer $\varpi_{0}$ of $\mathcal{O}_{K}$ inside $E$. Let
$\widehat{K}_{\infty}$ be the $p$-adic completion of $\cup_{n}K(\varpi_{n})$,
we know $\widehat{K}_{\infty}$ is perfectoid. Use
$\overline{R}_{0}[\\![u]\\!]$ to denote
$A/(p)=R/(\varpi)=R_{0}/(p)[\\![u]\\!]$, and let
$\overline{R}_{0}[\\![u]\\!]_{\rm perf}^{\wedge}$ to be the $u$-adic
completion of the direct perfection of $\overline{R}_{0}[\\![u]\\!]$, it can
be checked directly that $(\overline{R}_{0}[\\![u]\\!]_{\rm
perf}^{\wedge}[1/u],\overline{R}_{0}[\\![u]\\!]_{\rm perf}^{\wedge})$ is a
perfectoid affinoid $\widehat{K}_{\infty}^{\flat}$-algebra, by tilt
equivalence, there is a corresponded perfectoid affinoid
$\widehat{K}_{\infty}$-algebra. More explicitly, let
$\tilde{R}_{\infty}=W(\overline{R}_{0}[\\![u]\\!]_{\rm
perf}^{\wedge})\otimes_{W(\mathcal{O}_{\widehat{K}_{\infty}}^{\flat}),\theta}\mathcal{O}_{\widehat{K}_{\infty}}$.
Then $\tilde{R}_{\infty}$ is naturally an $R$-algebra, and we claim it is a
quasi-syntomic cover of $R$.

To show this, by [Kim14, §7.1.2], we have

$\tilde{R}_{\infty}=(R_{0}\widehat{\otimes}_{W}\mathcal{O}_{\widehat{K}_{\infty}})\widehat{\otimes}_{\mathbb
Z_{p}}\mathbb Z_{p}\langle T_{i}^{-p^{\infty}}\rangle$

where $T_{i}\in R_{0}$ is any lift of a $p$-basis of $R_{0}/(p)$. We have
$\mathcal{O}_{K}\to\mathcal{O}_{\widehat{K}_{\infty}}$ is a quasi-syntomic
cover so by (2) of [BMS19, Lemma 4.16], $R\to
R_{0}\widehat{\otimes}_{W}\mathcal{O}_{\widehat{K}_{\infty}}$ is also a quasi-
syntomic cover. And we have $S=\mathbb Z_{p}\langle
T_{i}^{-p^{\infty}}\rangle$ is a quasi-syntomic ring, this can be seen by
constructing a perfectoid quasi-syntomic covering of it, so by Lemma 4.34 of
$loc.cit.$, we have the complex $\mathbb{L}_{S/\mathbb Z_{p}}\in D(S)$ has
$p$-complete Tor amplitude in $[-1,0]$. In particular, $\mathbb
Z_{p}\to\mathbb Z_{p}\langle T_{i}^{-p^{\infty}}\rangle$ is also a quasi-
syntomic cover, so applying (1) in Lemma 4.16 of $loc.cit.$,
$R\to\tilde{R}_{\infty}$ is a quasi-syntomic perfectoid cover.

The boundedness of $A^{(2)}$ and $A^{(3)}$ is from (2) in Corollary 2.2.8. ∎

######  Corollary 4.1.9.

Assume the the base $X=\mathop{\rm Spf}\nolimits(R)$ satisfies the condition
in §2, and let $A$, $A^{(2)}$ and $A^{(3)}$ be defined as in §2.1, then a
prismatic $F$-crystal
$(\mathfrak{M}_{{{\mathbbl{\Delta}}}},\varphi_{\mathfrak{M}_{{{\mathbbl{\Delta}}}}})$
in finite locally free $\mathcal{O}_{{\mathbbl{\Delta}}}$-modules of height
$h$ over $X$ is the same as a Kisin module
$(\mathfrak{M},\varphi_{\mathfrak{M}})$ of height $h$ over $A$ with a descent
datum

$f:\mathfrak{M}\otimes_{A,i_{1}}A^{(2)}\simeq\mathfrak{M}\otimes_{A,i_{2}}A^{(2)}$

that compatible with the $\varphi$-structure and satisfies the cocycle
condition over $A^{(3)}$.

######  Theorem 4.1.10.

([BS21, Theorem 1.2]) Let $T$ be a crystalline representation of $G_{K}$ over
a $\mathbb Z_{p}$-lattice of Hodge-Tate weights in $[0,h]$, then there is a
prismatic $F$-crystal $\mathfrak{M}_{{{\mathbbl{\Delta}}}}(T)$ over
$X_{{\mathbbl{\Delta}}}$ of height $h$ over $X$ such that
$\mathfrak{M}_{{{\mathbbl{\Delta}}}}((A,E))$ is the Kisin module associated to
$T$. Moreover, the association of
$T\mapsto\mathfrak{M}_{{{\mathbbl{\Delta}}}}(T)$ induces an equivalence of the
above two categories.

We will prove this theorem in §4.3.

######  Remark 4.1.11.

Theorem 4.1.10 was first established by Bhatt-Scholze in [BS21, Theorem 1.2].
The harder direction of [BS21, Theorem 1.2] is to show for all $\mathbb
Z_{p}$-lattices inside crystalline representations of $G_{K}$, one can attach
a prismatic $F$-crystal. Using the theory of $(\varphi,\hat{G})$-modules, we
have shown in §3.2, given a crystalline representation of $G_{K}$ over a
$\mathbb Z_{p}$-lattices $T$, we can attach a Kisin module $\mathfrak{M}$ and
a descent data222Strictly speaking, §3.2 only constructs an isomorphism but
have not checked that it satisfies cocycle condition, which will be proved in
§4.3.

$f_{\tilde{\tau}}:\mathfrak{M}\otimes_{A,i_{1}}A^{(2)}[\frac{1}{p}]\simeq\mathfrak{M}\otimes_{A,i_{2}}A^{(2)}[\frac{1}{p}]$

comes from the $\tau$-action. We just show this is a $\varphi$-equivariant
isomorphism, and we need to show it gives rise to a descent data over
$A^{(2)}$. As we have mentioned in Remark 3.2.4, we can not find a direct ring
theoretic proof of this. Our idea is to use result of [Wu21] or [BS21,
Corollary 3.7]: the underlying Galois representation $T$ gives a descent data
over $A^{(2)}[\frac{1}{E}]^{\wedge}_{p}$. To finish our proof, we need to
compare this descent data with $f_{\tilde{\tau}}$ over
$A^{(2)}[\frac{1}{E}]^{\wedge}_{p}[\frac{1}{p}]$. This lead us to develop a
“prismatic” $(\varphi,\tau)$-module theory in the next subsection, where we
will have Lemma 4.2.12 and Lemma 4.2.16 to help us compare descent data over
$A^{(2)}[\frac{1}{E}]^{\wedge}_{p}$ and
$A^{(2)}[\frac{1}{E}]^{\wedge}_{p}[\frac{1}{p}]$ via an evaluation map to
$W(\mathcal{O}_{\hat{L}}^{\flat})$.

###  4.2. $(\varphi,\tau)$-modules and prismatic $F$-crystals

In this subsection, we make some preparations to prove Proposition 3.2.2 and
Theorem 4.1.10. So we restrict to the case that $R=\mathcal{O}_{K}$ is a
complete DVR with perfect residue field.

######  Definition 4.2.1.

An étale $\varphi$-module over $A[1/E]^{\wedge}_{p}$ is a pair
$(\mathcal{M},\varphi_{\mathcal{M}})$ such that $\mathcal{M}$ is a finite free
module over $A[1/E]^{\wedge}_{p}$, and $\varphi_{\mathcal{M}}$ is an
isomorphism

$\varphi_{\mathcal{M}}:\varphi^{\ast}\mathcal{M}:=A[1/E]^{\wedge}_{p}\otimes_{\varphi,A[1/E]^{\wedge}_{p}}\mathcal{M}\simeq\mathcal{M}$

of $A[1/E]^{\wedge}_{p}$-modules. And we define an étale $\varphi$-module over
$A[1/E]^{\wedge}_{p}[1/p]$ to be a $\varphi$-module over
$A[1/E]^{\wedge}_{p}[1/p]$ such that it is obtained from an étale
$\varphi$-module over $A[1/E]^{\wedge}_{p}$ by base change.

An étale $\varphi$-module over $A[1/E]^{\wedge}_{p}$ (resp.
$A[1/E]^{\wedge}_{p}[1/p]$) with descent data is a triple
$(\mathcal{M},\varphi_{\mathcal{M}},c)$, such that
$(\mathcal{M},\varphi_{\mathcal{M}})$ is an étale $\varphi$-module over
$A[1/E]^{\wedge}_{p}$ (resp. $A[1/E]^{\wedge}_{p}[1/p]$), and $c$ is an
isomorphism

$c:\mathcal{M}\otimes_{A[1/E]^{\wedge}_{p},i_{1}}B^{(2)}\simeq\mathcal{M}\otimes_{A[1/E]^{\wedge}_{p},i_{2}}B^{(2)}$
$(\text{resp.
}c:\mathcal{M}\otimes_{A[1/E]^{\wedge}_{p}[1/p],i_{1}}B^{(2)}[1/p]\simeq\mathcal{M}\otimes_{A[1/E]^{\wedge}_{p}[1/p],i_{2}}B^{(2)}[1/p])$

that compatible with the $\varphi$-structure and satisfies the cocycle
condition over $B^{(3)}$ (resp. $B^{(3)}[\frac{1}{p}]$). Here for $j=1,2$,
$i_{j}:A[1/E]^{\wedge}_{p}\to B^{(2)}$ is the map induced from
$i_{j}:(A,(E))\to(A^{(2)},(E))$.

######  Remark 4.2.2.

It is the main result in [Wu21] and [BS21, §2] that there is an equivalence of
the category of lattices in representations of $G_{K}$ and the category of
prismatic $F$-crystals in finite locally free
$\mathcal{O}_{{\mathbbl{\Delta}}}[1/I]^{\wedge}_{p}$-modules over
$\mathcal{O}_{K}$. Also by [BS21, Proposition 2.7], one can show prismatic
$F$-crystals in finite locally free
$\mathcal{O}_{{\mathbbl{\Delta}}}[1/I]^{\wedge}_{p}$-modules is the same as
étale $\varphi$-modules over $A[1/E]^{\wedge}_{p}$ with descent data.

The aim of this subsection is to use the ideas in [Wu21] and [KL19, §5.5] show
that étale $\varphi$-modules over $A[1/E]^{\wedge}_{p}$ (resp.
$A[1/E]^{\wedge}_{p}[1/p]$) with descent data are equivalence to
$\mathrm{Rep}_{\mathbb Z_{p}}(G_{K})$ (resp. $\mathrm{Rep}_{\mathbb
Q_{p}}(G_{K})$). More importantly, for all $\gamma\in\hat{G}$, we will
construct an evaluation at $\gamma$ map

$e_{\gamma}:B^{(2)}\to W(\hat{L}^{\flat})$

and use it to study $\varphi$-equivariant morphisms between finite free
$B^{(2)}$ and $B^{(2)}[1/p]$-modules. We will see the evaluation at $\tau$ map
will play a crucial role in our proof of Proposition 3.2.2 and the Theorem
4.1.10 below.

Recall in §3.3, we define
$L=\bigcup\limits_{n=1}^{\infty}K_{\infty}(\zeta_{p^{n}})$,
$\hat{G}:=\mathop{\rm Gal}\nolimits(L/K)$ and $H_{K}:=\mathop{\rm
Gal}\nolimits(L/K_{\infty})$. Moreover, we define $\widehat{K}_{1^{\infty}}$
to be the $p$-adic completion of $\cup_{n\geq 0}K(\zeta_{p^{n}})$, and we let
$\hat{L}$ to be the $p$-adic completion of $L$. It is clear that
$A[1/E]_{p}^{\wedge}\subset W(\hat{L}^{\flat})^{H_{K}}$. Recall the following
definition and theorem in [Car13]:

######  Theorem 4.2.3.

An étale $(\varphi,\tau)$-module is a triple
$(\mathcal{M},\varphi_{\mathcal{M}},\hat{G})$ where

  * •

$(\mathcal{M},\varphi_{\mathcal{M}})$ is an étale $\varphi$-module over
$A[1/E]^{\wedge}_{p}$;

  * •

$\hat{G}$ is a continuous $W(\hat{L}^{\flat})$-semi-linear $\hat{G}$-action on
$\hat{\mathcal{M}}:=W(\hat{L}^{\flat})\otimes_{A[1/E]^{\wedge}_{p}}\mathcal{M}$,
and $\hat{G}$ commutes with $\varphi_{\mathcal{M}}$;

  * •

regarding $\mathcal{M}$ as an $A[1/E]^{\wedge}_{p}$-submodule of
$\hat{\mathcal{M}}$, we have $\mathcal{M}\subset\hat{\mathcal{M}}^{H_{K}}$.

Then there is an anti-equivalence of the category of étale
$(\varphi,\tau)$-modules and $\mathrm{Rep}_{\mathbb Z_{p}}(G_{K})$, such that
if $T$ corresponds to $(\mathcal{M},\varphi_{\mathcal{M}},\hat{G})$, then

$T^{\vee}=(\hat{\mathcal{M}}\otimes_{W(\hat{L}^{\flat})}W(\mathbb
C_{p}^{\flat}))^{\varphi=1}.$

One of the basic facts used in the theory of étale $(\varphi,\tau)$-modules
developed in [Car13] is that $\mathop{\rm
Gal}\nolimits(\hat{L}/\widehat{K}_{1^{\infty}})\simeq\mathbb Z_{p}$, and we
write $\tau$ to be a topological generator of $\mathop{\rm
Gal}\nolimits(\hat{L}/K_{1^{\infty}})$ determined by
$\tau(\varpi_{n})=\zeta_{p^{n}}\varpi_{n}$ as the discussion before Corollary
3.3.4. Also $\hat{G}$ is topologically generated by $\tau$ and $H_{K}$, so in
particular, the $\hat{G}$-action on $\hat{\mathcal{M}}$ is determined by the
action of $\tau$ on $\mathcal{M}$ inside $\hat{\mathcal{M}}$. As discussed
before, we will provides a direct correspondence of the category of étale
$(\varphi,\tau)$-modules and the category of étale $\varphi$-modules over
$A[1/E]^{\wedge}_{p}$ with descent data. Moreover, we will construct an
evaluation at $\tau$ map:

$e_{\tau}:B^{(2)}\to W(\hat{L}^{\flat}),$

and show that the $\tau$-action on $\mathcal{M}$ inside $\hat{\mathcal{M}}$ is
given by the base change of the descent data along $e_{\tau}$.

######  Remark 4.2.4.

In [Wu21, Theorem 5.2], they prove a similar equivalence but for étale
$(\varphi,\Gamma)$-modules. The theory of étale $(\varphi,\Gamma)$-module is
defined for the cyclotomic tower $K_{1^{\infty}}$ over $K$ while the theory of
étale $(\varphi,\tau)$-modules is defined using the Kummer tower $K_{\infty}$.
We will use a lot of ideas and results developed in [Wu21] when proving our
claims in this subsection. The main difficulty in our situation is that the
Kummer tower $K_{\infty}$ is not a Galois tower over $K$. To deal with this,
we have to use the idea in [KL19, §5.5]. Roughly speaking, we will take the
Galois closure $L$ of $K_{\infty}$, then prove results over $\hat{L}$, then
descent back to $K_{\infty}$ using the fact
$\widehat{K}_{\infty}=\hat{L}^{H_{K}}$.

One should be able to construct the evaluation map in the content of [Wu21]
the same way as we define in this subsection. This map will give a more direct
correspondence of the descent data and the $\Gamma$-actions on étale
$(\varphi,\Gamma)$-modules.

By [BS22, Lem 3.9], any prism $(B,J)$ admits a map into its perfection
$(B_{\mathop{\rm perf}\nolimits},JB_{\mathop{\rm perf}\nolimits})$. The
following theorem ([BS22, Thm 3.10]) is the key to understand perfect prisms.

######  Theorem 4.2.5.

$(A,I)\to A/I$ induces an equivalence of the category of perfect prisms over
$\mathcal{O}_{K}$ with the category of integral perfectoid rings over
$\mathcal{O}_{K}$.

Let $(A,(E))$ be the Breuil-Kisin prism defined in §2.1, we have

######  Lemma 4.2.6.

$A_{\mathop{\rm perf}\nolimits}\simeq
W(\mathcal{O}_{\widehat{K}_{\infty}}^{\flat})$.

###### Proof.

Exactly the same as the proof of [Wu21, Lemma 2.17] ∎

######  Lemma 4.2.7.

Let $\mathop{\rm Perfd}\nolimits_{K}$ be the category of perfectoid
$K$-algebras, then $\mathop{\rm Perfd}\nolimits_{K}$ admits finite non-empty
coproducts.

###### Proof.

Let $R$ and $S$ be two perfectoid $K$-algebras, it follows from [KL15,
Corollary 3.6.18] that the uniform completion $(R\otimes_{K}S)^{u}$ of the
tensor product $(R\otimes_{K}S)$ is again a perfectoid $K$-algebra, and it is
easy to show this is the coproduct of $R$ and $S$ in the category of
perfectoid $K$-algebras. ∎

For $i\in\mathbb N_{>0}$, let $(A^{(i)},(E))$ (resp.
$(A_{\mathrm{inf}}(\mathcal{O}_{\hat{L}})^{(i)},(E))$) denote the $i$-th self-
coproduct of $(A,(E))$ (resp. $(A_{\mathrm{inf}}(\mathcal{O}_{\hat{L}}),(E))$)
in the category of prisms over $\mathcal{O}_{K}$, where
$A_{\mathrm{inf}}(\mathcal{O}_{\hat{L}}):=W(\mathcal{O}_{\hat{L}}^{\flat})$.
The following is a description of $(A^{(i)})_{\mathop{\rm
perf}\nolimits}[1/E]^{\wedge}_{p}$ and
$(A_{\mathrm{inf}}(\mathcal{O}_{\hat{L}})^{(i)})_{\mathop{\rm
perf}\nolimits}[1/E]^{\wedge}_{p}$.

######  Lemma 4.2.8.

Let $\widehat{K}_{\infty}^{(i)}$ (resp. $\hat{L}^{(i)}$) be the $i$-th self-
coproduct of $\widehat{K}_{\infty}$ (resp. $\hat{L}$) in $\mathop{\rm
Perfd}\nolimits_{K}$, then $(A^{(i)})_{\mathop{\rm
perf}\nolimits}[1/E]^{\wedge}_{p}\simeq
W((\widehat{K}_{\infty}^{(i)})^{\flat})$ (resp.
$(A_{\mathrm{inf}}(\mathcal{O}_{\hat{L}})^{(i)})_{\mathop{\rm
perf}\nolimits}[1/E]^{\wedge}_{p}\simeq W((\hat{L}^{(i)})^{\flat})$).

###### Proof.

We will only prove the lemma for $(A^{(i)})_{\mathop{\rm
perf}\nolimits}[1/E]^{\wedge}_{p}$, and the case for
$(A_{\mathrm{inf}}(\mathcal{O}_{\hat{L}})^{(i)})_{\mathop{\rm
perf}\nolimits}[1/E]^{\wedge}_{p}$ is similar.

We use similar arguments as in [Wu21, Lemma 5.3]. Fix $i$, first we can show
$(A^{(i)})_{\mathop{\rm perf}\nolimits}$ is the $i$-th self-coproduct of
$(A_{\mathop{\rm perf}\nolimits},(E))$ in the category of perfect prisms over
$\mathcal{O}_{K}$, i.e. $(A^{(i)})_{\mathop{\rm
perf}\nolimits}=(A_{\mathop{\rm perf}\nolimits})^{(i)}_{\mathop{\rm
perf}\nolimits}$. By Theorem 4.2.5, Lemma 4.2.6 and [Wu21, Proposition 2.15],
if we let $S=(A^{(i)})_{\mathop{\rm perf}\nolimits}/E$, then $S[1/p]$ is the
$i$-th self-coproduct of $\widehat{K}_{\infty}$ in the category of perfectoid
$K$-algebras. Now we have

$(A^{(i)})_{\mathop{\rm perf}\nolimits}[1/E]^{\wedge}_{p}\simeq
W(S^{\flat})[1/[\varpi^{\flat}]]^{\wedge}_{p}=W(S^{\flat}[1/\varpi^{\flat}])=W((S[1/p])^{\flat})\simeq
W((\widehat{K}_{\infty}^{(i)})^{\flat}).$

∎

######  Remark 4.2.9.

There is another way to view $\widehat{K}_{\infty}^{(i)}$ in terms of diamonds
over $\mathrm{Spd}(K,\mathcal{O}_{K})$ which is used in the proof of [Wu21,
Lemma 5.3], that there exist a ring of integral elements
$\widehat{K}_{\infty}^{(i),+}$ in $\widehat{K}_{\infty}^{(i)}$, such that we
have

(10)
$\mathrm{Spa}(\widehat{K}_{\infty}^{(i)},\widehat{K}_{\infty}^{(i),+})^{\diamond}\simeq\underbrace{\mathrm{Spa}(\widehat{K}_{\infty},\widehat{K}_{\infty}^{+})^{\diamond}\times_{\mathrm{Spd}(K,\mathcal{O}_{K})}\ldots\times_{\mathrm{Spd}(K,\mathcal{O}_{K})}\mathrm{Spa}(\widehat{K}_{\infty},\widehat{K}_{\infty}^{+})^{\diamond}}_{i\text{-copies
of }\mathrm{Spa}(K_{\infty},K_{\infty}^{+})^{\diamond}}.$

And similar results hold for $\hat{L}$. Using this description and the fact
that functor from perfectoid spaces over $\mathrm{Spa}(K,\mathcal{O}_{K})$ to
diamonds over $\mathrm{Spd}(K,\mathcal{O}_{K})$ is an equivalence, we have
$\hat{L}^{(i)}$ has a natural action of $\hat{G}^{i}$ coming from the action
on the diamond spectrum. Since $\hat{L}^{H_{K}}=\widehat{K}_{\infty}$, we have

$\mathrm{Spa}(\widehat{K}_{\infty}^{(i)},\widehat{K}_{\infty}^{(i),+})^{\diamond}\simeq\left(\mathrm{Spa}(\hat{L},\mathcal{O}_{\hat{L}})^{\diamond}\times\ldots\times_{\mathrm{Spd}(K,\mathcal{O}_{K})}\mathrm{Spa}(\hat{L},\mathcal{O}_{\hat{L}})^{\diamond}\right)^{H_{K}^{i}}\simeq(\mathrm{Spa}(\hat{L}^{(i)},\hat{L}^{(i),+})^{\diamond})^{H_{K}^{i}}.$

That is, $(\hat{L}^{(i)})^{H_{K}^{i}}=\widehat{K}_{\infty}^{(i)}$.

Now we use ideas in [Wu21] and [KL19, §5.5] to study étale $\varphi$-modules
over $A[1/E]^{\wedge}_{p}$ with descent data. We will show this category is
the same as generalized $(\varphi,\Gamma)$-modules in the work of Kedlaya-Liu.
The following is a quick review of Example 5.5.6 and 5.5.7 in [KL19].

Firstly, one has $\hat{L}^{(i)}\simeq\mathop{\rm
Cont}\nolimits(\hat{G}^{i-1},\hat{L})$, here $\mathop{\rm Cont}\nolimits$
means the set of continuous functions. One can see this fact from the proof of
[Wu21, Theorem 5.6]. When $i=2$, we choose the two canonical maps
$i_{1},i_{2}:\hat{L}\to\hat{L}^{(2)}$, corresponds to
$j_{1},j_{2}:\hat{L}\to\mathop{\rm Cont}\nolimits(\hat{G},\hat{L})$ given by

(11) $j_{1}(x):\gamma\mapsto\gamma(x)\quad\text{ and }\quad
j_{2}(x):\gamma\mapsto x.$

From Remark 4.2.9, there is a natural action of $\hat{G}^{2}$ on
$\hat{L}^{(2)}$. One can check this corresponds to the $\hat{G}^{2}$-action on
$\mathop{\rm Cont}\nolimits(\hat{G},\hat{L})$ given by:

$(\sigma_{1},\sigma_{2})(f)(\gamma)=\sigma_{2}f(\sigma_{2}^{-1}\gamma\sigma_{1}).$

######  Remark 4.2.10.

We interchange the roles of $j_{1}$ and $j_{2}$ comparing with the isomorphism
defined in [KL19, Example 5.5.6], so the $\hat{G}^{2}$-action is different
from that in Example 5.5.7 of $loc.cit.$, we will see this definition is more
convenient when relating the descent data with the semilinear group actions.

One can show $\mathop{\rm Cont}\nolimits(\hat{G},-)$ commutes with tilting and
the Witt vector functor, as been discussed in [Wu21, Lemma 5.3], so in
particular, we have

$W((\hat{L}^{(i)})^{\flat})\simeq\mathop{\rm
Cont}\nolimits(\hat{G}^{i-1},W(\hat{L}^{\flat})).$

For $i=2$, we still use $j_{1}$ and $j_{2}$ to represent the two canonical
maps from $W(\hat{L}^{\flat})$ to $\mathop{\rm
Cont}\nolimits(\hat{G},W(\hat{L}^{\flat}))$ that comes from (11). The above
isomorphism also is compatible with the action of $\hat{G}^{2}$, so we have