(12) $W((\widehat{K}_{\infty}^{(2)})^{\flat})\simeq\mathop{\rm
Cont}\nolimits(\hat{G},W(\hat{L}^{\flat}))^{H_{K}^{2}}$

We prove the following lemma for our later use.

Now let $\mathcal{M}$ be an étale $\varphi$-module over
$W(\widehat{K}_{\infty}^{\flat})$ with a descent data:

$\psi:\mathcal{M}\otimes_{W(\widehat{K}_{\infty}^{\flat}),j_{1}}W((\widehat{K}_{\infty}^{(2)})^{\flat})\simeq\mathcal{M}\otimes_{W(\widehat{K}_{\infty}^{\flat}),j_{2}}W((\widehat{K}_{\infty}^{(2)})^{\flat})$

as étale $\varphi$-modules over $W((\widehat{K}_{\infty}^{(2)})^{\flat})$ and
satisfies cocycle condition over $W((\widehat{K}_{\infty}^{(3)})^{\flat})$.
Using (12), we have $\psi$ is the same as a descent data:

(13)
$\hat{\psi}:{\mathcal{M}}\otimes_{W(\widehat{K}_{\infty}^{\flat}),j_{1}}\mathop{\rm
Cont}\nolimits(\hat{G},W(\hat{L}^{\flat}))^{H_{K}^{2}}\simeq{\mathcal{M}}\otimes_{W(\widehat{K}_{\infty}^{\flat}),j_{2}}\mathop{\rm
Cont}\nolimits(\hat{G},W(\hat{L}^{\flat}))^{H_{K}^{2}}.$

For each $\gamma\in\hat{G}$, we have an evaluation map
$\tilde{e}_{\gamma}:\mathop{\rm Cont}\nolimits(\hat{G},W(\hat{L}^{\flat}))\to
W(\hat{L}^{\flat})$ given by evaluating at $\gamma$. Using (11), one can check
$\tilde{e}_{\gamma}\circ j_{2}:W(\widehat{K}_{\infty}^{\flat})\to
W(\hat{L}^{\flat})$ is given by the natural embedding and
$\tilde{e}_{\gamma}\circ j_{1}:W(\widehat{K}_{\infty}^{\flat})\to
W(\hat{L}^{\flat})$ is given by $x\mapsto\gamma(x)$. So for each
$\gamma\in\hat{G}$, if we tensor (13) against the evaluation map
$\tilde{e}_{\gamma}$, we get an isomorphism:

$\psi_{\gamma}:{\mathcal{M}}\otimes_{W(\widehat{K}_{\infty}^{\flat}),\gamma}W(\hat{L}^{\flat})\simeq{\mathcal{M}}\otimes_{W(\widehat{K}_{\infty}^{\flat})}W(\hat{L}^{\flat}).$

And similar to the classical Galois descent theory, the cocycle condition for
$\psi$ implies $\\{\psi_{\gamma}\\}_{\gamma}$ satisfies

$\psi_{\sigma\gamma}=\psi_{\sigma}\circ\sigma^{\ast}\psi_{\gamma}.$

Hence $\\{\psi_{\gamma}\\}_{\gamma}$ defines a continuous semilinear action of
$\hat{G}$ on
$\hat{\mathcal{M}}:=\mathcal{M}\otimes_{W(\widehat{K}_{\infty}^{\flat})}W(\hat{L}^{\flat})$.
One can check for $\gamma\in H_{K}$, we have the composition

$W(\widehat{K}_{\infty}^{\flat})\xrightarrow{j_{k}}W((\widehat{K}_{\infty}^{(2)})^{\flat})\to\mathop{\rm
Cont}\nolimits(\hat{G},W(\hat{L}^{\flat}))\xrightarrow{\tilde{e}_{\gamma}}W(\hat{L}^{\flat})$

is the natural embedding $W(\widehat{K}_{\infty}^{\flat})\hookrightarrow
W(\hat{L}^{\flat})$ for $k=1,2$. And using the cocycle condition, one can show
$\psi_{\gamma}=\mathop{\rm id}\nolimits$ for $\gamma\in H_{K}$, so in
particular, $\mathcal{M}\subset\hat{\mathcal{M}}^{H_{K}}$. Conversely, given a
semilinear action of $\hat{G}$ on $\hat{\mathcal{M}}$ such that
$\mathcal{M}\subset\hat{\mathcal{M}}^{H_{K}}$, $\\{\psi_{\gamma}\\}_{\gamma}$
defines a descent data $\psi$ over $\mathop{\rm
Cont}\nolimits(\hat{G},W(\hat{L}^{\flat}))^{H_{K}^{2}}$ if and only if the
semilinear action is continuous. In summary, we have

######  Theorem 4.2.11.

  1. (1)

The category of étale $\varphi$-modules over $A[1/E]^{\wedge}_{p}$ with
descent data over $A^{(2)}[1/E]^{\wedge}_{p}$ is equivalent to the category of
étale $(\varphi,\tau)$-modules over $A[1/E]^{\wedge}_{p}$;

  2. (2)

Given a descent data $f$ of an étale $\varphi$-module $\mathcal{M}$ over
$A[1/E]^{\wedge}_{p}$, and $\gamma\in\hat{G}$, we can define the evaluation
$f_{\gamma}$ of $f$ at $\gamma$, defined by the base change of $f$ along

$e_{\gamma}:A^{(2)}[1/E]^{\wedge}_{p}\to(A^{(2)})_{\mathop{\rm
perf}\nolimits}[1/E]^{\wedge}_{p}\xrightarrow{\tilde{e}_{\gamma}}W(\hat{L}^{\flat}),$

which defines an isomorphism:

$f_{\gamma}:\mathcal{M}\otimes_{A[1/E]^{\wedge}_{p},\tilde{\iota}_{\gamma}}W(\hat{L}^{\flat})\simeq\mathcal{M}\otimes_{A[1/E]^{\wedge}_{p}}W(\hat{L}^{\flat})$

where $\tilde{\iota}_{\gamma}:A[1/E]^{\wedge}_{p}\to
W(\hat{L}^{\flat})\xrightarrow{\gamma}W(\hat{L}^{\flat})$. Suppose that
$(\mathcal{M},f)$ corresponds to a $\mathbb Z_{p}$-representation $T$ of
$G_{K}$, then $f_{\gamma}$ corresponds to the semilinear action of $\gamma$ on
$\mathcal{M}$ inside
$\mathcal{M}\otimes_{A[1/E]^{\wedge}_{p}}W(\mathbb{C}_{p}^{\flat})\simeq
T^{\vee}\otimes W(\mathbb{C}_{p}^{\flat})$. Moreover, two descent data $f,g$
are equal if and only if $f_{\tau}=g_{\tau}$.

###### Proof.

The discussion above the theorem establishes the equivalence between the
category of étale $\varphi$-modules over $A_{\mathop{\rm
perf}\nolimits}[1/E]^{\wedge}_{p}$ with descent data over
$(A^{(2)})_{\mathop{\rm perf}\nolimits}[1/E]^{\wedge}_{p}$ is equivalent to
the category of étale $(\varphi,\tau)$-modules over $A[1/E]^{\wedge}_{p}$. Now
(1) follows [Wu21, Theorem 4.6] which shows that the category of étale
$\varphi$-modules over $B[\frac{1}{I}]^{\wedge}_{p}$ is equivalent to the
category of étale $\varphi$-modules over $B_{\mathop{\rm
perf}\nolimits}[\frac{1}{I}]^{\wedge}_{p}$ for bounded prism $(B,I)$
satisfying $\varphi(I)\mod p$ is generated by a non-zero divisor in $B/p$.
Then it just remains to prove the last statement in (2). Actually one can
check (2) by chasing all the functors used in (1), and use the fact that for
any étale $(\varphi,\tau)$-module, the $\hat{G}$-action on $\hat{\mathcal{M}}$
is determined by the $\tau$-action on $\mathcal{M}$. However, this can also
been seen directly from the following lemma. ∎

######  Lemma 4.2.12.

Given two finite free étale $\varphi$-modules $\mathcal{M},\mathcal{N}$ over
$A^{(2)}[1/E]^{\wedge}_{p}$ and two morphisms $f,g:\mathcal{M}\to\mathcal{N}$
of étale $\varphi$-modules over $A^{(2)}[1/E]^{\wedge}_{p}$. Let
$f_{\tau},g_{\tau}$ be the base changes of $f,g$ along the map

$e_{\tau}:A^{(2)}[1/E]^{\wedge}_{p}\to(A^{(2)})_{\mathop{\rm
perf}\nolimits}[1/E]^{\wedge}_{p}\simeq\mathop{\rm
Cont}\nolimits\Big{(}\hat{G},W\big{(}(\hat{L}^{(2)})^{\flat}\big{)}\Big{)}^{H_{K}^{2}}\xrightarrow{\tilde{e}_{\tau}}W(\hat{L}^{\flat}).$

Then $f=g$ if and only if $f_{\tau}=g_{\tau}$.

###### Proof.

We take the natural base change of $f$ and $g$ along
$A^{(2)}[1/E]^{\wedge}_{p}\to(A^{(2)})_{\mathop{\rm
perf}\nolimits}[1/E]^{\wedge}_{p}$, we get two morphisms $\psi$ and
$\psi^{\prime}$ between étale $\varphi$-modules over $(A^{(2)})_{\mathop{\rm
perf}\nolimits}[1/E]^{\wedge}_{p}$. Since the base change functor between
étale $\varphi$-modules over $A^{(2)}[1/E]^{\wedge}_{p}$ and
$(A^{(2)})_{\mathop{\rm perf}\nolimits}[1/E]^{\wedge}_{p}$ is an equivalence
of categories, it reduces to show that $\psi=\psi^{\prime}$ if and only if
their base change along

$\tilde{e}_{\tau}:(A^{(2)})_{\mathop{\rm
perf}\nolimits}[1/E]^{\wedge}_{p}\simeq\mathop{\rm
Cont}\nolimits\Big{(}\hat{G},W\big{(}(\hat{L}^{(2)})^{\flat}\big{)}\Big{)}^{H_{K}^{2}}\xrightarrow{}W(\hat{L}^{\flat})$

is equal. Since $\mathcal{M}$ and $\mathcal{N}$ are finite free, it is enough
to show the evaluation map:

$\tilde{e}_{\tau}:\mathop{\rm
Cont}\nolimits\Big{(}\hat{G},W\big{(}(\hat{L}^{(2)})^{\flat}\big{)}\Big{)}^{H_{K}^{2}}\to
W\big{(}(\hat{L}^{(2)})^{\flat}\big{)}$

is injective. Suppose $h\in\mathop{\rm
Cont}\nolimits\Big{(}\hat{G},W\big{(}(\hat{L}^{(2)})^{\flat}\big{)}\Big{)}^{H_{K}^{2}}$
satisfies $h(\tau)=0$, then

$(\sigma_{1},\sigma_{2})(h)(\tau)=\sigma_{2}h(\sigma_{2}^{-1}\tau\sigma_{1})=0$

for $(\sigma_{1},\sigma_{2})\in H_{K}^{2}$. Since $\hat{G}$ is topologically
generated by $H_{K}$ and $\tau$, we get $h\equiv 0$. ∎

Now we give the $\mathbb Q$-isogeny versions of Theorem 4.2.11 and Lemma
4.2.12. Recall that the étale $(\varphi,\tau)$-modules over
$A[1/E]^{\wedge}_{p}[\frac{1}{p}]$ is equivalent to the category of $\mathbb
Q_{p}$-representations of $G_{K}$, and recall the following definition of
étale $(\varphi,\tau)$-modules over $B[1/J]^{\wedge}_{p}[\frac{1}{p}]$ for a
prism $(B,J)\in X_{{\mathbbl{\Delta}}}$.

######  Definition 4.2.13.

An (globally) étale $\varphi$-module $\mathcal{M}$ over
$B[1/J]^{\wedge}_{p}[\frac{1}{p}]$ is a (finite projective) $\varphi$-module
over $B[1/J]^{\wedge}_{p}[\frac{1}{p}]$ that arises by base extension from an
étale $\varphi$-module $B[1/J]^{\wedge}_{p}$.

From this definition, we immediately deduce the following result from [Wu21,
Theorem 4.6]

######  Proposition 4.2.14.

For any prism $(B,J)\in X_{{\mathbbl{\Delta}}}$ satisfying $\varphi(J)\mod p$
is generated by a non-zero divisor in $B/p$, the base change functor defined
by $B[1/J]^{\wedge}_{p}[\frac{1}{p}]\to B_{\mathop{\rm
perf}\nolimits}[1/J]^{\wedge}_{p}[\frac{1}{p}]$ induces an equivalence between
the category of étale $\varphi$-modules over
$B[1/J]^{\wedge}_{p}[\frac{1}{p}]$ and the category of étale $\varphi$-modules
over $B_{\mathop{\rm perf}\nolimits}[1/J]^{\wedge}_{p}[\frac{1}{p}]$.

And similar to Theorem 4.2.11 and Lemma 4.2.12, we have

######  Theorem 4.2.15.

The category of étale $\varphi$-modules over
$A[1/E]^{\wedge}_{p}[\frac{1}{p}]$ with descent data over
$A^{(2)}[1/E]^{\wedge}_{p}[\frac{1}{p}]$ is equivalent to the category of
étale $(\varphi,\tau)$-modules over $A[1/E]^{\wedge}_{p}[\frac{1}{p}]$.
Moreover,

$\mathop{\rm
Cont}\nolimits\big{(}\hat{G},W(\hat{L}^{\flat})[\frac{1}{p}]\big{)}^{H_{K}^{2}}\simeq
W(\widehat{K}_{\infty}^{(2)})^{\flat}[\frac{1}{p}].$

For $\gamma\in\hat{G}$, we can define the evaluation map

$\tilde{e}_{\gamma}:\mathop{\rm
Cont}\nolimits\big{(}\hat{G},W(\hat{L}^{\flat})[\frac{1}{p}]\big{)}\to
W(\hat{L}^{\flat})[\frac{1}{p}].$

And given a descent data $f$ of an étale $\varphi$-module $\mathcal{M}$ over
$A[1/E]^{\wedge}_{p}[\frac{1}{p}]$, and $\gamma\in\hat{G}$, we can define the
evaluation $f_{\gamma}$ of $f$ at $\gamma$, defined by the base change of $f$
along

$e_{\gamma}:A^{(2)}[1/E]^{\wedge}_{p}[\frac{1}{p}]\to(A^{(2)})_{\mathop{\rm
perf}\nolimits}[1/E]^{\wedge}_{p}[\frac{1}{p}]\xrightarrow{\tilde{e}_{\gamma}}W(\hat{L}^{\flat})[\frac{1}{p}],$

which defines an isomorphism:

$\mathcal{M}\otimes_{A[1/E]^{\wedge}_{p}[1/p],\tilde{\iota}_{\gamma}}W(\hat{L}^{\flat})[\frac{1}{p}]\simeq\mathcal{M}\otimes_{A[1/E]^{\wedge}_{p}[1/p]}W(\hat{L}^{\flat})[\frac{1}{p}]$

where $\tilde{\iota}_{\gamma}:A[1/E]^{\wedge}_{p}[\frac{1}{p}]\to
W(\hat{L}^{\flat})[\frac{1}{p}]\xrightarrow{\gamma}W(\hat{L}^{\flat})[\frac{1}{p}]$.
If $(\mathcal{M},f)$ corresponds to a $\mathbb Q_{p}$-representation $V$ of
$G_{K}$, then $f_{\gamma}$ corresponds to the semilinear action of $\gamma$ on
$\mathcal{M}$ inside $V^{\vee}\otimes W(\mathbb{C}_{p}^{\flat})[1/p]$.
Moreover, two descent data $f,g$ are equal if and only if $f_{\tau}=g_{\tau}$.

######  Lemma 4.2.16.

Given two finite free étale $\varphi$-modules $\mathcal{M},\mathcal{N}$ over
$A^{(2)}[1/E]^{\wedge}_{p}[\frac{1}{p}]$ and two morphisms
$f,g:\mathcal{M}\to\mathcal{N}$ of étale $\varphi$-modules over
$A^{(2)}[1/E]^{\wedge}_{p}[\frac{1}{p}]$. Let $f_{\tau},g_{\tau}$ be the base
changes of $f,g$ along the map

$e_{\tau}:A^{(2)}[1/E]^{\wedge}_{p}[\frac{1}{p}]\to(A^{(2)})_{\mathop{\rm
perf}\nolimits}[1/E]^{\wedge}_{p}[\frac{1}{p}]\simeq\mathop{\rm
Cont}\nolimits\Big{(}\hat{G},W\big{(}(\hat{L}^{(2)})^{\flat}[\frac{1}{p}]\big{)}\Big{)}^{H_{K}^{2}}\xrightarrow{\tilde{e}_{\tau}}W(\hat{L}^{\flat})[\frac{1}{p}].$

Then $f=g$ if and only if $f_{\tau}=g_{\tau}$.

###### Proof.

The proofs are exactly the same as the proof of Theorem 4.2.11 and Lemma
4.2.12, plus the following fact that

$\mathop{\rm
Cont}\nolimits\big{(}\hat{G},W(\hat{L}^{\flat})[\frac{1}{p}]\big{)}=\mathop{\rm
Cont}\nolimits\big{(}\hat{G},W(\hat{L}^{\flat})\big{)}[\frac{1}{p}],$

which can be shown by the compactness of $\hat{G}$. ∎

###  4.3. Proofs of Proposition 3.2.2 and Theorem 4.1.10

We keep the assumption that $R=\mathcal{O}_{K}$ is a mixed characteristic
complete DVR with perfect residue field in this subsection, and keep our
notations in §2.1.

Let us first prove Proposition 3.2.2 using Lemma 2.3.2 and results in §4.2.
First, we give a different interpretation of the “evaluation map”:

$e_{\gamma}:A^{(2)}[1/E]^{\wedge}_{p}\to(A^{(2)})_{\mathop{\rm
perf}\nolimits}[1/E]^{\wedge}_{p}\simeq\mathop{\rm
Cont}\nolimits\Big{(}\hat{G},W\big{(}(\hat{L}^{(2)})^{\flat}\big{)}\Big{)}^{H_{K}^{2}}\xrightarrow{\tilde{e}_{\gamma}}W(\hat{L}^{\flat})$

in Theorem 4.2.11 when restricted on $A^{(2)}$ . Recall that we fix a
compatible system $\\{\varpi_{n}\\}_{n}$ of $p^{n}$-th roots of a uniformizer
$\varpi\in\mathcal{O}_{K}$, this defines a map of prisms
$\iota:(A,(E))\to(A_{\mathrm{inf}},(E))$ maps $u$ to $[{\varpi}^{\flat}]$, and
given a $\gamma\in G_{K}$, we define $\iota_{\gamma}$ to be the composition of
$\iota$ with $\gamma:(A_{\mathrm{inf}},(E))\to(A_{\mathrm{inf}},(E))$ where
the second map is defined as $a\mapsto\gamma(a)$. Since $(E)\subset
A_{\mathrm{inf}}$ is equal to $\mathop{\rm Ker}\nolimits(\theta)$ and $\theta$
is $G_{K}$-equivariant, $\gamma$ is a well-defined map of $\delta$-pairs. By
the universal property of $A^{(2)}$, we can define a map of prisms
$\iota_{\gamma}^{(2)}:(A^{(2)},(E))\to(A_{\mathrm{inf}},(E))$ so that the
following diagram commutes:

(14)
${(A,(E))}$${(A^{(2)},(E))}$${(A,(E))}$${(A_{\mathrm{inf}},(E))}$$\scriptstyle{i_{1}}$$\scriptstyle{\iota_{\gamma}}$$\scriptstyle{\iota^{(2)}_{\gamma}}$$\scriptstyle{i_{2}}$$\scriptstyle{\iota}$

We have $\iota^{(2)}_{\gamma}$ induces a morphism
$\tilde{\iota}^{(2)}_{\gamma}:A^{(2)}[1/E]^{\wedge}_{p}\to W(\mathbb
C_{p}^{\flat})$. We claim for all $\gamma\in G_{K}$,
$\tilde{\iota}^{(2)}_{\gamma}$ is the same as the

$A^{(2)}[1/E]^{\wedge}_{p}\to(A^{(2)})_{\mathop{\rm
perf}\nolimits}[1/E]^{\wedge}_{p}\simeq\mathop{\rm
Cont}\nolimits\Big{(}\hat{G},W\big{(}(\hat{L}^{(2)})^{\flat}\big{)}\Big{)}^{H_{K}^{2}}\xrightarrow{\tilde{e}_{\gamma}}W(\hat{L}^{\flat})\hookrightarrow
W(\mathbb C_{p}^{\flat}).$

To see this, by the universal property of direct perfection, we have (14)
factorizes as:

${(A,(E))}$${(A^{(2)},(E))}$${(A,(E))}$${(A_{\mathop{\rm
perf}\nolimits},(E))}$${((A^{(2)})_{\mathop{\rm
perf}\nolimits},(E))}$${(A_{\mathop{\rm
perf}\nolimits},(E))}$${(A_{\mathrm{inf}},(E))}$$\scriptstyle{i_{1}}$$\scriptstyle{i_{2}}$$\scriptstyle{i^{\prime}_{1}}$$\scriptstyle{\iota^{\prime}_{\gamma}}$$\scriptstyle{\iota^{\prime(2)}_{\gamma}}$$\scriptstyle{i^{\prime}_{2}}$$\scriptstyle{\iota^{\prime}}$

So $\tilde{\iota}^{(2)}_{\gamma}$ has a factorization

$A^{(2)}[1/E]^{\wedge}_{p}\to(A^{(2)})_{\mathop{\rm
perf}\nolimits}[1/E]^{\wedge}_{p}\to W(\mathbb C_{p}^{\flat}).$

We just need to check $\iota^{\prime(2)}_{\tau}$ induces the evaluation map

$(A^{(2)})_{\mathop{\rm perf}\nolimits}[1/E]^{\wedge}_{p}\simeq\mathop{\rm
Cont}\nolimits\Big{(}\hat{G},W\big{(}(\hat{L}^{(2)})^{\flat}\big{)}\Big{)}^{H_{K}^{2}}\xrightarrow{\tilde{e}_{\tau}}W(\hat{L}^{\flat})\xhookrightarrow{}W(\mathbb{C}_{p}^{\flat}).$

And this follows from the isomorphism of $(A^{(2)})_{\mathop{\rm
perf}\nolimits}[1/E]^{\wedge}_{p}\simeq W((K^{(2)}_{\infty})^{\flat})$, then
one check directly for $j_{1},j_{2}$ defined in (11), $\tilde{e}_{\gamma}\circ
j_{1}:A_{\mathop{\rm perf}\nolimits}[1/E]^{\wedge}_{p}\to W(\hat{L}^{\flat})$
is equal to the map induced from $\iota^{\prime}_{\gamma}$ and
$\tilde{e}_{\gamma}\circ j_{2}:A_{\mathop{\rm
perf}\nolimits}[1/E]^{\wedge}_{p}\to W(\hat{L}^{\flat})$ is equal to the map
induced from $\iota^{\prime}$. In particular, we have a commutative diagram:

(15)
${A^{(2)}}$${A_{\mathrm{inf}}}$${A^{(2)}[1/E]^{\wedge}_{p}}$${(A^{(2)})_{\mathop{\rm
perf}\nolimits}[1/E]^{\wedge}_{p}}$${W(\hat{L}^{\flat})}$${W(\mathbb
C_{p}^{\flat}).}$$\scriptstyle{\iota^{(2)}_{\gamma}}$$\scriptstyle{\tilde{e}_{\gamma}}$

Now we can prove Proposition 3.2.2.

###### Proof of Proposition 3.2.2.

First we pick $\gamma=\tilde{\tau}$ that is a preimage of $\tau$ under the map
$G_{K}\to\hat{G}$, we have $\gamma(u)-u=Ez$ and $\iota^{(2)}_{\gamma}$ defined
as above is the embedding defined in §2.4 by Remark 2.4.2. In particular,
composing the embedding $A^{(2)}\hookrightarrow A_{\mathrm{inf}}$ defined in
§2.4 with $A_{\mathrm{inf}}\hookrightarrow W(\mathbb C_{p}^{\flat})$, one get
the evaluation map

$(A^{(2)})_{\mathop{\rm perf}\nolimits}[1/E]^{\wedge}_{p}\simeq\mathop{\rm
Cont}\nolimits\Big{(}\hat{G},W\big{(}(\hat{L}^{(2)})^{\flat}\big{)}\Big{)}^{H_{K}^{2}}\xrightarrow{\tilde{e}_{\tau}}W(\hat{L}^{\flat})\xhookrightarrow{}W(\mathbb{C}_{p}^{\flat}).$

restricted on $A^{(2)}$.

Keep the notations as in §3.2, and let
$\mathcal{M}_{A_{\mathrm{inf}}}=W(\mathbb
C_{p}^{\flat})\otimes_{A}\mathfrak{M}$ and
$\mathcal{M}_{A}\simeq\mathfrak{M}\otimes_{A}A[1/E]^{\wedge}_{p}$. By Theorem
4.2.11 and Theorem 4.2.3, recall we use
$B^{(2)}=A^{(2)}[\frac{1}{E}]^{\wedge}_{p}$ and $B^{(2)}_{\mathop{\rm
st}\nolimits}=A^{(2)}_{\mathop{\rm st}\nolimits}[\frac{1}{E}]^{\wedge}_{p}$ to
simplify our notations, we have there is a descent data

$c:\mathcal{M}_{A}\otimes_{A[1/E]^{\wedge}_{p},\tilde{i}_{1}}B^{(2)}\to\mathcal{M}_{A}\otimes_{A[1/E]^{\wedge}_{p},\tilde{i}_{2}}B^{(2)}$

of $\mathcal{M}_{A}$ over $B^{(2)}$ that corresponds to the representation
$T$. And the semilinear action of $\gamma=\tilde{\tau}$ on
$\mathcal{M}_{A_{\mathrm{inf}}}$ is given by the evaluation $c_{\tau}$, that
is, we have the linearization of the $\tilde{\tau}$-action is defined by

$c_{\tau}:W(\mathbb
C_{p}^{\flat})\otimes_{\tilde{\iota}_{\gamma},A[1/E]^{\wedge}_{p}}\mathcal{M}_{A}\simeq
W(\mathbb
C_{p}^{\flat})\otimes_{\tilde{\iota},A[1/E]^{\wedge}_{p}}\mathcal{M}_{A}.$

By base change $c$ along $B^{(2)}\to B^{(2)}[\frac{1}{p}]$, we get a
$B^{(2)}[\frac{1}{p}]$-linear $\varphi$-equivariant morphism:

$c^{\prime}:\mathcal{M}_{A}\otimes_{A[1/E]^{\wedge}_{p},\tilde{i}_{1}}B^{(2)}[\frac{1}{p}]\to\mathcal{M}_{A}\otimes_{A[1/E]^{\wedge}_{p},\tilde{i}_{2}}B^{(2)}[\frac{1}{p}].$

On the other hand, from the discussions after Proposition 3.2.2,
$\tilde{\tau}$-action also defines a $\varphi$-equivariant morphism

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

We will see in Proposition 4.3.1 below that $f_{\tilde{\tau}}$ actually
descents to a $B^{(2)}[1/p]$-linear morphism. Assuming this fact, then if we
base change $f_{\tilde{\tau}}$ along $A^{(2)}[\frac{1}{p}]\to W(\mathbb
C_{p}^{\flat})[\frac{1}{p}]$, we will have $f_{\tilde{\tau}}\otimes W(\mathbb
C_{p}^{\flat})[\frac{1}{p}]=c_{\tau}$ since the way we define
$f_{\tilde{\tau}}$ is by taking the ${\tilde{\tau}}$-action. From the
discussion at the beginning of the proof and Lemma 4.2.16, we have
$f_{\tilde{\tau}}=c^{\prime}$ as a $B^{(2)}[\frac{1}{p}]$-linear isomorphism
between
$\mathcal{M}_{A}\otimes_{A[1/E]^{\wedge}_{p},\tilde{i}_{1}}B^{(2)}[\frac{1}{p}]$
and
$\mathcal{M}_{A}\otimes_{A[1/E]^{\wedge}_{p},\tilde{i}_{2}}B^{(2)}[\frac{1}{p}]$.

We fix a basis $\\{e_{i}\\}$ of $\mathfrak{M}$, for $j=1,2$ let
$\\{e^{j}_{i}\\}$ be the basis of
$\mathcal{M}_{A}\otimes_{A,\tilde{i^{\prime}}_{j}}B^{(2)}[\frac{1}{p}]$
defined by $e^{j}_{i}=e_{i}\otimes 1$ and the tensor is via $A\to
A[1/E]^{\wedge}_{p}\xrightarrow{\tilde{i}_{j}}B^{(2)}[1/p]$. So we can
interpret $f_{\tilde{\tau}}=c^{\prime}$ as matrix using this two basis, this
matrix is $X_{\tilde{\tau}}$ from this definition, so it has coefficients
inside $A_{\mathop{\rm st}\nolimits}^{(2)}[\frac{1}{p}]$ by the discussion
before Proposition 3.2.2. On the other hand, $X_{\tilde{\tau}}$ has
coefficients in $B^{(2)}\subset B_{\mathop{\rm st}\nolimits}^{(2)}$ since
$c^{\prime}$ is defined by the $B^{(2)}$-linear map $c$. So by Lemma 2.3.2, we
have $X_{\tilde{\tau}}$ has coefficients inside $A_{\mathop{\rm
st}\nolimits}^{(2)}$. The same argument shows when $T$ is crystalline, then
$X_{\tilde{\tau}}$ has coefficients inside $A^{(2)}$. ∎

######  Proposition 4.3.1.

Base change along $B^{(2)}\to A^{(2)}_{\mathop{\rm
st}\nolimits}[1/E]^{\wedge}_{p}$ defines an equivalence of categories of étale
$\varphi$-modules over $B^{(2)}$ and $A^{(2)}_{\mathop{\rm
st}\nolimits}[1/E]^{\wedge}_{p}$ and an equivalence of categories of étale
$\varphi$-modules over $B^{(2)}[1/p]$ and $A^{(2)}_{\mathop{\rm
st}\nolimits}[1/E]^{\wedge}_{p}[1/p]$.

###### Proof.

By [Wu21, Theorem 4.6], we just need to show the same result after
perfections, we will show $(A^{(2)})_{\mathop{\rm
perf}\nolimits}=(A^{(2)}_{\mathop{\rm st}\nolimits})_{\mathop{\rm
perf}\nolimits}$ in Lemma 5.0.13 using the logarithmic prismatic site. ∎

Now, let us prove Theorem 4.1.10 by first producing a functor $\mathcal{T}$
from prismatic $F$-crystals in finite
$\mathcal{O}_{{\mathbbl{\Delta}}}$-modules to lattices inside a crystalline
representation. For prism $A$, we use $i_{k}:A\to A^{(2)}$ or $A^{(3)}$ for
natural map from $A$ to $k$-th factor of $A^{(2)}$ or $A^{(3)}$. The notation
$i_{kl}:A^{(2)}\to A^{(3)}$ has the similar meaning.

By Corollary 4.1.9, given a prismatic $F$-crystal
${\mathfrak{M}}_{{\mathbbl{\Delta}}}$, we obtain a Kisin module
$(\mathfrak{M},\varphi_{\mathfrak{M}})$ of height $h$ together with descent
data
$f:\mathfrak{M}\otimes_{A,i_{1}}A^{(2)}\to\mathfrak{M}\otimes_{A,i_{2}}A^{(2)}$
so that $f$ satisfies the following cocycle condition $i_{13}\otimes
f=(i_{23}\otimes f)\circ(i_{12}\otimes f)$, where $i_{kl}\otimes f$ is the
base change of $f$ along $i_{kl}$, and $f$ also compatible with the
$\varphi$-structure on the both sides of $f$. Note that the existence of $f$
follows from the crystal property of $\mathfrak{M}_{{\mathbbl{\Delta}}}$:

(16)
$f:\mathfrak{M}\otimes_{A,i_{1}}A^{(2)}\simeq\mathfrak{M}_{{\mathbbl{\Delta}}}((A^{(2)},(E)))\simeq\mathfrak{M}\otimes_{A,i_{2}}A^{(2)}$

We let $\mathcal{M}=\mathfrak{M}\otimes_{A}A[1/E]^{\wedge}_{p}$ and
$c=f\otimes_{A^{(2)}}B^{(2)}$, then $(\mathcal{M},c)$ is an étale
$\varphi$-module with descent data, which corresponds to a $\mathbb
Z_{p}$-representation of $G_{K}$. Moreover the semilinear action of $G_{K}$ on
$\mathfrak{M}\otimes_{A}W(\mathbb{C}_{p}^{\flat})$ comes from
$\\{c_{\gamma}\\}_{\gamma\in G_{K}}$ using the evaluation maps. If we define

$f_{\gamma}:A_{\mathrm{inf}}\otimes_{\iota_{\gamma},A}\mathfrak{M}\to
A_{\mathrm{inf}}\otimes_{\iota,A}\mathfrak{M}$

as the base change of $f$ along $\iota_{\gamma}^{(2)}$, then by (15), we have
$c_{\gamma}=f_{\gamma}$. The $G_{K}$-semilinear action commutes with $\varphi$
as $f$ does. For any $\gamma\in G_{K}$, we have $\gamma(A)\subset
W(k)[\\![u,\epsilon-1]\\!]\subset A^{(2)}_{\mathop{\rm st}\nolimits}\subset
A_{\mathrm{inf}}$. Therefore, the $G_{K}$-action on the
$A_{\mathrm{inf}}\otimes_{A}\mathfrak{M}$ defined the above factors through
$A^{(2)}_{\mathop{\rm st}\nolimits}\otimes_{A}\mathfrak{M}$. We claim that
$G_{K}$-action on $\widehat{\mathfrak{M}}:=A^{(2)}_{\mathop{\rm
st}\nolimits}\otimes_{A}\mathfrak{M}$ defines a $(\varphi,\hat{G})$-module
which corresponds to a crystalline representation.

First, for $\gamma\in G_{\infty}$, $\gamma(A)=A$ in $A_{\mathrm{inf}}$, we
conclude $\iota^{(2)}_{\gamma}:A^{(2)}\to A_{\mathrm{inf}}$ satisfies
$\iota^{(2)}_{\gamma}\circ i_{1}=\iota^{(2)}_{\gamma}\circ i_{2}$. In
particular, for any $\gamma\in G_{\infty}$ and $j=1,2$, using (16) and the
crystal property of $\mathfrak{M}_{{\mathbbl{\Delta}}}$, $f_{\gamma}$ comes
from the base change of (16) along $\iota^{(2)}_{\gamma}:A^{(2)}\to
A_{\mathrm{inf}}$, in particular, we have

$f_{\gamma}:\mathfrak{M}\otimes_{A,\iota^{(2)}_{\gamma}\circ
i_{1}}A_{\mathrm{inf}}\simeq\mathfrak{M}_{{\mathbbl{\Delta}}}((A_{\mathrm{inf}},\mathop{\rm
Ker}\nolimits\theta))\simeq\mathfrak{M}\otimes_{A,\iota^{(2)}_{\gamma}\circ
i_{2}}A_{\mathrm{inf}}.$

Since $\iota^{(2)}_{\gamma}\circ i_{1}=\iota^{(2)}_{\gamma}\circ i_{2}$, we
have $f_{\gamma}={\rm id}$ which means
$\mathfrak{M}\subset(\widehat{\mathfrak{M}})^{G_{\infty}}$. Similarly, $G_{K}$
acts on $\widehat{\mathfrak{M}}/I_{+}$ corresponds the base change of $f$
along

$A^{(2)}\xrightarrow{\iota^{(2)}_{\gamma}}A_{\mathrm{inf}}\to W(\bar{k})$

where the last arrow is the reduction modulo $W(\mathfrak{m})$ ($\mathfrak{m}$
is the maximal ideal of $\mathcal{O}_{\mathbb C_{p}}^{\flat}$). One can check
for all $\gamma\in G_{K}$ and $j=1,2$, we have

$A\xrightarrow{i_{j}}A^{(2)}\xrightarrow{\iota^{(2)}_{\gamma}}A_{\mathrm{inf}}\to
W(\bar{k})$

are all equal to $A\to W(k)\hookrightarrow W(\overline{k})$ with the first
arrow given by $u\mapsto 0$. The above map induces a morphism of prisms
$(A,(E))\to(W(k),(p))$, then using (16) and the crystal condition of
$\mathfrak{M}_{{\mathbbl{\Delta}}}$, we can similarly prove that $G_{K}$ acts
on $\widehat{\mathfrak{M}}/I_{+}$-trivially, so
$(\mathfrak{M},\varphi_{\mathfrak{M}},G_{K})$ is a $(\varphi,\hat{G})$-module.
Furthermore, $\widehat{T}(\widehat{\mathfrak{M}})$ is crystalline by Corollary
3.3.4 and Theorem 3.2.1.

######  Remark 4.3.2.

In §5, we will consider a category consisting of modules with descent data,
and similar arguments about the triviality of the Galois actions can be shown
directly using the cocycle condition of the descent data. We summarize this
fact in the following easy fact.

######  Lemma 4.3.3.

Let $q:(A^{(2)},(E))\to(B,J)$ be a map of prisms satisfying $q\circ
i_{1}=q\circ i_{2}$, then for any descent data $f$ over $A^{(2)}$, the base
change of $f$ along $q$ is the identity map.

To show the fully faithfulness of this functor, first let $(\mathfrak{M},f)$,
$(\mathfrak{M}^{\prime},f^{\prime})$ be two Kisin modules with descent data
$f,f^{\prime}$ respectively. Suppose that there exists a map
$\alpha:\mathcal{T}((\mathfrak{M},f))\to\mathcal{T}((\mathfrak{M}^{\prime},f^{\prime}))$
as lattices of crystalline representations, then from our construction of
$\mathcal{T}$ and Theorem 3.3.3, $\alpha$ is induced from a map
$\hat{\alpha}:(\mathfrak{M},\varphi_{\mathfrak{M}},\hat{G}_{\mathfrak{M}})\to(\mathfrak{M}^{\prime},\varphi_{\mathfrak{M}^{\prime}},\hat{G}_{\mathfrak{M}^{\prime}})$
between $(\varphi,\hat{G})$-modules. The faithfulness of $\mathcal{T}$ follows
the fact that $A\to A[1/E]^{\wedge}_{p}$ induces a fully faithful functor
between Kisin modules over $A$ and étale $\varphi$-modules over
$A[1/E]^{\wedge}_{p}$ from [Kis06, Proposition 2.1.12]. On the other hand,
$\hat{\alpha}$ gives morphisms
$\hat{\alpha}_{1}:\mathfrak{M}\otimes_{A,i_{1}}A^{(2)}\to\mathfrak{M}^{\prime}\otimes_{A,i_{1}}A^{(2)}$
and
$\hat{\alpha}_{2}:\mathfrak{M}\otimes_{A,i_{2}}A^{(2)}\to\mathfrak{M}^{\prime}\otimes_{A,i_{2}}A^{(2)}$.
If we view $A$ and $A^{(2)}$ as subrings of $A_{\mathrm{inf}}$ using diagram
(14), then the following diagram commutes by the fact that
$\hat{\alpha}:\widehat{\mathfrak{M}}\to\widehat{\mathfrak{M}^{\prime}}$ is
compatible with $\tau$-action.

${\mathfrak{M}\otimes_{A,i_{1}}A^{(2)}}$${\mathfrak{M}\otimes_{A,i_{2}}A^{(2)}}$${\mathfrak{M}^{\prime}\otimes_{A,i_{1}}A^{(2)}}$${\mathfrak{M}^{\prime}\otimes_{A,i_{2}}A^{(2)}}$$\scriptstyle{f}$$\scriptstyle{\hat{\alpha}_{1}}$$\scriptstyle{\hat{\alpha}_{2}}$$\scriptstyle{f^{\prime}}$

Thus we produces a morphism between $(\mathfrak{M},f)$ and
$(\mathfrak{M}^{\prime},f^{\prime})$, i.e. $\mathcal{T}$ is also full.

It remains to show the functor $\mathcal{T}$ is essential surjective. Given a
lattice $T$ in a crystalline representation of $G_{K}$, let $\mathfrak{M}$ be
the corresponded Kisin module, it suffices to construct a descent data of
$\mathfrak{M}$ over $A^{(2)}$. We have shown in our proof of Proposition 3.2.2
that if we view $A^{(2)}$ as a subring of $A_{\mathrm{inf}}$ via
$\iota^{(2)}_{\tilde{\tau}}$, then $X_{\tilde{\tau}}$ defines a
$\varphi$-equivariant isomorphism
$f:\mathfrak{M}\otimes_{A,i_{1}}A^{(2)}\simeq\mathfrak{M}\otimes_{A,i_{2}}A^{(2)}$
of $A^{(2)}$-modules. We also show the base change of $f$ along $A^{(2)}\to
B^{(2)}$ is equal to the descent data $c$ of the étale $\varphi$-module
$\mathcal{M}_{A}=\mathfrak{M}\otimes_{A}A[1/E]^{\wedge}_{p}$ that corresponds
to $G_{K}$-action on $T$. In particular,
$c:\mathfrak{M}\otimes_{A,i_{1}}B^{(2)}\simeq\mathfrak{M}\otimes_{A,i_{2}}B^{(2)}$
satisfies the cocycle condition. By Lemma 2.3.2, $A^{(2)}$ (resp. $A^{(3)}$)
injects into $B^{(2)}$ (resp. $B^{(3)}$), so we have $f$ also satisfies the
cocycle condition. In particular, $(\mathfrak{M},f)$ together produce a
primatic $F$-crystals in finite free $\mathcal{O}_{{\mathbbl{\Delta}}}$-module
by Corollary 4.1.9.

######  Remark 4.3.4.

Given an étale $\varphi$-module
$(\mathcal{M}_{A},\varphi_{\mathcal{M}_{A}},c)$ over $A[1/E]^{\wedge}_{p}$
with descent datum $c$, we call
$(\mathcal{M}_{A},\varphi_{\mathcal{M}_{A}},c)$ is _of finite $E$-height_ if
$\mathcal{M}_{A}$ is of finite $E$-height, i.e., if there is a finite free
Kisin module $(\mathfrak{M},\varphi_{\mathfrak{M}})$ of finite height and
defined over $A$ such that
$\mathfrak{M}\otimes_{A}A[1/E]^{\wedge}_{p}\simeq\mathcal{M}_{A}$ as
$\varphi$-modules. Since $(\mathcal{M}_{A},\varphi_{\mathcal{M}_{A}})$ is the
étale $\varphi$-module for $T|_{G_{\infty}}$, our definition of finite
$E$-height is compatible with the one given by Kisin under the equivalence in
(1) of Theorem 4.2.11.

We expect same arguments in the proof of Proposition 3.2.2 will be used to
study representations of finite $E$-height. Similar result has been studied
using the theory of $(\varphi,\tau)$-modules by Caruso. For example, in the
proof of [Car13, Lemma 2.23], Caruso shows for representations of finite
$E$-height, the $\tau$-actions descents to $\mathfrak{S}_{u\text{-np},\tau}$,
which is a subring of $A_{\mathrm{inf}}$ closely related to
$\tilde{\iota}^{(2)}_{\tilde{\tau}}(B^{(2)})\cap A_{\mathrm{inf}}$, where
$\tilde{\tau}$ is a preimage of $\tau$ in $G_{K}$.

######  Remark 4.3.5.

We can also establish the compatibility of our Theorem 4.1.10, the theory of
Kisin and [BS21, Theorem 1.2]. Given a lattice $T$ in a crystalline
representation of $G_{K}$ with non-negative Hodge-Tate weight, and let
$\mathfrak{M}$ be the Kisin module corresponds to $T$ in [Kis06], and let
$\mathfrak{M}_{{{\mathbbl{\Delta}}}}$ (reso.
$\mathfrak{M}^{\prime}_{{{\mathbbl{\Delta}}}}$) be the prismatic $F$-crystal
corresponds to $T^{\vee}$ under [BS21, Theorem 1.2] (resp. $T$ under Theorem
4.1.10). Note that we need to take $T^{\vee}$ since in the work of Bhatt-
Scholze, the equivalence is covariant. By our construction of
$\mathfrak{M}^{\prime}_{{{\mathbbl{\Delta}}}}$, we have
$\mathfrak{M}^{\prime}_{{{\mathbbl{\Delta}}}}((A,(E)))\simeq\mathfrak{M}$. By
[BS21, Remark 7.11],
$\mathfrak{M}_{{{\mathbbl{\Delta}}}}((A,(E)))\simeq\mathfrak{M}$. Next we need
to show the descent data over $A^{(2)}$ constructed respectively are the same.
By Corollary 2.4.5, we just need to show they are the same as descent data of
étale $\varphi$-modules over $A^{(2)}[1/E]^{\wedge}_{p}$, but they are the
same by our $\tau$-evaluation criteria in Lemma 4.2.12.

##  5\. Logarithmic prismatic $F$-crystals and semi-stable representations

In this section, we will propose a possible generalization of Theorem 4.1.10
to semi-stable representations using the absolute logarithmic prismatic site.
The main reference of this subsection is [Kos21]. We will restrict ourselves
to the base ring $R=\mathcal{O}_{K}$, a complete DVR with perfect residue
field. And we give $R$ the log structure associated to the prelog structure
$\alpha:\mathbb N\to R$ such that $\alpha(1)=\varpi$ is a uniformizer in $R$,
i.e., let $D=\\{\varpi=0\\}$, then the log structure on $X=\mathop{\rm
Spf}\nolimits(R)$ is defined by

$M_{X}=M_{D}\hookrightarrow\mathcal{O}_{X}\text{ where
}M_{D}(U):=\\{f\in\mathcal{O}_{X}(U)\,|\,f|_{U\backslash
D}\in\mathcal{O}^{\times}(U\backslash D)\\}.$

Let us introduce the absolute logarithmic site over $(X,M_{X})$.

######  Definition 5.0.1.

[Kos21, Definition 2.2 and Definition 3.3]

  1. (1)

A $\delta_{\log}$-ring is a tuple $(A,\delta,\alpha:M\to A,\delta_{\log}:M\to
A)$, where $(A,\delta)$ is a $\delta$-pair and $\alpha$ is a prelog-structure
on $A$. And $\delta_{\log}$ satisfies:

     * •

$\delta_{\log}(e)=0$,

     * •

$\delta(\alpha(m))=\alpha(m)^{p}\delta_{\log}(m)$,

     * •

$\delta_{\log}(mn)=\delta_{\log}(m)+\delta_{\log}(n)+p\delta_{\log}(m)\delta_{\log}(n)$
for all $m,n\in M$. And we will simply denote it by $(A,M)$ if this is no
confusion. Morphisms are morphisms of $\delta$-pairs that compatible with the
perlog structure and $\delta_{\log}$-stucture.

  2. (2)

A $\delta_{\log}$-triple is $(A,I,M)$ such that $(A,I)$ is a $\delta$-pair and
$(A,M)$ is a $\delta_{\log}$-ring.

  3. (3)

A $\delta_{\log}$-triple $(A,I,M)$ is a prelog prism if $(A,I)$ is a prism,
and it is bounded if $(A,I)$ is bounded.

  4. (4)

A bounded prelog prism is a log prism if it is $(p,I)$-adically log-affine
(cf. [Kos21, Definition 3.3]).

  5. (5)

A bounded (pre)log prism is integral if $M$ is an integral monoid.

  6. (6)

A $\delta_{\log}$-triple $(A,I,M)$ is said to be over $(R,\mathbb N)$ if $A/I$
is an $R$-algebra and there is a map $M\to\mathbb N$ of monoids such that the
following diagram commutes.

${M}$${A}$${\mathbb N}$${R}$${A/I}$

All $\delta_{\log}$-triples over $(R,\mathbb N)$ form a category. Similarly,
we can define the category of prelog prisms over $(R,\mathbb N)$ and the
category of bounded log prisms over $(R,\mathbb N)^{a}$.

######  Remark 5.0.2.

If $A$ is an integral domain, or more general if $\alpha(M)$ consists of non-
zero divisors, then $\delta_{\log}$ is uniquely determined by $\delta$ if
exists. In particular, morphisms between such $\delta_{\log}$-rings are just
morphisms of $\delta$-rings.

######  Remark 5.0.3.

Note that in this paper, for a $\delta$-pair $(A,I)$, we always assume $A$ is
$(p,I)$-adic complete, but in [Kos21], non-$(p,I)$-adic completed
$\delta_{\log}$-triples are also been studied. By Lemma 2.10 of loc.cit., we
can always take the $(p,I)$-adic completions of the $\delta$-pair $(A,I)$ and
the $\delta_{\log}$-structure will be inherited.

######  Proposition 5.0.4.

[Kos21, Corollary 2.15] Given a bounded prelog prism $(A,I,M)$, one can
associate it with a log prism

$(A,I,M)^{a}=(A,I,M^{a})$

######  Remark 5.0.5.

When we deal with log prisms in this paper, we will always take it as the log
prism associated with some prelog prism. And by the above proposition, we know
taking the associated log prism does not change the underlying $\delta$-pair.
Moreover, it is a general fact that $(A,I,M)^{a}$ is integral if $(A,I,M)$ is
a integral.

######  Definition 5.0.6.

The absolute logarithmic prismatic site
$(X,M_{X})_{{{\mathbbl{\Delta}}}_{\log}}$ is the opposite of the category
whose objects are

  1. (1)

bounded log prisms $(A,I,M_{A})$ with integral log structure,

  2. (2)

maps of formal schemes $f_{A}:\mathop{\rm Spf}\nolimits(A/IA)\to X$,

  3. (3)

the map $f_{A}$ satisfies

$(\mathop{\rm Spf}\nolimits(A/IA),f_{A}^{\ast}M_{X})\to(\mathop{\rm
Spf}\nolimits(A),M_{A})^{a}$

defines an exact closed immersion of log formal schemes.

A morphism $(A,I,M_{A})\to(B,I,M_{B})$ is a cover if and only if $A\to B$ is
$(p,I)$-complete faithfully flat and the pullback induces an isomorphism on
log structure. We define the structure sheaf
$\mathcal{O}_{{{\mathbbl{\Delta}}}_{\log}}$ on
$(X,M_{X})_{{{\mathbbl{\Delta}}}_{\log}}$ by $(A,I,M_{A})\mapsto A$.

There is a variant of the about definition that we will also use in this
subsection, we define $(X,M_{X})_{{{\mathbbl{\Delta}}}_{\log}}^{\mathop{\rm
perf}\nolimits}$ be the full subcategory of
$(X,M_{X})_{{{\mathbbl{\Delta}}}_{\log}}$ whose objects are $(A,I,M_{A})$ with
$A$ perfect.

######  Remark 5.0.7.

Our definition of the absolute logarithmic prismatic site is different from
[Kos21, Definition 4.1]. First, we need to consider the absolute prismatic
site, not the relative one. Furthermore, we use the $(p,I)$-complete
faithfully flat topology compared with the $(p,I)$-complete étale topology.
Also we require the log-structures to be integral.

######  Proposition 5.0.8.

$(X,M_{X})_{{{\mathbbl{\Delta}}}_{\log}}$ forms a site.

###### Proof.

Similar to [BS22, Corollary 3.12], we need to show for a given diagram

${(C,I,M_{C})}$${(A,I,M_{A})}$${(B,I,M_{B})}$$\scriptstyle{c}$$\scriptstyle{b}$

in $(X,M_{X})_{{{\mathbbl{\Delta}}}_{\log}}$ such that $b$ is a cover, then
the pushout of $b$ along $c$ is a covering. From the argument in $loc.cit.$,
we known for the underlying prisms, the pushout of $b$ along $c$ is the
$(p,I)$-completed tensor product $D=C\widehat{\otimes}_{A}B$, and $(D,I)$ is a
bounded prism covers $(C,I)$ in the $(p,I)$-complete faithful flat topology.
And we give $D$ the log structure $M_{D}$ defined by viewing $\mathop{\rm
Spf}\nolimits(D)$ as the fiber product via [Ogu18, Proposition 2.1.2], then
$(C,M_{C})\to(D,M_{D})$ is strict morphism by Remark 2.1.3 of $loc.cit.$, so
in particular, $M_{D}$ is integral since $M_{C}$ is. For the same reason,

$(\mathop{\rm Spf}\nolimits(D/ID),f_{D}^{\ast}M_{X})\to(\mathop{\rm
Spf}\nolimits(D),M_{D})^{a}$

is strict since it is the base change of a strict morphism. It is an exact
closed immersion since pushout of a surjective map of monoids is again
surjective. ∎

######  Example 5.0.9.

[Kos21, Example 3.4]

  1. (1)

Let $(A,(E))$ be the Breuil-Kisin prism, then we can define a perlog structure
to $(A,(E))$ given by $\mathbb N\to A;n\mapsto u^{n}$, one have
$(A,(E),\mathbb N)^{a}$ is in $(X,M_{X})_{{{\mathbbl{\Delta}}}_{\log}}$, where
(3) in Definition 5.0.6 follows from the prelog structures $\mathbb N\to R\to
A/(E)$ and $\mathbb N\to A\to A/(E)$ induce the same log structure.

  2. (2)

For any prism $(B,J)$ over $(A,(E))$, it has a natural prelog structure
$\mathbb N\to A\to B$, and similar to $(1)$, $(B,J,\mathbb N)^{a}$ is in
$(X,M_{X})_{{{\mathbbl{\Delta}}}_{\log}}$.

  3. (3)

A special case of (2) is that $(B,J)=(A_{\mathop{\rm perf}\nolimits},(E))$,
the perfection of $(A,(E))$. One has the prelog structure in (2) can be
directly defined as $1\mapsto[\varpi^{\flat}]$. And $(A,(E),\mathbb
N)^{a}\to(B,J,\mathbb N)^{a}$ is a covering of log prisms in
$(X,M_{X})_{{{\mathbbl{\Delta}}}_{\log}}$.

Actually, all logarithmic structures of log prisms in
$(X,M_{X})_{{{\mathbbl{\Delta}}}_{\log}}$ is the log structure associated to a
prelog structure defined by $\mathbb N$. We thank Teruhisa Koshikawa for
letting us know the following lemma.

######  Lemma 5.0.10.

For any log prism $(B,J,M_{B})$ inside
$(X,M_{X})_{{{\mathbbl{\Delta}}}_{\log}}$, $(B,M_{B})^{a}$ admits a chart
$\mathbb N\to B$ defined by $n\mapsto u_{B}^{n}$ for some $u_{B}\in B$
satisfying $u_{B}\equiv\varpi\mod J$.

###### Proof.

For any log prism $(B,J,M_{B})$ inside
$(X,M_{X})_{{{\mathbbl{\Delta}}}_{\log}}$, we have

$(\mathop{\rm Spf}\nolimits(B/J),f_{B}^{\ast}M_{X})\to(\mathop{\rm
Spf}\nolimits(B),M_{B})^{a}$

defines an exact closed immersion of log formal schemes. So by the proof of
[Kos21, Proposition 3.7], if we let $N^{a}_{B/J}:=\Gamma(\mathop{\rm
Spf}\nolimits(B/J),\underline{\mathbb N}^{a})$ for the prelog structure
$\mathbb N\to\mathcal{O}_{K}\to B/J$ induced from the given prelog structure
on $\mathcal{O}_{K}$, then the fiber product $M_{B}\times_{N^{a}_{B/J}}\mathbb
N$ is a chart for $(B,M_{B})^{a}$. Moreover, since we assume $M_{B}$ to be
integral, we have $(\mathop{\rm
Spf}\nolimits(B/J),f_{B}^{\ast}M_{X})\to(\mathop{\rm
Spf}\nolimits(B),M_{B})^{a}$ is a log thickening with ideal $J$ in the sense
of [Ogu18, Definition 2.1.1.], and one can show
$M_{B}\times_{N^{a}_{B/J}}\mathbb N\simeq\mathbb N\times(1+J)$. Now
$(1+J)^{\times}=(1+J)$, so

$\mathbb N\to\mathbb N\times(1+J)\simeq M_{B}\times_{N^{a}_{B/J}}\mathbb N\to
B$

is also a chart for $(B,M_{B})^{a}$. And the prelog structure given by
$n\mapsto u_{B}^{n}$ for some $u_{B}\in B$ satisfying the image of $u_{B}$ in
$B/J$ coincides with the image of $\varpi$ under $\mathcal{O}_{K}\to B/J$. ∎

In the rest of this subsection, we will try to generalize results we proved in
§4.1-§4.3 for the logarithmic prismatic site.

######  Lemma 5.0.11.

  1. (1)

For
$(A,I_{A},M_{A})^{a},(B,I_{B},M_{B})^{a}\in(X,M_{X})_{{{\mathbbl{\Delta}}}_{\log}}$
such that $M_{A},M_{B}$ are integral and $(A,M_{A})\to(A/I_{A},\mathbb N)$ and
$(B,M_{B})\to(B/I_{B},\mathbb N)$ are exact surjective, there is a prelog
prism $(C,I_{C},M_{C})$ with integral log structure that is universal in the
sense that the diagram

${(A,I_{A},M_{A})}$${(C,I_{C},M_{C})}$${(B,I_{B},M_{B})}$

is initial in the category of diagrams

${(A,I_{A},M_{A})}$${(D,I_{D},M_{D})}$${(B,I_{B},M_{B})}$

of prelog prisms over $(R,\mathbb N)$, and $(D,M_{D})\to(D/I_{D},\mathbb N)$
is an exact surjective.

  2. (2)

If $(C,I_{C})$ in (1) is bounded, then $(C,I_{C},M_{C})^{a}$ is the product of
$(A,I_{A},M_{A})^{a}$ and $(B,I_{B},M_{B})^{a}$ inside
$(X,M_{X})_{{{\mathbbl{\Delta}}}_{\log}}$.

  3. (3)

If $(A,I_{A},M_{A})^{a},(B,I_{B},M_{B})^{a}$ in (1) are in
$(X,M_{X})_{{{\mathbbl{\Delta}}}_{\log}}^{\mathop{\rm perf}\nolimits}$, and
let $(C_{\mathop{\rm perf}\nolimits},I_{C})$ be the perfection of $(C,I_{C})$
defined in (1). Let $(C_{\mathop{\rm perf}\nolimits},I_{C},M_{C})$ be the
prelog prism with prelog structure induced from $C$. Then $(C_{\mathop{\rm
perf}\nolimits},I_{C},M_{C})^{a}$ is the product of $(A,I_{A},M_{A})^{a}$ and
$(B,I_{B},M_{B})^{a}$ in $(X,M_{X})_{{{\mathbbl{\Delta}}}_{\log}}^{\mathop{\rm
perf}\nolimits}$.

###### Proof.

Let
$(A,I_{A},M_{A}),(B,I_{B},M_{B})\in(X,M_{X})_{{{\mathbbl{\Delta}}}_{\log}}$,
define $C_{0}$ to be the $(p,I_{A},I_{B})$-adic completion of
$A\otimes_{W(k)}B$ and let $J$ be the kernel of

$C_{0}\to A/I_{A}\widehat{\otimes}_{R}B/I_{B}.$

Then $(C_{0},J,M_{A}\times M_{B})$ is a $\delta_{\log}$-triple over
$(A,I_{A},M_{A})$. And we have $(C_{0},J,M_{A}\times M_{B})\to(C_{0}/J,\mathbb
N)$ is surjective. Then we can apply [Kos21, Proposition 3.6] to get a
universal prelog prism $(C,I_{C},M_{C})$ over $(A,I_{A},M_{A})$ and
$(B,I_{B},M_{B})$ and satisfies $(C,M_{C})\to(C/J,\mathbb N)$ is exact
surjective. Just recall in the proof of [Kos21, Proposition 3.6], we first
construct a $\delta_{\log}$-triple $(C^{\prime},J^{\prime},M_{C}^{\prime})$
which is universal in the sense that it is a $\delta_{\log}$-triple over both
$(A,I_{A},M_{A})$ and $(B,I_{B},M_{B})$ satisfying $C^{\prime}/J^{\prime}$ is
over $A/I_{A}$ and $B/I_{B}$ as $R$-algebra and
$(C^{\prime},M_{C}^{\prime})\to(C^{\prime}/J^{\prime},\mathbb N)$ is exact
surjective. Then we take the prismatic envelope with respect to
$(A,I_{A})\to(C^{\prime},J^{\prime})$ to get $(C,I_{C})$. Then we can check
such $(C,I_{C},M_{C})$ satisfies the universal property. For (2), when
$(C,I_{C})$ is bounded, the fact that $(C,I_{C},M_{C})^{a}$ is the product of
$(A,I_{A},M_{A})^{a}$ and $(B,I_{B},M_{B})^{a}$ inside
$(X,M_{X})_{{{\mathbbl{\Delta}}}_{\log}}$ follows from Proposition 3.7 of
$loc.cit.$. For (3), we have $(C_{\mathop{\rm perf}\nolimits},I_{C})$ is
automatic bounded, and one can check $(C_{\mathop{\rm perf}\nolimits},I_{C})$
is universal using exactly the same proof of Proposition 3.7 of $loc.cit.$. ∎

We thank Koji Shimizu for the following lemma on $A^{(2)}_{\mathop{\rm
st}\nolimits}$.

######  Lemma 5.0.12.

Let $(A,I,\mathbb N)^{a}$ be the Breuil-Kisin prism defined in $(1)$ of
Example 5.0.9, then the self-product (resp. self-triple product) of
$(A,I,\mathbb N)^{a}$ in $(X,M_{X})_{{{\mathbbl{\Delta}}}_{\log}}$ exist.
Moreover, if we let $(A^{\langle 2\rangle},I,M^{2})^{a}$ (resp. $(A^{\langle
3\rangle},I,M^{3})^{a}$) be self-product (resp. self-triple product) of
$(A,I,\mathbb N)^{a}$, then $A^{\langle i\rangle}\simeq A^{(i)}_{\mathop{\rm
st}\nolimits}$ for $i=2,3$.

###### Proof.

By our construction in Lemma 5.0.11, $(A^{\langle 2\rangle},I,M)$ is the
prelog prismatic envelope $(C,I_{C},M_{C})$ with respect to

$(A,(E),\mathbb N)\to(C_{0},J,\mathbb N^{2})\text{ and }(C_{0}/J,\mathbb
N^{2})\to(R,\mathbb N)$

where $C_{0}=W[\\![u,v]\\!]$, $J=(E(u),u-v)$ with the prelog structure given
by $\beta:(1,0)\mapsto u,(0,1)\mapsto v$. The prelog prismatic envelope is
constructed using the technique of exactification: consider
$\pi:(C_{0},\mathbb N^{2})\to(R=C/J,\mathbb N)$ where the map between log
structures is given by $\pi_{\log}:\mathbb N\times\mathbb N\to\mathbb
N;(m,n)\mapsto m+n$, here $\pi_{\log}$ is surjective but not exact, so to
constructsthe exactification of $\pi:(C,\mathbb N^{2})\to(R,\mathbb N)$ (cf.
[Kos21, Construction 2.18]), first we have the exactification of $\pi_{\log}$
is

$\alpha:M^{2}\to\mathbb N\quad\text{ given by }\quad(m,n)\mapsto m+n,$

where $M^{2}=\\{(m,n)\in\mathbb Z\times\mathbb Z\,|\,m+n\in\mathbb N\\}$.
Since $M^{2}$ is generated by $(-1,1)$, $(1,-1)$, $(0,1)$ and $(1,0)$, one has
the exactification of $\pi$ is

$\Big{(}W(k)[\\![u,v]\\!]\big{[}\frac{v}{u},\frac{u}{v}\big{]}^{\wedge}_{(p,J^{\prime})},J^{\prime},M^{2};\alpha:(1,0)\mapsto{u},(0,1)\mapsto
v,(1,-1)\mapsto\frac{u}{v},(-1,1)\mapsto\frac{v}{u}\Big{)}$

where $J^{\prime}:=\mathop{\rm
ker}\nolimits(W(k)[\\![u,v]\\!]\big{[}\frac{v}{u},\frac{u}{v}\big{]}\to R)$.

We have the $(p,J^{\prime})$-adic completion of
$W(k)[\\![u,v]\\!]\big{[}\frac{v}{u},\frac{u}{v}\big{]}$ is
$W(k)[\\![u,\frac{v}{u}-1]\\!]$. Then take prismatic envelope of
$(A,(E))\to(W(k)[\\![u,\frac{v}{u}-1]\\!],(E,\frac{v}{u}-1)).$ One can check

$W(k)[\\![u,\frac{v}{u}-1]\\!]\big{\\{}\frac{v/u-1}{E(u)}\big{\\}}^{\wedge}_{\delta}\simeq
A_{\mathop{\rm st}\nolimits}^{(2)}$

directly from the definition of $A_{\mathop{\rm st}\nolimits}^{(2)}$.

Similarly, we can show $A^{\langle 3\rangle}\simeq A^{(3)}_{\mathop{\rm
st}\nolimits}$ which is also bounded. ∎

The following is one of our key observations.

######  Lemma 5.0.13.

We have $(A^{\langle 2\rangle})_{\mathop{\rm
perf}\nolimits}\simeq(A^{(2)})_{\mathop{\rm perf}\nolimits}$.

###### Proof.

Let $u_{1},u_{2}$ be the image of $u$ under the two natural maps
$i_{j}:A_{\mathop{\rm perf}\nolimits}\to(A^{(2)})_{\mathop{\rm
perf}\nolimits}$ for $j=1,2$. We claim that $u_{2}/u_{1}$ is inside
$(A^{(2)})_{\mathop{\rm perf}\nolimits}$.

Firstly, we have already shown $A_{\mathop{\rm perf}\nolimits}\simeq
W(\widehat{\mathcal{O}}_{K_{\infty}}^{\flat})$ and $u=[\varpi^{\flat}]$, here
$\varpi^{\flat}=(\varpi_{n})$ with $\\{\varpi_{n}\\}_{n\geq 0}$ being a
compatible system of $p^{n}$-th roots of $\varpi$ inside
$\mathcal{O}_{\widehat{K}_{\infty}}$, and
$(\varpi_{n})\in\mathcal{O}_{\widehat{K}_{\infty}}^{\flat}$ via the
identification $\mathcal{O}_{\widehat{K}_{\infty}}^{\flat}\simeq\lim_{x\mapsto
x^{p}}\mathcal{O}_{\widehat{K}_{\infty}}$. Let $S=(A^{(2)})_{\mathop{\rm
perf}\nolimits}/(E)$, this is an integral perfectoid ring over
$\mathcal{O}_{K}$ in the sense of [BMS18]. We have
$S^{\flat}\simeq(A^{(2)})_{\mathop{\rm perf}\nolimits}/(p)$. For $j=1,2$,
define $\varpi_{j}^{\flat}=u_{j}\mod(p)\in S^{\flat}$, then we have
$u_{j}=[\varpi_{j}^{\flat}]$ for $j=1,2$.

Recall in § 2.1, we have $z=\frac{y-x}{E(x)}$ in $A^{(2)}$. Since $E(x)\equiv
x^{e}\mod p$, we have $x(1+x^{e-1}z)\equiv y\mod p$. If we denote
$\iota:A^{(2)}\to(A^{(2)})_{\mathop{\rm perf}\nolimits}$ the natural map, then
$\iota(x)=u_{1}$ and $\iota(y)=u_{2}$ in our definition, and
$u_{1}(1+u_{1}^{e-1}\iota(z))\equiv u_{2}\mod p$ inside
$S^{\flat}=A^{(2)}_{\mathop{\rm perf}\nolimits}/(p)$. This is the same as
$\varpi_{1}^{\flat}\mu=\varpi_{2}^{\flat}$ with
$\mu=(1+u_{1}^{e-1}\iota(z))\mod p$ in $S^{\flat}$. So we have
$[\mu]u_{1}=[\mu][\varpi_{1}^{\flat}]=[\varpi_{2}^{\flat}]=u_{2}$, which
proves our claim.

Now by symmetry, $u_{1}/u_{2}$ is also inside $(A^{(2)})_{\mathop{\rm
perf}\nolimits}$, so $u_{1}/u_{2}$ is a unit in $(A^{(2)})_{\mathop{\rm
perf}\nolimits}$. So we can give $(A^{(2)})_{\mathop{\rm perf}\nolimits}$ a
prelog structure

$\alpha:M^{2}\to(A^{(2)})_{\mathop{\rm perf}\nolimits}\text{ with
}(1,-1)\mapsto\frac{u_{1}}{u_{2}},(-1,1)\mapsto\frac{u_{2}}{u_{1}},(1,0)\mapsto{u_{1}},(0,1)\mapsto{u_{2}}$

with the monoid $M^{2}$ defined as in the proof of Lemma 5.0.12, then
$((A^{(2)})_{\mathop{\rm perf}\nolimits},(E),M^{2})^{a}$ is in
$X_{{{\mathbbl{\Delta}}}_{\log}}^{\mathop{\rm perf}\nolimits}$.

One can check the maps
$i_{1},i_{2}:(A,(E))\to(A^{(2)},(E))\to((A^{(2)})_{\mathop{\rm
perf}\nolimits},(E))$ induce $i_{1},i_{2}:(A_{\mathop{\rm
perf}\nolimits},(E),\mathbb N)\to((A^{(2)})_{\mathop{\rm
perf}\nolimits},(E),M^{2})$ of prelog prisms. So by Lemma 5.0.12, there is a
unique map $(A^{\langle 2\rangle},I,M^{2})\to((A^{(2)})_{\mathop{\rm
perf}\nolimits},(E),M^{2})$, which factors through $((A^{\langle
2\rangle})_{\mathop{\rm perf}\nolimits},(E),M^{2})$. So it induces a map
$((A^{\langle 2\rangle})_{\mathop{\rm
perf}\nolimits},(E),M^{2})\to((A^{(2)})_{\mathop{\rm
perf}\nolimits},(E),M^{2})$ inside
$X_{{{\mathbbl{\Delta}}}_{\log}}^{\mathop{\rm perf}\nolimits}$. On the other
hand, by the universal property of $A^{(2)}$, we know there is a map
$(A^{(2)})_{\mathop{\rm perf}\nolimits}\to(A^{\langle 2\rangle})_{\mathop{\rm
perf}\nolimits}$ fits into the coproduct diagram in
$X_{{{\mathbbl{\Delta}}}}^{\mathop{\rm perf}\nolimits}$, which is the full
subcategory of $X_{{\mathbbl{\Delta}}}$ containing perfect prisms.

One can check the composition $\eta:((A^{(2)})_{\mathop{\rm
perf}\nolimits},(E))\to((A^{\langle 2\rangle})_{\mathop{\rm
perf}\nolimits},(E))\to((A^{(2)})_{\mathop{\rm perf}\nolimits},(E))$ satisfies
$\eta\circ i_{j}=i_{j}\circ\eta$ for $i_{1},i_{2}:(A_{\mathop{\rm
perf}\nolimits},(E))\to((A^{(2)})_{\mathop{\rm perf}\nolimits},(E))$. Such a
map is unique inside $X_{{{\mathbbl{\Delta}}}}^{\mathop{\rm perf}\nolimits}$,
so $\eta=\mathop{\rm id}\nolimits_{((A^{(2)})_{\mathop{\rm
perf}\nolimits},(E))}$.

On the other hand, the composition

$\eta^{\prime}:((A^{\langle 2\rangle})_{\mathop{\rm
perf}\nolimits},(E),M^{2})^{a}\to((A^{(2)})_{\mathop{\rm
perf}\nolimits},(E),M^{2})^{a}\to((A^{\langle 2\rangle})_{\mathop{\rm
perf}\nolimits},(E),M^{2})^{a}$

satisfies $\eta\circ i^{\prime}_{j}=i^{\prime}_{j}\circ\eta$ for
$i^{\prime}_{1},i^{\prime}_{2}:(A_{\mathop{\rm perf}\nolimits},(E),\mathbb
N)^{a}\to((A^{\langle 2\rangle})_{\mathop{\rm perf}\nolimits},(E),M^{2})^{a}$
induced from $i^{\prime}_{1},i^{\prime}_{2}:(A,(E),\mathbb N)\to(A^{\langle
2\rangle},(E),M^{2})$. Such map is also unique inside
$X_{{{\mathbbl{\Delta}}}_{\log}}^{\mathop{\rm perf}\nolimits}$, so
$\eta^{\prime}=\mathop{\rm id}\nolimits_{((A^{\langle 2\rangle})_{\mathop{\rm
perf}\nolimits},(E),M^{2})^{a}}$. So in particular we have $(A^{\langle
2\rangle})_{\mathop{\rm perf}\nolimits}\simeq(A^{(2)})_{\mathop{\rm
perf}\nolimits}$. ∎

######  Theorem 5.0.14.

The category of étale $\varphi$-module over $A[1/E]^{\wedge}_{p}$ with a
descent data over $A_{\mathop{\rm st}\nolimits}^{(2)}[1/E]^{\wedge}_{p}$ is
equivalent to the category of lattice in representations of $G_{K}$. Moreover,
for all $\gamma\in\hat{G}$, we can define the evaluation map

$e_{\gamma}:A_{\mathop{\rm st}\nolimits}^{(2)}[1/E]^{\wedge}_{p}\to
W(\hat{L}^{\flat})$

such that Lemma 4.2.12 is still valid. Moreover, the $\mathbb Q$-isogeney
version of this theorem also holds.

######  Remark 5.0.15.

The above theorem should be related to the étale comparison theorem in the log
prismatic settings, which has not been studied in [Kos21] yet.

Moreover, we have a log version of Lemma 4.1.8 also holds. We thank Teruhisa
Koshikawa for hints of the following result.

######  Proposition 5.0.16.

The sheaf represented by $(A,(E),\mathbb N)^{a}$ covers the final object
$\ast$ in in $\mathop{\rm
Shv}\nolimits((X,M_{X})_{{{\mathbbl{\Delta}}}_{\log}})$.

###### Proof.

For any log prism $(B,J,M_{B})$, by Lemma 5.0.10, we can assume
$(B,J,M_{B})^{a}=(B,J,\mathbb N)^{a}$, with prelog structure defined by
$n\mapsto u_{B}^{n}$ with $u_{B}\equiv\varpi\mod J$.

Using deformation theory, we have there is a unique $W(k)$-algebra structure
for $B$, and we define
$C=B[\\![u]\\!][\frac{u_{B}}{u},\frac{u}{u_{B}}]\\{\frac{u_{B}/u-1}{J}\\}^{\wedge}_{\delta}$,
where the completion is taken for the $(p,J)$-adic topology. Similar to the
proof of Lemma 5.0.12, we have $(C,JC,\mathbb N)^{a}$ is the product of
$(A,(E),\mathbb N)^{a}$ and $(B,J,\mathbb N)^{a}$ inside
$(X,M_{X})_{{{\mathbbl{\Delta}}}_{\log}}$. Moreover, we have $B\to C$ is
$(p,J)$-complete flat by [BS22, Proposition 3.13]. It remains to show that
$(B,J)\to(C,J)$ is a covering, i.e., $B\to C$ is $(p,J)$-complete faithfully
flat. Let

$C^{nc}:=B[\\![u]\\!][\frac{u_{B}}{u},\frac{u}{u_{B}}]\\{\frac{u_{B}/u-1}{J}\\}_{\delta}$

be the non-complete version of $C$ that we have the $(p,J)$-adic completion of
$C^{nc}$ is $C$. Now we just need to show the flat ring map $B/(p,J)\to
C/(p,J)=C^{nc}/(p,J)$ is also faithful.

We claim that $C/(p,J)$ is free over $B/(p,J)$. One has $JC=E(u)C$, and
$(p,J)=(p,E)=(p,J,E)$ in $C$. So $C/(p,J)=C^{nc}/(p,J)$ is equal to

$B[\\![u]\\!][\frac{u_{B}}{u},\frac{u}{u_{B}}][\delta^{i}(z),i\geq
0]/\left(p,J,E,Ez=\frac{u_{B}}{u}-1,\delta^{i}(\frac{u_{B}}{u}-1))=\delta^{i}(Ez),i\geq
1\right).$

After modulo $(p,J)$, the above is the direct limit of

$B/(p,J)[\delta^{i}(z)]/\left(\delta^{i}(\frac{u_{B}}{u}-1))=\delta^{i}(Ez)\mod(p,E,J)\right)$

for $i\geq 0$.

Now we use Lemma 2.2.4 to compute
$\delta^{i}(\frac{u_{B}}{u}-1)=\delta^{i}(Ez)\mod(p,E,J)$. We keep the
notations in Lemma 2.2.4, by induction, we have $b_{n}=0\mod(p,E)$. Using that
$a_{p}^{(j)}\in A_{0}^{\times}$,
$\delta^{i}(\frac{u_{B}}{u}-1)=\delta^{i}(Ez)\mod(p,E,J)$ gives a relation
$(z_{i-1})^{p}=\sum\limits_{j=0}^{p-1}\tilde{a}_{j}^{(i)}(z_{i-1})^{j}$ where
$z_{i}=\mathfrak{z}_{i}\mod(p,J,E)$ and $\tilde{a}_{j}^{(i)}\in
B/(p,J)[z_{0},z_{1},\dots,z_{i-2}]$. In summary, we have

$C/(p,J)=B/(p,J)[z_{i},i\geq
0]\Bigg{/}\left((z_{i})^{p}-\sum\limits_{j=0}^{p-1}\tilde{a}_{j}^{(i)}(z_{i})^{j},i\geq
1\right)$

which is free over $B/(p,J)$. ∎

######  Definition 5.0.17.

  1. (1)

A prismatic crystal over $(X,M_{X})_{{{\mathbbl{\Delta}}}_{\log}}$ in finite
locally free $\mathcal{O}_{{{\mathbbl{\Delta}}}_{\log}}$-modules is a finite
locally free $\mathcal{O}_{{{\mathbbl{\Delta}}}_{\log}}$-module
$\mathfrak{M}_{{{\mathbbl{\Delta}}}}$ such that for all morphisms
$f:(A,I,M_{A})\to(B,J,M_{B})$ of log prisms, it induces an isomorphism:

$f^{\ast}\mathfrak{M}_{{{\mathbbl{\Delta}}},A}:=\mathfrak{M}_{{{\mathbbl{\Delta}}}}((A,I,M_{A}))\otimes_{A}B\simeq\mathfrak{M}_{{{\mathbbl{\Delta}}},B}:=\mathfrak{M}_{{{\mathbbl{\Delta}}}}((B,J,M_{B}))$

  2. (2)

A prismatic $F$-crystal over $(X,M_{X})_{{{\mathbbl{\Delta}}}_{\log}}$ of
height $h$ (in finite locally free
$\mathcal{O}_{{{\mathbbl{\Delta}}}_{\log}}$-modules) is a prismatic crystal
$\mathfrak{M}_{{{\mathbbl{\Delta}}}}$ in finite locally free
$\mathcal{O}_{{{\mathbbl{\Delta}}}_{\log}}$-modules together with a
$\varphi_{\mathcal{O}_{{{\mathbbl{\Delta}}}_{\log}}}$-semilinear endomorphism
$\varphi_{\mathfrak{M}_{{{\mathbbl{\Delta}}}}}$ of the
$\mathcal{O}_{{{\mathbbl{\Delta}}}_{\log}}$-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 $\mathcal{I}_{{\mathbbl{\Delta}}}^{h}$.

In particular, with help of Theorem 5.0.14 and Proposition 5.0.16, a direct
translation of proofs in §4.3 with $A^{(2)}$ replaced by $A^{(2)}_{\mathop{\rm
st}\nolimits}$ shows the following theorem.

######  Theorem 5.0.18.

The category of prismatic $F$-crystals over
$(X,M_{X})_{{{\mathbbl{\Delta}}}_{\log}}$ of height $h$ is equivalent to the
category of lattices in semi-stable representations of $G_{K}$ with Hodge-Tate
weights between $0$ and $h$.

##  6\. Some discussions on base rings

In this section, we show that our base ring assumed at the beginning of §2
covers many situations of base rings used in [Kim14] and [Bri08].

Let $K$ be complete DVR with perfect residue field $k$, and let
$K_{0}=W[\frac{1}{p}]$ with $W=W(k)$, fix a uniformizer
$\varpi\in\mathcal{O}_{K}$ and $E(u)\in W[u]$ a minimal polynomial of $\varpi$
over $K_{0}$. Let $R$ be a normal domain and satisfies that $R$ is a
$p$-complete flat $\mathcal{O}_{K}$-algebra that is complete with respect to
$J$-adic topology, for an ideal $J=(\varpi,{t_{1}},\ldots,{t_{d}})$ of $R$
containing $\varpi$. We also assume $\overline{R}=R/(\varpi)$ is a finite
generated $k$-algebra with _finite $p$-basis_ discussed in [dJ95, §1.1].

######  Lemma 6.0.1 ([Kim14] Lemma 2.3.1 and lemma 2.3.4).

  1. (1)

In the above setting, there is a $p$-adic formally smooth flat $W$-algebra
$R_{0}$ equipped with a Frobenius lift $\varphi_{0}$ such that
$\overline{R}:=R_{0}/(p)$. Moreover let $J_{0}$ be the preimage of
$\overline{J}$ inside $R_{0}$, then $R_{0}$ is $J_{0}$-adically complete, and
under this topology, $R_{0}$ is formally smooth.

  2. (2)

$R_{0}/(p)\xrightarrow{\sim}R/(\varpi)$ lifts to a $W$-algebra morphism
$R_{0}\to R$ and the induced $\mathcal{O}_{K}$-algebra morphism
$\mathcal{O}_{K}\otimes_{W}R_{0}\to R$ is an isomorphism. Moreover this
isomorphism is continuous with respect to the $J_{0}$-adic topology.

Let $(R_{0},\varphi_{R_{0}})$ denote a flat $W$-lift of $R/(\varpi)$ obtained
from the above lemma. And we will have $J_{0}=(p,t_{1},\ldots,t_{d})\in
R_{0}$, and we write
$\overline{J}=(\overline{t_{1}},\ldots,\overline{t_{d}})\subset\overline{R}$.

######  Definition 6.0.2.

Let $R_{0}$ be a $p$-complete $\mathbb Z_{p}$-algebra, we say $R_{0}$
satisfies the “refined almost étalenes” assumption, or simply RAE assumption,
if $\hat{\Omega}_{R_{0}}=\oplus_{i=1}^{m}R_{0}dT_{i}$ with $T_{i}\in
R_{0}^{\times}$. Where $\hat{\Omega}_{R_{0}}$ is the module of of $p$-adically
continuous Kähler differentials.

The following are examples of $R_{0}$ and $R$ which satisfy assumptions of
Lemma 6.0.1 and RAE assumption.

######  Example 6.0.3.

  1. (1)

If $R/(\varpi)$ is a completed noetherian regular local ring with residue
field $k$, then Cohen structure theorem implies
$R/(\varpi)=k[\\![\overline{x_{1}},\ldots,\overline{x_{d}}]\\!]$. In this
case, $R_{0}=W[\\![x_{1},\ldots,x_{d}]\\!]$ and
$J_{0}=(p,x_{1},\ldots,x_{d})$. Then $R=W[\\![x_{1},\ldots,x_{d}]\\!][u]/E$,
with $E\in W[u]$ is a Eisenstein polynomial.

  2. (2)

Let $R_{0}=W(k)\langle t_{1}^{\pm 1},\dots,t_{m}^{\pm 1}\rangle$ and
$J_{0}=(p)$, in this example, $\overline{R}=k[\overline{t}_{1}^{\pm
1},\dots,\overline{t}_{m}^{\pm 1}]$ is not local.

  3. (3)

An unramified complete DVR $(R_{0},p)$ with residue field $k$ so that
$[k:k^{p}]<\infty$.

  4. (4)

Note the the Frobenius liftings in Lemma 6.0.1 is not unique. In (2) we can
choose $\varphi_{R_{0}}(t_{i})=t_{i}^{p}$. In (1), we can choose the
$\varphi_{R_{0}}(x_{i})=x_{i}^{p}$ or
$\varphi_{R_{0}}(x_{i})=(x_{i}+1)^{p}-1$.

Let $R_{0}$ be $p$-complete algebra which satisfies the RAE assumption, Set
$\breve{R}_{0}=W\langle t_{1},\dots,t_{m}\rangle$ and $f:\breve{R}_{0}\to
R_{0}$ by sending $t_{i}$ to $T_{i}$.

######  Proposition 6.0.4.

Assume that $R_{0}$ is a $p$-complete integral domain which admits finite
$p$-basis and satisfies RAE assumption. Then $f$ is formally étale
$p$-adically.

###### Proof.

We thanks for Wansu Kim providing the following proof. By standard technique
using [Ill71, Ch.III, Corollaire 2.1.3.3] (e.g., see the proof in [Kim14, Lem.
2.3.1]), it suffices to show that the cotangent complex $\mathbb
L_{R_{0}/\breve{R}_{0}}$ is acyclic. Since both $R_{0}$ and $\breve{R}_{0}$
are $\mathbb Z_{p}$-flat, it suffice to show that $\mathbb
L_{R_{1}/\breve{R}_{1}}$ is acyclic where $R_{1}=R_{0}/pR_{0}$ and
$\breve{R}_{1}=\breve{R}_{0}/p\breve{R}_{0}$. Since $R_{0}$ has finite
$p$-basis, by [dJ95, Lem. 1.1.2], $\mathbb L_{R_{1}/k}\simeq\Omega_{R_{1}/k}$.
Note that maps $k\to\breve{R}_{1}\to R_{1}$ induces a fiber sequence

$\mathbb L_{\breve{R}_{1}/k}\otimes^{\mathbb L}_{\breve{R}_{1}}R_{1}\to\mathbb
L_{R_{1}/k}\to\mathbb L_{R_{1}/\breve{R}_{1}}$

Since that $\mathbb L_{\breve{R}_{1}/k}\simeq\Omega_{\breve{R}_{1}/k}$ and
$\Omega_{\breve{R}_{1}/k}\simeq\Omega_{R_{1}/k}$ by RAE condition, we conclude
that $\mathbb L_{R_{1}/\breve{R}_{1}}=0$ as required. ∎

Let us end with a discussion about our base rings and the base rings used in
[Bri08]. As explained in the beginning of [Bri08, Chap. 2], his base ring
$R_{0}$ in [Bri08] is obtained from $W\langle t_{1}^{\pm 1},\ldots,t_{m}^{\pm
1}\rangle$ by a finite number of iterations of certain operations and is also
assumed to satisfy certain properties. By Prop. 2.0.2 _loc. cit._ , we see
that $R_{0}$ has finite $p$-basis and satisfies RAE assumption. So the base
ring $R_{0}$ in [Bri08] also satisfies the requirement that $f:W\langle
t_{1},\ldots,t_{m}\rangle\to R_{0}$ is formally étale by Proposition 6.0.4.

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# Incorporating Annotator Uncertainty into Representations of Discourse
Relations

S. Magalí López Cortez Cassandra L. Jacobs
Department of Linguistics
University at Buffalo
Buffalo, NY, USA
<EMAIL_ADDRESS>

###### Abstract

Annotation of discourse relations is a known difficult task, especially for
non-expert annotators. In this paper, we investigate novice annotators’
uncertainty on the annotation of discourse relations on spoken conversational
data. We find that dialogue context (single turn, pair of turns within
speaker, and pair of turns across speakers) is a significant predictor of
confidence scores. We compute distributed representations of discourse
relations from co-occurrence statistics that incorporate information about
confidence scores and dialogue context. We perform a hierarchical clustering
analysis using these representations and show that weighting discourse
relation representations with information about confidence and dialogue
context coherently models our annotators’ uncertainty about discourse relation
labels.

##  1 Introduction

Discourse relations (DRs) are those relations such as Elaboration,
Explanation, Narration, which hold between discourse units. The task of
labeling DRs is known to pose difficulties for annotators (Spooren and Degand,
2010), as sometimes more than one interpretation may be possible (Scholman et
al., 2022; Webber, 2013).

Recent studies have shown that allowing for multiple labels in annotation can
improve the performance of discourse parsers (Yung et al., 2022). Scholman et
al. (2022) test different label aggregation methods in a crowdsourced corpus
annotated by 10 workers and find that probability distributions over labels
better capture ambiguous interpretations of discourse relations than majority
class labels. (1) shows an example from their corpus, where the relation
between the second and third sentences (in italics and bold, respectively),
was interpreted as Conjunction by four annotators and Result by five
annotators.

It is logical that our attention is focused on cities. Cities are home to 80%
of the 500 million or so inhabitants of the EU. It is in cities that the great
majority of jobs, companies and centres of education are located. (adapted
from DiscoGeM, Europarl genre; Scholman et al., 2022, italics and bolding are
ours.)

Annotating the discourse relation between these two sentences with both
Conjunction and Result captures different possible interpretations of the
relation between these segments. For example, the two sentences may contain
two conjoined facts about cities, but can also be perceived as describing a
causal relation between the first and second sentence (i.e., as cities are
home to the largest part of the population, most jobs, companies and
educational institutions are located there).

In this work, we investigate which relations are distributionally similar or
co-occurring in multilabel annotations of spontaneous conversations. We are
particularly interested in how novice annotators interpret discourse relation
categories when annotating spoken conversational data. We collect annotations
of DRs from Switchboard telephone conversations (Godfrey et al., 1992),
allowing for multiple labels, and ask for confidence scores. We find that
confidence scores vary significantly across dialogue contexts (single turn vs.
pairs of turns produced by the same speaker vs. pairs of turns produced by
different speakers). We incorporate information about these three dialogue
context types and confidence scores into distributed representations of
discourse relations. A clustering analysis shows that discourse relations that
tend to occur across speakers cluster together, while discourse relations
which tend to occur within a speaker, either in the same turn or different
turns, form their own cluster.

##  2 Annotation of Discourse Relations

Our analyses are built on the dataset collected in López Cortez and Jacobs
(2023), who selected 19 conversations from Switchboard111We discarded the
annotations from one conversation because the annotators did not follow the
guidelines., a corpus consisting of telephone conversations between pairs of
participants about a variety of topics (e.g. recycling, movies, child care).
We chose this corpus because it contains informal, spontaneous dialogues, and
because it has been used within linguistics in various studies on conversation
(Jaeger and Snider, 2013; Reitter and Moore, 2014).

###  2.1 Discourse Units

An initial set of turns for annotation was selected by using spaCy’s
dependency parser (Honnibal et al., 2020, version 3.3.1) to select turns with
two or more ROOT or VERB tags. We define a turn as each segment of dialogue
taken from Switchboard. We note that an utterance produced by one speaker (A)
may take place during a continuous utterance by another speaker (B).
Switchboard splits A’s utterance into two turns in these cases. We return to
this point in the Discussion.

We manually segmented these turns into elementary discourse units (EDUs). The
main criteria for segmenting turns into EDUs was that the unit performs some
basic discourse function (Asher and Lascarides, 2003). By default, finite
clauses are considered EDUs, as well as comment words like “Really?” or
acknowledgments such as “Uh-huh” or “Yeah.” Cases of interruptions and repairs
were segmented if they constituted a turn in Switchboard, as in example (2.1),
and when they contained a verb, as in example (2.1). Cases of repetition as in
(2.1) were not considered separate EDUs. We segmented disfluencies (“uh”) and
some non-verbal communication (“[laughter]”) but we did not select these for
discourse relation labeling.

B: || So you don’t see too many thrown out around the || [laughter] ||
streets. ||
A: || Really ||
B: || Or even bottles. ||

B: || I think, || uh, || I wonder || if that worked. ||

A: || What kind of experience do you, do you have, then with child care? ||

Because many EDUs are very short, we selected pairs of elementary discourse
units and complex discourse units (CDUs) for discourse relation annotation.
CDUs consist of two or more EDUs that constitute an argument to a discourse
relation Asher and Lascarides (2003). We use the term discourse units (DUs) to
refer to both EDUs and CDUs.

###  2.2 Dialogue Contexts

We manually selected items for annotation across three different contexts:
within a single turn, across two turns within a speaker, and across two
immediately adjacent turns (two speakers). (2.2) shows an example for each
context kind, with the first DU in italics and the second in bold. Example
(2.2) shows two discourse units within a speaker’s turn. (2.2) shows two
discourse units uttered by the same speaker but that span across two different
turns, interrupted by one turn. We did not include any constraint for the
length of the interrupting turn. (2.2) shows two DUs uttered by speakers in
adjacent turns. We leave for future work the annotation of pairs of discourse
units that may have a longer-distance relation with more turns in between DUs.

A: || and they discontinued them || because people were coming and dumping
their trash in them. || B: || No, || I just, I noticed || in Iowa and other
cities like that, it’s a nickel per aluminum can. ||
A: || Oh. ||
B: || So you don’t see too many thrown out around the || [laughter] ||
streets. A: || We live in the Saginaw area. ||
B: || Saginaw? ||

###  2.3 Taxonomy of Discourse Relations

The DRs chosen to annotate our corpus were adapted from the STAC corpus manual
(Asher et al., 2012, 2016). STAC is a corpus of strategic multi-party chat
conversations in an online game. Table 1 shows the taxonomy used. We selected
11 DRs based on a pilot annotation by the first author, and added an “Other”
category for relations not included in the list of labels. We focused on a
small taxonomy to minimize the number of choices presented to our novice
annotators. We refer readers to López Cortez and Jacobs (2023) for details and
examples of each relation in the taxonomy. Future work will include revising
the taxonomy used.

Acknowledgement | Elaboration
---|---
Background | Explanation
Clarification Question | Narration
Comment | Question-Answer Pair
Continuation | Result
Contrast | Other
Table 1: Taxonomy of discourse relations.

###  2.4 Annotation Procedure

The annotation of discourse relations was done by students enrolled in a
Computational Linguistics class. Students were divided into 19 teams of
approximately 5 members each, and each team was assigned a conversation. The
annotation was performed individually, but teams then discussed their work and
wrote a report together. Annotators were trained using written guidelines, a
quiz-like game, and a live group annotation demo.

We used the annotation interface Prodigy Montani and Honnibal (2018). Each
display presented the two target discourse units plus two context turns before
and two after. Annotators also had access to the entire conversation
throughout the annotation task. Below the text, the screen showed a multiple
choice list of discourse relations plus the “Other” category. We allowed for
the selection of multiple labels following previous findings that allowing for
multiple labels better captures ambiguous interpretations of discourse
relations Scholman et al. (2022) and improves the performance of discourse
parsers Yung et al. (2022).

Each display also asked for confidence scores in the range 1-5, corresponding
to least to most confident. We did not pursue label-specific confidence scores
but rather the confidence in the label(s) as a whole in the interest of
minimizing annotator overhead. The results of this work show that per-label
confidence scores or a slider-based approach may be informative and is a topic
for future work. We include an example annotation item in Appendix C.

##  3 Dialogue Context as a Predictor of Confidence Scores

First we sought to understand how discourse relations and dialogue context (as
defined above) influence annotator confidence. Because our confidence ratings
data has multiple observations for each annotator, each team and each DU, it
is hierarchical and thus benefits from being analyzed using hierarchical mixed
effects models. Due to the ordinal nature of the ratings data, we use the
cumulative link approach (CLMM; Liddell and Kruschke, 2018; Howcroft and
Rieser, 2021) rather than model confidence scores as real-valued in linear
regression. We first built a null model containing only random intercepts by
annotator and compared it to a model containing an additional fixed effect and
random slope by annotator for dialogue context type: single turn, across turns
within speaker and across speakers ($kind$, dummy coded). A likelihood ratio
test revealed a significant improvement in fit by adding kind as a predictor
($\chi^{2}(7)=126.64,p<0.001$). Adding random intercepts for DU pairs to
account for annotation difficulty across DU pairs also led to a significant
improvement in model fit beyond the model containing dialogue context kind
($\chi^{2}(1)=195.01,p<0.001$). This suggests that our annotators’ confidence
scores are sensitive to the context of DU pairs.

Figure 1: Confidence scores per context kind across discourse relations. qap
stands for Question-Answer Pair and clarificationq for Clarification Question.

Figure 1 shows mean confidence scores per context kind across discourse
relations. Confidence scores within a speaker both across and within turns
received similar confidence ratings ($\beta=-0.13$, $z=-0.56$, $p=$ n.s.222Not
statistically significant.), while annotators were significantly more
confident for relation annotation across speakers ($\beta=0.63$, $z=3.05$,
$p<.01$). The CLMM revealed that annotators used confidence scores between 3
and 5 overall, except for the label “Other”, for which they selected lower
confidence scores. Background received lower confidence scores overall.
Continuation, Contrast and Narration received higher scores for contexts
within speaker. Comment and Result received higher scores for turns across
speakers and single turn. For Elaboration and Explanation, mean confidence
scores are very similar across the three contexts, with slightly higher scores
for single turn and pairs of turns within speaker. Acknowledgment,
Clarification Question (“clarificationq”) and Question-Answer Pair (“qap”)
received higher scores for turns across speakers, which makes sense given the
dialogic nature of these relations. However, these relations also received
rather high confidence scores for single turn and pairs of turns within
speaker, which is a bit surprising. We suspect this might be due to the
context turns included for each pair of DUs, which might have led annotators
to choose relations between discourse units other than for the pair of
highlighted DUs. Future analysis will look closer at this aspect.

##  4 Distributed Representations from Discourse Relation Annotations

To model the similarity between discourse relations as perceived by
annotators, we computed embedding representations of discourse relations. We
extracted each $n$ individual annotation containing relation-confidence
$(r,c)$ tuples selected by a given annotator for a pair of DUs. We concatenate
bag-of-relation vectors with one-hot encoded features representing the
dialogue context kind, and multiply the count vector of annotated relations
(either 1 or 0 for each relation) by the confidence score (1-5) for that pair
of DUs. This weighting learns more from high confidence; an ideal reweighting
may be possible with additional parameter search, possibly in conjunction with
the CLMM outputs.

For an $n\times 1$ confidence ratings matrix $C$, an $n\times 12$ bag-of-
relations matrix $R$, an $n\times 3$ discourse context matrix $D$ for each
annotation, we obtain an annotation matrix $A=C\times(R|D)$. We then obtain a
square co-occurrence matrix $O$ such that $O=A\cdot A^{T}$, which we factorize
using Principal Component Analysis (without shifting the intercept following
Levy and Goldberg, 2014). Each relation is thus represented as a vector that
consolidates co-occurrences between all relations within a single annotator
that are weighted by confidence score. We then projected these embeddings into
two dimensions with UMAP McInnes et al. (2018) and performed a hierarchical
clustering analysis over the resulting coordinates due to the greater
discriminability afforded by continuous distance metrics.

(a) The coordinates obtained with UMAP for all discourse relations plotted in
two-dimensional space. The plot on the left shows the unweighted embedding
representations and the figure on the right shows the weighted embedding
representations.

(b) Dendrogram showing hierarchical clustering of Discourse Relations built
from UMAP coordinates. qap stands for Question-Answer Pair and clarificationq
for Clarification Question.

Figure 2: Dimensionality reduction and clustering of relation embeddings.

Informally, the UMAP coordinates appear more gradient in the representational
space when confidence was included (right panel) than when it was not included
(left panel). When context is not included, the UMAP coordinates primarily
represent the frequency of labels in our corpus, which we include in Appendix
A. We visualize the UMAP coordinates in Figure 2(a). Figure 2(b) shows a
dendrogram with the output clusters, colored according to the optimal number
of clusters ($k=2$), calculated using average silhouette widths Levshina
(2022). There are two large clusters, one of which contains two sub-clusters
with Background and Continuation, on the one hand, and Elaboration and
Explanation on the other. In the other large cluster, Acknowledgement and
Comment form a sub-cluster. These are very common relations between pairs of
turns across speakers. Clarification Question and Question-Answer Pair form
another sub-cluster, also common relations between pairs of turns across
speakers, in close proximity to the Other label, which received a sub-cluster
of its own. Narration and Contrast and Result, form the last sub-clusters,
which we suspect is due in part to the frequencies of these relations Schnabel
et al. (2015). We include a dendrogram with the output clusters of a
hierarchical clustering analysis performed with base bag-of-relations vectors
(without context kind and confidence scores weight) in Figure 3 in Appendix B
for comparison.

Currently, we provide these results as a proof of concept of the feasibility
and interpretability of noisy labels produced by novice annotators.
Importantly, annotations weighted by confidence produce coherent clusters of
discourse relations. We envision applications of DR embeddings to several
domains including dialogue generation, such that appropriate responses to
input are partially conditioned on a latent or mixed combination of DRs.

##  5 Related Work

Annotation of discourse relations is usually done within Rhetorical Structure
Theory (Mann and Thompson, 1987), as in the RST-DT (Carlson et al., 2003) and
GUM (Zeldes, 2017) corpora, within Segmented Discourse Representation Theory
(SDRT, Asher and Lascarides, 2003), as in the STAC (Asher et al., 2016) and
Molweni (Li et al., 2020) corpora, or within the Penn Discourse Treebank
framework (Prasad et al., 2008, 2014, 2018). We use a taxonomy adapted from
SDRT, in particular, the STAC corpus.

Annotators are usually trained to identify discourse relations using the
framework’s taxonomy. Some recent alternatives to explicitly collecting
annotation of DRs include crowdsourcing by eliciting connectives (Yung et al.,
2019; Scholman et al., 2022) or question-answer pairs (Pyatkin et al., 2020)
rather than relations. In this work, we wanted to investigate how annotators
perceive discourse relation categories, and therefore a connective insertion
task would only provide indirect evidence. We train annotators on DR labeling
and ask annotators to choose from a set of discourse relation labels. We allow
for multiple labels to investigate what relations are more confusable or
perceived as co-occurring (Marchal et al., 2022).

##  6 Discussion and Future Work

In this study, we collected multiple annotations of discourse relations from a
subset of the Switchboard corpus, together with confidence scores. We found
that dialogue context had a significant effect on confidence scores. We
computed embedding representations of DRs using co-occurrence statistics and
weighted the vectors using context type and confidence scores, and found that
these representations coherently model our annotators uncertainty about
discourse relation labels.

Discourse units that occur across turns as defined by Switchboard do not
necessarily occur across continuous utterances from the speaker’s point-of-
view. Obtaining information about whether same-speaker pairs of discourse
units fall into the same or different utterances may help to explain
additional variance in annotator confidence.

Additionally, in this work, we investigated annotators’ confidence on the
annotation of adjacent turns. In future work, we plan to annotate discourse
relations across longer-distance discourse units and to allow for hierarchical
annotation. We expect that annotation confidence will also vary across longer-
distance units and across different depths of annotation.

In the future, we plan to use this information to run a larger scale
annotation study of the Switchboard corpus to analyze discourse relation
patterns in spoken dialogues.

## Limitations

This work is limited by the size of the dataset and the taxonomy used in the
annotation task. While we found that our annotators perceived some of the
categories as more similar or confusable, future work can examine annotators’
uncertainty in a larger set of discourse relations. The selection of DUs for
annotation was also non-exhaustive. In future work, we plan to expand the
selection procedure so that we include more distantly related DUs. We also
note that the frequency of discourse relation labels and individual
differences in confidence levels among annotators may bias the
representations. We plan to look into these potential biases in future work.

## Ethics Statement

We are not aware of ethical issues associated with the texts used in this
work. Students participated in the annotation task as part of course credit
but annotation decisions were not associated with their performance in the
course.

## Acknowledgements

We would like to thank Jürgen Bohnemeyer and three anonymous reviewers for
feedback on a previous version of this paper. We also thank the students who
participated in the annotation task.

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##  Appendix A Frequencies of Discourse Relation Labels

Discourse Relation | Count
---|---
Elaboration | 636
Continuation | 554
Acknowledgement | 494
Explanation | 383
Comment | 265
Background | 252
Narration | 249
Question-Answer Pair | 248
Contrast | 191
Clarification Question | 179
Result | 124
Other | 106
Table 2: Raw counts of discourse relation labels in our corpus from most to
least frequent.

##  Appendix B Clustering without Context and Confidence Weighting

Figure 3: Dendrogram showing hierarchical clustering of Discourse Relations
built from UMAP coordinates without context kind and confidence scores
weighting. qap stands for Question-Answer Pair and clarificationq for
Clarification Question. The two main clusters align with the two-dimensional
plot of the unweighted UMAP coordinates in Figure 2(a)

##  Appendix C Annotation Interface

Figure 4: Example annotation task. EDUs to be annotated and discourse
relations. Figure 5: Example annotation task. Discourse relations and
confidence score.