##### Step 2: $T_{A}\otimes_{A}D$ and $T_{B}\otimes_{B}D$ (resp.
$U_{A}\otimes_{A}D$ and $U_{B}\otimes_{B}D$) are dense in $T_{D}$ (resp.
$U_{D}$) for the $\mathfrak{m}_{\infty}$-adic topology.

We only prove the density of $T_{A}\otimes_{A}D$ in $T_{D}$ since the argument
for the others follows along the same lines.

Let $t\in T_{D}=T\cap(M_{D}\otimes_{A\otimes D}\pazocal{B}_{\infty}(D))$. Let
$(m_{1},...,m_{s})$ be generators of $M_{A}$ as an $A\otimes A$-module. $t$
can be written as a sum $\sum_{i=1}^{r}{m_{i}\otimes b_{i}}$ with coefficients
$b_{i}\in\pazocal{B}_{\infty}(D)$. For $i\in\\{1,...,r\\}$, let
$(b_{i,n})_{n\in\mathbb{Z}}$ be a sequence in
$\pazocal{B}_{\infty}(A)\otimes_{A}D$, such that $b_{i,n}=0$ for $n\ll 0$,
satisfying $b_{i}-b_{i,n}\in\mathfrak{m}_{\infty}^{n}\pazocal{A}_{\infty}(D)$
for all $n\in\mathbb{Z}$. In particular, $(b_{i,n})_{n\in\mathbb{Z}}$
converges to $b_{i}$ when $n$ tens to infinity. For $n\in\mathbb{Z}$, we set:

$t_{n}:=\sum_{i=1}^{s}{m_{i}\otimes b_{i,n}}\in(M_{A}\otimes_{A\otimes
A}\pazocal{B}_{\infty}(A))\otimes_{A}D.$

Then, $t-t_{n}$ belongs to $\mathfrak{m}_{\infty}^{n}\Xi$ where $\Xi$ is the
$\pazocal{A}_{\infty}(D)$-module generated by $(m_{1},...,m_{s})$. For $n$
large enough, $\mathfrak{m}_{\infty}^{n}\Xi\subset T$, hence $t-t_{n}\in T$
and $t_{n}\in T$. We deduce that $t_{n}\in T_{A}\otimes_{A}D$ for large value
of $n$ and that $(t_{n})_{n\in\mathbb{Z}}$ converges to $t$ when $n$ goes to
infinity. We conclude that $T_{A}\otimes_{A}D$ is dense in $T_{D}$.

##### Steps $1\&2\Longrightarrow$ compatibility.

Because $T_{A}$ and $T_{B}$ are finitely generated over
$\pazocal{A}_{\infty}(A)$ and $\pazocal{A}_{\infty}(B)$ respectively,
$T_{A}\otimes_{\pazocal{A}_{\infty}(A)}\pazocal{A}_{\infty}(D)$ coincides with
the completion of $T_{A}\otimes_{A}D$ and
$T_{B}\otimes_{\pazocal{A}_{\infty}(B)}\pazocal{A}_{\infty}(D)$ with the
completion of $T_{B}\otimes_{B}D$ (by [Bou07, (AC)§.3 Thm. 3.4.3]). Therefore,
the multiplication maps are isomorphisms:

$T_{A}\otimes_{\pazocal{A}_{\infty}(A)}\pazocal{A}_{\infty}(D)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}T_{D}\stackrel{{\scriptstyle\sim}}{{\longleftarrow}}T_{B}\otimes_{\pazocal{A}_{\infty}(B)}\pazocal{A}_{\infty}(D),$
$U_{A}\otimes_{\pazocal{A}_{\infty}(A)}\pazocal{A}_{\infty}(D)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}U_{D}\stackrel{{\scriptstyle\sim}}{{\longleftarrow}}U_{B}\otimes_{\pazocal{A}_{\infty}(B)}\pazocal{A}_{\infty}(D).$

##### Step $3$: the glueing.

We consider the morphisms of formal schemes over
$\operatorname{Spf}\pazocal{O}_{\infty}$

$\operatorname{Spf}\pazocal{A}_{\infty}(A)=\operatorname{Spf}\pazocal{O}_{\infty}\hat{\otimes}A\stackrel{{\scriptstyle\hat{i}}}{{\longrightarrow}}\operatorname{Spf}\pazocal{O}_{\infty}\hat{\times}C\stackrel{{\scriptstyle\hat{j}}}{{\longleftarrow}}\operatorname{Spf}\pazocal{O}_{\infty}\hat{\otimes}B=\operatorname{Spf}\pazocal{A}_{\infty}(B).$

By the Beauville-Laszlo Theorem [BL95], there exists a unique pair of coherent
sheaves $(\pazocal{M},\pazocal{N})$ of $\pazocal{O}_{C\times C}$-modules such
that

$\pazocal{M}(\operatorname{Spec}A\otimes A)=M_{A}$ | $\pazocal{N}(\operatorname{Spec}A\otimes A)=N_{A}$
---|---
$\pazocal{M}(\operatorname{Spec}A\otimes B)=M_{B}$ | $\pazocal{N}(\operatorname{Spec}A\otimes B)=N_{B}$
$(\hat{i}^{*}\pazocal{M})(\operatorname{Spf}\pazocal{O}_{\infty}\hat{\otimes}A)=T_{A}$ | $(\hat{i}^{*}\pazocal{N})(\operatorname{Spf}\pazocal{O}_{\infty}\hat{\otimes}A)=U_{A}$
$(\hat{j}^{*}\pazocal{M})(\operatorname{Spf}\pazocal{O}_{\infty}\hat{\otimes}B)=T_{B}$ | $(\hat{j}^{*}\pazocal{N})(\operatorname{Spf}\pazocal{O}_{\infty}\hat{\otimes}B)=U_{B}$

Since, for each row of the above table, the left-hand side is canonically a
submodule of the right-hand side, we have $\pazocal{M}\subset\pazocal{N}$.
Because these inclusions become equalities away from $\Delta$, we deduce that
the cokernel of the inclusion $\pazocal{M}\subset\pazocal{N}$ is supported at
$\Delta$. The glueing of (5.3) results in a morphism
$\tau_{\pazocal{M}}:\tau^{*}\pazocal{M}\to\pazocal{N}$. Finally, we recall
that there exists $r\in B$ invertible in $D$ and vanishing at $\infty$ such
that $\tau_{M}(\tau^{*}M_{B})\subset rN_{B}$ and thus
$\tau_{M}(\tau^{*}T_{B})\subset rU_{B}$. Hence, the image of
$\tau_{\pazocal{M}}$ lands in $\pazocal{N}(-\infty_{C})$. ∎

######  Remark 5.9.

It is noteworthy that the converse of Theorem 5.6 do hold: if $\underline{M}$
admits a $C\times C$-shtuka model, then all the weights of $\underline{M}$ are
non-positive. Indeed, one easily shows that the existence of such a shtuka
model implies the existence of a stable $\pazocal{A}_{\infty}(K)$-lattice in
$M\otimes_{A\otimes K}\pazocal{B}_{\infty}(K)$. Such an event happens only if
the slopes of the $\infty$-isocrystal attached to $\underline{M}$ has non-
negative slopes, i.e. $\underline{M}$ only has non-positive weights.

We fix $\underline{\pazocal{M}}$ a $C\times C$-shtuka model of
$\underline{M}$. Let $i:\operatorname{Spec}\pazocal{A}_{\infty}(A)\to C\times
C$. We denote by:

$\displaystyle L_{A}$
$\displaystyle:=(i^{*}\pazocal{M})(\operatorname{Spf}\pazocal{A}_{\infty}(A))=(i^{*}\pazocal{N})(\operatorname{Spf}\pazocal{A}_{\infty}(A))$
$\displaystyle L$
$\displaystyle:=L_{A}\otimes_{\pazocal{A}_{\infty}(A)}\pazocal{A}_{\infty}(K)$

(for the first equality, we used that $\Delta$ is not supported at
$\operatorname{Spf}\pazocal{A}_{\infty}(A)$). $\tau_{M}$ induces an
$\pazocal{O}_{\infty}$-linear endormorphism of $L$ (resp. $L_{A}$). The next
lemma records the additional pleasant feature of shtuka models when the
weights of $\underline{M}$ are all negative (this will be used later in
Subsection 6.2).

######  Lemma 5.10.

Assume that all the weights of $\underline{M}$ are negative. Then, the
morphism $\operatorname{id}-\tau_{\pazocal{M}}$ induces an
$\pazocal{O}_{\infty}$-linear automorphism of $L$ and $L_{A}$.

###### Proof.

The statement for $L_{A}$ implies the one for $L$. Because the weights of
$\underline{M}$ are negative, there is, by Lemma 5.8, an
$\pazocal{A}_{\infty}(K)$-lattice $T$ in $M\otimes_{A\otimes
K}\pazocal{B}_{\infty}(K)$ and two positive integers $h$ and $d$ such that
$\tau_{M}^{h}(\tau^{h*}T)=\mathfrak{m}_{\infty}^{d}T$.

To show that $\operatorname{id}-\tau_{\pazocal{M}}$ is injective on $L_{A}$,
let $x$ be an element of $\ker(\operatorname{id}-\tau_{\pazocal{M}}|L_{A})$.
We may assume without loss that $x\in T$. For all positive integer $n$,

$x=\tau_{M}^{nh}(\tau^{nh*}x)\in\mathfrak{m}_{\infty}^{nd}T.$

Because $d>0$, $x=0$.
We turn to surjectivity. Let $T^{\prime}$ be the
$\pazocal{A}_{\infty}(K)$-lattice generated by the elements of $T$,
$\tau_{M}(\tau^{*}T)$, …, and of $\tau_{M}^{h-1}(\tau^{(h-1)*}T)$. Then
$T^{\prime}$ is stable by $\tau_{M}$. Let $y\in L_{A}$ and let $k\geq 0$ be
such that $\pi_{\infty}^{k}y\in T$. For all $n\geq 0$, we have

$\tau_{M}^{nh}(\tau^{nh*}y)\in\mathfrak{m}_{\infty}^{nd-k}T^{\prime}$

and, in particular, for all $u\in\\{0,1,...,h-1\\}$,

$\tau_{M}^{nh+u}(\tau^{(nh+u)*}y)\in\mathfrak{m}_{\infty}^{nd-k}T^{\prime}.$

Therefore, the series

$\sum_{t=0}^{\infty}{\tau_{M}^{t}(\tau^{t*}y)}=\sum_{n=0}^{\infty}{\left(\sum_{u=0}^{h-1}{\tau_{M}^{nh+u}(\tau^{(nh+u)*}y)}\right)}$

converges in $L_{A}$ to $f$ satisfying $f-\tau_{M}(\tau^{*}f)=y$. ∎

###  5.3 Shtuka models and extensions of mixed Hodge-Pink structures

Let $\underline{M}$ be a mixed and rigid analytically trivial $A$-motive over
$K$ whose weights are all non-positive. Let
$(\pazocal{N},\pazocal{M},\tau_{\pazocal{M}})$ be a $C\times C$-shtuka model
for $\underline{M}$, whose existence is ensured by Theorem 5.6. Let
$\iota:\pazocal{M}\to\pazocal{N}$ be the inclusion of sheaves. We consider the
inclusion of ringed spaces

$\operatorname{Spf}\pazocal{A}_{\infty}(\pazocal{O}_{\infty})=\operatorname{Spf}\pazocal{O}_{\infty}\hat{\otimes}\pazocal{O}_{\infty}\longrightarrow
C\times C$ (5.4)

and denote respectively $\hat{\pazocal{N}}$ and $\hat{\pazocal{M}}$ the
pullback of $\pazocal{N}$ and $\pazocal{M}$ through (5.4). Finally, denote by
$\hat{\pazocal{N}}_{\infty}$ and $\hat{\pazocal{M}}_{\infty}$ the finitely
generated $\pazocal{A}_{\infty}(\pazocal{O}_{\infty})$-modules:

$\hat{\pazocal{N}}_{\infty}:=\hat{\pazocal{N}}(\operatorname{Spf}\pazocal{O}_{\infty}\hat{\otimes}\pazocal{O}_{\infty}),\quad\hat{\pazocal{M}}_{\infty}:=\hat{\pazocal{M}}(\operatorname{Spf}\pazocal{O}_{\infty}\hat{\otimes}\pazocal{O}_{\infty}).$

The aim of this subsection is to prove that there is an exact sequence of
$K_{\infty}$-vector spaces (Corollary 5.19):

$0\longrightarrow\underline{M}_{B}^{+}\otimes_{A}K_{\infty}\longrightarrow\frac{\hat{\pazocal{N}}_{\infty}}{(\iota-\tau_{\pazocal{M}})(\hat{\pazocal{M}}_{\infty})}\otimes_{\pazocal{O}_{\infty}}K_{\infty}\longrightarrow\operatorname{Ext}^{1,\text{ha}}_{\infty}(\mathbbm{1}^{+},\mathscr{H}^{+}_{K_{\infty}}(\underline{M}))\longrightarrow
0.$

This above sequence appeared to the author as the most miracle part of the
proof of Theorem 4.4. A suprising feature is that this property does not
depend on the choice of the shtuka model. The reader will have no trouble to
notice how much this subsection relies on ideas from V. Lafforgue in [Laf09,
§4].

We start by a proposition.

######  Proposition 5.11.

There is an isomorphism of $K_{\infty}$-vector spaces

$\displaystyle\frac{\hat{\pazocal{N}}_{\infty}}{(\iota-\tau_{\pazocal{M}})(\hat{\pazocal{M}}_{\infty})}\otimes_{\pazocal{O}_{\infty}}K_{\infty}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}(\hat{\pazocal{N}}_{\infty}/\hat{\pazocal{M}}_{\infty})\otimes_{\pazocal{O}_{\infty}}K_{\infty}.$

We split the proof of Proposition 5.11 into several lemmas.

######  Lemma 5.12.

There exists an injective $\pazocal{A}_{\infty}(\pazocal{O}_{\infty})$-linear
morphism
$\iota^{\prime}:\hat{\pazocal{N}}_{\infty}\to\hat{\pazocal{M}}_{\infty}$ and a
positive integer $e$ such that $\iota^{\prime}\iota$ and $\iota\iota^{\prime}$
coincide with the multiplication by $(\pi_{\infty}\otimes
1-1\otimes\pi_{\infty})^{e}$ on $\hat{\pazocal{M}}_{\infty}$ and
$\hat{\pazocal{N}}_{\infty}$ respectively.

###### Proof.

Let
$\mathfrak{d}:=\pazocal{O}(\Delta)(\operatorname{Spf}\pazocal{O}_{\infty}\hat{\otimes}\pazocal{O}_{\infty})$
as an ideal of $\pazocal{A}_{\infty}(\pazocal{O}_{\infty})$. The cokernel of
the inclusion $\iota:\hat{\pazocal{M}}_{\infty}\to\hat{\pazocal{N}}_{\infty}$
is $\mathfrak{d}$-torsion. It is also finitely generated, and since
$\pi_{\infty}\otimes 1-1\otimes\pi_{\infty}\in\mathfrak{d}$, there exists
$e\geq 0$ such that $(\pi_{\infty}\otimes
1-1\otimes\pi_{\infty})^{e}v\in\hat{\pazocal{M}}_{\infty}$ for all
$v\in\hat{\pazocal{N}}_{\infty}$. We let
$\iota^{\prime}:\hat{\pazocal{N}}_{\infty}\to\hat{\pazocal{M}}_{\infty}$ be
the multiplication by $(\pi_{\infty}\otimes 1-1\otimes\pi_{\infty})^{e}$ and
the lemma follows. ∎

######  Lemma 5.13.

Let $t$ be a positive integer. Then, $\iota-\tau_{\pazocal{M}}$ and $\iota$
respectively induce isomorphisms of $K_{\infty}$-vector spaces:

$\left(\frac{\hat{\pazocal{M}}_{\infty}}{(1\otimes\pi_{\infty})^{t}\hat{\pazocal{M}}_{\infty}}\right)\otimes_{\pazocal{O}_{\infty}}K_{\infty}\stackrel{{\scriptstyle\iota-\tau_{\pazocal{M}}}}{{\longrightarrow}}\left(\frac{\hat{\pazocal{N}}_{\infty}}{(1\otimes\pi_{\infty})^{t}\hat{\pazocal{N}}_{\infty}}\right)\otimes_{\pazocal{O}_{\infty}}K_{\infty},$
$\left(\frac{\hat{\pazocal{M}}_{\infty}}{(1\otimes\pi_{\infty})^{t}\hat{\pazocal{M}}_{\infty}}\right)\otimes_{\pazocal{O}_{\infty}}K_{\infty}\stackrel{{\scriptstyle\iota}}{{\longrightarrow}}\left(\frac{\hat{\pazocal{N}}_{\infty}}{(1\otimes\pi_{\infty})^{t}\hat{\pazocal{N}}_{\infty}}\right)\otimes_{\pazocal{O}_{\infty}}K_{\infty}.$

###### Proof.

Let $\iota^{\prime}$ and $e\geq 0$ be as in Lemma 5.12. The multiplication by

$\left(\sum_{k=0}^{t-1}{\pi_{\infty}^{-(k+1)}\otimes\pi_{\infty}^{k}}\right)^{e}=\left(\frac{1-\pi_{\infty}^{-t}\otimes\pi^{t}_{\infty}}{\pi_{\infty}\otimes
1-1\otimes\pi_{\infty}}\right)^{e}\equiv(\pi_{\infty}\otimes
1-1\otimes\pi_{\infty})^{-e}$

on
$(\hat{\pazocal{M}}_{\infty}/(1\otimes\pi_{\infty})^{t}\hat{\pazocal{M}}_{\infty})\otimes_{\pazocal{O}_{\infty}}K_{\infty}$
defines an inverse of $\iota^{\prime}\iota$. The same argument shows that
$\iota\iota^{\prime}$ is an automorphism of
$(\hat{\pazocal{N}}_{\infty}/(1\otimes\pi_{\infty})^{t}\hat{\pazocal{N}}_{\infty})\otimes_{\pazocal{O}_{\infty}}K_{\infty}$.

On the other-hand, we have
$(\iota^{\prime}\tau_{\pazocal{M}})^{k}(\tau^{*k}\hat{\pazocal{M}}_{\infty})\subset(1\otimes\pi_{\infty})\hat{\pazocal{M}}_{\infty}$
for $k$ large enough. Hence, $(\iota^{\prime}\tau_{\pazocal{M}})$ is nilpotent
on
$\hat{\pazocal{M}}_{\infty}/(1\otimes\pi_{\infty})^{t}\hat{\pazocal{M}}_{\infty}$,
and so is $(\iota^{\prime}\iota)^{-1}(\iota^{\prime}\tau_{\pazocal{M}})$. In
particular,

$\iota^{\prime}(\iota-\tau_{\pazocal{M}})=(\iota^{\prime}\iota)(\operatorname{id}-(\iota^{\prime}\iota)^{-1}(\iota^{\prime}\tau_{\pazocal{M}}))$

is an isomorphism. It follows that $\iota-\tau_{\pazocal{M}}$ is injective and
$\iota^{\prime}$ surjective. Since $\iota\iota^{\prime}$ is invertible,
$\iota^{\prime}$ is injective. We deduce that $\iota-\tau_{\pazocal{M}}$,
$\iota^{\prime}$ and thus $\iota$ are isomorphisms. ∎

######  Lemma 5.14.

Let $t$ be a non-negative integer. Then, the canonical maps

$\frac{(1\otimes\pi_{\infty})^{t}\hat{\pazocal{N}}_{\infty}}{(\iota-\tau_{\pazocal{M}})((1\otimes\pi_{\infty})^{t}\hat{\pazocal{M}}_{\infty})}\otimes_{\pazocal{O}_{\infty}}K_{\infty}\longrightarrow\frac{\hat{\pazocal{N}}_{\infty}}{(\iota-\tau_{\pazocal{M}})(\hat{\pazocal{M}}_{\infty})}\otimes_{\pazocal{O}_{\infty}}K_{\infty},$
(5.5)
$\frac{(1\otimes\pi_{\infty})^{t}\hat{\pazocal{N}}_{\infty}}{\iota((1\otimes\pi_{\infty})^{t}\hat{\pazocal{M}}_{\infty})}\otimes_{\pazocal{O}_{\infty}}K_{\infty}\longrightarrow\frac{\hat{\pazocal{N}}_{\infty}}{\iota(\hat{\pazocal{M}}_{\infty})}\otimes_{\pazocal{O}_{\infty}}K_{\infty},$
(5.6)

are isomorphisms of $K_{\infty}$-vector spaces.

###### Proof.

In the category of $\pazocal{O}_{\infty}$-vector spaces, we have a diagram
exact on lines and commutative on squares:

${0}$${(1\otimes\pi_{\infty})^{t}\hat{\pazocal{M}}_{\infty}}$${\hat{\pazocal{M}}_{\infty}}$${\hat{\pazocal{M}}_{\infty}/(1\otimes\pi_{\infty})^{t}\hat{\pazocal{M}}_{\infty}}$${0}$${0}$${(1\otimes\pi_{\infty})^{t}\hat{\pazocal{N}}_{\infty}}$${\hat{\pazocal{N}}_{\infty}}$${\hat{\pazocal{N}}_{\infty}/(1\otimes\pi_{\infty})^{t}\hat{\pazocal{N}}_{\infty}}$${0}$$\scriptstyle{\iota-\tau_{\pazocal{M}}}$$\scriptstyle{\iota-\tau_{\pazocal{M}}}$$\scriptstyle{\iota-\tau_{\pazocal{M}}}$
(5.7)

By Lemma 5.13, the third vertical arrow once tensored with $K_{\infty}$ over
$\pazocal{O}_{\infty}$ is an isomorphism. The first isomorphism then follows
from the Snake Lemma. The second one follows from the very same argument, with
$\iota$ in place of $\iota-\tau_{\pazocal{M}}$. ∎

######  Lemma 5.15.

For $t$ large, we have
$(\iota-\tau_{\pazocal{M}})((1\otimes\pi_{\infty})^{t}\hat{\pazocal{M}}_{\infty})=\iota((1\otimes\pi_{\infty})^{t}\hat{\pazocal{M}}_{\infty})$.

###### Proof.

Let $\iota^{\prime}$ and $e\geq 0$ be as in Lemma 5.12. We chose $t$ such that
$(q-1)t>e$. For $s\geq t$, let
$\hat{\pazocal{M}}_{s}:=(1\otimes\pi_{\infty})^{s}\hat{\pazocal{M}}_{\infty}$.
$(\hat{\pazocal{M}}_{s})_{s\geq t}$ forms a decreasing family of
$\pazocal{A}_{\infty}(\pazocal{O}_{\infty})$-modules for the inclusion. It
suffices to show that

$\iota^{\prime}(\iota-\tau_{\pazocal{M}})(\hat{\pazocal{M}}_{t})=(\iota^{\prime}\iota)(\hat{\pazocal{M}}_{t}).$
(5.8)

By our assumption on $t$, we have
$(\iota^{\prime}\iota)^{-1}\tau_{\pazocal{M}}(\hat{\pazocal{M}}_{s})\subset\hat{\pazocal{M}}_{s+1}$
for all $s\geq t$. Hence, the endomorphism
$\operatorname{id}-(\iota^{\prime}\iota)^{-1}\tau_{\pazocal{M}}$ of
$\hat{\pazocal{M}}_{t}$ becomes an automorphism over the completion of
$\hat{\pazocal{M}}_{t}$ with respect to the $(1\otimes\pi_{\infty})$-adic
topology (equivalently, the topology which makes
$(\hat{\pazocal{M}}_{s})_{s\geq n}$ a neighbourhood of $0$ for all $n\geq t$).
To conclude, it suffices to show that $\hat{\pazocal{M}}_{t}$ is already
complete for this topology. Because $\hat{\pazocal{M}}_{t}$ is Noetherian, we
have

$\widehat{(\hat{\pazocal{M}}_{t})}_{(1\otimes\pi_{\infty})}\cong\hat{\pazocal{M}}_{t}\otimes_{\pazocal{A}_{\infty}(\pazocal{O}_{\infty})}\widehat{\pazocal{A}_{\infty}(\pazocal{O}_{\infty})}_{(1\otimes\pi_{\infty})},$

and it suffices to show that $\pazocal{A}_{\infty}(\pazocal{O}_{\infty})$ is
complete for the $(1\otimes\pi_{\infty})$-adic topology. We have the
identifications

$\pazocal{A}_{\infty}(\pazocal{O}_{\infty})=(\mathbb{F}_{\infty}\otimes\pazocal{O}_{\infty})[\\![\pi_{\infty}\otimes
1]\\!]=(\mathbb{F}_{\infty}\otimes\mathbb{F}_{\infty})[\\![1\otimes\pi_{\infty},\pi_{\infty}\otimes
1]\\!]$

which allows us to conclude that $\pazocal{A}_{\infty}(\pazocal{O}_{\infty})$
is complete for the $(1\otimes\pi_{\infty})$-adic topology. ∎

###### Proof of Proposition 5.11.

The desired isomorphism results of the composition

${\displaystyle\frac{\hat{\pazocal{N}}_{\infty}}{(\iota-\tau_{\pazocal{M}})(\hat{\pazocal{M}}_{\infty})}\otimes_{\pazocal{O}_{\infty}}K_{\infty}}$${\displaystyle\frac{(1\otimes\pi_{\infty})^{t}\hat{\pazocal{N}}_{\infty}}{(\iota-\tau_{\pazocal{M}})((1\otimes\pi_{\infty})^{t}\hat{\pazocal{M}}_{\infty})}\otimes_{\pazocal{O}_{\infty}}K_{\infty}}$${\displaystyle\frac{\hat{\pazocal{N}}_{\infty}}{\iota(\hat{\pazocal{M}}_{\infty})}\otimes_{\pazocal{O}_{\infty}}K_{\infty}}$${\displaystyle\frac{(1\otimes\pi_{\infty})^{t}\hat{\pazocal{N}}_{\infty}}{\iota((1\otimes\pi_{\infty})^{t}\hat{\pazocal{M}}_{\infty})}\otimes_{\pazocal{O}_{\infty}}K_{\infty}}$$\scriptstyle{\eqref{eq:quotient-
iota-tau}}$$\scriptstyle{\sim}$Lemma
5.15$\scriptstyle{\wr}$$\scriptstyle{\sim}$$\scriptstyle{\eqref{eq:quotient-
iota}}$

For $v\in\hat{\pazocal{N}}_{\infty}\otimes_{\pazocal{O}_{\infty}}K_{\infty}$,
the dashed morphism maps

$v+(\iota-\tau_{\pazocal{M}})(\hat{\pazocal{M}}_{\infty})\otimes_{\pazocal{O}_{\infty}}K_{\infty}\longmapsto
v^{\prime}+\iota(\hat{\pazocal{M}}_{\infty})\otimes_{\pazocal{O}_{\infty}}K_{\infty},$

where $v^{\prime}$ is any element of
$\hat{\pazocal{N}}_{\infty}\otimes_{\pazocal{O}_{\infty}}K_{\infty}$
satisfying

$v^{\prime}-v\in\iota(\hat{\pazocal{M}}_{\infty})\otimes_{\pazocal{O}_{\infty}}K_{\infty}+(\iota-\tau_{\pazocal{M}})(\hat{\pazocal{M}}_{\infty})\otimes_{\pazocal{O}_{\infty}}K_{\infty}.$

∎

Recall that the morphism $\nu:A\to K_{\infty}[\\![\mathfrak{j}]\\!]$,
$a\mapsto a\otimes 1$, extends to $K_{\infty}$ (e.g. [Gaz22, Lem. 5.1]). We
record:

######  Lemma 5.16.

The kernel $\mathfrak{v}$ of $\nu\otimes\operatorname{id}:K_{\infty}\otimes
K_{\infty}\to K_{\infty}[\\![\mathfrak{j}]\\!]$ corresponds to the ideal
generated by the set $\\{f\otimes 1-1\otimes
f~{}|~{}f\in\mathbb{F}_{\infty}\\}$.

###### Proof.

Let $d_{\infty}:=[\mathbb{F}_{\infty}:\mathbb{F}]$. For
$i\in\mathbb{Z}/d_{\infty}\mathbb{Z}$, we consider the ideal of
$K_{\infty}\otimes K_{\infty}$ given by

$\mathfrak{d}^{(i)}=\langle\\{f\otimes 1-1\otimes
f^{q^{i}}~{}|~{}f\in\mathbb{F}_{\infty}\\}\rangle.$

It is the kernel of the map $K_{\infty}\otimes K_{\infty}\to K_{\infty}$,
$a\otimes b\mapsto ab^{q^{i}}$, hence is a maximal ideal. For
$f\in\mathbb{F}_{\infty}$, the polynomial
$\prod_{i\in\mathbb{Z}/d_{\infty}\mathbb{Z}}{(x-f^{q^{i}})}$ belongs to
$\mathbb{F}[x]$, and thus the product of the $\mathfrak{d}^{(i)}$ is zero. By
the chinese remainders Theorem, we have

$K_{\infty}\otimes K_{\infty}=K_{\infty}\otimes
K_{\infty}/\mathfrak{d}^{(0)}\mathfrak{d}^{(1)}\cdots\mathfrak{d}^{(d_{\infty}-1)}=\prod_{i\in\mathbb{Z}/d_{\infty}\mathbb{Z}}{K_{\infty}\otimes
K_{\infty}/\mathfrak{d}^{(i)}}$

which is a product of $d_{\infty}$ fields. Because $\mathfrak{v}$ is a prime
ideal of $K_{\infty}\otimes K_{\infty}$, we have
$\mathfrak{v}=\mathfrak{d}^{(i)}$ for some $i$. If $f\in\mathbb{F}_{\infty}$,
then $f\otimes 1-1\otimes f$ belongs to $\mathfrak{v}$ by definition. We
deduce that $i=0$. ∎

We are almost in position to prove the main result of this section.

######  Theorem 5.17.

Let $(\pazocal{M},\pazocal{N},\tau_{\pazocal{M}})$ be a $C\times C$-shtuka
model for $\underline{M}$. Then, there is an isomorphism of
$K_{\infty}$-vector spaces

$\displaystyle\frac{\hat{\pazocal{N}}(\operatorname{Spf}\pazocal{O}_{\infty}\hat{\otimes}\pazocal{O}_{\infty})}{(\iota-\tau_{\pazocal{M}})\hat{\pazocal{M}}(\operatorname{Spf}\pazocal{O}_{\infty}\hat{\otimes}\pazocal{O}_{\infty})}\otimes_{\pazocal{O}_{\infty}}K_{\infty}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\displaystyle\frac{(M+\tau_{M}(\tau^{*}M))\otimes_{A\otimes
K}K_{\infty}[\\![\mathfrak{j}]\\!]}{M\otimes_{A\otimes
K}K_{\infty}[\\![\mathfrak{j}]\\!]}$

where the $K_{\infty}$-vector space structure on the right-hand side is given
through $\nu$.

We begin with two preliminary lemmas concerning the ring
$\pazocal{B}_{\infty}(\pazocal{O}_{\infty})$.

######  Lemma 5.18.

Let $\mathfrak{d}\subset\pazocal{O}_{\infty}\otimes\pazocal{O}_{\infty}$ be
the ideal generated by elements of the form $a\otimes 1-1\otimes a$ for
$a\in\pazocal{O}_{\infty}$. The canonical morphism

$\displaystyle\frac{\pazocal{O}_{\infty}\otimes\pazocal{O}_{\infty}}{\mathfrak{d}^{m}}\longrightarrow\displaystyle\frac{\pazocal{A}_{\infty}(\pazocal{O}_{\infty})}{\mathfrak{d}^{m}\pazocal{A}_{\infty}(\pazocal{O}_{\infty})}$

is an isomorphism for all $m\geq 1$.

###### Proof.

The sequence of $\pazocal{O}_{\infty}\otimes\pazocal{O}_{\infty}$-modules
$0\to\mathfrak{d}\to\pazocal{O}_{\infty}\otimes\pazocal{O}_{\infty}\to\pazocal{O}_{\infty}\to
0$ is exact, and taking completion with respect to the $\pi_{\infty}\otimes
1$-adic topology on $\pazocal{O}_{\infty}\otimes\pazocal{O}_{\infty}$ and the
$\pi_{\infty}$-adic topology on $\pazocal{O}_{\infty}$ reads:

$0\longrightarrow\mathfrak{d}\pazocal{A}_{\infty}(\pazocal{O}_{\infty})\longrightarrow\pazocal{A}_{\infty}(\pazocal{O}_{\infty})\longrightarrow
K_{\infty}\longrightarrow 0$

(e.g. (0315)) and the case $m=1$ follows. Before treating the general
$m$-case, observe that $\mathfrak{d}/\mathfrak{d}^{2}$ is the
$\pazocal{O}_{\infty}$-module $\Omega^{1}_{\pazocal{O}_{\infty}/\mathbb{F}}$
of Kähler differentials. In particular, $\mathfrak{d}/\mathfrak{d}^{2}$ is a
free $\pazocal{O}_{\infty}$-module of rank $1$. We deduce that for any
$r\in\mathfrak{d}\setminus\mathfrak{d}^{2}$, the multiplication by $r$ induces
an isomorphism of one-ranked $\pazocal{O}_{\infty}$-modules:

$(\pazocal{O}_{\infty}\otimes\pazocal{O}_{\infty})/\mathfrak{d}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathfrak{d}/\mathfrak{d}^{2}.$

It follows that
$\mathfrak{d}=\mathfrak{d}^{2}+r\cdot\pazocal{O}_{\infty}\otimes\pazocal{O}_{\infty}$
and hence $\mathfrak{d}^{m-1}=\mathfrak{d}^{m}+r\mathfrak{d}^{m-1}$ for all
$m\geq 1$. From Nakayama’s Lemma, $\mathfrak{d}^{m-1}\neq\mathfrak{d}^{m}$ and
we deduce from the sequence of isomorphisms

$\mathfrak{d}/\mathfrak{d}^{2}\stackrel{{\scriptstyle\times
r}}{{\to}}\mathfrak{d}^{2}/\mathfrak{d}^{3}\stackrel{{\scriptstyle\times
r}}{{\to}}\cdots\stackrel{{\scriptstyle\times
r}}{{\to}}\mathfrak{d}^{m-1}/\mathfrak{d}^{m}$

that $\mathfrak{d}^{m-1}/\mathfrak{d}^{m}$ is free of rank $1$ over
$\pazocal{O}_{\infty}$. It follows that there is an exact sequence

$0\longrightarrow\mathfrak{d}^{m}\longrightarrow\mathfrak{d}^{m-1}\longrightarrow\pazocal{O}_{\infty}\longrightarrow
0.$

Similarly, taking completions yields:

$0\longrightarrow\mathfrak{d}^{m}\pazocal{A}_{\infty}(\pazocal{O}_{\infty})\longrightarrow\mathfrak{d}^{m-1}\pazocal{A}_{\infty}(\pazocal{O}_{\infty})\longrightarrow\pazocal{O}_{\infty}\longrightarrow
0.$

Hence, for all $m\geq 1$, the canonical map

$\mathfrak{d}^{m-1}/\mathfrak{d}^{m}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathfrak{d}^{m-1}\pazocal{A}_{\infty}(\pazocal{O}_{\infty})/\mathfrak{d}^{m}\pazocal{A}_{\infty}(\pazocal{O}_{\infty}).$
(5.9)

is an isomorphism.

Back to the proof of the lemma, where we so far only proved the case $m=1$.
The general $m$-case follows by induction using the Snake Lemma on the diagram

${\mathfrak{d}^{m-1}/\mathfrak{d}^{m}}$${\pazocal{O}_{\infty}\otimes\pazocal{O}_{\infty}/\mathfrak{d}^{m}}$${\pazocal{O}_{\infty}\otimes\pazocal{O}_{\infty}/\mathfrak{d}^{m-1}}$${\mathfrak{d}^{m-1}\pazocal{A}_{\infty}(\pazocal{O}_{\infty})/\mathfrak{d}^{m}\pazocal{A}_{\infty}(\pazocal{O}_{\infty})}$${\pazocal{A}_{\infty}(\pazocal{O}_{\infty})/\mathfrak{d}^{m}\pazocal{A}_{\infty}(\pazocal{O}_{\infty})}$${\pazocal{A}_{\infty}(\pazocal{O}_{\infty})/\mathfrak{d}^{m-1}\pazocal{A}_{\infty}(\pazocal{O}_{\infty})}$$\scriptstyle{\wr}$$\scriptstyle{\eqref{eq:isomorphis-
m-m-1}}$$\scriptstyle{\wr~{}\text{hypothesis}}$

where our induction hypothesis implies that the middle vertical map is an
isomorphism. ∎

###### Proof of Theorem 5.17.

Let $\operatorname{Spec}A\sqcup\operatorname{Spec}B\to C$ be an affine Zariski
cover of $C$. Let $N_{B}:=\pazocal{N}(\operatorname{Spec}B\otimes B)$ and
$M_{B}:=\pazocal{M}(\operatorname{Spec}B\otimes B)$. From the sheaf property,
we have an isomorphism of $K_{\infty}\otimes K_{\infty}$-modules:

$\frac{N_{A}}{M_{A}}\otimes_{A\otimes A}(K_{\infty}\otimes
K_{\infty})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\frac{N_{B}}{M_{B}}\otimes_{B\otimes
B}(K_{\infty}\otimes K_{\infty}).$ (5.10)

By definition, there exists a positive integer $m$ for which the right-hand
side of (5.10) is annihilated by $\mathfrak{d}^{m}$, where
$\mathfrak{d}:=\ker(\pazocal{O}_{\infty}\otimes\pazocal{O}_{\infty}\to\pazocal{O}_{\infty})$.
Hence,

$\frac{N_{B}}{M_{B}}\otimes_{B\otimes B}(K_{\infty}\otimes
K_{\infty})\cong\frac{N_{B}}{M_{B}}\otimes_{B\otimes
B}\left(\frac{K_{\infty}\otimes
K_{\infty}}{\mathfrak{d}^{m}}\right)\cong\frac{N_{B}}{M_{B}}\otimes_{B\otimes
B}\left(\frac{K_{\infty}\otimes\pazocal{O}_{\infty}}{\mathfrak{d}^{m}}\right).$
(5.11)

The last isomorphism above comes from the fact the any element of the form
$(1\otimes a)\in K_{\infty}\otimes\pazocal{O}_{\infty}$ is coprime to
$\mathfrak{d}^{m}$, thus invertible in
$K_{\infty}\otimes\pazocal{O}_{\infty}/\mathfrak{d}^{m}$. Applying Lemma 5.18,
one gets

$\displaystyle\frac{N_{B}}{M_{B}}\otimes_{B\otimes B}(K_{\infty}\otimes
K_{\infty})$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:d-torsion}}}{{\cong}}\frac{N_{B}}{M_{B}}\otimes_{B\otimes
B}\left(\frac{K_{\infty}\otimes\pazocal{O}_{\infty}}{\mathfrak{d}^{m}}\right)$
$\displaystyle\cong\frac{N_{B}}{M_{B}}\otimes_{B\otimes
B}\left(\frac{\pazocal{B}_{\infty}(\pazocal{O}_{\infty})}{\mathfrak{d}^{m}}\right)$
$\displaystyle\cong\frac{N_{B}}{M_{B}}\otimes_{B\otimes
B}\pazocal{B}_{\infty}(\pazocal{O}_{\infty})$
$\displaystyle\cong\frac{\hat{\pazocal{N}}_{\infty}}{\hat{\pazocal{M}}_{\infty}}\otimes_{\pazocal{O}_{\infty}}K_{\infty}.$
(5.12)

On the other-hand, the left-hand side of (5.10) is annihilated by both
$\mathfrak{j}^{m}$ and $\mathfrak{v}^{m}$ for a large enough integer $m$,
where $\mathfrak{v}$ is the ideal of $K_{\infty}\otimes K_{\infty}$ generated
by $\\{f\otimes 1-1\otimes f~{}|~{}f\in\mathbb{F}_{\infty}\\}$: by Lemma 5.16,
$\mathfrak{v}=\ker(K_{\infty}\otimes K_{\infty}\to
K_{\infty}[\\![\mathfrak{j}]\\!])$. Taking $m$ to be a large enough power of
$q$, we can even assume $\mathfrak{v}=\mathfrak{v}^{m}$, so that

$\displaystyle\frac{N_{A}}{M_{A}}\otimes_{A\otimes A}(K_{\infty}\otimes
K_{\infty})$ $\displaystyle\cong\frac{N_{A}}{M_{A}}\otimes_{A\otimes
A}\left(\frac{K_{\infty}\otimes
K_{\infty}}{(\mathfrak{v}+\mathfrak{j}^{m})K_{\infty}\otimes
K_{\infty}}\right)$ $\displaystyle\cong\frac{N_{A}}{M_{A}}\otimes_{A\otimes
A}\left(\frac{K_{\infty}[\\![\mathfrak{j}]\\!]}{\mathfrak{j}^{m}K_{\infty}[\\![\mathfrak{j}]\\!]}\right)$
$\displaystyle\cong\frac{N_{A}}{M_{A}}\otimes_{A\otimes
A}K_{\infty}[\\![\mathfrak{j}]\\!]$
$\displaystyle\cong\frac{N_{A}\otimes_{A\otimes
A}K_{\infty}[\\![\mathfrak{j}]\\!]}{M_{A}\otimes_{A\otimes
A}K_{\infty}[\\![\mathfrak{j}]\\!]}.$ (5.13)

The composition of (5.3), (5.10) and (5.3) gives the desired isomorphism:

$\frac{\hat{\pazocal{N}}_{\infty}}{\hat{\pazocal{M}}_{\infty}}\otimes_{\pazocal{O}_{\infty}}K_{\infty}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\frac{(M+\tau_{M}(\tau^{*}M))\otimes_{A\otimes
K}K_{\infty}[\\![\mathfrak{j}]\\!]}{M\otimes_{A\otimes
K}K_{\infty}[\\![\mathfrak{j}]\\!]}.$

Pre-composition with the isomorphism of Proposition 5.11 gives the desired
isomorphism. ∎

As announced, we have:

######  Corollary 5.19.

There is an exact sequence of $K_{\infty}$-vector spaces:

$0\longrightarrow\underline{M}_{B}^{+}\otimes_{A}K_{\infty}\longrightarrow\frac{\hat{\pazocal{N}}_{\infty}}{(\iota-\tau_{\pazocal{M}})(\hat{\pazocal{M}}_{\infty})}\otimes_{\pazocal{O}_{\infty}}K_{\infty}\longrightarrow\operatorname{Ext}^{1,\operatorname{ha}}_{\infty}(\mathbbm{1}^{+},\mathscr{H}^{+}_{K_{\infty}}(\underline{M}))\longrightarrow
0.$

###### Proof.

This is a consequence of the combination of Theorem 5.17 and Theorem 3.28. ∎

##  6 Proof of the main theorems

###  6.1 Cohomological computations

In this subsection, we recall general facts on coherent cohomology of schemes.
This will subsequently be applied to sthuka models in the next subsection to
achieve the proof of Theorems 4.1 and 4.4. We refer to [Wei94] for the
definitions of homological algebra (cones, distinguished triangles, derived
categories, etc.)

To fix the setting, we consider the following commutative square in the
category of schemes over $C$:

${\operatorname{Spec}K_{\infty}\times
C}$${\operatorname{Spec}\pazocal{O}_{\infty}\times
C}$${\operatorname{Spec}K\times C}$${C\times
C}$$\scriptstyle{i}$$\scriptstyle{j}$$\scriptstyle{q}$$\scriptstyle{p}$ (6.1)

The next result is classical:

######  Proposition 6.1.

Let $\pazocal{F}$ be a sheaf of $\pazocal{O}_{C\times C}$-modules. In the
derived category of $K_{\infty}$-modules, there is a quasi-isomorphism

$R\Gamma(\operatorname{Spec}A\times C,\pazocal{F})\otimes_{A}K_{\infty}\cong
R\Gamma(\operatorname{Spec}\pazocal{O}_{\infty}\times
C,q^{*}\pazocal{F})\otimes_{\pazocal{O}_{\infty}}K_{\infty}$

which is functorial in $\pazocal{F}$.

Let now $S$ be a scheme and let $T$ be a separated scheme over $S$. Let $U$,
$V$ and $W$ be affine schemes over $S$ which insert in a commutative diagram
of $S$-schemes

${U}$${T}$${W}$${V}$$\scriptstyle{i}$$\scriptstyle{k}$$\scriptstyle{j}$

such that $\\{U\to T,V\to T\\}$ forms a covering of $T$.

For $\pazocal{F}$ a sheaf of $\pazocal{O}_{T}$-modules, we denote by
$S(\pazocal{F})$ the sequence of $\pazocal{O}_{T}$-modules:

$0\longrightarrow\pazocal{F}\longrightarrow i_{*}i^{*}\pazocal{F}\oplus
j_{*}j^{*}\pazocal{F}\longrightarrow k_{*}k^{*}\pazocal{F}\longrightarrow 0$

where the morphisms are given by the adjunction unit (note that the data of
$S(\pazocal{F})$ is functorial in $\pazocal{F}$). The next lemma is of
fundamental importance for our cohomological computations:

######  Lemma 6.2.

Assume that $S(\pazocal{O}_{T})$ is exact. Then, for any finite locally free
sheaf $\pazocal{F}$ of $\pazocal{O}_{T}$-modules, $S(\pazocal{F})$ is exact.
In particular, the natural map

$R\Gamma(T,\pazocal{F})\longrightarrow\left[\pazocal{F}(U)\oplus\pazocal{F}(V)\longrightarrow\pazocal{F}(W)\right],$
(6.2)

where the right-hand side is a complex concentrated in degrees $0$ and $1$, is
a quasi-isomorphism.

###### Proof.

We show that $S(\pazocal{F})$ is an exact sequence (the second assertion
follows, since applying $R\Gamma(T,-)$ to $S(\pazocal{F})$ yields the
distinguished triangle computing (6.2)). To prove exactness of
$S(\pazocal{F})$, first note that $i$, $j$ and $k$ are affine morphisms
because $T$ is separated (01SG). Thus, the pushforward functors appearing in
$S(\pazocal{F})$ are naturally isomorphic to their right-derived functor
(0G9R). Thereby, $S(\pazocal{F})$ is naturally isomorphic in
$\operatorname{D}_{qc}(T)$, the derived category of quasi-coherent sheaves
over $T$, to the triangle

$\pazocal{F}\longrightarrow Ri_{*}i^{*}\pazocal{F}\oplus
Rj_{*}j^{*}\pazocal{F}\longrightarrow
Rk_{*}k^{*}\pazocal{F}\longrightarrow[1]$ (6.3)

and it is sufficient to show that the latter is distinguished. Yet, because
$\pazocal{F}$ is finite locally-free, the projection formula (01E8) implies
that (6.3) is naturally isomorphic to

$\pazocal{F}\otimes_{\pazocal{O}_{T}}^{\mathbf{L}}\pazocal{O}_{T}\longrightarrow\pazocal{F}\otimes_{\pazocal{O}_{T}}^{\mathbf{L}}(Ri_{*}\pazocal{O}_{U}\oplus
Rj_{*}\pazocal{O}_{V})\longrightarrow\pazocal{F}\otimes_{\pazocal{O}_{T}}^{\mathbf{L}}Rk_{*}\pazocal{O}_{W}\longrightarrow[1]$

Because $\pazocal{F}$ is locally-free, the functor
$\pazocal{F}\otimes_{\pazocal{O}_{T}}^{\mathbf{L}}-$ is exact on
$\operatorname{D}_{qc}(T)$ and it suffices to show the distinguishness of

$\pazocal{O}_{T}\longrightarrow Ri_{*}\pazocal{O}_{U}\oplus
Rj_{*}\pazocal{O}_{V}\longrightarrow Rk_{*}\pazocal{O}_{W}\longrightarrow[1].$

But because $\pazocal{O}_{U}=i^{*}\pazocal{O}_{T}$,
$\pazocal{O}_{V}=j^{*}\pazocal{O}_{T}$ and
$\pazocal{O}_{W}=k^{*}\pazocal{O}_{T}$, this follows from our assumption that
$S(\pazocal{O}_{T})$ is exact. We conclude that (6.3) is distinguished. ∎

Assuming that $T$ is a smooth variety555By variety over $k$, we mean that $T$
is integral and that $T\to\operatorname{Spec}k$ is separated and of finite
type. over a field allows us to relax the "locally free" assumption in Lemma
6.2 to "coherent".

######  Proposition 6.3.

Let $k$ be a field and assume that $S=\operatorname{Spec}k$. Assume further
that $T$ is a smooth variety over $k$, and that $i$, $j$ and $k$ are flat. Let
$\pazocal{F}$ be a coherent sheaf on $X$. Then, $S(\pazocal{F})$ is exact. In
particular, the natural map

$R\Gamma(T,\pazocal{F})\longrightarrow\left[\pazocal{F}(U)\oplus\pazocal{F}(V)\longrightarrow\pazocal{F}(W)\right]$

is a quasi-isomorphism.

###### Proof.

Choose a resolution of $\pazocal{F}$ by finite locally free sheaves
$0\to\pazocal{F}_{n}\to\cdots\to\pazocal{F}_{0}\to\pazocal{F}\to 0$. Because
$i$ (resp. $j$, $k$) is flat, $i^{*}$ (resp. $j^{*}$, $k^{*}$) is an exact
functor on quasi-coherent sheaves. Because it is affine, $i_{*}$ (resp.
$j_{*}$, $k_{*}$) is an exact functor on quasi-coherent sheaves. Thereby, for
all $s\in\\{0,...,n\\}$, the sequence $S(\pazocal{F}_{s})$ is exact by Lemma
6.2. Using the $n\times n$-Lemma in the abelian category of quasi-coherent
sheaves of $\pazocal{O}_{T}$-modules, we deduce that $S(\pazocal{F})$ is
exact. ∎

The main result of this subsection is:

######  Theorem 6.4.

Assume the setting of Proposition 6.3. Let $\pazocal{F}^{\prime}$ be a
coherent sheaf of $\pazocal{O}_{T}$-module and let
$f:\pazocal{F}\to\pazocal{F}^{\prime}$ be a morphism of sheaves of abelian
groups. Then, the rows and the lines of the following diagram

${R\Gamma(T,\pazocal{F})}$${\pazocal{F}(U)\oplus\pazocal{F}(V)}$${\pazocal{F}(W)}$${\left[1\right]}$${R\Gamma(T,\pazocal{F}^{\prime})}$${\pazocal{F}^{\prime}(U)\oplus\pazocal{F}^{\prime}(V)}$${\pazocal{F}^{\prime}(W)}$${\left[1\right]}$${\operatorname{cone}(f_{T})}$${\operatorname{cone}(f_{U})\oplus\operatorname{cone}(f_{V})}$${\operatorname{cone}(f_{W})}$${\left[1\right]}$${\left[1\right]}$${\left[1\right]}$${\left[1\right]}$$\scriptstyle{f_{T}}$$\scriptstyle{f_{U}\oplus
f_{V}}$$\scriptstyle{f_{W}}$ (6.4)

form distinguished triangles in the derived category of abelian groups, where
$f_{Y}:=R\Gamma(Y,f)$ (for $Y\in\\{T,U,V,W\\}$).

###### Proof.

We lift the first two lines in the category of chain complexes: by Lemma 6.2,
the diagram

${0}$${\pazocal{F}}$${i_{*}i^{*}\pazocal{F}\oplus
j_{*}j^{*}\pazocal{F}}$${k_{*}k^{*}\pazocal{F}}$${0}$${0}$${\pazocal{F}^{\prime}}$${i_{*}i^{*}\pazocal{F}^{\prime}\oplus
j_{*}j^{*}\pazocal{F}^{\prime}}$${k_{*}k^{*}\pazocal{F}^{\prime}}$${0}$$\scriptstyle{f}$$\scriptstyle{i_{*}i^{*}f\oplus
j_{*}j^{*}f}$$\scriptstyle{k_{*}k^{*}f}$ (6.5)

is exact on lines and commutative on squares in the category of quasi-coherent
sheaves of $\pazocal{O}_{T}$-modules. From (013T) we can find injective
resolutions $\pazocal{F}\to I^{\bullet}_{1}$, $i_{*}i^{*}\pazocal{F}\oplus
j_{*}j^{*}\pazocal{F}\to I^{\bullet}_{2}$ and $k_{*}k^{*}\pazocal{F}\to
I^{\bullet}_{3}$ (respectively $\pazocal{F}^{\prime}\to J^{\bullet}_{1}$,
$i_{*}i^{*}\pazocal{F}^{\prime}\oplus j_{*}j^{*}\pazocal{F}^{\prime}\to
J^{\bullet}_{2}$ and $k_{*}k^{*}\pazocal{F}^{\prime}\to J^{\bullet}_{3}$) such
that

${0}$${I^{\bullet}_{1}}$${I^{\bullet}_{2}}$${I^{\bullet}_{3}}$${0}$${0}$${J^{\bullet}_{1}}$${J^{\bullet}_{2}}$${J^{\bullet}_{3}}$${0}$$\scriptstyle{i_{1}}$$\scriptstyle{i_{2}}$$\scriptstyle{i_{3}}$

is an injective resolution of the whole diagram (6.5). Completing the vertical
maps into distinguished triangles gives:

${0}$${I^{\bullet}_{1}}$${I^{\bullet}_{2}}$${I^{\bullet}_{3}}$${0}$${0}$${J^{\bullet}_{1}}$${J^{\bullet}_{2}}$${J^{\bullet}_{3}}$${0}$${0}$${\operatorname{cone}(i_{1})}$${\operatorname{cone}(i_{2})}$${\operatorname{cone}(i_{3})}$${0}$${\left[1\right]}$${\left[1\right]}$${\left[1\right]}$$\scriptstyle{i_{1}}$$\scriptstyle{i_{2}}$$\scriptstyle{i_{3}}$
(6.6)

where the rows are distinguished triangles. The third line is a direct sum of
exact sequences and therefore is exact. The horizontal exact sequences
transform to distinguished triangles in the derived category of abelian
module. This concludes. ∎

Under Noetherianity assumptions, Theorem 6.4 can be extended to the case of
formal schemes. Our main reference is [FK18, §I]. From now on, we assume that
$T$, $U$, $V$ and $W$ are Noetherian schemes over $S$. Let $T^{\prime}\subset
T$, $U^{\prime}\subset U$, $V^{\prime}\subset V$ and $W^{\prime}\subset W$ be
closed subschemes such that $i^{-1}(T^{\prime})=U^{\prime}$,
$j^{-1}(T^{\prime})=V^{\prime}$, and
$p^{-1}(U^{\prime})=W^{\prime}=q^{-1}(V^{\prime})$. It follows that
$k^{-1}(T^{\prime})=W^{\prime}$. Let $\hat{T}$, $\hat{U}$, $\hat{V}$ and
$\hat{W}$ be the formal completions along the corresponding closed subschemes
[FK18, §I.1.4]. We obtain a commutative diagram of formal schemes

${\hat{U}}$${\hat{T}}$${\hat{W}}$${\hat{V}}$$\scriptstyle{\hat{i}}$$\scriptstyle{\hat{k}}$$\scriptstyle{\hat{j}}$

Given an adically quasi-coherent sheaf666e.g. the formal completion of a
quasi-coherent sheaf with respect to a closed subscheme of finite presentation
is adically quasi-coherent by [FK18, §I, Prop.3.1.5] $\pazocal{F}$ of
$\pazocal{O}_{\hat{T}}$-modules [FK18, §I,Def.3.1.3], we consider the sequence

$\hat{S}(\pazocal{F}):\quad
0\to\pazocal{F}\to\hat{i}_{*}\hat{i}^{*}\pazocal{F}\oplus\hat{j}_{*}\hat{j}^{*}\pazocal{F}\to\hat{k}_{*}\hat{k}^{*}\pazocal{F}\to
0.$

######  Lemma 6.5.

Let $\pazocal{F}$ be a quasi-coherent sheaf on $T$. Then
$\widehat{S(\pazocal{F})}\cong\hat{S}(\hat{\pazocal{F}})$, where
$\pazocal{G}\mapsto\hat{\pazocal{G}}$ denotes the formal completion functor
along $T^{\prime}$. In particular, if $S(\pazocal{F})$ is exact, then
$\hat{S}(\hat{\pazocal{F}})$ is exact.

###### Proof.

This almost follows from the flat-base change Theorem (02KH). Indeed, the
diagram

${\hat{U}}$${U}$${\hat{T}}$${T}$$\scriptstyle{\hat{i}}$$\scriptstyle{f_{U}}$$\scriptstyle{i}$$\scriptstyle{f_{T}}$

where $f_{U}$ and $f_{T}$ are the canonical maps, is Cartesian. Because $i$ is
affine, $i$ is quasi-compact and quasi-separated (01S7). On the other-hand,
$f_{T}$ is flat and the flat-base change Theorem applies. It states that for
any quasi-coherent sheaf $\pazocal{G}$ of $\pazocal{O}_{U}$-modules, the
natural map

$f_{T}^{*}Ri_{*}\pazocal{G}\longrightarrow R\hat{i}_{*}(f_{U}^{*}\pazocal{G})$

is a quasi-isomorphism in the derived category of $\pazocal{O}_{U}$-modules.
Because $i$ is affine, the functors $R\hat{i}_{*}$ and $i_{*}$ are isomorphic
on the category of coherent sheaves (0G9R). Similarly, but in the setting of
formal geometry, $\hat{i}$ is also affine [FK18, §I,Def.4.1.1], and the formal
analogue of the previous argument [FK18, §I,Thm.7.1.1] reads that the functors
$R\hat{i}_{*}$ and $\hat{i_{*}}$ are isomorphic on the category of adically
quasi-coherent sheaves. Therefore, in the derived category of
$\pazocal{O}_{U}$-modules, we have an isomorphism

$f_{T}^{*}i_{*}\pazocal{G}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\hat{i}_{*}f_{U}^{*}\pazocal{G}.$

Applied to $\pazocal{G}=i^{*}\pazocal{F}$ for a quasi-coherent $\pazocal{F}$
on $T$, we obtain
$f_{T}^{*}i_{*}i^{*}\pazocal{F}\cong\hat{i}_{*}\hat{i}^{*}f_{T}^{*}\pazocal{F}$
functorially in $\pazocal{F}$. In other words,

$\widehat{i_{*}i^{*}\pazocal{F}}\cong\hat{i}_{*}\hat{i}^{*}\hat{\pazocal{F}}.$

The very same argument for $j$ and $k$ in place of $i$ yields respectively
$\widehat{j_{*}j^{*}\pazocal{F}}\cong\hat{j}_{*}\hat{j}^{*}\hat{\pazocal{F}}$
and
$\widehat{k_{*}k^{*}\pazocal{F}}\cong\hat{k}_{*}\hat{k}^{*}\hat{\pazocal{F}}$.
It follows that $\widehat{S(\pazocal{F})}\cong\hat{S}(\hat{\pazocal{F}})$.
Since the formal completion functor is exact, $\hat{S}(\hat{\pazocal{F}})$ is
exact if $S(\pazocal{F})$ is. ∎

Thanks to Lemma 6.5, the proof of Theorem 6.4 blithely applies to the formal
situation:

######  Theorem 6.6.

Assume the setting of Theorem 6.4. Then, each rows and each lines of the
following diagram

${R\Gamma(\hat{T},\hat{\pazocal{F}})}$${\hat{\pazocal{F}}(\hat{U})\oplus\hat{\pazocal{F}}(\hat{V})}$${\hat{\pazocal{F}}(\hat{W})}$${\left[1\right]}$${R\Gamma(\hat{T},\hat{\pazocal{F}}^{\prime})}$${\hat{\pazocal{F}}^{\prime}(\hat{U})\oplus\hat{\pazocal{F}}^{\prime}(\hat{V})}$${\hat{\pazocal{F}}^{\prime}(\hat{W})}$${\left[1\right]}$${\operatorname{cone}(\hat{f}_{\hat{T}})}$${\operatorname{cone}(\hat{f}_{\hat{U}})\oplus\operatorname{cone}(\hat{f}_{\hat{V}})}$${\operatorname{cone}(\hat{f}_{\hat{W}})}$${\left[1\right]}$${\left[1\right]}$${\left[1\right]}$${\left[1\right]}$$\scriptstyle{\hat{f}_{\hat{T}}}$$\scriptstyle{\hat{f}_{\hat{U}}\oplus\hat{f}_{\hat{V}}}$$\scriptstyle{\hat{f}_{\hat{W}}}$

form distinguished triangles in the derived category of abelian groups, where
$\hat{f}_{\hat{Y}}:=R\Gamma(\hat{Y},\hat{f})$ (for $Y\in\\{T,U,V,W\\}$).

###  6.2 Proof of Theorems 4.1 and 4.4

Let $\underline{M}$ be a rigid analytically trivial $A$-motive over $K$. We
now assemble the ingredients collected in the last subsections to end the
proof of Theorems 4.1 and 4.4. As promised, the complex $G_{\underline{M}}$
admits an interpretation in terms of the Zariski cohomology of $C$-shtuka
models of $\underline{M}$ (Definition 5.1):

######  Proposition 6.7.

Let $(\pazocal{N},\pazocal{M},\tau_{M})$ be a $C$-shtuka model for
$\underline{M}$. Let $\iota$ denotes the inclusion of $\pazocal{M}$ in
$\pazocal{N}$. There is a quasi-isomorphism of $A$-module complexes

$G_{\underline{M}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{cone}\left(R\Gamma(\operatorname{Spec}A\times
C,\pazocal{M})\stackrel{{\scriptstyle\iota-\tau_{\pazocal{M}}}}{{\longrightarrow}}R\Gamma(\operatorname{Spec}A\times
C,\pazocal{N})\right).$

The first part of Theorem 4.1 follows from the above:

######  Corollary 6.8.

The $A$-modules
$\operatorname{Ext}_{A,\infty}^{1,\operatorname{reg}}(\mathbbm{1},\underline{M})$
and $\operatorname{Cl}(\underline{M})$ are finitely generated.

###### Proof.

As $\operatorname{Spec}A\times C$ is proper over $\operatorname{Spec}A$, both
$R\Gamma(\operatorname{Spec}A\times C,\pazocal{M})$ and
$R\Gamma(\operatorname{Spec}A\times C,\pazocal{N})$ are perfect complexes of
$A$-modules. By Proposition 6.7, so is $G_{\underline{M}}$. We conclude by
Proposition 4.6. ∎

###### Proof of Proposition 6.7.

The main ingredients are the cohomological preliminaries of Section 6.1. We
consider the particular setting of $S=\operatorname{Spec}\mathbb{F}$ and of
the commutative diagram of $S$-schemes

${\operatorname{Spec}\pazocal{O}_{\infty}\langle
A\rangle}$${(\operatorname{Spec}A)\times
C}$${\operatorname{Spec}K_{\infty}\langle
A\rangle}$${\operatorname{Spec}A\otimes
A}$$\scriptstyle{i}$$\scriptstyle{p}$$\scriptstyle{k}$$\scriptstyle{q}$$\scriptstyle{j}$

Because $A$ is geometrically irreducible over $\mathbb{F}$,
$(\operatorname{Spec}A)\times C$ is a smooth variety over $\mathbb{F}$. To use
the results of Section 6.1, one requires the next two lemmas.

######  Lemma 6.9.

The morphisms $i$, $j$, $k$ are flat.

###### Proof.

We consider the affine open cover $(\operatorname{Spec}A\otimes
A)\cup(\operatorname{Spec}A\otimes B)$ of $(\operatorname{Spec}A)\times C$. We
first show that $i$ is flat. We have $i^{-1}(\operatorname{Spec}A\otimes
A)=\operatorname{Spec}K_{\infty}\langle A\rangle$ and
$i^{-1}(\operatorname{Spec}A\otimes
B)=\operatorname{Spec}\pazocal{O}_{\infty}\langle A\rangle$. The morphism
$A\otimes B\to\pazocal{O}_{\infty}\langle A\rangle$ is flat (because it is the
completion of the Noetherian ring $A\otimes B$ at the ideal
$\mathfrak{m}_{\infty}\subset B$) and thus, so is $A\otimes A\to
K_{\infty}\langle A\rangle$. By (01U5), $i$ is flat.

We have $j^{-1}(\operatorname{Spec}A\otimes B)=\operatorname{Spec}A\otimes D$,
where $D\subset K$ is the sub-$\mathbb{F}$-algebra such that
$\operatorname{Spec}D=\operatorname{Spec}A\cap\operatorname{Spec}B$. The
inclusion $B\to D$ is a localization, and hence $A\otimes B\to A\otimes D$ is
flat. Thereby, $j$ is flat.

Because $K_{\infty}\langle A\rangle\cong
K_{\infty}\otimes_{\pazocal{O}_{\infty}}\pazocal{O}_{\infty}\langle A\rangle$,
$p$ is flat. Since compositions of flat morphisms are flat, $k=i\circ p$ is
flat. ∎

######  Lemma 6.10.

For $T=(\operatorname{Spec}A)\times C$, the sequence $0\to\pazocal{O}_{T}\to
i_{*}i^{*}\pazocal{O}_{T}\oplus j_{*}j^{*}\pazocal{O}_{T}\to
k_{*}k^{*}\pazocal{O}_{T}\to 0$ is exact.

###### Proof.

We need to show that the complex $Z:=[\pazocal{O}_{\infty}\langle
A\rangle\oplus(A\otimes A)\to K_{\infty}\langle A\rangle]$, where the morphism
is the difference of the canonical inclusions, represents the sheaf cohomology
in the Zariski topology of $\pazocal{O}_{\operatorname{Spec}A\times C}$, the
latter being quasi-isomorphic to

$R\Gamma(\operatorname{Spec}A\times C,\pazocal{O}_{\operatorname{Spec}A\times
C})=[(A\otimes B)\oplus(A\otimes A)\to A\otimes D].$

Let $(t_{i})_{i\geq 0}$ be a (countable) basis of $A$ over $\mathbb{F}$. Any
element $f$ in $K_{\infty}\langle A\rangle$ can be represented uniquely by a
converging series

$f=\sum_{i=0}^{\infty}{t_{i}\otimes f_{i}},\quad t_{i}\in
K_{\infty},~{}t_{i}\to 0~{}(i\to\infty).$

Elements of $\pazocal{O}_{\infty}\langle A\rangle$ are the ones for which
$f_{i}\in\pazocal{O}_{\infty}$ ($\forall i\geq 0$) and elements of $A\otimes
A$ are the ones for which $f_{i}\in A$ ($\forall i\geq 0$) and $f_{i}=0$ for
$i$ large enough. Therefore, it is clear that $\pazocal{O}_{\infty}\langle
A\rangle\cap(A\otimes A)$ is $A\otimes(\pazocal{O}_{\infty}\cap A)$. Yet,
$\pazocal{O}_{\infty}\cap A$ is the constant field of $C$, showing that
$H^{0}(Z)=H^{0}(\operatorname{Spec}A\times
C,\pazocal{O}_{\operatorname{Spec}A\times C})$.

Because $K_{\infty}=\pazocal{O}_{\infty}+A+D$, the canonical map

$\frac{A\otimes D}{A\otimes B+A\otimes A}\to\frac{K_{\infty}\langle
A\rangle}{\pazocal{O}_{\infty}\langle A\rangle+A\otimes A}$

is surjective. Because $(A\otimes D)\cap\pazocal{O}_{\infty}\langle
A\rangle\subset A\otimes B+A\otimes A$, it is also injective. It follows that
$H^{1}(Z)=H^{1}(\operatorname{Spec}A\times
C,\pazocal{O}_{\operatorname{Spec}A\times C})$. ∎

Now, let $\underline{\pazocal{M}}=(\pazocal{N},\pazocal{M},\tau_{M})$ be a
$C$-shtuka model for $\underline{M}$. We have

$\displaystyle(j^{*}\pazocal{M})(\operatorname{Spec}A\otimes A)$
$\displaystyle=\pazocal{M}(\operatorname{Spec}A\otimes A)=M_{A},$
$\displaystyle(j^{*}\pazocal{N})(\operatorname{Spec}A\otimes A)$
$\displaystyle=\pazocal{N}(\operatorname{Spec}A\otimes A)=N_{A},$
$\displaystyle(k^{*}\pazocal{M})(\operatorname{Spec}K_{\infty}\langle
A\rangle)$
$\displaystyle=(k^{*}\pazocal{N})(\operatorname{Spec}K_{\infty}\langle
A\rangle)=M\otimes_{A\otimes K}K_{\infty}\langle A\rangle.$

Theorem 6.4 yields a morphism of distinguished triangles

${R\Gamma(\operatorname{Spec}A\times
C,\pazocal{M})}$${(i^{*}\pazocal{M})(\operatorname{Spec}\pazocal{O}_{\infty}\langle
A\rangle)}$${\displaystyle\frac{M\otimes_{A\otimes K}K_{\infty}\langle
A\rangle}{M_{A}}}$${\left[1\right]}$${R\Gamma(\operatorname{Spec}A\times
C,\pazocal{N})}$${(i^{*}\pazocal{N})(\operatorname{Spec}\pazocal{O}_{\infty}\langle
A\rangle)}$${\displaystyle\frac{M\otimes_{A\otimes K}K_{\infty}\langle
A\rangle}{N_{A}}}$${\left[1\right]}$${\operatorname{cone}(\iota-\tau_{\pazocal{M}}|\operatorname{Spec}A\times
C)}$${0}$${G_{\underline{M}}\left[1\right]}$${\left[1\right]}$$\scriptstyle{\iota-\tau_{\pazocal{M}}}$$\scriptstyle{\operatorname{id}-\tau_{\pazocal{M}}}$$\scriptstyle{\operatorname{id}-\tau_{M}}$

where the cone of the middle upper vertical morphism is zero by Lemma 5.3. The
third row is a distinguished triangle, and the proposition follows. ∎

Theorems 4.1 (second part) and 4.4 will follow from the study of the
cohomology of a $C\times C$-shtuka model of $\underline{\pazocal{M}}$ at
$\operatorname{Spf}\pazocal{O}_{\infty}\hat{\times}C$. The latter corresponds
to the completion of the Noetherian scheme $C\times C$ at the closed subscheme
$\\{\infty\\}\times C$. The argument given here is a refinement of the one
given in the proof of Proposition 6.7 where we use $C\times C$-shtuka models
instead of $C$-shtuka models. To ensure the existence of a $C\times C$-shtuka
model, we now assume that all the weights of $\underline{M}$ are negative
(Theorem 5.6).

We apply the results of Section 6.1 under a different setting. We consider the
commutative square of schemes over $\operatorname{Spec}\pazocal{O}_{\infty}$:

${\operatorname{Spec}\pazocal{O}_{\infty}\otimes
A}$${(\operatorname{Spec}\pazocal{O}_{\infty})\times
C}$${\operatorname{Spec}\pazocal{O}_{\infty}\otimes
K_{\infty}}$${\operatorname{Spec}\pazocal{O}_{\infty}\otimes\pazocal{O}_{\infty}}$$\scriptstyle{i}$$\scriptstyle{k}$$\scriptstyle{j}$

Similarly to Lemma 6.9, one shows that $i$, $j$ and $k$ are flat morphisms.
For the sake of compatibility of notations with the previous subsection, we
let $T=\operatorname{Spec}\pazocal{O}_{\infty}\times C$,
$U=\operatorname{Spec}(\pazocal{O}_{\infty}\otimes A)$,
$V=\operatorname{Spec}(\pazocal{O}_{\infty}\otimes\pazocal{O}_{\infty})$ and
$W=\operatorname{Spec}(\pazocal{O}_{\infty}\otimes K_{\infty})$. Consider the
respective closed subschemes $T^{\prime}=\\{\infty\\}\times C$,
$U^{\prime}=\\{\infty\\}\times\operatorname{Spec}A$,
$V=\\{\infty\\}\times\operatorname{Spec}\pazocal{O}_{\infty}$ and
$W=\\{\infty\\}\times\operatorname{Spec}K_{\infty}$ and the formal completions
$\hat{T}=\operatorname{Spf}\pazocal{O}_{\infty}\hat{\times}C$,
$\hat{U}=\operatorname{Spf}\pazocal{O}_{\infty}\hat{\times}\operatorname{Spec}A$,
$\hat{V}=\operatorname{Spf}\pazocal{O}_{\infty}\hat{\times}\operatorname{Spec}\pazocal{O}_{\infty}$
and
$\hat{W}=\operatorname{Spf}\pazocal{O}_{\infty}\hat{\times}\operatorname{Spec}K_{\infty}$.
We obtain the commutative square of formal schemes over
$\operatorname{Spf}\pazocal{O}_{\infty}$:

${\operatorname{Spf}\pazocal{O}_{\infty}\hat{\otimes}A}$${(\operatorname{Spf}\pazocal{O}_{\infty})\hat{\times}C}$${\operatorname{Spf}\pazocal{O}_{\infty}\hat{\otimes}K_{\infty}}$${\operatorname{Spf}\pazocal{O}_{\infty}\hat{\otimes}\pazocal{O}_{\infty}}$$\scriptstyle{\hat{i}}$$\scriptstyle{\hat{k}}$$\scriptstyle{\hat{j}}$

We let $q:\operatorname{Spec}\pazocal{O}_{\infty}\times C\to C\times C$ be the
inclusion of schemes. To the morphism of sheaves
$\tau_{\pazocal{M}}:\tau^{*}(q^{*}\pazocal{M})\to(q^{*}\pazocal{N})$ on
$(\operatorname{Spec}\pazocal{O}_{\infty})\times C$, one associates
functorially the morphism of the formal coherent sheaves
$\tau_{M}:\tau^{*}\hat{\pazocal{M}}\to\hat{\pazocal{N}}$ on the formal
spectrum $(\operatorname{Spf}\pazocal{O}_{\infty})\hat{\times}C$. Because both
$q^{*}\pazocal{M}$ and $q^{*}\pazocal{N}$ are coherent sheaves, their formal
completion corresponds to their pullback along the completion morphism of
locally ringed spaces:

$(\operatorname{Spf}\pazocal{O}_{\infty})\hat{\times}C\longrightarrow(\operatorname{Spec}\pazocal{O}_{\infty})\times
C.$

Recall that $\hat{\pazocal{N}}_{\infty}$ and $\hat{\pazocal{M}}_{\infty}$ were
the respective $\pazocal{A}_{\infty}(\pazocal{O}_{\infty})$-modules
$\hat{\pazocal{N}}(\operatorname{Spf}\pazocal{O}_{\infty}\hat{\otimes}\pazocal{O}_{\infty})$
and
$\hat{\pazocal{M}}(\operatorname{Spf}\pazocal{O}_{\infty}\hat{\otimes}\pazocal{O}_{\infty})$.
Let also $L$ and $L_{A}$ be given respectively by

$\displaystyle L$
$\displaystyle:=\hat{\pazocal{M}}(\operatorname{Spf}\pazocal{O}_{\infty}\hat{\otimes}K)=\hat{\pazocal{N}}(\operatorname{Spf}\pazocal{O}_{\infty}\hat{\otimes}K)$
$\displaystyle L_{A}$
$\displaystyle:=\hat{\pazocal{M}}(\operatorname{Spf}\pazocal{O}_{\infty}\hat{\otimes}A)=\hat{\pazocal{N}}(\operatorname{Spf}\pazocal{O}_{\infty}\hat{\otimes}A).$

Note that $L$ defines an $\pazocal{A}_{\infty}(K)$-lattice stable by
$\tau_{M}$ in $\pazocal{I}_{\infty}(M):=M\otimes_{A\otimes
K}\pazocal{B}_{\infty}(K)$.

By Theorem 6.6 we have a morphism of distinguished triangles:

${R\Gamma(\operatorname{Spf}\pazocal{O}_{\infty}\hat{\times}C,\hat{\pazocal{M}})}$${\hat{\pazocal{M}}_{\infty}}$${\displaystyle\frac{L\otimes_{\pazocal{A}_{\infty}(K)}\pazocal{A}_{\infty}(K_{\infty})}{L_{A}}}$${\left[1\right]}$${R\Gamma(\operatorname{Spf}\pazocal{O}_{\infty}\hat{\times}C,\hat{\pazocal{N}})}$${\hat{\pazocal{N}}_{\infty}}$${\displaystyle\frac{L\otimes_{\pazocal{A}_{\infty}(K)}\pazocal{A}_{\infty}(K_{\infty})}{L_{A}}}$${\left[1\right]}$$\scriptstyle{\iota-\tau_{\pazocal{M}}}$$\scriptstyle{\iota-\tau_{\pazocal{M}}}$$\scriptstyle{\operatorname{id}-\tau_{M}}$
(6.7)

The third vertical arrow is an isomorphism by Lemma 5.10. The next lemma shows
that the middle one is injective once tensored along $\pazocal{O}_{\infty}\to
K_{\infty}$:

######  Lemma 6.11.

The morphism
$\iota-\tau_{\pazocal{M}}:\hat{\pazocal{M}}_{\infty}\otimes_{\pazocal{O}_{\infty}}K_{\infty}\to\hat{\pazocal{N}}_{\infty}\otimes_{\pazocal{O}_{\infty}}K_{\infty}$
is injective.

###### Proof.

For $t>0$, $a\geq 0$ and $x\in\hat{\pazocal{M}}_{\infty}$, we have

$(\iota-\tau_{\pazocal{M}})((1\otimes\pi_{\infty})^{t+a}x)\equiv(1\otimes\pi_{\infty})^{t+a}\iota(x)\pmod{(1\otimes\pi_{\infty})^{t+a+1}\hat{\pazocal{N}}_{\infty}}.$

In particular, the first vertical arrow in diagram 5.7 is injective. The lemma
then follows from Lemma 5.13 together with the snake Lemma. ∎

Theorem 5.17 together with Lemma 6.11 implies the existence of a quasi-
isomorphism

$\operatorname{cone}([\hat{\pazocal{M}}_{\infty}\stackrel{{\scriptstyle\iota-\tau_{\pazocal{M}}}}{{\longrightarrow}}\hat{\pazocal{N}}_{\infty}])\otimes_{\pazocal{O}_{\infty}}K_{\infty}\cong\frac{(M+\tau_{M}(\tau^{*}M))\otimes_{A\otimes
K}K_{\infty}[\\![\mathfrak{j}]\\!]}{M\otimes_{A\otimes
K}K_{\infty}[\\![\mathfrak{j}]\\!]}.$

We then deduce from Theorem 6.6 that

$\operatorname{cone}\left(R\Gamma(\operatorname{Spf}\pazocal{O}_{\infty}\hat{\times}C,\hat{\pazocal{M}})\stackrel{{\scriptstyle\iota-\tau_{\pazocal{M}}}}{{\longrightarrow}}R\Gamma(\operatorname{Spf}\pazocal{O}_{\infty}\hat{\times}C,\hat{\pazocal{N}})\right)\otimes_{\pazocal{O}_{\infty}}K_{\infty}\\\
\cong\displaystyle\frac{(M+\tau_{M}(\tau^{*}M))\otimes_{A\otimes
K}K_{\infty}[\\![\mathfrak{j}]\\!]}{M\otimes_{A\otimes
K}K_{\infty}[\\![\mathfrak{j}]\\!]}.$ (6.8)

Because $(\operatorname{Spec}A)\times C\to\operatorname{Spec}A$ is proper,
Grothendieck’s comparison Theorem [Gro63, Thm. 4.1.5] provides natural quasi-
isomorphisms

$R\Gamma(\operatorname{Spec}\pazocal{O}_{\infty}\times C,\pazocal{F})\cong
R\Gamma(\operatorname{Spf}\pazocal{O}_{\infty}\hat{\times}C,\hat{\pazocal{F}})$

for $\pazocal{F}$ being either $q^{*}\pazocal{M}$ or $q^{*}\pazocal{N}$. This
allows us to rewrite (6.8) as

$\operatorname{cone}\left(R\Gamma(\operatorname{Spec}\pazocal{O}_{\infty}\times
C,q^{*}\pazocal{M})\stackrel{{\scriptstyle\iota-\tau_{\pazocal{M}}}}{{\longrightarrow}}R\Gamma(\operatorname{Spec}\pazocal{O}_{\infty}\times
C,q^{*}\pazocal{N})\right)\otimes_{\pazocal{O}_{\infty}}K_{\infty}\\\
\cong\displaystyle\frac{(M+\tau_{M}(\tau^{*}M))\otimes_{A\otimes
K}K_{\infty}[\\![\mathfrak{j}]\\!]}{M\otimes_{A\otimes
K}K_{\infty}[\\![\mathfrak{j}]\\!]},$

and we use Proposition 6.1 to obtain

$\operatorname{cone}\left(R\Gamma(\operatorname{Spec}A\times
C,\pazocal{M})\stackrel{{\scriptstyle\iota-\tau_{\pazocal{M}}}}{{\longrightarrow}}R\Gamma(\operatorname{Spec}A\times
C,\pazocal{N})\right)\otimes_{A}K_{\infty}\\\
\cong\displaystyle\frac{(M+\tau_{M}(\tau^{*}M))\otimes_{A\otimes
K}K_{\infty}[\\![\mathfrak{j}]\\!]}{M\otimes_{A\otimes
K}K_{\infty}[\\![\mathfrak{j}]\\!]}.$

From Proposition 6.7, we deduce

$G_{\underline{M}}\otimes_{A}K_{\infty}\cong\displaystyle\frac{(M+\tau_{M}(\tau^{*}M))\otimes_{A\otimes
K}K_{\infty}[\\![\mathfrak{j}]\\!]}{M\otimes_{A\otimes
K}K_{\infty}[\\![\mathfrak{j}]\\!]}.$

In particular, $G_{\underline{M}}\otimes_{A}K_{\infty}$ sits in degree $0$.
Therefore, we obtain the second part of Theorem 4.1:

######  Proposition 6.12.

The $A$-module
$\operatorname{H}^{1}(G_{\underline{M}})\cong\operatorname{Cl}(\underline{M})$
is torsion, and thus finite.

It remains to prove Theorem 4.4. We first introduce a definition (see the next
page).

######  Definition 6.13.

We denote by $\rho(\underline{M})$ the isomorphism of $K_{\infty}$-vector
spaces

$\displaystyle\frac{\left\\{\xi\in M\otimes_{A\otimes K}K_{\infty}\langle
A\rangle~{}|~{}\xi-\tau_{M}(\tau^{*}\xi)\in
N_{A}\right\\}}{M_{A}}\otimes_{A}K_{\infty}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\displaystyle\frac{(M+\tau_{M}(\tau^{*}M))\otimes_{A\otimes
K}K_{\infty}[\\![\mathfrak{j}]\\!]}{M\otimes_{A\otimes
K}K_{\infty}[\\![\mathfrak{j}]\\!]}$

obtained by the vertical composition of the quasi-isomorphisms of complexes of
$K_{\infty}$-vector spaces:

${\displaystyle\frac{\left\\{\xi\in M\otimes_{A\otimes K}K_{\infty}\langle
A\rangle~{}|~{}\xi-\tau_{M}(\tau^{*}\xi)\in
N_{A}\right\\}}{M_{A}}\otimes_{A}K_{\infty}}$${G_{\underline{M}}\otimes_{A}K_{\infty}}$${\operatorname{cone}\left[R\Gamma(\operatorname{Spec}A\times
C,\pazocal{M})\stackrel{{\scriptstyle\iota-\tau_{\pazocal{M}}}}{{\longrightarrow}}R\Gamma(\operatorname{Spec}A\times
C,\pazocal{N})\right]\otimes_{A}K_{\infty}}$${\operatorname{cone}\left[R\Gamma(\operatorname{Spec}\pazocal{O}_{\infty}\times
C,q^{*}\pazocal{M})\stackrel{{\scriptstyle\iota-\tau_{\pazocal{M}}}}{{\longrightarrow}}R\Gamma(\operatorname{Spec}\pazocal{O}_{\infty}\times
C,q^{*}\pazocal{N})\right]\otimes_{\pazocal{O}_{\infty}}K_{\infty}}$${\operatorname{cone}\left[R\Gamma(\operatorname{Spf}\pazocal{O}_{\infty}\hat{\times}C,\hat{\pazocal{M}})\stackrel{{\scriptstyle\iota-\tau_{\pazocal{M}}}}{{\longrightarrow}}R\Gamma(\operatorname{Spf}\pazocal{O}_{\infty}\hat{\times}C,\hat{\pazocal{N}})\right]\otimes_{\pazocal{O}_{\infty}}K_{\infty}}$${\operatorname{cone}\left[\hat{\pazocal{M}}_{\infty}\stackrel{{\scriptstyle\operatorname{id}-\tau_{M}}}{{\longrightarrow}}\hat{\pazocal{N}}_{\infty}\right]\otimes_{\pazocal{O}_{\infty}}K_{\infty}}$${\displaystyle\frac{\hat{\pazocal{N}}_{\infty}}{(\iota-\tau_{\pazocal{M}})(\hat{\pazocal{M}}_{\infty})}\otimes_{\pazocal{O}_{\infty}}K_{\infty}}$${\displaystyle\frac{(M+\tau_{M}(\tau^{*}M))\otimes_{A\otimes
K}K_{\infty}[\\![\mathfrak{j}]\\!]}{M\otimes_{A\otimes
K}K_{\infty}[\\![\mathfrak{j}]\\!]}}$$\scriptstyle{\wr~{}\operatorname{Cl}(\underline{M})\otimes_{A}K_{\infty}=0}$$\scriptstyle{\wr~{}\text{Proposition}~{}\ref{prop:ge-
in-terms-of-
cohomology}}$$\scriptstyle{\wr~{}\text{Proposition}~{}\ref{prop:cech-change-
of-
coefficients}}$$\scriptstyle{\wr~{}\text{Grothendieck's~{}comparison~{}Theorem}}$$\scriptstyle{\wr~{}\eqref{eq:diagram-
completed-cohomology}~{}\text{and~{}Lemma}~{}\ref{lem:id-tau-isomorphism-on-
TO}}$$\scriptstyle{\wr~{}\text{Lemma}~{}\ref{lem:injectivity-of-V-
to-W}}$$\scriptstyle{\wr~{}\text{Theorem}~{}\ref{thm:shtuka-model-to-hodge-
pink-additive}}$

###### Proof of Theorem 4.4.

We have an exact sequence of $K_{\infty}$-vector spaces:

$0\longrightarrow(\underline{M}_{B})^{+}_{K_{\infty}}\to\displaystyle\frac{\left\\{\xi\in
M\otimes_{A\otimes K}K_{\infty}\langle
A\rangle~{}|~{}\xi-\tau_{M}(\tau^{*}\xi)\in
N_{A}\right\\}}{M_{A}}\otimes_{A}K_{\infty}\\\
\longrightarrow\operatorname{Ext}^{1,\text{reg}}_{A,\infty}(\mathbbm{1},\underline{M})\otimes_{A}K_{\infty}\longrightarrow
0.$ (6.9)

On the other-hand, by Theorem 3.28, we have an exact sequence of
$K_{\infty}$-vector spaces:

$0\longrightarrow(\underline{M}_{B})^{+}_{K_{\infty}}\longrightarrow\displaystyle\frac{(M+\tau_{M}(\tau^{*}M))\otimes_{A\otimes
K}K_{\infty}[\\![\mathfrak{j}]\\!]}{M\otimes_{A\otimes
K}K_{\infty}[\\![\mathfrak{j}]\\!]}\longrightarrow\operatorname{Ext}^{1,\text{ha}}_{\infty}(\mathbbm{1}^{+},\mathscr{H}^{+}(\underline{M}))\longrightarrow
0.$

Theorem 4.4 then follows from the fact that $\rho(\underline{M})$ is an
isomorphism. ∎

From the proof of Theorem 4.4, we see that $\rho(\underline{M})$ induces an
isomorphism of $K_{\infty}$-vector spaces:

$\bar{\rho}(\underline{M}):\operatorname{Ext}^{1,\text{reg}}_{A,\infty}(\mathbbm{1},\underline{M})\otimes_{A}K_{\infty}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{Ext}^{1,\text{ha}}_{\infty}(\mathbbm{1}^{+},\mathscr{H}^{+}_{K_{\infty}}(\underline{M})).$

We call $\bar{\rho}(\underline{M})$ the Mock regulator of $\underline{M}$. The
terminology mock is here to emphasize the fact that
$\bar{\rho}(\underline{M})$ does not concide with
$\textrm{{Reg}}(\underline{M})$ in general. However, we believe that it still
have a role to play:

######  Conjecture 6.14.

The map $\rho(\underline{M})$ \- hence $\bar{\rho}(\underline{M})$ \- is
natural in $\underline{M}$.

Observe that Conjecture 6.14 is not obvious, as the construction of
$\rho(\underline{M})$ depends a priori on the choice of a shtuka model for
$\underline{M}$, an operation that is highly non canonical. It would have the
following consequence: let $(A\mathbf{Mot}_{K}^{\operatorname{rat}})_{<0}$
denotes the full subcategory of $A\mathbf{Mot}_{K}^{\operatorname{rat}}$
consisting of objects having all their weights negative. Then, the two
functors

$\underline{M}\longmapsto\operatorname{Ext}^{1,\operatorname{reg}}_{A,\infty}(\mathbbm{1},\underline{M})\otimes_{A}K_{\infty},\quad\underline{M}\longmapsto\operatorname{Ext}^{1,\operatorname{ha}}_{\infty}(\mathbbm{1}^{+},\mathscr{H}^{+}_{K_{\infty}}(\underline{M})),$

from $(A\mathbf{Mot}_{K}^{\operatorname{rat}})_{<0}$ to the category of
$K_{\infty}$-vector spaces, are isomorphic.

## References

  * [And86] Greg W. Anderson.  $t$-motives.  Duke Math. J., 53(2):457–502, 1986.
  * [And09] Yves André.  Slope filtrations.  Confluentes Math., 1(1):1–85, 2009.
  * [ANDTR22] Bruno Anglès, Tuan Ngo Dac, and Floric Tavares Ribeiro.  A class formula for admissible Anderson modules.  Invent. Math., 229(2):563–606, 2022.
  * [Bei84] A. A. Beilinson.  Higher regulators and values of $L$-functions.  In Current problems in mathematics, Vol. 24, Itogi Nauki i Tekhniki, pages 181–238. Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984.
  * [Bei86] A. A. Beilinson.  Notes on absolute Hodge cohomology.  In Applications of algebraic $K$-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983), volume 55 of Contemp. Math., pages 35–68. Amer. Math. Soc., Providence, RI, 1986.
  * [BGHP20] Gebhard Böckle, David Goss, Urs Hartl, and Matthew Papanikolas, editors.  $t$-motives: Hodge structures, transcendence and other motivic aspects.  EMS Series of Congress Reports. EMS Publishing House, Berlin, [2020] ©2020.
  * [BH07] Gebhard Böckle and Urs Hartl.  Uniformizable families of $t$-motives.  Trans. Amer. Math. Soc., 359(8):3933–3972, 2007.
  * [BL95] Arnaud Beauville and Yves Laszlo.  Un lemme de descente.  C. R. Acad. Sci. Paris Sér. I Math., 320(3):335–340, 1995.
  * [Bou07] N. Bourbaki.  Éléments de mathématique. Alèbre & Algèbre commutative.  Springer-Verlag, Berlin, 2007.  Reprint of the 1998 original.
  * [Car35] Leonard Carlitz.  On certain functions connected with polynomials in a Galois field.  Duke Math. J., 1(2):137–168, 1935.
  * [Car80] James A. Carlson.  Extensions of mixed Hodge structures.  In Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, pages 107–127. Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980.
  * [Dd74] V. G. Drinfel′ d.  Elliptic modules.  Mat. Sb. (N.S.), 94(136):594–627, 656, 1974.
  * [Del89] P. Deligne.  Le groupe fondamental de la droite projective moins trois points.  In Galois groups over ${\bf Q}$ (Berkeley, CA, 1987), volume 16 of Math. Sci. Res. Inst. Publ., pages 79–297. Springer, New York, 1989.
  * [Fan15] Jiangxue Fang.  Special $L$-values of abelian $t$-modules.  J. Number Theory, 147:300–325, 2015.
  * [FK18] Kazuhiro Fujiwara and Fumiharu Kato.  Foundations of rigid geometry. I.  EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich, 2018.
  * [Gaz22] Quentin Gazda.  On the integral part of $A$-motivic cohomology, 2022.
  * [GM21] Quentin Gazda and Andreas Maurischat.  Special functions and Gauss-Thakur sums in higher rank and dimension.  Journal für die reine und angewandte Mathematik (Crelles Journal), 2021(773):231–261, 2021.
  * [GM22a] Quentin Gazda and Andreas Maurischat.  Motivic cohomology of Carlitz twists, 2022.
  * [GM22b] Quentin Gazda and Andreas Maurischat.  Residue of special functions of anderson $A$-modules at the characteristic graph, 2022.
  * [GL11] Alain Genestier and Vincent Lafforgue.  Théorie de Fontaine en égales caractéristiques.  Ann. Sci. Éc. Norm. Supér. (4), 44(2):263–360, 2011.
  * [Gro63] A. Grothendieck.  Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. II.  Inst. Hautes Études Sci. Publ. Math., (17):91, 1963.
  * [Har09] Urs Hartl.  A dictionary between Fontaine-theory and its analogue in equal characteristic.  J. Number Theory, 129(7):1734–1757, 2009.
  * [Har11] Urs Hartl.  Period spaces for Hodge structures in equal characteristic.  Ann. of Math. (2), 173(3):1241–1358, 2011.
  * [Har19] Urs Hartl.  Isogenies of Abelian Anderson $A$-modules and $A$-motives.  Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 19(4):1429–1470, 2019\.
  * [HJ20] Urs Hartl and Ann-Kristin Juschka.  Pink’s theory of Hodge structures and the Hodge conjecture over function fields.  In $t$-motives: Hodge structures, transcendence and other motivic aspects, EMS Ser. Congr. Rep., pages 31–182. EMS Publ. House, Berlin, [2020] ©2020.
  * [Kat73] Nicholas Katz.  Une nouvelle formule de congruence pour la fonction $\zeta$.  In Lecture Notes in Mathematics (340), Éxposé XXII SGA7 t. II. 1973.
  * [Laf09] Vincent Lafforgue.  Valeurs spéciales des fonctions $L$ en caractéristique $p$.  J. Number Theory, 129(10):2600–2634, 2009.
  * [Mor18] Maxim Mornev.  Shtuka cohomology and Goss $L$-values.  PhD thesis, 2018.
  * [Nek94] Jan Nekovář.  Beĭlinson’s conjectures.  In Motives (Seattle, WA, 1991), volume 55 of Proc. Sympos. Pure Math., pages 537–570. Amer. Math. Soc., Providence, RI, 1994.
  * [Pin97] Richard Pink.  Hodge structures over function fields, 1997.
  * [PS08] Chris A. M. Peters and Joseph H. M. Steenbrink.  Mixed Hodge structures, volume 52 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics].  Springer-Verlag, Berlin, 2008.
  * [Ros02] Michael Rosen.  Number theory in function fields, volume 210 of Graduate Texts in Mathematics.  Springer-Verlag, New York, 2002.
  * [Sch91] Anthony J. Scholl.  Remarks on special values of $L$-functions.  In $L$-functions and arithmetic (Durham, 1989), volume 153 of London Math. Soc. Lecture Note Ser., pages 373–392. Cambridge Univ. Press, Cambridge, 1991.
  * [Ser68] Jean-Pierre Serre.  Corps locaux.  Publications de l’Université de Nancago, No. VIII. Hermann, Paris, 1968.  Deuxième édition.
  * [Tae09] Lenny Taelman.  Artin $t$-motifs.  J. Number Theory, 129(1):142–157, 2009.
  * [Tae10] Lenny Taelman.  A Dirichlet unit theorem for Drinfeld modules.  Math. Ann., 348(4):899–907, 2010.
  * [Tae12] Lenny Taelman.  Special $L$-values of Drinfeld modules.  Ann. of Math. (2), 175(1):369–391, 2012.
  * [Wei94] Charles A. Weibel.  An introduction to homological algebra, volume 38 of Cambridge Studies in Advanced Mathematics.  Cambridge University Press, Cambridge, 1994.