#  On the Integral Part of $A$-Motivic Cohomology
Quentin Gazda111Affiliation: Max Planck Institut für Mathematik, Vivatsgasse
7, 53111 Bonn, Germany. Mail<EMAIL_ADDRESS>
Abstract

The deepest arithmetic invariants attached to an algebraic variety defined
over a number field $F$ are conjecturally captured by the integral part of its
motivic cohomology. There are essentially two ways of defining it when $X$ is
a smooth projective variety: one is via the $K$-theory of a regular integral
model, the other is through its $\ell$-adic realization. Both approaches are
conjectured to coincide.

This paper initiates the study of motivic cohomology for global fields of
positive characteristic, hereafter named $A$-motivic cohomology, where
classical mixed motives are replaced by mixed Anderson $A$-motives. Our main
objective is to set the definitions of the integral part and the good
$\ell$-adic part of the $A$-motivic cohomology using Gardeyn’s notion of
maximal models as the analogue of regular integral models of varieties. Our
main result states that the integral part is contained in the good $\ell$-adic
part. As opposed to what is expected in the number field setting, we show that
the two approaches do not match in general.

We conclude this work by introducing the submodule of regulated extensions of
mixed Anderson $A$-motives, for which we expect the two approaches to match,
and solve some particular cases of this expectation.

###### Contents

  1. 1 Introduction
    1. 1.1 The number field picture
    2. 1.2 The function field picture
    3. 1.3 Plan of the paper
  2. 2 Anderson $A$-motives and their extension modules
    1. 2.1 Definition of $A$-motives
    2. 2.2 Extension modules in $\pazocal{M}_{R}$
    3. 2.3 Extensions having good reduction
  3. 3 Mixed $A$-motives and their extension modules
    1. 3.1 Mixed Anderson $A$-motives
    2. 3.2 Extension modules of mixed $A$-motives
  4. 4 Models and the integral part of $A$-motivic cohomology
    1. 4.1 Models of Frobenius spaces
    2. 4.2 Models of $A$-motives over a local function field
    3. 4.3 Models of $A$-motives over a global function field
    4. 4.4 The integral part of $A$-motivic cohomology
  5. 5 Regulated extensions
    1. 5.1 A particular extension of $\mathbbm{1}$ by itself
    2. 5.2 Hodge poylgons of $A$-motives
    3. 5.3 Regulated extensions having good reduction

##  1 Introduction

###  1.1 The number field picture

The idea of mixed motives and motivic cohomology has been gradually formulated
by Deligne, Beilinson and Lichtenbaum and aims to extend Grothendieck’s
philosophy of pure motives. Before discussing the function fields side,
subject of the paper, let us first present the classical setting to derive
some motivations.

The theory, mostly conjectural, starts with a number field $F$. The
hypothetical landscape portrays a $\mathbb{Q}$-linear Tannakian category
$\pazocal{M}\pazocal{M}_{F}$ of mixed motives over $F$, equipped with several
realization functors having $\pazocal{M}\pazocal{M}_{F}$ as source (see [Del,
§1]). Among them, the $\ell$-adic realization functor $V_{\ell}$, for a prime
number $\ell$, takes values in the category of continuous $\ell$-adic
representations of the absolute Galois group $G_{F}$ of $F$.

It is expected that reasonable cohomology theories factor through the category
$\pazocal{M}\pazocal{M}_{F}$. For instance, the $\ell$-adic realization should
recover the étale cohomology of algebraic varieties with coefficients in
$\mathbb{Q}_{\ell}$ in the following way: for all integer $i$, one foresees
the existence of a functor $h^{i}$, from the category of algebraic varieties
over $F$ to $\pazocal{M}\pazocal{M}_{F}$, making the following diagram of
categories commute:

${\\{\text{Varieties}/F\\}}$${\pazocal{M}\pazocal{M}_{F}}$${\operatorname{Rep}_{\mathbb{Q}_{\ell}}(G_{F})}$$\scriptstyle{h^{i}}$$\scriptstyle{X\mapsto
H^{i}_{\text{\'{e}t}}(X\times_{F}F^{s}{,}\mathbb{Q}_{\ell})}$$\scriptstyle{V_{\ell}}$

According to Deligne [Del, §1.3], the category $\pazocal{M}\pazocal{M}_{F}$
should admit a weight filtration in the sense of Jannsen [Jan, Def 6.3], which
would coincide with the classical weight filtration of varieties. The weights
of a mixed motive $M$ would then be defined as the breaks of its weight
filtration.

From the Tannakian formalism, $\pazocal{M}\pazocal{M}_{F}$ admits a tensor
operation, extending the fiber product on varieties, and we fix $\mathbbm{1}$
a neutral object. Let $M$ be a mixed motive over $F$. According to Beilinson
[Bei1, §0.3] (see also [Andr, Def 17.2.11]), the motivic cohomology of $M$ is
defined as the complex

$\operatorname{RHom}_{\pazocal{M}\pazocal{M}_{F}}(\mathbbm{1},M)$

in the derived category of $\mathbb{Q}$-vector spaces. Its $i$th cohomology is
the $\mathbb{Q}$-vector space
$\operatorname{Ext}^{i}_{\pazocal{M}\pazocal{M}_{F}}(\mathbbm{1},M)$, the
space of $i$-fold extensions of $\mathbbm{1}$ by $M$ in
$\pazocal{M}\pazocal{M}_{F}$. We quote from [Sch, §2] and [Del, §1.3]
respectively:

######  Conjecture.

We expect that:

  1. (C$1$)

for $i\not\in\\{0,1\\}$,
$\operatorname{Ext}^{i}_{\pazocal{M}\pazocal{M}_{F}}(\mathbbm{1},M)=0$,

  2. (C$2$)

if the weights of $M$ are non-negative,
$\operatorname{Ext}^{1}_{\pazocal{M}\pazocal{M}_{F}}(\mathbbm{1},M)=0$.

Let us focus on the $\mathbb{Q}$-vector space of $1$-fold extensions
$\operatorname{Ext}^{1}_{\pazocal{M}\pazocal{M}_{F}}(\mathbbm{1},M)$. A
subspace thereof of fundamental importance is the space of extensions having
everywhere good reduction. In the literature, we encounter two definitions
which are expected to meet. Let us first describe its local constructions.

##### Via the $\ell$-adic realization:

Let $F_{\mathfrak{p}}$ be the local field of $F$ at a finite place
$\mathfrak{p}$, and let $M_{\mathfrak{p}}$ be a mixed motive over
$F_{\mathfrak{p}}$. Let $G_{\mathfrak{p}}$ be the absolute Galois group of
$F_{\mathfrak{p}}$, and let $I_{\mathfrak{p}}$ be its inertia subgroup. Given
a prime number $\ell$, one predicts that the $\ell$-adic realization
$V_{\ell}$ is an exact functor. This allows to construct a $\mathbb{Q}$-linear
morphism, called the $\ell$-adic realization map of $M_{\mathfrak{p}}$,

$r_{M,\ell,\mathfrak{p}}:\operatorname{Ext}^{1}_{\pazocal{M}\pazocal{M}_{F_{\mathfrak{p}}}}(\mathbbm{1}_{\mathfrak{p}},M_{\mathfrak{p}})\longrightarrow
H^{1}(G_{\mathfrak{p}},V_{\ell}M_{\mathfrak{p}})$

which maps the class of an exact sequence $[E_{\mathfrak{p}}]:0\to
M_{\mathfrak{p}}\to E_{\mathfrak{p}}\to\mathbbm{1}_{\mathfrak{p}}\to 0$ in
$\pazocal{M}\pazocal{M}_{F_{\mathfrak{p}}}$ to the class of the continuous
cocycle $c:G_{\mathfrak{p}}\to V_{\ell}M_{\mathfrak{p}}$ associated to the
class of the exact sequence $[V_{\ell}E_{\mathfrak{p}}]:0\to
V_{\ell}M_{\mathfrak{p}}\to V_{\ell}E_{\mathfrak{p}}\to
V_{\ell}\mathbbm{1}_{\mathfrak{p}}\to 0$ in
$\operatorname{Rep}_{\mathbb{Q}_{\ell}}(G_{\mathfrak{p}})$.

Suppose $\ell$ does not divide $\mathfrak{p}$. Following Scholl [Sch], we say
that
$[E_{\mathfrak{p}}]\in\operatorname{Ext}^{1}_{\pazocal{M}\pazocal{M}_{F_{\mathfrak{p}}}}(\mathbbm{1},M)$
has good reduction if $r_{M,\ell,\mathfrak{p}}([E_{\mathfrak{p}}])$ splits as
a representation of $I_{\mathfrak{p}}$ (that is, $[V_{\ell}E_{\mathfrak{p}}]$
is zero in $H^{1}(I_{\mathfrak{p}},V_{\ell}M_{\mathfrak{p}})$). In [Sch, §2
Rmk], Scholl conjectures:

######  Conjecture.

We expect that:

  1. (C$3$)

The property that $[E_{\mathfrak{p}}]$ has good reduction is independent of
the prime $\ell$.

We then define
$\operatorname{Ext}^{1}_{\text{good}}(\mathbbm{1},M_{\mathfrak{p}})_{\ell}$ as
the subspace of
$\operatorname{Ext}^{1}_{\pazocal{M}\pazocal{M}_{F}}(\mathbbm{1},M_{\mathfrak{p}})$
consisting of extensions having good reduction. By (C$3$), it should not
depend on $\ell$: this is the $\mathfrak{p}$-integral part of the motivic
cohomology of $\underline{M}_{\mathfrak{p}}$.

##### Via the $K$-theory of regular models:

An other conjectural way of defining the $\mathfrak{p}$-integral part of
motivic cohomology uses its expected link with $K$-theory. Following Beilinson
[Bei1] in the case where $M_{\mathfrak{p}}$ is of the form $h^{i-1}(X)(n)$ for
a smooth projective variety $X$ over $F_{\mathfrak{p}}$ and two integers $n$,
$i\geq 1$, there should be a natural isomorphism of $\mathbb{Q}$-vector spaces
(see loc. cit. for details):

$\operatorname{Ext}^{1}_{\pazocal{M}\pazocal{M}_{F_{\mathfrak{p}}}}(\mathbbm{1},M_{\mathfrak{p}})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}(K_{2n-i}(X)\otimes_{\mathbb{Z}}\mathbb{Q})^{(n)}.$
(1.1)

Assume that $X$ has a regular model $\mathscr{X}$ over
$\pazocal{O}_{\mathfrak{p}}$ (i.e. $\mathscr{X}$ is regular over
$\operatorname{Spec}\pazocal{O}_{\mathfrak{p}}$ and
$\mathscr{X}\times_{\operatorname{Spec}\pazocal{O}_{\mathfrak{p}}}\operatorname{Spec}F_{\mathfrak{p}}=X$).
Then, we define
$\operatorname{Ext}^{1}_{\pazocal{O}_{\mathfrak{p}}}(\mathbbm{1},M_{\mathfrak{p}})$
to be the inverse image of

$\operatorname{image}\left((K_{2n-i}(\mathscr{X})\otimes_{\mathbb{Z}}\mathbb{Q})^{(n)}\longrightarrow(K_{2n-i}(X)\otimes_{\mathbb{Z}}\mathbb{Q})^{(n)}\right)$

through (1.1). By [Bei1, Lem. 8.3.1], this does not depend on the choice of
the model $\mathscr{X}$. The next conjecture supersedes (C$3$):

######  Conjecture.

We expect that

  1. (C$4$)

For any prime $\ell$ from which $\mathfrak{p}$ is not above,
$\operatorname{Ext}^{1}_{\pazocal{O}_{\mathfrak{p}}}(\mathbbm{1},M_{\mathfrak{p}})=\operatorname{Ext}^{1}_{\text{good}}(\mathbbm{1},M_{\mathfrak{p}})_{\ell}$.

The global version of the integral part of the motivic cohomology of a mixed
motive $M$ over $F$ is recovered as follows. The motive $M$ induces a motive
$M_{\mathfrak{p}}$ over $F_{\mathfrak{p}}$ by localization. Assuming
conjecture (C$3$), we say that an extension $[E]$ (of $\mathbbm{1}$ by
$\underline{M}$) has everywhere good reduction if, for all $\mathfrak{p}$,
$[E_{\mathfrak{p}}]$ belongs to
$\operatorname{Ext}^{1}_{\pazocal{M}\pazocal{M}_{F_{\mathfrak{p}}}}(\mathbbm{1}_{\mathfrak{p}},M_{\mathfrak{p}})_{\ell}$
for some prime $\ell$ not dividing $\mathfrak{p}$. We denote by
$\operatorname{Ext}^{1}_{\text{good}}(\mathbbm{1},M)$ the subspace of
$\operatorname{Ext}^{1}_{\pazocal{M}\pazocal{M}_{F}}(\mathbbm{1},M)$
consisting of extensions having everywhere good reduction.

Similarly, in the case where $M=h^{i-1}(X)(n)$ for a smooth projective variety
$X$ over $F$, we let $\operatorname{Ext}^{1}_{\pazocal{O}_{F}}(\mathbbm{1},M)$
be the subspace of extensions $[E]$ such that $[E_{\mathfrak{p}}]$ belongs to
$\operatorname{Ext}^{1}_{\pazocal{O}_{\mathfrak{p}}}(\mathbbm{1}_{\mathfrak{p}},M_{\mathfrak{p}})$
for all finite places $\mathfrak{p}$ of $F$. In virtue of the previous
conjectures, we should have:

$\operatorname{Ext}^{1}_{\text{good}}(\mathbbm{1},M)=\operatorname{Ext}^{1}_{\pazocal{O}_{F}}(\mathbbm{1},M)\cong(K_{2n-i}(\mathscr{X})\otimes_{\mathbb{Z}}\mathbb{Q})^{(n)},$

where $\mathscr{X}$ is a regular model of $X$ over $\pazocal{O}_{F}$.

The space $\operatorname{Ext}^{1}_{\pazocal{O}_{F}}(\mathbbm{1},M)$ is at the
heart of Beilinson’s conjectures, the next expectation being the starting
point thereof:

######  Conjecture.

We expect that

  1. (C$5$)

The space $\operatorname{Ext}^{1}_{\pazocal{O}_{F}}(\mathbbm{1},M)$ has finite
dimension over $\mathbb{Q}$.

###  1.2 The function field picture

Despite its intrisic obscurities, Motivic cohomology remains a difficult
subject also because its definition sits on a completely conjectural
framework. The present paper grew out as an attempt to understand the
analogous picture in function fields arithmetic. There, the theory looks more
promising using Anderson $A$-motives, instead of classical motives, whose
definition is well-established. This parallele has been drawn by many authors
and led to celebrated achievements. The analogue of the Tate conjecture [Tag]
[Tam], of Grothendieck’s periods conjecture [Pap] and of the Hodge conjecture
[HarJu] are now theorems on the function fields side. The recent volume [tMo]
records some of these feats. Counterparts of Motivic cohomology in function
fields arithmetic have not been studied yet, although recent works of Taelman
[Tae2] [Tae4] and Mornev [Mo1] strongly suggest the pertinence of such a
project.

#### The setting

Let $\mathbb{F}$ be a finite field, $q$ its number of elements, and let
$(C,\pazocal{O}_{C})$ be a geometrically irreducible smooth projective curve
over $\mathbb{F}$. We let $K$ be the function field of $C$ and we fix a closed
point $\infty$ on $C$. The $\mathbb{F}$-algebra:

$A:=\Gamma(C\setminus\\{\infty\\},\pazocal{O}_{C})$

has $K$ for fraction field. We let $K_{\infty}$ be the completion of $K$ with
respect to the valution $v_{\infty}$ associated to $\infty$, and let
$K_{\infty}^{s}$ be the separable closure of $K_{\infty}$. The analogy with
number fields that should guide us in this text is:

Number fields: | $\mathbb{Z}$ | $\subset$ | $\mathbb{Q}$ | $\subset$ | $\mathbb{R}$ | $\subset$ | $\mathbb{C}$
---|---|---|---|---|---|---|---
| $\wr$ |  | $\wr$ |  | $\wr$ |  | $\wr$
Function fields: | $A$ | $\subset$ | $K$ | $\subset$ | $K_{\infty}$ | $\subset$ | $K_{\infty}^{s}$

Let $F$ be a finite extension of $K$ (that is, a global function field over
$K$) and let $\pazocal{O}_{F}$ denote the integral closure of $A$ in $F$.

The analogy with number fields disappears when one considers the tensor
product $A\otimes F$, which is at the heart of the definition of Anderson
$A$-motives (unlabeled fiber and tensor products are over $\mathbb{F}$). We
consider the ring endomorphism $\tau$ of $A\otimes F$ which acts as the
identity on $A$ and as the $q$-Frobenius on $F$. $A\otimes F$ is a Dedekind
domain, and we let $\mathfrak{j}$ be its maximal ideal generated by the set
$\\{a\otimes 1-1\otimes a|a\in A\\}$.

Following [And], an Anderson $A$-motive $\underline{M}$ over $F$ is a pair
$(M,\tau_{M})$ where $M$ designates a finite locally free $A\otimes F$-module
of constant rank, and where $\tau_{M}:(\tau^{*}M)[\mathfrak{j}^{-1}]\to
M[\mathfrak{j}^{-1}]$ is an $(A\otimes F)[\mathfrak{j}^{-1}]$-linear
isomorphism (see Definition 2.1). We let $\pazocal{M}_{F}$ denote the category
of Anderson $A$-motives with obvious morphisms. $\pazocal{M}_{F}$ is known to
be $A$-linear, rigid monoidal, and is exact in the sense of Quillen (together
with the class of exact sequences defined in Definition 2.5) but not abelian
([HarJu, §2.3] or Section 2.1). Let $\mathbbm{1}$ in $\pazocal{M}_{F}$ be a
neutral object for the tensor operation.

#### Extensions of $A$-motives

The category $\pazocal{M}_{F}$, or rather full subcategories of it, will play
the role of the category of Grothendieck’s motives. Guided by this, the next
theorem already describes the analogue of motivic cohomology in an explicit
manner, and is the starting point of our research (see Theorem 2.13). Let
$\underline{M}$ be an $A$-motive over $F$.

######  Theorem A.

The complex
$\left[M\stackrel{{\scriptstyle\operatorname{id}-\tau_{M}}}{{\longrightarrow}}M[\mathfrak{j}^{-1}]\right]$
of $A$-modules placed in degree $0$ and $1$ represents the complex
$\operatorname{RHom}_{\pazocal{M}_{F}}(\mathbbm{1},\underline{M})$.

We immediately deduce that
$\operatorname{Ext}^{i}_{\pazocal{M}_{F}}(\mathbbm{1},\underline{M})$ vanishes
for $i>1$. For $i=1$, the $A$-module of $1$-fold extensions admit the
following explicit description. There is a natural surjective morphism

$\iota:M[\mathfrak{j}^{-1}]\longrightarrow\operatorname{Ext}^{1}_{\pazocal{M}_{F}}(\mathbbm{1},\underline{M})$
(1.2)

which maps $m\in M[\mathfrak{j}^{-1}]$ to the class of the extension of
$\mathbbm{1}$ by $\underline{M}$ given by $\left[M\oplus(A\otimes
F),\left(\begin{smallmatrix}\tau_{M}&m\\\ 0&1\end{smallmatrix}\right)\right]$
(Section 2.2). The kernel of $\iota$ being $(\operatorname{id}-\tau_{M})(M)$,
we recover the isomorphism provided by Theorem A.

######  Remark.

Extension groups in the full subcategory of $\pazocal{M}_{F}$ consisting of
effective $A$-motives (see Definition 2.3) were already determined in the
existing literature (see e.g. [Tae3], [Tae4], [PapRa]). The novelty of Theorem
A is to consider the whole category $\pazocal{M}_{F}$.

To pursue the analogy with number fields, we now present the notion of weights
and mixedness for Anderson $A$-motives. In the case $A=\mathbb{F}[t]$ or
$\deg(\infty)=1$ and over a complete algebraically closed base field, the
corresponding definitions were carried out respectively by Taelman [Tae1] and
Hartl-Juschka [HarJu]. We completed this picture in the most general way (over
any $A$-field and without any restriction on $\deg(\infty)$).

To an Anderson $A$-motive $\underline{M}$ over $F$, we attach an isocrystal
$\pazocal{I}_{\infty}(\underline{M})$ over $F$ at $\infty$ (in the sense of
[Mor2]). The term isocrystal is borrowed from $p$-adic Hodge theory, where the
function field setting allows to apply the non-archimedean theory at the
infinite point $\infty$ of $C$ as well. Following [And, 1.9], we call
$\underline{M}$ pure of weight $\mu$ if its associated isocrystal is pure of
slope $-\mu$ (Definition 3.21). More generally, we call $\underline{M}$ mixed
if there exists rational numbers $\mu_{1}<...<\mu_{s}$ together with a finite
ascending filtration of $\underline{M}$ by saturated sub-$A$-motives:

$0=W_{\mu_{0}}\underline{M}\subsetneq
W_{\mu_{1}}\underline{M}\subsetneq\cdots\subsetneq
W_{\mu_{s}}\underline{M}=\underline{M}$

for which the successive quotients
$W_{\mu_{i}}\underline{M}/W_{\mu_{i-1}}\underline{M}$ are pure of weight
$\mu_{i}$ (Definition 3.23). We show in Proposition 3.28 that such a
filtration - when it exists - is unique, as well as numbers $\mu_{i}$ which
are then called the weights of $\underline{M}$. As it was observed by Hartl-
Juschka ([HarJu, Ex. 2.3.13]), there exist non mixed $A$-motives. Nonetheless,
it is always possible to define the weights of a (not necessarily mixed)
$A$-motive via the Dieudonné-Manin decomposition of isocrystals (see
Definition 3.20).

We let $\pazocal{M}\pazocal{M}_{F}$ be the full subcategory of
$\pazocal{M}_{F}$ whose objects are mixed Anderson $A$-motives over $F$
(Section 3). The main results of Section 3 are condensed in:

######  Theorem B.

Let $\underline{M}$ be an object of $\pazocal{M}\pazocal{M}_{F}$. If all the
weights of $\underline{M}$ are non-positive, then every extension of
$\mathbbm{1}$ by $\underline{M}$ is mixed, that is:

$\operatorname{Ext}^{1}_{\pazocal{M}\pazocal{M}_{F}}(\mathbbm{1},\underline{M})=\operatorname{Ext}^{1}_{\pazocal{M}_{F}}(\mathbbm{1},\underline{M}).$

If all the weights of $\underline{M}$ are positive, then an extension of
$\mathbbm{1}$ by $\underline{M}$ is mixed if and only if its class is torsion,
that is:

$\operatorname{Ext}^{1}_{\pazocal{M}\pazocal{M}_{F}}(\mathbbm{1},\underline{M})=\operatorname{Ext}^{1}_{\pazocal{M}_{F}}(\mathbbm{1},\underline{M})^{\text{tors}}.$

Furthermore, for $i>1$,
$\operatorname{Ext}^{i}_{\pazocal{M}\pazocal{M}_{F}}(\mathbbm{1},\underline{M})$
is a torsion module for all $\underline{M}$.

######  Remark.

Although the category of classical mixed motives is expected to be
$\mathbb{Q}$-linear, the category $\pazocal{M}\pazocal{M}_{F}$ of mixed
Anderson $A$-motives over $F$ is $A$-linear. To obtain a $K$-linear category,
it might be convenient to introduce $\pazocal{M}\pazocal{M}^{\text{iso}}_{F}$
whose objects are the ones of $\pazocal{M}\pazocal{M}_{F}$ and whose Hom-
spaces are given by
$\operatorname{Hom}_{\pazocal{M}\pazocal{M}_{F}}(-,-)\otimes_{A}K$. In the
literature, $\pazocal{M}\pazocal{M}^{\text{iso}}_{F}$ is called the category
of mixed $A$-motives over $F$ up to isogenies [Har3], [HarJu]. Theorem B
implies that
$\operatorname{Ext}^{i}_{\pazocal{M}\pazocal{M}^{\text{iso}}_{F}}(\mathbbm{1},\underline{M})=0$
for $i>1$ and
$\operatorname{Ext}^{1}_{\pazocal{M}\pazocal{M}^{\text{iso}}_{F}}(\mathbbm{1},\underline{M})=0$
if the weights of $\underline{M}$ are positive. This reveals that the analogue
of the number fields conjecture (C$1$) and (C$2$) are true for function
fields. Note, however, that contrary to what is expected for number fields,
the full subcategory of pure $A$-motives is not semi-simple. Hence, we cannot
expect any $1$-fold Yoneda extension of two pure $A$-motives to split, even if
they have the same weight.

#### The good $\ell$-adic part

Let $\mathfrak{p}$ be a finite place of $F$ (i.e. not above $\infty$),
$F_{\mathfrak{p}}$ the associated local field, $F_{\mathfrak{p}}^{s}$ a
separable closure and
$G_{\mathfrak{p}}=\operatorname{Gal}(F_{\mathfrak{p}}^{s}|F_{\mathfrak{p}})$
the absolute Galois group of $F_{\mathfrak{p}}$ equipped with the profinite
topology. Given a maximal ideal $\ell$ in $A$ from which $\mathfrak{p}$ is not
above, there is an $\ell$-adic realization functor from
$\pazocal{M}_{F_{\mathfrak{p}}}$ to the category of continuous
$\pazocal{O}_{\ell}$-linear representations of $G_{\mathfrak{p}}$. For
$\underline{M}_{\mathfrak{p}}=(M_{\mathfrak{p}},\tau_{M})$ an object of
$\pazocal{M}_{F_{\mathfrak{p}}}$, it is given by the
$\pazocal{O}_{\ell}$-module

$T_{\ell}\underline{M}_{\mathfrak{p}}=\varprojlim_{n}\\{m\in(M_{\mathfrak{p}}\otimes_{F_{\mathfrak{p}}}F_{\mathfrak{p}}^{s})/\ell^{n}(M_{\mathfrak{p}}\otimes_{F_{\mathfrak{p}}}F_{\mathfrak{p}}^{s})~{}|~{}m=\tau_{M}(\tau^{*}m)\\}$

where $G_{\mathfrak{p}}$ acts compatibly on the right of the tensor
$M_{\mathfrak{p}}\otimes_{F_{\mathfrak{p}}}F_{\mathfrak{p}}^{s}$ (Definition
2.16).

We prove in Corollary 2.20 that $T_{\ell}$ is exact. This paves the way for
introducing extensions with good reduction, as Scholl did in the number fields
setting. Let $I_{\mathfrak{p}}\subset G_{\mathfrak{p}}$ be the inertia
subgroup. We consider the $\ell$-adic realization map restricted to
$I_{\mathfrak{p}}$:

$r_{\underline{M},\ell,\mathfrak{p}}:\operatorname{Ext}^{1}_{\pazocal{M}_{F_{\mathfrak{p}}}}(\mathbbm{1}_{\mathfrak{p}},\underline{M}_{\mathfrak{p}})\longrightarrow
H^{1}(I_{\mathfrak{p}},T_{\ell}\underline{M}_{\mathfrak{p}})$ (1.3)

(we refer to Subsection 2.3). Mimicking Scholl’s approach, we say that an
extension $[\underline{E}_{\mathfrak{p}}]$ of $\mathbbm{1}_{\mathfrak{p}}$ by
$\underline{M}_{\mathfrak{p}}$ has good reduction if
$[\underline{E}_{\mathfrak{p}}]$ lies in the kernel of (1.3). As in the number
field setting, we expect this definition to be independent of $\ell$. We let
$\operatorname{Ext}^{1}_{\text{good}}(\mathbbm{1}_{\mathfrak{p}},\underline{M}_{\mathfrak{p}})_{\ell}$
denote the kernel of $r_{\underline{M},\ell,\mathfrak{p}}$ (Definition 2.22).

#### The integral part

Gardeyn in [Gar2] has introduced a notion of maximal models for
$\tau$-sheaves. Inspired by Gardeyn’s work, we developed the notion of maximal
integral models of $A$-motives (Section 4). They form the function field
analogue of Néron models of abelian varieties, or more generally, of regular
models of varieties. Let $\pazocal{O}_{\mathfrak{p}}$ be the valuation ring of
$F_{\mathfrak{p}}$.

######  Definition (Definition 4.16).

Let $\underline{M}_{\mathfrak{p}}$ and $\underline{M}$ be $A$-motives over
$F_{\mathfrak{p}}$ and $F$ respectively.

  1. 1.

An $\pazocal{O}_{\mathfrak{p}}$-model for $\underline{M}_{\mathfrak{p}}$ is a
finite sub-$A\otimes\pazocal{O}_{\mathfrak{p}}$-module $L$ of
$M_{\mathfrak{p}}$ which generates $M_{\mathfrak{p}}$ over $F_{\mathfrak{p}}$,
and such that $\tau_{M}(\tau^{*}L)\subset L[\mathfrak{j}^{-1}]$.

  2. 2.

An $\pazocal{O}_{F}$-model for $\underline{M}$ is a finite
sub-$A\otimes\pazocal{O}_{F}$-module $L$ of $M$ which generates $M$ over $F$,
and such that $\tau_{M}(\tau^{*}L)\subset L[\mathfrak{j}^{-1}]$.

We say that $L$ is maximal if $L$ is not strictly contained in any other
models.

As opposed to [Gar2, Def 2.1 & 2.3], we do not ask for an
$\pazocal{O}_{\mathfrak{p}}$-model (resp. $\pazocal{O}_{F}$-model) to be
locally free. We show that this is implicit for maximal ones using Bourbaki’s
flatness criterion (Proposition 4.43). Compared to Gardeyn, our exposition is
therefore simplified and avoids the use of a technical lemma due to L.
Lafforgue [Gar2, §2.2]. Our next result should be compared with [Gar2, Prop
2.13] (see Propositions 4.41, 4.43 in the text).

######  Proposition.

A maximal $\pazocal{O}_{\mathfrak{p}}$-model $M_{\pazocal{O}_{\mathfrak{p}}}$
for $\underline{M}_{\mathfrak{p}}$ (resp. $\pazocal{O}_{F}$-model
$M_{\pazocal{O}}$ for $\underline{M}$) exists and is unique. It is locally
free over $A\otimes\pazocal{O}_{\mathfrak{p}}$ (resp.
$A\otimes\pazocal{O}_{F}$).

We let
$\operatorname{Ext}^{1}_{\pazocal{O}_{\mathfrak{p}}}(\mathbbm{1}_{\mathfrak{p}},\underline{M}_{\mathfrak{p}})$
be the image of $M_{\pazocal{O}_{\mathfrak{p}}}[\mathfrak{j}^{-1}]$ through
$\iota$ (1.2) (Definition 4.46). Our main result (repeated from Theorem 4.48)
is the next:

######  Theorem C.

Let $\underline{M}_{\mathfrak{p}}$ be an $A$-motive over $F_{\mathfrak{p}}$
and let $\ell$ be a maximal ideal of $A$ from which $\mathfrak{p}$ is not
above. Then,
$\operatorname{Ext}^{1}_{\pazocal{O}_{\mathfrak{p}}}(\mathbbm{1}_{\mathfrak{p}},\underline{M}_{\mathfrak{p}})$
is a sub-$A$-module of
$\operatorname{Ext}^{1}_{\text{good}}(\mathbbm{1}_{\mathfrak{p}},\underline{M}_{\mathfrak{p}})_{\ell}$.

Surprisingly enough, we cannot claim equality in general. In Subsection 5.1,
in the simplest case of the neutral $A$-motive, we construct for some $\ell$
and $\mathfrak{p}$ an explicit extension in
$\operatorname{Ext}^{1}_{\text{good}}(\mathbbm{1}_{\mathfrak{p}},\mathbbm{1}_{\mathfrak{p}})_{\ell}$
which does not belong to
$\operatorname{Ext}^{1}_{\pazocal{O}_{\mathfrak{p}}}(\mathbbm{1}_{\mathfrak{p}},\mathbbm{1}_{\mathfrak{p}})$.

In Subsection 4.4, we define the global version of the above. Namely, let
$\underline{M}$ be an $A$-motive over $F$. The $A$-motive $\underline{M}$
defines an $A$-motive $\underline{M}_{\mathfrak{p}}$ over $F_{\mathfrak{p}}$
by extending the base field. We let:

$\operatorname{Ext}^{1}_{\pazocal{O}_{F}}(\mathbbm{1},\underline{M})\stackrel{{\scriptstyle\text{def}}}{{=}}\bigcap_{\mathfrak{p}}\left\\{[\underline{E}]\in\operatorname{Ext}^{1}_{\pazocal{M}_{F}}(\mathbbm{1},\underline{M})~{}|~{}[\underline{E}_{\mathfrak{p}}]\in\operatorname{Ext}^{1}_{\pazocal{O}_{\mathfrak{p}}}(\mathbbm{1}_{\mathfrak{p}},\underline{M}_{\mathfrak{p}})\right\\}$

where the intersection is indexed over all the finite places of $F$. Our
second main result (repeated from Theorem 4.50) is the following:

######  Theorem D.

The $A$-module
$\operatorname{Ext}^{1}_{\pazocal{O}_{F}}(\mathbbm{1},\underline{M})$ equals
the image of $M_{\pazocal{O}_{F}}[\mathfrak{j}^{-1}]$ through $\iota$. In
addition, $\iota$ induces a natural isomorphism of $A$-modules:

$\frac{M_{\pazocal{O}}[\mathfrak{j}^{-1}]}{(\operatorname{id}-\tau_{M})(M_{\pazocal{O}})}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{Ext}^{1}_{\pazocal{O}}(\mathbbm{1},\underline{M}).$

#### Regulated $A$-motives

We are facing two main issues to pursue our analogy: the counterpart of
Conjecture (C$4$) does not hold true, and more seriously, neither is the
counterpart of (C$5$): the $A$-module
$\operatorname{Ext}^{1}_{\pazocal{O}_{F}}(\mathbbm{1},\underline{M})$ is
typically not finitely generated. Those facts suggest that the category
$\pazocal{M}_{F}$ – even $\pazocal{M}\pazocal{M}_{F}$ – is too huge to held a
convincing motivic cohomology theory. We end this text by presenting a
conjectural picture aiming to answer the analogue of (C$4$) and (C$5$).

######  Definition (c.f. 5.2 for details).

Let $0\to\underline{M}\to\underline{E}\to\underline{N}\to 0$ be an exact
sequence of $A$-motives over $F_{\mathfrak{p}}$. We say that $[\underline{E}]$
is regulated if the Hodge polygon of the Hodge-Pink structure attached to
$\underline{M}\oplus\underline{N}$ matches the one of $\underline{E}$ (see
[Pin, §6]). We denote by $\operatorname{Ext}^{1,\text{reg}}$ the submodule of
regulated extensions.

We took inspiration for this definition from the work of Pink, more precisely
from his notion of Hodge additivity [Pin, §. 6]. The above definition is
motivated by the observation that
$\operatorname{Ext}^{1}_{\pazocal{O}_{F}}(\mathbbm{1},\underline{M})$ is non
finitely generated whenever $\underline{M}$ is non zero. One reason,
consequence of Theorem D, stems from the fact that elements of
$M_{\pazocal{O}_{F}}[\mathfrak{j}^{-1}]$ – hence the resulting extensions of
$\mathbbm{1}$ by $\underline{M}$ obtained from $\iota$ – might have an
arbitrary large pole at $\mathfrak{j}$. The notion of regulated extensions
exactly prevents this to happen: in Corollary 5.10, we prove that $\iota$
induces an isomorphism of $A$-modules:

$\frac{M+\tau_{M}(\tau^{*}M)}{(\operatorname{id}-\tau_{M})(M)}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{Ext}^{1,\text{reg}}_{\pazocal{M}_{F_{\mathfrak{p}}}}(\mathbbm{1},\underline{M}).$

We strongly expect the following to hold for $A$-motives
$\underline{M}_{\mathfrak{p}}$ (see Conjecture 5.13):

######  Conjecture.

Let $\ell$ be a maximal ideal of $A$ from which $\mathfrak{p}$ is not above.
Then,

$\operatorname{Ext}^{1,\text{reg}}_{\pazocal{O}_{\mathfrak{p}}}(\mathbbm{1}_{\mathfrak{p}},\underline{M}_{\mathfrak{p}})=\operatorname{Ext}^{1,\text{reg}}_{\text{good}}(\mathbbm{1}_{\mathfrak{p}},\underline{M}_{\mathfrak{p}})_{\ell}.$

In particular, the module
$\operatorname{Ext}^{1,\text{reg}}_{\text{good}}(\mathbbm{1}_{\mathfrak{p}},\underline{M}_{\mathfrak{p}})_{\ell}$
does not depend on $\ell$.

We conclude this text by solving some particular cases of the above conjecture
(Subsection 5.3). The general case, however, remains open.

While
$\operatorname{Ext}^{1,\text{reg}}_{\pazocal{O}_{F}}(\mathbbm{1},\underline{M})$
is still not finitely generated over $A$ in general, a version of Conjecture
(C$5$) involving the infinite places holds. This will be the subject of
another work.

##### Acknowledgment:

This text consists of the - revised - first half of the author’s PhD thesis.
To that extent, I wish to reiterate my profound gratitude to Professors
Gebhard Böckle and Urs Hartl, and my advisor Professor Federico Pellarin. I am
also grateful to Max Planck Institute for Mathematics in Bonn for its
hospitality and financial support.

###  1.3 Plan of the paper

The paper is organized as follows.

In the begining of Section 2, Subsection 2.1, we review the usual set up
(notations, definitions, basic properties) of $A$-motives over an arbitrary
$A$-algebra or field. We follow [HarJu] and [Har3] as a guideline, though the
former reference is concerned with the particular choice of a closed point
$\infty$ of degree one and over a complete algebraically closed field. Most of
the results on $A$-motives extend without changes to our larger setting. In
Subsection 2.2, $A$-Motivic cohomology in $\pazocal{M}_{F}$ is introduced. We
describe the extension modules in $\pazocal{M}_{F}$ and obtain Theorem A as
Theorem 2.13 in the text. In Subsection 2.3, we recall the definition and main
properties of the $\ell$-adic realization functor for $A$-motives, and
introduce extensions having good reduction with respect to $\ell$ in
Definition 2.22.

Section 3 is concerned with mixed $A$-motives. In the beginning of Subsection
3.1 we recall, and add some new material, to the theory of function fields
isocrystals in the steps of [Mor2]. The main ingredient, used later on in
Subsection 3.1 to define the category of mixed $A$-motives, is the existence
and uniqueness of the slope filtration (extending [Har2, Prop. 1.5.10] to
general coefficient rings $A$). We focus on extension modules in the category
$\pazocal{M}\pazocal{M}_{F}$ in Subsection 3.2 where we deduce Theorem B from
Propositions 3.38, 3.40 and Theorem 3.44.

In Section 4, we develop the notion of maximal integral models of $A$-motives
over local and global function fields. It splits into four subsections. In
Subsection 4.1, we present integral models of Frobenius spaces over local
function fields. The theory is much easier than the one for $A$-motives,
introduced over a local function field in Subsection 4.2 and over a global
function field in Subsection 4.3. Although our definition of integral model is
inspired by Gardeyn’s work in the context of $\tau$-sheaves [Gar2], our
presentation is simpler as we removed the locally free assumption. That
maximal integral models are locally free is automatic, as we show in
Propositions 4.26 and 4.43. The chief aim of this section, however, is
Subsection 4.4 where we use the results of the previous ones to prove Theorems
C and D (respectively Theorems 4.48 and 4.50 in the text).

In our last Section 5, we introduce the notion of regulated extensions of
$A$-motives with an eye toward understanding the lack of equality in Theorem
C, highlighted in Subsection 5.1. We recall the definitions of Hodge-Pink
structures and Hodge polygon, as introduced in [Pin], in Subsection 5.2. Those
are used to define regulated extensions in Definition 5.7. We conclude this
text by Subsection 5.3, where we present a general hope that
$\operatorname{Ext}^{1,\text{reg}}_{\pazocal{O}_{\mathfrak{p}}}(\mathbbm{1}_{\mathfrak{p}},\underline{M}_{\mathfrak{p}})$
and
$\operatorname{Ext}^{1,\text{reg}}_{\text{good}}(\mathbbm{1}_{\mathfrak{p}},\underline{M}_{\mathfrak{p}})_{\ell}$
match. We then prove some particular cases of this expectation, namely when
$\underline{M}$ is effective, pure of weight $0$ and has good reduction
(Theorem 5.14), or when $\underline{M}$ is a $p$-tensor power of Carlitz’s
twist $\underline{A}(1)$ (Theorem 5.18).

##  2 Anderson $A$-motives and their extension modules

Let $\mathbb{F}$ be a finite field of cardinality $q$. By convention,
throughout this text unlabeled tensor products and fiber products are over
$\mathbb{F}$. Let $(C,\pazocal{O}_{C})$ be a geometrically irreducible smooth
projective curve over $\mathbb{F}$, and fix a closed point $\infty$ on $C$.
Let $A:=\Gamma(C\setminus\\{\infty\\},\pazocal{O}_{C})$ be the ring of regular
functions on $C\setminus\\{\infty\\}$ and let $K$ be the function field of
$C$.

###  2.1 Definition of $A$-motives

This subsection is devoted to define and recall the main properties of
Anderson $A$-motives. We begin with a paragraph of notations.

Let $R$ be a commutative $\mathbb{F}$-algebra and let $\kappa:A\to R$ be an
$\mathbb{F}$-algebra morphism. $R$ will be referred to as the base algebra and
$\kappa$ as the characteristic morphism. The kernel of $\kappa$ is called the
characteristic of $(R,\kappa)$. We consider the ideal
$\mathfrak{j}=\mathfrak{j}_{\kappa}$ of $A\otimes R$ generated by the set
$\\{a\otimes 1-1\otimes\kappa(a)|a\in A\\}$; $\mathfrak{j}$ is equivalently
defined as the kernel of $A\otimes R\to R$, $a\otimes f\mapsto\kappa(a)f$. The
ideal $\mathfrak{j}$ is maximal if and only if $R$ is a field, and is a prime
ideal if and only if $R$ is a domain.

Let $\operatorname{Quot}(A\otimes R)$ be the localization of $A\otimes R$ at
its non-zero-divisors (if $A\otimes R$ is an integral domain,
$\operatorname{Quot}(A\otimes R)$ is the field of fractions of $A\otimes R$).

Let $M$ be an $A\otimes R$-module. For $n\in\mathbb{Z}$, we denote
$\mathfrak{j}^{-n}M$ the submodule of $M\otimes_{A\otimes
R}\operatorname{Quot}(A\otimes R)$ consisting of elements $m$ for which
$(a\otimes 1-1\otimes\kappa(a))^{n}m\in M$ for all $a\in
A\setminus\mathbb{F}$. We then set

$M[\mathfrak{j}^{-1}]:=\bigcup_{n\geq 0}\mathfrak{j}^{-n}M.$

Let $\tau:A\otimes R\to A\otimes R$ be the $A$-linear morphism given by
$a\otimes r\mapsto a\otimes r^{q}$ on elementary tensors. Let $\tau^{*}M$
denotes the pull-back of $M$ by $\tau$ ([Bou, A.II.§5]). That is, $\tau^{*}M$
is the $A\otimes R$-module

$(A\otimes R)\otimes_{\tau,A\otimes R}M$

where the subscript $\tau$ signifies that the relation
$(a\otimes_{\tau}bm)=(a\tau(b)\otimes_{\tau}m)$ holds for $a,b\in A\otimes R$,
and where the $A\otimes R$-module structure on $\tau^{*}M$ corresponds to
$b\cdot(a\otimes_{\tau}m):=(ba\otimes_{\tau}m)$. We let
$\mathbf{1}:\tau^{*}(A\otimes R)\to A\otimes R$ be the $A\otimes R$-linear
morphism which maps $(a\otimes r)\otimes_{\tau}(b\otimes
s)\in\tau^{*}(A\otimes R):=(A\otimes R)_{\tau,A\otimes R}(A\otimes R)$ to
$ab\otimes rs^{q}\in A\otimes R$.

The next definition takes its roots in the work of Anderson [And], though this
version is borrowed from [Har3, Def. 2.1]:

######  Definition 2.1.

An Anderson $A$-motive $\underline{M}$ (over $R$)  is a pair $(M,\tau_{M})$
where $M$ is a locally free $A\otimes R$-module of finite constant rank and
where $\tau_{M}:(\tau^{*}M)[\mathfrak{j}^{-1}]\to M[\mathfrak{j}^{-1}]$ is an
isomorphism of $(A\otimes R)[\mathfrak{j}^{-1}]$-modules.
In all the following, we shall more simply write $A$-motive instead of
Anderson $A$-motive. The rank of $\underline{M}$ is the (constant) rank of $M$
over $A\otimes R$.
A morphism $(M,\tau_{M})\to(N,\tau_{N})$ of $A$-motives (over $R$) is an
$A\otimes R$-linear morphism $f:M\to N$ such that
$f\circ\tau_{M}=\tau_{N}\circ\tau^{*}f$. We let $\pazocal{M}_{R}$ be the
$A$-linear category of $A$-motives over $R$.

######  Remark 2.2.

$A$-motives as in Definition 2.1 are called abelian $A$-motives by several
authors (see e.g. [BroPa]). The word abelian refers to the assumption that the
underlying $A\otimes R$-module is finite locally free. Dropping this
assumption is not a good strategy in our work, as too many analogies with
number fields motives would fail to hold.

######  Definition 2.3.

An $A$-motive $\underline{M}=(M,\tau_{M})$ (over $R$) is called effective if
$\tau_{M}(\tau^{*}M)\subset M$. We let $\pazocal{M}_{R}^{\text{eff}}$ be the
full subcategory of $\pazocal{M}_{R}$ whose objects are effective $A$-motives.

Let $\mathbbm{1}$ be the unit $A$-motive over $R$ defined as $(A\otimes
R,\textbf{1})$. The biproduct of two $A$-motives $\underline{M}$ and
$\underline{N}$, denoted $\underline{M}\oplus\underline{N}$, is defined to be
the $A$-motive whose underlying $A\otimes R$-module is $M\oplus N$ and whose
$\tau$-linear morphism is $\tau_{M}\oplus\tau_{N}$. Their tensor product,
denoted $\underline{M}\otimes\underline{N}$, is defined to be
$(M\otimes_{A\otimes R}N,\tau_{M}\otimes\tau_{N})$. The tensor operation
admits $\mathbbm{1}$ as a neutral object. The dual of $\underline{M}$ is
defined to be the $A$-motive whose underlying $A\otimes R$-module is
$M^{\vee}:=\operatorname{Hom}_{A\otimes R}(M,A\otimes R)$ and where
$\tau_{M^{\vee}}$ is defined as

$\tau_{M^{\vee}}:(\tau^{*}M^{\vee})[\mathfrak{j}^{-1}]=(\tau^{*}M)^{\vee}[\mathfrak{j}^{-1}]\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}M^{\vee}[\mathfrak{j}^{-1}],\quad
h\longmapsto h\circ\tau_{M}^{-1}$

(we refer to [HarJu, §.2.3] for more details). Given $S$ an $R$-algebra, there
is a base-change functor $\pazocal{M}_{R}\to\pazocal{M}_{S}$ mapping
$\underline{M}=(M,\tau_{M})$ to
$\underline{M}_{S}:=(M\otimes_{R}S,\tau_{M}\otimes_{R}\operatorname{id}_{S})$.
The restriction functor
$\operatorname{Res}_{S/R}:\pazocal{M}_{S}\to\pazocal{M}_{R}$ maps an
$A$-motive $\underline{M}$ over $S$ to $\underline{M}$ seen as an $A$-motive
over $R$. Given two $A$-motives $\underline{M}$ and $\underline{N}$ over $R$
and $S$ respectively, we have

$\operatorname{Hom}_{\pazocal{M}_{R}}(\underline{M},\operatorname{Res}_{S/R}\underline{N})=\operatorname{Hom}_{\pazocal{M}_{S}}(\underline{M}_{S},\underline{N}).$

In other words, the base-change functor is left-adjoint to the restriction
functor.

######  Example 2.4 (Carlitz’s motive).

Let $C=\mathbb{P}_{\mathbb{F}}^{1}$ be the projective line over $\mathbb{F}$
and let $\infty$ be the closed point of coordinates $[0:1]$. If $t$ is any
element in
$\Gamma(\mathbb{P}^{1}_{\mathbb{F}}\setminus\\{\infty\\},\pazocal{O}_{\mathbb{P}^{1}})$
whose order of vanishing at $\infty$ is $1$, we have an identification
$A=\mathbb{F}[t]$. For an $\mathbb{F}$-algebra $R$, the tensor product
$A\otimes R$ is identified with $R[t]$. The morphism $\tau$ acts on $p(t)\in
R[t]$ by raising its coefficients to the $q$th-power. It is rather common to
denote by $p(t)^{(1)}$ the polynomial $\tau(p(t))$. Let $\kappa:A\to R$ be an
injective $\mathbb{F}$-algebra morphism and let $\theta=\kappa(t)$. The ideal
$\mathfrak{j}\subset R[t]$ is principal, generated by $(t-\theta)$.

The Carlitz $\mathbb{F}[t]$-motive $\underline{C}$ over $R$ is defined by the
couple $(R[t],\tau_{\mathsf{C}})$ where $\tau_{C}$ maps $\tau^{*}p(t)$ to
$(t-\theta)p(t)^{(1)}$. Its $n$th tensor power
$\underline{C}^{n}:=\underline{C}^{\otimes n}$ is isomorphic to the
$\mathbb{F}[t]$-motive whose underlying module is $R[t]$ and where
$\tau_{C^{n}}$ maps $\tau^{*}p(t)$ to $(t-\theta)^{n}p(t)^{(1)}$. We let
$\underline{A}(n):=\underline{C}^{-n}=(\underline{C}^{n})^{\vee}$.

For $A=\mathbb{F}[t]$, $\underline{A}(1)$ plays the role of the number fields’
Tate motive $\mathbb{Z}(1)$ and, more generally, $\underline{A}(n)$ plays the
role of $\mathbb{Z}(n)$.

The category $\pazocal{M}_{R}$ of $A$-motives over $R$ is generally not
abelian, even if $R=F$ is a field. This comes from the fact that a morphism in
$\pazocal{M}_{F}$ might not admit a cokernel. However, there is a notion of
exact sequences in the category $\pazocal{M}_{R}$ which we borrow from [HarJu,
Rmk. 2.3.5(b)]:

######  Definition 2.5.

We say that a sequence
$0\to\underline{M}^{\prime}\to\underline{M}\to\underline{M}^{\prime\prime}\to
0$ in $\pazocal{M}_{R}$ is exact if its underlying sequence of $A\otimes
R$-modules is exact.

The next proposition appears and is discussed in [HarJu, Rmk. 2.3.5(b)] and
will allow us to consider extension modules (Subsection 3.2). Although stated
in the case where $R$ is a particular $A$-algebra and $\deg(\infty)=1$, it
extends without changes to our setting:

######  Proposition 2.6.

The category $\pazocal{M}_{R}$ together with the notion of exact sequences as
in Definition 2.5 is exact in the sense of Quillen [Qui1, §2].

To remedy to the non-abelian feature, we shall introduce next the category
$\pazocal{M}^{\text{iso}}_{R}$ of $A$-motives up to isogeny (over $R$) (see
Definition 2.9), which is abelian when $R=F$ is a field. We first discuss the
notion of saturation.

######  Definition 2.7.

Let $\underline{M}=(M,\tau_{M})$ be an Anderson $A$-motive over $R$. A
submotive of $\underline{M}$ is an $A$-motive $\underline{N}=(N,\tau_{N})$
such that $N\subset M$ and
$\tau_{N}=\tau_{M}|_{\tau^{*}N[\mathfrak{j}^{-1}]}$.

We set $\underline{N}^{\text{sat}}$ to be the submotive of $\underline{M}$
whose underlying $A\otimes R$-module is

$N^{\text{sat}}:=\\{n\in M~{}|~{}\exists a\in A\otimes R,~{}an\in N\\}$

and call it the saturation of $\underline{N}$ in $\underline{M}$. We say that
$\underline{N}$ is saturated in $\underline{M}$ if
$\underline{N}=\underline{N}^{\text{sat}}$.

Following [Har3, Def. 5.5, Thm. 5.12], we have the next:

######  Definition 2.8.

A morphism $f:\underline{M}\to\underline{N}$ in $\pazocal{M}_{R}$ is an
isogeny if one of the following equivalent conditions is satisfied.

  1. $(a)$

$f$ is injective and $\operatorname{coker}(f:M\to N)$ is a finite locally free
$R$-module,

  2. $(b)$

$M$ and $N$ have the same rank and $\operatorname{coker}f$ is finite locally
free over $R$,

  3. $(c)$

$M$ and $N$ have the same rank and $f$ is injective,

  4. $(d)$

there exists $0\neq a\in A$ such that $f$ induces an isomorphism of $(A\otimes
R)[a^{-1}]$-modules $M[a^{-1}]\stackrel{{\scriptstyle\sim}}{{\to}}N[a^{-1}]$,

  5. $(e)$

there exists $0\neq a\in A$ and $g:\underline{N}\to\underline{M}$ in
$\pazocal{M}_{R}$ such that $f\circ g=a\operatorname{id}_{N}$ and $g\circ
f=a\operatorname{id}_{M}$.

If an isogeny between $\underline{M}$ and $\underline{N}$ exists,
$\underline{M}$ and $\underline{N}$ are said to beisogenous.

As a consequence of those equivalent definitions, a submotive of an $A$-motive
$\underline{M}$ is isogenous to its saturation in $\underline{M}$. This
motivates the definition of the category of $A$-motives up to isogeny (see
[HarJu, Def. 2.3.1]).

######  Definition 2.9.

Let $\pazocal{M}^{\text{iso}}_{R}$ be the $K$-linear category whose objects
are those of $\pazocal{M}_{R}$ and where the hom-sets of two objects
$\underline{M}$ and $\underline{N}$ is given by the $K$-vector space

$\operatorname{Hom}_{\pazocal{M}^{\text{iso}}_{R}}(\underline{M},\underline{N}):=\operatorname{Hom}_{\pazocal{M}_{R}}(\underline{M},\underline{N})\otimes_{A}K.$

We call the objects of $\pazocal{M}^{\text{iso}}_{R}$ the $A$-motives over $R$
up to isogeny.

An isogeny in $\pazocal{M}_{R}$ then becomes an isomorphism in
$\pazocal{M}^{\text{iso}}_{R}$. According to [HarJu, Prop. 2.3.4], the
category $\pazocal{M}^{\text{iso}}_{F}$ is abelian.

###  2.2 Extension modules in $\pazocal{M}_{R}$

In this subsection, we are concerned with the computation of extension modules
in the category $\pazocal{M}_{F}$. Theorem A of the introduction is proved
below (Theorem 2.13).

Let $R$ be an $\mathbb{F}$-algebra and let $\underline{M}$ and $\underline{N}$
be two $A$-motives over $R$. The morphisms from $\underline{N}$ to
$\underline{M}$ in $\pazocal{M}_{R}$ are precisely the $A\otimes R$-linear map
of the underlying modules $f:N\to M$ such that
$\tau_{M}\circ\tau^{*}f=f\circ\tau_{N}$. The module of $0$-fold extensions is
described by the homomorphisms:

$\operatorname{Ext}^{0}_{\pazocal{M}_{R}}(\underline{N},\underline{M})=\operatorname{Hom}_{\pazocal{M}_{R}}(\underline{N},\underline{M})=\\{f\in\operatorname{Hom}_{A\otimes
R}(N,M)~{}|~{}\tau_{M}\circ\tau^{*}f=f\circ\tau_{N}\\}.$

As we saw in Proposition 2.6, $\pazocal{M}_{R}$ possesses exact sequences in
the sense of Quillen which turns it into an $A$-linear exact category. It
allows us to consider higher Yoneda extension $A$-modules
$\operatorname{Ext}^{i}_{\pazocal{M}_{R}}(\underline{N},\underline{M})$ (for
$i\geq 0$) of two $A$-motives $\underline{M}$ and $\underline{N}$. The next
proposition computes the first extension group.

######  Proposition 2.10.

Let $\underline{M}$ and $\underline{N}$ be $A$-motives over $R$. There is a
surjective morphism of $A$-modules, functorial in both $\underline{N}$ and
$\underline{M}$,

$\operatorname{Hom}_{A\otimes
R}(\tau^{*}N,M)[\mathfrak{j}^{-1}]\twoheadrightarrow\operatorname{Ext}^{1}_{\pazocal{M}_{R}}(\underline{N},\underline{M})$

whose kernel is
$\left\\{f\circ\tau_{N}-\tau_{M}\circ\tau^{*}f~{}|~{}f\in\operatorname{Hom}_{A\otimes
R}(N,M)\right\\}$. It is given by mapping a morphism
$u\in\operatorname{Hom}_{A\otimes R}(\tau^{*}N,M)[\mathfrak{j}^{-1}]$ to the
class of the extension $[M\oplus N,\left(\begin{smallmatrix}\tau_{M}&u\\\
0&\tau_{N}\end{smallmatrix}\right)]$ in
$\operatorname{Ext}^{1}_{\pazocal{M}_{R}}(\underline{N},\underline{M})$.

###### Proof.

Let
$[\underline{E}]:0\to\underline{M}\stackrel{{\scriptstyle\iota}}{{\to}}\underline{E}\stackrel{{\scriptstyle\pi}}{{\to}}\underline{N}\to
0$ be an exact sequence in $\pazocal{M}_{R}$, that is an exact sequence of the
underlying $A\otimes R$-modules with commuting $\tau$-action. Because $N$ is a
projective module, there exists $s:N\to E$ a section of the underlying short
exact sequence of $A\otimes R$-modules. We let $\xi:=\iota\oplus s:M\oplus
N\to E$. We have an equivalence:

${0}$${\underline{M}}$${\underline{E}}$${\underline{N}}$${0}$${0}$${\underline{M}}$${(M\oplus
N,\xi^{-1}\circ\tau_{E}\circ\xi)}$${\underline{N}}$${0}$$\scriptstyle{\iota}$$\scriptstyle{\pi}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\xi}$$\scriptstyle{\operatorname{id}}$

Because $\xi^{-1}\circ\tau_{M}\circ\xi$ is an isomorphism from
$\tau^{*}M[\mathfrak{j}^{-1}]\oplus\tau^{*}N[\mathfrak{j}^{-1}]$ to
$M[\mathfrak{j}^{-1}]\oplus N[\mathfrak{j}^{-1}]$ which restricts to
$\tau_{M}$ on the left and to $\tau_{N}$ on the right, there exists
$u\in\operatorname{Hom}_{A\otimes
R}(\tau^{*}N[\mathfrak{j}^{-1}],M[\mathfrak{j}^{-1}])=\operatorname{Hom}_{A\otimes
R}(\tau^{*}N,M)[\mathfrak{j}^{-1}]$ such that
$\xi^{-1}\circ\tau_{E}\circ\xi=\left(\begin{smallmatrix}\tau_{M}&u\\\
0&\tau_{N}\end{smallmatrix}\right)$. We have just shown that the map

$\iota:\operatorname{Hom}_{A\otimes
R}(\tau^{*}N,M)[\mathfrak{j}^{-1}]\to\operatorname{Ext}^{1}_{\pazocal{M}_{R}}(\underline{N},\underline{M}),\quad
u\mapsto[M\oplus N,\left(\begin{smallmatrix}\tau_{M}&u\\\
0&\tau_{N}\end{smallmatrix}\right)]$

is onto. Note that $\iota(0)$ corresponds to the class of the split extension.
Further, $\iota(u+v)$ corresponds to the Baer sum of $\iota(u)$ and
$\iota(v)$. In addition, given the exact sequence $[\underline{E}]$ and $a\in
A$, the pullback of multiplication by $a$ on $N$ and $\pi$ gives another
extension which defines $a\cdot[\underline{E}]$. If
$[\underline{E}]=\iota(u)$, it is formal to check that
$a\cdot[\underline{E}]=\iota(au)$. As such, $\iota$ is a surjective $A$-module
morphism. To find its kernel, it suffices to determine whenever $\iota(u)$ is
equivalent to the split extension. This happens if and only if there is a
commutative diagram in $\pazocal{M}_{R}$ of the form

${0}$${\underline{M}}$${\underline{M}\oplus\underline{N}}$${\underline{N}}$${0}$${0}$${\underline{M}}$${\left[M\oplus
N,\left(\begin{smallmatrix}\tau_{M}&&u\\\
0&&\tau_{N}\end{smallmatrix}\right)\right]}$${\underline{N}}$${0}$$\scriptstyle{\operatorname{id}_{M}}$$\scriptstyle{h}$$\scriptstyle{\operatorname{id}_{N}}$

where $h$ is a morphism in $\pazocal{M}_{R}$. Since the diagram commutes in
the category of $A\otimes R$-modules, it follows that $h$ is of the form
$\left(\begin{smallmatrix}\operatorname{id}_{M}&f\\\
0&\operatorname{id}_{N}\end{smallmatrix}\right)$ for an $A\otimes R$-linear
map $f:N\to M$. Because it is a diagram in $\pazocal{M}_{R}$, it further
requires commuting $\tau$-action, that is:

$\begin{pmatrix}\tau_{M}&u\\\
0&\tau_{N}\end{pmatrix}\tau^{*}\begin{pmatrix}\operatorname{id}_{M}&f\\\
0&\operatorname{id}_{N}\end{pmatrix}=\begin{pmatrix}\operatorname{id}_{M}&f\\\
0&\operatorname{id}_{N}\end{pmatrix}\begin{pmatrix}\tau_{M}&0\\\
0&\tau_{N}\end{pmatrix}.$

The above equation amounts to $u=f\circ\tau_{N}-\tau_{M}\circ\tau^{*}f$, and
hence

$\ker(\iota)=\left\\{f\circ\tau_{N}-\tau_{M}\circ\tau^{*}f~{}|~{}f\in\operatorname{Hom}_{A\otimes
R}(N,M)\right\\}.$

This concludes. ∎

######  Corollary 2.11.

Suppose that $R$ is Noetherian. Let $\underline{N}$ be an $A$-motive over $R$
and let $f:\underline{M}\to\underline{M}^{\prime\prime}$ be an epimorphism
morphism in $\pazocal{M}_{R}$. Then, the induced map
$\operatorname{Ext}^{1}_{\pazocal{M}_{R}}(\underline{N},\underline{M})\to\operatorname{Ext}^{1}_{\pazocal{M}_{R}}(\underline{N},\underline{M}^{\prime\prime})$
is onto.

###### Proof.

As $R$ is Noetherian, so is $A\otimes R$. Because $\tau^{*}N$ is finite
locally-free over $A\otimes R$, it is projective. That $f$ is an epimorphism
means that $f$ is a surjective morphism of the underlying modules. The induced
morphism

$\operatorname{Hom}_{A\otimes
R}(\tau^{*}N,M)[\mathfrak{j}^{-1}]\to\operatorname{Hom}_{A\otimes
R}(\tau^{*}N,M^{\prime\prime})[\mathfrak{j}^{-1}]$

is therefore surjective, and we conclude by Proposition 2.10. ∎

Let $\underline{N}$ be an $A$-motive over $R$. The functor
$\operatorname{Hom}_{\pazocal{M}_{R}}(\underline{N},-)$ from the category
$\pazocal{M}_{R}$ to the category $\mathsf{Mod}_{A}$ of $A$-modules is left-
exact and therefore right-derivable. Because $\pazocal{M}_{R}$ is an exact
category, the higher extensions modules
$\operatorname{Ext}^{i}_{\pazocal{M}_{R}}(\underline{N},\underline{M})$ are
computed by the cohomology of
$\operatorname{RHom}_{\pazocal{M}_{R}}(\underline{N},\underline{M})$. This
implies that, given a short exact sequence in $\pazocal{M}_{R}$

$0\longrightarrow\underline{M}^{\prime}\longrightarrow\underline{M}\longrightarrow\underline{M}^{\prime\prime}\longrightarrow
0,$

we derive a long-exact sequence of $A$-modules:

$\operatorname{Hom}_{\pazocal{M}_{R}}(\underline{N},\underline{M}^{\prime})\hookrightarrow\operatorname{Hom}_{\pazocal{M}_{R}}(\underline{N},\underline{M})\to\operatorname{Hom}_{\pazocal{M}_{R}}(\underline{N},\underline{M}^{\prime\prime})\to\operatorname{Ext}^{1}_{\pazocal{M}_{R}}(\underline{N},\underline{M}^{\prime})\to...$

We deduce the following from Corollary 2.11.

######  Proposition 2.12.

Suppose that $R$ is Noetherian. The modules
$\operatorname{Ext}^{i}_{\pazocal{M}_{R}}(\underline{N},\underline{M})$ vanish
for $i>1$. In particular, the cohomology of
$\operatorname{RHom}_{\pazocal{M}_{R}}(\underline{N},\underline{M})$ is
represented by the complex of $A$-modules

$\left[\operatorname{Hom}_{A\otimes
R}(N,M)\stackrel{{\scriptstyle\tau_{N}^{\vee}-\tau_{M}}}{{\longrightarrow}}\operatorname{Hom}_{A\otimes
R}(\tau^{*}N,M)[\mathfrak{j}^{-1}]\right]$ (2.1)

placed in degree $0$ and $1$.

###### Proof.

Let $C$ be the complex (2.1). We have

$\displaystyle\operatorname{Ext}^{0}_{\pazocal{M}_{R}}(\underline{N},\underline{M})$
$\displaystyle=\operatorname{Hom}_{\pazocal{M}_{R}}(\underline{N},\underline{M})=\\{f\in\operatorname{End}_{A\otimes
R}(M,N)~{}|~{}f\circ\tau_{N}=\tau_{M}\circ\tau^{*}f\\}$
$\displaystyle=\ker(\tau_{N}^{\vee}-\tau_{M})=H^{0}(C).$

By Proposition 2.10,
$\operatorname{Ext}^{1}_{\pazocal{M}_{R}}(\underline{N},\underline{M})\cong
H^{1}(C)$ and, by Corollary 2.11, the functor
$\operatorname{Ext}^{1}_{\pazocal{M}_{R}}(N,-)$ is right-exact. It follows
that
$\operatorname{Ext}^{i}_{\pazocal{M}_{R}}(\underline{N},\underline{M})\cong
H^{i}(C)=0$ for $i\geq 2$ ([PetSt][Lem. A.33]). ∎

Let $\underline{N}$ be an nonzero $A$-motive over $R$. The canonical morphism
of $A$-motives

$\mathbf{1}_{N}:\mathbbm{1}\to\underline{N}\otimes\underline{N}^{\vee}=\operatorname{Hom}(\underline{N},\underline{N}),\quad
a\longmapsto a\cdot\operatorname{id}_{\underline{N}}$

induces functorial isomorphisms for all $i\geq 0$:

$\operatorname{Ext}^{i}_{\pazocal{M}_{R}}(\underline{N},\underline{M})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{Ext}^{i}_{\pazocal{M}_{R}}(\mathbbm{1},\underline{M}\otimes\underline{N}^{\vee}).$
(2.2)

In particular, there is no loss of generality in considering extension modules
of the form
$\operatorname{Ext}^{i}_{\pazocal{M}_{R}}(\mathbbm{1},\underline{M})$. From
now on, we will be interested mainly in extension modules of the latter form.
We shall restate the main results of this section in this case (repeated from
Theorem A of the introduction in the case of a global function field).

######  Theorem 2.13.

Suppose that $R$ is Noetherian, and let $\underline{M}$ be an $A$-motive over
$R$. The cohomology of
$\operatorname{RHom}_{\pazocal{M}_{R}}(\mathbbm{1},\underline{M})$ is computed
by the cohomology of the complex of $A$-modules

$\left[M\stackrel{{\scriptstyle\operatorname{id}-\tau_{M}}}{{\longrightarrow}}M[\mathfrak{j}^{-1}]\right]$

placed in degree $0$ and $1$. Further, there is an $A$-linear surjective
natural morphism

$\iota:M[\mathfrak{j}^{-1}]\twoheadrightarrow\operatorname{Ext}^{1}_{\pazocal{M}_{R}}(\mathbbm{1},\underline{M})$

whose kernel is $(\operatorname{id}-\tau_{M})(M)$, and which is given
explicitly by mapping $m\in M[\mathfrak{j}^{-1}]$ to the class of the
extension

$0\to\underline{M}\to\left[M\oplus(A\otimes
R),\left(\begin{smallmatrix}\tau_{M}&m\cdot\mathbf{1}\\\
0&\mathbf{1}\end{smallmatrix}\right)\right]\to\mathbbm{1}\to 0.$

######  Remark 2.14.

From
$\operatorname{Hom}_{\pazocal{M}^{\text{iso}}_{R}}(-,-)=\operatorname{Hom}_{\pazocal{M}_{R}}(-,-)\otimes_{A}K$
(c.f. Definition 2.9), the extension spaces of $\mathbbm{1}$ by
$\underline{M}$ in the $K$-linear category $\pazocal{M}^{\text{iso}}_{R}$ are
computed by the complex

$\left[M\otimes_{A}K\stackrel{{\scriptstyle\operatorname{id}-\tau_{M}}}{{\longrightarrow}}M[\mathfrak{j}^{-1}]\otimes_{A}K\right].$

###  2.3 Extensions having good reduction

We now introduce the function field analogue of the $\ell$-adic realization
functor (Definition 2.16), and show that it is exact (Proposition 2.19). It
will allow us to define extensions with good reduction next (Definition 2.22).

For the rest of this section, let $\ell$ be a maximal ideal of $A$, and denote
by $\pazocal{O}_{\ell}$ the completed local ring of $A$ at $\ell$. We let $F$
be a field containing $K$ and let $\kappa:A\to F$ be the inclusion. Let
$F^{s}$ be a separable closure of $F$ and denote by
$G_{F}=\operatorname{Gal}(F^{s}|F)$ the absolute Galois group of $F$ equipped
with the profinite topology. The group $G_{F}$ acts $A$-linearly on the left-
hand side of the tensor $A\otimes F^{s}$, and this action extends by
continuity to an $\pazocal{O}_{\ell}$-linear action of the algebra

$\pazocal{A}_{\ell}(F^{s}):=(A\hat{\otimes}F^{s})_{\ell}=\varprojlim_{n}~{}(A\otimes
F^{s})/\ell^{n}(A\otimes F^{s})$

leaving $\pazocal{A}_{\ell}(F):=(A\hat{\otimes}F)_{\ell}$ invariant. If
$\mathbb{F}_{\ell}$ denotes the residue field of $\pazocal{O}_{\ell}$ and
$\pi$ a uniformizer, we have an identification
$\pazocal{A}_{\ell}(F)=(\mathbb{F}_{\ell}\otimes F)[\\![\pi]\\!]$.

######  Remark 2.15.

In the function field/number field dictionary, the assignment
$R\mapsto\pazocal{A}(R)$ is akin to the $p$-typical Witt vectors construction
$R\mapsto W(R)$ (e.g. [Har1, §1.1]). In this analogy, $R\mapsto\pazocal{B}(R)$
would be akin to $R\mapsto W(R)[p^{-1}]$.

Let $\underline{M}=(M,\tau_{M})$ be an $A$-motive over $F$ of rank $r$. Let
$\underline{M}_{F^{s}}=(M_{F^{s}},\tau_{M})$ be the $A$-motive over $F^{s}$
obtained from $\underline{M}$ by base-change. $G_{F}$ acts
$\pazocal{O}_{\ell}$-linearly on:

$\widehat{(M_{F^{s}})}_{\ell}:=\varprojlim_{n}~{}(M\otimes_{F}F^{s})/\ell^{n}(M\otimes_{F}F^{s})=M\otimes_{A\otimes
F}\pazocal{A}_{\ell}(F^{s})$

and leaves the submodule $\widehat{M}_{\ell}=M\otimes_{A\otimes
F}\pazocal{A}_{\ell}(F)$ invariant. Following [HarJu, §2.3.5], we define:

######  Definition 2.16.

The $\ell$-adic realization $T_{\ell}\underline{M}$ of $\underline{M}$
consists of the $\pazocal{O}_{\ell}$-module

$T_{\ell}\underline{M}:=\left\\{m\in\widehat{(M_{F^{s}})}_{\ell}~{}|~{}m=\tau_{M}(\tau^{*}m)\right\\}$

together with the compatible action of $G_{F}$ it inherits as a submodule of
$\widehat{(M_{F^{s}})}_{\ell}$.

######  Remark 2.17.

In [Mor2], Mornev extended this construction to the situation where $\ell$ is
the closed point $\infty$.

The next lemma is well-known in the case of $\tau$-sheaves (e.g. [TagWa,
Prop.6.1]).

######  Lemma 2.18.

The map
$T_{\ell}\underline{M}\otimes_{\pazocal{O}_{\ell}}\pazocal{A}_{\ell}(F^{s})\to\widehat{(M_{F^{s}})}_{\ell}$,
$\omega\otimes f\mapsto\omega\cdot f$ is an isomorphism of
$\pazocal{A}_{\ell}(F^{s})$-modules. In particular, the
$\pazocal{O}_{\ell}$-module $T_{\ell}\underline{M}$ is free of rank $r$ and
the action of $G_{F}$ on $T_{\ell}\underline{M}$ is continuous.

###### Proof.

Let $n\geq 1$. In the ring $A\otimes F$, the ideals $\ell^{n}$ and
$\mathfrak{j}$ are coprime. Hence, the following composition of $A\otimes
F$-linear map is a well-defined isomorphism:

$\varphi_{n}:\tau^{*}(M/\ell^{n}M)\cong\frac{(\tau^{*}M)}{\ell^{n}(\tau^{*}M)}\cong\frac{(\tau^{*}M)[\mathfrak{j}^{-1}]}{\ell^{n}(\tau^{*}M)[\mathfrak{j}^{-1}]}\stackrel{{\scriptstyle\tau_{M}}}{{\longrightarrow}}\frac{M[\mathfrak{j}^{-1}]}{\ell^{n}M[\mathfrak{j}^{-1}]}\cong
M/\ell^{n}M.$

The data of $\varphi_{n}$ induces a semi-simple $q$-linear map222For $k$ a
field containing $\mathbb{F}$ and $V$ a $k$-vector space, an
$\mathbb{F}$-linear endomorphism $f$ of $V$ is q-linear if $f(rv)=r^{q}f(v)$
for all $r\in k$ and $v\in V$. (in the sense of [Kat1, §1]) on the finite
dimensional $F^{s}$-vector space:

$(M\otimes_{F}F^{s})/\ell^{n}(M\otimes_{F}F^{s})=\widehat{(M_{F^{s}})}_{\ell}/\ell^{n}\widehat{(M_{F^{s}})}_{\ell}.$

By [Kat1, Prop. 1.1], the multiplication map

$\\{m\in\widehat{(M_{F^{s}})}_{\ell}/\ell^{n}\widehat{(M_{F^{s}})}_{\ell}~{}|~{}\tau_{M}(\tau^{*}m)=m\\}\otimes_{\mathbb{F}}F^{s}\to\widehat{(M_{F^{s}})}_{\ell}/\ell^{n}\widehat{(M_{F^{s}})}_{\ell}$
(2.3)

is an isomorphism. Taking the inverse limit of (2.3) over all $n$ yields the
desired isomorphism.

As $\widehat{(M_{F^{s}})}_{\ell}$ is free of rank $r$ over
$\pazocal{A}_{\ell}(F^{s})$, the same is true for
$\widehat{(M_{F^{s}})}_{\ell}/\ell^{n}\widehat{(M_{F^{s}})}_{\ell}$ over
$(A/\ell^{n})\otimes F^{s}$. The isomorphism (2.3) implies that the
$A/\ell^{n}$-module

$[\widehat{(M_{F^{s}})}_{\ell}/\ell^{n}\widehat{(M_{F^{s}})}_{\ell}]^{\tau_{M}=1}:=\\{m\in\widehat{(M_{F^{s}})}_{\ell}~{}|~{}\tau_{M}(\tau^{*}m)=m\\}$

is free of rank $r$ over $A/\ell^{n}$. Their projective limit
$T_{\ell}\underline{M}$ is thus a free $\pazocal{O}_{\ell}$-module of rank
$r$.

By definition, the action of $G_{F}$ on $T_{\ell}\underline{M}$ is continuous
if, and only if, the induced action of $G_{F}$ on
$T_{\ell}\underline{M}/\ell^{n}T_{\ell}\underline{M}$ factors through a finite
quotient for all $n$. Let $\textbf{t}=\\{t_{1},...,t_{s}\\}$ be a basis of the
finite dimensional $F$-vector space
$\widehat{M}_{\ell}/\ell^{n}\widehat{M}_{\ell}$. Let $F_{M}$ be the matrix of
$\tau_{M}$ written in the basis $\tau^{*}\textbf{t}$ and t. Let
$\bm{\omega}=\\{\omega_{1},...,\omega_{s}\\}$ be a basis of
$T_{\ell}\underline{M}/\ell^{n}T_{\ell}\underline{M}$ over $\mathbb{F}$. By
(2.3), $\bm{\omega}$ is a basis of
$\widehat{(M_{F^{s}})}_{\ell}/\ell^{n}\widehat{(M_{F^{s}})}_{\ell}$ over
$F^{s}$, and we let $w_{ij}\in F^{s}$ be the coefficients of $\bm{\omega}$
expressed in $\mathbf{t}$, that is, for $i\in\\{1,...,s\\}$,
$\omega_{i}=\sum{w_{ij}t_{j}}$. We let $E_{n}$ denote the Galois closure of
the finite separable extension $F(w_{ij}|(i,j)\in\\{1,...,s\\}^{2})$ of $F$ in
$F^{s}$. Then,

$T_{\ell}\underline{M}/\ell^{n}T_{\ell}\underline{M}=\\{m\in(M\otimes_{F}E_{n})/\ell^{n}(M\otimes_{F}E_{n})~{}|~{}\tau_{M}(\tau^{*}m)=m\\}.$

That is, the action of $G_{F}$ factors through
$\operatorname{Gal}(E_{n}|F)=G_{F}/\operatorname{Gal}(F^{s}|E_{n})$, as
desired. ∎

######  Proposition 2.19.

The following sequence of $\pazocal{O}_{\ell}[G_{F}]$-modules is exact

$0\to
T_{\ell}\underline{M}\longrightarrow\widehat{(M_{F^{s}})}_{\ell}\stackrel{{\scriptstyle\operatorname{id}-\tau_{M}}}{{\longrightarrow}}\widehat{(M_{F^{s}})}_{\ell}\longrightarrow
0.$

###### Proof.

Everything is clear but the surjectivity of $\operatorname{id}-\tau_{M}$. Let
$\pi$ be a uniformizer of $\pazocal{O}_{\ell}$ and let $\mathbb{F}_{\ell}$ be
its residue field. Let $f=\sum_{n\geq 0}{a_{n}\pi^{n}}$ be a series in
$\pazocal{A}_{\ell}(F^{s})=(\mathbb{F}_{\ell}\otimes F^{s})[\\![\pi]\\!]$. Let
$b_{n}\in\mathbb{F}_{\ell}\otimes F^{s}$ be such that
$[\operatorname{id}_{\mathbb{F}_{\ell}}\otimes(\operatorname{id}-\operatorname{Frob}_{q})](b_{n})=a_{n}$
(which exists as $F^{s}$ is separably closed), and let $g$ be the series
$\sum_{n\geq 0}{b_{n}\pi^{n}}$ in $\pazocal{A}_{\ell}(F^{s})$. For $\omega\in
T_{\ell}\underline{M}$, we have

$(\operatorname{id}-\tau_{M})\left(\omega\cdot g\right)=\omega\cdot f.$

It follows that any element in
$\widehat{(M_{F^{s}})}_{\ell}=M\otimes_{A\otimes F}\pazocal{A}_{\ell}(F^{s})$
of the form $\omega\cdot f$ is in the image of $\operatorname{id}-\tau_{M}$.
By the first part of Lemma 2.18, those elements generates
$\widehat{(M_{F^{s}})}_{\ell}$. We conclude that $\operatorname{id}-\tau_{M}$
is surjective. ∎

We obtain the following:

######  Corollary 2.20.

The functor $\underline{M}\mapsto T_{\ell}\underline{M}$, from
$\pazocal{M}_{F}$ to the category of continuous $\pazocal{O}_{\ell}$-linear
$G_{F}$-representations, is exact.

###### Proof.

Let
$S:0\to\underline{M}^{\prime}\to\underline{M}\to\underline{M}^{\prime\prime}\to
0$ be an exact sequence in $\pazocal{M}_{F}$. The underlying sequence of
$A\otimes F$-modules is exact, and because $\pazocal{A}_{\ell}(F^{s})$ is flat
over $A\otimes F$, the sequence of $\pazocal{A}_{\ell}(F^{s})$-modules
$\widehat{(S_{F^{s}})}_{\ell}$ is exact. In particular, the next commutative
diagram of $\pazocal{O}_{\ell}$-modules has exact rows:

${0}$${\widehat{(M^{\prime}_{F^{s}})}_{\ell}}$${\widehat{(M_{F^{s}})}_{\ell}}$${\widehat{(M^{\prime\prime}_{F^{s}})}_{\ell}}$${0}$${0}$${\widehat{(M^{\prime}_{F^{s}})}_{\ell}}$${\widehat{(M_{F^{s}})}_{\ell}}$${\widehat{(M^{\prime\prime}_{F^{s}})}_{\ell}}$${0}$$\scriptstyle{\operatorname{id}-\tau_{M^{\prime}}}$$\scriptstyle{\operatorname{id}-\tau_{M}}$$\scriptstyle{\operatorname{id}-\tau_{M^{\prime\prime}}}$

and the Snake Lemma together with Proposition 2.19 yields that $T_{\ell}S$ is
exact. ∎

Let $\underline{M}$ be an $A$-motive over $F$. From Corollary 2.20, the
functor $T_{\ell}$ induces an $A$-linear morphism:

$\operatorname{Ext}^{1}_{\pazocal{M}_{F}}(\mathbbm{1},\underline{M})\longrightarrow
H^{1}(G_{F},T_{\ell}\underline{M})$ (2.4)

into the first continuous cohomology group of $G_{F}$ with values in
$T_{\ell}\underline{M}$. The next paragraph is devoted to the explicit
determination of (2.4).

Let $[\underline{E}]:0\to\underline{M}\to\underline{E}\to\mathbbm{1}\to 0$ be
a class in
$\operatorname{Ext}^{1}_{\pazocal{M}_{F}}(\mathbbm{1},\underline{M})$ of the
form $\iota(m)$ for some $m\in M[\mathfrak{j}^{-1}]$ (Theorem 2.13). The
$\ell$-adic realization $T_{\ell}\underline{E}$ of $\underline{E}$ is the
$\pazocal{O}_{\ell}[G_{F}]$-module consisting of solutions $\xi\oplus
a\in\widehat{(M_{F^{s}})}_{\ell}\oplus\pazocal{A}_{\ell}(F^{s})$ of the
equation

$\begin{pmatrix}\tau_{M}&m\cdot\mathbf{1}\\\
0&\mathbf{1}\end{pmatrix}\begin{pmatrix}\tau^{*}\xi\\\
\tau^{*}a\end{pmatrix}=\begin{pmatrix}\xi\\\ a\end{pmatrix}$

(see Definition 2.16). The above equality amounts to $a\in\pazocal{O}_{\ell}$
and $\xi-\tau_{M}(\tau^{*}\xi)=am$. A splitting of $[T_{\ell}\underline{E}]$
as a sequence of $\pazocal{O}_{\ell}$-modules corresponds to the choice of a
particular solution $\xi_{m}\in\widehat{(M_{F^{s}})}_{\ell}$ of
$\xi-\tau_{M}(\tau^{*}\xi)=m$ (whose existence is provided by Proposition
2.19). We then have

$T_{\ell}\underline{M}\oplus\pazocal{O}_{\ell}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}T_{\ell}\underline{E},\quad(\omega,a)\longmapsto(\omega+a\xi_{m},a).$

It follows that the morphism (2.4) maps $[\underline{E}]$ to the class of the
cocycle $(\sigma\mapsto\xi_{m}^{\sigma}-\xi_{m})$, where $\xi_{m}$ is any
solution in $\widehat{(M_{F^{s}})}_{\ell}$ of the equation
$\xi-\tau_{M}(\tau^{*}\xi)=m$. In other words, we have almost proved:

######  Proposition 2.21.

There is a commutative diagram of $A$-modules:

${\operatorname{Ext}^{1}_{\pazocal{M}_{F}}(\mathbbm{1},\underline{M})}$${H^{1}(G_{F},T_{\ell}\underline{M})}$${\displaystyle\frac{M[\mathfrak{j}^{-1}]}{(\operatorname{id}-\tau_{M})(M)}}$${\displaystyle\frac{\widehat{M}_{\ell}}{(\operatorname{id}-\tau_{M})(\widehat{M}_{\ell})}}$$\scriptstyle{\eqref{eq:Tell-
exact-on-Ext}}$$\scriptstyle{\iota}$$\scriptstyle{\wr}$$\scriptstyle{\wr}$

where the right vertical morphism maps the class of $f\in\widehat{M}_{\ell}$
to the class of the cocycle $\sigma\mapsto\xi-\,^{\sigma}\xi$, $\xi$ begin any
solution in $\widehat{(M_{F^{s}})}_{\ell}$ of $f=\xi-\tau_{M}(\tau^{*}\xi)$.

###### Proof.

The only remaining fact to check is that the right vertical morphism is an
isomorphism. Applying the functor of $G_{F}$-invariants to the short exact
sequence of Proposition 2.19, we obtain a long exact sequence of cohomology:

$\widehat{M}_{\ell}\stackrel{{\scriptstyle\operatorname{id}-\tau_{M}}}{{\longrightarrow}}\widehat{M}_{\ell}\longrightarrow
H^{1}(G_{F},T_{\ell}\underline{M})\longrightarrow
H^{1}(G_{F},\widehat{(M_{F^{s}})}_{\ell}).$

Therefore, it is sufficient to prove that
$H^{1}(G_{F},\widehat{(M_{F^{s}})}_{\ell})$ vanishes. We have
$\widehat{(M_{F^{s}})}_{\ell}=M\otimes_{A\otimes F}\pazocal{A}_{\ell}(F^{s})$,
$G_{F}$ acting on the right-hand side of the tensor. Hence, it is enough to
prove that $H^{1}(G_{F},\pazocal{A}_{\ell}(F^{s}))$ vanishes. The latter stems
easily from the identification
$\pazocal{A}_{\ell}(F^{s})=(\mathbb{F}_{\ell}\otimes F^{s})[\\![\pi]\\!]$. ∎

For the remaining of this section, let us assume that $F=F_{\mathfrak{p}}$ is
a local function field with valuation ring $\pazocal{O}_{\mathfrak{p}}$ and
maximal ideal $\mathfrak{p}$. Let $F_{\mathfrak{p}}^{\text{ur}}$ be the
maximal unramified extension of $F_{\mathfrak{p}}$ in $F_{\mathfrak{p}}^{s}$.
Let $I_{\mathfrak{p}}$ be the inertia subgroup of
$G_{\mathfrak{p}}=G_{F_{\mathfrak{p}}}$.

######  Definition 2.22.

Let $\underline{M}$ be an $A$-motive over $F_{\mathfrak{p}}$, and let $\ell$
be a maximal ideal in $A$. We say that an extension $[\underline{E}]$ of
$\mathbbm{1}$ by $\underline{M}$ has good reduction with respect to $\ell$ if
$[\underline{E}]$ stands in the kernel of
$\operatorname{Ext}^{1}_{\pazocal{M}_{F}}(\mathbbm{1},\underline{M})\to
H^{1}(I_{\mathfrak{p}},T_{\ell}\underline{M})$.

From Proposition 2.21 together with the fact that
$I_{\mathfrak{p}}=\operatorname{Gal}(F^{s}_{\mathfrak{p}}|F_{\mathfrak{p}}^{\text{ur}})$,
we easily derive:

######  Proposition 2.23.

Let $m\in M[\mathfrak{j}^{-1}]$. The following are equivalent:

  1. $(a)$

The extension $\iota(m)$ has good reduction with respect to $\ell$,

  2. $(b)$

The equation $\xi-\tau_{M}(\tau^{*}\xi)=m$ admits a solution $\xi$ in
$\widehat{(M_{F^{\text{ur}}_{\mathfrak{p}}})}_{\ell}$.

######  Remark 2.24.

If $\kappa(A)\subset\pazocal{O}_{\mathfrak{p}}$, then Definition 2.22 should
presumably be independent of $\ell$, as long as
$\kappa(\ell)\pazocal{O}_{\mathfrak{p}}=\pazocal{O}_{\mathfrak{p}}$. The
analogous statement is a conjecture in the number field setting (c.f. [Sch, §2
Rmk]). So far, I have no clue on how to prove this statement.

##  3 Mixed $A$-motives and their extension modules

We discuss here the notion of mixedness for Anderson $A$-motives. In the case
where $A$ is a polynomial algebra, the definition of pure $t$-motives is
traced back to the work of Anderson [And, 1.9]. The definition of mixed
$A$-motives appeared only recently in the work of Hartl-Juschka [HarJu, §3]
under the condition that the place $\infty$ has degree one and the base field
$R=F$ is algebraically closed. Our presentation deals with the most general
case: arbitrary curve $C$ and place $\infty$, and over an arbitrary base field
$F$. Compared to [Tae1] or [HarJu], the new difficulty is to deal with non
perfect fields $F$, since then, the slope filtration for isocrystals does not
necessarily split.

###  3.1 Mixed Anderson $A$-motives

#### Isocrystals over a field

In this subsection, we present some materials on function fields isocrystals
following [Mor2]. Our objective is to prove existence and uniqueness of the
slope filtration with pure subquotients having increasing slopes. The general
theory of slope filtrations has been developed in [Andr], and the results of
interest for us on isocrystals appear in [Har2] in the case $A$ is a
polynomial algebra. The new account of this subsection is the adaptation of
[Har2, Prop 1.5.10] to allow more general ring $A$ (see Theorem 3.9). This
will be used to define weights and study mixedness later on (Definition 3.20).

We begin with some general notations. Let $R$ be a Noetherian
$\mathbb{F}$-algebra, and let $k$ be a finite field extension of $\mathbb{F}$.
Let $E$ be the field of Laurent series over $k$ in the formal variable $\pi$,
$\pazocal{O}$ the subring of $E$ consisting of power series over $k$ and
$\mathfrak{m}$ the maximal ideal of $\pazocal{O}$. Explicitly
$E=k(\\!(\pi)\\!)$, $\pazocal{O}=k[\\![\pi]\\!]$ and
$\mathfrak{m}=\pi\pazocal{O}$. In the sequel, $E$ will correspond to the local
field of $(C,\pazocal{O}_{C})$ at a closed point of $C$.

Extending the notation introduced in Subsection 2.3 in the context of the
$\ell$-adic realization functor, we denote by $\pazocal{A}(R)$ the completion
of the ring $\pazocal{O}\otimes R$ at its ideal $\mathfrak{m}\otimes R$:

$\pazocal{A}(R)=\varprojlim_{n}(\pazocal{O}\otimes R)/(\mathfrak{m}^{n}\otimes
R)$

and we let $\pazocal{B}(R)$ be the ring
$E\otimes_{\pazocal{O}}\pazocal{A}(R)$. Throughout the previous
identifications, we readily check that $\pazocal{A}(R)=(k\otimes
R)[\\![\pi]\\!]$ and $\pazocal{B}(R)=(k\otimes R)(\\!(\pi)\\!)$. Let
$\tau:\pazocal{O}\otimes R\to\pazocal{O}\otimes R$, be the
$\pazocal{O}$-linear map induced by $a\otimes r\mapsto a\otimes r^{q}$. We
shall also denote by $\tau$ its continuous extension to $\pazocal{A}(R)$ or
$\pazocal{B}(R)$. Similarly, we denote by $\mathbf{1}$ the canonical $A\otimes
R$-linear morphisms $\tau^{*}\pazocal{A}(R)\to\pazocal{A}(R)$ and
$\tau^{*}\pazocal{B}(R)\to\pazocal{B}(R)$.

For the remaining of this subsection, we assume that $R=F$ is a field.

######  Definition 3.1.

An isocrystal $\underline{D}$ over $F$ is a pair $(D,\varphi_{D})$ where $D$
is a free $\pazocal{B}(F)$-module of finite rank and $\varphi_{D}:\tau^{*}D\to
D$ is a $\pazocal{B}(F)$-linear isomorphism.
A morphism $(D,\varphi_{D})\to(C,\varphi_{C})$ of isocrystals is a
$\pazocal{B}(F)$-linear morphism of the underlying modules $f:B\to C$ such
that $f\circ\varphi_{D}=\varphi_{C}\circ\tau^{*}f$. We let $\pazocal{I}_{F}$
be the category of isocrystals over $F$.

######  Remark 3.2.

Pursuing the analogy of Remark 2.15, iscorystals are the analogue of the
eponymous object in $p$-adic Hodge theory (we refer to [Har1, §3.5]). In both
settings, such objects carry a slope filtration (see Theorem 3.9 for the
function fields one). For number fields, isocrystals are only defined at
finite places, whereas function fields isocrystals are defined regardless of
the finiteness of the place. In the next subsection, we use the slope
filtration at $\infty$ in order to define weights.

We define the rank $\operatorname{rk}\underline{D}$ of $\underline{D}$ to be
the rank of $D$ over $\pazocal{B}(F)$. If $\underline{D}$ is nonzero, let b be
a basis of $D$ and let $U$ denote the matrix of $\varphi$ expressed in
$\tau^{*}\textbf{b}$ and b. A different choice of basis $\textbf{b}^{\prime}$
leads to a matrix $U^{\prime}$ such that $U=\tau(P)U^{\prime}P^{-1}$ for a
certain invertible matrix $P$ with coefficients in $\pazocal{B}(F)$. As such,
the valuation of $\det U$ in $\pi$ is independent of b. We denote it by
$\deg\underline{D}$ and we name it the degree of $\underline{D}$. We define
the slope of $\underline{D}$ to be the rational number
$\lambda(\underline{D})=\delta\deg\underline{D}/\operatorname{rk}\underline{D}$,
where $\delta$ is the degree of $k$ over $\mathbb{F}$.

From [Mor2, Prop. 4.1.1], the category $\pazocal{I}_{F}$ is abelian. We can
therefore consider exact sequences in $\pazocal{I}_{F}$. The degree and rank
are additive in short exact sequences, and the association
$\underline{D}\mapsto-\lambda(\underline{D})$ defines a slope function for
$\pazocal{I}_{F}$ in the sense of [Andr, Def. 1.3.1]. The second point of the
next definition should be compared with [Andr, Def. 1.3.6]:

######  Definition 3.3.

Let $\underline{D}=(D,\varphi)$ be an isocrystal over $F$.

  1. 1.

A subisocrystal of $\underline{D}$ is an isocrystal
$\underline{G}=(G,\varphi_{G})$ for which $G\subset D$,
$\varphi_{G}=\varphi_{D}|_{\tau^{*}G}$. The quotient of $\underline{D}$ by
$\underline{G}$ is the pair $(D/G,\varphi_{D})$ (this is indeed an isocrystal
by [Mor2, Prop. 4.1.1]).

  2. 2.

The isocrystal $\underline{D}$ is semistable (resp. isoclinic) if, for any
nonzero subisocrystal $\underline{D}^{\prime}$ of $\underline{D}$,
$\lambda(\underline{D}^{\prime})\geq\lambda(\underline{D})$ (resp.
$\lambda(\underline{D}^{\prime})=\lambda(\underline{D})$).

Semistability and isoclinicity are related to the notion of purity, borrowed
from [Mor2, Def. 3.4.6], that we next recall. We first require the definition
of $\pazocal{A}(F)$-lattices:

######  Definition 3.4.

Let $D$ be a free $\pazocal{B}(F)$-module of finite rank. An
$\pazocal{A}(F)$-lattice in $D$ is a sub-$\pazocal{A}(F)$-module of finite
type of $D$ which generates $D$ over $E$.

Note that any $\pazocal{A}(F)$-lattice $L$ in $D$ is free, and that its rank
is the rank of $D$ over $\pazocal{B}(F)$. We denote by
$\langle\varphi_{D}L\rangle$ the sub-$\pazocal{A}(F)$-module
$\varphi_{D}(\tau^{*}L)$ in $D$: it is again an $\pazocal{A}(F)$-lattice in
$D$ since $\varphi_{D}$ is an isomorphism. We define
$\langle\varphi_{D}^{n}L\rangle$ inductively to be the
$\pazocal{A}(F)$-lattice
$\langle\varphi_{D}\langle\varphi_{D}^{n-1}L\rangle\rangle$. To include the
$n=0$-case, we agree that $\langle\varphi_{D}^{0}L\rangle=L$.

######  Definition 3.5.

A nonzero isocrystal $(D,\varphi_{D})$ over $F$ is said to be pure of slope
$\lambda$ if there exist an $\pazocal{A}(F)$-lattice $L$ in $D$ and integers
$s$ and $r>0$ such that
$\langle\varphi_{D}^{r\delta}L\rangle=\mathfrak{m}^{s}L$ and $\lambda=s/r$. By
convention, the zero isocrystal is pure with no slope.

######  Example 3.6.

Let $D$ be the free $\pazocal{B}(F)$-module of rank $r\geq 1$ with basis
$\\{e_{0},...,e_{r-1}\\}$ and let $\varphi_{D}:\tau^{*}D\to D$ be the unique
linear map such that $\varphi_{D}(\tau^{*}e_{i-1})=e_{i}$ for $1\leq i<r$ and
$\varphi_{D}(\tau^{*}e_{r-1})=\pi^{s}e_{0}$. Then $(D,\varphi_{D})$ is a pure
isocrystal of slope $\delta s/r$ with
$\pazocal{A}(F)e_{0}\oplus\cdots\oplus\pazocal{A}(F)e_{s-1}$ for
$\pazocal{A}(F)$-lattice.

The following lemma relates the definition of slopes from purity and from
slope functions. It also implies that one can refer to the slope of a pure
isocrystal:

######  Lemma 3.7.

If $\underline{D}$ is a pure isocrystal of slope $\lambda$, then
$\lambda(\underline{D}^{\prime})=\lambda$ for any nonzero sub-isocrystal
$\underline{D}^{\prime}$ of $\underline{D}$. In particular, $\underline{D}$ is
isoclinic (hence semistable).

###### Proof.

Assume there exists an $\pazocal{A}(F)$-lattice $L$ in $D$ such that
$\langle\varphi^{r\delta}L\rangle=\mathfrak{m}^{s}L$ for integers $r>0$ and
$d$ such that $\lambda=s/r$. If $\underline{D}^{\prime}=(D^{\prime},\varphi)$
is a nonzero subisocrystal of $\underline{D}$, then $L^{\prime}=L\cap
D^{\prime}$ is an $\pazocal{A}(F)$-lattice in $D^{\prime}$ such that
$\langle\varphi^{r\delta}L^{\prime}\rangle=\mathfrak{m}^{s}L^{\prime}$. As
$L^{\prime}$ is nonzero, let $\\{t_{1},...,t_{\ell}\\}$ be a basis of
$L^{\prime}$ over $\pazocal{A}(F)$. We have

$(\det\varphi)^{r\delta}(t_{1}\wedge\cdots\wedge
t_{\ell})=\mathfrak{m}^{s\ell}(t_{1}\wedge\cdots\wedge
t_{\ell})\quad\text{in}~{}\bigwedge^{\ell}L^{\prime}.$

Hence
$r\delta\deg\underline{D}^{\prime}=s\operatorname{rk}\underline{D}^{\prime}$,
which yields $\lambda(\underline{D}^{\prime})=\lambda$. ∎

######  Definition 3.8.

A slope filtration for $\underline{D}$ is an increasing sequence of sub-
isocrystals of $\underline{D}$

$0=\underline{D}_{0}\subsetneq\underline{D}_{1}\subsetneq\cdots\subsetneq\underline{D}_{s}=\underline{D},$

satisfying:

  1. $(i)$

for all $i\in\\{1,...,s\\}$, $\underline{D}_{i}/\underline{D}_{i-1}$ is
semistable,

  2. $(ii)$

we have
$\lambda(\underline{D}_{1})<\lambda(\underline{D}_{2}/\underline{D}_{1})<\cdots<\lambda(\underline{D}_{s}/\underline{D}_{s-1})$.

######  Theorem 3.9.

Let $\underline{D}$ be an isocrystal over $F$. Then $\underline{D}$ carries a
unique slope filtration:

$0=\underline{D}_{0}\subsetneq\underline{D}_{1}\subsetneq\cdots\subsetneq\underline{D}_{s}=\underline{D}.$
(3.1)

In addition, for all $i\in\\{1,...,s\\}$, the quotients
$\underline{D}_{i}/\underline{D}_{i-1}$ are pure isocrystals.

######  Remark 3.10.

The proof presented below relies on [Har2, Prop. 1.5.10] which already uses
Dieudonné-Manin classification (in the case $A=\mathbb{F}[t]$). It would be
much more satisfactory to prove the equivalence between semistability and
isoclinicity directly, so that Theorem 3.9 would follow from André’s theory.

###### Proof of Theorem 3.9.

The existence and uniqueness of the slope filtration follows from [Andr, Thm
1.4.7] applied to the slope function
$\underline{D}\mapsto-\lambda(\underline{D})$ on the abelian category
$\pazocal{I}_{F}$.

We focus on the second part of the statement. If $\delta=1$, then
$\pazocal{A}(F)$ is identified with $F[\\![\pi]\\!]$ and Theorem 3.9 is proved
in [Har2, Prop. 1.5.10]. We now explain how the general case follows from the
above. Let $\mathbb{G}$ be the finite field extension of $\mathbb{F}$
corresponding to

$\mathbb{G}:=\\{f\in\bar{\mathbb{F}}\cap F~{}|~{}f^{q^{\delta}}=f\\}.$

Let $\phi:\mathbb{G}\to F$ denote the inclusion. This defines an embedding of
$\mathbb{G}$ in $k$, the residue field of $E$. Let $\pazocal{A}_{\phi}(F)$ be
the completion of $\pazocal{O}\otimes_{\mathbb{G}}F$ at the ideal
$\mathfrak{m}\otimes_{\mathbb{G}}F$. In the theory of isocrystals over $F$
with $\mathbb{G}$ in place of $\mathbb{F}$, $\pazocal{A}_{\phi}(F)$ appears in
place of $\pazocal{A}(F)$ and $\delta=1$. In [Mor2, §4.2], Mornev defines an
additive functor

$[\phi]^{*}:\left(\pazocal{A}(F)-\text{isocrystals}\right)\longrightarrow\left(\pazocal{A}_{\phi}(F)-\text{isocrystals}\right)$

which, by [Mor2, Prop 4.2.2] (see also [BorHa, Prop 8.5]), is an equivalence
of categories such that $[\phi]^{*}(\underline{D})$ is a pure isocrystals of
slope $\lambda$ if $\underline{D}$ is. Let

$[\phi]_{*}:\left(\pazocal{A}_{\phi}(F)-\text{isocrystals}\right)\longrightarrow\left(\pazocal{A}(F)-\text{isocrystals}\right)$

be a quasi-inverse of $[\phi]^{*}$ and let
$\ell:[\phi]_{*}[\phi]^{*}\stackrel{{\scriptstyle\sim}}{{\to}}\operatorname{id}$
be a natural transformation.

Let $\underline{D}$ be an $\pazocal{A}(F)$-isocrystal. We only need to prove
existence of (3.1) with pure subquotients since uniqueness follows from [Andr,
Thm 1.4.7]. By [Har2, Prop. 1.5.10], there exists an increasing sequence of
sub-$\pazocal{A}_{\phi}(F)$-isocrystals of $[\phi]^{*}\underline{D}$:

$0=\underline{G}_{0}\subsetneq\underline{G}_{1}\subsetneq\underline{G}_{2}\subsetneq\cdots\subsetneq\underline{G}_{s}=[\phi]^{*}\underline{D}$

the subquotients $\underline{G}_{i}/\underline{G}_{i-1}$ being pure of slopes
$\lambda_{i}$ with $\lambda_{1}<\cdots<\lambda_{s}$. Applying $[\phi]_{*}$ and
then $\ell$, we obtain

$0=\underline{D}_{0}\subsetneq\underline{D}_{1}\subsetneq\underline{D}_{2}\subsetneq\cdots\subsetneq\underline{D}_{s}=\underline{D}$
(3.2)

with $\underline{D}_{i}:=\ell([\phi]_{*}[\phi]^{*}\underline{D}_{i})$ for all
$i\in\\{0,1,...,s\\}$. We claim that the isocrystals
$\underline{D}_{i}/\underline{D}_{i-1}$ are pure of slope $\lambda_{i}$.
Indeed, we have

$\underline{D}_{i}/\underline{D}_{i-1}\cong[\phi]_{*}\underline{G}_{i}/[\phi]_{*}\underline{G}_{i-1}\cong[\phi]_{*}(\underline{G}_{i}/\underline{G}_{i-1})$

where the last isomorphism comes from the fact that $[\phi]_{*}$ is an exact
functor (any equivalence of categories is exact). Because
$\underline{G}_{i}/\underline{G}_{i-1}$ is pure of slope $\lambda_{i}$,
$\underline{D}_{i}/\underline{D}_{i-1}$ is also pure of slope $\lambda_{i}$.
We conclude that (3.2) is the slope filtration for $\underline{D}$ and
satisfies the assumption of the theorem. ∎

Let $\underline{D}$ be an isocrystal over $F$. It is useful to rewrite the
slope filtration of $\underline{D}$ as $(\underline{D}_{\lambda_{i}})_{1\leq
i\leq s}$ for rational numbers $\lambda_{1}<\cdots<\lambda_{s}$, where the
successive quotients
$\underline{D}_{\lambda_{i}}/\underline{D}_{\lambda_{i-1}}$ are pure of slope
$\lambda_{i}$. We let $D_{\lambda_{i}}$ be the underlying module of
$\underline{D}_{\lambda_{i}}$. For $\lambda\in\mathbb{Q}$, let
$\underline{D}_{\lambda}$ be the subisocrystal of $\underline{D}$ whose
underlying module is

$D_{\lambda}:=\bigcup_{\lambda_{i}\leq\lambda}D_{\lambda_{i}}.$

We also let

$\operatorname{Gr}_{\lambda}\underline{D}:=\underline{D}_{\lambda}/\bigcup_{\lambda^{\prime}<\lambda}\underline{D}_{\lambda^{\prime}},$

the symbol $\cup$ being understood as the isocrystal whose underlying module
is given by the union.

######  Corollary 3.11.

For all $\lambda\in\mathbb{Q}$, the assignment
$\pazocal{I}_{F}\to\pazocal{I}_{F}$,
$\underline{D}\mapsto\underline{D}_{\lambda}$ defines an exact functor.
Equivalently, any morphism $f:\underline{D}\to\underline{C}$ of isocrystals
over $F$ is strict with respect to the slope filtration, that is:

$\forall\lambda\in\mathbb{Q},\quad f(D_{\lambda})=f(D)\cap C_{\lambda}.$

###### Proof.

This follows at once from Theorem 3.9 and Lemma 3.7 that any semistable
isocrystal is isoclinic. Hence, the corollary follows from [Andr, Thm 1.5.9].
∎

We observe that the slope filtration is not split in general. However it does
when the ground field $F$ is perfect:

######  Theorem 3.12.

If $F$ is perfect, the slope filtration of $\underline{D}$ splits, i.e.
$\underline{D}$ decomposes along a direct sum

$\underline{D}\cong\bigoplus_{\lambda\in\mathbb{Q}}\operatorname{Gr}_{\lambda}\underline{D}.$

######  Remark 3.13.

The proof is similar to the argument given for Theorem 3.9: the corresponding
result for $\delta=1$ is proven in [Har2, Prop 1.5.10] and the general
$\delta$-case is easily deduced from [Mor2, Prop 4.2.2].

######  Remark 3.14.

The above theorem is the Dieudonné-Manin decomposition for isocrystals. When
$F$ is algebraically closed, given $\lambda\in\mathbb{Q}$ there exists a
unique (up to isomorphisms) simple and pure isocrystal
$\underline{S}_{\lambda}$ of slope $\lambda$ (see [Mor2, Prop 4.3.4]). Any
pure isocrystal of slope $\lambda$ decomposes as a direct sum of
$\underline{S}_{\lambda}$ (see [Mor2, Prop 4.3.7]) and together with Theorem
3.12 yields the Dieudonné-Manin classification (see [Lau]). It does not hold
for any $F$, even separably closed, as noticed by Mornev in [Mor2, Rmk 4.3.5]
(see also Example 3.26).

#### Isocrystals attached to $A$-motives

We now explain how to attach isocrystals to $A$-motives over fields. This
construction (Definition 3.16) is required next in the definition of weights
of $A$-motives (Definition 3.20).

Let $R$ be a Noetherian $\mathbb{F}$-algebra and let $\kappa:A\to R$ be an
$\mathbb{F}$-algebra morphism. We choose the rings $\pazocal{A}(R)$ and
$\pazocal{B}(R)$ of the previous paragraph in the following way. Given a
closed point $x$ on $C$, we let $O_{x}\subset K$ be the associated discrete
valuation ring of maximal ideal $\mathfrak{m}_{x}$. We denote
$\pazocal{O}_{x}$ the completion of $O_{x}$ and $K_{x}$ the completion of $K$.
We let $\mathbb{F}_{x}$ denote the residue field of $x$ (of finite dimension
over $\mathbb{F}$, its dimension being the degree of $x$). We let
$\pazocal{A}_{x}(R)$ and $\pazocal{B}_{x}(R)$ be the completions of
$\pazocal{O}_{x}\otimes R$ and $K_{x}\otimes R$ for the
$\mathfrak{m}_{x}$-adic topology.

Recall that $\mathfrak{j}_{\kappa}$ is the ideal of $A\otimes R$ generated by
$\\{a\otimes 1-1\otimes\kappa(a)~{}|~{}a\in A\\}$.

######  Lemma 3.15.

We have
$\mathfrak{j}_{\kappa}\pazocal{B}_{\infty}(R)=\pazocal{B}_{\infty}(R)$. For
$x$ a closed point of $C$ distinct from $\infty$ such that
$\kappa(\mathfrak{m}_{x})R=R$, we have
$\mathfrak{j}_{\kappa}\pazocal{A}_{x}(R)=\pazocal{A}_{x}(R)$.

###### Proof.

We prove the first assertion. Let $a$ be a non constant element of $A$ so that
$a^{-1}\in\mathfrak{m}_{\infty}$. Then $a\otimes
1-1\otimes\kappa(a)\in\mathfrak{j}$ is invertible with $-\sum_{n\geq
0}{a^{-(n+1)}\otimes\kappa(a)^{n}}$ as inverse, where the infinite sum
converges in $\pazocal{A}_{\infty}(R)\subset\pazocal{B}_{\infty}(R)$.

To prove the second assertion, let $\ell\in\mathfrak{m}_{x}$ be such that
$\kappa(\ell)$ is invertible in $R$. Then $\ell\otimes
1-1\otimes\kappa(\ell)\in\mathfrak{j}$ is invertible with $-\sum_{n\geq
0}{\ell^{n}\otimes\kappa(\ell)^{-(n+1)}}$ as inverse, where the infinite sum
converges in $\pazocal{A}_{x}(R)$. ∎

In order to use the results of the previous paragraph, we now assume that
$R=F$ is a field and that $x$ is a closed point of $C$ distinct from
$\ker\kappa$. Thanks to Lemma 3.15, for an $A$-motive $(M,\tau_{M})$,
$\tau_{M}\otimes_{A\otimes F}\mathbf{1}$ defines an isomorphism
$\tau^{*}(M\otimes_{A\otimes F}\pazocal{B}_{x}(F))\to M\otimes_{A\otimes
F}\pazocal{B}_{x}(F)$. This motivates the following construction:

######  Definition 3.16.

Let $\underline{M}=(M,\tau_{M})$ be an $A$-motive over $F$ and let $x$ be a
closed point of $C$. We let $\pazocal{I}_{x}(M)$ be the
$\pazocal{B}_{x}(F)$-module $M\otimes_{A\otimes R}\pazocal{B}_{x}(F)$. Let
$\pazocal{I}_{x}(\underline{M})$ be the pair
$(\pazocal{I}_{x}(M),\tau_{M}\otimes\mathbf{1})$.

The above definition is motivated by the following:

######  Proposition 3.17.

Let $\underline{M}=(M,\tau_{M})$ be a nonzero $A$-motive over $F$.

  1. $(i)$

$\pazocal{I}_{x}(\underline{M})$ is an isocrystal over $F$.

  2. $(ii)$

If $x\neq\infty$, $\pazocal{I}_{x}(\underline{M})$ is pure of slope $0$.

###### Proof.

Because $M$ is locally free of constant rank and $\pazocal{B}_{x}(F)$ is a
finite product of fields, $\pazocal{I}_{x}(M)$ is a free
$\pazocal{B}_{x}(F)$-module. Thus, point $(i)$ follows from Lemma 3.15. To
prove $(ii)$, it suffices to note that $L=M\otimes_{A\otimes
F}\pazocal{A}_{x}(F)$ is an $\pazocal{A}_{x}(F)$-lattice in
$M\otimes_{A\otimes F}\pazocal{B}_{x}(F)$ such that
$\langle\tau_{M}L\rangle=L$. ∎

######  Proposition 3.18.

The assignment
$\pazocal{I}_{x}:\underline{M}\mapsto\pazocal{I}_{x}(\underline{M})$ defines
an exact functor from the category of $A$-motives to the category of
isocrystals at $x$ over $F$.

###### Proof.

It suffices to prove that $\pazocal{B}_{x}(F)$ is flat over $A\otimes F$.
First note that $\pazocal{O}_{x}\otimes F$ is Noetherian, so that its
completion $\pazocal{A}_{x}(F)$ for the $\mathfrak{m}_{x}$-adic topology is
flat over it (this is [Bou, AC.III.§4,Thm 3(iii)]). Tensoring by $K_{x}$ over
$\pazocal{O}_{x}$, we obtain that $\pazocal{B}_{x}(F)$ is flat over
$K_{x}\otimes F$. Yet the latter is flat over $A\otimes F$, which concludes. ∎

The following lemma will be important in the next subsection:

######  Lemma 3.19.

Let $\underline{M}$ be an $A$-motive over $F$ and let $\underline{P}$ be a
sub-$A$-motive of $\underline{M}$. Then
$\pazocal{I}_{x}(\underline{P})=\pazocal{I}_{x}(\underline{P}^{\text{sat}})$.
If $\underline{Q}$ is a sub-$A$-motive of $\underline{M}$ such that
$\pazocal{I}_{x}(\underline{P})=\pazocal{I}_{x}(\underline{Q})$ inside
$\pazocal{I}_{x}(\underline{M})$, then
$\underline{P}^{\text{sat}}=\underline{Q}^{\text{sat}}$.

###### Proof.

The inclusion $\underline{P}\subset\underline{P}^{\text{sat}}$ is an isogeny
and therefore its cokernel is $A$-torsion (see [Har3, Thm. 5.12]).
Consequently,
$\pazocal{I}_{x}(\underline{P})=\pazocal{I}_{x}(\underline{P}^{\text{sat}})$.
We prove the second part. The $A\otimes F$-modules $P$, $P^{\text{sat}}$, $Q$,
$Q^{\text{sat}}$ and $(P\cap Q)^{\text{sat}}=P^{\text{sat}}\cap
Q^{\text{sat}}$ are locally-free of the same rank, as they become equal once
$\pazocal{I}_{x}$ is applied. Note that $(P\cap
Q)^{\text{sat}}=P^{\text{sat}}\cap Q^{\text{sat}}$ is again endowed with an
$A$-motive structure and hence so is the quotient $A\otimes F$-module
$P^{\text{sat}}/(P\cap Q)^{\text{sat}}$. The underlying $A\otimes F$-module is
locally-free and has rank $0$; hence it is zero. The inclusion
$P^{\text{sat}}\cap Q^{\text{sat}}\to P^{\text{sat}}$ is therefore an
isomorphism which yields $P^{\text{sat}}\subset Q^{\text{sat}}$. We conclude
by exchanging the roles of $P$ and $Q$ in the above argument to obtain the
converse inclusion. ∎

#### Weight filtration and mixedness

As before, let $F$ be a field containing $\mathbb{F}$, let $\kappa:A\to F$ be
an $\mathbb{F}$-algebra morphism and let $\underline{M}$ be an $A$-motive over
$F$. Let $\underline{D}:=\pazocal{I}_{\infty}(\underline{M})$ be the
isocrystal at the place $\infty$ attached to $\underline{M}$. By Theorem 3.9,
$\underline{D}$ carries a unique slope filtration:

$0=\underline{D}_{0}\subsetneq\underline{D}_{1}\subsetneq\cdots\subsetneq\underline{D}_{s}=\underline{D}$
(3.3)

with ascending slopes $\lambda_{1}<\cdots<\lambda_{s}$, where
$\lambda_{i}:=\lambda(\underline{D}_{i}/\underline{D}_{i-1})$.

######  Definition 3.20.

We call the set $w(\underline{M}):=\\{-\lambda_{i}~{}|~{}1\leq i\leq s\\}$ the
set of weights of $\underline{M}$.

The next definition dates back to the seminal paper of Anderson [And, 1.9].

######  Definition 3.21.

We call $\underline{M}$ pure of weight $\mu$ if $\underline{D}$ is pure of
slope $-\mu$. Equivalently, $\underline{M}$ is pure of weight $\mu$ if
$w(\underline{M})=\\{\mu\\}$.

The sign convention - weights opposed to slopes - is made to fit with the
number field picture:

######  Example 3.22.

Using notations of Example 2.4, the Carlitz twist $\underline{A}(1)$ over $F$
is pure of weight $-1$ and, more generally, $\underline{A}(n)$ is pure of
weight $-n$. It is analogous to the number field case, where the motive
$\mathbb{Z}(n)$ is pure of weight $-2n$ (the factor $2$, reflecting the degree
$[\mathbb{C}:\mathbb{R}]$, could be removed by renormalizing the weight
filtration using half-integral numbers).

In analogy with number fields, we define mixedness for $A$-motives as follows:

######  Definition 3.23.

We call $\underline{M}$ mixed if there exist rational numbers
$\mu_{1}<...<\mu_{s}$ and an increasing finite filtration by saturated
sub-$A$-motives of $\underline{M}$:

$0=W_{\mu_{0}}\underline{M}\subsetneq
W_{\mu_{1}}\underline{M}\subsetneq\cdots\subsetneq
W_{\mu_{s}}\underline{M}=\underline{M},$ (3.4)

for which the successive quotients
$W_{\mu_{i}}\underline{M}/W_{\mu_{i-1}}\underline{M}$ are pure of weight
$\mu_{i}$.

Before pursuing on the properties of mixedness, let us explain why being mixed
is a very restrictive condition over unperfect base fields. Suppose that
$\underline{M}$ is a mixed $A$-motive and consider a filtration as in (3.4).
The functor $\pazocal{I}_{\infty}$ is exact (Proposition 3.18), and applying
it to (3.4) yields a finite filtration of
$\underline{D}=\pazocal{I}_{\infty}(\underline{M})$ by subisocrystals:

$0=\underline{D}^{0}\subsetneq\underline{D}^{1}\subsetneq\cdots\subsetneq\underline{D}^{s}=\underline{D}$
(3.5)

whose successive quotients $\underline{D}^{i}/\underline{D}^{i-1}$ are pure
isocrystals of slope $-\mu_{i}$. Note that the slopes of this filtration are
decreasing, hence (3.5) is not the slope filtration of $\underline{D}$.

######  Proposition 3.24.

If $\underline{M}$ is mixed, the slope filtration of
$\pazocal{I}_{\infty}(\underline{M})$ splits.

We begin with a lemma on isocrystals:

######  Lemma 3.25.

Let
$S:0\to\underline{D}^{\prime}\to\underline{D}\to\underline{D}^{\prime\prime}\to
0$ be an exact sequence of isocrystals with
$\mu(\underline{D}^{\prime})>\mu(\underline{D}^{\prime\prime})$. Then $S$
splits.

###### Proof.

Thanks to the formula
$\operatorname{Ext}^{1}_{\pazocal{I}_{F}}(\underline{D}^{\prime\prime},\underline{D}^{\prime})\cong\operatorname{Ext}^{1}_{\pazocal{I}_{F}}(\mathbbm{1},\underline{D}^{\prime}\otimes(\underline{D}^{\prime\prime})^{\vee})$
(where the tensor product and duals are defined in the usual way), we may
assume $\underline{D}^{\prime\prime}=\mathbbm{1}$ and
$\mu(\underline{D}^{\prime})>0$.

We claim that $D^{\prime}$ contains an $\pazocal{A}(F)$-lattice $L$ satisfying
$\langle\varphi^{h}L\rangle\subset\mathfrak{m}L$ for some $h>0$ large enough.
Indeed, this is clearly true if $\underline{D}^{\prime}$ is pure. If $F$ is
perfect, then $\underline{D}^{\prime}$ decomposes as a direct sum of pure
subisocrystals of positive slopes, and the claim follows from the pure-case.
In general, let $F^{\prime}$ denote the perfection of $F$, and let
$L^{\prime}$ be an $\pazocal{A}(F^{\prime})$-lattice in
$D\otimes_{\pazocal{A}(F)}\pazocal{A}(F^{\prime})$ as in the claim. Then, one
checks that $L:=L^{\prime}\cap D$ is again an $\pazocal{A}(F)$-lattice in $D$
which is as desired.

Observe that the extensions of $\mathbbm{1}$ by
$\underline{D}^{\prime}=(D^{\prime},\varphi^{\prime})$ are parametrized up to
equivalence by the $\pazocal{O}$-module
$D^{\prime}/\operatorname{im}(\operatorname{id}-\varphi^{\prime})$. Now, it
follows from the previous claim that $\operatorname{id}-\varphi^{\prime}$ is
surjective on $D^{\prime}$. This concludes. ∎

###### Proof of Proposition 3.24.

Using induction on Lemma 3.25, we obtain that the filtration (3.5) splits. By
reordering the pure parts, we build the filtration

$0\subsetneq\underline{D}^{s}/\underline{D}^{s-1}\subsetneq(\underline{D}^{s}/\underline{D}^{s-1})\oplus(\underline{D}^{s-1}/\underline{D}^{s-2})\subsetneq\cdots\subsetneq\bigoplus_{i=1}^{s}\underline{D}^{i}/\underline{D}^{i-1}$

which is a split finite filtration of $\underline{D}$ having increasing
slopes. By uniqueness, it coincides with the slope filtration of
$\underline{D}$. ∎

Yet, over unperfect fields, the slope filtration might not be split, even for
isocrystals coming from $A$-motives:

######  Example 3.26.

We suppose that $A=\mathbb{F}[t]$ and denote by $\pi$ the uniformizer $t^{-1}$
in $\pazocal{B}_{\infty}(F)$, then identified with $F(\\!(\pi)\\!)$. Assume
that $F$ is not perfect, and let $\alpha\in F$ be such that $-\alpha$ does not
have any $q$th root in $F$. Consider the $t$-motive $\underline{M}$ whose
underlying module is $F[t]e_{0}\oplus F[t]e_{1}$, and where $\tau_{M}$ acts by
$\tau_{M}(\tau^{*}e_{0})=e_{0}$ and $\tau_{M}(\tau^{*}e_{1})=\alpha
e_{0}+(t-\theta)^{-1}e_{1}$. $\underline{M}$ inserts in a short exact sequence
of $t$-motives:

$0\to\mathbbm{1}\cdot e_{0}\to\underline{M}\to\underline{A}(1)\cdot e_{1}\to
0.$ (3.6)

We claim that $\underline{M}$ does not possess a weight filtration as in
(3.4).

By the above discussion it is enough to prove that the slope filtration of
$\pazocal{I}_{\infty}(\underline{M})$ does not split. More precisely, let us
show that $\pazocal{I}_{\infty}(\mathbbm{1})\cdot e_{0}$ is the only non-zero
strict subisocrystal of $\pazocal{I}_{\infty}(\underline{M})$.

Let $\underline{L}\neq 0$ be a strict subisocrystal of
$\pazocal{I}_{\infty}(\underline{M})$. $\underline{L}$ must have rank $1$ over
$F(\\!(\pi)\\!)$, and we let $\ell_{0}e_{0}+\ell_{1}e_{1}$ be a generator of
its underlying module. Let $f\in F(\\!(\pi)\\!)$ be a non zero element such
that
$\tau_{M}(\tau^{*}(\ell_{0}e_{0}+\ell_{1}e_{1}))=f(\ell_{0}e_{0}+\ell_{1}e_{1})$.
This yields the equation:

$\begin{pmatrix}1&\alpha\\\
0&\pi(1-\theta\pi)^{-1}\end{pmatrix}\begin{pmatrix}\tau(\ell_{0})\\\
\tau(\ell_{1})\end{pmatrix}=f\begin{pmatrix}\ell_{0}\\\
\ell_{1}\end{pmatrix}.$

Because $\underline{L}\neq 0$, the first row implies $\ell_{0}\neq 0$. If
$\ell_{1}\neq 0$, as $(1-\theta\pi)$ is a unit in $F[\\![\pi]\\!]$ the bottom
row imposes $v_{\pi}(f)=1$, whereas the first row reads

$\tau(\ell_{0})+\alpha\tau(\ell_{1})=f\ell_{0}.$ (3.7)

Because $v_{\pi}(f)=1$, (3.7) also implies that
$v_{\pi}(\ell_{0})=v_{\pi}(\ell_{1})$. In particular, if $l_{0}$ and $l_{1}$
are the first nonzero coefficients in $F$ of $\ell_{0}$ and $\ell_{1}$ in
$F(\\!(\pi)\\!)$ respectively, (3.7) gives

$\alpha=-(l_{0}/l_{1})^{q}.$

This is in contradiction with our assumption. Hence $\ell_{1}=0$ and
$f=\tau(\ell_{0})/\ell_{0}$. Therefore, $\underline{L}$ coincides with
$\pazocal{I}_{\infty}(\mathbbm{1})\cdot e_{0}$.

In this example, $\underline{M}$ is an extension of $\underline{A}(1)$ by
$\mathbbm{1}$, two $t$-motives of respective weights $-1<0$. When the weights
goes in ascending orders, such examples do not appear anymore (see Proposition
3.38).

######  Remark 3.27.

Observe that the converse of Proposition 3.24 does not hold: there are non-
mixed $A$-motives over perfect base fields (e.g. [HarJu, Ex. 2.3.13]). For a
partial converse, we refer to Proposition 3.30 below.

######  Proposition-Definition 3.28.

If $\underline{M}$ is mixed, a filtration $W=(W_{\mu_{i}}\underline{M})_{1\leq
i\leq s}$ as in (3.4) is unique. Further, the set $\\{\mu_{1},...,\mu_{s}\\}$
equals $w(\underline{M})$.

  1. $(i)$

For all $i\in\\{1,...,s\\}$, we let $W_{\mu_{i}}M$ be the underlying module of
$W_{\mu_{i}}\underline{M}$.

  2. $(ii)$

For all $\mu\in\mathbb{Q}$, we set

$\displaystyle W_{\mu}M$
$\displaystyle:=\bigcup_{\mu_{i}\leq\mu}{W_{\mu_{i}}M},\quad
W_{\mu}\underline{M}:=(W_{\mu}M,\tau_{M}),$ $\displaystyle W_{<\mu}M$
$\displaystyle:=\bigcup_{\mu_{i}<\mu}{W_{\mu_{i}}M},\quad
W_{<\mu}\underline{M}:=(W_{<\mu}M,\tau_{M}),$

and
$\operatorname{Gr}_{\mu}\underline{M}:=W_{\mu}\underline{M}/W_{<\mu}\underline{M}$.
Both $\operatorname{Gr}_{\mu}\underline{M}$ and $W_{\mu}\underline{M}$, as
well as $W_{<\mu}\underline{M}$, define mixed $A$-motives over $F$ for all
$\mu\in\mathbb{Q}$.

  3. $(iii)$

We call $(W_{\mu}\underline{M})_{\mu\in\mathbb{Q}}$ the weight filtration of
$\underline{M}$.

  4. $(iv)$

We let $\pazocal{M}\pazocal{M}_{F}$ (resp.
$\pazocal{M}\pazocal{M}^{\text{iso}}_{F}$) be the subcategory of
$\pazocal{M}_{F}$ (resp. $\pazocal{M}^{\text{iso}}_{F}$) whose objects are
mixed, and whose morphisms $f:\underline{M}\to\underline{N}$ preserves the
weight filtration:

$\forall\mu\in\mathbb{Q}:\quad f(W_{\mu}M)\subset W_{\mu}N.$

######  Remark 3.29.

We shall prove below (Corollary 3.31) that every morphism of $A$-motives
$f:\underline{M}\to\underline{N}$, $\underline{M}$ and $\underline{N}$ being
mixed, preserves the weight filtation. In other words, that
$\pazocal{M}\pazocal{M}_{F}$ is a full subcategory of $\pazocal{M}_{F}$.

###### Proof of Proposition 3.28.

Let $\underline{D}=\pazocal{I}_{\infty}(\underline{M})$. By Proposition 3.24,
we have a canonical decomposition:

$\underline{D}=\bigoplus_{i=1}^{s}\operatorname{Gr}_{\lambda_{i}}\underline{D},$
(3.8)

$\operatorname{Gr}_{\lambda_{i}}\underline{D}$ being canonically identified
with a pure subisocrystal of $\underline{D}$ of slope $\lambda_{i}$. On the
other-hand, let $W$ be as in (3.4). By uniqueness of the decomposition (3.8),
we have equalities of subisocrystals of $\underline{D}$:

$\forall\mu\in\mathbb{Q}:\quad\pazocal{I}_{\infty}(W_{\mu}\underline{M})=\bigoplus_{\lambda_{i}\geq-\mu}\operatorname{Gr}_{\lambda_{i}}\underline{D}.$
(3.9)

We conclude by Lemma 3.19 that the above identity determines $W$ uniquely. The
fact that $w(\underline{M})=\\{\mu_{1},...,\mu_{s}\\}$ also follows. ∎

Next, we suggest a criterion for mixedness:

######  Proposition 3.30.

Suppose that the slope filtration of $\underline{D}$ splits. For all
$\mu\in\mathbb{Q}$, let $\pazocal{I}_{\infty}(\underline{M})^{\mu}$ be the
subisocrystal of $\underline{D}$ corresponding to
$\bigoplus_{\lambda_{i}\geq-\mu}\operatorname{Gr}_{\lambda_{i}}\underline{D}$.
Denote by $\pazocal{I}_{\infty}(M)^{\mu}$ its underlying module. Then, we
have:

$\forall\mu\in\mathbb{Q}:\quad\operatorname{rank}_{A\otimes
F}(\pazocal{I}_{\infty}(M)^{\mu}\cap
M)\leq\operatorname{rank}\pazocal{I}_{\infty}(\underline{M})^{\mu}$

with equality if and only if $\underline{M}$ is mixed. In the latter case,
$W_{\mu}M=\pazocal{I}_{\infty}(M)^{\mu}\cap M$.

###### Proof.

First note that, for all $\mu$, the couple
$\underline{M}_{\mu}:=(\pazocal{I}_{\infty}(M)^{\mu}\cap M,\tau_{M})$ defines
a saturated sub-$A$-motive of $\underline{M}$. Furthermore, since the
underlying module of $\pazocal{I}_{\infty}(\underline{M}_{\mu})$ corresponds
to the completion of $\pazocal{I}_{\infty}(M)^{\mu}\cap(M\otimes_{A}K)$ for
the $\infty$-adic topology, $\pazocal{I}_{\infty}(\underline{M}_{\mu})$
defines a subisocrystal of $\pazocal{I}_{\infty}(\underline{M})^{\mu}$. The
rank being additive in short exact sequences in the category of isocrystals,
we get the desired inequality:

$\operatorname{rank}_{A\otimes F}(\pazocal{I}_{\infty}(M)^{\mu}\cap
M)=\operatorname{rank}\underline{M}_{\mu}=\operatorname{rank}\pazocal{I}_{\infty}(\underline{M}_{\mu})\leq\operatorname{rank}\pazocal{I}_{\infty}(\underline{M})^{\mu}.$

If this is an equality for all $\mu$, then we obtain
$\pazocal{I}_{\infty}(\underline{M}_{\mu})=\pazocal{I}_{\infty}(\underline{M})^{\mu}$
and deduce that the family $(\underline{M}_{\mu})_{\mu}$ satisfies the
requested property of Definition 3.23.

Conversely, if $\underline{M}$ is mixed, the following sequence of inclusions
holds:

$(W_{\mu}M)\otimes_{A}K\subset\pazocal{I}_{\infty}(M)^{\mu}\cap(M\otimes_{A}K)\subset\pazocal{I}_{\infty}(M)^{\mu}.$

Note that the left-hand side forms a dense subset of the right-hand side for
the $\infty$-adic topology. Hence, the first inclusion is an inclusion of a
dense subset. Taking the completion, we obtain
$\pazocal{I}_{\infty}(W_{\mu}M)=\pazocal{I}_{\infty}(\pazocal{I}_{\infty}(M)^{\mu}\cap
M)=\pazocal{I}_{\infty}(\underline{M})^{\mu}$. The corresponding ranks are
thus equal.

To conclude, as both $W_{\mu}M$ and $\pazocal{I}_{\infty}(M)_{\mu}\cap M$ are
saturated submodules of $M$, we deduce the equality
$W_{\mu}M=\pazocal{I}_{\infty}(M)^{\mu}\cap M$ from Lemma 3.19. ∎

As an important corollary of the above formula for $W$, we obtain:

######  Corollary 3.31.

Let $f:\underline{M}\to\underline{N}$ be a morphism of $A$-motives,
$\underline{M}$ and $\underline{N}$ being mixed. Then, $f$ preserves the
weight filtration:

$\forall\mu\in\mathbb{Q}:\quad f(W_{\mu}M)\subset W_{\mu}N.$

In particular, for all $\mu\in\mathbb{Q}$, the assignment
$\underline{M}\mapsto W_{\mu}\underline{M}$ defines a functor from the
category $\pazocal{M}\pazocal{M}_{F}$ to itself.

###### Proof.

Let $\mu\in\mathbb{Q}$. The claim follows from the following two easy
observations:

  1. 1.

If $\underline{M}$ is mixed, then so is $W_{\mu}\underline{M}$,

  2. 2.

If $f:\underline{M}\to\underline{N}$ is a morphism of mixed $A$-motives, then
the associated morphism of isocrystals maps $\pazocal{I}_{\infty}(M)^{\mu}$ to
$\pazocal{I}_{\infty}(N)^{\mu}$, and by Proposition 3.30 we have
$f(W_{\mu}M)\subset W_{\mu}N$.

∎

######  Remark 3.32.

We end this subsection by describing how weights behave under linear algebra
type operations. Proofs are presented in [HarJu, Prop. 2.3.11] and extend
without change to our larger setting. First note that $\mathbbm{1}$ is a pure
$A$-motive over $F$ of weight $0$. Given two mixed $A$-motives $\underline{M}$
and $\underline{N}$, their biproduct $\underline{M}\oplus\underline{N}$ is
again mixed with weight filtration
$W_{\mu}(\underline{M}\oplus\underline{N})=W_{\mu}\underline{M}\oplus
W_{\mu}\underline{N}$ ($\mu\in\mathbb{Q}$). Their tensor product
$\underline{M}\otimes\underline{N}$ is also mixed, with $\lambda$-part of its
weight filtration being:

$W_{\lambda}(\underline{M}\otimes\underline{N})=\left(\sum_{\mu+\nu=\lambda}{W_{\mu}\underline{M}\otimes
W_{\nu}\underline{N}}\right)^{\text{sat}}.$

We took the saturation $A$-motive to ensure that the above is a saturated
sub-$A$-motive of $\underline{M}\otimes\underline{N}$. The dual
$\underline{M}^{\vee}$ is mixed, and the $\mu$-part of its weight filtration
$W_{\mu}\underline{M}$ has for underlying module $W_{\mu}M^{\vee}=\\{m\in
M^{\vee}|\forall\lambda<-\mu:~{}m(W_{\lambda}M)=0\\}^{\text{sat}}$. In
general, given $\underline{M}$ and $\underline{N}$ two $A$-motives over $F$
(without regarding whether $\underline{M}$ or $\underline{N}$ are mixed) and
an exact sequence
$0\to\underline{M}^{\prime}\to\underline{M}\to\underline{M}^{\prime\prime}\to
0$ in $\pazocal{M}_{F}$, we have

$w(\underline{0})$ | $=$ | $\emptyset$
---|---|---
$w(\underline{M}^{\vee})$ | $=$ | $-w(\underline{M})$
$w(\underline{M}\oplus\underline{N})$ | $=$ | $w(\underline{M})\cup w(\underline{N})$
$w(\underline{M})$ | $=$ | $w(\underline{M}^{\prime})\cup w(\underline{M}^{\prime\prime})$
$w(\underline{M}\otimes\underline{N})$ | $=$ | $\\{w+v~{}|~{}w\in w(\underline{M}),~{}v\in w(\underline{N})\\}$

###  3.2 Extension modules of mixed $A$-motives

As before, let $F$ be a field containing $\mathbb{F}$ and consider an
$\mathbb{F}$-algebra morphism $\kappa:A\to F$. In this subsection, we are
concerned with extension modules in the category $\pazocal{M}\pazocal{M}_{F}$.
The next proposition shows that they are well-defined.

######  Proposition 3.33.

The category $\pazocal{M}\pazocal{M}_{F}$ together with the notion of exact
sequences of Definition 2.5 is an exact subcategory of $\pazocal{M}_{F}$.

To prove Proposition 3.33, one has to check all Quillen’s axioms (e.g. [Qui1,
§2]). The only non-straightforward one is that the class of exact sequences of
mixed $A$-motives is closed under pushouts and pullbacks. This follows from
the next result:

######  Proposition 3.34.

Let
$0\to\underline{M}^{\prime}\to\underline{M}\to\underline{M}^{\prime\prime}\to
0$ be an exact sequence of $A$-motives over $F$. If $\underline{M}$ is mixed
and one of $\underline{M}^{\prime}$ or $\underline{M}^{\prime\prime}$ is
mixed, then all three are mixed.

We begin by an intermediate result on isocrystals:

######  Lemma 3.35.

Let
$0\to\underline{D}^{\prime}\to\underline{D}\to\underline{D}^{\prime\prime}\to
0$ be an exact sequence of isocrystals over $F$. If the slope filtration of
$\underline{D}$ and one of $\underline{D}^{\prime}$ or
$\underline{D}^{\prime\prime}$ split, then all three are split.

###### Proof.

Assume that the slope filtration of $\underline{D}$ and
$\underline{D}^{\prime}$ are split. Then $\underline{D}$ decomposes as a
finite direct sum of pure subisocrystals
$\operatorname{Gr}_{\lambda}\underline{D}$. We have a commutative diagram of
subisocrystals of $\underline{D}$:

${\underline{D}}$${\displaystyle\bigoplus_{\lambda\in\mathbb{Q}}\operatorname{Gr}_{\lambda}\underline{D}}$${\underline{D}^{\prime}}$${\displaystyle\bigoplus_{\lambda\in\mathbb{Q}}(\underline{D}^{\prime}\cap\operatorname{Gr}_{\lambda}\underline{D})}$$\scriptstyle{\sim}$$\scriptstyle{\iota}$
(3.10)

where the symbol $\cap$ is understood to denote the subisocrystal whose
underlying module is given by the intersection. We claim that $\iota$ is an
isomorphism. Firstly, it is injective as one checks by a diagram chase. By
assumption, $\underline{D}^{\prime}$ also decomposes as a direct sum of
subisocrystals $\operatorname{Gr}_{\lambda}\underline{D}^{\prime}$ which, when
non zero, are pure of slope $\lambda$. By functoriality of the slope
filtration, the inclusion
$\operatorname{Gr}_{\lambda}\underline{D}^{\prime}\hookrightarrow\underline{D}^{\prime}\hookrightarrow\underline{D}$
factors through $\operatorname{Gr}_{\lambda}\underline{D}$. Hence,

$\operatorname{rk}\underline{D}^{\prime}=\sum_{\lambda\in\mathbb{Q}}{\operatorname{rk}\operatorname{Gr}_{\lambda}\underline{D}^{\prime}}\leq\sum_{\lambda\in\mathbb{Q}}{\operatorname{rk}(\underline{D}^{\prime}\cap\operatorname{Gr}_{\lambda}\underline{D})}\leq\operatorname{rk}\underline{D}^{\prime}$

where the last inequality follows from the injectivity of $\iota$. All the
above inequalities are thus equalities, from which we deduce that $\iota$ is
an isomorphism.

Therefore, we can pursue the square (3.10) into exact sequences:

${0}$${\displaystyle\bigoplus_{\lambda\in\mathbb{Q}}(\underline{D}^{\prime}\cap\operatorname{Gr}_{\lambda}\underline{D})}$${\displaystyle\bigoplus_{\lambda\in\mathbb{Q}}\operatorname{Gr}_{\lambda}\underline{D}}$${\displaystyle\bigoplus_{\lambda\in\mathbb{Q}}\operatorname{Gr}_{\lambda}\underline{D}/(\underline{D}^{\prime}\cap\operatorname{Gr}_{\lambda}\underline{D})}$${0}$${0}$${\underline{D}^{\prime}}$${\underline{D}}$${\underline{D}^{\prime\prime}}$${0}$$\scriptstyle{\iota}$$\scriptstyle{\wr}$

where the dashed arrow is the only one making the right-hand square commute.
By the snake Lemma, it is an isomorphism. In particular,
$\underline{D}^{\prime\prime}$ decomposes as a direct sum of pure
subisocrystals, and by uniqueness, we deduce that its slope filtration splits.

The argument for the second part of the statement is similar enough to the
first one to be skipped. ∎

###### Proof of Proposition 3.34.

Suppose $\underline{M}$ and $\underline{M}^{\prime}$ (resp.
$\underline{M}^{\prime\prime}$) are mixed. For $\mu\in\mathbb{Q}$, let
$W_{\mu}M^{\prime}:=W_{\mu}M\cap M^{\prime}$ and
$W_{\mu}M^{\prime\prime}:=f(W_{\mu}M)^{\text{sat}}$, where $f$ is the
epimorphism $\underline{M}\twoheadrightarrow\underline{M}^{\prime\prime}$.
They are canonically endowed with $A$-motive structures denoted by
$W_{\mu}\underline{M}^{\prime}$ and $W_{\mu}\underline{M}^{\prime\prime}$
respectively, and our task is to prove that
$(W_{\mu}\underline{M}^{\prime})_{\mu}$ and
$(W_{\mu}\underline{M}^{\prime\prime})_{\mu}$ satisfy the property of
Definition 3.4.

By Lemma 3.35, the slope filtrations of $\pazocal{I}_{\infty}(\underline{N})$
for
$\underline{N}\in\\{\underline{M},\underline{M}^{\prime},\underline{M}^{\prime\prime}\\}$
are split, and, for $\mu\in\mathbb{Q}$, we denote by
$\pazocal{I}_{\infty}(\underline{N})^{\mu}$ the direct sum of the
subisocrystals of $\pazocal{I}_{\infty}(\underline{N})$ which are pure of
slope $\geq-\mu$.

Since $\pazocal{B}_{\infty}(F)$ is flat over $A\otimes F$, the functor
$-\otimes_{A\otimes F}\pazocal{B}_{\infty}(F)$ commutes with finite
intersections (this is [Bou, §I.2, Prop. 6]):

$\pazocal{I}_{\infty}(W_{\mu}M^{\prime})=(W_{\mu}M\cap
M^{\prime})\otimes_{A\otimes
F}\pazocal{B}_{\infty}(F)=\pazocal{I}_{\infty}(M)^{\mu}\cap\pazocal{I}_{\infty}(M^{\prime})=\pazocal{I}_{\infty}(M^{\prime})^{\mu}$

where the last equality follows from the fact that the category of isocrystals
has strict morphisms for the slope filtration (Corollary 3.11). Similarly,

$\displaystyle\pazocal{I}_{\infty}(W_{\mu}M^{\prime\prime})$
$\displaystyle=f(W_{\mu}M)^{\text{sat}}\otimes_{A\otimes
F}\pazocal{B}_{\infty}(F)=f(\pazocal{I}_{\infty}(M)^{\mu})=\pazocal{I}_{\infty}(M^{\prime\prime})^{\mu}.$

This shows that $\underline{M}^{\prime}$ and $\underline{M}^{\prime\prime}$
are both mixed with respective weight filtrations
$(W_{\mu}\underline{M}^{\prime})_{\mu\in\mathbb{Q}}$ and
$(W_{\mu}\underline{M}^{\prime\prime})_{\mu\in\mathbb{Q}}$. ∎

######  Remark 3.36.

If the field $F$ is perfect, it suffices for $\underline{M}$ is mixed to
obtain the mixedness of $\underline{M}^{\prime}$ and
$\underline{M}^{\prime\prime}$ (e.g. [HarJu, Prop 2.3.11(c)]). This is because
the slope filtration always split and Lemma 3.35 is not needed anymore.

######  Corollary 3.37.

For all $\mu\in\mathbb{Q}$, the functor $\underline{M}\mapsto
W_{\mu}\underline{M}$ is exact on $\pazocal{M}\pazocal{M}_{F}^{\text{iso}}$.

###### Proof.

Let $0\to\underline{M}^{\prime}\to\underline{M}\stackrel{{\scriptstyle
f}}{{\to}}\underline{M}^{\prime\prime}\to 0$ be an exact sequence of mixed
$A$-motives. By the proof of Proposition 3.34, we have
$W_{\mu}M^{\prime}=W_{\mu}M\cap M^{\prime}$. Hence, the sequence $0\to
W_{\mu}M^{\prime}\to W_{\mu}M\to W_{\mu}M^{\prime\prime}\to 0$ is left-exact.
We also have $W_{\mu}M^{\prime\prime}=f(W_{\mu}\underline{M})^{\text{sat}}$,
so that the $A$-motive $\underline{\operatorname{im}}(f|W_{\mu}\underline{M})$
is isogeneous to $W_{\mu}\underline{M}^{\prime\prime}$. In particular, the
sequence $0\to W_{\mu}\underline{M}^{\prime}\to W_{\mu}\underline{M}\to
W_{\mu}\underline{M}^{\prime\prime}\to 0$ is exact in
$\pazocal{M}\pazocal{M}_{F}^{\text{iso}}$. ∎

Thanks to Proposition 3.33, we can consider Yoneda’s extension modules of two
mixed $A$-motives in $\pazocal{M}\pazocal{M}_{F}$. By Corollary 3.31, we have
an equality

$\operatorname{Ext}^{0}_{\pazocal{M}\pazocal{M}_{F}}(\underline{M},\underline{N})=\operatorname{Ext}^{0}_{\pazocal{M}_{F}}(\underline{M},\underline{N}),$

but this is not true for higher extension modules.
$\operatorname{Ext}^{1}_{\pazocal{M}\pazocal{M}_{F}}(\underline{M},\underline{N})$
can be interpreted as a submodule of
$\operatorname{Ext}^{i}_{\pazocal{M}_{F}}(\underline{M},\underline{N})$, but
in general $\operatorname{Ext}^{i}_{\pazocal{M}\pazocal{M}_{F}}$ is not even a
submodule of $\operatorname{Ext}^{i}_{\pazocal{M}_{F}}$ ($i>1$). To that
regard, we shall show that
$\operatorname{Ext}^{2}_{\pazocal{M}\pazocal{M}_{F}}(\mathbbm{1},\underline{M})$
can be non zero (see Remark 3.43).

######  Proposition 3.38.

Let $0\to\underline{M}^{\prime}\to\underline{M}\stackrel{{\scriptstyle
p}}{{\to}}\underline{M}^{\prime\prime}\to 0$ be an exact sequence of
$A$-motives in $\pazocal{M}_{F}$. If $\underline{M}^{\prime}$ and
$\underline{M}^{\prime\prime}$ are mixed, and if the smallest weight of
$\underline{M}^{\prime\prime}$ is bigger than the biggest weight of
$\underline{M}^{\prime}$, then $\underline{M}$ is mixed.

###### Proof.

By Proposition 2.10, there exists $u\in\operatorname{Hom}_{A\otimes
F}(\tau^{*}M^{\prime\prime},M^{\prime})[\mathfrak{j}^{-1}]$ and a diagram

${0}$${\underline{M}^{\prime}}$${\underline{M}}$${\underline{M}^{\prime\prime}}$${0}$${0}$${\underline{M}^{\prime}}$${(M^{\prime}\oplus
M^{\prime\prime},\left(\begin{smallmatrix}\tau_{M^{\prime}}&&u\\\
0&&\tau_{M^{\prime\prime}}\end{smallmatrix}\right))}$${\underline{M}^{\prime\prime}}$${0}$$\scriptstyle{\iota}$$\scriptstyle{p}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{\xi}$$\scriptstyle{\operatorname{id}}$

which commutes in $\pazocal{M}_{F}$. From the the weight assumption on
$\underline{M}^{\prime}$ and $\underline{M}^{\prime\prime}$, $u$ automatically
respects the weight filtration: for all $\mu\in\mathbb{Q}$,
$u(\tau^{*}W_{\mu}M^{\prime\prime})\subset W_{\mu}M^{\prime}$. In particular,

$\underline{M}_{\mu}:=\xi\left(W_{\mu}M^{\prime}\oplus
W_{\mu}M^{\prime\prime},\begin{pmatrix}\tau_{M^{\prime}}&u\\\
0&\tau_{M^{\prime\prime}}\end{pmatrix}\right)$

defines a sub-$A$-motive of $\underline{M}$ inserting in a short exact
sequence in $\pazocal{M}_{F}$:

$0\longrightarrow
W_{\mu}\underline{M}^{\prime}\longrightarrow\underline{M}_{\mu}\longrightarrow
W_{\mu}\underline{M}^{\prime\prime}\longrightarrow 0.$ (3.11)

Similarly, $\underline{M}_{<\mu}:=\xi\left(W_{<\mu}M^{\prime}\oplus
W_{<\mu}M^{\prime\prime},\left(\begin{smallmatrix}\tau_{M^{\prime}}&u\\\
0&\tau_{M^{\prime\prime}}\end{smallmatrix}\right)\right)$ defines a
sub-$A$-motive of $\underline{M}_{\mu}$. By (3.11), we obtain an exact
sequence of $A$-motives:

$0\longrightarrow\operatorname{Gr}_{\mu}\underline{M}^{\prime}\longrightarrow\underline{M}_{\mu}/\underline{M}_{<\mu}\longrightarrow\operatorname{Gr}_{\mu}\underline{M}^{\prime\prime}\longrightarrow
0.$ (3.12)

The extremal terms of (3.12) are pure of weight $\mu$ and thus so is the
middle term. We deduce that the increasing sequence
$(\underline{M}_{\mu})_{\mu}$ of sub-$A$-motives of $\underline{M}$ satisfies
the condition of Definition 3.23. Therefore $\underline{M}$ is mixed. ∎

######  Remark 3.39.

Contrary to the number fields situation, the full subcategory of
$\pazocal{M}\pazocal{M}_{F}$ consisting of pure $A$-motives over $F$ is not
semi-simple. This follows easily from the equality
$\operatorname{Ext}^{1}_{\pazocal{M}\pazocal{M}_{F}}(\underline{N},\underline{M})=\operatorname{Ext}^{1}_{\pazocal{M}_{F}}(\underline{N},\underline{M})$
for two pure motives of the same weight.

Proposition 3.38 implies that
$\operatorname{Ext}^{1}_{\pazocal{M}_{F}}(\underline{M}^{\prime\prime},\underline{M}^{\prime})=\operatorname{Ext}^{1}_{\pazocal{M}\pazocal{M}_{F}}(\underline{M}^{\prime\prime},\underline{M}^{\prime})$
when the weights of $\underline{M}^{\prime\prime}$ are bigger than the biggest
weight of $\underline{M}^{\prime}$. In general this is not true. In this
direction we record:

######  Proposition 3.40.

Let
$0\to\underline{M}^{\prime}\to\underline{M}\to\underline{M}^{\prime\prime}\to
0$ be an exact sequence of $A$-motives where $\underline{M}^{\prime}$ and
$\underline{M}^{\prime\prime}$ are mixed. We assume that all the weights of
$\underline{M}^{\prime\prime}$ are strictly smaller than the smallest weight
of $\underline{M}^{\prime}$. Then, the sequence is torsion in
$\operatorname{Ext}^{1}_{\pazocal{M}_{F}}(\underline{M}^{\prime\prime},\underline{M}^{\prime})$
if, and only if, $\underline{M}$ is mixed.

###### Proof of Proposition 3.40.

Taking
$\underline{N}:=\underline{M}^{\prime}\otimes(\underline{M}^{\prime\prime})^{\vee}$,
we can assume that the exact sequence is of the form
$(S):~{}0\to\underline{N}\to\underline{E}\to\mathbbm{1}\to 0$, $\underline{N}$
having positive weights. In view of Theorem 2.13, we may assume that
$\underline{E}=\iota(u)$ for some $u\in M[\mathfrak{j}^{-1}]$. Note that $0$
is a weight of $\underline{E}$, the smallest.

If $\underline{E}$ is mixed, then $\underline{E}$ contains a sub-$A$-motive
$\underline{L}=(L,\tau_{M})$ of weight $0$ which is isomorphic to
$\mathbbm{1}$. Let $(m\oplus a)\in M\oplus(A\otimes F)$ be a generator of $L$
over $A\otimes F$. We have

$\begin{pmatrix}\tau_{M}&u\\\ 0&1\end{pmatrix}\begin{pmatrix}\tau^{*}m\\\
\tau^{*}a\end{pmatrix}=\begin{pmatrix}m\\\ a\end{pmatrix}.$

This amounts to $a\in A$ and
$au\in\operatorname{im}(\operatorname{id}-\tau_{M})$, and then that
$a[\underline{E}]=0$ in
$\operatorname{Ext}^{1}_{\pazocal{M}_{F}}(\mathbbm{1},\underline{N})$.
Conversely, if there exists a nonzero $a\in A$ such that $a[\underline{E}]$ is
split, Theorem 2.13 implies that there exists $m\in N$ such that
$au=m-\tau_{M}(\tau^{*}m)$. The nonzero $A\otimes F$-module $L$ generated by
$m\oplus a$ together with $\tau_{M}$ defines a sub-$A$-module of
$\underline{E}$ isomorphic to $\mathbbm{1}$. For all $\mu\in\mathbb{Q}$, we
define $A\otimes F$-modules

$E_{\mu}:=W_{\mu}M+\mathbf{1}_{\mu\geq 0}L,\quad
E_{<\mu}:=W_{<\mu}M+\mathbf{1}_{\mu>0}L$

where $(W_{\mu}\underline{M})_{\mu\in\mathbb{Q}}$ is the weight filtration of
$\underline{M}$, and where $\mathbf{1}_{\mu\in S}$ is the indicator function
of the set $S$. It is easy to see that
$\underline{E}_{\mu}:=(E_{\mu},\tau_{M})$ and
$\underline{E}_{<\mu}:=(E_{<\mu},\tau_{M})$ define sub-$A$-motives of
$\underline{E}$ such that

$\underline{E}_{\mu}/\underline{E}_{<\mu}\cong\begin{cases}0~{}&\text{ if
}~{}\mu<0,\\\ \underline{L}~{}&\text{ if }~{}\mu=0,\\\
\operatorname{Gr}_{\mu}\underline{M}~{}&\text{ if }~{}\mu>0.\end{cases}$

As desired, we have constructed an increasing sequence
$(\underline{E}_{\mu})_{\mu}$ of sub-$A$-motives of $\underline{E}$ satisfying
the property of Definition 3.23. Hence, $\underline{E}$ is mixed. ∎

######  Remark 3.41.

Under the same hypothesis, Proposition 3.40 can be rephrased into

$\operatorname{Ext}^{1}_{\pazocal{M}_{F}}(\underline{M}^{\prime\prime},\underline{M}^{\prime})^{\text{tors}}=\operatorname{Ext}^{1}_{\pazocal{M}\pazocal{M}_{F}}(\underline{M}^{\prime\prime},\underline{M}^{\prime}).$

In particular, the $K$-vector space
$\operatorname{Ext}^{1}_{\pazocal{M}\pazocal{M}^{\text{iso}}_{F}}(\underline{M}^{\prime\prime},\underline{M}^{\prime})$
vanishes. The latter is only conjectured to be true in the number fields
setting ([Del, §1.3]).

######  Remark 3.42.

Nevertheless,
$\operatorname{Ext}^{1}_{\pazocal{M}\pazocal{M}_{F}}(\mathbbm{1},\underline{M})$
is generally non zero for $\underline{M}$ having positive weights. In the
notations of Example 2.4, let $n$ be a positive integer that is a power of the
characteristic $p$, and consider the $A$-motive $\underline{A}(-n)$ over a
field extension $F$ of $K$ that contains a $(q-1)$-st root $\eta$ of
$(-1/\theta)^{n}$. We claim that
$\operatorname{Ext}^{1}_{\pazocal{M}\pazocal{M}_{F}}(\mathbbm{1},\underline{A}(-n))$
is non zero, equivalently that
$\operatorname{Ext}^{1}_{\pazocal{M}_{F}}(\mathbbm{1},\underline{A}(-n))$ has
non zero torsion. Indeed, let $[\underline{E}]:=\iota(\eta^{q})$. Then,

$-t^{n}\cdot[\underline{E}]=\iota(-t^{n}\eta^{q})=\iota(\eta-(t-\theta)^{n}\eta^{q})=0.$

On the other-hand, $[\underline{E}]$ is non zero: for degree reasons, there
does not exist $p(t)\in F[t]$ such that
$\eta^{q}=p(t)-(t-\theta)^{n}p(t)^{(1)}$.

######  Remark 3.43.

A variation of the above argument shows that
$\operatorname{Ext}^{2}_{\pazocal{M}\pazocal{M}_{F}}(\mathbbm{1},\underline{A}(-1))$
is non zero for certain fields $F$. Assume that:

  1. 1.

$F$ contains a $(q-1)$-st root $\eta$ of $(-1/\theta)^{p}$,

  2. 2.

$F$ does not contain of $(q-1)$-st root of $-1/\theta$

  3. 3.

there exists $\beta\in F$ for which the polynomial $\theta X^{q}+X+\beta$ does
not split over $F$.

For $\eta,\beta\in F$ as above, let $\alpha:=\beta/\eta^{q}\in F$. Consider
the extension $\underline{M}$ of $\underline{A}(-p)$ by $\underline{A}(-1)$
given by

$\underline{M}=\left[F[t]^{2},\begin{pmatrix}(t-\theta)\mathbf{1}&\alpha\mathbf{1}\\\
0&(t-\theta)^{p}\mathbf{1}\end{pmatrix}\right].$

As $\underline{A}(-p)$ and $\underline{A}(-1)$ are pure of weight $p$ and $1$
respectively, $\underline{M}$ is mixed by Proposition 3.38, with weights
$\\{1,p\\}$. We claim that the $A$-linear morphism

$\operatorname{Ext}_{\pazocal{M}\pazocal{M}_{F}}^{1}(\mathbbm{1},f):\operatorname{Ext}^{1}_{\pazocal{M}\pazocal{M}_{F}}(\mathbbm{1},\underline{M})\longrightarrow\operatorname{Ext}^{1}_{\pazocal{M}\pazocal{M}_{F}}(\mathbbm{1},\underline{A}(-p)),$
(3.13)

induced by the epimorphism $f:\underline{M}\to\underline{A}(-p)$, is not
surjective. By the long-exact sequence of $\operatorname{Ext}$-modules, it
implies that
$\operatorname{Ext}^{2}_{\pazocal{M}\pazocal{M}_{F}}(\mathbbm{1},\underline{\ker}f)=\operatorname{Ext}^{2}_{\pazocal{M}\pazocal{M}_{F}}(\mathbbm{1},\underline{A}(-1))$
is non zero.

Because $\underline{M}$ has positive weights, by Proposition 3.40, (3.13) can
be rewritten as:

$\left(\frac{F[t,(t-\theta)^{-1}]^{\oplus
2}}{(\operatorname{id}-\tau_{M})(F[t]^{\oplus
2})}\right)^{\text{tors}}\longrightarrow\left(\frac{F[t,(t-\theta)^{-1}]}{(\operatorname{id}-(t-\theta)^{p}\tau)(F[t])}\right)^{\text{tors}}.$
(3.14)

In Remark 3.41, we showed that the class of $\eta^{q}$ defines a nonzero
element in the right-hand side of (3.14). If the latter was surjective, there
would exist $x\in F[t,(t-\theta)^{-1}]$ such that the class of $(x,\eta^{q})$
belongs to the right-hand side of (3.14). That is, there exist $a\in
A=\mathbb{F}[t]$ nonzero and $(f,g)\in F[t]^{\oplus 2}$ such that

$a\cdot\begin{pmatrix}x\\\ \eta^{q}\end{pmatrix}=\begin{pmatrix}f\\\
g\end{pmatrix}-\begin{pmatrix}(t-\theta)&\alpha\\\
0&(t-\theta)^{p}\end{pmatrix}\begin{pmatrix}f^{(1)}\\\ g^{(1)}\end{pmatrix}.$
(3.15)

As $-t^{p}\eta^{q}=\eta-(t-\theta)^{p}\eta^{q}$, we obtain from the bottom row
that $t^{p}$ divides $a$ and that $g=-(a/t^{p})\eta^{q}$. Evaluating the top
row at $t=0$, we get:

$0=f(0)+\theta f(0)^{q}+\alpha\cdot s\eta^{q}$ (3.16)

where $s\in\mathbb{F}$ is the evaluation of $a/t^{p}$ at $t=0$. If $s=0$, then
$g(0)=0$ and $f(0)=0$ by 2. Hence, dividing (3.15) by the correct power of
$t$, we can assume without loss that $s\neq 0$. Yet, dividing (3.16) by $s$
and using that $\alpha\eta^{q}=\beta$, we obtain a root of $\theta
X^{q}+X+\beta$ in $F$. This contradicts our assumption 3.

Although $\operatorname{Ext}^{2}_{\pazocal{M}\pazocal{M}_{F}}$ might be non
zero, we prove next that it is always torsion:

######  Theorem 3.44.

Let $\underline{M}$ be a mixed $A$-motive over $F$. For $i>1$, the $A$-module
$\operatorname{Ext}^{i}_{\pazocal{M}\pazocal{M}_{F}}(\mathbbm{1},\underline{M})$
is torsion.

###### Proof.

We begin with a very general remark. Given three objects $A$, $B$ and $C$ in
an abelian category $\pazocal{A}$, Yoneda’s cup product:

$\cup:\operatorname{Ext}^{1}_{\pazocal{A}}(A,B)\times\operatorname{Ext}^{1}_{\pazocal{A}}(B,C)\longrightarrow\operatorname{Ext}^{2}_{\pazocal{A}}(A,C)$
(3.17)

admits the following two descriptions:

  1. 1.

Given $e=[0\to B\stackrel{{\scriptstyle i}}{{\to}}E\to A\to
0]\in\operatorname{Ext}^{1}_{\pazocal{A}}(A,B)$ and $f=[0\to C\to
F\stackrel{{\scriptstyle j}}{{\to}}B\to
0]\in\operatorname{Ext}^{1}_{\pazocal{A}}(B,C)$, we obtain $e\cup
f\in\operatorname{Ext}^{2}_{\pazocal{A}}(A,C)$ as the class of the long-exact
sequence:

$0\to C\to F\stackrel{{\scriptstyle i\circ j}}{{\to}}E\to A\to 0.$

  2. 2.

Applying the functor $\operatorname{Ext}^{1}_{\pazocal{A}}(A,-)$ to a short
exact sequence associated to $e$, we obtain a connecting homomorphism:

$\delta_{e}:\operatorname{Ext}^{1}_{\pazocal{A}}(A,B)\longrightarrow\operatorname{Ext}^{2}_{\pazocal{A}}(A,C).$

Then, $e\cup f=\delta_{e}(f)$.

It follows from 1 that for any element $g$ in
$\operatorname{Ext}^{2}_{\pazocal{A}}(A,C)$, there exists an object $B$ in
$\pazocal{A}$ such that $g=e\cup f$ for