The setting

Let $\mathbb {F}$ be a finite field, let $q$ be its number of elements, and let $(C,\mathcal {O}_C)$ 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:

\begin{align*} A:=\mathcal{O}_C(C\setminus\{\infty\}) \end{align*}

has $K$ as its field of fractions. We let $K_{\infty }$ be the completion of $K$ with respect to the valuation $v_{\infty }$ associated to $\infty$, and let $K_{\infty }^s$ be the separable closure of $K_{\infty }$. The analogy with number fields that guide us in this text is as follows.

Let $R$ be an $A$-algebra. The analogy with number fields disappears when one considers the tensor product $A\otimes R$, 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 $A\otimes R$ which acts as the identity on $A$ and as the $q$-Frobenius on $R$. We let $\mathfrak {j}$ be the ideal of $A\otimes R$ generated by the set $\{a\otimes 1-1\otimes a|a\in A\}$.

Following [Reference AndersonAnd86], an Anderson $A$-motive $\underline {M}$ over $R$ is a pair $(M,\tau _M)$ where $M$ denotes a finite projective $A\otimes R$-module of constant rank, and where $\tau _M:(\tau ^*M)[\mathfrak {j}^{-1}]\to M[\mathfrak {j}^{-1}]$ is an $(A\otimes R)[\mathfrak {j}^{-1}]$-linear isomorphism (Definition 2.3). We let $\mathscr {M}_R$ denote the category of Anderson $A$-motives with obvious morphisms. $\mathscr {M}_R$ is known to be $A$-linear, rigid monoidal, and is exact in the sense of Quillen (Proposition 2.15) but not abelian ([Reference Hartl and JuschkaHJ20, § 2.3] or § 2.1). Let $\mathbb {1}$ in $\mathscr {M}_R$ be a neutral object for the tensor operation.

Extensions of $A$-motives

The category $\mathscr {M}_R$, 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 2.21). Let $\underline {M}$ be an $A$-motive over $R$.

We immediately deduce that $\operatorname {Ext}^i_{\mathscr {M}_R}(\mathbb {1},\underline {M})$ vanishes for $i>1$. For $i=1$, the $A$-module of degree one extensions admit the following explicit description. There is a natural surjective morphism

(1.2)\begin{equation} \iota:M[\mathfrak{j}^{-1}]\longrightarrow\operatorname{Ext}^1_{\mathscr{M}_F}(\mathbb {1},\underline{M}), \end{equation}

which maps $m\in M[\mathfrak {j}^{-1}]$ to the class of the extension of $\mathbb {1}$ by $\underline {M}$ whose middle term is the $A$-motive given by $\big [M\oplus (A\otimes R),\big (\begin{smallmatrix}\tau _M & m \\ 0 & 1\end{smallmatrix}\big )\big ]$ (§ 2.2). The kernel of $\iota$ being $(\operatorname {id}-\tau _M)(M)$, we recover the isomorphism provided by Theorem A.

To pursue the analogy with number fields, we now present the notion of weights and mixedness for Anderson $A$-motives over fields. 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 [Reference TaelmanTae09] and Hartl and Juschka [Reference Hartl and JuschkaHJ20]. We completed this picture in the most general way (over any $A$-field and without any restriction on $\deg (\infty )$).

Let $R=F$ be a field. To an Anderson $A$-motive $\underline {M}$ over $F$, we attach an isocrystal $\operatorname {I}_{\infty }(\underline {M})$ at $\infty$ (in the sense of [Reference MornevMor21]). The name isocrystal is borrowed from $p$-adic Hodge theory, where the function field setting allows the application of the non-archimedean theory at the infinite point $\infty$ of $C$ as well. Following [Reference AndersonAnd86, 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 $\mu _1<\cdots <\mu _s$ together with a finite ascending filtration of $\underline {M}$ by saturated sub-$A$-motives:

\begin{align*} 0=W_{\mu_0}\underline{M}\subsetneq W_{\mu_1}\underline{M}\subsetneq \cdots \subsetneq W_{\mu_s}\underline{M}=\underline{M} \end{align*}

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 the numbers $\mu _i$ which are then called the weights of $\underline {M}$. As was observed by Hartl and Juschka [Reference Hartl and JuschkaHJ20, Example 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 denote by $\mathscr {M}\mathscr {M}_F$ the full subcategory of $\mathscr {M}_F$ whose objects are mixed Anderson $A$-motives over $F$. The main results of § 3 are gathered in the next theorem.

Theorem B Let $\underline {M}$ be an object of $\mathscr {M}\mathscr {M}_F$. If all the weights of $\underline {M}$ are non-positive, then every extension of $\mathbb {1}$ by $\underline {M}$ is mixed; that is,

\begin{align*} \operatorname{Ext}^1_{\mathscr{M}\mathscr{M}_F}(\mathbb {1},\underline{M})=\operatorname{Ext}^1_{\mathscr{M}_F}(\mathbb {1},\underline{M}). \end{align*}

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

\begin{align*} \operatorname{Ext}^1_{\mathscr{M}\mathscr{M}_F}(\mathbb {1},\underline{M})=\operatorname{Ext}^1_{\mathscr{M}_F}(\mathbb {1},\underline{M})^{\operatorname{tors}}. \end{align*}

Furthermore, for $i>1$, $\operatorname {Ext}^i_{\mathscr {M}\mathscr {M}_F}(\mathbb {1},\underline {M})$ is a torsion module for all $\underline {M}$.

The good $\ell$-adic part

To present the theory of good and integral extensions, we now assume that $F$ is a finite field extension of $K$ (namely, a global function field). Let $\mathfrak {p}$ be a finite place of $F$ (i.e. not above $\infty$), $F_{\mathfrak {p}}$ the associated local function field, $F_{\mathfrak {p}}^s$ a separable closure of $F_{\mathfrak {p}}$ 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 $A$ which does not lie under $\mathfrak {p}$, there is an $\ell$-adic realization functor from $\mathscr {M}_{F_{\mathfrak {p}}}$ to the category of continuous $\mathcal {O}_{\ell }$-linear representations of $G_{\mathfrak {p}}$. Given an object $\underline {M}_{\mathfrak {p}}=(M_{\mathfrak {p}},\tau _M)$ of $\mathscr {M}_{F_{\mathfrak {p}}}$, it is defined as the $\mathcal {O}_{\ell }$-module

\begin{align*} \operatorname{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)\}, \end{align*}

where $G_{\mathfrak {p}}$ acts compatibly on the right of the tensor $M_{\mathfrak {p}}\otimes _{F_{\mathfrak {p}}} F_{\mathfrak {p}}^{s}$ (Definition 2.24).

We prove in Corollary 2.28 that $\operatorname {T}_{\ell }$ is exact. This paves the way for introducingextensions 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 $I_{\mathfrak {p}}$:

(1.3)\begin{equation} r_{\underline{M},\ell,\mathfrak{p}}:\operatorname{Ext}^1_{\mathscr{M}_{F_{\mathfrak{p}}}}(\mathbb {1}_{\mathfrak{p}},\underline{M}_{\mathfrak{p}})\longrightarrow \operatorname{H}^1(I_{\mathfrak{p}},\operatorname{T}_{\ell}\underline{M}_{\mathfrak{p}}) \end{equation}

(we refer to § 2.3). Mimicking Scholl's approach, we say that an extension $[\underline {E}_{\mathfrak {p}}]$ of $\mathbb {1}_{\mathfrak {p}}$ 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 $\ell$, although this is presumably out of our reach. We let $\operatorname {Ext}^1_{\textrm {good}}(\mathbb {1}_{\mathfrak {p}},\underline {M}_{\mathfrak {p}})_{\ell }$ denote the kernel of $r_{\underline {M},\ell,\mathfrak {p}}$ (Definition 2.30).

The integral part

Gardeyn [Reference GardeynGar03] 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 (§ 4). They form the function field analogue of Néron models of abelian varieties, or more generally, of regular models of varieties.

Let $\underline {M}$ be an $A$-motive over $F$, and denote by $\underline {M}_{\mathfrak {p}}$ the $A$-motive of $F_{\mathfrak {p}}$ obtained from $\underline {M}$ by base-change along $F\subset F_{\mathfrak {p}}$. Let $\mathcal {O}_{\mathfrak {p}}$ be the valuation ring of $F_{\mathfrak {p}}$ and let also $\mathcal {O}_F$ denote the integral closure of $A$ in $F$.

As opposed to [Reference GardeynGar03, Definitions 2.1 and 2.3], we do not ask for an $\mathcal {O}_{\mathfrak {p}}$-model (respectively, $\mathcal {O}_F$-model) to be locally free. We show that this is automatic for maximal ones (Theorem 4.32). Compared with Gardeyn, our exposition is therefore simplified and avoids the use of a technical lemma due to Lafforgue [Reference GardeynGar03, § 2.2]. Our next result should be compared with [Reference GardeynGar03, Proposition 2.13] (see Proposition 4.51 and Theorems 4.53 and 4.55 in the text).

Theorem C A maximal $\mathcal {O}_{\mathfrak {p}}$-model $M_{\mathcal {O}_{\mathfrak {p}}}$ for $\underline {M}_{\mathfrak {p}}$ (respectively, $\mathcal {O}_F$-model $M_{\mathcal {O}_F}$ for $\underline {M}$) exists and is unique. It is locally free over $A\otimes \mathcal {O}_{\mathfrak {p}}$ (respectively, $A\otimes \mathcal {O}_F$). In addition, the maximal local and global models are related by canonical isomorphisms:

\begin{align*} M_{\mathcal{O}_F}\cong \bigcap_{\mathfrak{p}}(M\cap M_{\mathcal{O}_{\mathfrak{p}}}) \quad \textrm{and} \quad M_{\mathcal{O}_{\mathfrak{p}}}\cong M_{\mathcal{O}_F}\otimes_{\mathcal{O}_F}\mathcal{O}_{\mathfrak{p}}, \end{align*}

where the intersection is taken over finite places $\mathfrak {p}$ of $F$.

Along the way, we also prove a good reduction criterion for $A$-motives, in the style of Néron–Ogg–Shafarevič (Proposition 4.49).

Then, we call $\mathfrak {p}$-integral any extension of $\mathbb {1}$ by $\underline {M}$ which arises as an element of $\iota (M_{\mathcal {O}_{\mathfrak {p}}}[\mathfrak {j}^{-1}])$ as in (1.2). We let $\operatorname {Ext}^1_{\mathcal {O}_{\mathfrak {p}}}(\mathbb {1}_{\mathfrak {p}},\underline {M}_{\mathfrak {p}})$ be the module of $\mathfrak {p}$-integral extensions (Definition 4.58). Our main result (repeated from Theorem 4.60) is next.

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

In § 4.4, we define the global version of the above. Namely, the $A$-motive $\underline {M}$ defines an $A$-motive $\underline {M}_{F_{\mathfrak {p}}}$ over $F_{\mathfrak {p}}$ by extending the base field. We let $\operatorname {Ext}^1_{\mathcal {O}_F}(\mathbb {1},\underline {M})$ be the module of integral extensions, i.e. of extensions that are $\mathfrak {p}$-integral for all $\mathfrak {p}$ after base-change along $F\subset F_{\mathfrak {p}}$ (Definition 4.58). Our second main result (repeated from Theorem 4.62) is as follows.

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

\begin{align*} \frac{M_{\mathcal{O}_F}[\mathfrak{j}^{-1}]}{(\operatorname{id}-\tau_M)(M_{\mathcal{O}_F})}\stackrel{\sim}{\longrightarrow} \operatorname{Ext}^1_{\mathcal{O}_F}(\mathbb {1},\underline{M}). \end{align*}

Regulated extensions of $A$-motives

We are facing two main issues to pursue our analogy: the counterpart of conjecture (C4) does not hold true and, more seriously, neither is the counterpart of conjecture (C5): the $A$-module $\operatorname {Ext}^1_{\mathcal {O}_F}(\mathbb {1},\underline {M})$ is typically not finitely generated. Those facts suggest that the category $\mathscr {M}_F$–and also $\mathscr {M}\mathscr {M}_F$–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 conjectures (C4) and (C5).

Pink, in the context of function fields Hodge structures [Reference PinkPin97, § 6], is facing a similar issue. There, he introduced the notion of Hodge additivity, whose counterpart for $A$-motives is as follows.

Pink's intuition for introducing Hodge additivity is driven by the inexact feature of the operation assigning to a Hodge–Pink structure its ‘classical’ Hodge structure (i.e. the data of its Hodge filtration). Although Pink was concerned with the computation of Hodge groups, for our purpose this prevents the well-definedness of a regulator map from the module of extensions of $A$-motives to that of extensions of ‘classical’ Hodge structures (see Remark 5.2). Regulated extensions are designed to resolve this point, hence the naming. The notion of regulation has applications in sequels to this text [Reference GazdaGaz22, Reference Gazda and MaurischatGM23].

In Corollary 5.5, we prove that $\iota$ induces an isomorphism of $A$-modules:

\begin{align*} \frac{M+\tau_{M}(\tau^*M)}{(\operatorname{id}-\tau_M)(M)}\stackrel{\sim}{\longrightarrow} \operatorname{Ext}^{1,\textrm{reg}}_{\mathscr{M}_{F}}(\mathbb {1},\underline{M}). \end{align*}

Although $\operatorname {Ext}^{1,\textrm {reg}}_{\mathcal {O}_F}(\mathbb {1},\underline {M})$ is still not finitely generated over $A$ in general, a version of conjecture (C5) involving the infinite places holds (we refer the reader to [Reference GazdaGaz22, Theorem 4.1]). Concerning the analogue of conjecture (C4), we strongly suspect regulated extensions to also correct the integral$\neq$good extensions phenomenon; namely we expect the following to hold for $A$-motives $\underline {M}_{\mathfrak {p}}$ over $F_{\mathfrak {p}}$ (see Conjecture 5.8).

Conjecture Let $\ell$ be a maximal ideal of $A$ not under $\mathfrak {p}$. Then,

\begin{align*} \operatorname{Ext}^{1,\textrm{reg}}_{\mathcal{O}_{\mathfrak{p}}}(\mathbb {1}_{\mathfrak{p}},\underline{M}_{\mathfrak{p}})=\operatorname{Ext}^{1,\textrm{reg}}_{\textrm{good}}(\mathbb {1}_{\mathfrak{p}},\underline{M}_{\mathfrak{p}})_{\ell}. \end{align*}

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

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