Isogenies of abelian Anderson A-modules and A-motives

TL;DR

This paper explores the theory of abelian Anderson A-modules and A-motives, establishing dual isogenies, equivalences of isogenies, and analyzing torsion submodules and their relation to local shtukas in the function field setting.

Contribution

It introduces a duality theory for isogenies of abelian t-modules and connects torsion submodules to local shtukas, extending the understanding of Anderson modules and t-motives over rings.

Findings

Every isogeny has a dual isogeny in the opposite direction.

A morphism is an isogeny if and only if the associated t-motives are isogenous.

Torsion submodules relate to local shtukas and p-divisible groups analogs.

Abstract

As a generalization of Drinfeld modules, Greg Anderson introduced abelian t-modules and t-motives over a perfect field. In this article we study relative versions of these over rings. We investigate isogenies among them. Our main results state that every isogeny possesses a dual isogeny in the opposite direction, and that a morphism between abelian t-modules is an isogeny if and only if the corresponding morphism between their associated t-motives is an isogeny. We also study torsion submodules of abelian t-modules which in general are non-reduced group schemes. They can be obtained from the associated t-motive via the finite shtuka correspondence of Drinfeld and Abrashkin. The inductive limits of torsion submodules are the function field analogs of p-divisible groups. These limits correspond to the local shtukas attached to the t-motives associated with the abelian t-modules. In this sense the theory of abelian t-modules is captured by the theory of t-motives.

Equations

$$\begin{array}[]{rll}A\ni a\colon&M\longrightarrow M,&m\mapsto m\circ\varphi_{a}\\[5.69054pt] R\ni b\colon&M\longrightarrow M,&m\mapsto b\circ m\end{array}$$

$$M/gM⟶N/gN⟶\mathrm{coker}f⟶0$$

$$0⟶{\mathrm{Tor}}_1^R(k,\mathrm{coker}f)⟶M{\otimes }_{R}k⟶N{\otimes }_{R}k⟶(\mathrm{coker}f){\otimes }_{R}k⟶0.$$

$$({\sigma }^{\ast }{A}_R^{\oplus n_1}){|}_{\mathrm{Spec}{A}_R\setminus V(\mathcal{J})}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$E\underset{f,\,E^{\prime},f}{\times}E\;\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{$\scriptstyle\sim$}$\longrightarrow$}}}\;E\underset{R}{\times}\ker f$$

$$R\{\tau\}\;:=\;\bigl{\{}\,\textstyle\sum\limits_{i=0}^{n}b_{i}\tau^{i}\colon n\in{\mathbb{N}}_{0},b_{i}\in R\,\bigr{\}}\quad\text{with}\quad\tau b\;=\;b^{q}\tau$$

$$M(\underline{E})$$

$$m$$

$$0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\operatorname{coker}\tau_{M}\;\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{$\scriptstyle\sim$}$\longrightarrow$}}}\;\operatorname{Hom}_{R}(\operatorname{Lie}E,R),\quad m{\rm\;mod\;}\tau_{M}(\sigma^{*}M)\;\longmapsto\;\operatorname{Lie}m\,.$$

$$\operatorname{Hom}_{R}({\underline{E\!}\,},{\underline{E\!}\,}^{\prime})\;\longrightarrow\;\operatorname{Hom}_{R}\bigl{(}{\underline{M\!}\,},{\underline{M\!}\,}^{\prime}\bigr{)},\quad f\;\longmapsto(m^{\prime}\mapsto m^{\prime}\circ f)$$

$$R^{\prime }[t]$$

$${\Phi }_a:=\sum _{i=0}^{n(a)}{b}_i(a){\tau }^i:=\alpha \circ {\phi }_a\circ {\alpha }^{-1}\in {\mathrm{End}}_{R^{\prime }\text{-groups},{\mathbb{F}}_q\text{-lin}}({\mathbb{G}}_{a,R^{\prime }})=R^{\prime }\{\tau \},$$

$${\underline{V}}^{\tau }(R^{\prime }):=\{v\otimes V{\otimes }_R{R}^{\prime }:v=F_V({\sigma }_V^{\ast }v)\}.$$

$${\Delta }^{♭}:B^{♭}⟶(B{\otimes }_{R}B{)}^{♭}:=R[\underline{X}\otimes 1,1\otimes \underline{X}]/(I\otimes 1+1\otimes I)(I_{R[\underline{X}]}\otimes 1+1\otimes I_{R[\underline{X}]})$$

$${\ell }_{\mathcal{G}/\mathrm{Spec}R}^{{}^{\bullet }}:0⟶(I/I^2){\otimes }_{B,e_B}R\stackrel{d}{\to }{\Omega }_{R[\underline{X}]/R}^1{\otimes }_{R[\underline{X}],e_{R[\underline{X}]}}R⟶0$$

$$\operatorname{Dr}_{q}({\underline{V\!}\,})\enspace:=\enspace\operatorname{Spec}\;\bigl{(}{\textstyle\bigoplus\limits_{n\geq 0}}\operatorname{Sym}^{n}_{R}V\bigr{)}/I$$

$$\operatorname{Dr}_{q}({\underline{V\!}\,})^{\flat}\enspace:=\enspace\operatorname{Spec}\;\bigl{(}{\textstyle\bigoplus\limits_{n\geq 0}}\operatorname{Sym}^{n}_{R}V\bigr{)}/(I\cdot I_{0}),$$

$$M_q(G):={\mathrm{Hom}}_{R\text{-groups},{\mathbb{F}}_q\text{-lin}}(G,{\mathbb{G}}_{a,R})$$

$$\bigl{(}{\textstyle\bigoplus\limits_{n\geq 0}}\operatorname{Sym}_{R}^{n}M_{q}({\underline{E\!}\,})\bigr{)}\,\big{/}\,\bigl{(}m^{\otimes q}-F_{M_{q}({\underline{E\!}\,})}(\sigma_{M_{q}({\underline{E\!}\,})}^{*}m)\colon m\in M_{q}({\underline{E\!}\,})\bigr{)}\;\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{$\scriptstyle\sim$}$\longrightarrow$}}}\;\Gamma(E,{\cal{O}}_{E})\,,\quad m\mapsto m^{*}(\xi)$$

$$0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\mathrm{rk}\underline{M}={\mathrm{rk}}_{A_R}M={\mathrm{rk}}_{A_k}(M{\otimes }_{R}k)={\mathrm{rk}}_{A_k}(N{\otimes }_{R}k)={\mathrm{rk}}_{A_R}N=\mathrm{rk}\underline{N}.$$

$${\mathrm{rk}}_R\mathrm{coker}{\tau }_M={\dim }_k\mathrm{coker}({\tau }_M\otimes {\mathrm{id}}_k)={\dim }_k\mathrm{coker}({\tau }_N\otimes {\mathrm{id}}_k)={\mathrm{rk}}_R\mathrm{coker}{\tau }_N.$$

$$\bigl{(}\operatorname{coker}{\underline{M\!}\,}(f),\tau_{\operatorname{coker}{\underline{M\!}\,}(f)}\bigr{)}\;\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{$\scriptstyle\sim$}$\longrightarrow$}}}\;{\underline{M\!}\,}_{q}(\ker f)$$

$$\operatorname{Dr}_{q}\bigl{(}\operatorname{coker}{\underline{M\!}\,}(f)\bigr{)}\;\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{$\scriptstyle\sim$}$\longrightarrow$}}}\;\ker f\,.$$

$$\mathrm{coker}\underline{M}(f)⟶{\underline{M}}_q(\ker f),m\mathrm{mod}\mathrm{im}\underline{M}(f)\mapsto m\circ \iota .$$

$$\operatorname{Dr}_{q}\bigl{(}{\underline{M\!}\,}_{q}(\ker f)\bigr{)}\longrightarrow\operatorname{Dr}_{q}(\operatorname{coker}{\underline{M\!}\,}(f))\lhook\joinrel\longrightarrow\ker f\lhook\joinrel\longrightarrow E\,.$$

$$k[x_1,\dots ,x_d]/(f^{\ast }(y_{\ell }):1\le \ell \le d)\cong k[z_{i,n}:1\le i\le d,\le n\le N]/I\text{with}$$

$$\displaystyle I\;=\;\bigl{(}f^{*}(y_{i}),\,\textstyle z_{i,N}^{q}-\sum\limits_{n=0}^{N}b_{i,n}\!\cdot\!z_{i,n},\,z_{i,n}^{q}-z_{i,n+1}\colon 1\leq i\leq d,0\leq n<N\bigr{)}\,.$$

$$k[x_{1},\ldots,x_{d}]/(f^{*}(y_{\ell})\colon 1\leq\ell\leq d)\;\cong\;k[\tilde{z}_{1},\ldots,\tilde{z}_{r}]\big{/}\bigl{(}\tilde{z}_{i}^{q}-\textstyle\sum\limits_{j=1}^{r}c_{ij}\tilde{z}_{j}\colon 1\leq i\leq r\bigr{)}\;=:\;B\,.$$

$$\underset{i=1}{\overset{d}{⨁}}\underset{0\le n}{⨁}{R}^{\prime }\cdot {\tau }^n{m}_i\cong \underline{M}(\underline{E}{\times }_R{R}^{\prime })\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

Full text

Mathematics Subject Classification (2010): 11G09, (14K02, 13A35, 14L05)

1. 1 Introduction
2. 2 $A$-Motives
3. 3 Abelian Anderson $A$-modules
4. 4 Review of the finite shtuka equivalence
5. 5 Isogenies
6. 6 Torsion points
7. 7 Divisible local Anderson modules
8. References

1 Introduction

As a generalization of Drinfeld modules [Dri74], Greg Anderson [And86] introduced abelian $t$-modules and $t$-motives over a perfect field. In this article we study relative versions of these over rings and generalize them to abelian Anderson $A$-modules and $A$-motives. The upshot of our results is that the entire theory of abelian Anderson $A$-modules is contained in the theory of $A$-motives. More precisely, let ${\mathbb{F}}_{q}$ be a finite field with $q$ elements, let $C$ be a smooth projective geometrically irreducible curve over ${\mathbb{F}}_{q}$ and let $Q={\mathbb{F}}_{q}(C)$ be its function field. Let $\infty\in C$ be a closed point and let $A=\Gamma(C\smallsetminus\{\infty\},{\cal{O}}_{C})$ be the ring of functions which are regular outside $\infty$. Let $(R,\gamma)$ be an $A$-ring, that is a commutative unitary ring together with a ring homomorphism $\gamma\colon A\to R$. We consider the ideal ${\cal{J}}:=(a\otimes 1-1\otimes\gamma(a)\colon a\in A)=\ker(\gamma\otimes\operatorname{\,id}_{R}\colon A_{R}\to R)\subset A_{R}:=A\otimes_{{\mathbb{F}}_{q}}R$ and the endomorphism $\sigma:=\operatorname{\,id}_{A}\otimes\operatorname{Frob}_{q,R}\colon a\otimes b\mapsto a\otimes b^{q}$ of $A_{R}$. For an $A_{R}$-module $M$ we set $\sigma^{*}M:=M\otimes_{A_{R},\,\sigma}A_{R}=M\otimes_{R,\,\operatorname{Frob}_{q,R}}R$, and for an element $m\in M$ we write $\sigma_{M}^{*}m:=m\otimes 1\in\sigma^{*}M$.

Definition 1.1**.**

An effective $A$-motive of rank $r$ over an $A$-ring $R$ is a pair ${\underline{M\!}\,}=(M,\tau_{M})$ consisting of a locally free $A_{R}$-module $M$ of rank $r$ and an $A_{R}$-homomorphism $\tau_{M}\colon\sigma^{*}M\to M$ whose cokernel is annihilated by ${\cal{J}}^{n}$ for some positive integer $n$. We say that ${\underline{M\!}\,}$ has dimension $d$ if $\operatorname{coker}\tau_{M}$ is a locally free $R$-module of rank $d$ and annihilated by ${\cal{J}}^{d}$. We write $\operatorname{rk}{\underline{M\!}\,}=r$ and $\dim{\underline{M\!}\,}=d$ for the rank and the dimension of ${\underline{M\!}\,}$.

A morphism $f\colon(M,\tau_{M})\to(N,\tau_{N})$ between effective $A$-motives is an $A_{R}$-homomorphism $f\colon M\to N$ which satisfies $f\circ\tau_{M}=\tau_{N}\circ\sigma^{*}f$.

Note that $\tau_{M}$ is always injective and $\operatorname{coker}\tau_{M}$ is always a finite locally free $R$-module by Proposition 2.3 below. We give some explanations for this definition in Section 2 and also define non-effective $A$-motives. If $R$ is a perfect field, $A={\mathbb{F}}_{q}[t]$ and in addition, $M$ is finitely generated over the non-commutative polynomial ring R\{\tau\}:=\bigl{\{}\,\textstyle\sum\limits_{i=0}^{n}b_{i}\tau^{i}\colon n\in{\mathbb{N}}_{0},b_{i}\in R\,\bigr{\}} with $\tau b=b^{q}\tau$, which acts on $m\in M$ via $\tau\colon m\mapsto\tau_{M}(\sigma_{M}^{*}m)$, then $(M,\tau_{M})$ is a $t$-motive in the sense of Anderson [And86, § 1.2].

Next let us define abelian Anderson $A$-modules. In Section 3 we give some explanations on the terminology in the following

Definition 1.2**.**

Let $d$ and $r$ be positive integers. An abelian Anderson $A$-module of rank $r$ and dimension $d$ over $R$ is a pair ${\underline{E\!}\,}=(E,\varphi)$ consisting of a smooth affine group scheme $E$ over $\operatorname{Spec}R$ of relative dimension $d$, and a ring homomorphism $\varphi\colon A\to\operatorname{End}_{R\text{-groups}}(E),\,a\mapsto\varphi_{a}$ such that

1. (a)

there is a faithfully flat ring homomorphism $R\to R^{\prime}$ for which $E\times_{R}\operatorname{Spec}R^{\prime}\cong{\mathbb{G}}_{a,R^{\prime}}^{d}$ as ${\mathbb{F}}_{q}$-module schemes, where ${\mathbb{F}}_{q}$ acts on $E$ via $\varphi$ and ${\mathbb{F}}_{q}\subset A$, 2. (b)

\bigl{(}\operatorname{Lie}\varphi_{a}-\gamma(a)\bigr{)}^{d}=0 on $\operatorname{Lie}E$ for all $a\in A$, 3. (c)

the set $M:=M({\underline{E\!}\,}):=M_{q}({\underline{E\!}\,}):=\operatorname{Hom}_{R\text{-groups},{\mathbb{F}}_{q}\text{\rm-lin}}(E,{\mathbb{G}}_{a,R})$ of ${\mathbb{F}}_{q}$-equivariant homomorphisms of $R$-group schemes is a locally free $A_{R}$-module of rank $r$ under the action given on $m\in M$ by

[TABLE]

A morphism $f\colon(E,\varphi)\to(E^{\prime},\varphi^{\prime})$ between abelian Anderson $A$-modules is a homomorphism of group schemes $f\colon E\to E^{\prime}$ over $R$ which satisfies $\varphi^{\prime}_{a}\circ f=f\circ\varphi_{a}$ for all $a\in A$.

In particular, if $R$ is a perfect field and $A={\mathbb{F}}_{q}[t]$, then an abelian Anderson $A$-module is nothing else than an abelian $t$-module in the sense of Anderson [And86, § 1.1]. When $q$ is not a prime and $R$ is not a field, we do not know the answer to the following

Question 1.3**.**

If we weaken Definition 1.2(a) and only require that there is an isomorphism of group schemes $E\times_{\operatorname{Spec}R}\operatorname{Spec}R^{\prime}\cong{\mathbb{G}}_{a,R^{\prime}}^{d}$, do we get an equivalent definition?

For general $A$ and $R$, the abelian Anderson $A$-modules of dimension $1$ over $R$ are precisely the Drinfeld $A$-modules over $R$; see Definition 3.7 and Theorem 3.9. Anderson’s anti-equivalence [And86, Theorem 1] between abelian $t$-modules and $t$-motives directly generalizes to the following

Theorem 3.5. *If ${\underline{E\!}\,}=(E,\varphi)$ is an abelian Anderson $A$-module then ${\underline{M\!}\,}({\underline{E\!}\,})=(M,\tau_{M})$ with $\tau_{M}\colon\sigma^{*}M\to M$, $\sigma_{M}^{*}m\mapsto\operatorname{Frob}_{q,{\mathbb{G}}_{a,R}}\!\circ\,m$ is an effective $A$-motive of the same rank and dimension as ${\underline{E\!}\,}$. The contravariant functor ${\underline{E\!}\,}\mapsto{\underline{M\!}\,}({\underline{E\!}\,})$ is fully faithful. Its essential image consists of all effective $A$-motives ${\underline{M\!}\,}=(M,\tau_{M})$ over $R$ for which there exists a faithfully flat ring homomorphism $R\to R^{\prime}$ such that $M\otimes_{R}R^{\prime}$ is a finite free left $R^{\prime}\{\tau\}$-module under the map $\tau\colon M\to M,\,m\mapsto\tau_{M}(\sigma_{M}^{*}m)$. *

The main purpose of this article is to study isogenies and their (co-)kernels over arbitrary $A$-rings $R$. Here a morphism $f\colon{\underline{E\!}\,}\to{\underline{E\!}\,}^{\prime}$ between abelian Anderson $A$-modules over $R$ is an isogeny if it is finite and surjective. On the other hand, a morphism $f\in\operatorname{Hom}_{R}({\underline{M\!}\,},{\underline{N\!}\,})$ between $A$-motives over $R$ is an isogeny if $f$ is injective and $\operatorname{coker}f$ is finite and locally free as $R$-module. We give equivalent characterizations in Propositions 5.2, 5.4 and 5.8. The following are our two main results.

Theorem 5.9. Let $f\in\operatorname{Hom}_{R}({\underline{E\!}\,},{\underline{E\!}\,}^{\prime})$ be a morphism between abelian Anderson $A$-modules and let ${\underline{M\!}\,}(f)\in\operatorname{Hom}_{R}({\underline{M\!}\,}^{\prime},{\underline{M\!}\,})$ be the associated morphism between the associated effective $A$-motives ${\underline{M\!}\,}={\underline{M\!}\,}({\underline{E\!}\,})$ and ${\underline{M\!}\,}^{\prime}={\underline{M\!}\,}({\underline{E\!}\,}^{\prime})$. Then

1. (a)

$f$* is an isogeny if and only if ${\underline{M\!}\,}(f)$ is an isogeny.* 2. (c)

If $f$ is an isogeny, then $\ker f$ and $\operatorname{coker}{\underline{M\!}\,}(f)$ correspond to each other under the finite shtuka equivalence which we review in Section 4.

Corollary 5.15. *If $f\in\operatorname{Hom}_{R}({\underline{M\!}\,},{\underline{N\!}\,})$ is an isogeny between $A$-motives then there is an element $0\neq a\in A$ and an isogeny $g\in\operatorname{Hom}_{R}({\underline{N\!}\,},{\underline{M\!}\,})$ with $f\circ g=a\cdot\operatorname{\,id}_{\underline{N\!}\,}$ and $g\circ f=a\cdot\operatorname{\,id}_{\underline{M\!}\,}$. The same is true for abelian Anderson $A$-modules. *

This leads to the following result about torsion points in Section 6. Let $(0)\neq{\mathfrak{a}}\subset A$ be an ideal and let ${\underline{E\!}\,}=(E,\varphi)$ be an abelian Anderson $A$-module over $R$. The ${\mathfrak{a}}$-torsion submodule ${\underline{E\!}\,}[{\mathfrak{a}}]$ of ${\underline{E\!}\,}$ is the closed subscheme of $E$ defined by ${\underline{E\!}\,}[{\mathfrak{a}}](S)\,=\,\{\,P\in E(S)\colon\varphi_{a}(P)=0\text{ for all }a\in{\mathfrak{a}}\,\}$ on any $R$-algebra $S$.

Theorem 6.4. ${\underline{E\!}\,}[{\mathfrak{a}}]$* is a finite locally free group scheme over $R$. It is étale over $R$ if and only if ${\mathfrak{a}}+{\cal{J}}=A_{R}$. If ${\underline{M\!}\,}={\underline{M\!}\,}({\underline{E\!}\,})$ is the associated $A$-motive then ${\underline{E\!}\,}[{\mathfrak{a}}]$ and ${\underline{M\!}\,}/{\mathfrak{a}}{\underline{M\!}\,}$ correspond to each other under the finite shtuka equivalence reviewed in Section 4. *

If ${\mathfrak{a}}+{\cal{J}}=A_{R}$ and $\bar{s}=\operatorname{Spec}\Omega$ is a geometric base point of $\operatorname{Spec}R$, then we also prove in Theorem 6.6 that ${\underline{E\!}\,}[{\mathfrak{a}}](\Omega)$ is a free $A/{\mathfrak{a}}$-module of rank $r$ which carries a continuous action of the étale fundamental group $\pi_{1}^{\rm\acute{e}t}(\operatorname{Spec}R,\bar{s})$.

In the final Section 7 we turn towards the case where ${\mathfrak{p}}\subset A$ is a maximal ideal and where all elements of $\gamma({\mathfrak{p}})\subset R$ are nilpotent. In this case, we can associate with an $A$-motive ${\underline{M\!}\,}$ over $R$ a local shtuka ${\underline{\hat{M}\!}\,}_{\mathfrak{p}}({\underline{M\!}\,})$; see Example 7.2 and with an abelian Anderson $A$-module ${\underline{E\!}\,}$ a divisible local Anderson module ${\underline{E\!}\,}[{\mathfrak{p}}^{\infty}]:=\displaystyle\lim_{\longrightarrow}{\underline{E\!}\,}[{\mathfrak{p}}^{n}]$ in the sense of [HS15]; see Definition 7.3 and Theorem 7.6. If ${\underline{M\!}\,}={\underline{M\!}\,}({\underline{E\!}\,})$ then ${\underline{\hat{M}\!}\,}_{\mathfrak{p}}({\underline{M\!}\,})$ and ${\underline{E\!}\,}[{\mathfrak{p}}^{\infty}]$ correspond to each other under the local shtuka equivalence from [HS15]; see Theorems 7.4 and 7.6.

Acknowledgments. The author acknowledges support of the DFG (German Research Foundation) in form of SFB 878.

Notation

Throughout this article we denote by

[TABLE]

Note that $\gamma$ makes $R$ into an ${\mathbb{F}}_{q}$-algebra. Further note that ${\cal{J}}$ is a locally free $A_{R}$-module of rank $1$. Indeed, ${\cal{J}}=I\otimes_{A_{A}}A_{R}$ for the ideal $I:=(a\otimes 1-1\otimes a\colon a\in A)\subset A_{A}=A\otimes_{{\mathbb{F}}_{q}}A$. The latter is a locally free $A_{A}$-module of rank $1$ by Nakayama’s lemma, because $I\otimes_{A_{A}}A_{A}/I=I/I^{2}=\Omega^{1}_{A/{\mathbb{F}}_{q}}$ is a locally free module of rank $1$ over $A_{A}/I=A$.

We will sometimes reduce from the ring $A$ to the polynomial ring ${\mathbb{F}}_{q}[t]$ by applying the following

Lemma 1.4**.**

Let $a\in A\smallsetminus{\mathbb{F}}_{q}$ and let ${\mathbb{F}}_{q}[t]$ be the polynomial ring in the variable $t$. Then the homomorphism ${\mathbb{F}}_{q}[t]\to A,\,t\mapsto a$ makes $A$ into a finite free ${\mathbb{F}}_{q}[t]$-module of rank equal to $-[{\mathbb{F}}_{\infty}:{\mathbb{F}}_{q}]\operatorname{ord}_{\infty}(a)$, where $\operatorname{ord}_{\infty}$ is the normalized valuation of the discrete valuation ring ${\cal{O}}_{C,\infty}$.

Proof.

If $\operatorname{ord}_{\infty}(a)=0$ then $a$ would have no pole on the curve $C$, hence would be constant. Since $C$ is geometrically irreducible this would imply $a\in{\mathbb{F}}_{q}$ which was excluded. Therefore $a$ is non-constant and defines a finite surjective morphism of curves $f\colon C\to{\mathbb{P}}^{1}_{{\mathbb{F}}_{q}}$ with $\operatorname{Spec}A\to\operatorname{Spec}{\mathbb{F}}_{q}[t]={\mathbb{P}}^{1}_{{\mathbb{F}}_{q}}\smallsetminus\{\infty^{\prime}\}$, where $\infty^{\prime}\in{\mathbb{P}}^{1}_{{\mathbb{F}}_{q}}$ is the pole of $t$. By [GW10, Proposition 15.31] its degree can be computed in the fiber $f^{-1}(\infty^{\prime})=\{\infty\}$ as $\deg f=[{\mathbb{F}}_{\infty}:{\mathbb{F}}_{\infty^{\prime}}]\!\cdot\!e_{f}(\infty)$ where ${\mathbb{F}}_{\infty^{\prime}}={\mathbb{F}}_{q}$ and $e_{f}(\infty)=\operatorname{ord}_{\infty}f^{*}(\tfrac{1}{t})=-\operatorname{ord}_{\infty}(a)$ is the ramification index of $f$ at $\infty$. Since $\operatorname{Spec}A=f^{-1}(\operatorname{Spec}{\mathbb{F}}_{q}[t])$ we conclude that $A$ is a finite (locally) free ${\mathbb{F}}_{q}[t]$-module of rank $-[{\mathbb{F}}_{\infty}:{\mathbb{F}}_{q}]\operatorname{ord}_{\infty}(a)$. ∎

2 $A$-Motives

We keep the notation introduced in the introduction and generalize Definition 1.1 to not necessarily effective $A$-motives.

Definition 2.1**.**

An $A$-motive of rank $r$ over an $A$-ring $R$ is a pair ${\underline{M\!}\,}=(M,\tau_{M})$ consisting of a locally free $A_{R}$-module $M$ of rank $r$ and an isomorphism outside the zero locus $\operatorname{V}({\cal{J}})$ of ${\cal{J}}$ between the induced finite locally free sheaves \tau_{M}\colon\sigma^{*}M|_{\operatorname{Spec}A_{R}\smallsetminus\operatorname{V}({\cal{J}})}\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{\scriptstyle\sim}\longrightarrow}}}M|_{\operatorname{Spec}A_{R}\smallsetminus\operatorname{V}({\cal{J}})}.

A morphism $f\colon(M,\tau_{M})\to(N,\tau_{N})$ between $A$-motives is an $A_{R}$-homomorphism $f\colon M\to N$ which satisfies $f\circ\tau_{M}=\tau_{N}\circ\sigma^{*}f$. We write $\operatorname{Hom}_{R}({\underline{M\!}\,},{\underline{N\!}\,})$ for the $A$-module of morphisms between ${\underline{M\!}\,}$ and ${\underline{N\!}\,}$. The elements of $\operatorname{QHom}_{R}({\underline{M\!}\,},{\underline{N\!}\,}):=\operatorname{Hom}_{R}({\underline{M\!}\,},{\underline{N\!}\,})\otimes_{A}Q$ are called quasi-morphisms. We also set $\operatorname{End}_{R}({\underline{M\!}\,}):=\operatorname{Hom}_{R}({\underline{M\!}\,},{\underline{M\!}\,})$ and $\operatorname{QEnd}_{R}({\underline{M\!}\,}):=\operatorname{QHom}_{R}({\underline{M\!}\,},{\underline{M\!}\,})=\operatorname{End}_{R}({\underline{M\!}\,})\otimes_{A}Q$.

To explain the relation between Definitions 1.1 and 2.1 we begin with a

Lemma 2.2**.**

Let $f\colon M\to N$ be a homomorphism between finite locally free $A_{R}$-modules $M$ and $N$ of the same rank, and assume that $\operatorname{coker}f$ is a finitely generated $R$-module, then $f$ is injective and $\operatorname{coker}f$ is a finite locally free $R$-module.

Proof.

To make the proof more transparent, we choose an element $t\in A\smallsetminus{\mathbb{F}}_{q}$. Then $A$ is a finite free ${\mathbb{F}}_{q}[t]$-module by Lemma 1.4, and $M$ and $N$ are finite locally free modules over $R[t]$. Also $t$ acts as an endomorphism of the finite $R$-module $\operatorname{coker}f$. By the Cayley-Hamilton Theorem [Eis95, Theorem 4.3] there is a monic polynomial $g\in R[t]$ which annihilates $\operatorname{coker}f$. This implies on the one hand that

[TABLE]

is exact, and therefore $\operatorname{coker}f$ is an $R$-module of finite presentation, because $R[t]/(g)$ is a finite free $R$-module of rank $\deg_{t}g$. On the other hand it implies that $M[\tfrac{1}{g}]\twoheadrightarrow N[\tfrac{1}{g}]$ is an epimorphism, whence an isomorphism by [GW10, Corollary 8.12], because $M$ and $N$ are finite locally free over $R[t]$ of the same rank. Since $g$ is a non-zero divisor on $R[t]$ and thus also on $M$, the localization map $M\to M[\tfrac{1}{g}]$ is injective, and hence also $f$ is injective.

We obtain the exact sequence $0\to M\to N\to\operatorname{coker}f\to 0$, which yields for every maximal ideal ${\mathfrak{m}}\subset R$ with residue field $k=R/{\mathfrak{m}}$ the exact sequence

[TABLE]

Again the $k[t]$-modules $M\otimes_{R}k$ and $N\otimes_{R}k$ are locally free of the same rank and $(\operatorname{coker}f)\otimes_{R}k$ is a torsion $k[t]$-module, annihilated by $g$. Since $k[t]$ is a PID, this implies that $M\otimes_{R}k\to N\otimes_{R}k$ is injective and so $\operatorname{Tor}_{1}^{R}(k,\operatorname{coker}f)=(0)$. Since $\operatorname{coker}f$ is finitely presented, it is locally free of finite rank by Nakayama’s Lemma; e.g. [Eis95, Exercise 6.2]. ∎

For the next proposition note that ${\cal{J}}$ is an invertible sheaf on $\operatorname{Spec}A_{R}$ as we remarked before Lemma 1.4.

Proposition 2.3**.**

1. (a)

Let $(M,\tau_{M})$ be an $A$-motive. Then there exist integers $e,e^{\prime}\in{\mathbb{Z}}$ such that ${\cal{J}}^{e}\cdot\tau_{M}(\sigma^{*}M)\subset M$ and ${\cal{J}}^{e^{\prime}}\cdot\tau_{M}^{-1}(M)\subset\sigma^{*}M$. For any such $e,e^{\prime}$ the induced $A_{R}$-homomorphism $\tau_{M}\colon{\cal{J}}^{e}\cdot\sigma^{*}M\to M$ is injective, and the quotient $M/\tau_{M}({\cal{J}}^{e}\cdot\sigma^{*}M)$ is a locally free $R$-module of finite rank, which is annihilated by ${\cal{J}}^{e+e^{\prime}}$. 2. (b)

An $A$-motive $(M,\tau_{M})$ is an effective $A$-motive, if and only if $\tau_{M}(\sigma^{*}M)\subset M$. 3. (c)

Let $(M,\tau_{M})$ be an effective $A$-motive over $R$. Then $(M,\tau_{M}|_{\operatorname{Spec}A_{R}\smallsetminus\operatorname{V}({\cal{J}})})$ is an $A$-motive. Moreover, $\tau_{M}\colon\sigma^{*}M\to M$ is injective and $\operatorname{coker}\tau_{M}$ is a finite locally free $R$-module. 4. (d)

Let ${\underline{M\!}\,}=(M,\tau_{M})$ be an effective $A$-motive over a field $k$. Then ${\underline{M\!}\,}$ has dimension $\dim_{k}\operatorname{coker}\tau_{M}$.

Proof.

(a) Working locally on affine subsets of $\operatorname{Spec}A_{R}$ we may assume that ${\cal{J}}$ is generated by a non-zero divisor $h\in{\cal{J}}$. By [EGA, I, Théorème 1.4.1(d1)] we obtain for every generator $m$ of the $A_{R}$-module $\sigma^{*}M$ an integer $n$ such that locally $h^{n}\cdot\tau_{M}(m)\in M$. Taking $e$ as the maximum of the $n$ when $m$ runs through a finite generating system of $\sigma^{*}M$, yields ${\cal{J}}^{e}\cdot\tau_{M}(\sigma^{*}M)\subset M$. The inclusion ${\cal{J}}^{e^{\prime}}\cdot\tau_{M}^{-1}(M)\subset\sigma^{*}M$ is proved analogously.

Let $e$ and $e^{\prime}$ be any integers with $\tau_{M}({\cal{J}}^{e}\cdot\sigma^{*}M)\subset M$ and $\tau_{M}^{-1}({\cal{J}}^{e^{\prime}}\cdot M)\subset\sigma^{*}M$, whence ${\cal{J}}^{e+e^{\prime}}\cdot M\subset\tau_{M}({\cal{J}}^{e}\cdot\sigma^{*}M)$. Then $M/\tau_{M}({\cal{J}}^{e}\cdot\sigma^{*}M)$ is annihilated by ${\cal{J}}^{e+e^{\prime}}$, and hence a finite module over $A_{R}/{\cal{J}}^{e+e^{\prime}}$ and over $R$. Therefore $\tau_{M}\colon{\cal{J}}^{e}\cdot\sigma^{*}M\to M$ is injective, and the quotient $M/\tau_{M}({\cal{J}}^{e}\cdot\sigma^{*}M)$ is a finite locally free $R$-module by Lemma 2.2.

(c) Since ${\cal{J}}^{n}\cdot\operatorname{coker}\tau_{M}=(0)$, the map $\tau_{M}|_{\operatorname{Spec}A_{R}\smallsetminus\operatorname{V}({\cal{J}})}$ is an epimorphism between locally free sheaves of the same rank, and hence an isomorphism by [GW10, Corollary 8.12]. Thus ${\underline{M\!}\,}$ is an $A$-motive and the remaining assertions follow from (a). Also (b) follows directly.

(d) Set $d:=\dim_{k}\operatorname{coker}\tau_{M}$. Since every $h\in{\cal{J}}$ which generates ${\cal{J}}$ locally on $\operatorname{Spec}A_{k}$ is nilpotent on the $k$-vector space $\operatorname{coker}\tau_{M}$, it satisfies $h^{d}=0$ by the Cayley-Hamilton theorem from linear algebra. We conclude that ${\cal{J}}^{d}\cdot\operatorname{coker}\tau_{M}=(0)$ and ${\underline{M\!}\,}$ has dimension $d$. ∎

Proposition 2.4**.**

1. (a)

If $S$ is an $R$-algebra, then ${\underline{M\!}\,}=(M,\tau_{M})\longmapsto{\underline{M\!}\,}\otimes_{R}S:=(M\otimes_{R}S,\tau_{M}\otimes\operatorname{\,id}_{S})$ defines a functor from (effective) $A$-motives of rank $r$ (and dimension $d$) over $R$ to (effective) $A$-motives of rank $r$ (and dimension $d$) over $S$. 2. (b)

Every $A$-motive over $R$ and every morphism $f\in\operatorname{Hom}({\underline{M\!}\,},{\underline{N\!}\,})$ between $A$-motives over $R$ can be defined over a subring $R^{\prime}$ of $R$, which via $\gamma\colon A\to R^{\prime}\subset R$ is a finitely generated $A$-algebra, hence noetherian.

Proof.

(a) This is obvious.

(b) Every $A$-motive ${\underline{M\!}\,}=(M,\tau_{M})$ has a presentation of the form $A_{R}^{\oplus n_{1}}\xrightarrow{\;U\,}A_{R}^{\oplus n_{0}}\xrightarrow{\;\,\rho\;}M\longrightarrow 0$. Since $M$ is locally free over $A_{R}$, there is a section $s$ of the epimorphism $\rho$. It corresponds to an endomorphism $S$ of $A_{R}^{\oplus n_{0}}$ with $SU=0$ such that there is a map $W\colon A_{R}^{\oplus n_{0}}\to A_{R}^{\oplus n_{1}}$ with $S-\operatorname{Id}=UW$. The isomorphism $\tau_{M}$ lifts to diagram

[TABLE]

Likewise $\tau_{M}^{-1}$ lifts to a similar diagram with vertical morphism $T^{\prime}_{0}$ and $T^{\prime}_{1}$. The equations $\tau_{M}\circ\tau_{M}^{-1}=\operatorname{\,id}$ and $\tau_{M}^{-1}\circ\tau_{M}=\operatorname{\,id}$ imply the existence of matrices $V$ and $V^{\prime}$ in $A_{R}^{n_{1}\times n_{0}}|_{\operatorname{Spec}A_{R}\smallsetminus\operatorname{V}({\cal{J}})}$ with $T_{0}\circ T^{\prime}_{0}-\operatorname{Id}=U\circ V$ and $T^{\prime}_{0}\circ T_{0}-\operatorname{Id}=\sigma^{*}U\circ V^{\prime}$. Let $R^{\prime}\subset R$ be the $A$-algebra generated by the finitely many elements of $R$ which occur in the entries of the matrices $U$, $S$, $W$, $T_{0}$, $T_{1}$, $T^{\prime}_{0}$, $T^{\prime}_{1}$, $V$ and $V^{\prime}$. Define $M^{\prime}$ as the $A_{R^{\prime}}$-module which is the cokernel of $U\in A_{R^{\prime}}^{n_{0}\times n_{1}}$, and define $\tau_{M^{\prime}}\colon\sigma^{*}M^{\prime}|_{\operatorname{Spec}A_{R}\smallsetminus\operatorname{V}({\cal{J}})}\to M^{\prime}|_{\operatorname{Spec}A_{R}\smallsetminus\operatorname{V}({\cal{J}})}$ and $\tau_{M^{\prime}}^{-1}\colon M^{\prime}|_{\operatorname{Spec}A_{R}\smallsetminus\operatorname{V}({\cal{J}})}\to\sigma^{*}M^{\prime}|_{\operatorname{Spec}A_{R}\smallsetminus\operatorname{V}({\cal{J}})}$ as the $A_{R^{\prime}}$-homomorphisms given by diagram (2.1) and its analog for $\tau_{M}^{-1}$. Then $M^{\prime}$ is via $S$ a direct summand of $A_{R^{\prime}}^{\oplus n_{0}}$, hence a finite locally free $A_{R^{\prime}}$-module, and $\tau_{M^{\prime}}$ and $\tau_{M^{\prime}}^{-1}$ are inverse to each other. It follows from diagram (2.1) that $M^{\prime}\otimes_{R^{\prime}}R=M$ and $\tau_{M^{\prime}}\otimes\operatorname{\,id}_{R}=\tau_{M}$.

Finally, the assertion for the morphism $f\in\operatorname{Hom}_{R}({\underline{M\!}\,},{\underline{N\!}\,})$ follows from a diagram similar to (2.1) for $f$ instead of $\tau_{M}$. ∎

We end this section with the following observation.

Proposition 2.5**.**

Let ${\underline{M\!}\,}$ and ${\underline{N\!}\,}$ be $A$-motives over $R$ and let $f\in\operatorname{Hom}_{R}({\underline{M\!}\,},{\underline{N\!}\,})$ be a morphism. Then the set $X$ of points $s\in\operatorname{Spec}R$ such that $f\otimes\operatorname{\,id}_{\kappa(s)}=0$ in \operatorname{Hom}_{\kappa(s)}\bigl{(}{\underline{M\!}\,}\otimes_{R}\kappa(s),{\underline{N\!}\,}\otimes_{R}\kappa(s)\bigr{)} is open and closed, but possibly empty. Let $\operatorname{Spec}{\widetilde{R}}\subset\operatorname{Spec}R$ be this set, then $f\otimes\operatorname{\,id}_{{\widetilde{R}}}=0$ in \operatorname{Hom}_{{\widetilde{R}}}\bigl{(}{\underline{M\!}\,}\otimes_{R}{\widetilde{R}},{\underline{N\!}\,}\otimes_{R}{\widetilde{R}}\bigr{)}. In particular if $\operatorname{Spec}R$ is connected and $S\neq(0)$ is an $R$-algebra, then the map $\operatorname{Hom}_{R}({\underline{M\!}\,},{\underline{N\!}\,})\to\operatorname{Hom}_{S}({\underline{M\!}\,}\otimes_{R}S,{\underline{N\!}\,}\otimes_{R}S),\,f\mapsto f\otimes\operatorname{\,id}_{S}$ is injective.

Proof.

We fix an element $t\in A\smallsetminus{\mathbb{F}}_{q}$. Then $A$ is a finite free ${\mathbb{F}}_{q}[t]$-module. By Proposition 2.3 we can find integers $e,e^{\prime}$ with ${\cal{J}}^{e}\cdot\tau_{N}(\sigma^{*}N)\subset N$ and ${\cal{J}}^{e^{\prime}}\cdot\tau_{M}^{-1}(M)\subset\sigma^{*}M$, such that $d:=e+e^{\prime}$ is a power of $q$. We obtain morphisms $(t-\gamma(t))^{e}\tau_{N}\colon\sigma^{*}N\to N$ and $(t-\gamma(t))^{e^{\prime}}\tau_{M}^{-1}\colon M\to\sigma^{*}M$. So the equation $f\circ\tau_{M}=\tau_{N}\circ\sigma^{*}f$ implies $(t^{d}-\gamma(t)^{d})f=(t-\gamma(t))^{e}\tau_{N}\circ\sigma^{*}f\circ(t-\gamma(t))^{e^{\prime}}\tau_{M}^{-1}$. We view $M$ and $N$ as modules over $R[t]$ and replace $A_{R}$ by $R[t]$. Since $M$ and $N$ are finite projective $R[t]$-modules there are split epimorphisms $R[t]^{\oplus n^{\prime}}\twoheadrightarrow M$ and $R[t]^{\oplus n}\twoheadrightarrow N$. Then $R[t]^{\oplus n^{\prime}}\twoheadrightarrow M\xrightarrow{\;f\,}N\hookrightarrow R[t]^{\oplus n}$ is given by a matrix $F\in R[t]^{n\times n^{\prime}}$ whose entries are polynomials in $t$. Let $I\subset R$ be the ideal generated by the coefficients of all these polynomials. A prime ideal ${\mathfrak{p}}\subset R$ belongs to the set $X$ if and only if $I\subset{\mathfrak{p}}$. In particular $X=\operatorname{V}(I)\subset\operatorname{Spec}R$ is closed.

On the other hand, we consider the map $R[t]^{\oplus n}\twoheadrightarrow\sigma^{*}N\xrightarrow{\,(t-\gamma(t))^{e}\tau_{N}}N\hookrightarrow R[t]^{\oplus n}$ as a matrix $T\in R[t]^{n\times n}$ and the map $R[t]^{\oplus n^{\prime}}\twoheadrightarrow M\xrightarrow{\,(t-\gamma(t))^{e^{\prime}}\tau_{M}^{-1}}\sigma^{*}M\hookrightarrow R[t]^{\oplus n^{\prime}}$ as a matrix $V\in R[t]^{n^{\prime}\times n^{\prime}}$. The formula $(t^{d}-\gamma(t)^{d})f=(t-\gamma(t))^{e}\tau_{N}\circ\sigma^{*}f\circ(t-\gamma(t))^{e^{\prime}}\tau_{M}^{-1}$ implies $(t^{d}-\gamma(t)^{d})F=T\,\sigma(F)\,V$, and it follows that the entries of the matrix $(t^{d}-\gamma(t)^{d})F$ are polynomials in $t$ whose coefficients lie in $I^{q}$. If $\sum_{i=0}^{\ell}b_{i}t^{i}$ is an entry of $F$ then $(t^{d}-\gamma(t)^{d})\sum_{i=0}^{\ell}b_{i}t^{i}=\sum_{i=0}^{\ell+d}(b_{i-d}-\gamma(t)^{d}b_{i})t^{i}$ is the corresponding entry of $(t^{d}-\gamma(t)^{d})F$ and all $b_{i-d}-\gamma(t)^{d}b_{i}\in I^{q}$. By descending induction on $i=\ell+d,\ldots,0$ we see that all $b_{i}\in I^{q}$. It follows that the finitely generated ideal $I\subset R$ satisfies $I=I^{q}$. By Nakayama’s lemma [Eis95, Corollary 4.7] there is an element $b\in 1+I$ such that $b\!\cdot\!I=(0)$. Now let ${\mathfrak{p}}\subset R$ be a prime ideal which lies in $X$, that is $I\subset{\mathfrak{p}}$. Then ${\mathfrak{p}}$ lies in the open subset $\operatorname{Spec}R[\tfrac{1}{b}]\subset\operatorname{Spec}R$ on which $F=0$ and hence $f=0$. In particular $X\subset\operatorname{Spec}R[\tfrac{1}{b}]\subset X$. Therefore $X$ is open and closed and $f=0$ on $X$.

Now let $\operatorname{Spec}R$ be connected and $S\neq(0)$ be an $R$-algebra. Let $f\in\operatorname{Hom}_{R}({\underline{M\!}\,},{\underline{N\!}\,})$ be such that $f\otimes\operatorname{\,id}_{S}=0$ in $\operatorname{Hom}_{S}({\underline{M\!}\,}\otimes_{R}S,{\underline{N\!}\,}\otimes_{R}S)$. Let $s\in\operatorname{Spec}S$ be a point and let $s^{\prime}\in\operatorname{Spec}R$ be its image. Then $f\otimes\operatorname{\,id}_{\kappa(s^{\prime})}=0$ and the set $X$ from above is non-empty. Since it is open and closed and $\operatorname{Spec}R$ is connected, it follows that $X=\operatorname{Spec}R$ and $f=0$. This proves the injectivity. ∎

Corollary 2.6**.**

Let ${\underline{M\!}\,}$ and ${\underline{N\!}\,}$ be $A$-motives over $R$ with $\operatorname{Spec}R$ connected. Then $\operatorname{Hom}_{R}({\underline{M\!}\,},{\underline{N\!}\,})$ is a finite projective $A$-module of rank less or equal to $(\operatorname{rk}{\underline{M\!}\,})\!\cdot\!(\operatorname{rk}{\underline{N\!}\,})$.

Proof.

If $R=k$ is a field and ${\underline{M\!}\,}$ and ${\underline{N\!}\,}$ are effective, the result is due to Anderson [And86, Corollary 1.7.2]. For general $R$ we apply Proposition 2.5 with $S=R/{\mathfrak{m}}$ for ${\mathfrak{m}}\subset R$ a maximal ideal, and use that over the Dedekind ring $A$ every submodule of a finite projective module is itself finite projective. ∎

3 Abelian Anderson $A$-modules

We recall Definition 1.2 of abelian Anderson $A$-modules from the introduction. Let us give some explanations. All group schemes in this article are assumed to be commutative.

Definition 3.1**.**

Let ${\cal{O}}$ be a commutative unitary ring. An ${\cal{O}}$-module scheme over $R$ is a commutative group scheme $E$ over $R$ together with a ring homomorphism ${\cal{O}}\to\operatorname{End}_{R}(E)$.

For a group scheme $E$ over $\operatorname{Spec}R$ we let $E^{n}:=E\times_{R}\ldots\times_{R}E$ be the $n$-fold fiber product over $R$. We denote by $e\colon\operatorname{Spec}R\to E$ its zero section and by $\operatorname{Lie}E:=\operatorname{Hom}_{R}(e^{*}\Omega^{1}_{E/R},R)$ the tangent space of $E$ along $e$. If $E$ is smooth over $\operatorname{Spec}R$, then $\operatorname{Lie}E$ is a locally free $R$-module of rank equal to the relative dimension of $E$ over $R$. In particular $\operatorname{Lie}E^{n}=(\operatorname{Lie}E)^{\oplus n}$. For a homomorphism $f\colon E\to E^{\prime}$ of group schemes over $\operatorname{Spec}R$ we denote by $\operatorname{Lie}f\colon\operatorname{Lie}E\to\operatorname{Lie}E^{\prime}$ the induced morphism of $R$-modules. Also we define the kernel of $f$ as the $R$-group scheme $\ker f:=E\underset{f,\,E^{\prime},e^{\prime}}{\times}\operatorname{Spec}R$ where $e^{\prime}\colon\operatorname{Spec}R\to E^{\prime}$ is the zero section. There is a canonical isomorphism

[TABLE]

given on $T$-valued points $P,Q\in E(T)$ for any $R$-scheme $T$ by $(P,Q)\mapsto(P,Q-P)$. If $P\in E(k)$ for a field $k$ and $P^{\prime}=f(P)\in E^{\prime}(k)$, pulling back (3.1) under $P\colon\operatorname{Spec}k\to E$ yields an isomorphism of the fiber $\operatorname{Spec}k\underset{P^{\prime},\,E^{\prime},f}{\times}E$ of $f$ over $P^{\prime}$ with $\operatorname{Spec}k\times_{R}\ker f$.

On ${\mathbb{G}}_{a,R}=\operatorname{Spec}R[x]$ the elements $b\in R$, and in particular $\gamma(a)\in R$ for $a\in{\mathbb{F}}_{q}$, act via $b^{*}\colon R[x]\to R[x],\,x\mapsto bx$. This makes ${\mathbb{G}}_{a,R}$ into an ${\mathbb{F}}_{q}$-module scheme. In addition let $\tau:=\operatorname{Frob}_{q,{\mathbb{G}}_{a,R}}$ be the relative $q$-Frobenius endomorphism of ${\mathbb{G}}_{a,R}=\operatorname{Spec}R[x]$ given by $x\mapsto x^{q}$. It satisfies $\operatorname{Lie}\tau=0$ and $\tau\circ b=b^{q}\circ\tau$. We let

[TABLE]

be the non-commutative polynomial ring in $\tau$ over $R$. For an element $f=\sum_{i}b_{i}\tau^{i}\in R\{\tau\}$ we set $f(x):=\sum_{i}b_{i}x^{q^{i}}$.

Lemma 3.2**.**

There is an isomorphism of $R$-modules R\{\tau\}^{d^{\prime}\times d}\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{\scriptstyle\sim}\longrightarrow}}}\operatorname{Hom}_{R\text{\rm-groups},{\mathbb{F}}_{q}\text{\rm-lin}}({\mathbb{G}}_{a,R}^{d},{\mathbb{G}}_{a,R}^{d^{\prime}}), which sends the matrix $F=(f_{ij})_{i,j}$ to the ${\mathbb{F}}_{q}$-equivariant morphism $f\colon{\mathbb{G}}_{a,R}^{d}\to{\mathbb{G}}_{a,R}^{d^{\prime}}$ of group schemes over $R$ with $f^{*}(y_{i})=\sum_{j}f_{ij}(x_{j})$ where ${\mathbb{G}}_{a,R}^{d}=\operatorname{Spec}R[x_{1},\ldots,x_{d}]$ and ${\mathbb{G}}_{a,R}^{d^{\prime}}=\operatorname{Spec}R[y_{1},\ldots,y_{d^{\prime}}]$. Under this isomorphism the map $f\mapsto\operatorname{Lie}f$ is given by the map $R\{\tau\}^{d^{\prime}\times d}\to R^{d^{\prime}\times d},\,F=\sum_{n}F_{n}\tau^{n}\mapsto F_{0}$.

Proof.

This is straight forward to prove using Lucas’s theorem [Luc78] on congruences of binomial coefficients which states that $\left(\begin{smallmatrix}pn+t\\ pm+s\end{smallmatrix}\right)\equiv\left(\begin{smallmatrix}n\\ m\end{smallmatrix}\right)\left(\begin{smallmatrix}t\\ s\end{smallmatrix}\right){\rm\;mod\;}p$ for all $n,m,t,s\in{\mathbb{N}}_{0}$, and implies that $\left(\begin{smallmatrix}n\\ i\end{smallmatrix}\right)\equiv 0{\rm\;mod\;}p$ for all $0<i<n$ if and only if $n=p^{e}$ for an $e\in{\mathbb{N}}_{0}$. ∎

Remark 3.3**.**

The affine group scheme $E$ and its multiplication map $\Delta\colon E\times_{R}E\to E$ are described by its coordinate ring $B_{E}:=\Gamma(E,{\cal{O}}_{E})$ together with the comultiplication $\Delta^{*}\colon B_{E}\to B_{E}\otimes_{R}B_{E}$. If we write ${\mathbb{G}}_{a,R}=\operatorname{Spec}R[\xi]$ the map

[TABLE]

is an isomorphism of $A_{R}$-modules. Choosing an element $\lambda\in{\mathbb{F}}_{q}$ with ${\mathbb{F}}_{q}={\mathbb{F}}_{p}(\lambda)$ we obtain an exact sequence of $R$-modules

[TABLE]

This shows that for every flat $R$-algebra $R^{\prime}$ we have $M({\underline{E\!}\,})\otimes_{R}R^{\prime}=M({\underline{E\!}\,}\times_{R}\operatorname{Spec}R^{\prime})$, because $\Gamma(E\times_{R}R^{\prime},{\cal{O}}_{E\times R^{\prime}})=B_{E}\otimes_{R}R^{\prime}$. In particular, if $R^{\prime}$ satisfies condition (a) of Definition 1.2 then $M({\underline{E\!}\,})\otimes_{R}R^{\prime}\cong R^{\prime}\{\tau\}^{1\times d}$ by Lemma 3.2.

From this we see that for any $R$-algebra $S$ the tensor product of the sequence (3.3) with $S$ stays exact and $M({\underline{E\!}\,})\otimes_{R}S=M({\underline{E\!}\,}\times_{\operatorname{Spec}R}\operatorname{Spec}S)$. Namely, we choose a faithfully flat morphism $R\to R^{\prime}$ as in Definition 1.2(a) and tensor (3.3) with $S\otimes_{R}R^{\prime}$. This tensor product stays exact by Lemma 3.2 because $M({\underline{E\!}\,})\otimes_{R}R^{\prime}\cong R^{\prime}\{\tau\}^{1\times d}$. Since $S\to S\otimes_{R}R^{\prime}$ is faithfully flat, already the tensor product of (3.3) with $S$ was exact.

Definition 3.4**.**

If ${\underline{E\!}\,}$ is an abelian Anderson $A$-module we consider in addition on $M({\underline{E\!}\,})$ the map $\tau\colon m\mapsto\operatorname{Frob}_{q,{\mathbb{G}}_{a,R}}\!\circ\,m$. Since $\tau(bm)=b^{q}\tau(m)$ the map $\tau$ is $\sigma$-semilinear and induces an $A_{R}$-linear map $\tau_{M}\colon\sigma^{*}M\to M$. We set {\underline{M\!}\,}({\underline{E\!}\,}):=\bigl{(}M({\underline{E\!}\,}),\tau_{M}) and call it the (effective) $A$-motive associated with ${\underline{E\!}\,}$.

This definition is justified by the following relative version of Anderson’s theorem [And86, Theorem 1].

Theorem 3.5**.**

If ${\underline{E\!}\,}=(E,\varphi)$ is an abelian Anderson $A$-module of rank $r$ and dimension $d$ then ${\underline{M\!}\,}({\underline{E\!}\,})=(M,\tau_{M})$ is an effective $A$-motive of rank $r$ and dimension $d$. There is a canonical isomorphism of $R$-modules

[TABLE]

The contravariant functor ${\underline{E\!}\,}\mapsto{\underline{M\!}\,}({\underline{E\!}\,})$ is fully faithful. Its essential image consists of all effective $A$-motives ${\underline{M\!}\,}=(M,\tau_{M})$ over $R$ of some dimension $d$, for which there exists a faithfully flat ring homomorphism $R\to R^{\prime}$ such that $M\otimes_{R}R^{\prime}$ is a finite free left $R^{\prime}\{\tau\}$-module under the map $\tau\colon M\to M,\,m\mapsto\tau_{M}(\sigma_{M}^{*}m)$.

Proof.

We first establish the isomorphism (3.4). If $m=\tau_{M}(\sum_{i}m_{i}\otimes b_{i})=\sum_{i}b_{i}\circ\operatorname{Frob}_{q,{\mathbb{G}}_{a,R}}\!\circ\,m_{i}$ with $m_{i}\in M$ and $b_{i}\in R$, then $\operatorname{Lie}m=0$ because $\operatorname{Lie}\operatorname{Frob}_{q,{\mathbb{G}}_{a,R}}=0$. So the map (3.4) is well defined. To prove that it is an isomorphism one can apply a faithfully flat base change $R\to R^{\prime}$, see [EGA, § 0${}_{\text{I}}$.6.6], such that $E\otimes_{R}R^{\prime}\cong{\mathbb{G}}_{a,R^{\prime}}^{d}$ and $\operatorname{Lie}E\otimes_{R}R^{\prime}\cong(R^{\prime})^{\oplus d}$. Then $M\otimes_{R}R^{\prime}\cong R^{\prime}\{\tau\}^{1\times d}$ by Remark 3.3, and the inverse map is given by the natural inclusion $(R^{\prime})^{1\times d}\subset R^{\prime}\{\tau\}^{1\times d},F_{0}\mapsto F_{0}\tau^{0}$.

As a consequence, $\operatorname{coker}\tau_{M}$ is a locally free $R$-module of rank equal to $d=\dim{\underline{E\!}\,}$ and annihilated by ${\cal{J}}^{d}$ because of condition (b) in Definition 1.2. This implies $\operatorname{coker}\tau_{M}|_{\operatorname{Spec}A_{R}\smallsetminus\operatorname{V}({\cal{J}})}=(0)$, and therefore the morphism $\tau_{M}\colon\sigma^{*}M|_{\operatorname{Spec}A_{R}\smallsetminus\operatorname{V}({\cal{J}})}\to M|_{\operatorname{Spec}A_{R}\smallsetminus\operatorname{V}({\cal{J}})}$ is surjective. By [GW10, Corollary 8.12] it is an isomorphism, because $M$ and $\sigma^{*}M$ are finite locally free over $A_{R}$ of the same rank. Thus ${\underline{M\!}\,}({\underline{E\!}\,})$ is an $A$-motive and even an effective $A$-motive of dimension $d$ by Proposition 2.3.

Let ${\underline{E\!}\,}=(E,\varphi)$ and ${\underline{E\!}\,}^{\prime}=(E^{\prime},\varphi^{\prime})$ be two abelian Anderson $A$-modules over $R$ and let ${\underline{M\!}\,}={\underline{M\!}\,}({\underline{E\!}\,})$ and ${\underline{M\!}\,}^{\prime}={\underline{M\!}\,}({\underline{E\!}\,}^{\prime})$ be the associated effective $A$-motives. To prove that the map

[TABLE]

is bijective, we again apply a faithfully flat base change $R\to R^{\prime}$, such that $E\otimes_{R}R^{\prime}\cong{\mathbb{G}}_{a,R^{\prime}}^{d}$ and $E^{\prime}\otimes_{R}R^{\prime}\cong{\mathbb{G}}_{a,R^{\prime}}^{d^{\prime}}$. Then \operatorname{Hom}_{R^{\prime}}({\underline{E\!}\,}\otimes_{R}R^{\prime},{\underline{E\!}\,}^{\prime}\otimes_{R}R^{\prime})\cong\bigl{\{}F\in R^{\prime}\{\tau\}^{d^{\prime}\times d}\colon\varphi^{\prime}_{a}\circ F=F\circ\varphi_{a}\;\forall\,a\in A\bigr{\}} by Lemma 3.2. Also ${\underline{M\!}\,}({\underline{E\!}\,})\otimes_{R}R^{\prime}\cong R^{\prime}\{\tau\}^{1\times d}$ and ${\underline{M\!}\,}({\underline{E\!}\,}^{\prime})\otimes_{R}R^{\prime}\cong R^{\prime}\{\tau\}^{1\times d^{\prime}}$. The condition $h\circ\tau_{M^{\prime}}=\tau_{M}\circ\sigma^{*}h$ on an element h\in\operatorname{Hom}_{R^{\prime}}\bigl{(}{\underline{M\!}\,}({\underline{E\!}\,}^{\prime})\otimes_{R}R^{\prime},{\underline{M\!}\,}({\underline{E\!}\,})\otimes_{R}R^{\prime}\bigr{)} implies that $h\colon R^{\prime}\{\tau\}^{1\times d^{\prime}}\to R^{\prime}\{\tau\}^{1\times d}$ is a homomorphism of left $R^{\prime}\{\tau\}$-modules, hence given by multiplication on the right by a matrix $H\in R^{\prime}\{\tau\}^{d^{\prime}\times d}$. Then m^{\prime}\circ\varphi^{\prime}_{a}\circ H=h\bigl{(}(a\otimes 1)\cdot m^{\prime})=(a\otimes 1)\cdot h(m^{\prime})=m^{\prime}\circ H\circ\varphi_{a} implies $\varphi^{\prime}_{a}\circ H=H\circ\varphi_{a}$ for all $a\in A$. It follows that the map (3.5) is bijective over $R^{\prime}$. So every h\in\operatorname{Hom}_{R}\bigl{(}{\underline{M\!}\,}({\underline{E\!}\,}^{\prime}),{\underline{M\!}\,}({\underline{E\!}\,})\bigr{)} gives rise to a morphism $f^{\prime}\in\operatorname{Hom}_{R^{\prime}}({\underline{E\!}\,}\otimes_{R}R^{\prime},{\underline{E\!}\,}^{\prime}\otimes_{R}R^{\prime})$ which carries a descent datum because $h$ was defined over $R$. Since by [BLR90, § 6.1, Theorem 6(a)] the descent of morphisms relative to the faithfully flat morphism $R\to R^{\prime}$ is effective, $f^{\prime}$ descends to the desired $f\in\operatorname{Hom}_{R}({\underline{E\!}\,},{\underline{E\!}\,}^{\prime})$. This shows that the functor ${\underline{E\!}\,}\mapsto{\underline{M\!}\,}({\underline{E\!}\,})$ is fully faithful.

Let ${\underline{M\!}\,}=(M,\tau_{M})$ be an effective $A$-motive of dimension $d$ over $R$ for which there exists a faithfully flat ring homomorphism $R\to R^{\prime}$ such that $M\otimes_{R}R^{\prime}\cong R^{\prime}\{\tau\}^{1\times d}$. Observe that $\operatorname{coker}(\tau_{M}\otimes\operatorname{\,id}_{R^{\prime}})\cong(R^{\prime}\{\tau\}/R^{\prime}\{\tau\}\tau)^{1\times d}=(R^{\prime})^{1\times d}$. For all $a\in A$ we have $\tau\cdot(a\otimes 1)m=\sigma(a\otimes 1)\cdot\tau(m)=(a\otimes 1)\tau m$. Therefore the map $m\mapsto(a\otimes 1)m$ is a homomorphism of left $R^{\prime}\{\tau\}$-modules, and hence given by $(a\otimes 1)m=m\cdot\varphi^{\prime}_{a}$ for a matrix $\varphi^{\prime}_{a}\in R^{\prime}\{\tau\}^{d\times d}$. Then ${\underline{E\!}\,}^{\prime}:=(E^{\prime}={\mathbb{G}}_{a,R^{\prime}}^{d},\varphi^{\prime}\colon A\to R^{\prime}\{\tau\}^{d\times d},a\mapsto\varphi^{\prime}_{a})$ satisfies ${\underline{M\!}\,}({\underline{E\!}\,}^{\prime})={\underline{M\!}\,}\otimes_{R}R^{\prime}$. Again \bigl{(}a\otimes 1-1\otimes\gamma(a)\bigr{)}^{d}=0 on $\operatorname{coker}\tau_{M}$ implies \bigl{(}\operatorname{Lie}\varphi^{\prime}_{a}-\gamma(a)\bigr{)}^{d}=0 on $\operatorname{Lie}E^{\prime}$. So ${\underline{E\!}\,}^{\prime}$ is an abelian Anderson $A$-module over $R^{\prime}$ with ${\underline{M\!}\,}(E^{\prime})\cong{\underline{M\!}\,}\otimes_{R}R^{\prime}$. Consider the ring $R^{\prime\prime}:=R^{\prime}\otimes_{R}R^{\prime}$ and the two maps $p_{1},p_{2}\colon R^{\prime}\to R^{\prime\prime}$ given by $p_{1}(b^{\prime})=b^{\prime}\otimes 1$ and $p_{2}(b^{\prime})=1\otimes b^{\prime}$. The canonical isomorphism $p_{1}^{*}({\underline{M\!}\,}\otimes_{R}R^{\prime})=p_{2}^{*}({\underline{M\!}\,}\otimes_{R}R^{\prime})$ induces an isomorphism $p_{1}^{*}{\underline{E\!}\,}^{\prime}\cong p_{2}^{*}{\underline{E\!}\,}^{\prime}$ which is a descend datum on ${\underline{E\!}\,}^{\prime}$ relative to $R\to R^{\prime}$. Since faithfully flat descend on affine schemes is effective by [BLR90, § 6.1, Theorem 6(b)] there exists a group scheme $E$ over $R$ with a ring homomorphism $\varphi\colon A\to\operatorname{End}_{R\text{-groups}}(E)$ such that $(E,\varphi)\otimes_{R}R^{\prime}\cong{\underline{E\!}\,}^{\prime}$. By [EGA, IV2, Proposition 2.7.1 and IV4, Corollaire 17.7.3] the group scheme $E$ is affine and smooth over $R$ and hence $(E,\varphi)$ is an abelian Anderson $A$-module with ${\underline{M\!}\,}(E,\varphi)\cong{\underline{M\!}\,}$. ∎

The theorem implies the following

Corollary 3.6**.**

The assertions of Proposition 2.5 and Corollary 2.6 also hold for abelian Anderson $A$-modules. ∎

An important class of examples are Drinfeld modules. We recall their definition from [Dri74, § 5] and [Saï97, § 1].

Definition 3.7**.**

A Drinfeld $A$-module of rank $r\in{\mathbb{N}}_{>0}$ over $R$ is a pair ${\underline{E\!}\,}=(E,\varphi)$ consisting of a smooth affine group scheme $E$ over $\operatorname{Spec}R$ of relative dimension $1$ and a ring homomorphism $\varphi\colon A\to\operatorname{End}_{R\text{-groups}}(E),\,a\mapsto\varphi_{a}$ satisfying the following conditions:

1. (a)

Zariski-locally on $\operatorname{Spec}R$ there is an isomorphism \alpha\colon E\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{\scriptstyle\sim}\longrightarrow}}}{\mathbb{G}}_{a,R} of ${\mathbb{F}}_{q}$-module schemes such that 2. (b)

the coefficients of $\Phi_{a}:=\alpha\circ\varphi_{a}\circ\alpha^{-1}=\sum\limits_{i\geq 0}b_{i}(a)\tau^{i}\in\operatorname{End}_{R\text{-groups},{\mathbb{F}}_{q}\text{\rm-lin}}({\mathbb{G}}_{a,R})=R\{\tau\}$ satisfy $b_{0}(a)=\gamma(a)$, $b_{r(a)}(a)\in R^{\scriptscriptstyle\times}$ and $b_{i}(a)$ is nilpotent for all $i>r(a):=-r\,[{\mathbb{F}}_{\infty}:{\mathbb{F}}_{q}]\operatorname{ord}_{\infty}(a)$.

If $b_{i}(a)=0$ for all $i>r(a)$ we say that ${\underline{E\!}\,}$ is in standard form.

It is well known that every Drinfeld $A$-module over $R$ can be put in standard form; see [Dri74, § 5] or [Mat96, § 4.2]. This is a consequence of the following lemma of Drinfeld [Dri74, Propositions 5.1 and 5.2] which we will need again below. For the convenience of the reader we recall the proof.

Lemma 3.8**.**

1. (a)

Let $b=\sum_{i=0}^{n}b_{i}\tau^{i}\in R\{\tau\}$ and let $r$ be a positive integer such that $b_{r}\in R^{\scriptscriptstyle\times}$ and $b_{i}$ is nilpotent for all $i>r$. Then there is a unique unit $c=\sum_{i\geq 0}c_{i}\tau^{i}\in R\{\tau\}^{\scriptscriptstyle\times}$ with $c_{0}=1$ and $c_{i}$ nilpotent for $i>0$, such that $c^{-1}bc=\sum_{i=0}^{r}b^{\prime}_{i}\tau^{i}$ with $b^{\prime}_{r}\in R^{\scriptscriptstyle\times}$. 2. (b)

Let $\operatorname{Spec}R$ be connected and let $b=\sum_{i=0}^{m}b_{i}\tau^{i}$ and $c=\sum_{i=0}^{n}c_{i}\tau^{i}\in R\{\tau\}$ with $m,n>0$ and $b_{m},c_{n}\in R^{\scriptscriptstyle\times}$. Let $d\in R\{\tau\}\smallsetminus\{0\}$ satisfy $db=cd$. Then $m=n$ and $d=\sum_{i=0}^{r}d_{i}\tau^{i}$ with $d_{r}\in R^{\scriptscriptstyle\times}$.

Proof.

(a) was also reproved in [Lau96, Lemma 1.1.2] and [Mat96, Proposition 1.4].

(b) We write $d=\sum_{i=0}^{r}d_{i}\tau^{i}$ with $d_{r}\neq 0$. The equation $db=cd$ implies $\sum_{j}(d_{i-j}b_{j}^{\,q^{i-j}}-c_{j}d_{i-j}^{\;q^{j}})=0$ for all $i$, where the sum runs over $j=\max\{0,i-r\},\ldots,\min\{i,\max\{m,n\}\}$. We now distinguish three cases.

If $m>n$ then $i=m+r$ yields $d_{r}b_{m}^{\,q^{r}}=0$, whence $d_{r}=0$ which is a contradiction.

If $m<n$ then $i=n+r$ yields $c_{n}d_{r}^{\,q^{n}}=0$, whence $d_{r}\in{\mathfrak{p}}$ for every prime ideal ${\mathfrak{p}}\subset R$. For $n+r>i\geq n$ we obtain $c_{n}d_{i-n}^{\,q^{n}}=\sum\limits_{0\leq j<n}(d_{i-j}b_{j}^{\,q^{i-j}}-c_{j}d_{i-j}^{\;q^{j}})$ and by descending induction on $i$ it follows that $d_{i-n}\in{\mathfrak{p}}$ for every prime ideal ${\mathfrak{p}}\subset R$ for all $i-n=r,\ldots,0$. So the ideal $I:=(d_{i}\colon 0\leq i\leq r)\subset R$ is contained in every prime ideal ${\mathfrak{p}}\subset R$. Now $i=m+r$ yields $d_{r}b_{m}^{\,q^{r}}=\sum\limits_{j=m}^{m+r}c_{j}d_{m+r-j}^{\;q^{j}}$, whence $d_{r}\in I^{q}$. For $m+r>i\geq m$ we obtain $d_{i-m}b_{m}^{\,q^{i-m}}=\sum\limits_{0\leq j<m}d_{i-j}b_{j}^{\,q^{i-j}}-\sum\limits_{0\leq j\leq n}c_{j}d_{i-j}^{\;q^{j}}$ and by descending induction on $i$ it follows that $d_{i-m}\in I^{q}$ for all $i-m=r,\ldots,0$. Therefore the finitely generated ideal $I$ satisfies $I=I^{q}$ and by Nakayama’s lemma [Eis95, Corollary 4.7] there is an element $f\in 1+I$ such that $f\!\cdot\!I=(0)$. Since $I\subset{\mathfrak{p}}$ for all prime ideals ${\mathfrak{p}}\subset R$, the element $1-f$ is a unit in $R$ and $I=0$. Therefore $d_{i}=0$ for all $i$ which is a contradiction.

If $m=n$ then $c_{m}d_{r}^{q^{m}}=d_{r}b_{m}^{q^{r}}$ and we consider the ideal $I=(d_{r})\subset R$. Again $I=I^{q^{m}}$ and by [Eis95, Corollary 4.7] there is an element $f\in 1+I$ such that $f\!\cdot\!d_{r}=0$. Now assume that $d_{r}\in{\mathfrak{p}}$ for some prime ideal ${\mathfrak{p}}\subset R$. Then $f\notin{\mathfrak{p}}$, whence ${\mathfrak{p}}\in\operatorname{Spec}R[\tfrac{1}{f}]\subset\operatorname{Spec}R$ and $d_{r}=0$ on the open neighborhood $\operatorname{Spec}R[\tfrac{1}{f}]$ of ${\mathfrak{p}}$. Since the set of prime ideals ${\mathfrak{p}}\subset R$ with $d_{r}\in{\mathfrak{p}}$ is closed in $\operatorname{Spec}R$ and the latter is connected, it follows that $d_{r}=0$ on all of $\operatorname{Spec}R$. This is a contradiction and so our assumption was false. In particular $d_{r}$ is not contained in any prime ideal and so $d_{r}\in R^{\scriptscriptstyle\times}$ as desired. ∎

Theorem 3.9**.**

The abelian Anderson $A$-modules of dimension $1$ and rank $r$ over $R$ are precisely the Drinfeld $A$-modules of rank $r$ over $R$.

Proof.

Let ${\underline{E\!}\,}$ be a Drinfeld $A$-module of rank $r$ over $R$. Choose a Zariski covering as in Definition 3.7(a) such that ${\underline{E\!}\,}$ is in standard form. Since $\operatorname{Spec}R$ is quasi-compact this Zariski covering can be refined to a covering by finitely many affines. Their disjoint union is of the form $\operatorname{Spec}R^{\prime}$ and the ring homomorphism $R\to R^{\prime}$ is faithfully flat. So ${\underline{E\!}\,}$ satisfies conditions (a) and (b) of Definition 1.2. Choose an element $t\in A\smallsetminus{\mathbb{F}}_{q}$. Then $A$ is a finite free ${\mathbb{F}}_{q}[t]$-module of rank equal to $-[{\mathbb{F}}_{\infty}:{\mathbb{F}}_{q}]\operatorname{ord}_{\infty}(t)$ by Lemma 1.4. Writing $\Phi_{t}=\sum_{i=0}^{r(t)}b_{i}(t)\tau^{i}$ with $r(t)=-r\,[{\mathbb{F}}_{\infty}:{\mathbb{F}}_{q}]\operatorname{ord}_{\infty}(t)$ and $b_{r(t)}(t)\in(R^{\prime})^{\scriptscriptstyle\times}$, we make the following

[TABLE]

By Remark 3.3 and Lemma 3.2 we have $M({\underline{E\!}\,})\otimes_{R}R^{\prime}=M({\underline{E\!}\,}\times_{\operatorname{Spec}R}\operatorname{Spec}R^{\prime})=R^{\prime}\{\tau\}$. We prove by induction on $n$ that for every $c=\sum_{i=0}^{n}c_{i}\tau^{i}\in R^{\prime}\{\tau\}=M({\underline{E\!}\,})$ there are uniquely determined elements $f_{\ell}(t)\in R^{\prime}[t]$ such that $c=\sum_{\ell=0}^{r(t)-1}f_{\ell}(t)\!\cdot\!\tau^{\ell}$. If $n<r(t)$ then we take $f_{\ell}(t)=c_{\ell}$. If $n\geq r(t)$, dividing $c$ by $\Phi_{t}$ on the right produces uniquely determined $g=\sum_{i=0}^{n-r(t)}g_{i}\tau^{i}$ and $h=\sum_{\ell=0}^{r(t)-1}h_{\ell}\tau^{\ell}\in R^{\prime}\{\tau\}$ with $c=g\Phi_{t}+h$. Namely, starting with $g_{i}=0$ for $i>n-r(t)$ we can and must take g_{i}=b_{r(t)}^{-q^{i}}\bigl{(}c_{i+r(t)}-\sum\limits_{j=i+1}^{i+r(t)}g_{j}\,b_{i+r(t)-j}^{q^{j}}\bigr{)} for $i=n-r(t),\ldots,0$ and $h_{\ell}=c_{\ell}-\sum\limits_{j=0}^{\ell}g_{j}\,b_{\ell-j}^{q^{j}}$ for $\ell=r(t)-1,\ldots,1$. The induction hypothesis implies $g=\sum\limits_{\ell=0}^{r(t)-1}\tilde{f}_{\ell}(t)\!\cdot\!\tau^{\ell}$. Now $f_{\ell}(t):=\tilde{f}_{\ell}(t)\!\cdot\!t+h_{\ell}$ satisfies $c=\sum_{\ell=0}^{r(t)-1}f_{\ell}(t)\!\cdot\!\tau^{\ell}$. This proves the claim.

By faithfully flat descent [EGA, IV2, Proposition 2.5.2] with respect to $R[t]\to R^{\prime}[t]$ and by the claim, $M({\underline{E\!}\,})$ is finite, locally free over $R[t]$ and in particular flat over $R$. We next show that it is finitely presented over $A_{R}$. Namely, let $(m_{i})_{i\in I}$ be a finite generating system of $M({\underline{E\!}\,})$ over $R[t]$. Using it as a generating system over $A_{R}$ we obtain an epimorphism $\rho\colon A_{R}^{I}\twoheadrightarrow M({\underline{E\!}\,})$. Since $A_{R}$ is a finite free $R[t]$-module, also $A_{R}^{I}$ is a finite free $R[t]$-module and so the kernel of $\rho$ is a finitely generated $R[t]$-module, whence a finitely generated $A_{R}$-module. This shows that $M({\underline{E\!}\,})$ is a finitely presented $A_{R}$-module. From [EGA, IV3, Théorème 11.3.10] it follows that $M({\underline{E\!}\,})$ is finite locally free over $A_{R}$, because for every point $s\in\operatorname{Spec}R$ the finite $A_{\kappa(s)}$-module $M({\underline{E\!}\,})\otimes_{R}\kappa(s)$ is a free $\kappa(s)[t]$-module and hence a torsion free and flat $A_{\kappa(s)}$-module. Its rank is $r$ as can be computed by comparing the ranks of $A_{R^{\prime}}$ and $M({\underline{E\!}\,})\otimes_{R}R^{\prime}$ over $R^{\prime}[t]$. This proves that ${\underline{E\!}\,}$ is an abelian Anderson $A$-module of dimension $1$ and rank $r$ over $R$.

Conversely let ${\underline{E\!}\,}=(E,\varphi)$ be an abelian Anderson $A$-module of dimension $1$ and rank $r$ over $R$. Let $R\to R^{\prime}$ be a faithfully flat ring homomorphism and let \alpha\colon E\times_{R}\operatorname{Spec}R^{\prime}\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{\scriptstyle\sim}\longrightarrow}}}{\mathbb{G}}_{a,R^{\prime}} be an isomorphism of ${\mathbb{F}}_{q}$-module schemes as in Definition 1.2(a). For $a\in A$ write

[TABLE]

where $n(a)\in{\mathbb{N}}_{0}$ and $b_{i}(a)\in R^{\prime}$. For $a\in{\mathbb{F}}_{q}$ we obtain $\Phi_{a}=\gamma(a)\!\cdot\!\tau^{0}$. For $t:=a\in A\smallsetminus{\mathbb{F}}_{q}$ we consider $A$ as a finite free ${\mathbb{F}}_{q}[t]$-module of rank $-[{\mathbb{F}}_{\infty}:{\mathbb{F}}_{q}]\operatorname{ord}_{\infty}(a)$ by Lemma 1.4. Then $M({\underline{E\!}\,})$ is a finite locally free $R[t]$-module of rank $r(a):=-r\,[{\mathbb{F}}_{\infty}:{\mathbb{F}}_{q}]\operatorname{ord}_{\infty}(a)$ by condition (c) of Definition 1.2. Let ${\mathfrak{p}}\subset R^{\prime}$ be a prime ideal, set $k=\operatorname{Frac}(R^{\prime}/{\mathfrak{p}})$, and consider the abelian Anderson $A$-module ${\underline{E\!}\,}\times_{R}\operatorname{Spec}k$ over $k$ and the free $k[t]$-module $M({\underline{E\!}\,})\otimes_{R}k=M({\underline{E\!}\,}\times_{R}\operatorname{Spec}k)$ of rank $r(a)$. By an argument similarly to our claim (3.6) we see that \deg_{\tau}\bigl{(}\Phi_{a}\otimes_{R^{\prime}}1_{k}\bigr{)}=r(a), that is $b_{r(a)}(a)\otimes 1_{k}\in k^{\scriptscriptstyle\times}$ and $b_{i}(a)\otimes 1_{k}=0$ for all $i>r(a)$. This implies that $b_{r(a)}(a)\in(R^{\prime})^{\scriptscriptstyle\times}$ and $b_{i}(a)$ is nilpotent for all $i>r(a)$ by [Eis95, Corollary 2.12]. By Lemma 3.8(a) we may change the isomorphism $\alpha$ such that $\Phi_{a}=\sum_{i=0}^{r(a)}b_{i}(a)\tau^{i}$ with $b_{r(a)}(a)\in(R^{\prime})^{\scriptscriptstyle\times}$ for one $a\in A$, and by Lemma 3.8(b) this then holds for all $a\in A$, because $\Phi_{a}\Phi_{b}=\Phi_{ab}=\Phi_{b}\Phi_{a}$. By condition (b) of Definition 1.2 we have $b_{0}(a)=\gamma(a)$. Thus ${\underline{E\!}\,}\times_{R}\operatorname{Spec}R^{\prime}$ is a Drinfeld $A$-module of rank $r$ over $R^{\prime}$ in standard form.

It remains to show that we can replace the faithfully flat covering $\operatorname{Spec}R^{\prime}\to\operatorname{Spec}R$ by a Zariski covering. For this purpose consider $R^{\prime\prime}:=R^{\prime}\otimes_{R}R^{\prime}$ and the two projections $pr_{i}\colon\operatorname{Spec}R^{\prime\prime}\to\operatorname{Spec}R^{\prime}$ onto the $i$-th factor for $i=1,2$. Then $h:=\sum_{i\geq 0}h_{i}\tau^{i}:=pr_{2}^{*}\alpha\circ pr_{1}^{*}\alpha^{-1}\in R^{\prime\prime}\{\tau\}^{\scriptscriptstyle\times}$ satisfies $h_{0}\in(R^{\prime\prime})^{\scriptscriptstyle\times}$ and $h_{i}$ is nilpotent for all $i>0$; see [Mat96, Proposition 1.4]. By Lemma 3.8(b) the equation $pr_{2}^{*}\Phi_{a}\circ h=h\circ pr_{1}^{*}\Phi_{a}$ implies that $h_{i}=0$ for all $i>0$ and $h=h_{0}\in(R^{\prime\prime})^{\scriptscriptstyle\times}\subset R^{\prime\prime}\{\tau\}^{\scriptscriptstyle\times}$. The cocycle ${\underline{h}}:=(\operatorname{Spec}R^{\prime}\to\operatorname{Spec}R,\,h)$ defines an element in the Čech cohomology group $\operatorname{\check{H}}^{1}_{\it fpqc}(\operatorname{Spec}R,{\mathbb{G}}_{m})$. By Hilbert 90, see [Mil80, Proposition III.4.9] we have $\operatorname{\check{H}}^{1}_{\it fpqc}(\operatorname{Spec}R,{\mathbb{G}}_{m})=\operatorname{\check{H}}^{1}_{Zar}(\operatorname{Spec}R,{\mathbb{G}}_{m})$. This means that there is a Zariski covering $\operatorname{Spec}{\widetilde{R}}\to\operatorname{Spec}R$, where $\operatorname{Spec}{\widetilde{R}}=\coprod_{i}\operatorname{Spec}{\widetilde{R}}_{i}$ is a disjoint union of open affine subschemes $\operatorname{Spec}{\widetilde{R}}_{i}\subset\operatorname{Spec}R$, and a unit $\tilde{h}=(\tilde{h}_{ij})_{i,j}\in({\widetilde{R}}\otimes_{R}{\widetilde{R}})^{\scriptscriptstyle\times}=\prod_{i,j}({\widetilde{R}}_{i}\otimes_{R}{\widetilde{R}}_{j})^{\scriptscriptstyle\times}$, such that $(\operatorname{Spec}{\widetilde{R}}\to\operatorname{Spec}R,\,\tilde{h})={\underline{h}}$. Let ${\widetilde{E}}$ be the smooth affine group and ${\mathbb{F}}_{q}$-module scheme over $\operatorname{Spec}R$ with \beta_{i}\colon{\widetilde{E}}|_{\operatorname{Spec}{\widetilde{R}}_{i}}\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{\scriptstyle\sim}\longrightarrow}}}{\mathbb{G}}_{a,{\widetilde{R}}_{i}} and $\beta_{j}=\tilde{h}_{ij}\circ\beta_{i}$ on $\operatorname{Spec}{\widetilde{R}}_{i}\otimes_{R}{\widetilde{R}}_{j}$. Then over $\operatorname{Spec}R^{\prime}\otimes_{R}{\widetilde{R}}=\coprod_{i}\operatorname{Spec}R^{\prime}\otimes_{R}{\widetilde{R}}_{i}$ we have an isomorphism \tilde{\alpha}:=(\beta_{i}^{-1}\circ\alpha)_{i}\colon E\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{\scriptstyle\sim}\longrightarrow}}}{\widetilde{E}}. Let $p_{i}\colon\operatorname{Spec}(R^{\prime}\otimes_{R}{\widetilde{R}})\otimes_{R}(R^{\prime}\otimes_{R}{\widetilde{R}})\to\operatorname{Spec}R^{\prime}\otimes_{R}{\widetilde{R}}$ be the projection onto the $i$-th factor for $i=1,2$. Then $p_{2}^{*}\tilde{\alpha}\circ p_{1}^{*}\tilde{\alpha}^{-1}=(\tilde{h}_{ij}^{-1}h)_{i,j}=1$. This shows that $\tilde{\alpha}$ descends to an isomorphism \tilde{\alpha}\colon E\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{\scriptstyle\sim}\longrightarrow}}}{\widetilde{E}} over $\operatorname{Spec}R$ by [BLR90, § 6.1, Theorem 6(a)]. On $\operatorname{Spec}{\widetilde{R}}_{i}$, now \beta_{i}\circ\tilde{\alpha}\colon E\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{\scriptstyle\sim}\longrightarrow}}}{\mathbb{G}}_{a,{\widetilde{R}}_{i}} is an isomorphism of ${\mathbb{F}}_{q}$-module schemes. Moreover ${\widetilde{\Phi}}_{a}:=\beta_{i}\tilde{\alpha}\circ\varphi_{a}\circ\tilde{\alpha}^{-1}\beta_{i}^{-1}\in{\widetilde{R}}_{i}\{\tau\}$ satisfies ${\widetilde{\Phi}}_{a}\otimes 1_{R^{\prime}}=\Phi_{a}\otimes 1_{{\widetilde{R}}_{i}}$ in $(R^{\prime}\otimes_{R}{\widetilde{R}}_{i})\{\tau\}\supset{\widetilde{R}}_{i}\{\tau\}$ and by what we proved for $\Phi_{a}$ above, this implies that ${\underline{E\!}\,}$ is a Drinfeld $A$-module of rank $r$ over $R$ which by ${\widetilde{R}}$ and $(\beta_{i}\circ\tilde{\alpha})_{i}$ is put in standard form. ∎

4 Review of the finite shtuka equivalence

In preparation for our main results in Sections 5 and 6 we need to recall Drinfeld’s functor [Dri87, § 2] and the equivalence it defines between finite ${\mathbb{F}}_{q}$-shtukas and finite locally free strict ${\mathbb{F}}_{q}$-module schemes; see also [Abr06], [Tag95, § 1], [Lau96, § B.3] and [HS15, §§ 3-5].

Definition 4.1**.**

A finite ${\mathbb{F}}_{q}$-shtuka over $R$ is a pair ${\underline{V\!}\,}=(V,F_{V})$ consisting of a finite locally free $R$-module $V$ on $R$ and an $R$-module homomorphism $F_{V}\colon\sigma^{\ast}V\to V$. A morphism $f\colon(V,F_{V})\to(V^{\prime},F_{V^{\prime}})$ of finite ${\mathbb{F}}_{q}$-shtukas is an $R$-module homomorphism $f\colon V\to V^{\prime}$ satisfying $f\circ F_{V}=F_{V^{\prime}}\circ\sigma^{*}f$.

We say that $F_{V}$ is nilpotent if there is an integer $n$ such that $F_{V}^{n}:=F_{V}\circ\sigma^{*}F_{V}\circ\ldots\circ\sigma^{(n-1)*}F_{V}=0$. A finite ${\mathbb{F}}_{q}$-shtuka over $R$ is called étale if $F_{V}$ is an isomorphism. If ${\underline{V\!}\,}=(V,F_{V})$ is étale, we define for any $R$-algebra $R^{\prime}$ the $\tau$-invariants of ${\underline{V\!}\,}$ over $R^{\prime}$ as the ${\mathbb{F}}_{q}$-vector space

[TABLE]

Recall that an $R$-group scheme $G=\operatorname{Spec}B$ is finite locally free if $B$ is a finite locally free $R$-module. By [EGA, Inew, Proposition 6.2.10] this is equivalent to $G$ being finite, flat and of finite presentation over $\operatorname{Spec}R$. Every finite locally free $R$-group scheme $G=\operatorname{Spec}B$ is a relative complete intersection by [SGA 3, III.4.15]. This means that locally on $\operatorname{Spec}R$ we can choose a presentation $B=R[X_{1},...,X_{n}]/I$ where the ideal $I$ is generated by a regular sequence; compare [EGA, IV4, Proposition 19.3.7]. The zero section $e\colon\operatorname{Spec}R\to G$ defines an augmentation $e_{B}:=e^{*}\colon B\twoheadrightarrow R$ of the $R$-algebra $B$. Set $I_{B}:=\ker e_{B}$. For the polynomial ring $R[{\underline{X\!}\,}]=R[X_{1},\ldots,X_{n}]$ set $I_{R[{\underline{X\!}\,}]}=(X_{1},\ldots,X_{n})$ and $e_{R[{\underline{X\!}\,}]}\colon R[{\underline{X\!}\,}]\twoheadrightarrow R,\,X_{\nu}\mapsto 0$. Faltings [Fal02] and Abrashkin [Abr06] consider the deformation $B^{\flat}:=R[{\underline{X\!}\,}]/(I\!\cdot\!I_{R[{\underline{X\!}\,}]})$ and the canonical epimorphism $B^{\flat}\twoheadrightarrow B$. They remark that there is a unique morphism

[TABLE]

lifting the comultiplication $\Delta\colon B\to B\otimes_{R}B$ and satisfying $(\operatorname{\,id}_{B^{\flat}}\otimes e_{B}^{\flat})\circ\Delta^{\flat}=\operatorname{\,id}_{B^{\flat}}=(e_{B}^{\flat}\otimes\operatorname{\,id}_{B^{\flat}})\circ\Delta^{\flat}$, where $e_{B}^{\flat}\colon B^{\flat}\twoheadrightarrow R$ is the augmentation map; see [Abr06, § 1.2] or [HS15, Remark after Definition 3.5]. It satisfies $\Delta^{\flat}(x)-x\otimes 1-1\otimes x\,\in\,I_{B^{\flat}}\otimes I_{B^{\flat}}$ for all $x\in I_{B^{\flat}}$. Set ${\cal{G}}=(G,G^{\flat}):=(\operatorname{Spec}B,\operatorname{Spec}B^{\flat})$. The co-Lie complex of ${\cal{G}}$ over $\operatorname{Spec}$R (that is, the fiber at the zero section of $G$ of the cotangent complex; see [Ill72, § VII.3.1]) is the complex of finite locally free $R$-modules of rank $n$

[TABLE]

concentrated in degrees $-1$ and [math] with $d$ being the differential map. Note that $(I/{I^{2}})\otimes_{B,\,e_{B}}R=\ker(B^{\flat}\twoheadrightarrow B)$ and $\Omega^{1}_{R[{\underline{X\!}\,}]/R}\otimes_{R[{\underline{X\!}\,}],\,e_{R[{\underline{X\!}\,}]}}R=\ker(e_{B}^{\flat})/\ker(e_{B}^{\flat})^{2}$ can be computed from $(B,B^{\flat})$. Up to homotopy equivalence it only depends on $G$ and not on the presentation $B=R[{\underline{X\!}\,}]/I$. The co-Lie module of $G$ over $R$ is defined as $\omega_{G}:=\operatorname{H}^{0}(\ell^{\,{}^{\bullet}}_{{\cal{G}}/\operatorname{Spec}R}):=\operatorname{coker}d$. We can now recall the definition of strict ${\mathbb{F}}_{q}$-module schemes from Faltings [Fal02] and Abrashkin [Abr06]; see also [HS15, § 4].

Definition 4.2**.**

Let $(G,[\,.\,])$ be a pair, where $G=\operatorname{Spec}B$ is an affine flat commutative group scheme over $R$ which is a relative complete intersection and where $[\,.\,]\colon{\mathbb{F}}_{q}\to\operatorname{End}_{R\text{-groups}}(G),\,a\mapsto[a]$ is a ring homomorphism. Then $(G,[\,.\,])$ is called a strict ${\mathbb{F}}_{q}$-module scheme if there exists a presentation $B=R[{\underline{X\!}\,}]/I$ and a lift $[\,.\,]^{\flat}\colon{\mathbb{F}}_{q}\to\operatorname{End}_{R\text{-algebras}}(B^{\flat}),\,a\mapsto[a]^{\flat}$ of the ${\mathbb{F}}_{q}$-action on $G$, such that the induced action on $\ell^{\,{}^{\bullet}}_{{\cal{G}}/\operatorname{Spec}R}$ is equal to the scalar multiplication via $\gamma\colon{\mathbb{F}}_{q}\to R$, and such that $[1]^{\flat}=\operatorname{\,id}_{B^{\flat}}$ and $[0]^{\flat}=e_{B}^{\flat}$, as well as $[a\tilde{a}]^{\flat}=[a]^{\flat}\circ[\tilde{a}]^{\flat}$ and $[a+\tilde{a}]^{\flat}=m\circ([a]^{\flat}\otimes[\tilde{a}]^{\flat})\circ\Delta^{\flat}$, where $m\colon(B\otimes_{R}B)^{\flat}\to B^{\flat}$ is induced by the multiplication map $B^{\flat}\otimes_{R}B^{\flat}\to B^{\flat}$ in the ring $B^{\flat}$ and the homomorphism $[a]^{\flat}\otimes[\tilde{a}]^{\flat}\colon B^{\flat}\otimes_{R}B^{\flat}\to B^{\flat}\otimes_{R}B^{\flat}$ induces a homomorphism $(B\otimes_{R}B)^{\flat}\to(B\otimes_{R}B)^{\flat}$ denoted again by $[a]^{\flat}\otimes[\tilde{a}]^{\flat}$. If $G$ is finite locally free, such a lift $a\mapsto[a]^{\flat}$ then exists for every presentation and is uniquely determined by [HS15, Lemmas 4.4 and 4.7].

Example 4.3**.**

The group scheme ${\mathbb{G}}_{a,R}^{d}$ is a strict ${\mathbb{F}}_{q}$-module scheme for any $d$, because we can choose $B=R[X_{1},\ldots,X_{d}]$ and so $I=(0)$ and $B^{\flat}=B$, and $a\in{\mathbb{F}}_{q}$ acts as $[a]^{*}X_{i}=a\cdot X_{i}$. Moreover, every ${\mathbb{F}}_{q}$-linear group homomorphism ${\mathbb{G}}_{a,R}^{d}\to{\mathbb{G}}_{a,R}^{d^{\prime}}$ is strict in the sense of [Fal02, Definition 1], meaning that the homomorphism lifts to a homomorphism between the $B^{\flat}$ which is equivariant for the ${\mathbb{F}}_{q}$-action via $[\,.\,]^{\flat}$.

Lemma 4.4**.**

Let $G$ be a finite locally free group scheme over $R$, let ${\mathbb{F}}_{q}\to\operatorname{End}_{R\text{\rm-groups}}(G)$ be a ring homomorphism, and let $R\to R^{\prime}$ be a faithfully flat ring homomorphism. Then $G$ is a strict ${\mathbb{F}}_{q}$-module scheme if and only if $G\times_{R}R^{\prime}$ is.

Proof.

Let $pr\colon\operatorname{Spec}R^{\prime}\to\operatorname{Spec}R$ be the induced morphism and let $pr_{i}\colon\operatorname{Spec}R^{\prime}\otimes_{R}R^{\prime}\to\operatorname{Spec}R^{\prime}$ be the projection onto the $i$-th factor. Let $G=\operatorname{Spec}B$, let $R^{\prime}[{\underline{X\!}\,}]\twoheadrightarrow B\otimes_{R}R^{\prime}$ be a presentation, and let {\mathbb{F}}_{q}\to\operatorname{End}_{R\text{-algebras}}\bigl{(}(B\otimes_{R}R^{\prime})^{\flat}\bigr{)}, $a\mapsto[a]^{\flat}$ be a lift of the ${\mathbb{F}}_{q}$-action on $G$ as in Definition 4.2, which makes $G\times_{R}R^{\prime}$ into a strict ${\mathbb{F}}_{q}$-module scheme over $R^{\prime}$. Moreover, let $f\colon R[{\underline{Y\!}\,}]\twoheadrightarrow B$ be an arbitrary presentation and let {\widetilde{{\cal{G}}}}=\bigl{(}\operatorname{Spec}B,\,\operatorname{Spec}R[{\underline{Y\!}\,}]/({\underline{Y\!}\,})\!\cdot\!\ker(f)\bigr{)} be the corresponding deformation. By [HS15, Lemmas 4.4 and 4.7] there exists a unique lift $a\mapsto{\widetilde{[a]}}{}^{\flat}$ on the deformation ${\widetilde{{\cal{G}}}}\times_{R}R^{\prime}=pr^{*}{\widetilde{{\cal{G}}}}$. By the uniqueness the two lifts $pr_{1}^{*}{\widetilde{[a]}}{}^{\flat}$ and $pr_{2}^{*}{\widetilde{[a]}}{}^{\flat}$ on the deformation $pr_{1}^{*}\,pr^{*}{\widetilde{{\cal{G}}}}=pr_{2}^{*}\,pr^{*}{\widetilde{{\cal{G}}}}$ coincide. By faithfully flat descent [BLR90, § 6.1, Theorem 6] this lift descends to a lift on the deformation ${\widetilde{{\cal{G}}}}$, which makes $G$ into a strict ${\mathbb{F}}_{q}$-module scheme over $R$. ∎

To explain the equivalence between finite ${\mathbb{F}}_{q}$-shtukas and finite locally free strict ${\mathbb{F}}_{q}$-module schemes over $R$ we recall Drinfeld’s functor.

Definition 4.5**.**

Let ${\underline{V\!}\,}=(V,F_{V})$ be a pair consisting of a (not necessarily finite locally free) $R$-module $V$ and a morphism $F_{V}\colon\sigma^{*}V\to V$ of $R$-modules. Following Drinfeld [Dri87, § 2] we define

[TABLE]

where the ideal $I$ is generated by the elements $v^{\otimes q}-F_{V}(\sigma_{V}^{*}v)$ for all $v\in V$. (Here $v^{\otimes q}$ lives in $\operatorname{Sym}^{q}V$ and $F_{V}(\sigma_{V}^{*}v)$ in $\operatorname{Sym}^{1}V$.) Then $\operatorname{Dr}_{q}({\underline{V\!}\,})$ is a group scheme over $R$ via the comultiplication $\Delta\colon v\mapsto v\otimes 1+1\otimes v$ and an ${\mathbb{F}}_{q}$-module scheme via $[a]\colon v\mapsto av$ for $a\in{\mathbb{F}}_{q}$. It has a canonical deformation

[TABLE]

where $I_{0}=\bigoplus_{n\geq 1}\,\operatorname{Sym}^{n}_{R}V$ is the ideal generated by the $v\in V$. This deformation is equipped with the comultiplication $\Delta^{\flat}\colon v\mapsto v\otimes 1+1\otimes v$ and the ${\mathbb{F}}_{q}$-action $[a]^{\flat}\colon v\mapsto av$. We set $\operatorname{{\cal{D}}{\it r}}_{q}({\underline{V\!}\,}):=(\operatorname{Dr}_{q}({\underline{V\!}\,}),\operatorname{Dr}_{q}({\underline{V\!}\,})^{\flat})$. On its co-Lie complex $[a]$ acts by scalar multiplication with $a$ because $(av)^{\otimes q}-F_{V}(\sigma_{V}^{*}(av))=a^{q}(v^{\otimes q}-F_{V}(\sigma_{V}^{*}v))$. Therefore $\operatorname{Dr}_{q}({\underline{V\!}\,})$ is a finite locally free strict ${\mathbb{F}}_{q}$-module scheme if $V$ is a finite locally free $R$-module. Every morphism $(V,F_{V})\to(W,F_{W})$, that is, every $R$-homomorphism $f\colon V\to W$ with $f\circ F_{V}=F_{W}\circ\sigma^{*}f$, induces a morphism $\operatorname{Dr}_{q}(f)\colon\operatorname{Dr}_{q}(W,F_{W})\to\operatorname{Dr}_{q}(V,F_{V})$. So $\operatorname{Dr}_{q}$ is a contravariant functor. If $f$ is surjective then $\operatorname{Dr}_{q}(f)$ is a closed immersion.

Conversely, with a (not necessarily finite locally free) ${\mathbb{F}}_{q}$-module scheme $G$ over $R$ we associate the pair {\underline{M\!}\,}_{q}(G):=\bigl{(}M_{q}(G),F_{M_{q}(G)}\bigr{)} consisting of the $R$-module

[TABLE]

and the $R$-homomorphism $F_{M_{q}(G)}\colon\sigma^{\ast}M_{q}(G)\to M_{q}(G)$ which is induced from $M_{q}(G)\to M_{q}(G)$, $m\mapsto\operatorname{Frob}_{q,{\mathbb{G}}_{a,R}}\!\circ\,m$. Every morphism of ${\mathbb{F}}_{q}$-module schemes $f\colon G\to G^{\prime}$ induces an $R$-homomorphism ${\underline{M\!}\,}_{q}(G^{\prime})\to{\underline{M\!}\,}_{q}(G),\,m^{\prime}\mapsto m^{\prime}\circ f$. Note that by an argument as in Remark 3.3 we have ${\underline{M\!}\,}_{q}(G)\otimes_{R}S={\underline{M\!}\,}_{q}(G\times_{\operatorname{Spec}R}\operatorname{Spec}S)$ for every $R$-algebra $S$.

There is a natural morphism ${\underline{V\!}\,}\to{\underline{M\!}\,}_{q}(\operatorname{Dr}_{q}({\underline{V\!}\,})),\,v\mapsto f_{v}$, where $f_{v}\colon\operatorname{Dr}_{q}({\underline{V\!}\,})\to{\mathbb{G}}_{a,R}=\operatorname{Spec}R[\xi]$ is given by $f_{v}^{*}(\xi)=v$. There is also a natural morphism of group schemes $G\to\operatorname{Dr}_{q}({\underline{M\!}\,}_{q}(G))$ given by $\bigoplus\limits_{n\geq 0}\operatorname{Sym}^{n}_{R}M_{q}(G)/I\to\Gamma(G,{\cal{O}}_{G}),\,m\mapsto m^{*}(\xi)$, which is well defined because $F_{M_{q}(G)}(\sigma^{\ast}m)^{*}(\xi)=(\operatorname{Frob}_{q,{\mathbb{G}}_{a,R}}\!\circ\,m)^{*}(\xi)=m^{*}(\xi^{q})=(m^{*}(\xi))^{q}$.

Example 4.6**.**

For example if ${\underline{E\!}\,}=(E,\varphi)$ is an abelian Anderson $A$-module of dimension $d$, then ${\underline{M\!}\,}_{q}({\underline{E\!}\,})=(M_{q}({\underline{E\!}\,}),F_{M_{q}({\underline{E\!}\,})})$ was denoted ${\underline{M\!}\,}({\underline{E\!}\,})=(M({\underline{E\!}\,}),\tau_{M({\underline{E\!}\,})})$ in Definition 1.2. There is a canonical isomorphism {\underline{E\!}\,}\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{\scriptstyle\sim}\longrightarrow}}}\operatorname{Dr}_{q}({\underline{M\!}\,}_{q}({\underline{E\!}\,})) which is constructed as follows. We set ${\mathbb{G}}_{a,R}=\operatorname{Spec}R[\xi]$ and consider for each $m\in M_{q}({\underline{E\!}\,})=\operatorname{Hom}_{R\text{-groups},{\mathbb{F}}_{q}\text{\rm-lin}}(E,{\mathbb{G}}_{a,R})$ the element $m^{*}(\xi)\in\Gamma(E,{\cal{O}}_{E})$. We claim that

[TABLE]

is an isomorphism of $R$-algebras. To prove that it is an isomorphism we may apply a faithfully flat base change $R\to R^{\prime}$ over which we have an ${\mathbb{F}}_{q}$-linear isomorphism \alpha\colon E\otimes_{R}R^{\prime}\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{\scriptstyle\sim}\longrightarrow}}}{\mathbb{G}}_{a,R^{\prime}}^{d}=\operatorname{Spec}R^{\prime}[x_{1},\ldots,x_{d}]. Let $m_{i}:=pr_{i}\circ\alpha\in M_{q}({\underline{E\!}\,})\otimes_{R}R^{\prime}$ where $pr_{i}\colon{\mathbb{G}}_{a,R^{\prime}}^{d}\to{\mathbb{G}}_{a,R^{\prime}}$ is the projection onto the $i$-th factor. Then $M_{q}({\underline{E\!}\,})\otimes_{R}R^{\prime}=\bigoplus_{i=0}^{d}R^{\prime}\{\tau\}\!\cdot\!m_{i}$ by Remark 3.3 and the inverse of (4.3) sends $\alpha^{*}(x_{i})$ to $m_{i}$. This is indeed the inverse, because (4.3) sends each of the generators $\tau^{j}m_{i}=\operatorname{Frob}_{q^{j},{\mathbb{G}}_{a,R}}\!\circ m_{i}$ of the $R^{\prime}$-module $M_{q}({\underline{E\!}\,})\otimes_{R}R^{\prime}$ to $(\operatorname{Frob}_{q^{j},{\mathbb{G}}_{a,R}}\!\circ m_{i})^{*}(\xi)=m_{i}^{*}(\xi^{q^{j}})=\alpha^{*}(x_{i})^{q^{j}}$, and this inverse sends it back to $m_{i}^{\otimes q^{j}}=\operatorname{Frob}_{q^{j},{\mathbb{G}}_{a,R}}\!\circ m_{i}=\tau^{j}m_{i}$.

The following theorem goes back to Abrashkin [Abr06, Theorem 2]. Statements (b)–(d) were proved in [HS15, Theorem 5.2].

Theorem 4.7**.**

1. (a)

The contravariant functors $\operatorname{Dr}_{q}$ and ${\underline{M\!}\,}_{q}$ are mutually quasi-inverse anti-equivalences between the category of finite ${\mathbb{F}}_{q}$-shtukas over $R$ and the category of finite locally free strict ${\mathbb{F}}_{q}$-module schemes over $R$. Both functors are ${\mathbb{F}}_{q}$-linear and exact.

Let ${\underline{V\!}\,}=(V,F_{V})$ be a finite ${\mathbb{F}}_{q}$-shtuka over $R$ and let $G=\operatorname{Dr}_{q}({\underline{V\!}\,})$. Then

1. (b)

*the ${\mathbb{F}}_{q}$-module scheme $\operatorname{Dr}_{q}({\underline{V\!}\,})$ is étale over $R$ if and only if ${\underline{V\!}\,}$ is étale. * 2. (c)

the natural morphisms ${\underline{V\!}\,}\to{\underline{M\!}\,}_{q}(\operatorname{Dr}_{q}({\underline{V\!}\,})),\,v\mapsto f_{v}$ and $G\to\operatorname{Dr}_{q}({\underline{M\!}\,}_{q}(G))$ are isomorphisms. 3. (d)

the co-Lie complex $\ell^{\,{}^{\bullet}}_{\operatorname{{\cal{D}}{\it r}}_{q}({\underline{V\!}\,})/S}$ is canonically isomorphic to the complex  $0\to\sigma^{*}V\xrightarrow{\;F_{V}}V\to 0$.

5 Isogenies

Definition 5.1**.**

A morphism $f\in\operatorname{Hom}_{R}({\underline{E\!}\,},{\underline{E\!}\,}^{\prime})$ between two abelian Anderson $A$-modules ${\underline{E\!}\,}$ and ${\underline{E\!}\,}^{\prime}$ over $R$ is an isogeny if $f\colon E\to E^{\prime}$ is finite and surjective. If there exists an isogeny between ${\underline{E\!}\,}$ and ${\underline{E\!}\,}^{\prime}$ then they are called isogenous. (Being isogenous is an equivalence relation; see Corollary 5.16 below.)

An isogeny $f\colon{\underline{E\!}\,}\to{\underline{E\!}\,}^{\prime}$ is separable if $f$ is étale, or equivalently if the group scheme $\ker f$ is étale over $R$. Indeed, since $f$ is flat by Proposition 5.2(b) it suffices to see that all fibers of $f$ over $E^{\prime}$ are étale by [BLR90, § 2.4, Proposition 8]. Now all fibers are isomorphic to $\ker f$ by the remarks after (3.1).

We recall the following well known criterion for being an isogeny. For the convenience of the reader we include a proof.

Proposition 5.2**.**

Let $f\colon E\to E^{\prime}$ be a morphism between two affine, smooth $R$-group schemes $E$ of relative dimension $d$ and $E^{\prime}$ of relative dimension $d^{\prime}$, such that the fibers of $E^{\prime}$ over all points of $\operatorname{Spec}R$ are connected. Then the following are equivalent:

1. (a)

$f$* is finite and faithfully flat, that is flat and surjective; see [EGA, 0${}_{\text{I}}$.6.7.8]**,* 2. (b)

$\ker f$* is finite and $f$ is flat,* 3. (c)

$\ker f$* is finite and $f$ is surjective,* 4. (d)

$\ker f$* is finite and $d=d^{\prime}$,* 5. (e)

$\ker f$* is finite and $f$ is an epimorphism of sheaves for the fpqc-topology.*

If $R=k$ is a field, then these conditions are equivalent to

1. (f)

$f$* is surjective and $d=d^{\prime}$.*

Proof.

We show that (a) implies all other conditions. This is obvious for (b), (c) and (e). To prove that $d=d^{\prime}$ let ${\mathfrak{m}}\subset R$ be a maximal ideal and consider the base change to $k=R/{\mathfrak{m}}$. Then $f\times\operatorname{\,id}_{k}\colon E\times_{R}k\to E^{\prime}\times_{R}k$ is a finite surjective morphism, and hence $d=\dim E\times_{R}k=\dim E^{\prime}\times_{R}k=d^{\prime}$; see [Eis95, Corollary 9.3].

Conversely, clearly (e)$\Longrightarrow$(c). We now show (f)$\Longrightarrow$(c) and (b)$\Longrightarrow$(c)$\Longrightarrow$(d)$\Longrightarrow$(b)$\Longrightarrow$(a). Generally note that by the remarks after (3.1) all non-empty fibers of $f$ are isomorphic to $\ker f$.

First assume (f) and note that when $R=k$ is a field, the ring $\Gamma(E^{\prime},{\cal{O}}_{E^{\prime}})$ is an integral domain by our assumptions on $E^{\prime}$. The surjectivity of $f$ implies that $f^{*}\colon\Gamma(E^{\prime},{\cal{O}}_{E^{\prime}})\hookrightarrow\Gamma(E,{\cal{O}}_{E})$ is injective of relative transcendence degree $d-d^{\prime}=0$. Since all fibers of $f$ are isomorphic to $\ker f$, [Eis95, Corollary 14.6] implies that $\ker f$ is finite over $\operatorname{Spec}k$ and (c) holds.

We next show for general $R$ that (b) implies (c). Namely, $f$ is of finite presentation by [EGA, IV1, Proposition 1.6.2(v)], because $E$ and $E^{\prime}$ are of finite presentation over $R$. Therefore (b) implies that $f$ is universally open by [EGA, IV2, Théorème 2.4.6]. In particular $(f\times\operatorname{\,id}_{k})(E\times_{R}k)\subset E^{\prime}\times_{R}k$ is open for every point $\operatorname{Spec}k\to\operatorname{Spec}R$ of $\operatorname{Spec}R$. Since $E^{\prime}\times_{R}k$ was assumed to be connected, it possesses no proper open subgroup, and hence $f\times\operatorname{\,id}_{k}$ is surjective. This establishes (c).

To prove that (c) implies (d) again consider the morphism $f\times\operatorname{\,id}_{k}\colon E\times_{R}k\to E^{\prime}\times_{R}k$ over a point $\operatorname{Spec}k\to\operatorname{Spec}R$ of $\operatorname{Spec}R$. Since $f\times\operatorname{\,id}_{k}$ is surjective, $f^{*}\otimes\operatorname{\,id}_{k}\colon\Gamma(E^{\prime},{\cal{O}}_{E^{\prime}})\otimes_{R}k\,\hookrightarrow\,\Gamma(E,{\cal{O}}_{E})\otimes_{R}k$ is injective, because otherwise its kernel would define a proper closed subscheme of $E^{\prime}\times_{R}k$ through which $f\times\operatorname{\,id}_{k}$ factors. Since all fibers of $f$ are isomorphic to $\ker f$, and hence finite, [Eis95, Corollary 13.5] shows that $d^{\prime}\,=\,\dim\Gamma(E^{\prime},{\cal{O}}_{E^{\prime}})\otimes_{R}k\,=\,\dim\Gamma(E,{\cal{O}}_{E})\otimes_{R}k\,=\,d$.

We prove the implication (d)$\Longrightarrow$(b). Consider the fiber $f\times\operatorname{\,id}_{k}\colon E\times_{R}k\to E^{\prime}\times_{R}k$ over a point $\operatorname{Spec}k\to\operatorname{Spec}R$ of $\operatorname{Spec}R$ and the inclusion \bigl{(}\Gamma(E^{\prime},{\cal{O}}_{E^{\prime}})\otimes_{R}k\bigr{)}/\ker(f^{*}\otimes\operatorname{\,id}_{k})\,\lhook\joinrel\longrightarrow\,\Gamma(E,{\cal{O}}_{E})\otimes_{R}k. Since all fibers of $f$ are finite, [Eis95, Corollary 13.5] implies \dim\Gamma(E^{\prime},{\cal{O}}_{E^{\prime}})\otimes_{R}k\,=\,d^{\prime}\,=\,d\,=\,\dim\Gamma(E,{\cal{O}}_{E})\otimes_{R}k\,=\,\dim\bigl{(}\Gamma(E^{\prime},{\cal{O}}_{E^{\prime}})\otimes_{R}k\bigr{)}/\ker(f^{*}\otimes\operatorname{\,id}_{k}). It follows that $\ker(f^{*}\otimes\operatorname{\,id}_{k})=(0)$ and $f^{*}\otimes\operatorname{\,id}_{k}\colon\Gamma(E^{\prime},{\cal{O}}_{E^{\prime}})\otimes_{R}k\,\hookrightarrow\,\Gamma(E,{\cal{O}}_{E})\otimes_{R}k$ is injective. Let ${\mathfrak{m}}\subset\Gamma(E,{\cal{O}}_{E})\otimes_{R}k$ be a maximal ideal. Then $(f^{*}\otimes\operatorname{\,id}_{k})^{-1}({\mathfrak{m}})\subset\Gamma(E^{\prime},{\cal{O}}_{E^{\prime}})\otimes_{R}k$ is a maximal ideal by [Eis95, Theorem 4.19]. Since the fiber of $f$ over ${\mathfrak{m}}$ is finite, [Eis95, Theorem 18.16(b)] implies that $f\otimes\operatorname{\,id}_{k}$ is flat at ${\mathfrak{m}}$. Since $E$ and $E^{\prime}$ are smooth over $R$ it follows from [EGA, IV3, Théorème 11.3.10] that $f$ is flat.

Finally we show that (b) and (c) together imply (a). By (b) and (c) the morphism $f\colon E\to E^{\prime}$ is faithfully flat. Whether $f$ is finite can by [EGA, IV2, Proposition 2.7.1] be tested after the faithfully flat base change $E\to E^{\prime}$. By (3.1) the finiteness of the projection $E\times_{E^{\prime}}E\to E$ onto the first factor follows from the finiteness of $\ker f$ over $\operatorname{Spec}R$. This proves (a). ∎

Corollary 5.3**.**

Let $f\in\operatorname{Hom}_{R}({\underline{E\!}\,},{\underline{E\!}\,}^{\prime})$ be an isogeny. Then

1. (a)

the kernel $\ker f$ of $f$ is a finite locally free group scheme and a strict ${\mathbb{F}}_{q}$-module scheme over $R$. 2. (b)

$E^{\prime}$* is the quotient $E/\ker f$.*

Proof.

(a) Since $f$ is flat of finite presentation by [EGA, IV1, Proposition 1.6.2(v)], $\ker f$ is flat of finite presentation over $R$. Since it is also finite, it is finite locally free. Over a faithfully flat $R$-algebra $R^{\prime}$ both $E$ and $E^{\prime}$ become isomorphic to powers of ${\mathbb{G}}_{a,R^{\prime}}$ and hence are strict ${\mathbb{F}}_{q}$-module schemes by Example 4.3. Therefore $(\ker f)\otimes_{R}R^{\prime}$ is a strict ${\mathbb{F}}_{q}$-module scheme over $R^{\prime}$ by [Fal02, Proposition 2] and $\ker f$ is a strict ${\mathbb{F}}_{q}$-module scheme over $R$ by Lemma 4.4.

(b) This follows from [SGA 3, Théorème V.4.1]. ∎

Proposition 5.4**.**

1. (a)

If ${\underline{E\!}\,}$ and ${\underline{E\!}\,}^{\prime}$ are Drinfeld $A$-modules over $R$ with $\operatorname{Spec}R$ connected and $f\in\operatorname{Hom}_{k}({\underline{E\!}\,},{\underline{E\!}\,}^{\prime})$, then $f$ is an isogeny if and only if $f\neq 0$. 2. (b)

If this is the case then $f$ is separable if and only if $\operatorname{Lie}f\in R^{\scriptscriptstyle\times}$.

Proof.

(a) Let $f\colon{\underline{E\!}\,}\to{\underline{E\!}\,}^{\prime}$ be an isogeny, then $f\neq 0$ because the zero morphism is not surjective. Conversely let $f\neq 0$. By Proposition 5.2(d) we must show that $\ker f$ is finite. This question is local on $\operatorname{Spec}R$, so we may assume that $E=E^{\prime}={\mathbb{G}}_{a,R}$ and that ${\underline{E\!}\,}=(E,\varphi)$ and ${\underline{E\!}\,}^{\prime}=(E^{\prime},\psi)$ are in standard form. Let $t\in A\smallsetminus{\mathbb{F}}_{q}$, and hence $\deg_{\tau}\varphi_{t}>0$ and $\deg_{\tau}\psi_{t}>0$. By Lemma 3.8(b) applied to $f\circ\varphi_{t}=\psi_{t}\circ f$ we have $f=\sum_{i=0}^{n}f_{i}\tau^{i}\in R\{\tau\}$ with $f_{n}\in R^{\scriptscriptstyle\times}$. It follows that $\ker f=\operatorname{Spec}R[x]/(\sum_{i=0}^{n}f_{i}x^{q^{i}})$ which is finite over $R$.

(b) By the Jacobi criterion [BLR90, §2.2, Proposition 7], $\ker f=\operatorname{Spec}R[x]/(\sum_{i=0}^{n}f_{i}x^{q^{i}})$ is étale if and only if $\operatorname{Lie}f=f_{0}=\tfrac{\partial f(x)}{\partial x}\in R^{\scriptscriptstyle\times}$. ∎

Next we turn to $A$-motives.

Definition 5.5**.**

A morphism $f\in\operatorname{Hom}_{R}({\underline{M\!}\,},{\underline{N\!}\,})$ between $A$-motives over $R$ is an isogeny if $f$ is injective and $\operatorname{coker}f$ is finite and locally free as $R$-module. If there exists an isogeny between ${\underline{M\!}\,}$ and ${\underline{N\!}\,}$ then they are called isogenous. (Being isogenous is an equivalence relation; see Corollary 5.16 below.) A quasi-morphism $f\in\operatorname{QHom}_{R}({\underline{M\!}\,},{\underline{N\!}\,})$ which is of the form $g\otimes c$ for an isogeny $g\in\operatorname{Hom}_{R}({\underline{M\!}\,},{\underline{N\!}\,})$ and a $c\in Q$ is called a quasi-isogeny.

If $f$ is an isogeny and ${\underline{M\!}\,}$ and ${\underline{N\!}\,}$ are effective, then the snake lemma yields the following commutative diagram with exact rows and columns

[TABLE]

Namely, by local freeness of $\operatorname{coker}f$ the upper row is again exact and identifies $\sigma^{*}(\operatorname{coker}f)$ with $\operatorname{coker}(\sigma^{*}f)$.

An isogeny $f\colon{\underline{M\!}\,}\to{\underline{N\!}\,}$ between effective $A$-motives is separable if $\tau_{\operatorname{coker}f}\colon\sigma^{*}(\operatorname{coker}f)\to\operatorname{coker}f$ is an isomorphism.

Remark 5.6**.**

If $f\in\operatorname{Hom}_{R}({\underline{M\!}\,},{\underline{N\!}\,})$ is an isogeny and $S$ is an $R$-algebra, then the base change $f\otimes\operatorname{\,id}_{S}\in\operatorname{Hom}_{S}({\underline{M\!}\,}\otimes_{R}S,{\underline{N\!}\,}\otimes_{R}S)$ of $f$ to $S$ is again an isogeny. This follows from the exact sequence $0\longrightarrow{\underline{M\!}\,}\xrightarrow{\enspace f\;}{\underline{N\!}\,}\longrightarrow\operatorname{coker}f\longrightarrow 0$ because $\operatorname{coker}f$ is a flat $R$-module.

Example 5.7**.**

For $0\neq a\in A$ the morphism $a\colon{\underline{M\!}\,}\to{\underline{M\!}\,}$ is an isogeny with $\operatorname{coker}a=M/aM$. Let ${\underline{M\!}\,}$ be effective. Then $a$ is separable if and only if $\ker(\tau_{\operatorname{coker}a})=\operatorname{coker}(\tau_{\operatorname{coker}a})=(0)$. That is, if and only if multiplication with $a$ is an automorphism of $\operatorname{coker}\tau_{M}$. Since $a-\gamma(a)$ is nilpotent on $\operatorname{coker}\tau_{M}$ this is the case if and only if $\gamma(a)\in R^{\scriptscriptstyle\times}$. For the corresponding result about abelian Anderson $A$-modules see Corollary 5.11.

Proposition 5.8**.**

Let ${\underline{M\!}\,}$ and ${\underline{N\!}\,}$ be $A$-motives over $R$. If ${\underline{M\!}\,}$ and ${\underline{N\!}\,}$ are isogenous then $\operatorname{rk}{\underline{M\!}\,}=\operatorname{rk}{\underline{N\!}\,}$, and if, moreover, ${\underline{M\!}\,}$ and ${\underline{N\!}\,}$ are effective, then $\operatorname{rk}_{R}\operatorname{coker}\tau_{M}=\operatorname{rk}_{R}\operatorname{coker}\tau_{N}$. Conversely assume $\operatorname{rk}{\underline{M\!}\,}=\operatorname{rk}{\underline{N\!}\,}$ and let $f\in\operatorname{Hom}_{R}({\underline{M\!}\,},{\underline{N\!}\,})$ be a morphism such that $\operatorname{coker}f$ is a finitely generated $R$-module. Then $f$ is an isogeny.

Proof.

Let $f\colon{\underline{M\!}\,}\to{\underline{N\!}\,}$ be an isogeny. Since $M$, respectively $\operatorname{coker}\tau_{M}$, are finite locally free over $A_{R}$, respectively over $R$, we can compute their ranks by choosing a maximal ideal ${\mathfrak{m}}\subset R$ and applying the base change from $R$ to $k=R/{\mathfrak{m}}$. Then $f\otimes\operatorname{\,id}_{k}$ is still an isogeny by Remark 5.6. Since $\operatorname{coker}(f\otimes\operatorname{\,id}_{k})$ is a torsion $A_{k}$-module it follows that

[TABLE]

If ${\underline{M\!}\,}$ and ${\underline{N\!}\,}$ are effective, we consider diagram (5.1) for the isogeny $f\otimes\operatorname{\,id}_{k}$. Since $\operatorname{coker}(f\otimes\operatorname{\,id}_{k})$ and $\sigma^{*}\operatorname{coker}(f\otimes\operatorname{\,id}_{k})$ are finite dimensional $k$-vector spaces of the same dimension, the right vertical column and the bottom row of diagram (5.1) imply that

[TABLE]

The converse follows from Lemma 2.2. ∎

After these preparations we are now able to formulate and prove our main theorem.

Theorem 5.9**.**

Let $f\in\operatorname{Hom}_{R}({\underline{E\!}\,},{\underline{E\!}\,}^{\prime})$ be a morphism between abelian Anderson $A$-modules and let ${\underline{M\!}\,}(f)\in\operatorname{Hom}_{R}({\underline{M\!}\,}^{\prime},{\underline{M\!}\,})$ be the associated morphism between the associated effective $A$-motives ${\underline{M\!}\,}={\underline{M\!}\,}({\underline{E\!}\,})$ and ${\underline{M\!}\,}^{\prime}={\underline{M\!}\,}({\underline{E\!}\,}^{\prime})$. Then

1. (a)

$f$* is an isogeny if and only if ${\underline{M\!}\,}(f)$ is an isogeny.* 2. (b)

$f$* is a separable isogeny if and only if ${\underline{M\!}\,}(f)$ is a separable isogeny.* 3. (c)

If $f$ is an isogeny there are canonical $A$-equivariant isomorphisms of finite ${\mathbb{F}}_{q}$-shtukas

[TABLE]

and of finite locally free $R$-group schemes

[TABLE]

Proof.

In the beginning we do neither assume that $f$ nor that ${\underline{M\!}\,}(f)$ is an isogeny. We denote by $\iota$ the inclusion $\ker f\hookrightarrow E$. Consider the $A_{R}$-homomorphism ${\underline{M\!}\,}({\underline{E\!}\,})\to{\underline{M\!}\,}_{q}(\ker f),\,m\mapsto m\circ\iota$, which is compatible with the Frobenii $\tau_{M({\underline{E\!}\,})}$ and $F_{M_{q}(\ker f)}$. Since $m={\underline{M\!}\,}(f)(m^{\prime})=m^{\prime}\circ f$ implies $m^{\prime}\circ f\circ\iota=0$, it factors over

[TABLE]

On the other hand we claim that there are $A$-equivariant morphisms

[TABLE]

where the last two are closed immersions. The first morphism is obtained from (5.2). Moreover, the epimorphism ${\underline{M\!}\,}({\underline{E\!}\,})\twoheadrightarrow\operatorname{coker}{\underline{M\!}\,}(f)$ induces by Example 4.6 an $A$-equivariant closed immersion \alpha\colon\operatorname{Dr}_{q}(\operatorname{coker}{\underline{M\!}\,}(f))\hookrightarrow\operatorname{Dr}_{q}\bigl{(}{\underline{M\!}\,}({\underline{E\!}\,})\bigr{)}={\underline{E\!}\,}. We compose it with $f\colon E\to E^{\prime}$ and show that the composition factors through the zero section $e^{\prime}\colon\operatorname{Spec}R\to E^{\prime}$. This will imply that $\alpha$ factors through $\ker f$. We can study this composition after a faithfully flat base change $R\to R^{\prime}$ over which we have an ${\mathbb{F}}_{q}$-linear isomorphism $\beta\colon E^{\prime}\otimes_{R}R^{\prime}\cong{\mathbb{G}}_{a,R^{\prime}}^{d^{\prime}}=\operatorname{Spec}R^{\prime}[y_{1},\ldots,y_{d}]$. Let $m^{\prime}_{i}:=pr_{i}\circ\beta\in M({\underline{E\!}\,}^{\prime})\otimes_{R}R^{\prime}$ where $pr_{i}\colon{\mathbb{G}}_{a,R^{\prime}}^{d}\to{\mathbb{G}}_{a,R^{\prime}}=\operatorname{Spec}R[\xi]$ is the projection onto the $i$-th factor. Then $pr_{i}^{*}(\xi)=y_{i}$ and $\alpha^{*}f^{*}\beta^{*}(y_{i})=\alpha^{*}f^{*}m^{\prime}_{i}{}^{*}(\xi)=\alpha^{*}\circ{\underline{M\!}\,}(f)(m^{\prime}_{i})^{*}(\xi)=0$ because ${\underline{M\!}\,}(f)(m^{\prime}_{i})=0$ in $\operatorname{coker}{\underline{M\!}\,}(f)$.

(a) Now assume that $f$ is an isogeny. Then $\ker f$ is a finite locally free group scheme over $R$, and a strict ${\mathbb{F}}_{q}$-module scheme by Corollary 5.3(a). So ${\underline{M\!}\,}_{q}(\ker f)$ is a finite locally free $R$-module by Theorem 4.7 and the morphism \operatorname{Dr}_{q}\bigl{(}{\underline{M\!}\,}_{q}(\ker f)\bigr{)}\to\ker f in (5.3) is an isomorphism. This shows that \operatorname{Dr}_{q}(\operatorname{coker}{\underline{M\!}\,}(f))\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{\scriptstyle\sim}\longrightarrow}}}\ker f. We next show that the map (5.2) is an isomorphism. Its cokernel is a finite $R$-module because ${\underline{M\!}\,}_{q}(\ker f)$ is. We apply again a faithfully flat base change $R\to R^{\prime}$ such that $E\otimes_{R}R^{\prime}\cong{\mathbb{G}}_{a,R^{\prime}}^{d}$ and $E^{\prime}\otimes_{R}R^{\prime}\cong{\mathbb{G}}_{a,R^{\prime}}^{d^{\prime}}$. Then $f$ is given by a matrix $F\in R^{\prime}\{\tau\}^{d^{\prime}\times d}$ by Lemma 3.2. By faithfully flat descent and by Nakayama’s lemma [Eis95, Corollaries 2.9 and 4.8] the map (5.2) will be surjective if for all maximal ideals ${\mathfrak{m}}^{\prime}\subset R^{\prime}$ its tensor product with $k:=R^{\prime}/{\mathfrak{m}}^{\prime}$ is surjective. By Remark 3.3 and its analog for ${\underline{M\!}\,}_{q}(\ker f)$ the tensor product of (5.2) with $k$ equals \operatorname{coker}{\underline{M\!}\,}(f\times\operatorname{\,id}_{k})\to{\underline{M\!}\,}_{q}\bigl{(}\ker(f\times\operatorname{\,id}_{k})\bigr{)}, where $f\times\operatorname{\,id}_{k}\colon{\underline{E\!}\,}\times_{R}k\to{\underline{E\!}\,}^{\prime}\times_{R}k$ is given by the matrix ${\,\overline{\!F}}:=F\otimes 1_{k}$. In particular $\ker(f\times\operatorname{\,id}_{k})=\operatorname{Spec}k[x_{1},\ldots,x_{d}]/(f^{*}(y_{\ell})\colon 1\leq\ell\leq d)$. Since $\ker f$ is finite, $k[x_{1},\ldots,x_{d}]/(f^{*}(y_{\ell})\colon 1\leq\ell\leq d)$ is a finite dimensional $k$-vector space. For fixed $i$ this implies that $\{x_{i},x_{i}^{q},x_{i}^{q^{2}},\ldots\}$ is linearly dependent and there is a positive integer $N$ and $b_{i,n}\in k$ such that $x_{i}^{q^{N+1}}=\sum\limits_{n=0}^{N}b_{i,n}\!\cdot\!x_{i}^{q^{n}}$ in $k[x_{1},\ldots,x_{d}]/(f^{*}(y_{\ell})\colon 1\leq\ell\leq d)$. We introduce the new variables $z_{i,n}:=x_{i}^{q^{n}}$ for $1\leq i\leq d$ and $0\leq n\leq N$. Then $f^{*}(y_{\ell})$ is a $k$-linear relation between the $z_{i,n}$. Furthermore

[TABLE]

Let $\tilde{z}_{1},\ldots,\tilde{z}_{r}$ be a $k$-basis of $(\bigoplus\limits_{i=1}^{d}\bigoplus\limits_{n=0}^{N}k\!\cdot\!z_{i,n})/(f^{*}(y_{\ell})\colon 1\leq\ell\leq d)$. Then there are elements $c_{ij}\in k$ for $1\leq i,j\leq r$ such that

[TABLE]

Moreover, the group law on $\ker f$ is given by the comultiplication $\Delta^{*}\colon B\to B\otimes_{k}B,\,\Delta^{*}({\widetilde{z}}_{i})={\widetilde{z}}_{i}\otimes 1+1\otimes{\widetilde{z}}_{i}$ and the ${\mathbb{F}}_{q}$-action is given by $\varphi_{\lambda}\colon B\to B,\,\varphi_{\lambda}^{*}({\widetilde{z}}_{i})=\gamma(\lambda)\!\cdot\!{\widetilde{z}}_{i}$.

We now are ready to compute {\underline{M\!}\,}_{q}\bigl{(}\ker(f\times\operatorname{\,id}_{k})\bigr{)} from (3.3). If ${\mathbb{G}}_{a,k}=\operatorname{Spec}k[\xi]$ then every element {\widetilde{m}}\in{\underline{M\!}\,}_{q}\bigl{(}\ker(f\times\operatorname{\,id}_{k})\bigr{)} satisfies ${\widetilde{m}}^{*}(\xi)=\sum\limits_{\ell_{i}\in\{0\ldots q-1\}}d_{\ell_{1},\ldots,\ell_{r}}\cdot\tilde{z}_{1}^{\ell_{1}}\cdot\ldots\cdot\tilde{z}_{r}^{\ell_{r}}$ with $d_{\ell_{1},\ldots,\ell_{r}}\in k$. Since the $\tilde{z}_{1}^{\ell_{1}}\cdot\ldots\cdot\tilde{z}_{r}^{\ell_{r}}$ form a $k$-basis of $B$, the conditions $\Delta^{*}{\widetilde{m}}^{*}(\xi)={\widetilde{m}}^{*}(\xi)\otimes 1+1\otimes{\widetilde{m}}^{*}(\xi)$ in $B\otimes_{k}B$ and $\varphi_{\lambda}^{*}{\widetilde{m}}^{*}(\xi)=m^{*}(\gamma(\lambda)\!\cdot\!\xi)=\gamma(\lambda)\!\cdot\!{\widetilde{m}}^{*}(\xi)$ in $B$ for $\lambda\in{\mathbb{F}}_{q}$ imply as in Lemma 3.2 that ${\widetilde{m}}^{*}(\xi)=d_{1,0\ldots 0}\!\cdot\!\tilde{z}_{1}+\ldots+d_{0\ldots 0,1}\!\cdot\!\tilde{z}_{r}$. Since $\tilde{z}_{i}$ is a $k$-linear combination of the $z_{j,n}=x_{j}^{q^{n}}$ the morphism $m\colon E\times_{R}k\to{\mathbb{G}}_{a,k}$ with $m^{*}(\xi)=d_{1,0\ldots 0}\!\cdot\!\tilde{z}_{1}+\ldots+d_{0\ldots 0,1}\!\cdot\!\tilde{z}_{r}$ belongs to ${\underline{M\!}\,}(E\times_{R}k)$ and maps to ${\widetilde{m}}$ under the map \operatorname{coker}{\underline{M\!}\,}(f\times\operatorname{\,id}_{k})\to{\underline{M\!}\,}_{q}\bigl{(}\ker(f\times\operatorname{\,id}_{k})\bigr{)}. This proves that (5.2) is surjective.

In order to show that (5.2) is injective let $m\in M({\underline{E\!}\,})$ be an element with $m\circ\iota=0$. By [SGA 3, Théorème V.4.1] the morphism $m\colon E\to{\mathbb{G}}_{a,R}$ factors through E/\ker f\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{\scriptstyle\sim}\longrightarrow}}}E^{\prime} (use Corollary 5.3(b)) in the form $m=m^{\prime}\circ f$ for an $m^{\prime}\in M({\underline{E\!}\,}^{\prime})$. This shows that $m{\rm\;mod\;}\operatorname{im}{\underline{M\!}\,}(f)=0$ in $\operatorname{coker}{\underline{M\!}\,}(f)$. All together we have proved that \operatorname{coker}{\underline{M\!}\,}(f)\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{\scriptstyle\sim}\longrightarrow}}}{\underline{M\!}\,}_{q}(\ker f) is a finite locally free $R$-module. Moreover, ${\underline{M\!}\,}(f)$ is injective, because if $m^{\prime}\in M({\underline{E\!}\,}^{\prime})$ satisfies $m^{\prime}\circ f={\underline{M\!}\,}(f)(m^{\prime})=0$ the surjectivity of $f$ implies $m^{\prime}=0$. More precisely, $f$ is an epimorphism of sheaves for the fpqc-topology by Proposition 5.2(e). Now the injectivity of ${\underline{M\!}\,}(f)$ follows from the left exactness of the functor $\operatorname{Hom}_{R\text{-groups},{\mathbb{F}}_{q}\text{\rm-lin}}({\,{\scriptscriptstyle\bullet}\,},{\mathbb{G}}_{a,R})$. This proves that ${\underline{M\!}\,}(f)$ is an isogeny, and it also proves (c).

Conversely assume that ${\underline{M\!}\,}(f)$ is an isogeny. Then $d:=\dim{\underline{E\!}\,}=\dim{\underline{E\!}\,}^{\prime}$ by Theorem 3.5 and Proposition 5.8. We prove that $\ker f$ is finite. For this purpose we apply a faithfully flat base change $R\to R^{\prime}$ such that $E\otimes_{R}R^{\prime}\cong{\mathbb{G}}_{a,R^{\prime}}^{d}=\operatorname{Spec}R^{\prime}[x_{1},\ldots,x_{d}]$ and $E^{\prime}\otimes_{R}R^{\prime}\cong{\mathbb{G}}_{a,R^{\prime}}^{d}=\operatorname{Spec}R[y_{1},\ldots,y_{d}]$. Also when we write ${\mathbb{G}}_{a,R^{\prime}}=\operatorname{Spec}R^{\prime}[\xi]$ then ${\underline{M\!}\,}({\underline{E\!}\,}\times_{R}R^{\prime})\cong\bigoplus\limits_{i=1}^{d}R^{\prime}\{\tau\}\!\cdot\!m_{i}$ and ${\underline{M\!}\,}({\underline{E\!}\,}^{\prime}\times_{R}R^{\prime})\cong\bigoplus\limits_{i=1}^{d}R^{\prime}\{\tau\}\!\cdot\!m^{\prime}_{i}$ where $m_{i}^{*}(\xi)=x_{i}$ and $m^{\prime}_{i}{}^{*}(\xi)=y_{i}$. Consider the epimorphism of $R$-modules

[TABLE]

Since $\operatorname{coker}{\underline{M\!}\,}(f\otimes\operatorname{\,id}_{R^{\prime}})$ is finite locally free over $R^{\prime}$, and hence projective, this epimorphism has a section $s$ whose image lies in $\bigoplus\limits_{i=1}^{d}\bigoplus\limits_{n=0}^{N}R^{\prime}\!\cdot\!\tau^{n}m_{i}$ for some $N$. It follows that $\tau^{N+1}m_{i}-s(\delta(\tau^{N+1}m_{i}))$ maps to zero in $\operatorname{coker}{\underline{M\!}\,}(f\otimes\operatorname{\,id}_{R^{\prime}})$. That is, there are elements $b_{i,j,n}\in R^{\prime}$ and ${\widetilde{m}}^{\prime}_{i}\in{\underline{M\!}\,}({\underline{E\!}\,}^{\prime}\times_{R}R^{\prime})$ with $\tau^{N+1}m_{i}\,-\,\sum\limits_{j=1}^{d}\sum\limits_{n=0}^{N}b_{i,j,n}\!\cdot\!\tau^{n}m_{j}\,=\,{\underline{M\!}\,}(f)({\widetilde{m}}^{\prime}_{i})$. Applying this equation to $\xi$ yields

[TABLE]

Thus $f\times\operatorname{\,id}_{R^{\prime}}\colon E\times_{R}R^{\prime}\to E^{\prime}\times_{R}R^{\prime}$ is finite. By faithfully flat descent [EGA, IV2, Proposition 2.7.1] also $f$ is finite. By Proposition 5.2(d) this proves that $f$ is an isogeny and establishes (a).

Finally (b) follows from (c) and Theorem 4.7(b). ∎

Corollary 5.10**.**

If ${\underline{E\!}\,}$ and ${\underline{E\!}\,}^{\prime}$ are isogenous abelian Anderson $A$-modules then $\operatorname{rk}{\underline{E\!}\,}=\operatorname{rk}{\underline{E\!}\,}^{\prime}$.

Proof.

This follows directly from Theorems 3.5, 5.9 and Proposition 5.8. ∎

Corollary 5.11**.**

Let ${\underline{E\!}\,}$ be an abelian Anderson $A$-module over $R$ and let $a\in A$. Then $\varphi_{a}\colon{\underline{E\!}\,}\to{\underline{E\!}\,}$ is an isogeny. It is separable if and only if $\gamma(a)\in R^{\scriptscriptstyle\times}$.

Proof.

The assertion follows from Theorem 5.9 and Example 5.7. The criterion for separability can also be proved without reference to $A$-motives; see our proof of Theorem 6.4(b) below. ∎

We next come to our second main result.

Theorem 5.12**.**

Let ${\underline{M\!}\,}$ and ${\underline{N\!}\,}$ be two $A$-motives over $R$ and let $f\in\operatorname{Hom}_{R}({\underline{M\!}\,},{\underline{N\!}\,})$ be a morphism. Then the following are equivalent:

1. (a)

$f$* is an isogeny,* 2. (b)

there is an element $0\neq a\in A$ such that $f$ induces an isomorphism of $A_{R}[\tfrac{1}{a}]$-modules M[\tfrac{1}{a}]\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{\scriptstyle\sim}\longrightarrow}}}N[\tfrac{1}{a}].

In particular, a quasi-morphism $f\in\operatorname{QHom}_{R}({\underline{M\!}\,},{\underline{N\!}\,})$ is a quasi-isogeny if and only if it induces an isomorphism f\colon M[\tfrac{1}{a}]\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{\scriptstyle\sim}\longrightarrow}}}N[\tfrac{1}{a}] for an element $a\in A\smallsetminus\{0\}$.

Proof.

(b)$\Longrightarrow$(a) Clearly $\operatorname{rk}{\underline{M\!}\,}=\operatorname{rk}{\underline{N\!}\,}$. Since $\operatorname{coker}f$ is a finitely generated $A_{R}$-module, $(\operatorname{coker}f)\otimes_{A}A[\tfrac{1}{a}]=(0)$ implies that $a^{n}\cdot\operatorname{coker}f=(0)$ for some positive integer $n$. Therefore, $\operatorname{coker}f$ is a finitely generated module over $A_{R}/(a^{n})=A/(a^{n})\otimes_{{\mathbb{F}}_{q}}R$, whence over $R$. So (a) follows from Proposition 5.8.

(a)$\Longrightarrow$(b) If $R$ is a field this was proved in [BH11, Corollary 5.4] and also follows from [Pap08, Proposition 3.4.5] and [Tae09, Proposition 3.1.2]. We generalize the proof to the relative situation.

1. If $f$ is an isogeny, then $\operatorname{coker}f$ is a finite locally free $R$-module, which we may assume to be free after passing to an open affine covering of $\operatorname{Spec}R$. Let $t\in A\smallsetminus{\mathbb{F}}_{q}$ and consider the finite flat homomorphism ${\widetilde{A}}:={\mathbb{F}}_{q}[t]\hookrightarrow A$ from Lemma 1.4, under which we view ${\underline{M\!}\,}$ and ${\underline{N\!}\,}$ as ${\widetilde{A}}$-motives by restriction of scalars. That is, we view $M$ and $N$ as locally free $R[t]$-modules of rank $\tilde{r}=\operatorname{rk}{\underline{M\!}\,}\cdot\operatorname{rk}_{{\widetilde{A}}}A$ and $\tau_{M}$ and $\tau_{N}$ as $R[t][\tfrac{1}{t-\gamma(t)}]$-isomorphisms. By multiplying both $\tau_{M}$ and $\tau_{N}$ with $(t-\gamma(t))^{e}$ for $e\gg 0$ we may assume that ${\underline{M\!}\,}$ and ${\underline{N\!}\,}$ are effective ${\widetilde{A}}$-motives without changing the isogeny $f\colon{\underline{M\!}\,}\to{\underline{N\!}\,}$. Let {\mathfrak{a}}=\operatorname{ann}_{R[t]}(\operatorname{coker}f)=\ker\bigl{(}R[t]\to\operatorname{End}_{R}(\operatorname{coker}f)\bigr{)} be the annihilator of $\operatorname{coker}f$. By the Cayley-Hamilton theorem [Eis95, Theorem 4.3] (applied with $I=R$), the monic characteristic polynomial $\chi_{t}$ of the endomorphism $t$ of $\operatorname{coker}f$ lies in ${\mathfrak{a}}$. This shows that $R[t]/{\mathfrak{a}}$ is a quotient of the finite $R$-module $R[t]/(\chi_{t})$. In particular the closed subscheme $V:=\operatorname{Spec}R[t]/{\mathfrak{a}}$ of ${\mathbb{A}}^{1}_{R}=\operatorname{Spec}R[t]$ is finite over $\operatorname{Spec}R$. On its open complement $f\colon M\to N$ is an isomorphism.

We now consider the exterior powers $\wedge^{\tilde{r}}M$ and $\wedge^{\tilde{r}}N$ of the $R[t]$-modules $M$ and $N$ and set ${\cal{L}}:=(\wedge^{\tilde{r}}M)^{\scriptscriptstyle\lor}\otimes\wedge^{\tilde{r}}N$. These are invertible $R[t]$-modules. The isogeny $f$ induces a global section $\wedge^{\tilde{r}}f$ of the invertible sheaf ${\cal{L}}$ on ${\mathbb{A}}^{1}_{R}$ which provides an isomorphism {\cal{O}}_{{\mathbb{A}}^{1}_{R}}\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{\scriptstyle\sim}\longrightarrow}}}{\cal{L}}, $1\mapsto\wedge^{\tilde{r}}f$ on ${\mathbb{A}}^{1}_{R}\smallsetminus V$. Likewise we obtain global sections $\wedge^{\tilde{r}}\sigma^{*}f$, resp. $\wedge^{\tilde{r}}\tau_{M}$, resp. $\wedge^{\tilde{r}}\tau_{N}$ of the invertible sheaves $\sigma^{*}{\cal{L}}$, resp. $(\wedge^{\tilde{r}}\sigma^{*}M)^{\scriptscriptstyle\lor}\otimes\wedge^{\tilde{r}}M$, resp. $(\wedge^{\tilde{r}}\sigma^{*}N)^{\scriptscriptstyle\lor}\otimes\wedge^{\tilde{r}}N$ by the effectivity assumption on ${\underline{M\!}\,}$ and ${\underline{N\!}\,}$. Diagram (5.1) implies that there is an equality of global sections

[TABLE]

of (\wedge^{\tilde{r}}\sigma^{*}M)^{\scriptscriptstyle\lor}\otimes\wedge^{\tilde{r}}N\;=\;{\cal{L}}\otimes(\wedge^{\tilde{r}}\sigma^{*}M)^{\scriptscriptstyle\lor}\otimes\wedge^{\tilde{r}}M\bigr{)}\;=\;\bigl{(}(\wedge^{\tilde{r}}\sigma^{*}N)^{\scriptscriptstyle\lor}\otimes\wedge^{\tilde{r}}N\bigr{)}\otimes\sigma^{*}{\cal{L}}.

Since $V$ is proper over $\operatorname{Spec}R$ and the projective line ${\mathbb{P}}^{1}_{R}$ is separated, the map $V\hookrightarrow{\mathbb{A}}^{1}_{R}\hookrightarrow{\mathbb{P}}^{1}_{R}$ is a closed immersion which does not meet $\{\infty\}\times_{{\mathbb{F}}_{q}}\operatorname{Spec}R$, where $\{\infty\}={\mathbb{P}}^{1}_{{\mathbb{F}}_{q}}\smallsetminus{\mathbb{A}}^{1}_{{\mathbb{F}}_{q}}$. Thus we may glue ${\cal{L}}$ with the trivial sheaf ${\cal{O}}_{{\mathbb{P}}^{1}_{R}\smallsetminus V}$ on ${\mathbb{P}}^{1}_{R}\smallsetminus V$ along the isomorphism {\cal{O}}_{{\mathbb{P}}^{1}_{R}}\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{\scriptstyle\sim}\longrightarrow}}}{\cal{L}}, $1\mapsto\wedge^{\tilde{r}}f$ over ${\mathbb{A}}^{1}_{R}\smallsetminus V$. In this way we obtain an invertible sheaf ${\,\overline{\!{\cal{L}}}}$ on the projective line ${\mathbb{P}}^{1}_{R}$. By replacing ${\,\overline{\!{\cal{L}}}}$ with ${\,\overline{\!{\cal{L}}}}\otimes{\cal{O}}_{{\mathbb{P}}^{1}_{R}}(m\cdot\infty)$ for a suitable integer $m$ we may achieve that ${\,\overline{\!{\cal{L}}}}$ has degree zero (see [BLR90, § 9.1, Proposition 2]) and induces an $R$-valued point of the relative Picard functor $\operatorname{Pic}^{0}_{{\mathbb{P}}^{1}/{\mathbb{F}}_{q}}$; cf. [BLR90, § 8.1]. Since $\operatorname{Pic}^{0}_{{\mathbb{P}}^{1}/{\mathbb{F}}_{q}}$ is trivial, [BLR90, § 8.1, Proposition 4] shows that ${\,\overline{\!{\cal{L}}}}\cong{\cal{K}}\otimes_{R}{\cal{O}}_{{\mathbb{P}}^{1}_{R}}$ for an invertible sheaf ${\cal{K}}$ on $\operatorname{Spec}R$. Replacing $\operatorname{Spec}R$ by an open affine covering which trivializes ${\cal{K}}$ we may assume that there is an isomorphism \alpha\colon{\cal{L}}\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{\scriptstyle\sim}\longrightarrow}}}R[t] of $R[t]$-modules. Let $h:=\alpha(\wedge^{\tilde{r}}f)\in R[t]$.

2. Let $d:=\operatorname{rk}_{R}\operatorname{coker}\tau_{M}$. We claim that locally on $\operatorname{Spec}R$ there is a positive integer $n_{0}$ and for every integer $n\geq n_{0}$ an isomorphism of $R[t]$-modules

[TABLE]

and similarly for ${\underline{N\!}\,}$. To prove the claim we apply Proposition 2.3(c) to the $A$-motive $\wedge^{\tilde{r}}{\underline{M\!}\,}$ and derive that $\wedge^{\tilde{r}}\tau_{M}\colon\wedge^{\tilde{r}}\sigma^{*}M\to\wedge^{\tilde{r}}M$ is injective $\operatorname{coker}\wedge^{\tilde{r}}\tau_{M}$ is a finite locally free $R$-module, annihilated by a power of $t-\gamma(t)$. Consider the exact sequence

[TABLE]

Choose an open affine covering of $\operatorname{Spec}R[t]$ which trivializes the locally free $R[t]$-module $\wedge^{\tilde{r}}M$. Pulling back this covering under the section \operatorname{Spec}R\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{\scriptstyle\sim}\longrightarrow}}}\operatorname{Spec}R[t]/(t-\gamma(t))\hookrightarrow\operatorname{Spec}R[t] gives an open affine covering of $\operatorname{Spec}R$ on which we may find an isomorphism \operatorname{coker}\wedge^{\tilde{r}}\tau_{M}\otimes_{R[t]}(\wedge^{\tilde{r}}M)^{\scriptscriptstyle\lor}\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{\scriptstyle\sim}\longrightarrow}}}\operatorname{coker}\wedge^{\tilde{r}}\tau_{M}. We replace $\operatorname{Spec}R$ by this open affine covering and even shrink it further in such a way that $\operatorname{coker}\wedge^{\tilde{r}}\tau_{M}$ becomes a free $R$-module. By [Eis95, Proposition 4.1(b)] the sequence (5.6) is then isomorphic to the sequence

[TABLE]

where $g\in R[t]$ is a monic polynomial of degree equal to $\operatorname{rk}_{R}(\operatorname{coker}\wedge^{\tilde{r}}\tau_{M})$. We now tensor sequence (5.7) over $R$ with $k:=\operatorname{Frac}(R/{\mathfrak{p}})$ where ${\mathfrak{p}}\subset R$ is a prime ideal. It remains exact because $\operatorname{coker}\wedge^{\tilde{r}}\tau_{M}$ is free. Since $k[t]$ is a principal ideal domain the elementary divisor theorem applied to

[TABLE]

allows to write $\tau_{M}\otimes\operatorname{\,id}_{k}$ as a diagonal matrix. This shows that $\operatorname{coker}\wedge^{\tilde{r}}\tau_{M}\otimes_{R}k$ is a $k$-vector space of dimension equal to $\operatorname{rk}_{R}(\operatorname{coker}\tau_{M})=:d$. Since $t-\gamma(t)$ is nilpotent on this vector space, the Cayley-Hamilton theorem from linear algebra implies $g{\rm\;mod\;}{\mathfrak{p}}=(t-\gamma(t))^{d}$. In particular the coefficients of the difference $g^{\prime}:=g-(t-\gamma(t))^{d}$ lie in every prime ideal of $R$, and hence are nilpotent by [Eis95, Corollary 2.12]. Therefore there is a positive integer $n_{0}$ with $(g^{\prime})^{q^{n_{0}}}=0$, whence $g^{q^{n}}=(t-\gamma(t))^{q^{n}d}$ for every $n\geq n_{0}$. The $q^{n}$-th tensor power of the isomorphism between (the left entries in) the sequences (5.6) and (5.7) provides the isomorphism in (5.5). This proves the claim.

3. Since $d=\operatorname{rk}_{R}\operatorname{coker}\tau_{M}=\operatorname{rk}_{R}\operatorname{coker}\tau_{N}$ by Proposition 5.8, equations (5.4) and (5.5) imply that for $n\gg 0$ there is an isomorphism \beta\colon\sigma^{*}{\cal{L}}^{\otimes q^{n}}\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{\scriptstyle\sim}\longrightarrow}}}{\cal{L}}^{\otimes q^{n}} of $R[t]$-modules sending $(t-\gamma(t))^{q^{n}}(\sigma^{*}\wedge^{\tilde{r}}f)^{\otimes q^{n}}$ to $(t-\gamma(t))^{q^{n}}(\wedge^{\tilde{r}}f)^{\otimes q^{n}}$ and hence $(\sigma^{*}\wedge^{\tilde{r}}f)^{\otimes q^{n}}$ to $(\wedge^{\tilde{r}}f)^{\otimes q^{n}}$ because $t-\gamma(t)$ is a non-zero divisor. In particular the isomorphism

[TABLE]

which is given by multiplication with a unit $u\in R[t]^{\scriptscriptstyle\times}$, sends $\sigma(h^{q^{n}})=\sigma^{*}\alpha^{\otimes q^{n}}(\wedge^{\tilde{r}}\sigma^{*}f)^{\otimes q^{n}}$ to $h^{q^{n}}=\alpha^{\otimes q^{n}}(\wedge^{\tilde{r}}f)^{\otimes q^{n}}$. We thus obtain the equation $h^{q^{n}}=u\cdot\sigma(h^{q^{n}})$ in $R[t]$.

By Lemma 5.13 below, $u=\sum_{i\geq 0}u_{i}t^{i}$ with $u_{0}\in R^{\scriptscriptstyle\times}$ and $u_{i}\in R$ nilpotent for all $i\geq 1$. Let $R^{\prime}=R[v_{0}]/(v_{0}^{q-1}u_{0}-1)$ be the finite étale $R$-algebra obtained by adjoining a $(q-1)$-th root $v_{0}$ of $u_{0}^{-1}$. Then there is a unit $v=\sum_{i\geq 1}v_{i}t^{i}\in R^{\prime}[t]^{\scriptscriptstyle\times}$ with $v=u\cdot\sigma(v)$. Indeed the latter amounts to the equations

[TABLE]

which have the solutions \tfrac{v_{i}}{v_{0}}=\sum_{n\geq 0}\bigl{(}\sum_{j\geq 1}\tfrac{u_{j}}{u_{0}}\,(\tfrac{v_{i-j}}{v_{0}})^{q}\bigr{)}^{q^{n}} because the $u_{j}$ are nilpotent. Therefore the element $v^{-1}h^{q^{n}}\in R^{\prime}[t]$ satisfies $\sigma(v^{-1}h^{q^{n}})=v^{-1}h^{q^{n}}$. Working on each connected component of $\operatorname{Spec}R^{\prime}$ separately, Lemma 5.14 below shows that $a:=v^{-1}h^{q^{n}}\in{\mathbb{F}}_{q}[t]\subset A$.

In the ring $R^{\prime}[t][\tfrac{1}{a}]$ the element $h$ becomes a unit. Therefore the map $\alpha^{-1}\circ h\colon R^{\prime}[t][\tfrac{1}{a}]\to{\cal{L}}[\tfrac{1}{a}]$, $1\mapsto\wedge^{\tilde{r}}f$ is an isomorphism. This implies that $\wedge^{\tilde{r}}f\colon\wedge^{\tilde{r}}M[\tfrac{1}{a}]\to\wedge^{\tilde{r}}N[\tfrac{1}{a}]$ is an isomorphism, and hence also $f\colon M[\tfrac{1}{a}]\to N[\tfrac{1}{a}]$ by Cramer’s rule (e.g. [Bou70, III.8.6, Formulas (21) and (22)]). Thus we have established (b) étale locally on $\operatorname{Spec}R$. Replacing $a$ by the product of all the finitely many elements $a$ obtained locally, establishes (b) globally on $\operatorname{Spec}R$.

4. To prove the statement about quasi-morphisms $f\in\operatorname{QHom}_{R}({\underline{M\!}\,},{\underline{N\!}\,})$ assume first, that $f$ induces an isomorphism f\colon M[\tfrac{1}{a}]\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{\scriptstyle\sim}\longrightarrow}}}N[\tfrac{1}{a}] for some $a\in A\smallsetminus\{0\}$. Then $g:=a^{n}\cdot f\in\operatorname{Hom}_{R}({\underline{M\!}\,},{\underline{N\!}\,})$ for $n\gg 0$, because $M$ is finitely generated. In particular $g$ is an isogeny and $f=g\otimes a^{-n}$ is a quasi-isogeny.

Conversely, if $f$ is a quasi-isogeny, that is $f=g\otimes c$ for an isogeny $g\in\operatorname{Hom}_{R}({\underline{M\!}\,},{\underline{N\!}\,})$ and a $c\in Q$, there is an element $a\in A\smallsetminus\{0\}$ such that g\colon M[\tfrac{1}{a}]\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{\scriptstyle\sim}\longrightarrow}}}N[\tfrac{1}{a}]. If $d$ is the denominator of $c$ it follows that f\colon M[\tfrac{1}{ad}]\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{\scriptstyle\sim}\longrightarrow}}}N[\tfrac{1}{ad}]. ∎

To finish the proof of Theorem 5.12 we must demonstrate the following two lemmas.

Lemma 5.13**.**

An element $u=\sum_{i\geq 0}u_{i}t^{i}\in R[t]$ is a unit in $R[t]$ if and only if $u_{0}\in R^{\scriptscriptstyle\times}$ and $u_{i}$ is nilpotent for all $i\geq 1$.

Proof.

If the $u_{i}$ satisfy the assertion then there is a positive integer $n$ such that $u_{i}^{q^{n}}=0$ for all $i\geq 1$. Therefore $u^{q^{n}}=u_{0}^{q^{n}}$ is a unit in $R[t]$ and so the same holds for $u$.

Conversely if $u$ is a unit then $u_{0}$ must be a unit in $R$. By [Eis95, Corollary 2.12] the kernel of the map $R\to\prod_{{\mathfrak{p}}\subset R}R/{\mathfrak{p}}$ where ${\mathfrak{p}}$ runs over all prime ideals of $R$, equals the nil-radical of $R$. Under this map $u$ is sent to a unit in each factor $R/{\mathfrak{p}}[t]$. Since $R/{\mathfrak{p}}$ is an integral domain, the $u_{i}$ for $i\geq 1$ must be sent to zero in each factor $R/{\mathfrak{p}}$. This shows that $u_{i}$ is nilpotent for $i\geq 1$. ∎

Lemma 5.14**.**

Assume that $R$ contains no idempotents besides [math] and $1$, that is $\operatorname{Spec}R$ is connected. Then $R^{\sigma}:=\{x\in R\colon x^{q}=x\}={\mathbb{F}}_{q}$.

Proof.

Let ${\mathfrak{m}}\subset R$ be a maximal ideal and let $\bar{x}\in R/{\mathfrak{m}}$ be the image of $x$. Then $\bar{x}^{q}=\bar{x}$ implies that $\bar{x}$ is equal to an element $\alpha\in{\mathbb{F}}_{q}\subset R/{\mathfrak{m}}$. Now $e:=(x-\alpha)^{q-1}$ satisfies $e^{2}=(x-\alpha)^{q-2}(x^{q}-\alpha^{q})=(x-\alpha)^{q-1}=e$, that is $e$ is an idempotent. Since $e\in{\mathfrak{m}}$ we cannot have $e=1$ and must have $e=0$. Therefore $x-\alpha=(x-\alpha)^{q}=(x-\alpha)\cdot e=0$ in $R$, that is $x=\alpha\in{\mathbb{F}}_{q}$. ∎

Corollary 5.15**.**

If $f\in\operatorname{Hom}_{R}({\underline{M\!}\,},{\underline{N\!}\,})$ is an isogeny between $A$-motives then there is an element $0\neq a\in A$ and an isogeny $g\in\operatorname{Hom}_{R}({\underline{N\!}\,},{\underline{M\!}\,})$ with $f\circ g=a\cdot\operatorname{\,id}_{\underline{N\!}\,}$ and $g\circ f=a\cdot\operatorname{\,id}_{\underline{M\!}\,}$. The same is true for abelian Anderson $A$-modules.

Proof.

Let $a\in A$ be the element from Theorem 5.12(b). As in the proof of (b)$\Longrightarrow$(a) of this theorem there is a positive integer $n$ such $a^{n}\!\cdot\!\operatorname{coker}f=(0)$. Therefore there is a map $g\colon N\to M$ with $g\circ f=a^{n}\!\cdot\!\operatorname{\,id}_{M}$ and $f\circ g=a^{n}\!\cdot\!\operatorname{\,id}_{N}$. This implies that $g$ is injective, because $a^{n}$ is a non-zero divisor on $N$. From

[TABLE]

and the injectivity of $f$ we conclude that $g\circ\tau_{N}=\tau_{M}\circ\sigma^{*}g$ and that $g\in\operatorname{Hom}_{R}({\underline{N\!}\,},{\underline{M\!}\,})$. By construction $g$ induces an isomorphism N[\tfrac{1}{a}]\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{\scriptstyle\sim}\longrightarrow}}}M[\tfrac{1}{a}] after inverting $a$. So $g$ is an isogeny by Theorem 5.12. The statement about abelian Anderson $A$-modules follows from Theorems 3.5 and 5.9. ∎

Corollary 5.16**.**

The relation of being isogenous is an equivalence relation for $A$-motives and for abelian Anderson $A$-modules.

Proof.

This follows from Theorem 5.12 and Corollary 5.15. ∎

Corollary 5.17**.**

Let $\gamma(A\smallsetminus\{0\})\subset R^{\scriptscriptstyle\times}$ and let $f\in\operatorname{Hom}_{R}({\underline{M\!}\,},{\underline{N\!}\,})$ be an isogeny between effective $A$-motives ${\underline{M\!}\,}$ and ${\underline{N\!}\,}$. Then $f$ is separable. The same is true for isogenies between abelian Anderson $A$-modules.

Proof.

Consider diagram (5.1) and set $K:=\operatorname{coker}(\tau_{\operatorname{coker}f})$. As in the proof of Theorem 5.12 there is an element $0\neq a\in A$ and a positive integer $n$ with $a^{n}\cdot\operatorname{coker}f=(0)$, and hence $a^{n}\cdot K=(0)$. Let $e$ be an integer with $q^{e}\geq\operatorname{rk}_{R}\operatorname{coker}\tau_{N}$ and $q^{e}\geq n$. Then $(a\otimes 1-1\otimes\gamma(a))^{q^{e}}\cdot\operatorname{coker}\tau_{N}=(0)$. Therefore

[TABLE]

Since $\gamma(a)\in R^{\scriptscriptstyle\times}$ we have $K=(0)$, and since $\operatorname{coker}f$ and $\sigma^{*}(\operatorname{coker}f)$ are finite locally free $R$ modules of the same rank, [GW10, Corollary 8.12] shows that $\tau_{\operatorname{coker}f}$ is an isomorphism, that is $f$ is separable. The statement about abelian Anderson $A$-modules follows from Theorem5.9(b). ∎

Corollary 5.18**.**

If $f\in\operatorname{Hom}_{R}({\underline{M\!}\,},{\underline{N\!}\,})$ and $g\in\operatorname{Hom}_{R}({\underline{N\!}\,},{\underline{M\!}\,})$ are isogenies between $A$-motives with $f\circ g=a\cdot\operatorname{\,id}_{\underline{N\!}\,}$ and $g\circ f=a\cdot\operatorname{\,id}_{\underline{M\!}\,}$ for an $a\in A$, then there is an isomorphism of $Q$-algebras \operatorname{QEnd}_{R}({\underline{M\!}\,})\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{\scriptstyle\sim}\longrightarrow}}}\operatorname{QEnd}_{R}({\underline{N\!}\,}) given by $h\otimes b\mapsto f\circ h\circ g\otimes\tfrac{b}{a}$ for $h\in\operatorname{End}_{R}({\underline{M\!}\,})$. ∎

Example 5.19**.**

Let $R$ be an $A$-ring of finite characteristic ${\mathfrak{p}}$, that is $\gamma\colon A\to R$ factors through ${\mathbb{F}}_{\mathfrak{p}}:=A/{\mathfrak{p}}$ for a maximal ideal ${\mathfrak{p}}\subset A$. Let $\ell\in{\mathbb{N}}_{>0}$ be divisible by $[{\mathbb{F}}_{\mathfrak{p}}:{\mathbb{F}}_{q}]$. Then $\sigma^{\ell*}({\cal{J}})=(a\otimes 1-1\otimes\gamma(a)^{q^{\ell}}\colon a\in A)={\cal{J}}\subset A_{R}$, because the elements $\gamma(a)\in{\mathbb{F}}_{\mathfrak{p}}$ satisfy $\gamma(a)^{q^{\ell}}=\gamma(a)$. Let ${\underline{M\!}\,}=(M,\tau_{M})$ be an $A$-motive over $R$. Then $\sigma^{\ell*}{\underline{M\!}\,}=(\sigma^{\ell*}M,\sigma^{\ell*}\tau_{M})$ is also an $A$-motive over $R$, because $\sigma^{\ell*}\tau_{M}$ is an isomorphism outside $\operatorname{V}(\sigma^{\ell*}{\cal{J}})=\operatorname{V}({\cal{J}})$. If ${\underline{M\!}\,}$ is effective, then the $A_{R}$-homomorphism

[TABLE]

satisfies $\tau_{M}\circ\sigma^{*}{\rm Fr}_{q^{\ell}\!,\,{\underline{M\!}\,}}={\rm Fr}_{q^{\ell}\!,\,{\underline{M\!}\,}}\circ\sigma^{\ell*}\tau_{M}$. Moreover, it is injective and its cokernel is a successive extension of the $\sigma^{i*}\operatorname{coker}\tau_{M}$ for $i=0,\ldots,\ell-1$, whence a finitely presented $R$-module. Therefore {\rm Fr}_{q^{\ell}\!,\,{\underline{M\!}\,}}\in\operatorname{Hom}_{R}\bigl{(}\sigma^{\ell*}{\underline{M\!}\,},{\underline{M\!}\,}) is an isogeny, called the $q^{\ell}$-Frobenius isogeny of ${\underline{M\!}\,}$. It is always inseparable, because the $\ell$-th power of $\tau_{M}$, which equals ${\rm Fr}_{q^{\ell}\!,\,{\underline{M\!}\,}}$ annihilates the cokernel of ${\rm Fr}_{q^{\ell}\!,\,{\underline{M\!}\,}}$.

If ${\underline{M\!}\,}$ is not effective, let $n\in{\mathbb{N}}_{>0}$ be such that ${\mathfrak{p}}^{n}=(a)$ is principal. Then $(a\otimes 1)\subset{\cal{J}}$ and $(a\otimes 1)\subset\sigma^{i*}{\cal{J}}$ for all $i$. This shows that

[TABLE]

is a quasi-isogeny in \operatorname{QHom}_{R}\bigl{(}\sigma^{\ell*}{\underline{M\!}\,},{\underline{M\!}\,}) by Theorem 5.12, called the $q^{\ell}$-Frobenius quasi-isogeny of ${\underline{M\!}\,}$.

Finally if $R=k$ is a field contained in ${\mathbb{F}}_{q^{\ell}}$ then $\sigma^{\ell*}{\underline{M\!}\,}={\underline{M\!}\,}$ and {\rm Fr}_{q^{\ell}\!,\,{\underline{M\!}\,}}\in\operatorname{QEnd}_{k}\bigl{(}{\underline{M\!}\,}), respectively {\rm Fr}_{q^{\ell}\!,\,{\underline{M\!}\,}}\in\operatorname{End}_{k}\bigl{(}{\underline{M\!}\,}) if ${\underline{M\!}\,}$ is effective. In this case, $A[\pi]$ lies in the center of $\operatorname{End}_{k}({\underline{M\!}\,})$ and $Q[\pi]$ lies in the center of $\operatorname{QEnd}_{k}({\underline{M\!}\,})$, because every $f\in\operatorname{End}_{k}({\underline{M\!}\,})$ satisfies $f\circ\tau_{M}=\tau_{M}\circ\sigma^{*}f$ and $\sigma^{\ell*}f=f$. If $k={\mathbb{F}}_{q^{\ell}}$, the center equals $A[\pi]$, respectively $Q[\pi]$, and the isogeny classes of $A$-motives are largely controlled by their Frobenius endomorphism; see [BH09, Theorems 8.1 and 9.1].

6 Torsion points

Definition 6.1**.**

Let $(0)\neq{\mathfrak{a}}=(a_{1},\ldots,a_{n})\subset A$ be an ideal and let ${\underline{E\!}\,}=(E,\varphi)$ be an abelian Anderson $A$-module over $R$. Then

[TABLE]

is called the ${\mathfrak{a}}$-torsion submodule of ${\underline{E\!}\,}$.

This definition is independent of the generators $(a_{1},\ldots,a_{n})$ of ${\mathfrak{a}}$ by the following

Lemma 6.2**.**

1. (a)

If $(a_{1},\ldots,a_{n})\subset(b_{1},\ldots,b_{m})\subset A$ are ideals then $\ker(\varphi_{b_{1},\ldots,b_{m}})\hookrightarrow\ker(\varphi_{a_{1},\ldots,a_{n}})$ is a closed immersion. 2. (b)

If $(a_{1},\ldots,a_{n})=(b_{1},\ldots,b_{m})$ then $\ker(\varphi_{b_{1},\ldots,b_{m}})=\ker(\varphi_{a_{1},\ldots,a_{n}})$. 3. (c)

For any $R$-algebra $S$ we have ${\underline{E\!}\,}[{\mathfrak{a}}](S)\,=\,\{\,P\in E(S)\colon\varphi_{a}(P)=0\text{ for all }a\in{\mathfrak{a}}\,\}$. 4. (d)

${\underline{E\!}\,}[{\mathfrak{a}}]$* is an $A/{\mathfrak{a}}$-module via $A/{\mathfrak{a}}\to\operatorname{End}_{R}({\underline{E\!}\,}[{\mathfrak{a}}]),\,\bar{b}\mapsto\varphi_{b}$.* 5. (e)

${\underline{E\!}\,}[{\mathfrak{a}}]$* is a finite $R$-group scheme of finite presentation.*

Proof.

(a) By assumption there are elements $c_{ij}\in A$ with $a_{i}=\sum_{j}c_{ij}b_{j}$. Therefore $\varphi_{a_{i}}=\sum_{j}\varphi_{c_{ij}}\varphi_{b_{j}}$ and the composition of $\varphi_{b_{1},\ldots,b_{m}}\colon E\to E^{m}$ followed by $(\varphi_{c_{ij}})_{i,j}\colon E^{m}\to E^{n}$ equals $\varphi_{a_{1},\ldots,a_{n}}\colon E\to E^{n}$. This proves (a) and clearly (a) implies (b).

To prove (c) let $P\colon\operatorname{Spec}S\to{\underline{E\!}\,}$ be an $S$-valued point in ${\underline{E\!}\,}(S)$ with $0=\varphi_{a}(P):=\varphi_{a}\circ P$ for all $a\in{\mathfrak{a}}$. If ${\mathfrak{a}}=(a_{1},\ldots,a_{n})$ then in particular $\varphi_{a_{i}}\circ P=0$ for $i=1,\ldots,n$. Therefore $P$ factors through $\ker\varphi_{a_{1},\ldots,a_{n}}={\underline{E\!}\,}[{\mathfrak{a}}]$.

Conversely let $P\colon\operatorname{Spec}S\to{\underline{E\!}\,}[{\mathfrak{a}}]$ be an $S$-valued point in ${\underline{E\!}\,}[{\mathfrak{a}}](S)$ and let $a\in{\mathfrak{a}}$. By (b) we may write ${\mathfrak{a}}=(a_{1},\ldots,a_{n})$ with $a_{1}=a$ to have ${\underline{E\!}\,}[{\mathfrak{a}}]=\ker\varphi_{a_{1},\ldots,a_{n}}$. Therefore $\varphi_{a}(P):=\varphi_{a}\circ P=0$. This proves (c).

(d) The relation $ab=ba$ in $A$ implies $\varphi_{a}\circ\varphi_{b}=\varphi_{b}\circ\varphi_{a}$. Using that the closed subscheme ${\underline{E\!}\,}[{\mathfrak{a}}]$ is uniquely determined by (c) it follows that the ring homomorphism $A\to\operatorname{End}_{R}({\underline{E\!}\,}[{\mathfrak{a}}]),\,b\mapsto\varphi_{b}|_{{\underline{E\!}\,}[{\mathfrak{a}}]}$ is well defined. If $b\in{\mathfrak{a}}$ then clearly $\varphi_{b}|_{{\underline{E\!}\,}[{\mathfrak{a}}]}=0$ and so this ring homomorphism factors through $A/{\mathfrak{a}}$.

(e) If ${\mathfrak{a}}=(a_{1},\ldots,a_{n})$ then ${\underline{E\!}\,}[{\mathfrak{a}}]=\ker\varphi_{a_{1},\ldots,a_{n}}$ is of finite presentation, because $\varphi_{a_{1},\ldots,a_{n}}$ is a morphism of finite presentation between the schemes $E$ and $E^{n}$ of finite presentation over $R$ by [EGA, IV1, Proposition 1.6.2]. The finiteness of ${\underline{E\!}\,}[{\mathfrak{a}}]$ follows for ${\mathfrak{a}}=(a)$ from Corollaries 5.11 and 5.3, and for general ${\mathfrak{a}}$ from (a) by considering some $(a)\subset{\mathfrak{a}}$. ∎

The following lemma is a version of the Chinese remainder theorem in our context.

Lemma 6.3**.**

Let $(0)\neq{\mathfrak{a}},{\mathfrak{b}}\subset A$ be two ideals with ${\mathfrak{a}}+{\mathfrak{b}}=A$.

1. (a)

For an abelian Anderson $A$-module ${\underline{E\!}\,}$ there is a canonical isomorphism {\underline{E\!}\,}[{\mathfrak{a}}]\times_{R}{\underline{E\!}\,}[{\mathfrak{b}}]\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{\scriptstyle\sim}\longrightarrow}}}{\underline{E\!}\,}[{\mathfrak{a}}{\mathfrak{b}}]. 2. (b)

For an effective $A$-motive ${\underline{M\!}\,}$ there is a canonical isomorphism {\underline{M\!}\,}/{\mathfrak{a}}{\mathfrak{b}}{\underline{M\!}\,}\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{\scriptstyle\sim}\longrightarrow}}}{\underline{M\!}\,}/{\mathfrak{a}}{\underline{M\!}\,}\oplus{\underline{M\!}\,}/{\mathfrak{b}}{\underline{M\!}\,} of finite ${\mathbb{F}}_{q}$-shtukas.

Proof.

By the Chinese remainder theorem there is an isomorphism A/{\mathfrak{a}}{\mathfrak{b}}\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{\scriptstyle\sim}\longrightarrow}}}A/{\mathfrak{a}}\times A/{\mathfrak{b}} whose inverse is given by $(x_{\mathfrak{a}},x_{\mathfrak{b}})\mapsto bx_{\mathfrak{a}}+ax_{\mathfrak{b}}$ for certain elements $a\in{\mathfrak{a}}$ and $b\in{\mathfrak{b}}$ which satisfy $a\equiv 1{\rm\;mod\;}{\mathfrak{b}}$ and $b\equiv 1{\rm\;mod\;}{\mathfrak{a}}$, and hence $a+b\equiv 1{\rm\;mod\;}{\mathfrak{a}}{\mathfrak{b}}$.

(b) follows directly from this, because ${\underline{M\!}\,}/{\mathfrak{a}}{\underline{M\!}\,}={\underline{M\!}\,}\otimes_{A}A/{\mathfrak{a}}$.

(a) By Lemma 6.2(a) the addition $\Delta$ on ${\underline{E\!}\,}[{\mathfrak{a}}{\mathfrak{b}}]$ defines a canonical morphism ${\underline{E\!}\,}[{\mathfrak{a}}]\times_{R}{\underline{E\!}\,}[{\mathfrak{b}}]\hookrightarrow{\underline{E\!}\,}[{\mathfrak{a}}{\mathfrak{b}}]\times_{R}{\underline{E\!}\,}[{\mathfrak{a}}{\mathfrak{b}}]\xrightarrow{\;\Delta\,}{\underline{E\!}\,}[{\mathfrak{a}}{\mathfrak{b}}]$. Its inverse is described as follows. The elements $a,b\in A$ from above satisfy $a{\mathfrak{b}}\subset{\mathfrak{a}}{\mathfrak{b}}$ and $b{\mathfrak{a}}\subset{\mathfrak{a}}{\mathfrak{b}}$. By Lemma 6.2(c) the endomorphism $\varphi_{a}$ of ${\underline{E\!}\,}[{\mathfrak{a}}{\mathfrak{b}}]$ factors through ${\underline{E\!}\,}[{\mathfrak{b}}]$ and $\varphi_{b}$ factors through ${\underline{E\!}\,}[{\mathfrak{a}}]$. So the inverse is the morphism $(\varphi_{b},\varphi_{a})\colon{\underline{E\!}\,}[{\mathfrak{a}}{\mathfrak{b}}]\to{\underline{E\!}\,}[{\mathfrak{a}}]\times_{R}{\underline{E\!}\,}[{\mathfrak{b}}]$. Indeed, for $x\in{\underline{E\!}\,}[{\mathfrak{a}}{\mathfrak{b}}]$, we compute $\varphi_{b}(x)+\varphi_{a}(x)=\varphi_{a+b}(x)=\varphi_{1}(x)=x$, because $a+b\equiv 1{\rm\;mod\;}{\mathfrak{a}}{\mathfrak{b}}$. On the other hand, for $x\in{\underline{E\!}\,}[{\mathfrak{a}}]$ and $y\in{\underline{E\!}\,}[{\mathfrak{b}}]$, we compute $\varphi_{b}(x+y)=\varphi_{b}(x)=x$ and $\varphi_{a}(x+y)=\varphi_{a}(y)=y$, because $b\equiv 1{\rm\;mod\;}{\mathfrak{a}}$ and $a\equiv 1{\rm\;mod\;}{\mathfrak{b}}$. ∎

Theorem 6.4**.**

Let ${\underline{E\!}\,}$ be an abelian Anderson $A$-module and let $(0)\neq{\mathfrak{a}}\subset A$ be an ideal.

1. (a)

Then ${\underline{E\!}\,}[{\mathfrak{a}}]$ is a finite locally free group scheme over $\operatorname{Spec}R$ and a strict ${\mathbb{F}}_{q}$-module scheme. 2. (b)

${\underline{E\!}\,}[{\mathfrak{a}}]$* is étale over $R$ if and only if $R\cdot\gamma({\mathfrak{a}})=R$, that is if and only if ${\mathfrak{a}}+{\cal{J}}=A_{R}$.* 3. (c)

If ${\underline{M\!}\,}={\underline{M\!}\,}({\underline{E\!}\,})$ is the associated effective $A$-motive then there are canonical $A$-equivariant isomorphisms

[TABLE]

Proof.

Since $A$ is a Dedekind domain, ${\mathfrak{a}}={\mathfrak{p}}_{1}^{e_{1}}\cdot\ldots\cdot{\mathfrak{p}}_{r}^{e_{r}}$ for prime ideals ${\mathfrak{p}}_{i}\in A$ and positive integers $e_{i}$. By Lemma 6.3 and the exactness of the functors $\operatorname{Dr}_{q}$ and ${\underline{M\!}\,}_{q}$, see Theorem 4.7(a), it suffices to treat the case ${\mathfrak{a}}={\mathfrak{p}}^{e}$. Let $A_{\mathfrak{p}}$ be the localization of $A$ at ${\mathfrak{p}}$. Since $A/{\mathfrak{p}}^{e}=A_{\mathfrak{p}}/{\mathfrak{p}}^{e}A_{\mathfrak{p}}$ there is an element $z\in A$ which is congruent modulo ${\mathfrak{a}}$ to a uniformizer of $A_{\mathfrak{p}}$. Moreover, since ${\underline{E\!}\,}[{\mathfrak{p}}^{e}]$ is an $A_{\mathfrak{p}}/{\mathfrak{p}}^{e}A_{\mathfrak{p}}$-module, every $\varphi_{s}$ with $s\in A\smallsetminus{\mathfrak{p}}$ is an automorphism of ${\underline{E\!}\,}[{\mathfrak{p}}^{e}]$. Let $0\leq n\leq e$. We denote the inclusion ${\underline{E\!}\,}[{\mathfrak{p}}^{n}]\hookrightarrow{\underline{E\!}\,}[{\mathfrak{p}}^{e}]$ of Lemma 6.2(a) by $i_{n,e}$. By Lemma 6.2(c) the endomorphism $\varphi_{z}^{e-n}$ of ${\underline{E\!}\,}[{\mathfrak{p}}^{e}]$ has kernel ${\underline{E\!}\,}[{\mathfrak{p}}^{e-n}]$ and factors through the closed subscheme ${\underline{E\!}\,}[{\mathfrak{p}}^{n}]$ via a morphism $j_{e,n}\colon{\underline{E\!}\,}[{\mathfrak{p}}^{e}]\to{\underline{E\!}\,}[{\mathfrak{p}}^{n}]$ with $\varphi_{z}^{e-n}=i_{n,e}\circ j_{e,n}$. We claim that $j_{e,n}$ is an epimorphism in the category of sheaves on the big fpqc-site over $\operatorname{Spec}R$, and we therefore have an exact sequence

[TABLE]

To prove the claim let $S$ be an $R$-algebra and let $P\colon\operatorname{Spec}S\to{\underline{E\!}\,}[{\mathfrak{p}}^{n}]$ be an $S$-valued point in ${\underline{E\!}\,}[{\mathfrak{p}}^{n}](S)$. Since $\varphi_{z^{e-n}}\colon{\underline{E\!}\,}\to{\underline{E\!}\,}$ is an isogeny by Corollary 5.11, hence an epimorphism of fpqc-sheaves by Proposition 5.2(e), there exists a faithfully flat $S$-algebra $S^{\prime}$ and a point $P^{\prime}\in E(S^{\prime})$ with $\varphi_{z^{e-n}}(P^{\prime})=P$. We have to show that $P^{\prime}\in{\underline{E\!}\,}[{\mathfrak{p}}^{e}](S^{\prime})$. For this purpose let $a\in{\mathfrak{p}}^{e}$. Then $\tfrac{a}{1}=\tfrac{c}{s}(\tfrac{z}{1})^{e}$ in $A_{\mathfrak{p}}$ for $c\in A,\,s\in A\smallsetminus{\mathfrak{p}}$. We compute

[TABLE]

because $z^{n}\in{\mathfrak{p}}^{n}$. This proves our claim and establishes the exactness of (6.1).

We now use that $A$ is a Dedekind domain with finite ideal class group. This means that for the prime ideal ${\mathfrak{p}}\subset A$ there are (arbitrarily large) integers $e$ such that ${\mathfrak{p}}^{e}=(a)$ is principal. Then ${\underline{E\!}\,}[{\mathfrak{p}}^{e}]=\ker\varphi_{a}$ is a finite locally free $R$-group scheme by Corollaries 5.11 and 5.3. If $0\leq n\leq e$ then we show that ${\underline{E\!}\,}[{\mathfrak{p}}^{n}]$ is flat over $R$. Namely, using the epimorphism $j_{e,n}\colon{\underline{E\!}\,}[{\mathfrak{p}}^{e}]\to{\underline{E\!}\,}[{\mathfrak{p}}^{n}]$ from (6.1) and the flatness of ${\underline{E\!}\,}[{\mathfrak{p}}^{e}]$ over $R$, the flatness of ${\underline{E\!}\,}[{\mathfrak{p}}^{n}]$ will follow from [EGA, IV3, Théorème 11.3.10] once we show that $j_{e,n}$ is flat in each fiber over a point of $\operatorname{Spec}R$. This follows from [DG70, § III.3, Corollaire 7.4] and so ${\underline{E\!}\,}[{\mathfrak{p}}^{n}]$ is flat over $R$ for all $n$. By Lemma 6.2(e) this proves that ${\underline{E\!}\,}[{\mathfrak{p}}^{n}]$ is a finite locally free group scheme over $\operatorname{Spec}R$. Moreover, it is a strict ${\mathbb{F}}_{q}$-module scheme by [Fal02, Proposition 2], because for ${\mathfrak{p}}^{n}=(a_{1},\ldots,a_{n})$ the morphism $\varphi_{a_{1},\ldots,a_{n}}$ is strict ${\mathbb{F}}_{q}$-linear by Example 4.3. So (a) is established.

If ${\mathfrak{a}}={\mathfrak{p}}^{e}=(a)$ we know from Theorem 5.9(c) applied to the isogeny $\varphi_{a}$ and $\operatorname{coker}{\underline{M\!}\,}(\varphi_{a})={\underline{M\!}\,}/a{\underline{M\!}\,}$ that (c) holds. If $0\leq n\leq e$ we use the exact sequence (6.1) and the fact that the functors $\operatorname{Dr}_{q}$ and ${\underline{M\!}\,}_{q}$ are exact by Theorem 4.7. Namely, multiplication with $z^{e-n}$ on ${\underline{M\!}\,}/a{\underline{M\!}\,}$ has cokernel ${\underline{M\!}\,}/{\mathfrak{p}}^{e-n}{\underline{M\!}\,}$ and image isomorphic to ${\underline{M\!}\,}/{\mathfrak{p}}^{n}{\underline{M\!}\,}$. We obtain an exact sequence of finite ${\mathbb{F}}_{q}$-shtukas

[TABLE]

with $\beta_{n,e}\circ\alpha_{e,n}=z^{e-n}$ on ${\underline{M\!}\,}/a{\underline{M\!}\,}$. Applying $\operatorname{Dr}_{q}$ to (6.2), using the exactness of $\operatorname{Dr}_{q}$, and that $\operatorname{Dr}_{q}({\underline{M\!}\,}/a{\underline{M\!}\,})={\underline{E\!}\,}[{\mathfrak{p}}^{e}]$ and $\operatorname{Dr}_{q}(z^{e-n})=\varphi_{z}^{e-n}$, proves $\operatorname{Dr}_{q}({\underline{M\!}\,}/{\mathfrak{p}}^{n}{\underline{M\!}\,})={\underline{E\!}\,}[{\mathfrak{p}}^{n}]$. Conversely applying ${\underline{M\!}\,}_{q}$ to (6.1), using the exactness of ${\underline{M\!}\,}_{q}$, and that ${\underline{M\!}\,}/a{\underline{M\!}\,}={\underline{M\!}\,}({\underline{E\!}\,}[{\mathfrak{p}}^{e}])$ and $z^{e-n}={\underline{M\!}\,}_{q}(\varphi_{z}^{e-n})$, proves ${\underline{M\!}\,}/{\mathfrak{p}}^{n}{\underline{M\!}\,}={\underline{M\!}\,}_{q}({\underline{E\!}\,}[{\mathfrak{p}}^{n}])$. This establishes (c) in general.

(b) Let $R\cdot\gamma({\mathfrak{a}})=R$, that is there are elements $a_{1},\ldots,a_{n}\in{\mathfrak{a}}$ and $b_{1},\ldots,b_{n}\in R$ with $\sum_{i=1}^{n}b_{i}\gamma(a_{i})=1$. Then the open subschemes $\operatorname{Spec}R[\tfrac{1}{\gamma(a_{i})}]\subset\operatorname{Spec}R$ cover $\operatorname{Spec}R$ and it suffices to check that ${\underline{E\!}\,}[{\mathfrak{a}}]$ is étale over $\operatorname{Spec}R[\tfrac{1}{\gamma(a_{i})}]$ for each $i$. But there ${\underline{E\!}\,}[{\mathfrak{a}}]$ is a closed subscheme of ${\underline{E\!}\,}[a_{i}]$ which is étale by Corollary 5.11. This shows that ${\underline{E\!}\,}[{\mathfrak{a}}]$ is unramified over $R$. Since it is flat by (a), it is étale as desired.

Conversely assume that $R\cdot\gamma({\mathfrak{a}})\subset{\mathfrak{m}}$ for a maximal ideal ${\mathfrak{m}}\subset R$ and set $k=R/{\mathfrak{m}}$. Over a field extension $k^{\prime}$ of $k$ we have $E\times_{R}k={\mathbb{G}}_{a,k^{\prime}}^{d}=\operatorname{Spec}k^{\prime}[x_{1},\ldots,x_{d}]$. We will show that ${\underline{E\!}\,}[{\mathfrak{a}}]\times_{R}k^{\prime}$ is not étale over $k^{\prime}$ by applying the Jacobi criterion [BLR90, §2.2, Proposition 7]. Let ${\mathfrak{a}}=(a_{1},\ldots,a_{n})$. Then {\underline{E\!}\,}[{\mathfrak{a}}]=\operatorname{Spec}k^{\prime}[x_{1},\ldots,x_{d}]/\bigl{(}\varphi_{a_{1}}^{*}(x_{1},\ldots,x_{d})\colon j=1,\ldots,n\bigr{)}. The Jacobi matrix is

[TABLE]

Since $\gamma(a_{i})=0$ in $k^{\prime}$ each $\operatorname{Lie}\varphi_{a_{i}}$ is a nilpotent $d\times d$ matrix. Since $\varphi_{a_{i}}\circ\varphi_{a_{j}}=\varphi_{a_{i}a_{j}}=\varphi_{a_{j}}\circ\varphi_{a_{i}}$ we have $\operatorname{Lie}\varphi_{a_{i}}(\ker\operatorname{Lie}\varphi_{a_{j}})\subset\ker\operatorname{Lie}\varphi_{a_{j}}$. Therefore all $\ker\operatorname{Lie}\varphi_{a_{i}}$ have a non-trivial intersection. This shows that the rank of the Jacobi matrix is less than $d$ and ${\underline{E\!}\,}[{\mathfrak{a}}]\times_{R}k^{\prime}$ is not étale over $k^{\prime}$. ∎

Proposition 6.5**.**

Let ${\underline{M\!}\,}=(M,\tau_{M})$ be an $A$-motive over $R$ of rank $r$ and let $(0)\neq{\mathfrak{a}}\subset A$ be an ideal with $R\cdot\gamma({\mathfrak{a}})=R$, that is ${\mathfrak{a}}+{\cal{J}}=A_{R}$. Let $\bar{s}=\operatorname{Spec}\Omega$ be a geometric base point of $\operatorname{Spec}R$. Then ${\underline{M\!}\,}/{\mathfrak{a}}{\underline{M\!}\,}$ is an étale finite ${\mathbb{F}}_{q}$-shtuka whose $\tau$-invariants $({\underline{M\!}\,}/{\mathfrak{a}}{\underline{M\!}\,})^{\tau}(\Omega)$, see (4.1), form a free $A/{\mathfrak{a}}$-module of rank $r$ which carries a continuous action of the étale fundamental group $\pi_{1}^{\rm\acute{e}t}(\operatorname{Spec}R,\bar{s})$.

Proof.

This result and its proof are due to Anderson [And86, Lemma 1.8.2] for $R$ a field. We let $G:=\operatorname{Res}_{A/{\mathfrak{a}}|{\mathbb{F}}_{q}}\operatorname{GL}_{r,A/{\mathfrak{a}}}$ be the Weil restriction with $G(R^{\prime})=\operatorname{GL}_{r}(A/{\mathfrak{a}}\otimes_{{\mathbb{F}}_{q}}R^{\prime})$ for all ${\mathbb{F}}_{q}$-algebras $R^{\prime}$. Then $G$ is a smooth connected affine group scheme over ${\mathbb{F}}_{q}$ by [CGP10, Proposition A.5.9]. Thus by Lang’s theorem [Lan56, Corollary on p. 557] the Lang map $L\colon G\to G,g\mapsto g\cdot\sigma^{*}g^{-1}$ is finite étale and surjective (although not a group homomorphism if $r>1$ and ${\mathfrak{a}}\neq A$).

Since ${\mathfrak{a}}+{\cal{J}}=A_{R}$ the isomorphism \tau_{M}\colon\sigma^{*}M|_{\operatorname{Spec}A_{R}\smallsetminus\operatorname{V}({\cal{J}})}\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{\scriptstyle\sim}\longrightarrow}}}M|_{\operatorname{Spec}A_{R}\smallsetminus\operatorname{V}({\cal{J}})} of ${\underline{M\!}\,}$ induces an isomorphism \tau_{M/{\mathfrak{a}}M}\colon\sigma^{*}M/{\mathfrak{a}}M\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{\scriptstyle\sim}\longrightarrow}}}M/{\mathfrak{a}}M and makes ${\underline{M\!}\,}/{\mathfrak{a}}{\underline{M\!}\,}$ into a finite ${\mathbb{F}}_{q}$-shtuka, which is étale. After passing to a covering of $\operatorname{Spec}R$ by open affine subschemes, we may assume that there is an isomorphism \alpha\colon(A/{\mathfrak{a}})^{r}\otimes_{{\mathbb{F}}_{q}}R\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{\scriptstyle\sim}\longrightarrow}}}M/{\mathfrak{a}}M and then $\alpha^{-1}\circ\tau_{M/{\mathfrak{a}}M}\circ\sigma^{*}\alpha$ is an element $b\in G(R)$ and corresponds to a morphism $b\colon\operatorname{Spec}R\to G$. The fiber product $\operatorname{Spec}R\underset{b,G,L}{\times}G$ is finite étale over $\operatorname{Spec}R$ and of the form $\operatorname{Spec}R^{\prime}$. The projection onto the second factor $G$ corresponds to an element $c\in G(R^{\prime})$ with $c\cdot\sigma^{*}c^{-1}=b$, that is $c=b\cdot\sigma^{*}c$. This implies $\alpha\circ c=\tau_{M/{\mathfrak{a}}M}\circ\sigma^{*}(\alpha\circ c)$, and thus $\alpha\circ c$ is an isomorphism (A/{\mathfrak{a}})^{r}\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{\scriptstyle\sim}\longrightarrow}}}({\underline{M\!}\,}/{\mathfrak{a}}{\underline{M\!}\,})^{\tau}(R^{\prime})\;:=\;\{\,m\otimes M/{\mathfrak{a}}M\otimes_{R}R^{\prime}\colon m=\tau_{M}(\sigma_{M}^{*}m)\,\}. The proposition follows from this. ∎

Theorem 6.6**.**

Let ${\underline{E\!}\,}$ be an abelian Anderson $A$-module over $R$ of rank $r$ and let ${\underline{M\!}\,}={\underline{M\!}\,}({\underline{E\!}\,})$ be its associated effective $A$-motive. Let $(0)\neq{\mathfrak{a}}\subset A$ be an ideal with $R\cdot\gamma({\mathfrak{a}})=R$, that is ${\mathfrak{a}}+{\cal{J}}=A_{R}$. Then for every $R$-algebra $R^{\prime}$ such that $\operatorname{Spec}R^{\prime}$ is connected, there is an isomorphism of $A/{\mathfrak{a}}$-modules

[TABLE]

In particular, if $\bar{s}=\operatorname{Spec}\Omega$ is a geometric base point of $\operatorname{Spec}R$, then ${\underline{E\!}\,}[{\mathfrak{a}}](\Omega)$ is a free $A/{\mathfrak{a}}$-module of rank $r$ which carries a continuous action of the étale fundamental group $\pi_{1}^{\rm\acute{e}t}(\operatorname{Spec}R,\bar{s})$.

Proof.

This result and its proof are due to Anderson [And86, Proposition 1.8.3] for $R$ a field. For general $R$ the proof was carried out in [BH07, Lemma 2.4 and Theorem 8.6]. The last statement follows from Proposition 6.5. ∎

7 Divisible local Anderson modules

In this section we consider the situation where ${\mathfrak{p}}\subset A$ is a maximal ideal and the elements of $\gamma({\mathfrak{p}})\subset R$ are nilpotent. Let $\hat{q}$ be the cardinality of the residue field ${\mathbb{F}}_{\mathfrak{p}}=A/{\mathfrak{p}}$ and $f=[{\mathbb{F}}_{\mathfrak{p}}:{\mathbb{F}}_{q}]$, that is $\hat{q}=q^{f}$. We fix a uniformizing parameter $z\in\operatorname{Frac}(A)$ at ${\mathfrak{p}}$. It defines an isomorphism {\mathbb{F}}_{\mathfrak{p}}{\mathchoice{\mbox{\rm[[}}{\mbox{\rm[[}}{\mbox{\scriptsize\rm[[}}{\mbox{\tiny\rm[[}}}z{\mathchoice{\mbox{\rm]]}}{\mbox{\rm]]}}{\mbox{\scriptsize\rm]]}}{\mbox{\tiny\rm]]}}}\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{\scriptstyle\sim}\longrightarrow}}}{\widehat{A}}_{\mathfrak{p}}:=\displaystyle\lim_{\longleftarrow}A/{\mathfrak{p}}^{n}. We consider the ${\mathfrak{p}}$-adic completion ${\widehat{A}}_{{\mathfrak{p}},R}:=\displaystyle\lim_{\longleftarrow}A_{R}/{\mathfrak{p}}^{n}=({\mathbb{F}}_{\mathfrak{p}}\otimes_{{\mathbb{F}}_{q}}R){\mathchoice{\mbox{\rm[[}}{\mbox{\rm[[}}{\mbox{\scriptsize\rm[[}}{\mbox{\tiny\rm[[}}}z{\mathchoice{\mbox{\rm]]}}{\mbox{\rm]]}}{\mbox{\scriptsize\rm]]}}{\mbox{\tiny\rm]]}}}$. By continuity the map $\gamma$ extends to a ring homomorphism $\gamma\colon{\widehat{A}}_{\mathfrak{p}}\to R$. We consider the ideals ${\mathfrak{a}}_{i}=(a\otimes 1-1\otimes\gamma(a)^{q^{i}}\colon a\in{\mathbb{F}}_{\mathfrak{p}})\subset{\widehat{A}}_{{\mathfrak{p}},R}$ for $i\in{\mathbb{Z}}/f{\mathbb{Z}}$. By the Chinese remainder theorem ${\widehat{A}}_{{\mathfrak{p}},R}$ decomposes

[TABLE]

and ${\widehat{A}}_{{\mathfrak{p}},R}/{\mathfrak{a}}_{i}$ is the subset of ${\widehat{A}}_{{\mathfrak{p}},R}$ on which $a\otimes 1$ acts as $1\otimes\gamma(a)^{q^{i}}$ for all $a\in{\mathbb{F}}_{\mathfrak{p}}$. Each factor is canonically isomorphic to $R{\mathchoice{\mbox{\rm[[}}{\mbox{\rm[[}}{\mbox{\scriptsize\rm[[}}{\mbox{\tiny\rm[[}}}z{\mathchoice{\mbox{\rm]]}}{\mbox{\rm]]}}{\mbox{\scriptsize\rm]]}}{\mbox{\tiny\rm]]}}}$. The factors are cyclically permuted by $\sigma$ because $\sigma({\mathfrak{a}}_{i})={\mathfrak{a}}_{i+1}$. In particular $\hat{\sigma}:=\sigma^{f}$ stabilizes each factor and acts on it via $\hat{\sigma}(z)=z$ and $\hat{\sigma}(b)=b^{\hat{q}}$ for $b\in R$. The ideal ${\cal{J}}:=(a\otimes 1-1\otimes\gamma(a)\colon a\in A)\subset A_{R}$ decomposes as follows ${\cal{J}}\!\cdot\!{\widehat{A}}_{{\mathfrak{p}},R}/{\mathfrak{a}}_{0}=(z-\gamma(z))$ and ${\cal{J}}\!\cdot\!{\widehat{A}}_{{\mathfrak{p}},R}/{\mathfrak{a}}_{i}=(1)$ for $i\neq 0$. In particular, ${\widehat{A}}_{{\mathfrak{p}},R}/{\mathfrak{a}}_{0}$ equals the ${\cal{J}}$-adic completion of $A_{R}$, as $\gamma(z)$ is nilpotent in $R$; compare also [AH14, Lemma 5.3]. We also set $R{\mathchoice{\mbox{\rm((}}{\mbox{\rm((}}{\mbox{\scriptsize\rm((}}{\mbox{\tiny\rm((}}}z{\mathchoice{\mbox{\rm))}}{\mbox{\rm))}}{\mbox{\scriptsize\rm))}}{\mbox{\tiny\rm))}}}:=R{\mathchoice{\mbox{\rm[[}}{\mbox{\rm[[}}{\mbox{\scriptsize\rm[[}}{\mbox{\tiny\rm[[}}}z{\mathchoice{\mbox{\rm]]}}{\mbox{\rm]]}}{\mbox{\scriptsize\rm]]}}{\mbox{\tiny\rm]]}}}[\tfrac{1}{z}]$.

Definition 7.1**.**

A local $\hat{\sigma}$-shtuka (or local shtuka) of rank $r$ over $R$ is a pair ${\underline{\hat{M}\!}\,}=(\hat{M},\tau_{\hat{M}})$ consisting of a locally free $R{\mathchoice{\mbox{\rm[[}}{\mbox{\rm[[}}{\mbox{\scriptsize\rm[[}}{\mbox{\tiny\rm[[}}}z{\mathchoice{\mbox{\rm]]}}{\mbox{\rm]]}}{\mbox{\scriptsize\rm]]}}{\mbox{\tiny\rm]]}}}$-module $\hat{M}$ of rank $r$, and an isomorphism \tau_{\hat{M}}\colon\hat{\sigma}^{\ast}\hat{M}[\frac{1}{z-\gamma(z)}]\stackrel{{\scriptstyle}}{{\mbox{\hskip 2.84526pt\raisebox{3.98337pt}{\scriptstyle\sim}\longrightarrow}}}\hat{M}[\frac{1}{z-\gamma(z)}]. If $\tau_{\hat{M}}(\hat{\sigma}^{*}\hat{M})\subset\hat{M}$ then ${\underline{\hat{M}\!}\,}$ is called effective, and if $\tau_{\hat{M}}(\hat{\sigma}^{*}\hat{M})=\hat{M}$ then ${\underline{\hat{M}\!}\,}$ is called étale.

A morphism of local shtukas $f\colon(\hat{M},\tau_{\hat{M}})\to(\hat{M}^{\prime},\tau_{\hat{M}^{\prime}})$ over $R$ is a morphism of $R{\mathchoice{\mbox{\rm[[}}{\mbox{\rm[[}}{\mbox{\scriptsize\rm[[}}{\mbox{\tiny\rm[[}}}z{\mathchoice{\mbox{\rm]]}}{\mbox{\rm]]}}{\mbox{\scriptsize\rm]]}}{\mbox{\tiny\rm]]}}}$-modules $f\colon\hat{M}\to\hat{M}^{\prime}$ which satisfies $\tau_{\hat{M}^{\prime}}\circ\hat{\sigma}^{\ast}f=f\circ\tau_{\hat{M}}$.

Example 7.2**.**

Let ${\underline{M\!}\,}=(M,\tau_{M})$ be an $A$-motive over $R$. We consider the ${\mathfrak{p}}$-adic completion ${\underline{M\!}\,}\otimes_{A_{R}}{\widehat{A}}_{{\mathfrak{p}},R}:=(M\otimes_{A_{R}}{\widehat{A}}_{{\mathfrak{p}},R}\,,\,\tau_{M}\otimes 1)=\displaystyle\lim_{\longleftarrow}{\underline{M\!}\,}/{\mathfrak{p}}^{n}{\underline{M\!}\,}$. We define the local $\hat{\sigma}$-shtuka at ${\mathfrak{p}}$ associated with ${\underline{M\!}\,}$ as {\underline{\hat{M}\!}\,}_{\mathfrak{p}}({\underline{M\!}\,}):=\bigl{(}M\otimes_{A_{R}}{\widehat{A}}_{{\mathfrak{p}},R}/{\mathfrak{a}}_{0}\,,\,(\tau_{M}\otimes 1)^{f}\bigr{)}, where $\tau_{M}^{f}:=\tau_{M}\circ\sigma^{*}\tau_{M}\circ\ldots\circ\sigma^{(f-1)*}\tau_{M}$. It equals the ${\cal{J}}$-adic completion of ${\underline{M\!}\,}$ and therefore is effective if and only if ${\underline{M\!}\,}$ is effective, because of Proposition 2.3. Of course if ${\mathbb{F}}_{\mathfrak{p}}={\mathbb{F}}_{q}$, and hence $\hat{q}=q$ and $\hat{\sigma}=\sigma$, we have ${\widehat{A}}_{{\mathfrak{p}},R}=R{\mathchoice{\mbox{\rm[[}}{\mbox{\rm[[}}{\mbox{\scriptsize\rm[[}}{\mbox{\tiny\rm[[}}}z{\mathchoice{\mbox{\rm]]}}{\mbox{\rm]]}}{\mbox{\scriptsize\rm]]}}{\mbox{\tiny\rm]]}}}$ and ${\underline{\hat{M}\!}\,}_{\mathfrak{p}}({\underline{M\!}\,})={\underline{M\!}\,}\otimes_{A_{R}}{\widehat{A}}_{{\mathfrak{p}},R}$.

Also for $f>1$ the local shtuka ${\underline{\hat{M}\!}\,}_{\mathfrak{p}}({\underline{M\!}\,})$ allows to recover ${\underline{M\!}\,}\otimes_{A_{R}}{\widehat{A}}_{{\mathfrak{p}},R}$ via the isomorphism

[TABLE]

because for $i\neq 0$ the equality ${\cal{J}}\!\cdot\!{\widehat{A}}_{{\mathfrak{p}},R}/{\mathfrak{a}}_{i}=(1)$ implies that $\tau_{M}\otimes 1$ is an isomorphism modulo ${\mathfrak{a}}_{i}$; see [HK16, Example 2.2] or [BH11, Propositions 8.8 and 8.5] for more details.

Let ${\underline{\hat{M}\!}\,}=(\hat{M},\tau_{\hat{M}})$ be an effective local shtuka over $R$. Set ${\underline{\hat{M}\!}\,}_{n}:=(\hat{M}_{n},\tau_{\hat{M}_{n}}):=(\hat{M}/z^{n}\hat{M},\tau_{\hat{M}}{\rm\;mod\;}z^{n})$ and $G_{n}:=\operatorname{Dr}_{\hat{q}}({\underline{\hat{M}\!}\,}_{n})$. Then $G_{n}$ is a finite locally free strict ${\mathbb{F}}_{\mathfrak{p}}$-module scheme over $R$ and ${\underline{\hat{M}\!}\,}_{n}={\underline{M\!}\,}_{\hat{q}}(G_{n})$ by Theorem 4.7. Moreover, $G_{n}$ inherits from ${\underline{\hat{M}\!}\,}_{n}$ an action of ${\mathbb{F}}_{\mathfrak{p}}[z]/(z^{n})$. The canonical epimorphisms ${\underline{\hat{M}\!}\,}_{n+1}\twoheadrightarrow{\underline{\hat{M}\!}\,}_{n}$ induce closed immersions $i_{n}\colon G_{n}\hookrightarrow G_{n+1}$. The inductive limit $\operatorname{Dr}_{\hat{q}}({\underline{\hat{M}\!}\,}):=\displaystyle\lim_{\longrightarrow}G_{n}$ in the category of sheaves on the big fppf-site of $\operatorname{Spec}R$ is a sheaf of ${\mathbb{F}}_{\mathfrak{p}}{\mathchoice{\mbox{\rm[[}}{\mbox{\rm[[}}{\mbox{\scriptsize\rm[[}}{\mbox{\tiny\rm[[}}}z{\mathchoice{\mbox{\rm]]}}{\mbox{\rm]]}}{\mbox{\scriptsize\rm]]}}{\mbox{\tiny\rm]]}}}$-modules that satisfies the following

Definition 7.3**.**

A ${\mathfrak{p}}$-divisible local Anderson module over $R$ is a sheaf of ${\mathbb{F}}_{\mathfrak{p}}{\mathchoice{\mbox{\rm[[}}{\mbox{\rm[[}}{\mbox{\scriptsize\rm[[}}{\mbox{\tiny\rm[[}}}z{\mathchoice{\mbox{\rm]]}}{\mbox{\rm]]}}{\mbox{\scriptsize\rm]]}}{\mbox{\tiny\rm]]}}}$-modules $G$ on the big fppf-site of $\operatorname{Spec}R$ such that

1. (a)

$G$ is ${\mathfrak{p}}$-torsion, that is $G=\displaystyle\lim_{\longrightarrow}G[z^{n}]$, where $G[z^{n}]:=\ker(z^{n}\colon G\to G)$, 2. (b)

$G$ is ${\mathfrak{p}}$-divisible, that is $z\colon G\to G$ is an epimorphism, 3. (c)

For every $n$ the ${\mathbb{F}}_{\mathfrak{p}}$-module $G[z^{n}]$ is representable by a finite locally free strict ${\mathbb{F}}_{\mathfrak{p}}$-module scheme over $R$ (Definition 4.2), and 4. (d)

there exist an integer $d\in{\mathbb{Z}}_{\geq 0}$, such that $(z-\gamma(z))^{d}=0$ on $\omega_{G}$ where $\omega_{G}:=\displaystyle\lim_{\longleftarrow}\omega_{G[z^{n}]}$ and $\omega_{G[z^{n}]}=e^{*}\Omega^{1}_{G[z^{n}]/\operatorname{Spec}R}$ is the pullback under the zero section $e\colon\operatorname{Spec}R\to G[z^{n}]$.

A morphism of ${\mathfrak{p}}$-divisible local Anderson modules over $R$ is a morphism of fppf-sheaves of ${\mathbb{F}}_{\mathfrak{p}}{\mathchoice{\mbox{\rm[[}}{\mbox{\rm[[}}{\mbox{\scriptsize\rm[[}}{\mbox{\tiny\rm[[}}}z{\mathchoice{\mbox{\rm]]}}{\mbox{\rm]]}}{\mbox{\scriptsize\rm]]}}{\mbox{\tiny\rm]]}}}$-modules. The category of divisible local Anderson modules is ${\mathbb{F}}_{\mathfrak{p}}{\mathchoice{\mbox{\rm[[}}{\mbox{\rm[[}}{\mbox{\scriptsize\rm[[}}{\mbox{\tiny\rm[[}}}z{\mathchoice{\mbox{\rm]]}}{\mbox{\rm]]}}{\mbox{\scriptsize\rm]]}}{\mbox{\tiny\rm]]}}}$-linear. It is shown in [HS15, Lemma 8.2] that $\omega_{G}$ is a finite locally free $R$-module and we define the dimension of $G$ as $\operatorname{rk}\omega_{G}$ . A ${\mathfrak{p}}$-divisible local Anderson module is called étale if $\omega_{G}=0$. Since $\omega_{G}$ surjects onto each $\omega_{G[z^{n}]}$, this is the case if and only if all $G[z^{n}]$ are étale, see [HS15, Lemma 3.7].

Conversely with a ${\mathfrak{p}}$-divisible local Anderson module $G$ over $R$ one associates the local shtuka ${\underline{M\!}\,}_{\hat{q}}(G):=\displaystyle\lim_{\longleftarrow}{\underline{M\!}\,}_{\hat{q}}(G[z^{n}])$. Multiplication with $z$ on $G$ gives $M_{\hat{q}}(G)$ the structure of an $R{\mathchoice{\mbox{\rm[[}}{\mbox{\rm[[}}{\mbox{\scriptsize\rm[[}}{\mbox{\tiny\rm[[}}}z{\mathchoice{\mbox{\rm]]}}{\mbox{\rm]]}}{\mbox{\scriptsize\rm]]}}{\mbox{\tiny\rm]]}}}$-module. In [HS15, Theorem 8.3] we proved the following

Theorem 7.4**.**

1. (a)

The two contravariant functors $\operatorname{Dr}_{\hat{q}}$ and ${\underline{M\!}\,}_{\hat{q}}$ are mutually quasi-inverse anti-equivalences between the category of effective local shtukas over $R$ and the category of ${\mathfrak{p}}$-divisible local Anderson modules over $R$. 2. (b)

Both functors are ${\mathbb{F}}_{\mathfrak{p}}{\mathchoice{\mbox{\rm[[}}{\mbox{\rm[[}}{\mbox{\scriptsize\rm[[}}{\mbox{\tiny\rm[[}}}z{\mathchoice{\mbox{\rm]]}}{\mbox{\rm]]}}{\mbox{\scriptsize\rm]]}}{\mbox{\tiny\rm]]}}}$-linear and map short exact sequences to short exact sequences. They preserve étale objects.

Let ${\underline{\hat{M}\!}\,}=(\hat{M},\tau_{\hat{M}})$ be an effective local shtuka over $S$ and let $G=\operatorname{Dr}_{\hat{q}}({\underline{\hat{M}\!}\,})$ be its associated ${\mathfrak{p}}$-divisible local Anderson module. Then

1. (c)

$G$* is a formal Lie group if and only if $\tau_{\hat{M}}$ is topologically nilpotent, that is $\operatorname{im}(\tau_{\hat{M}}^{n})\subset z\hat{M}$ for an integer $n$. * 2. (d)

the $R{\mathchoice{\mbox{\rm[[}}{\mbox{\rm[[}}{\mbox{\scriptsize\rm[[}}{\mbox{\tiny\rm[[}}}z{\mathchoice{\mbox{\rm]]}}{\mbox{\rm]]}}{\mbox{\scriptsize\rm]]}}{\mbox{\tiny\rm]]}}}$-modules $\omega_{\operatorname{Dr}_{\hat{q}}({\underline{\hat{M}\!}\,})}$ and $\operatorname{coker}\tau_{\hat{M}}$ are canonically isomorphic.

We now want to show that for an abelian Anderson $A$-module ${\underline{E\!}\,}$ over $R$ the local shtuka {\underline{\hat{M}\!}\,}_{\mathfrak{p}}\bigl{(}{\underline{M\!}\,}({\underline{E\!}\,})\bigr{)} corresponds to the ${\mathfrak{p}}$-power torsion of ${\underline{E\!}\,}$ as in the following

Definition 7.5**.**

Let ${\underline{E\!}\,}$ be an abelian Anderson $A$-module over $R$ and assume that the elements of $\gamma({\mathfrak{p}})\subset R$ are nilpotent. We define ${\underline{E\!}\,}[{\mathfrak{p}}^{\infty}]:=\displaystyle\lim_{\longrightarrow}{\underline{E\!}\,}[{\mathfrak{p}}^{n}]$ and call it the ${\mathfrak{p}}$-divisible local Anderson module associated with ${\underline{E\!}\,}$.

This definition is justified by the following

Theorem 7.6**.**

Let ${\underline{E\!}\,}=(E,\varphi)$ be an abelian Anderson $A$-module over $R$ and assume that the elements of $\gamma({\mathfrak{p}})\subset R$ are nilpotent. Then

1. (a)

all ${\underline{E\!}\,}[{\mathfrak{p}}^{n}]$ are finite locally free strict ${\mathbb{F}}_{\mathfrak{p}}$-module schemes, 2. (b)

${\underline{E\!}\,}[{\mathfrak{p}}^{\infty}]$* is a ${\mathfrak{p}}$-divisible local Anderson module over $R$,* 3. (c)

If ${\underline{M\!}\,}={\underline{M\!}\,}({\underline{E\!}\,})$ is the associated effective $A$-motive of ${\underline{E\!}\,}$ and ${\underline{\hat{M}\!}\,}:={\underline{\hat{M}\!}\,}_{\mathfrak{p}}({\underline{M\!}\,})={\underline{M\!}\,}\otimes_{A_{R}}{\widehat{A}}_{{\mathfrak{p}},R}/{\mathfrak{a}}_{0}$ is the local $\hat{\sigma}$-shtuka at ${\mathfrak{p}}$ associated with ${\underline{M\!}\,}$, then there are canonical isomorphisms

[TABLE]

Proof.

(a) By Lemma 4.4 we may test strictness after applying a faithfully flat base change to $R$ and assume that $E={\mathbb{G}}_{a,R}^{d}=\operatorname{Spec}R[x_{1},\ldots,x_{d}]=\operatorname{Spec}R[{\underline{X\!}\,}]$ and ${\underline{M\!}\,}({\underline{E\!}\,})=R\{\tau\}^{1\times d}$. We set $B:=\Gamma({\underline{E\!}\,}[{\mathfrak{p}}^{n}],{\cal{O}}_{{\underline{E\!}\,}[{\mathfrak{p}}^{n}]})$ and $I=\ker(R[{\underline{X\!}\,}]\twoheadrightarrow B)$ and $I_{0}=(x_{1},\ldots,x_{d})$, and consider the deformation $B^{\flat}=R[{\underline{X\!}\,}]/I\!\cdot\!I_{0}$. The endomorphisms $\varphi_{a}$ of $E$ for $a\in A$ satisfy $\varphi_{a}^{*}(I)\subset I$ and $\varphi_{a}^{*}(I_{0})\subset I_{0}$. This defines a lift $A\to\operatorname{End}_{R\text{-algebras}}(B^{\flat}),\,a\mapsto[a]^{\flat}:=\varphi_{a}^{*}$ compatible with addition and multiplication as in Definition 4.2.

Let $N\geq\dim{\underline{E\!}\,}$ be a positive integer which is a power of $\hat{q}$ such that $\gamma(a)^{N}=0$ for every $a\in{\mathfrak{p}}^{n}$. Choose $\lambda\in{\mathbb{F}}_{\mathfrak{p}}$ with ${\mathbb{F}}_{\mathfrak{p}}={\mathbb{F}}_{q}(\lambda)$ and let $g$ be the minimal polynomial of $\lambda$ over ${\mathbb{F}}_{q}$. Choose an element $t\in A$ with $t{\rm\;mod\;}{\mathfrak{p}}^{n}=\lambda$ in $A/{\mathfrak{p}}^{n}={\mathbb{F}}_{\mathfrak{p}}{\mathchoice{\mbox{\rm[[}}{\mbox{\rm[[}}{\mbox{\scriptsize\rm[[}}{\mbox{\tiny\rm[[}}}z{\mathchoice{\mbox{\rm]]}}{\mbox{\rm]]}}{\mbox{\scriptsize\rm]]}}{\mbox{\tiny\rm]]}}}/(z^{n})$. Then $g(t)\in{\mathfrak{p}}^{n}$, and hence $\gamma(g(t))^{N}=0$. On $\operatorname{Lie}E$ the equation $g(t^{N})=g(t)^{N}$ implies \operatorname{Lie}\varphi_{g(t^{N})}=\operatorname{Lie}\varphi_{g(t)}^{N}-\gamma(g(t))^{N}=\bigl{(}\operatorname{Lie}\varphi_{g(t)}-\gamma(g(t))\bigr{)}^{N}=0. So $\varphi_{g(t^{N})}\,\in\,\operatorname{End}_{R\text{-groups},{\mathbb{F}}_{q}\text{\rm-lin}}({\mathbb{G}}_{a,R}^{d})\,=\,R\{\tau\}^{d\times d}$ as a polynomial in $\tau$ has no constant term. This means that $\varphi_{g(t^{N})}^{*}(x_{i})\in I_{0}^{q}$. Moreover, since $g(t)\in{\mathfrak{p}}^{n}$ we have $\varphi_{g(t)}=0$ on ${\underline{E\!}\,}[{\mathfrak{p}}^{n}]$ and hence $\varphi_{g(t)}^{*}(x_{i})\in I$. Therefore $\varphi_{g(t^{\hat{q}N})}^{*}(I_{0})=\varphi_{g(t)}^{*}\circ\varphi_{g(t^{\hat{q}N-N-1})}^{*}\circ\varphi_{g(t^{N})}^{*}(I_{0})\subset\varphi_{g(t)}^{*}(I_{0}^{q})\subset\varphi_{g(t)}^{*}(I_{0})^{2}\subset I\!\cdot\!I_{0}$. In other words $[g(t^{\hat{q}N})]^{\flat}=[0]^{\flat}$ on $B^{\flat}$. This shows that the map ${\mathbb{F}}_{\mathfrak{p}}={\mathbb{F}}_{q}[t^{\hat{q}N}]/(g(t^{\hat{q}N}))\to\operatorname{End}_{R\text{-algebras}}(B^{\flat})$ lifts the action of ${\mathbb{F}}_{\mathfrak{p}}\subset{\mathbb{F}}_{\mathfrak{p}}{\mathchoice{\mbox{\rm[[}}{\mbox{\rm[[}}{\mbox{\scriptsize\rm[[}}{\mbox{\tiny\rm[[}}}z{\mathchoice{\mbox{\rm]]}}{\mbox{\rm]]}}{\mbox{\scriptsize\rm]]}}{\mbox{\tiny\rm]]}}}/(z^{n})$ on ${\underline{E\!}\,}[{\mathfrak{p}}^{n}]$ and is compatible with addition and multiplication.

We compute the induced action on the co-Lie complex $\ell^{\,{}^{\bullet}}_{{\cal{G}}/\operatorname{Spec}R}$ of ${\cal{G}}=(\operatorname{Spec}B,\operatorname{Spec}B^{\flat})$. In degree [math] we have $\ell^{\,0}_{{\cal{G}}/\operatorname{Spec}R}=\Omega^{1}_{R[{\underline{X\!}\,}]/R}\otimes_{R[{\underline{X\!}\,}],\,e_{R[{\underline{X\!}\,}]}}R=\bigoplus_{i=1}^{d}R\!\cdot\!x_{i}=I_{0}/I_{0}^{2}$. From $t-\lambda\in{\mathfrak{p}}^{n}$ we obtain $\gamma(t^{\hat{q}N})-\gamma(\lambda)=\gamma(t-\lambda)^{\hat{q}N}=0$ in $R$. On $\operatorname{Lie}E$ this implies $\operatorname{Lie}\varphi_{t^{\hat{q}N}}-\gamma(\lambda)=(\operatorname{Lie}\varphi_{t}-\gamma(t))^{\hat{q}N}=0$ and therefore $\varphi_{t^{\hat{q}N}}-\gamma(\lambda)\,\in\,\operatorname{End}_{R\text{-groups},{\mathbb{F}}_{q}\text{\rm-lin}}({\mathbb{G}}_{a,R}^{d})\,=\,R\{\tau\}^{d\times d}$ as a polynomial in $\tau$ has no constant term. This implies that \bigl{(}\varphi_{t^{\hat{q}N}}^{*}-\gamma(\lambda)\bigr{)}(I_{0})\subset I_{0}^{q}\subset I_{0}^{2}. We conclude that $t^{\hat{q}N}$ acts as the scalar $\gamma(\lambda)$ on $I_{0}/I_{0}^{2}$.

To compute the action of $t^{\hat{q}N}$ on $\ell^{\,-1}_{{\cal{G}}/\operatorname{Spec}R}$ we use that by Theorem 4.7(d), $\ell^{\,{}^{\bullet}}_{{\cal{G}}/\operatorname{Spec}R}$ is homotopically equivalent to the complex $0\to\sigma^{*}M/{\mathfrak{p}}^{n}\sigma^{*}M\xrightarrow{\;\tau_{M}\,}M/{\mathfrak{p}}^{n}M\to 0$ where ${\underline{M\!}\,}_{q}({\underline{E\!}\,}[{\mathfrak{p}}^{n}])={\underline{M\!}\,}/{\mathfrak{p}}^{n}{\underline{M\!}\,}$ and ${\underline{M\!}\,}={\underline{M\!}\,}({\underline{E\!}\,})=(M,\tau_{M})$; see Theorem 6.4(c). Since $t^{\hat{q}N}-\gamma(\lambda)=(t\otimes 1-1\otimes\gamma(t))^{\hat{q}N}=0$ on $\operatorname{coker}\tau_{M}$ there is an $A_{R}$-homomorphism $h\colon M\to\sigma^{*}M$ with h\,\tau_{M}=\bigl{(}t^{\hat{q}N}-\gamma(\lambda)\bigr{)}\!\cdot\!\operatorname{\,id}_{\sigma^{*}M} and \tau_{M}\,h=\bigl{(}t^{\hat{q}N}-\gamma(\lambda)\bigr{)}\!\cdot\!\operatorname{\,id}_{M}. This means that $t^{\hat{q}N}$ is homotopic to the scalar multiplication with $\gamma(\lambda)$ on $0\to\sigma^{*}M/{\mathfrak{p}}^{n}\sigma^{*}M\xrightarrow{\;\tau_{M}\,}M/{\mathfrak{p}}^{n}M\to 0$, and therefore also on $\ell^{\,{}^{\bullet}}_{{\cal{G}}/\operatorname{Spec}R}$. Let $h^{\prime}\colon I_{0}/I_{0}^{2}\to\ell^{\,-1}_{{\cal{G}}/\operatorname{Spec}R}=:\ell^{\,-1}$ be this homotopy, that is $(t^{\hat{q}N}-\gamma(\lambda))|_{\ell^{\,-1}}=h^{\prime}d$ and $(t^{\hat{q}N}-\gamma(\lambda))|_{I_{0}/I_{0}^{2}}=dh^{\prime}$. But we must show that $t^{\hat{q}N}$ and $\gamma(\lambda)$ are not only homotopic on $\ell^{\,{}^{\bullet}}_{{\cal{G}}/\operatorname{Spec}R}$, but equal.

Since $0=g(t^{\hat{q}N})=\prod_{i\in{\mathbb{Z}}/f{\mathbb{Z}}}(t^{\hat{q}N}-\gamma(\lambda)^{q^{i}})$ on $\ell^{\,{}^{\bullet}}_{{\cal{G}}/\operatorname{Spec}R}$, we can decompose $\ell^{\,-1}=\prod_{i\in{\mathbb{Z}}/f{\mathbb{Z}}}(\ell^{\,-1})_{i}$ where $(\ell^{\,-1})_{i}:=\ker(t^{\hat{q}N}-\gamma(\lambda)^{q^{i}}\colon\ell^{-1}\to\ell^{\,-1})$. Since the differential $d$ of $\ell^{\,{}^{\bullet}}_{{\cal{G}}/\operatorname{Spec}R}$ is an $R$-homomorphism and equivariant for the action of $t^{\hat{q}N}$, it maps $(\ell^{\,-1})_{i}$ into $\ker(t^{\hat{q}N}-\gamma(\lambda)^{q^{i}}\colon I_{0}/I_{0}^{2}\to I_{0}/I_{0}^{2})$ which is trivial for $i\neq 0$. This shows that $0=h^{\prime}d=t^{\hat{q}N}-\gamma(\lambda)=\gamma(\lambda^{q^{i}}-\lambda)$ on $(\ell^{\,-1})_{i}$, whence $(\ell^{\,-1})_{i}=(0)$ for $i\neq 0$, because $\gamma(\lambda^{q^{i}}-\lambda)\in R^{\scriptscriptstyle\times}$. We conclude that $\ell^{\,-1}=(\ell^{\,-1})_{0}$ and $t^{\hat{q}N}$ acts as the scalar $\gamma(\lambda)$ on $\ell^{\,-1}$. This proves that ${\underline{E\!}\,}[{\mathfrak{p}}^{n}]$ is a finite locally free strict ${\mathbb{F}}_{\mathfrak{p}}$-module scheme over $R$.

(b) By construction $\ker(z^{n}\colon{\underline{E\!}\,}[{\mathfrak{p}}^{\infty}]\to{\underline{E\!}\,}[{\mathfrak{p}}^{\infty}])={\underline{E\!}\,}[{\mathfrak{p}}^{n}]$ and ${\underline{E\!}\,}[{\mathfrak{p}}^{\infty}]$ is ${\mathfrak{p}}$-torsion. Using the epimorphism $j_{n+1,n}\colon{\underline{E\!}\,}[{\mathfrak{p}}^{n+1}]\twoheadrightarrow{\underline{E\!}\,}[{\mathfrak{p}}^{n}]$ from (6.1) with $i_{n,n+1}\circ j_{n+1,n}=\varphi_{z}$ we see that ${\underline{E\!}\,}[{\mathfrak{p}}^{\infty}]$ is ${\mathfrak{p}}$-divisible. In (a) we saw that ${\underline{E\!}\,}[{\mathfrak{p}}^{n}]$ is representable by a finite locally free strict ${\mathbb{F}}_{\mathfrak{p}}$-module scheme over $R$. It remains to verify condition (d) of Definition 7.3. Since ${\underline{E\!}\,}[{\mathfrak{p}}^{n}]\hookrightarrow{\underline{E\!}\,}$ is a closed immersion, $\omega_{{\underline{E\!}\,}[{\mathfrak{p}}^{n}]}$ is a quotient of $\omega_{\underline{E\!}\,}=\operatorname{Hom}_{R}(\operatorname{Lie}{\underline{E\!}\,},R)$. Since $A/{\mathfrak{p}}^{n}={\mathbb{F}}_{\mathfrak{p}}{\mathchoice{\mbox{\rm[[}}{\mbox{\rm[[}}{\mbox{\scriptsize\rm[[}}{\mbox{\tiny\rm[[}}}z{\mathchoice{\mbox{\rm]]}}{\mbox{\rm]]}}{\mbox{\scriptsize\rm]]}}{\mbox{\tiny\rm]]}}}/(z^{n})$, there is an element $a\in A$ with $z-a\in{\mathfrak{p}}^{n}$, whence $\varphi_{a}=\varphi_{z}$ on ${\underline{E\!}\,}[{\mathfrak{p}}^{n}]$. Therefore $(\operatorname{Lie}\varphi_{a}-\gamma(a))^{d}=0$ on $\operatorname{Lie}{\underline{E\!}\,}$ implies $(\varphi_{z}-\gamma(z))^{N}=(\varphi_{a}-\gamma(a))^{N}+\gamma(a-z)^{N}=0$ on $\omega_{{\underline{E\!}\,}[{\mathfrak{p}}^{n}]}$. It follows that $(\varphi_{z}-\gamma(z))^{N}=0$ on $\omega_{{\underline{E\!}\,}[{\mathfrak{p}}^{\infty}]}:=\displaystyle\lim_{\longleftarrow}\omega_{{\underline{E\!}\,}[{\mathfrak{p}}^{n}]}$, and that ${\underline{E\!}\,}[{\mathfrak{p}}^{\infty}]$ is a ${\mathfrak{p}}$-divisible local Anderson module over $R$.

(c) We have ${\underline{M\!}\,}_{q}({\underline{E\!}\,}[{\mathfrak{p}}^{n}])=\operatorname{Hom}_{R\text{-groups},{\mathbb{F}}_{q}\text{\rm-lin}}({\underline{E\!}\,}[{\mathfrak{p}}^{n}],{\mathbb{G}}_{a,R})={\underline{M\!}\,}/{\mathfrak{p}}^{n}{\underline{M\!}\,}$ and ${\underline{E\!}\,}[{\mathfrak{p}}^{n}]=\operatorname{Dr}_{q}({\underline{M\!}\,}/{\mathfrak{p}}^{n}{\underline{M\!}\,})$ by Theorem 6.4(c). This implies

[TABLE]

and ${\underline{E\!}\,}[{\mathfrak{p}}^{\infty}]\,=\,\displaystyle\lim_{\longrightarrow}\operatorname{Dr}_{q}({\underline{M\!}\,}/{\mathfrak{p}}^{n}{\underline{M\!}\,})\,=\,\operatorname{Dr}_{q}(\displaystyle\lim_{\longleftarrow}{\underline{M\!}\,}/{\mathfrak{p}}^{n}{\underline{M\!}\,})\,=\,\operatorname{Dr}_{q}({\underline{M\!}\,}\otimes_{A_{R}}{\widehat{A}}_{{\mathfrak{p}},R})$.

On ${\underline{E\!}\,}[{\mathfrak{p}}^{n}]$ every $\lambda\in{\mathbb{F}}_{\mathfrak{p}}$ acts as $\varphi_{\lambda}$ and on ${\mathbb{G}}_{a,R}$ as $\gamma(\lambda)$. Therefore

[TABLE]

where the second equality is due to the fact that ${\widehat{A}}_{{\mathfrak{p}},R}/{\mathfrak{a}}_{0}$ is the summand of ${\widehat{A}}_{{\mathfrak{p}},R}$ on which $\lambda\otimes 1$ acts as $1\otimes\gamma(\lambda)$ for all $\lambda\in{\mathbb{F}}_{\mathfrak{p}}$. This implies

[TABLE]

On the other hand, since ${\underline{E\!}\,}[{\mathfrak{p}}^{n}]$ is a finite locally free strict ${\mathbb{F}}_{\mathfrak{p}}$-module by (a), {\underline{E\!}\,}[{\mathfrak{p}}^{n}]=\operatorname{Dr}_{\hat{q}}\bigl{(}{\underline{M\!}\,}_{\hat{q}}({\underline{E\!}\,}[{\mathfrak{p}}^{n}])\bigr{)}=\operatorname{Dr}_{\hat{q}}({\underline{\hat{M}\!}\,}/{\mathfrak{p}}^{n}{\underline{\hat{M}\!}\,}) by Theorem 4.7(c), and so {\underline{E\!}\,}[{\mathfrak{p}}^{\infty}]=\displaystyle\lim_{\longrightarrow}\operatorname{Dr}_{\hat{q}}({\underline{\hat{M}\!}\,}/{\mathfrak{p}}^{n}{\underline{\hat{M}\!}\,})=\operatorname{Dr}_{\hat{q}}\bigl{(}{\underline{\hat{M}\!}\,}_{\mathfrak{p}}({\underline{M\!}\,})\bigr{)}. ∎