Isogenies of abelian Anderson A-modules and A-motives

Urs Hartl

June 22, 2017

Abstract

As a generalization of Drinfeld modules, Greg Anderson introduced abelian t-modules and tmotives 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 pdivisible 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. Mathematics Subject Classification (2010): 11G09, (14K02, 13A35, 14L05)

Contents

1 Introduction 1

2 A-Motives 4

3 Abelian Anderson A-modules 7

4 Review of the finite shtuka equivalence 12

5 Isogenies 15

6 Torsion points 23

7 Divisible local Anderson modules 27

References 30

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 𝔽_{q} be a finite field with q elements, let C be a smooth projective geometrically irreducible curve over 𝔽_{q} and let Q = 𝔽_{q}(C) be its function field. Let ∞ ∈ C be a closed point and let A = Γ(C ∖ {∞}, O_{C}) be the ring of functions which are regular outside ∞. Let (R, γ) be an A-ring, that is a commutative unitary ring together with a ring homomorphism γ: A → R. We consider the ideal J := (a ⊗ 1 − 1 ⊗ γ(a): a ∈ A) = ker(γ ⊗ id_{R}: A_{R} → R) ⊂ A_{R} := A ⊗_{𝔽_{q}} R and the endomorphism σ := id_{A} ⊗ Frob_{q,R}: a ⊗ b ↦ a ⊗ b^{q} of A_{R}. For an A_{R}-module M we set σ^{∗}M := M ⊗_{A_{R}, σ} A_{R} = M ⊗_{R, Frob_{q,R}} R, and for an element m ∈ M we write σ_{M}^{∗}m := m ⊗ 1 ∈ σ^{∗}M.

Definition 1.1. An effective A-motive of rank r over an A-ring R is a pair \underline{M} = (M, τ_{M}) consisting of a locally free A_{R}-module M of rank r and an A_{R}-homomorphism τ_{M}: σ^{∗}M → M whose cokernel is annihilated by J^{n} for some positive integer n. We say that \underline{M} has dimension d if coker τ_{M} is a locally free R-module of rank d and annihilated by J^{d}. We write rk \underline{M} = r and dim \underline{M} = d for the rank and the dimension of \underline{M}.

A morphism f: (M, τ_{M}) → (N, τ_{N}) between effective A-motives is an A_{R}-homomorphism f: M → N which satisfies f ∘ τ_{M} = τ_{N} ∘ σ^{∗}f.

Note that τ_{M} is always injective and coker τ_{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 = 𝔽_{q}[t] and in addition, M is finitely generated over the non-

commutative polynomial ring R{τ} := { ∑_{i=0}^{n} b_{i}τ^{i}: n ∈ ℕ_{0}, b_{i} ∈ R } with τb = b^{q}τ, which acts on

m ∈ M via τ: m ↦ τ_{M}(σ_{M}^{∗}m), then (M, τ_{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, φ) consisting of a smooth affine group scheme E over Spec R of relative dimension d, and a ring homomorphism φ: A → End_{R-groups}(E), a ↦ φ_{a} such that

(a) there is a faithfully flat ring homomorphism R → R′ for which E×_{R}Spec R′ ≅ 𝔾_{a,R′}^{d} as 𝔽_{q}-module schemes, where 𝔽_{q} acts on E via φ and 𝔽_{q} ⊂ A,

(b) (Lie φ_{a} − γ(a))^{d} = 0 on Lie E for all a ∈ A,

(c) the set M := M(\underline{E}) := M_{q}(\underline{E}) := Hom_{R-groups,𝔽_{q}-lin}(E, 𝔾_{a,R}) of 𝔽_{q}-equivariant homomorphisms of R-group schemes is a locally free A_{R}-module of rank r under the action given on m ∈ M by

\begin{matrix}A ∋ a: & M ⟶ M, & m ↦ m ∘ φ_{a} \\ R ∋ b: & M ⟶ M, & m ↦ b ∘ m \end{matrix}

A morphism f: (E, φ) → (E′, φ′) between abelian Anderson A-modules is a homomorphism of group schemes f: E → E′ over R which satisfies φ′_{a} ∘ f = f ∘ φ_{a} for all a ∈ A.

In particular, if R is a perfect field and A = 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 ×_{Spec R} Spec R′ ≅ 𝔾_{a,R′}^{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, φ) is an abelian Anderson A-module then \underline{M}(\underline{E}) = (M, τ_{M}) with τ_{M}: σ^{∗}M → M, σ_{M}^{∗}m ↦ Frob_{q,𝔾_{a,R}}∘ m is an effective A-motive of the same rank and dimension as \underline{E}. The contravariant functor \underline{E} ↦ \underline{M}(\underline{E}) is fully faithful. Its essential image consists of all effective A-motives \underline{M} = (M, τ_{M}) over R for which there exists a faithfully flat ring homomorphism R → R′ such that M ⊗_{R} R′ is a finite free left R′{τ}-module under the map τ: M → M, m ↦ τ_{M}(σ_{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: \underline{E} → \underline{E}′ between abelian Anderson A-modules over R is an isogeny if it is finite and surjective. On the other hand, a morphism f ∈ Hom_{R}(\underline{M}, \underline{N}) between A-motives over R is an isogeny if f is injective and 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 ∈ Hom_{R}(\underline{E}, \underline{E}′) be a morphism between abelian Anderson A-modules and let \underline{M}(f) ∈ Hom_{R}(\underline{M}′, \underline{M}) be the associated morphism between the associated effective A-motives \underline{M} = \underline{M}(\underline{E}) and \underline{M}′ = \underline{M}(\underline{E}′). Then

(a) f is an isogeny if and only if \underline{M}(f) is an isogeny.

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

Corollary 5.15. If f ∈ Hom_{R}(\underline{M}, \underline{N}) is an isogeny between A-motives then there is an element 0 ≠ a ∈ A and an isogeny g ∈ Hom_{R}(\underline{N}, \underline{M}) with f ∘ g = a · id_{\underline{N}} and g ∘ f = a · 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) ≠ 𝔞 ⊂ A be an ideal and let \underline{E} = (E, φ) be an abelian Anderson A-module over R. The 𝔞-torsion submodule \underline{E}[𝔞] of \underline{E} is the closed subscheme of E defined by \underline{E}[𝔞](S) = \{ P ∈ E(S): φ_{a}(P) = 0 for all a ∈ 𝔞 \} on any R-algebra S.

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

If 𝔞+J = A_{R} and s̄ = Spec Ω is a geometric base point of Spec R, then we also prove in Theorem 6.6 that \underline{E}[𝔞](Ω) is a free A/𝔞-module of rank r which carries a continuous action of the étale fundamental group π_{1}^{ét}(Spec R, s̄).

In the final Section 7 we turn towards the case where 𝔭 ⊂ A is a maximal ideal and where all elements of γ(𝔭) ⊂ R are nilpotent. In this case, we can associate with an A-motive \underline{M} over R a local shtuka \underline{M̂}_{𝔭}(\underline{M}); see Example 7.2 and with an abelian Anderson A-module \underline{E} a divisible local Anderson module \underline{E}[𝔭^{∞}] := lim_{⟶} \underline{E}[𝔭^{n}] in the sense of [HS15]; see Definition 7.3 and Theorem 7.6. If \underline{M} = \underline{M}(\underline{E})

then \underline{M̂}_{𝔭}(\underline{M}) and \underline{E}[𝔭^{∞}] 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

ℕ_{>0} and ℕ_{0} the positive, respectively the non-negative integers, 𝔽_{q} a finite field with q elements and characteristic p, C a smooth projective geometrically irreducible curve over 𝔽_{q}, Q := 𝔽_{q}(C) the function field of C, ∞ a fixed closed point of C, 𝔽_{∞} the residue field of the point ∞ ∈ C, A := Γ(C ∖ {∞}, O_{C}) the ring of regular functions on C outside ∞, (R, γ: A → R) an A-ring, that is a ring R with a ring homomorphism γ: A → R, A_{R} := A ⊗_{𝔽_{q}} R, σ := id_{A} ⊗ Frob_{q,R} the endomorphism of A_{R} with a ⊗ b ↦ a ⊗ b^{q} for a ∈ A and b ∈ R, σ^{∗}M := M ⊗_{R, Frob_{q,R}} R = M ⊗_{A_{R}, σ} A_{R} the Frobenius pullback for an A_{R}-module M, σ^{∗}V := V ⊗_{R, Frob_{q,R}} R the Frobenius pullback more generally for an R-module V, σ_{V}^{∗} v := v ⊗ 1 ∈ σ^{∗}V for an element v ∈ V, σ^{∗}f := f ⊗ id: σ^{∗}M → σ^{∗}N for a morphism f: M → N of A_{R}-modules, J := ker(γ ⊗ id_{R}: A_{R} → R) = (a ⊗ 1 − 1 ⊗ γ(a): a ∈ A) ⊂ A_{R}.

Note that γ makes R into an 𝔽_{q}-algebra. Further note that J is a locally free A_{R}-module of rank 1. Indeed, J = I ⊗_{A_{A}} A_{R} for the ideal I := (a ⊗ 1 − 1 ⊗ a: a ∈ A) ⊂ A_{A} = A ⊗_{𝔽_{q}} A. The latter is a locally free A_{A}-module of rank 1 by Nakayama’s lemma, because I ⊗_{A_{A}} A_{A}/I = I/I^{2} = Ω_{A/𝔽_{q}}^{1} 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 𝔽_{q}[t] by applying the following

Lemma 1.4. Let a ∈ A∖𝔽_{q} and let 𝔽_{q}[t] be the polynomial ring in the variable t. Then the homomorphism 𝔽_{q}[t] → A, t ↦ a makes A into a finite free 𝔽_{q}[t]-module of rank equal to −[𝔽_{∞} : 𝔽_{q}] ord_{∞}(a), where ord_{∞} is the normalized valuation of the discrete valuation ring O_{C,∞}.

Proof. If ord_{∞}(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 ∈ 𝔽_{q} which was excluded. Therefore a is non-constant and defines a finite surjective morphism of curves f: C → ℙ_{𝔽_{q}}^{1} with Spec A → Spec 𝔽_{q}[t] = ℙ_{𝔽_{q}}^{1} ∖{∞′}, where ∞′ ∈ ℙ_{𝔽_{q}}^{1} is the pole of t. By [GW10, Proposition 15.31] its degree can be computed in the fiber f^{−1}(∞′) = \{∞\} as deg f = [𝔽_{∞} : 𝔽_{∞′}]·e_{f}(∞) where 𝔽_{∞′} = 𝔽_{q} and e_{f}(∞) = ord_{∞} f^{∗}(\frac{1}{t}) = − ord_{∞}(a) is the ramification index of f at ∞. Since Spec A = f^{−1}(Spec 𝔽_{q}[t]) we conclude that A is a finite (locally) free 𝔽_{q}[t]-module of rank −[𝔽_{∞} : 𝔽_{q}] ord_{∞}(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, τ_{M}) consisting of a locally free A_{R}-module M of rank r and an isomorphism outside the zero locus V(J) of J between the induced finite locally free sheaves τ_{M}: σ^{∗}M|_{Spec A_{R}∖V(J)} ^{∼} ⟶ M|_{Spec A_{R}∖V(J)}.

A morphism f: (M, τ_{M}) → (N, τ_{N}) between A-motives is an A_{R}-homomorphism f: M → N which satisfies f ∘ τ_{M} = τ_{N} ∘ σ^{∗}f. We write Hom_{R}(\underline{M}, \underline{N}) for the A-module of morphisms between \underline{M} and \underline{N}. The elements of QHom_{R}(\underline{M}, \underline{N}) := Hom_{R}(\underline{M}, \underline{N}) ⊗_{A} Q are called quasi-morphisms. We also set End_{R}(\underline{M}) := Hom_{R}(\underline{M}, \underline{M}) and QEnd_{R}(\underline{M}) := QHom_{R}(\underline{M}, \underline{M}) = End_{R}(\underline{M}) ⊗_{A} Q.

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

Lemma 2.2. Let f: M → N be a homomorphism between finite locally free A_{R}-modules M and N of the same rank, and assume that coker f is a finitely generated R-module, then f is injective and coker f is a finite locally free R-module.

Proof. To make the proof more transparent, we choose an element t ∈ A ∖ 𝔽_{q}. Then A is a finite free 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 coker f. By the Cayley-Hamilton Theorem [Eis95, Theorem 4.3] there is a monic polynomial g ∈ R[t] which annihilates coker f. This implies on the one hand that

M/gM ⟶ N/gN ⟶ coker f ⟶ 0

is exact, and therefore 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[\frac{1}{g}] ↠ N[\frac{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 → M[\frac{1}{g}] is injective, and hence also f is injective.

We obtain the exact sequence 0 → M → N → coker f → 0, which yields for every maximal ideal 𝔪 ⊂ R with residue field k = R/𝔪 the exact sequence

0 ⟶ Tor_{1}^{R}(k, coker f) ⟶ M ⊗_{R} k ⟶ N ⊗_{R} k ⟶ (coker f) ⊗_{R} k ⟶ 0 .

Again the k[t]-modules M ⊗_{R} k and N ⊗_{R} k are locally free of the same rank and (coker f) ⊗_{R} k is a torsion k[t]-module, annihilated by g. Since k[t] is a PID, this implies that M ⊗_{R} k → N ⊗_{R} k is injective and so Tor_{1}^{R}(k, coker f) = (0). Since 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 J is an invertible sheaf on Spec A_{R} as we remarked before Lemma 1.4.

Proposition 2.3. (a) Let (M, τ_{M}) be an A-motive. Then there exist integers e, e′ ∈ ℤ such that J^{e} ·τ_{M}(σ^{∗}M) ⊂ M and J^{e′} ·τ_{M}^{−1}(M) ⊂ σ^{∗}M. For any such e, e′ the induced A_{R}-homomorphism τ_{M}: J^{e} · σ^{∗}M → M is injective, and the quotient M/τ_{M}(J^{e} · σ^{∗}M) is a locally free R-module of finite rank, which is annihilated by J^{e+e′}.

(b) An A-motive (M, τ_{M}) is an effective A-motive, if and only if τ_{M}(σ^{∗}M) ⊂ M.

(c) Let (M, τ_{M}) be an effective A-motive over R. Then (M, τ_{M}|_{Spec A_{R}∖V(J)}) is an A-motive. Moreover, τ_{M}: σ^{∗}M → M is injective and coker τ_{M} is a finite locally free R-module.

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

Proof. (a) Working locally on affine subsets of Spec A_{R} we may assume that J is generated by a non-zero divisor h ∈ J. By [EGA, I, Théorème 1.4.1(d1)] we obtain for every generator m of the A_{R}-module σ^{∗}M an integer n such that locally h^{n} · τ_{M}(m) ∈ M. Taking e as the maximum of the n when m runs through a finite generating system of σ^{∗}M, yields J^{e} · τ_{M}(σ^{∗}M) ⊂ M. The inclusion J^{e′} · τ_{M}^{−1}(M) ⊂ σ^{∗}M is proved analogously.

Let e and e′ be any integers with τ_{M}(J^{e}·σ^{∗}M) ⊂ M and τ_{M}^{−1}(J^{e′} ·M) ⊂ σ^{∗}M, whence J^{e+e′} ·M ⊂ τ_{M}(J^{e} · σ^{∗}M). Then M/τ_{M}(J^{e} · σ^{∗}M) is annihilated by J^{e+e′}, and hence a finite module over A_{R}/J^{e+e′} and over R. Therefore τ_{M}: J^{e} · σ^{∗}M → M is injective, and the quotient M/τ_{M}(J^{e} · σ^{∗}M) is a finite locally free R-module by Lemma 2.2.

(c) Since J^{n} ·coker τ_{M} = (0), the map τ_{M}|_{Spec A_{R}∖V(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} coker τ_{M}. Since every h ∈ J which generates J locally on Spec A_{k} is nilpotent on the k-vector space coker τ_{M}, it satisfies h^{d} = 0 by the Cayley-Hamilton theorem from linear algebra. We conclude that J^{d} · coker τ_{M} = (0) and \underline{M} has dimension d.

Proposition 2.4. (a) If S is an R-algebra, then \underline{M} = (M, τ_{M}) ⟼ \underline{M} ⊗_{R} S := (M ⊗_{R} S, τ_{M} ⊗ 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.

(b) Every A-motive over R and every morphism f ∈ Hom(\underline{M}, \underline{N}) between A-motives over R can be defined over a subring R′ of R, which via γ: A → R′ ⊂ R is a finitely generated A-algebra, hence noetherian.

Proof. (a) This is obvious.

(b) Every A-motive \underline{M} = (M, τ_{M}) has a presentation of the form A_{R}^{⊕n_{1}} ⟶^{U} A_{R}^{⊕n_{0}} ⟶^{ρ} M ⟶ 0. Since M is locally free over A_{R}, there is a section s of the epimorphism ρ. It corresponds to an endomorphism S of A_{R}^{⊕n_{0}} with SU = 0 such that there is a map W: A_{R}^{⊕n_{0}} → A_{R}^{⊕n_{1}} with S −Id = UW. The isomorphism τ_{M} lifts to diagram

[figure: (σ^{∗}A_{R}^{⊕n_{1}})|_{Spec A_{R}∖V(J)}; σ^{∗}U; T_{1}; (σ^{∗}A_{R}^{⊕n_{0}})|_{Spec A_{R}∖V(J)}; σ^{∗}ρ; T_{0}; σ^{∗}M|_{Spec A_{R}∖V(J)}; τ_{M}; 0; A_{R}^{⊕n_{1}}|_{Spec A_{R}∖V(J)}; U; A_{R}^{⊕n_{0}}|_{Spec A_{R}∖V(J)}; ρ; M|_{Spec A_{R}∖V(J)}; 0 .] (2.1)

Likewise τ_{M}^{−1} lifts to a similar diagram with vertical morphism T′_{0} and T′_{1}. The equations τ_{M} ∘τ_{M}^{−1} = id and τ_{M}^{−1} ∘τ_{M} = id imply the existence of matrices V and V′ in A_{R}^{n_{1}×n_{0}}|_{Spec A_{R}∖V(J)} with T_{0} ∘T′_{0} −Id = U ∘V and T′_{0}∘T_{0}−Id = σ^{∗}U ∘V′. Let R′ ⊂ 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′_{0}, T′_{1}, V and V′. Define M′ as the A_{R′}- module which is the cokernel of U ∈ A_{R′}^{n_{0}×n_{1}}, and define τ_{M′}: σ^{∗}M′|_{Spec A_{R}∖V(J)} → M′|_{Spec A_{R}∖V(J)} and τ_{M′}^{−1} : M′|_{Spec A_{R}∖V(J)} → σ^{∗}M′|_{Spec A_{R}∖V(J)} as the A_{R′}-homomorphisms given by diagram (2.1) and its analog for τ_{M}^{−1}. Then M′ is via S a direct summand of A_{R′}^{⊕n_{0}}, hence a finite locally free A_{R′}- module, and τ_{M′} and τ_{M′}^{−1} are inverse to each other. It follows from diagram (2.1) that M′ ⊗_{R′} R = M and τ_{M′} ⊗ id_{R} = τ_{M}.

Finally, the assertion for the morphism f ∈ Hom_{R}(\underline{M}, \underline{N}) follows from a diagram similar to (2.1) for f instead of τ_{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 ∈ Hom_{R}(\underline{M}, \underline{N}) be a morphism. Then the set X of points s ∈ Spec R such that f ⊗ id_{κ(s)} = 0 in Hom_{κ(s)}(\underline{M} ⊗_{R} κ(s), \underline{N} ⊗_{R} κ(s)) is open and closed, but possibly empty. Let Spec R̃ ⊂ Spec R be this set, then f ⊗ idR̃ = 0 in HomR̃(\underline{M} ⊗_{R} R̃, \underline{N} ⊗_{R} R̃). In particular if Spec R is connected and S ≠ (0) is an R-algebra, then the map Hom_{R}(\underline{M}, \underline{N}) → Hom_{S}(\underline{M} ⊗_{R} S, \underline{N} ⊗_{R} S), f ↦ f ⊗ id_{S} is injective.

Proof. We fix an element t ∈ A ∖ 𝔽_{q}. Then A is a finite free 𝔽_{q}[t]-module. By Proposition 2.3 we can find integers e, e′ with J^{e} · τ_{N}(σ^{∗}N) ⊂ N and J^{e′} · τ_{M}^{−1}(M) ⊂ σ^{∗}M, such that d := e + e′ is a power of q. We obtain morphisms (t − γ(t))^{e}τ_{N}: σ^{∗}N → N and (t − γ(t))^{e′}τ_{M}^{−1} : M → σ^{∗}M. So the equation f ∘ τ_{M} = τ_{N} ∘ σ^{∗}f implies (t^{d} − γ(t)^{d})f = (t − γ(t))^{e}τ_{N} ∘ σ^{∗}f ∘ (t − γ(t))^{e′}τ_{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]^{⊕n′} ↠ M and R[t]^{⊕n} ↠ N. Then R[t]^{⊕n′} ↠ M ⟶^{f} N ↪ R[t]^{⊕n} is given by a matrix F ∈ R[t]^{n×n′} whose entries are polynomials in t. Let I ⊂ R be the ideal generated by the coefficients of all these polynomials. A prime ideal 𝔭 ⊂ R belongs to the set X if and only if I ⊂ 𝔭. In particular X = V(I) ⊂ Spec R is closed.

On the other hand, we consider the map R[t]^{⊕n} ↠ σ^{∗}N ⟶^{(t−γ(t))^{e}τ_{N}} N ↪ R[t]^{⊕n} as a matrix

T ∈ R[t]^{n×n} and the map R[t]^{⊕n′} ↠ M ⟶^{(t−γ(t))^{e′}τ_{M}^{−1}} σ^{∗}M ↪ R[t]^{⊕n′} as a matrix V ∈ R[t]^{n′×n′}. The

formula (t^{d} − γ(t)^{d})f = (t − γ(t))^{e}τ_{N} ∘ σ^{∗}f ∘ (t − γ(t))^{e′}τ_{M}^{−1} implies (t^{d} − γ(t)^{d})F = T σ(F) V, and it follows that the entries of the matrix (t^{d} − γ(t)^{d})F are polynomials in t whose coefficients lie in I^{q}. If ∑_{i=0}^{ℓ} b_{i}t^{i} is an entry of F then (t^{d} − γ(t)^{d}) ∑_{i=0}^{ℓ} b_{i}t^{i} = ∑_{i=0}^{ℓ+d}(b_{i−d} − γ(t)^{d}b_{i})t^{i} is the corresponding entry of (t^{d} − γ(t)^{d})F and all b_{i−d} − γ(t)^{d}b_{i} ∈ I^{q}. By descending induction on i = ℓ + d, …, 0 we see that all b_{i} ∈ I^{q}. It follows that the finitely generated ideal I ⊂ R satisfies I = I^{q}. By Nakayama’s lemma [Eis95, Corollary 4.7] there is an element b ∈ 1 + I such that b·I = (0). Now let 𝔭 ⊂ R be a prime ideal which lies in X, that is I ⊂ 𝔭. Then 𝔭 lies in the open subset Spec R[\frac{1}{b}] ⊂ Spec R on which F = 0 and hence f = 0. In particular X ⊂ Spec R[\frac{1}{b}] ⊂ X. Therefore X is open and closed and f = 0 on X.

Now let Spec R be connected and S ≠ (0) be an R-algebra. Let f ∈ Hom_{R}(\underline{M}, \underline{N}) be such that f ⊗ id_{S} = 0 in Hom_{S}(\underline{M} ⊗_{R} S, \underline{N} ⊗_{R} S). Let s ∈ Spec S be a point and let s′ ∈ Spec R be its image. Then f ⊗ id_{κ(s′)} = 0 and the set X from above is non-empty. Since it is open and closed and Spec R is connected, it follows that X = Spec R and f = 0. This proves the injectivity.

Corollary 2.6. Let \underline{M} and \underline{N} be A-motives over R with Spec R connected. Then Hom_{R}(\underline{M}, \underline{N}) is a finite projective A-module of rank less or equal to (rk \underline{M})·(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/𝔪 for 𝔪 ⊂ 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 O be a commutative unitary ring. An O-module scheme over R is a commutative group scheme E over R together with a ring homomorphism O → End_{R}(E).

For a group scheme E over Spec R we let E^{n} := E ×_{R} … ×_{R} E be the n-fold fiber product over R. We denote by e: Spec R → E its zero section and by Lie E := Hom_{R}(e^{∗}Ω_{E/R}^{1}, R) the tangent space of E along e. If E is smooth over Spec R, then Lie E is a locally free R-module of rank equal to the relative dimension of E over R. In particular Lie E^{n} = (Lie E)^{⊕n}. For a homomorphism f: E → E′ of group schemes over Spec R we denote by Lie f: Lie E → Lie E′ the induced morphism of R-modules. Also we define the kernel of f as the R-group scheme ker f := E ×_{f, E′,e′} Spec R where e′: Spec R → E′

is the zero section. There is a canonical isomorphism

E ×_{f, E′,f} E ^{∼} ⟶ E ×_{R} ker f  (3.1)

given on T -valued points P, Q ∈ E(T) for any R-scheme T by (P, Q) ↦ (P, Q − P). If P ∈ E(k) for a field k and P′ = f(P) ∈ E′(k), pulling back (3.1) under P: Spec k → E yields an isomorphism of the fiber Spec k ×_{P′, E′,f} E of f over P′ with Spec k ×_{R} ker f.

On 𝔾_{a,R} = Spec R[x] the elements b ∈ R, and in particular γ(a) ∈ R for a ∈ 𝔽_{q}, act via b^{∗}: R[x] → R[x], x ↦ bx. This makes 𝔾_{a,R} into an 𝔽_{q}-module scheme. In addition let τ := Frob_{q,𝔾_{a,R}} be the relative q-Frobenius endomorphism of 𝔾_{a,R} = Spec R[x] given by x ↦ x^{q}. It satisfies Lie τ = 0 and τ ∘ b = b^{q} ∘ τ. We let

R{τ} := { ∑_{i=0}^{n} b_{i}τ^{i}: n ∈ ℕ_{0}, b_{i} ∈ R } with τb = b^{q}τ  (3.2)

be the non-commutative polynomial ring in τ over R. For an element f = ∑_{i} b_{i}τ^{i} ∈ R{τ} we set f(x) := ∑_{i} b_{i}x^{q^{i}}.

Lemma 3.2. There is an isomorphism of R-modules R{τ}^{d′×d} ^{∼} ⟶ Hom_{R-groups,𝔽_{q}-lin}(𝔾_{a,R}^{d}, 𝔾_{a,R}^{d′}), which sends the matrix F = (f_{ij})_{i,j} to the 𝔽_{q}-equivariant morphism f: 𝔾_{a,R}^{d} → 𝔾_{a,R}^{d′} of group schemes over R with f^{∗}(y_{i}) = ∑_{j} f_{ij}(x_{j}) where 𝔾_{a,R}^{d} = Spec R[x_{1}, …, x_{d}] and 𝔾_{a,R}^{d′} = Spec R[y_{1}, …, y_{d′}]. Under this isomorphism the map f ↦ Lie f is given by the map R{τ}^{d′×d} → R^{d′×d}, F = ∑_{n} F_{n}τ^{n} ↦ F_{0}.

Proof. This is straight forward to prove using Lucas’s theorem [Luc78] on congruences of binomial coefficients which states that ( \begin{matrix}pn+t \\ pm+s \end{matrix} ) ≡ ( \begin{matrix}n \\ m \end{matrix}) ( \begin{matrix}t \\ s \end{matrix}) mod p for all n, m, t, s ∈ ℕ_{0}, and implies that ( \begin{matrix}n \\ i \end{matrix}) ≡ 0 mod p for all 0 < i < n if and only if n = p^{e} for an e ∈ ℕ_{0}.

Remark 3.3. The affine group scheme E and its multiplication map Δ: E ×_{R} E → E are described by its coordinate ring B_{E} := Γ(E, O_{E}) together with the comultiplication Δ^{∗}: B_{E} → B_{E} ⊗_{R} B_{E}. If we write 𝔾_{a,R} = Spec R[ξ] the map

M(\underline{E}) ^{∼} ⟶ \{ x ∈ B_{E}: Δ^{∗}x = x ⊗ 1 + 1 ⊗ x and φ_{a}^{∗}x = γ(a)x for all a ∈ 𝔽_{q} \}
m ⟼ m^{∗}(ξ)

is an isomorphism of A_{R}-modules. Choosing an element λ ∈ 𝔽_{q} with 𝔽_{q} = 𝔽_{p}(λ) we obtain an exact sequence of R-modules

[figure: 0; M(\underline{E}); B_{E}; B_{E} ⊗_{R} B_{E} ⊕ B_{E}; m; m^{∗}(ξ) , x; (Δ^{∗}x − x ⊗ 1 − 1 ⊗ x , φ_{λ}^{∗}x − γ(λ)x)]  (3.3)

This shows that for every flat R-algebra R′ we have M(\underline{E}) ⊗_{R} R′ = M(\underline{E} ×_{R} Spec R′), because Γ(E ×_{R} R′, O_{E×R′}) = B_{E} ⊗_{R} R′. In particular, if R′ satisfies condition (a) of Definition 1.2 then M(\underline{E}) ⊗_{R} R′ ≅ R′\{τ\}^{1×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}) ⊗_{R} S = M(\underline{E} ×_{Spec R} Spec S). Namely, we choose a faithfully flat morphism R → R′ as in Definition 1.2(a) and tensor (3.3) with S ⊗_{R} R′. This tensor product stays exact by Lemma 3.2 because M(\underline{E}) ⊗_{R} R′ ≅ R′\{τ\}^{1×d}. Since S → S ⊗_{R} R′ 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 τ: m ↦ Frob_{q,𝔾_{a,R}}∘ m. Since τ(bm) = b^{q}τ(m) the map τ is σ-semilinear and induces an A_{R}-linear map τ_{M}: σ^{∗}M → M. We set \underline{M}(\underline{E}) := (M(\underline{E}), τ_{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, φ) is an abelian Anderson A-module of rank r and dimension d then \underline{M}(\underline{E}) = (M, τ_{M}) is an effective A-motive of rank r and dimension d. There is a canonical isomorphism of R-modules

coker τ_{M} ^{∼} ⟶ Hom_{R}(Lie E, R), m mod τ_{M}(σ^{∗}M) ⟼ Lie m .  (3.4)

The contravariant functor \underline{E} ↦ \underline{M}(\underline{E}) is fully faithful. Its essential image consists of all effective A-motives \underline{M} = (M, τ_{M}) over R of some dimension d, for which there exists a faithfully flat ring homomorphism R → R′ such that M ⊗_{R} R′ is a finite free left R′{τ}-module under the map τ: M → M, m ↦ τ_{M}(σ_{M}^{∗}m).

Proof. We first establish the isomorphism (3.4). If m = τ_{M}(∑_{i} m_{i} ⊗ b_{i}) = ∑_{i} b_{i} ∘ Frob_{q,𝔾_{a,R}}∘ m_{i} with m_{i} ∈ M and b_{i} ∈ R, then Lie m = 0 because Lie Frob_{q,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 → R′, see [EGA, § 0_{I}.6.6], such that E ⊗_{R} R′ ≅ 𝔾_{a,R′}^{d} and Lie E ⊗_{R} R′ ≅ (R′)^{⊕d}. Then M ⊗_{R} R′ ≅ R′{τ}^{1×d} by Remark 3.3, and the inverse map is given by the natural inclusion (R′)^{1×d} ⊂ R′{τ}^{1×d}, F_{0} ↦ F_{0}τ^{0}.

As a consequence, coker τ_{M} is a locally free R-module of rank equal to d = dim \underline{E} and annihilated by J^{d} because of condition (b) in Definition 1.2. This implies coker τ_{M}|_{Spec A_{R}∖V(J)} = (0), and therefore the morphism τ_{M}: σ^{∗}M|_{Spec A_{R}∖V(J)} → M|_{Spec A_{R}∖V(J)} is surjective. By [GW10, Corollary 8.12] it is an isomorphism, because M and σ^{∗}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, φ) and \underline{E}′ = (E′, φ′) be two abelian Anderson A-modules over R and let \underline{M} = \underline{M}(\underline{E}) and \underline{M}′ = \underline{M}(\underline{E}′) be the associated effective A-motives. To prove that the map

Hom_{R}(\underline{E}, \underline{E}′) ⟶ Hom_{R}(\underline{M}, \underline{M}′), f ⟼ (m′ ↦ m′ ∘ f)  (3.5)

is bijective, we again apply a faithfully flat base change R → R′, such that E ⊗_{R} R′ ≅ 𝔾_{a,R′}^{d} and E′ ⊗_{R} R′ ≅ 𝔾_{a,R′}^{d′}. Then Hom_{R′}(\underline{E} ⊗_{R} R′, \underline{E}′ ⊗_{R} R′) ≅ \{F ∈ R′\{τ\}^{d′×d}: φ′_{a} ∘ F = F ∘ φ_{a} ∀ a ∈ A\} by Lemma 3.2. Also \underline{M}(\underline{E}) ⊗_{R} R′ ≅ R′\{τ\}^{1×d} and \underline{M}(\underline{E}′) ⊗_{R} R′ ≅ R′\{τ\}^{1×d′}. The condition h ∘ τ_{M′} = τ_{M} ∘σ^{∗}h on an element h ∈ Hom_{R′}(\underline{M}(\underline{E}′)⊗_{R} R′, \underline{M}(\underline{E})⊗_{R} R′) implies that h: R′\{τ\}^{1×d′} → R′\{τ\}^{1×d} is a homomorphism of left R′{τ}-modules, hence given by multiplication on the right by a matrix H ∈ R′{τ}^{d′×d}. Then m′∘φ′_{a}∘H = h((a⊗1)·m′) = (a⊗1)·h(m′) = m′∘H ∘φ_{a} implies φ′_{a}∘H = H ∘φ_{a} for all a ∈ A. It follows that the map (3.5) is bijective over R′. So every h ∈ Hom_{R}(\underline{M}(\underline{E}′), \underline{M}(\underline{E})) gives rise to a morphism f′ ∈ Hom_{R′}(\underline{E} ⊗_{R} R′, \underline{E}′ ⊗_{R} R′) 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 → R′ is effective, f′ descends to the desired f ∈ Hom_{R}(\underline{E}, \underline{E}′). This shows that the functor \underline{E} ↦ \underline{M}(\underline{E}) is fully faithful.

Let \underline{M} = (M, τ_{M}) be an effective A-motive of dimension d over R for which there exists a faithfully flat ring homomorphism R → R′ such that M ⊗_{R} R′ ≅ R′{τ}^{1×d}. Observe that coker(τ_{M} ⊗ id_{R′}) ≅ (R′{τ}/R′{τ}τ)^{1×d} = (R′)^{1×d}. For all a ∈ A we have τ · (a ⊗ 1)m = σ(a ⊗ 1) · τ(m) = (a ⊗ 1)τm. Therefore the map m ↦ (a ⊗ 1)m is a homomorphism of left R′{τ}-modules, and hence given by (a ⊗ 1)m = m · φ′_{a} for a matrix φ′_{a} ∈ R′\{τ\}^{d×d}. Then \underline{E}′ := (E′ = 𝔾_{a,R′}^{d}, φ′: A → R′\{τ\}^{d×d}, a ↦ φ′_{a})

satisfies \underline{M}(\underline{E}′) = \underline{M} ⊗_{R} R′. Again (a ⊗ 1 − 1 ⊗ γ(a))^{d} = 0 on coker τ_{M} implies (Lie φ′_{a} − γ(a))^{d} = 0 on Lie E′. So \underline{E}′ is an abelian Anderson A-module over R′ with \underline{M}(E′) ≅ \underline{M} ⊗_{R} R′. Consider the ring R′′ := R′ ⊗_{R} R′ and the two maps p_{1}, p_{2} : R′ → R′′ given by p_{1}(b′) = b′ ⊗ 1 and p_{2}(b′) = 1 ⊗ b′. The canonical isomorphism p_{1}^{∗}(\underline{M} ⊗_{R} R′) = p_{2}^{∗}(\underline{M} ⊗_{R} R′) induces an isomorphism p_{1}^{∗}\underline{E}′ ≅ p_{2}^{∗}\underline{E}′ which is a descend datum on \underline{E}′ relative to R → R′. 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 φ: A → End_{R-groups}(E) such that (E, φ) ⊗_{R} R′ ≅ \underline{E}′. By [EGA, IV_{2}, Proposition 2.7.1 and IV_{4}, Corollaire 17.7.3] the group scheme E is affine and smooth over R and hence (E, φ) is an abelian Anderson A-module with \underline{M}(E, φ) ≅ \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 ∈ ℕ_{>0} over R is a pair \underline{E} = (E, φ) consisting of a smooth affine group scheme E over Spec R of relative dimension 1 and a ring homomorphism φ: A → End_{R-groups}(E), a ↦ φ_{a} satisfying the following conditions:

(a) Zariski-locally on Spec R there is an isomorphism α: E ^{∼} ⟶ 𝔾_{a,R} of 𝔽_{q}-module schemes such that

(b) the coefficients of Φ_{a} := α ∘ φ_{a} ∘ α^{−1} = ∑_{i≥0} b_{i}(a)τ^{i} ∈ End_{R-groups,𝔽_{q}-lin}(𝔾_{a,R}) = R{τ} satisfy

b_{0}(a) = γ(a), b_{r(a)}(a) ∈ R^{×} and b_{i}(a) is nilpotent for all i > r(a) := −r [𝔽_{∞} : 𝔽_{q}] ord_{∞}(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. (a) Let b = ∑_{i=0}^{n} b_{i}τ^{i} ∈ R{τ} and let r be a positive integer such that b_{r} ∈ R^{×} and b_{i} is nilpotent for all i > r. Then there is a unique unit c = ∑_{i≥0} c_{i}τ^{i} ∈ R{τ}^{×} with c_{0} = 1 and c_{i} nilpotent for i > 0, such that c^{−1}bc = ∑_{i=0}^{r} b′_{i}τ^{i} with b′_{r} ∈ R^{×}.

(b) Let Spec R be connected and let b = ∑_{i=0}^{m} b_{i}τ^{i} and c = ∑_{i=0}^{n} c_{i}τ^{i} ∈ R{τ} with m, n > 0 and b_{m}, c_{n} ∈ R^{×}. Let d ∈ R{τ} ∖ {0} satisfy db = cd. Then m = n and d = ∑_{i=0}^{r} d_{i}τ^{i} with d_{r} ∈ R^{×}.

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

(b) We write d = ∑_{i=0}^{r} d_{i}τ^{i} with d_{r} ≠ 0. The equation db = cd implies ∑_{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}, …, 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} ∈ 𝔭 for every prime ideal 𝔭 ⊂ R. For n+r > i ≥ n we obtain c_{n}d_{i−n}^{q^{n}} = ∑_{0≤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} ∈ 𝔭 for every prime ideal 𝔭 ⊂ R for all i − n = r, …, 0. So the ideal I := (d_{i}: 0 ≤ i ≤ r) ⊂ R

is contained in every prime ideal 𝔭 ⊂ R. Now i = m + r yields d_{r}b_{m}^{q^{r}} = ∑_{j=m}^{m+r} c_{j}d_{m+r−j}^{q^{j}}, whence d_{r} ∈ I^{q}.

For m+r > i ≥ m we obtain d_{i−m}b_{m}^{q^{i−m}} = ∑_{0≤j<m} d_{i−j}b_{j}^{q^{i−j}} − ∑_{0≤j≤n} c_{j}d_{i−j}^{q^{j}} and by descending induction

on i it follows that d_{i−m} ∈ I^{q} for all i − m = r, …, 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 ∈ 1 + I such that f·I = (0). Since I ⊂ 𝔭 for all prime ideals 𝔭 ⊂ 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}) ⊂ R. Again I = I^{q^{m}} and by [Eis95, Corollary 4.7] there is an element f ∈ 1 + I such that f ·d_{r} = 0. Now assume that d_{r} ∈ 𝔭 for some prime ideal 𝔭 ⊂ R. Then f ∉ 𝔭, whence 𝔭 ∈ Spec R[\frac{1}{f}] ⊂ Spec R and d_{r} = 0 on the open neighborhood Spec R[\frac{1}{f}] of 𝔭. Since the set of prime ideals 𝔭 ⊂ R with d_{r} ∈ 𝔭 is closed in Spec R and the latter is connected, it follows that d_{r} = 0 on all of 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} ∈ R^{×} 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 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 Spec R′ and the ring homomorphism R → R′ is faithfully flat. So \underline{E} satisfies conditions (a) and (b) of Definition 1.2. Choose an element t ∈ A∖𝔽_{q}. Then A is a finite free 𝔽_{q}[t]-module of rank equal to −[𝔽_{∞} : 𝔽_{q}] ord_{∞}(t) by Lemma 1.4. Writing Φ_{t} = ∑_{i=0}^{r(t)} b_{i}(t)τ^{i} with r(t) = −r [𝔽_{∞} : 𝔽_{q}] ord_{∞}(t) and b_{r(t)}(t) ∈ (R′)^{×}, we make the following

Claim. As an R′[t]-module M(\underline{E}) ⊗_{R} R′ = ⨁_{ℓ=0}^{r(t)−1} R′[t]·τ^{ℓ}.  (3.6)

By Remark 3.3 and Lemma 3.2 we have M(\underline{E}) ⊗_{R} R′ = M(\underline{E} ×_{Spec R} Spec R′) = R′\{τ\}. We prove by induction on n that for every c = ∑_{i=0}^{n} c_{i}τ^{i} ∈ R′\{τ\} = M(\underline{E}) there are uniquely determined elements f_{ℓ}(t) ∈ R′[t] such that c = ∑_{ℓ=0}^{r(t)−1} f_{ℓ}(t) · τ^{ℓ}. If n < r(t) then we take f_{ℓ}(t) = c_{ℓ}. If n ≥ r(t), dividing c by Φ_{t} on the right produces uniquely determined g = ∑_{i=0}^{n−r(t)} g_{i}τ^{i} and h = ∑_{ℓ=0}^{r(t)−1} h_{ℓ}τ^{ℓ} ∈ R′{τ} with c = gΦ_{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}}(c_{i+r(t)} − ∑_{j=i+1}^{i+r(t)} g_{j} b_{i+r(t)−j}^{q^{j}}) for i = n − r(t), …, 0 and h_{ℓ} = c_{ℓ} − ∑_{j=0}^{ℓ} g_{j} b_{ℓ−j}^{q^{j}} for

ℓ = r(t) − 1, …, 1. The induction hypothesis implies g = ∑_{ℓ=0}^{r(t)−1} f̃_{ℓ}(t)·τ^{ℓ}. Now f_{ℓ}(t) := f̃_{ℓ}(t)·t + h_{ℓ}

satisfies c = ∑_{ℓ=0}^{r(t)−1} f_{ℓ}(t)·τ^{ℓ}. This proves the claim.

By faithfully flat descent [EGA, IV_{2}, Proposition 2.5.2] with respect to R[t] → R′[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∈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 ρ: A_{R}^{I} ↠ 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 ρ 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, IV_{3}, Théorème 11.3.10] it follows that M(\underline{E}) is finite locally free over A_{R}, because for every point s ∈ Spec R the finite A_{κ(s)}-module M(\underline{E}) ⊗_{R} κ(s) is a free κ(s)[t]-module and hence a torsion free and flat A_{κ(s)}-module. Its rank is r as can be computed by comparing the ranks of A_{R′} and M(\underline{E}) ⊗_{R} R′ over R′[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, φ) be an abelian Anderson A-module of dimension 1 and rank r over R. Let R → R′ be a faithfully flat ring homomorphism and let α: E ×_{R} Spec R′ ^{∼} ⟶ 𝔾_{a,R′} be an isomorphism of 𝔽_{q}-module schemes as in Definition 1.2(a). For a ∈ A write

Φ_{a} := ∑_{i=0}^{n(a)}b_{i}(a)τ^{i} := α ∘ φ_{a} ∘ α^{−1} ∈ End_{R′-groups,𝔽_{q}-lin}(𝔾_{a,R′}) = R′{τ} ,

where n(a) ∈ ℕ_{0} and b_{i}(a) ∈ R′. For a ∈ 𝔽_{q} we obtain Φ_{a} = γ(a)·τ^{0}. For t := a ∈ A∖𝔽_{q} we consider A as a finite free 𝔽_{q}[t]-module of rank −[𝔽_{∞} : 𝔽_{q}] ord_{∞}(a) by Lemma 1.4. Then M(\underline{E}) is a finite locally free R[t]-module of rank r(a) := −r [𝔽_{∞} : 𝔽_{q}] ord_{∞}(a) by condition (c) of Definition 1.2. Let 𝔭 ⊂ R′ be a prime ideal, set k = Frac(R′/𝔭), and consider the abelian Anderson A-module \underline{E} ×_{R} Spec k over k and the free k[t]-module M(\underline{E}) ⊗_{R} k = M(\underline{E} ×_{R} Spec k) of rank r(a). By an argument similarly to our claim (3.6) we see that deg_{τ} (Φ_{a} ⊗_{R′} 1_{k}) = r(a), that is b_{r(a)}(a) ⊗ 1_{k} ∈ k^{×} and b_{i}(a) ⊗ 1_{k} = 0 for all i > r(a). This implies that b_{r(a)}(a) ∈ (R′)^{×} 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 α such that Φ_{a} = ∑_{i=0}^{r(a)} b_{i}(a)τ^{i} with b_{r(a)}(a) ∈ (R′)^{×} for one a ∈ A, and by Lemma 3.8(b) this then holds for all a ∈ A, because Φ_{a}Φ_{b} = Φ_{ab} = Φ_{b}Φ_{a}. By condition (b) of Definition 1.2 we have b_{0}(a) = γ(a). Thus \underline{E} ×_{R} Spec R′ is a Drinfeld A-module of rank r over R′ in standard form.

It remains to show that we can replace the faithfully flat covering Spec R′ → Spec R by a Zariski covering. For this purpose consider R′′ := R′ ⊗_{R} R′ and the two projections pr_{i}: Spec R′′ → Spec R′ onto the i-th factor for i = 1, 2. Then h := ∑_{i≥0} h_{i}τ^{i} := pr_{2}^{∗}α ∘ pr_{1}^{∗}α^{−1} ∈ R′′{τ}^{×} satisfies h_{0} ∈ (R′′)^{×} and h_{i} is nilpotent for all i > 0; see [Mat96, Proposition 1.4]. By Lemma 3.8(b) the equation pr_{2}^{∗}Φ_{a}∘h = h∘pr_{1}^{∗}Φ_{a} implies that h_{i} = 0 for all i > 0 and h = h_{0} ∈ (R′′)^{×} ⊂ R′′\{τ\}^{×}. The cocycle \underline{h} := (Spec R′ → Spec R, h) defines an element in the Čech cohomology group Ȟ_{fpqc}^{1}(Spec R, 𝔾_{m}). By Hilbert 90, see [Mil80, Proposition III.4.9] we have Ȟ_{fpqc}^{1}(Spec R, 𝔾_{m}) = Ȟ_{Zar}^{1}(Spec R, 𝔾_{m}). This means that there is a Zariski covering Spec R̃ → Spec R, where Spec R̃ = ∐_{i} Spec R̃_{i} is a disjoint union of open affine subschemes Spec R̃_{i} ⊂ Spec R, and a unit h̃ = (h̃_{ij})_{i,j} ∈ (R̃ ⊗_{R} R̃)^{×} = ∏_{i,j}(R̃_{i} ⊗_{R} R̃_{j})^{×}, such that (Spec R̃ → Spec R, h̃) = \underline{h}. Let Ẽ be the smooth affine group and 𝔽_{q}-module scheme over Spec R with β_{i}: Ẽ|_{Spec R̃_{i}} ^{∼} ⟶ 𝔾_{a,R̃_{i}} and β_{j} = h̃_{ij} ∘β_{i} on Spec R̃_{i} ⊗_{R} R̃_{j}. Then over Spec R′ ⊗_{R} R̃ = ∐_{i} Spec R′ ⊗_{R} R̃_{i} we have an isomorphism α̃ := (β_{i}^{−1}∘α)_{i}: E ^{∼} ⟶ Ẽ. Let p_{i}: Spec(R′⊗_{R} R̃)⊗_{R}(R′⊗_{R} R̃) → Spec R′⊗_{R} R̃ be the projection onto the i-th factor for i = 1, 2. Then p_{2}^{∗}α̃∘p_{1}^{∗}α̃^{−1} = (h̃_{ij}^{−1}h)_{i,j} = 1. This shows that α̃ descends to an isomorphism α̃: E ^{∼} ⟶ Ẽ over Spec R by [BLR90, § 6.1, Theorem 6(a)]. On Spec R̃_{i}, now β_{i}∘α̃: E ^{∼} ⟶ 𝔾_{a,R̃_{i}} is an isomorphism of 𝔽_{q}-module schemes. Moreover Φ̃_{a} := β_{i}α̃∘φ_{a}∘α̃^{−1}β_{i}^{−1} ∈ R̃_{i}{τ} satisfies Φ̃_{a} ⊗ 1_{R′} = Φ_{a} ⊗ 1R̃_{i} in (R′ ⊗_{R} R̃_{i}){τ} ⊃ R̃_{i}{τ} and by what we proved for Φ_{a} above, this implies that \underline{E} is a Drinfeld A-module of rank r over R which by R̃ and (β_{i} ∘ α̃)_{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 𝔽_{q}-shtukas and finite locally free strict 𝔽_{q}-module schemes; see also [Abr06], [Tag95, § 1], [Lau96, § B.3] and [HS15, §§ 3-5].

Definition 4.1. A finite 𝔽_{q}-shtuka over R is a pair \underline{V} = (V, F_{V}) consisting of a finite locally free Rmodule V on R and an R-module homomorphism F_{V} : σ^{∗}V → V. A morphism f: (V, F_{V}) → (V′, F_{V′}) of finite 𝔽_{q}-shtukas is an R-module homomorphism f: V → V′ satisfying f ∘ F_{V} = F_{V′} ∘ σ^{∗}f.

We say that F_{V} is nilpotent if there is an integer n such that F_{V}^{n} := F_{V} ∘σ^{∗}F_{V} ∘…∘σ^{(n−1)∗}F_{V} = 0. A finite 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′ the τ-invariants of \underline{V} over R′ as the 𝔽_{q}-vector space

\underline{V}^{τ} (R′) := \{ v ⊗ V ⊗_{R} R′: v = F_{V} (σ_{V}^{∗} v) \} .  (4.1)

Recall that an R-group scheme G = Spec B is finite locally free if B is a finite locally free Rmodule. By [EGA, I_{new}, Proposition 6.2.10] this is equivalent to G being finite, flat and of finite presentation over Spec R. Every finite locally free R-group scheme G = Spec B is a relative complete intersection by [SGA 3, III.4.15]. This means that locally on 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, IV_{4}, Proposition 19.3.7]. The zero section e: Spec R → G defines an augmentation e_{B} := e^{∗}: B ↠ R of the R-algebra B. Set I_{B} := ker e_{B}. For the polynomial ring R[\underline{X}] = R[X_{1}, …, X_{n}] set I_{R[\underline{X}]} = (X_{1}, …, X_{n}) and e_{R[\underline{X}]} : R[\underline{X}] ↠ R, X_{ν} ↦ 0. Faltings [Fal02] and Abrashkin [Abr06] consider the deformation B^{♭} := R[\underline{X}]/(I ·I_{R[\underline{X}]}) and the canonical epimorphism B^{♭} ↠ B. They remark that there is a unique morphism

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

lifting the comultiplication Δ: B → B ⊗_{R} B and satisfying ( id_{B^{♭}} ⊗e_{B}^{♭})∘Δ^{♭} = id_{B^{♭}} = (e_{B}^{♭} ⊗ id_{B^{♭}})∘Δ^{♭}, where e_{B}^{♭}: B^{♭} ↠ R is the augmentation map; see [Abr06, § 1.2] or [HS15, Remark after Definition 3.5]. It satisfies Δ^{♭}(x)−x⊗1−1⊗x ∈ I_{B^{♭}} ⊗I_{B^{♭}} for all x ∈ I_{B^{♭}}. Set G = (G, G^{♭}) := (Spec B, Spec B^{♭}). The co-Lie complex of G over SpecR (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

ℓG/ Spec R^{•}:  0 ⟶ (I/I^{2}) ⊗_{B, e_{B}} R ⟶^{d} Ω_{R[\underline{X}]/R}^{1} ⊗_{R[\underline{X}], e_{R[\underline{X}]}} R ⟶ 0  (4.2)

concentrated in degrees −1 and 0 with d being the differential map. Note that (I/I^{2}) ⊗_{B, e_{B}} R = ker(B^{♭} ↠ B) and Ω_{R[\underline{X}]/R}^{1} ⊗_{R[\underline{X}], e_{R[\underline{X}]}} R = ker(e_{B}^{♭})/ ker(e_{B}^{♭})^{2} can be computed from (B, B^{♭}). 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 ω_{G} := H^{0}(ℓG/ Spec R^{•}) := coker d. We can now recall the definition of strict 𝔽_{q}-module schemes from Faltings [Fal02] and Abrashkin [Abr06]; see also [HS15, § 4].

Definition 4.2. Let (G, [ .]) be a pair, where G = Spec B is an affine flat commutative group scheme over R which is a relative complete intersection and where [ .]: 𝔽_{q} → End_{R-groups}(G), a ↦ [a] is a ring homomorphism. Then (G, [ .]) is called a strict 𝔽_{q}-module scheme if there exists a presentation B = R[\underline{X}]/I and a lift [ .]^{♭}: 𝔽_{q} → End_{R-algebras}(B^{♭}), a ↦ [a]^{♭} of the 𝔽_{q}-action on G, such that the induced action on ℓG/ Spec R^{•} is equal to the scalar multiplication via γ: 𝔽_{q} → R, and such that [1]^{♭} = id_{B^{♭}} and [0]^{♭} = e_{B}^{♭}, as well as [aã]^{♭} = [a]^{♭} ∘ [ã]^{♭} and [a + ã]^{♭} = m ∘ ([a]^{♭} ⊗ [ã]^{♭}) ∘ Δ^{♭}, where m: (B ⊗_{R} B)^{♭} → B^{♭} is induced by the multiplication map B^{♭} ⊗_{R} B^{♭} → B^{♭} in the ring B^{♭} and the homomorphism [a]^{♭} ⊗ [ã]^{♭}: B^{♭} ⊗_{R} B^{♭} → B^{♭} ⊗_{R} B^{♭} induces a homomorphism (B ⊗_{R} B)^{♭} → (B ⊗_{R} B)^{♭} denoted again by [a]^{♭} ⊗ [ã]^{♭}. If G is finite locally free, such a lift a ↦ [a]^{♭} then exists for every presentation and is uniquely determined by [HS15, Lemmas 4.4 and 4.7].

Example 4.3. The group scheme 𝔾_{a,R}^{d} is a strict 𝔽_{q}-module scheme for any d, because we can choose B = R[X_{1}, …, X_{d}] and so I = (0) and B^{♭} = B, and a ∈ 𝔽_{q} acts as [a]^{∗}X_{i} = a · X_{i}. Moreover, every 𝔽_{q}-linear group homomorphism 𝔾_{a,R}^{d} → 𝔾_{a,R}^{d′} is strict in the sense of [Fal02, Definition 1], meaning that the homomorphism lifts to a homomorphism between the B^{♭} which is equivariant for the 𝔽_{q}-action via [ .]^{♭}.

Lemma 4.4. Let G be a finite locally free group scheme over R, let 𝔽_{q} → End_{R-groups}(G) be a ring homomorphism, and let R → R′ be a faithfully flat ring homomorphism. Then G is a strict 𝔽_{q}-module scheme if and only if G ×_{R} R′ is.

Proof. Let pr: Spec R′ → Spec R be the induced morphism and let pr_{i}: Spec R′ ⊗_{R} R′ → Spec R′ be the projection onto the i-th factor. Let G = Spec B, let R′[\underline{X}] ↠ B ⊗_{R} R′ be a presentation, and let 𝔽_{q} → End_{R-algebras}((B ⊗_{R} R′)^{♭}), a ↦ [a]^{♭} be a lift of the 𝔽_{q}-action on G as in Definition 4.2, which makes G ×_{R} R′ into a strict 𝔽_{q}-module scheme over R′. Moreover, let f: R[\underline{Y}] ↠ B be an arbitrary presentation and let G̃ = (Spec B, Spec R[\underline{Y}]/(\underline{Y})·ker(f)) be the corresponding deformation. By [HS15, Lemmas 4.4 and 4.7] there exists a unique lift a ↦ ˜[a]^{♭} on the deformation G̃×_{R} R′ = pr^{∗}G̃. By the uniqueness the two lifts pr_{1}^{∗}˜[a]^{♭} and pr_{2}^{∗}˜[a]^{♭} on the deformation pr_{1}^{∗} pr^{∗}G̃ = pr_{2}^{∗} pr^{∗}G̃ coincide. By faithfully flat descent [BLR90, § 6.1, Theorem 6] this lift descends to a lift on the deformation G̃, which makes G into a strict 𝔽_{q}-module scheme over R.

To explain the equivalence between finite 𝔽_{q}-shtukas and finite locally free strict 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} : σ^{∗}V → V of R-modules. Following Drinfeld [Dri87, § 2] we define

Dr_{q}(\underline{V}) := Spec ( ⨁_{n≥0} Sym_{R}^{n} V )/I

where the ideal I is generated by the elements v^{⊗q} − F_{V} (σ_{V}^{∗} v) for all v ∈ V. (Here v^{⊗q} lives in Sym^{q} V and F_{V} (σ_{V}^{∗} v) in Sym^{1} V.) Then Dr_{q}(\underline{V}) is a group scheme over R via the comultiplication Δ: v ↦ v ⊗ 1 + 1 ⊗ v and an 𝔽_{q}-module scheme via [a]: v ↦ av for a ∈ 𝔽_{q}. It has a canonical deformation

Dr_{q}(\underline{V})^{♭} := Spec ( ⨁_{n≥0} Sym_{R}^{n} V )/(I · I_{0}),

where I_{0} = ⨁_{n≥1} Sym_{R}^{n} V is the ideal generated by the v ∈ V. This deformation is equipped with the comultiplication Δ^{♭}: v ↦ v ⊗ 1 + 1 ⊗ v and the 𝔽_{q}-action [a]^{♭}: v ↦ av. We set Dr_{q}(\underline{V}) := (Dr_{q}(\underline{V}), Dr_{q}(\underline{V})^{♭}). On its co-Lie complex [a] acts by scalar multiplication with a because (av)^{⊗q} − F_{V} (σ_{V}^{∗} (av)) = a^{q}(v^{⊗q}−F_{V} (σ_{V}^{∗} v)). Therefore Dr_{q}(\underline{V}) is a finite locally free strict 𝔽_{q}-module scheme if V is a finite locally free R-module. Every morphism (V, F_{V}) → (W, F_{W}), that is, every R-homomorphism f: V → W with f ∘ F_{V} = F_{W} ∘ σ^{∗}f, induces a morphism Dr_{q}(f): Dr_{q}(W, F_{W}) → Dr_{q}(V, F_{V}). So Dr_{q} is a contravariant functor. If f is surjective then Dr_{q}(f) is a closed immersion.

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

M_{q}(G) := Hom_{R-groups,𝔽_{q}-lin}(G, 𝔾_{a,R})

and the R-homomorphism F_{M_{q}(G)} : σ^{∗}M_{q}(G) → M_{q}(G) which is induced from M_{q}(G) → M_{q}(G), m ↦ Frob_{q,𝔾_{a,R}}∘ m. Every morphism of 𝔽_{q}-module schemes f: G → G′ induces an R-homomorphism \underline{M}_{q}(G′) → \underline{M}_{q}(G), m′ ↦ m′ ∘ f. Note that by an argument as in Remark 3.3 we have \underline{M}_{q}(G) ⊗_{R} S = \underline{M}_{q}(G ×_{Spec R} Spec S) for every R-algebra S.

There is a natural morphism \underline{V} → \underline{M}_{q}(Dr_{q}(\underline{V})), v ↦ f_{v}, where f_{v}: Dr_{q}(\underline{V}) → 𝔾_{a,R} = Spec R[ξ] is given by f_{v}^{∗}(ξ) = v. There is also a natural morphism of group schemes G → Dr_{q}(\underline{M}_{q}(G)) given by ⨁_{n≥0} Sym_{R}^{n} M_{q}(G)/I → Γ(G, O_{G}), m ↦ m^{∗}(ξ), which is well defined because F_{M_{q}(G)}(σ^{∗}m)^{∗}(ξ) =

(Frob_{q,𝔾_{a,R}}∘ m)^{∗}(ξ) = m^{∗}(ξ^{q}) = (m^{∗}(ξ))^{q}.

Example 4.6. For example if \underline{E} = (E, φ) 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}), τ_{M(\underline{E})}) in Definition 1.2. There is a canonical isomorphism \underline{E} ^{∼} ⟶ Dr_{q}(\underline{M}_{q}(\underline{E})) which is constructed as follows. We set 𝔾_{a,R} = Spec R[ξ] and consider for each m ∈ M_{q}(\underline{E}) = Hom_{R-groups,𝔽_{q}-lin}(E, 𝔾_{a,R}) the element m^{∗}(ξ) ∈ Γ(E, O_{E}). We claim that

( ⨁_{n≥0} Sym_{R}^{n} M_{q}(\underline{E})) / (m^{⊗q} − F_{M_{q}(\underline{E})}(σ_{M_{q}(\underline{E})}^{∗}m): m ∈ M_{q}(\underline{E})) ^{∼} ⟶ Γ(E, O_{E}) , m ↦ m^{∗}(ξ) (4.3)

is an isomorphism of R-algebras. To prove that it is an isomorphism we may apply a faithfully flat base change R → R′ over which we have an 𝔽_{q}-linear isomorphism α: E ⊗_{R} R′ ^{∼} ⟶ 𝔾_{a,R′}^{d} = Spec R′[x_{1}, …, x_{d}]. Let m_{i} := pr_{i} ∘ α ∈ M_{q}(\underline{E}) ⊗_{R} R′ where pr_{i}: 𝔾_{a,R′}^{d} → 𝔾_{a,R′} is the projection onto the i-th factor. Then M_{q}(\underline{E}) ⊗_{R} R′ = ⨁_{i=0}^{d} R′\{τ\}·m_{i} by Remark 3.3 and the inverse of (4.3) sends α^{∗}(x_{i}) to m_{i}. This is indeed the inverse, because (4.3) sends each of the generators τ^{j}m_{i} = Frob_{q^{j},𝔾_{a,R}}∘m_{i} of the R′-module M_{q}(\underline{E}) ⊗_{R} R′ to (Frob_{q^{j},𝔾_{a,R}}∘m_{i})^{∗}(ξ) = m_{i}^{∗}(ξ^{q^{j}}) = α^{∗}(x_{i})^{q^{j}}, and this

inverse sends it back to m_{i}^{⊗q^{j}} = Frob_{q^{j},𝔾_{a,R}}∘m_{i} = τ^{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. (a) The contravariant functors Dr_{q} and \underline{M}_{q} are mutually quasi-inverse anti-equiva-lences between the category of finite 𝔽_{q}-shtukas over R and the category of finite locally free strict 𝔽_{q}-module schemes over R. Both functors are 𝔽_{q}-linear and exact.

Let \underline{V} = (V, F_{V}) be a finite 𝔽_{q}-shtuka over R and let G = Dr_{q}(\underline{V}). Then (b) the 𝔽_{q}-module scheme Dr_{q}(\underline{V}) is étale over R if and only if \underline{V} is étale.

(c) the natural morphisms \underline{V} → \underline{M}_{q}(Dr_{q}(\underline{V})), v ↦ f_{v} and G → Dr_{q}(\underline{M}_{q}(G)) are isomorphisms.

(d) the co-Lie complex ℓDr_{q}(\underline{V})/S^{•} is canonically isomorphic to the complex 0 → σ^{∗}V ⟶^{F_{V}} V → 0.

5 Isogenies

Definition 5.1. A morphism f ∈ Hom_{R}(\underline{E}, \underline{E}′) between two abelian Anderson A-modules \underline{E} and \underline{E}′ over R is an isogeny if f: E → E′ is finite and surjective. If there exists an isogeny between \underline{E} and \underline{E}′ then they are called isogenous. (Being isogenous is an equivalence relation; see Corollary 5.16 below.)

An isogeny f: \underline{E} → \underline{E}′ 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′ 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: E → E′ be a morphism between two affine, smooth R-group schemes E of relative dimension d and E′ of relative dimension d′, such that the fibers of E′ over all points of Spec R are connected. Then the following are equivalent:

(a) f is finite and faithfully flat, that is flat and surjective; see [EGA, 0_{I}.6.7.8],

(b) ker f is finite and f is flat,

(c) ker f is finite and f is surjective,

(d) ker f is finite and d = d′,

(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

(f) f is surjective and d = d′.

Proof. We show that (a) implies all other conditions. This is obvious for (b), (c) and (e). To prove that d = d′ let 𝔪 ⊂ R be a maximal ideal and consider the base change to k = R/𝔪. Then f× id_{k}: E×_{R}k → E′×_{R}k is a finite surjective morphism, and hence d = dim E×_{R}k = dim E′×_{R}k = d′; see [Eis95, Corollary 9.3].

Conversely, clearly (e)⟹(c). We now show (f)⟹(c) and (b)⟹(c)⟹(d)⟹(b)⟹(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 Γ(E′, O_{E′}) is an integral domain by our assumptions on E′. The surjectivity of f implies that f^{∗}: Γ(E′, O_{E′}) ↪ Γ(E, O_{E}) is injective of relative transcendence degree d − d′ = 0. Since all fibers of f are isomorphic to ker f, [Eis95, Corollary 14.6] implies that ker f is finite over Spec k and (c) holds.

We next show for general R that (b) implies (c). Namely, f is of finite presentation by [EGA, IV_{1}, Proposition 1.6.2(v)], because E and E′ are of finite presentation over R. Therefore (b) implies that f is universally open by [EGA, IV_{2}, Théorème 2.4.6]. In particular (f × id_{k})(E ×_{R} k) ⊂ E′ ×_{R} k is open for every point Spec k → Spec R of Spec R. Since E′ ×_{R} k was assumed to be connected, it possesses no proper open subgroup, and hence f × id_{k} is surjective. This establishes (c).

To prove that (c) implies (d) again consider the morphism f × id_{k}: E ×_{R} k → E′ ×_{R} k over a point Spec k → Spec R of Spec R. Since f × id_{k} is surjective, f^{∗} ⊗ id_{k}: Γ(E′, O_{E′})⊗_{R} k ↪ Γ(E, O_{E})⊗_{R} k is injective, because otherwise its kernel would define a proper closed subscheme of E′×_{R}k through which f × id_{k} factors. Since all fibers of f are isomorphic to ker f, and hence finite, [Eis95, Corollary 13.5] shows that d′ = dim Γ(E′, O_{E′}) ⊗_{R} k = dim Γ(E, O_{E}) ⊗_{R} k = d.

We prove the implication (d)⟹(b). Consider the fiber f × id_{k}: E ×_{R} k → E′ ×_{R} k over a point Spec k → Spec R of Spec R and the inclusion (Γ(E′, O_{E′}) ⊗_{R} k)/ ker(f^{∗} ⊗ id_{k}) ⟶ Γ(E, O_{E}) ⊗_{R} k. Since all fibers of f are finite, [Eis95, Corollary 13.5] implies dim Γ(E′, O_{E′}) ⊗_{R} k = d′ = d = dim Γ(E, O_{E}) ⊗_{R} k = dim(Γ(E′, O_{E′}) ⊗_{R} k)/ ker(f^{∗} ⊗ id_{k}). It follows that ker(f^{∗} ⊗ id_{k}) = (0) and f^{∗} ⊗ id_{k}: Γ(E′, O_{E′}) ⊗_{R} k ↪ Γ(E, O_{E}) ⊗_{R} k is injective. Let 𝔪 ⊂ Γ(E, O_{E}) ⊗_{R} k be a maximal ideal. Then (f^{∗} ⊗ id_{k})^{−1}(𝔪) ⊂ Γ(E′, O_{E′}) ⊗_{R} k is a maximal ideal by [Eis95, Theorem 4.19]. Since the fiber of f over 𝔪 is finite, [Eis95, Theorem 18.16(b)] implies that f ⊗ id_{k} is flat at 𝔪. Since E and E′ are smooth over R it follows from [EGA, IV_{3}, 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: E → E′ is faithfully flat. Whether f is finite can by [EGA, IV_{2}, Proposition 2.7.1] be tested after the faithfully flat base change E → E′. By (3.1) the finiteness of the projection E ×_{E′} E → E onto the first factor follows from the finiteness of ker f over Spec R. This proves (a).

Corollary 5.3. Let f ∈ Hom_{R}(\underline{E}, \underline{E}′) be an isogeny. Then

(a) the kernel ker f of f is a finite locally free group scheme and a strict 𝔽_{q}-module scheme over R.

(b) E′ is the quotient E/ ker f.

Proof. (a) Since f is flat of finite presentation by [EGA, IV_{1}, 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′ both E and E′ become isomorphic to powers of 𝔾_{a,R′} and hence are strict 𝔽_{q}-module schemes by Example 4.3. Therefore (ker f) ⊗_{R} R′ is a strict 𝔽_{q}-module scheme over R′ by [Fal02, Proposition 2] and ker f is a strict 𝔽_{q}-module scheme over R by Lemma 4.4.

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

Proposition 5.4. (a) If \underline{E} and \underline{E}′ are Drinfeld A-modules over R with Spec R connected and f ∈ Hom_{k}(\underline{E}, \underline{E}′), then f is an isogeny if and only if f ≠ 0.

(b) If this is the case then f is separable if and only if Lie f ∈ R^{×}.

Proof. (a) Let f: \underline{E} → \underline{E}′ be an isogeny, then f ≠ 0 because the zero morphism is not surjective. Conversely let f ≠ 0. By Proposition 5.2(d) we must show that ker f is finite. This question is local on Spec R, so we may assume that E = E′ = 𝔾_{a,R} and that \underline{E} = (E, φ) and \underline{E}′ = (E′, ψ) are in standard form. Let t ∈ A∖𝔽_{q}, and hence deg_{τ} φ_{t} > 0 and deg_{τ} ψ_{t} > 0. By Lemma 3.8(b) applied to f∘φ_{t} = ψ_{t}∘f we have f = ∑_{i=0}^{n} f_{i}τ^{i} ∈ R{τ} with f_{n} ∈ R^{×}. It follows that ker f = Spec R[x]/(∑_{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 = Spec R[x]/(∑_{i=0}^{n} f_{i}x^{q^{i}}) is étale if and only if Lie f = f_{0} = \frac{∂f(x)}{∂x} ∈ R^{×}.

Next we turn to A-motives.

Definition 5.5. A morphism f ∈ Hom_{R}(\underline{M}, \underline{N}) between A-motives over R is an isogeny if f is injective and 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 ∈ QHom_{R}(\underline{M}, \underline{N}) which is of the form g ⊗ c for an isogeny g ∈ Hom_{R}(\underline{M}, \underline{N}) and a c ∈ 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

[figure: 0; 0; ker(τ_{coker f}); 0; σ^{∗}M; σ^{∗}f; τ_{M}; σ^{∗}N; τ_{N}; σ^{∗}(coker f); τ_{coker f}; 0; 0; M; f; N; coker f; 0; 0; ker(τ_{coker f}); coker τ_{M}; coker τ_{N}; coker(τ_{coker f}); 0 .]  (5.1)

Namely, by local freeness of coker f the upper row is again exact and identifies σ^{∗}(coker f) with coker(σ^{∗}f).

An isogeny f: \underline{M} → \underline{N} between effective A-motives is separable if τ_{coker f}: σ^{∗}(coker f) → coker f is an isomorphism.

Remark 5.6. If f ∈ Hom_{R}(\underline{M}, \underline{N}) is an isogeny and S is an R-algebra, then the base change f ⊗ id_{S} ∈ Hom_{S}(\underline{M} ⊗_{R} S, \underline{N} ⊗_{R} S) of f to S is again an isogeny. This follows from the exact sequence 0 ⟶ \underline{M} ⟶^{f} \underline{N} ⟶ coker f ⟶ 0 because coker f is a flat R-module.

Example 5.7. For 0 ≠ a ∈ A the morphism a: \underline{M} → \underline{M} is an isogeny with coker a = M/aM. Let \underline{M} be effective. Then a is separable if and only if ker(τ_{coker a}) = coker(τ_{coker a}) = (0). That is, if and only if multiplication with a is an automorphism of coker τ_{M}. Since a − γ(a) is nilpotent on coker τ_{M} this is the case if and only if γ(a) ∈ R^{×}. 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 rk \underline{M} = rk \underline{N}, and if, moreover, \underline{M} and \underline{N} are effective, then rk_{R} coker τ_{M} = rk_{R} coker τ_{N}. Conversely assume rk \underline{M} = rk \underline{N} and let f ∈ Hom_{R}(\underline{M}, \underline{N}) be a morphism such that coker f is a finitely generated Rmodule. Then f is an isogeny.

Proof. Let f: \underline{M} → \underline{N} be an isogeny. Since M, respectively coker τ_{M}, are finite locally free over A_{R}, respectively over R, we can compute their ranks by choosing a maximal ideal 𝔪 ⊂ R and applying the base change from R to k = R/𝔪. Then f ⊗ id_{k} is still an isogeny by Remark 5.6. Since coker(f ⊗ id_{k}) is a torsion A_{k}-module it follows that

rk \underline{M} = rk_{A_{R}} M = rk_{A_{k}}(M ⊗_{R} k) = rk_{A_{k}}(N ⊗_{R} k) = rk_{A_{R}} N = rk \underline{N} .

If \underline{M} and \underline{N} are effective, we consider diagram (5.1) for the isogeny f ⊗ id_{k}. Since coker(f ⊗ id_{k}) and σ^{∗} coker(f ⊗ 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

rk_{R} coker τ_{M} = dim_{k} coker(τ_{M} ⊗ id_{k}) = dim_{k} coker(τ_{N} ⊗ id_{k}) = rk_{R} coker τ_{N} .

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 ∈ Hom_{R}(\underline{E}, \underline{E}′) be a morphism between abelian Anderson A-modules and let \underline{M}(f) ∈ Hom_{R}(\underline{M}′, \underline{M}) be the associated morphism between the associated effective A-motives \underline{M} = \underline{M}(\underline{E}) and \underline{M}′ = \underline{M}(\underline{E}′). Then (a) f is an isogeny if and only if \underline{M}(f) is an isogeny.

(b) f is a separable isogeny if and only if \underline{M}(f) is a separable isogeny.

(c) If f is an isogeny there are canonical A-equivariant isomorphisms of finite 𝔽_{q}-shtukas

(coker \underline{M}(f), τ_{coker \underline{M}(f)}) ^{∼} ⟶ \underline{M}_{q}(ker f)

and of finite locally free R-group schemes

Dr_{q}(coker \underline{M}(f)) ^{∼} ⟶ ker f .

Proof. In the beginning we do neither assume that f nor that \underline{M}(f) is an isogeny. We denote by ι the inclusion ker f ↪ E. Consider the A_{R}-homomorphism \underline{M}(\underline{E}) → \underline{M}_{q}(ker f), m ↦ m ∘ ι, which is compatible with the Frobenii τ_{M(\underline{E})} and F_{M_{q}(ker f)}. Since m = \underline{M}(f)(m′) = m′ ∘f implies m′ ∘f ∘ι = 0, it factors over

coker \underline{M}(f) ⟶ \underline{M}_{q}(ker f), m mod im \underline{M}(f) ↦ m ∘ ι .  (5.2)

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

Dr_{q}(\underline{M}_{q}(ker f)) ⟶ Dr_{q}(coker \underline{M}(f)) ⟶ ker f ⟶ E .  (5.3)

where the last two are closed immersions. The first morphism is obtained from (5.2). Moreover, the epimorphism \underline{M}(\underline{E}) ↠ coker \underline{M}(f) induces by Example 4.6 an A-equivariant closed immersion α: Dr_{q}(coker \underline{M}(f)) ↪ Dr_{q}(\underline{M}(\underline{E})) = \underline{E}. We compose it with f: E → E′ and show that the composition factors through the zero section e′: Spec R → E′. This will imply that α factors through ker f. We can study this composition after a faithfully flat base change R → R′ over which we have an 𝔽_{q}-linear isomorphism β: E′ ⊗_{R} R′ ≅ 𝔾_{a,R′}^{d′} = Spec R′[y_{1}, …, y_{d}]. Let m′_{i} := pr_{i} ∘ β ∈ M(\underline{E}′) ⊗_{R} R′ where pr_{i}: 𝔾_{a,R′}^{d} → 𝔾_{a,R′} = Spec R[ξ] is the projection onto the i-th factor. Then pr_{i}^{∗}(ξ) = y_{i} and α^{∗}f^{∗}β^{∗}(y_{i}) = α^{∗}f^{∗}m′_{i}^{∗}(ξ) = α^{∗} ∘ \underline{M}(f)(m′_{i})^{∗}(ξ) = 0 because \underline{M}(f)(m′_{i}) = 0 in 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 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 Dr_{q}(\underline{M}_{q}(ker f)) → ker f in (5.3) is an isomorphism. This shows that Dr_{q}(coker \underline{M}(f)) ^{∼} ⟶ 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 → R′ such that E ⊗_{R} R′ ≅ 𝔾_{a,R′}^{d} and E′ ⊗_{R} R′ ≅ 𝔾_{a,R′}^{d′}. Then f is given by a matrix F ∈ R′{τ}^{d′×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 𝔪′ ⊂ R′ its tensor product with k := R′/𝔪′ is surjective. By Remark 3.3 and its analog for \underline{M}_{q}(ker f) the tensor product of (5.2) with k equals coker \underline{M}(f × id_{k}) → \underline{M}_{q}(ker(f × id_{k})), where f × id_{k}: \underline{E} ×_{R} k → \underline{E}′ ×_{R} k is given by the matrix \overline{F} := F ⊗ 1_{k}. In particular ker(f × id_{k}) = Spec k[x_{1}, …, x_{d}]/(f^{∗}(y_{ℓ}): 1 ≤ ℓ ≤ d). Since ker f is finite, k[x_{1}, …, x_{d}]/(f^{∗}(y_{ℓ}): 1 ≤ ℓ ≤ d) is a finite dimensional k-vector space. For fixed i this implies that {x_{i}, x_{i}^{q}, x_{i}^{q^{2}}, …} is linearly dependent and there is a positive integer N and b_{i,n} ∈ k such that

x_{i}^{q^{N+1}} = ∑_{n=0}^{N} b_{i,n}·x_{i}^{q^{n}} in k[x_{1}, …, x_{d}]/(f^{∗}(y_{ℓ}): 1 ≤ ℓ ≤ d). We introduce the new variables z_{i,n} := x_{i}^{q^{n}}

for 1 ≤ i ≤ d and 0 ≤ n ≤ N. Then f^{∗}(y_{ℓ}) is a k-linear relation between the z_{i,n}. Furthermore

k[x_{1}, …, x_{d}]/(f^{∗}(y_{ℓ}): 1 ≤ ℓ ≤ d) ≅ k[z_{i,n}: 1 ≤ i ≤ d, ≤ n ≤ N]/I with

I = (f^{∗}(y_{i}), z_{i,N}^{q} − ∑_{n=0}^{N} b_{i,n}·z_{i,n}, z_{i,n}^{q} − z_{i,n+1} : 1 ≤ i ≤ d, 0 ≤ n < N) .

Let z̃_{1}, …, z̃_{r} be a k-basis of (⨁_{i=1}^{d} ⨁_{n=0}^{N} k·z_{i,n})/(f^{∗}(y_{ℓ}): 1 ≤ ℓ ≤ d). Then there are elements c_{ij} ∈ k for

1 ≤ i, j ≤ r such that

k[x_{1}, …, x_{d}]/(f^{∗}(y_{ℓ}): 1 ≤ ℓ ≤ d) ≅ k[z̃_{1}, …, z̃_{r}]/(z̃_{i}^{q} − ∑_{j=1}^{r} c_{ij}z̃_{j}: 1 ≤ i ≤ r) =: B .

Moreover, the group law on ker f is given by the comultiplication Δ^{∗}: B → B ⊗_{k} B, Δ^{∗}(z̃_{i}) = z̃_{i} ⊗ 1 + 1 ⊗ z̃_{i} and the 𝔽_{q}-action is given by φ_{λ}: B → B, φ_{λ}^{∗}(z̃_{i}) = γ(λ)·z̃_{i}.

We now are ready to compute \underline{M}_{q}(ker(f × id_{k})) from (3.3). If 𝔾_{a,k} = Spec k[ξ] then every

element m̃ ∈ \underline{M}_{q}(ker(f × id_{k})) satisfies m^{∗}(ξ) = ∑_{ℓ_{i}∈\{0...q−1\}} d_{ℓ_{1},...,ℓ_{r}} · z̃_{1}^{ℓ_{1}} · . . . · z̃_{r}^{ℓ_{r}} with d_{ℓ_{1},...,ℓ_{r}} ∈ k.

Since the z̃_{1}^{ℓ_{1}} · . . . · z̃_{r}^{ℓ_{r}} form a k-basis of B, the conditions Δ^{∗}m^{∗}(ξ) = m^{∗}(ξ) ⊗ 1 + 1 ⊗ m^{∗}(ξ) in B ⊗_{k} B and φ_{λ}^{∗}m^{∗}(ξ) = m^{∗}(γ(λ)·ξ) = γ(λ)·m^{∗}(ξ) in B for λ ∈ 𝔽_{q} imply as in Lemma 3.2 that m^{∗}(ξ) = d_{1,0...0}·z̃_{1} + … + d_{0...0,1}·z̃_{r}. Since z̃_{i} is a k-linear combination of the z_{j,n} = x_{j}^{q^{n}} the morphism m: E ×_{R} k → 𝔾_{a,k} with m^{∗}(ξ) = d_{1,0...0}·z̃_{1} + … + d_{0...0,1}·z̃_{r} belongs to \underline{M}(E ×_{R} k) and maps to m̃ under the map coker \underline{M}(f × id_{k}) → \underline{M}_{q}(ker(f × id_{k})). This proves that (5.2) is surjective.

In order to show that (5.2) is injective let m ∈ M(\underline{E}) be an element with m ∘ ι = 0. By [SGA 3, Théorème V.4.1] the morphism m: E → 𝔾_{a,R} factors through E/ ker f ^{∼} ⟶ E′ (use Corollary 5.3(b)) in the form m = m′ ∘ f for an m′ ∈ M(\underline{E}′). This shows that m mod im \underline{M}(f) = 0 in coker \underline{M}(f). All together we have proved that coker \underline{M}(f) ^{∼} ⟶ \underline{M}_{q}(ker f) is a finite locally free R-module. Moreover, \underline{M}(f) is injective, because if m′ ∈ M(\underline{E}′) satisfies m′ ∘ f = \underline{M}(f)(m′) = 0 the surjectivity of f implies m′ = 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 Hom_{R-groups,𝔽_{q}-lin}( • , 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}′ 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 → R′ such that E ⊗_{R} R′ ≅ 𝔾_{a,R′}^{d} = Spec R′[x_{1}, …, x_{d}] and E′ ⊗_{R} R′ ≅ 𝔾_{a,R′}^{d} = Spec R[y_{1}, …, y_{d}]. Also

when we write 𝔾_{a,R′} = Spec R′[ξ] then \underline{M}(\underline{E} ×_{R} R′) ≅ ⨁_{i=1}^{d} R′\{τ\}·m_{i} and \underline{M}(\underline{E}′ ×_{R} R′) ≅ ⨁_{i=1}^{d} R′\{τ\}·m′_{i}

where m_{i}^{∗}(ξ) = x_{i} and m′_{i}^{∗}(ξ) = y_{i}. Consider the epimorphism of R-modules

[figure: ⨁_{i=1}^{d} ⨁_{0≤n} R′·τ^{n}m_{i} ≅ \underline{M}(\underline{E} ×_{R} R′); δ; coker \underline{M}(f ⊗ id_{R′}) .]

Since coker \underline{M}(f ⊗ id_{R′}) is finite locally free over R′, and hence projective, this epimorphism has a

section s whose image lies in ⨁_{i=1}^{d} ⨁_{n=0}^{N} R′ ·τ^{n}m_{i} for some N. It follows that τ^{N+1}m_{i} − s(δ(τ^{N+1}m_{i}))

maps to zero in coker \underline{M}(f ⊗ id_{R′}). That is, there are elements b_{i,j,n} ∈ R′ and m′_{i} ∈ \underline{M}(\underline{E}′ ×_{R} R′) with

τ^{N+1}m_{i} − ∑_{j=1}^{d} ∑_{n=0}^{N} b_{i,j,n}·τ^{n}m_{j} = \underline{M}(f)(m′_{i}). Applying this equation to ξ yields

x_{i}^{q^{N+1}} − ∑_{j=1}^{d} ∑_{n=0}^{N} b_{i,j,n}·x_{j}^{q^{n}} = f^{∗}m′_{i}^{∗}(ξ) ∈ f^{∗}R′[y_{1}, …, y_{d}] ≅ f^{∗}Γ(E′, O_{E′}) ⊗_{R} R′ .

Thus f × id_{R′}: E ×_{R} R′ → E′ ×_{R} R′ is finite. By faithfully flat descent [EGA, IV_{2}, 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}′ are isogenous abelian Anderson A-modules then rk \underline{E} = rk \underline{E}′.

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 ∈ A. Then φ_{a}: \underline{E} → \underline{E} is an isogeny. It is separable if and only if γ(a) ∈ R^{×}.

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 ∈ Hom_{R}(\underline{M}, \underline{N}) be a morphism. Then the following are equivalent:

(a) f is an isogeny,

(b) there is an element 0 ≠ a ∈ A such that f induces an isomorphism of A_{R}[\frac{1}{a}]-modules M[\frac{1}{a}] ^{∼} ⟶ N[\frac{1}{a}].

In particular, a quasi-morphism f ∈ QHom_{R}(\underline{M}, \underline{N}) is a quasi-isogeny if and only if it induces an isomorphism f: M[\frac{1}{a}] ^{∼} ⟶ N[\frac{1}{a}] for an element a ∈ A ∖ \{0\}.

Proof. (b)⟹(a) Clearly rk \underline{M} = rk \underline{N}. Since coker f is a finitely generated A_{R}-module, (coker f) ⊗_{A} A[\frac{1}{a}] = (0) implies that a^{n} · coker f = (0) for some positive integer n. Therefore, coker f is a finitely generated module over A_{R}/(a^{n}) = A/(a^{n}) ⊗_{𝔽_{q}} R, whence over R. So (a) follows from Proposition 5.8.

(a)⟹(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 coker f is a finite locally free R-module, which we may assume to be free after passing to an open affine covering of Spec R. Let t ∈ A ∖ 𝔽_{q} and consider the finite flat homomorphism Ã := 𝔽_{q}[t] ↪ A from Lemma 1.4, under which we view \underline{M} and \underline{N} as Ã-motives by restriction of scalars. That is, we view M and N as locally free R[t]-modules of rank r̃ = rk \underline{M} · rkÃ A and τ_{M} and τ_{N} as R[t][\frac{1}{t−γ(t)}]-isomorphisms. By multiplying both τ_{M} and τ_{N} with (t−γ(t))^{e} for e ≫ 0

we may assume that \underline{M} and \underline{N} are effective Ã-motives without changing the isogeny f: \underline{M} → \underline{N}. Let 𝔞 = ann_{R[t]}(coker f) = ker(R[t] → End_{R}(coker f)) be the annihilator of coker f. By the Cayley-Hamilton theorem [Eis95, Theorem 4.3] (applied with I = R), the monic characteristic polynomial χ_{t} of the endomorphism t of coker f lies in 𝔞. This shows that R[t]/𝔞 is a quotient of the finite R-module R[t]/(χ_{t}). In particular the closed subscheme V := Spec R[t]/𝔞 of 𝔸_{R}^{1} = Spec R[t] is finite over Spec R. On its open complement f: M → N is an isomorphism.

We now consider the exterior powers ∧r̃M and ∧r̃N of the R[t]-modules M and N and set L := (∧r̃M)^{∨} ⊗ ∧r̃N. These are invertible R[t]-modules. The isogeny f induces a global section ∧r̃f of the invertible sheaf L on 𝔸_{R}^{1} which provides an isomorphism O_{𝔸_{R}^{1}} ^{∼} ⟶ L, 1 ↦ ∧r̃f on 𝔸_{R}^{1} ∖ V. Likewise we obtain global sections ∧r̃σ^{∗}f, resp. ∧r̃τ_{M}, resp. ∧r̃τ_{N} of the invertible sheaves σ^{∗}L, resp. (∧r̃σ^{∗}M)^{∨} ⊗ ∧r̃M, resp. (∧r̃σ^{∗}N)^{∨} ⊗ ∧r̃N by the effectivity assumption on \underline{M} and \underline{N}. Diagram (5.1) implies that there is an equality of global sections

∧r̃f ⊗ ∧r̃τ_{M} = ∧r̃τ_{N} ⊗ ∧r̃σ^{∗}f  (5.4)

of (∧r̃σ^{∗}M)^{∨} ⊗ ∧r̃N = L ⊗ (∧r̃σ^{∗}M)^{∨} ⊗ ∧r̃M) = ((∧r̃σ^{∗}N)^{∨} ⊗ ∧r̃N) ⊗ σ^{∗}L.

Since V is proper over Spec R and the projective line ℙ_{R}^{1} is separated, the map V ↪ 𝔸_{R}^{1} ↪ ℙ_{R}^{1} is a closed immersion which does not meet {∞} ×_{𝔽_{q}} Spec R, where {∞} = ℙ_{𝔽_{q}}^{1} ∖ 𝔸_{𝔽_{q}}^{1}. Thus we may glue L with the trivial sheaf O_{ℙ_{R}^{1}∖V} on ℙ_{R}^{1} ∖ V along the isomorphism O_{ℙ_{R}^{1}} ^{∼} ⟶ L, 1 ↦ ∧r̃f over 𝔸_{R}^{1} ∖ V. In this way we obtain an invertible sheaf \overline{L} on the projective line ℙ_{R}^{1}. By replacing \overline{L} with \overline{L}⊗O_{ℙ_{R}^{1}}(m·∞) for a suitable integer m we may achieve that \overline{L} has degree zero (see [BLR90, § 9.1, Proposition 2]) and induces an R-valued point of the relative Picard functor Pic_{ℙ^{1}/𝔽_{q}}^{0}; cf. [BLR90, § 8.1]. Since Pic_{ℙ^{1}/𝔽_{q}}^{0} is trivial, [BLR90, § 8.1, Proposition 4] shows that \overline{L} ≅ K ⊗_{R} O_{ℙ_{R}^{1}} for an invertible sheaf K on Spec R. Replacing Spec R by an open affine covering which trivializes K we may assume that there is an isomorphism α: L ^{∼} ⟶ R[t] of R[t]-modules. Let h := α(∧r̃f) ∈ R[t].

2. Let d := rk_{R} coker τ_{M}. We claim that locally on Spec R there is a positive integer n_{0} and for every integer n ≥ n_{0} an isomorphism of R[t]-modules

((∧r̃σ^{∗}M)^{∨} ⊗_{R[t]} ∧r̃M)^{⊗q^{n}} ^{∼} ⟶ R[t] with (∧r̃τ_{M})^{⊗q^{n}} ⟼ (t − γ(t))^{q^{n}d}  (5.5)

and similarly for \underline{N}. To prove the claim we apply Proposition 2.3(c) to the A-motive ∧r̃\underline{M} and derive that ∧r̃τ_{M}: ∧r̃ σ^{∗}M → ∧r̃M is injective coker ∧r̃τ_{M} is a finite locally free R-module, annihilated by a power of t − γ(t). Consider the exact sequence

0 ⟶ ∧r̃σ^{∗}M ⊗_{R[t]} (∧r̃M)^{∨} ⟶^{∧r̃τ_{M} ⊗ id_{(∧r̃M)^{∨}}} R[t] ⟶ coker ∧r̃τ_{M} ⊗_{R[t]} (∧r̃M)^{∨} ⟶ 0 .  (5.6)

Choose an open affine covering of Spec R[t] which trivializes the locally free R[t]-module ∧r̃M. Pulling back this covering under the section Spec R ^{∼} ⟶ Spec R[t]/(t − γ(t)) ↪ Spec R[t] gives an open affine covering of Spec R on which we may find an isomorphism coker ∧r̃τ_{M} ⊗_{R[t]} (∧r̃M)^{∨} ^{∼} ⟶ coker ∧r̃τ_{M}. We replace Spec R by this open affine covering and even shrink it further in such a way that coker ∧r̃τ_{M} becomes a free R-module. By [Eis95, Proposition 4.1(b)] the sequence (5.6) is then isomorphic to the sequence

\xymatrix{0 \ar[r] & R[t] \ar[r]^{g} & R[t] \ar[r] & coker ∧r̃τ_{M} \ar[r] & 0 ,}  (5.7)

where g ∈ R[t] is a monic polynomial of degree equal to rk_{R}(coker ∧r̃τ_{M}). We now tensor sequence (5.7) over R with k := Frac(R/𝔭) where 𝔭 ⊂ R is a prime ideal. It remains exact because coker ∧r̃τ_{M} is free. Since k[t] is a principal ideal domain the elementary divisor theorem applied to

\xymatrix{0 \ar[r] & σ^{∗}M ⊗_{R} k \ar[r]^{τ_{M} ⊗ id_{k}} & M ⊗_{R} k \ar[r] & coker τ_{M} ⊗_{R} k \ar[r] & 0}

allows to write τ_{M} ⊗ id_{k} as a diagonal matrix. This shows that coker ∧r̃τ_{M} ⊗_{R} k is a k-vector space of dimension equal to rk_{R}(coker τ_{M}) =: d. Since t − γ(t) is nilpotent on this vector space, the Cayley-Hamilton theorem from linear algebra implies g mod 𝔭 = (t − γ(t))^{d}. In particular the coefficients of the difference g′ := g − (t − γ(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′)^{q^{n_{0}}} = 0, whence g^{q^{n}} = (t − γ(t))^{q^{n}d} for every n ≥ 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 = rk_{R} coker τ_{M} = rk_{R} coker τ_{N} by Proposition 5.8, equations (5.4) and (5.5) imply that for n ≫ 0 there is an isomorphism β: σ^{∗}L^{⊗q^{n}} ^{∼} ⟶ L^{⊗q^{n}} of R[t]-modules sending (t − γ(t))^{q^{n}}(σ^{∗} ∧r̃ f)^{⊗q^{n}} to (t − γ(t))^{q^{n}}(∧r̃f)^{⊗q^{n}} and hence (σ^{∗} ∧r̃ f)^{⊗q^{n}} to (∧r̃f)^{⊗q^{n}} because t − γ(t) is a non-zero divisor. In particular the isomorphism

α^{⊗q^{n}} ∘ β ∘ (σ^{∗}α^{⊗q^{n}})^{−1} : R[t] ^{∼} ⟶ σ^{∗}L^{⊗q^{n}} ^{∼} ⟶ L^{⊗q^{n}} ^{∼} ⟶ R[t] ,

which is given by multiplication with a unit u ∈ R[t]^{×}, sends σ(h^{q^{n}}) = σ^{∗}α^{⊗q^{n}}(∧r̃σ^{∗}f)^{⊗q^{n}} to h^{q^{n}} = α^{⊗q^{n}}(∧r̃f)^{⊗q^{n}}. We thus obtain the equation h^{q^{n}} = u · σ(h^{q^{n}}) in R[t].

By Lemma 5.13 below, u = ∑_{i≥0} u_{i}t^{i} with u_{0} ∈ R^{×} and u_{i} ∈ R nilpotent for all i ≥ 1. Let R′ = 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 = ∑_{i≥1} v_{i}t^{i} ∈ R′[t]^{×} with v = u · σ(v). Indeed the latter amounts to the equations

v_{i} = ∑_{j=0}^{i} u_{j}v_{i−j}^{q}  and  \frac{v_{i}}{v_{0}} = (\frac{v_{i}}{v_{0}} )^{q} + ∑_{j≥1} \frac{u_{j}}{u_{0}} (\frac{v_{i−j}}{v_{0}} )^{q}

which have the solutions \frac{v_{i}}{v_{0}} = ∑_{n≥0}(∑_{j≥1} \frac{u_{j}}{u_{0}} (\frac{v_{i−j}}{v_{0}} )^{q})^{q^{n}} because the u_{j} are nilpotent. Therefore the element v^{−1}h^{q^{n}} ∈ R′[t] satisfies σ(v^{−1}h^{q^{n}}) = v^{−1}h^{q^{n}}. Working on each connected component of Spec R′ separately, Lemma 5.14 below shows that a := v^{−1}h^{q^{n}} ∈ 𝔽_{q}[t] ⊂ A.

In the ring R′[t][\frac{1}{a}] the element h becomes a unit. Therefore the map α^{−1} ∘ h: R′[t][\frac{1}{a}] → L[\frac{1}{a}], 1 ↦ ∧r̃f is an isomorphism. This implies that ∧r̃f: ∧r̃ M[\frac{1}{a}] → ∧r̃N[\frac{1}{a}] is an isomorphism, and hence also f: M[\frac{1}{a}] → N[\frac{1}{a}] by Cramer’s rule (e.g. [Bou70, III.8.6, Formulas (21) and (22)]). Thus we have established (b) étale locally on Spec R. Replacing a by the product of all the finitely many elements a obtained locally, establishes (b) globally on Spec R.

4. To prove the statement about quasi-morphisms f ∈ QHom_{R}(\underline{M}, \underline{N}) assume first, that f induces an isomorphism f: M[\frac{1}{a}] ^{∼} ⟶ N[\frac{1}{a}] for some a ∈ A ∖ \{0\}. Then g := a^{n} · f ∈ Hom_{R}(\underline{M}, \underline{N}) for n ≫ 0, because M is finitely generated. In particular g is an isogeny and f = g ⊗ a^{−n} is a quasi-isogeny.

Conversely, if f is a quasi-isogeny, that is f = g ⊗ c for an isogeny g ∈ Hom_{R}(\underline{M}, \underline{N}) and a c ∈ Q, there is an element a ∈ A ∖ \{0\} such that g: M[\frac{1}{a}] ^{∼} ⟶ N[\frac{1}{a}]. If d is the denominator of c it follows that f: M[\frac{1}{ad}] ^{∼} ⟶ N[\frac{1}{ad}].

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

Lemma 5.13. An element u = ∑_{i≥0} u_{i}t^{i} ∈ R[t] is a unit in R[t] if and only if u_{0} ∈ R^{×} and u_{i} is nilpotent for all i ≥ 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 ≥ 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 → ∏_{𝔭⊂R} R/𝔭 where 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/𝔭[t]. Since R/𝔭 is an integral domain, the u_{i} for i ≥ 1 must be sent to zero in each factor R/𝔭. This shows that u_{i} is nilpotent for i ≥ 1.

Lemma 5.14. Assume that R contains no idempotents besides 0 and 1, that is Spec R is connected. Then R^{σ} := {x ∈ R: x^{q} = x} = 𝔽_{q}.

Proof. Let 𝔪 ⊂ R be a maximal ideal and let x̄ ∈ R/𝔪 be the image of x. Then x̄^{q} = x̄ implies that x̄ is equal to an element α ∈ 𝔽_{q} ⊂ R/𝔪. Now e := (x − α)^{q−1} satisfies e^{2} = (x − α)^{q−2}(x^{q} − α^{q}) = (x − α)^{q−1} = e, that is e is an idempotent. Since e ∈ 𝔪 we cannot have e = 1 and must have e = 0. Therefore x − α = (x − α)^{q} = (x − α) · e = 0 in R, that is x = α ∈ 𝔽_{q}.

Corollary 5.15. If f ∈ Hom_{R}(\underline{M}, \underline{N}) is an isogeny between A-motives then there is an element 0 ≠ a ∈ A and an isogeny g ∈ Hom_{R}(\underline{N}, \underline{M}) with f ∘ g = a · id_{\underline{N}} and g ∘ f = a · id_{\underline{M}}. The same is true for abelian Anderson A-modules.

Proof. Let a ∈ A be the element from Theorem 5.12(b). As in the proof of (b)⟹(a) of this theorem there is a positive integer n such a^{n} · coker f = (0). Therefore there is a map g: N → M with g ∘ f = a^{n}· id_{M} and f ∘ g = a^{n}· id_{N}. This implies that g is injective, because a^{n} is a non-zero divisor on N. From

f ∘ g ∘ τ_{N} = a^{n}·τ_{N} = τ_{N} ∘ σ^{∗}a^{n}· id_{N} = τ_{N} ∘ σ^{∗}f ∘ σ^{∗}g = f ∘ τ_{M} ∘ g

and the injectivity of f we conclude that g∘τ_{N} = τ_{M} ∘σ^{∗}g and that g ∈ Hom_{R}(\underline{N}, \underline{M}). By construction g induces an isomorphism N[\frac{1}{a}] ^{∼} ⟶ M[\frac{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 γ(A ∖ \{0\}) ⊂ R^{×} and let f ∈ 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 := coker(τ_{coker f}). As in the proof of Theorem 5.12 there is an element 0 ≠ a ∈ A and a positive integer n with a^{n} · coker f = (0), and hence a^{n} · K = (0). Let e be an integer with q^{e} ≥ rk_{R} coker τ_{N} and q^{e} ≥ n. Then (a ⊗ 1 − 1 ⊗ γ(a))^{q^{e}} · coker τ_{N} = (0). Therefore

0 = (a ⊗ 1 − 1 ⊗ γ(a))^{q^{e}} ·K = (a^{q^{e}} ⊗ 1 − 1 ⊗ γ(a)^{q^{e}})·K = −γ(a)^{q^{e}} ·K .

Since γ(a) ∈ R^{×} we have K = (0), and since coker f and σ^{∗}(coker f) are finite locally free R modules of the same rank, [GW10, Corollary 8.12] shows that τ_{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 ∈ Hom_{R}(\underline{M}, \underline{N}) and g ∈ Hom_{R}(\underline{N}, \underline{M}) are isogenies between A-motives with f ∘ g = a · id_{\underline{N}} and g ∘ f = a · id_{\underline{M}} for an a ∈ A, then there is an isomorphism of Q-algebras QEnd_{R}(\underline{M}) ^{∼} ⟶ QEnd_{R}(\underline{N}) given by h ⊗ b ↦ f ∘ h ∘ g ⊗ \frac{b}{a} for h ∈ End_{R}(\underline{M}).

Example 5.19. Let R be an A-ring of finite characteristic 𝔭, that is γ: A → R factors through 𝔽_{𝔭} := A/𝔭 for a maximal ideal 𝔭 ⊂ A. Let ℓ ∈ ℕ_{>0} be divisible by [𝔽_{𝔭} : 𝔽_{q}]. Then σ^{ℓ∗}(J) = (a ⊗ 1 − 1 ⊗ γ(a)^{q^{ℓ}}: a ∈ A) = J ⊂ A_{R}, because the elements γ(a) ∈ 𝔽_{𝔭} satisfy γ(a)^{q^{ℓ}} = γ(a). Let \underline{M} = (M, τ_{M}) be an A-motive over R. Then σ^{ℓ∗}\underline{M} = (σ^{ℓ∗}M, σ^{ℓ∗}τ_{M}) is also an A-motive over R, because σ^{ℓ∗}τ_{M} is an isomorphism outside V(σ^{ℓ∗}J) = V(J). If \underline{M} is effective, then the A_{R}- homomorphism

Fr_{q^{ℓ}, \underline{M}} := τ_{M}^{ℓ} := τ_{M} ∘ σ^{∗}τ_{M} ∘ … ∘ σ^{(ℓ−1)∗}τ_{M}: σ^{ℓ∗}\underline{M} ⟶ \underline{M}  (5.8)

satisfies τ_{M} ∘ σ^{∗}Fr_{q^{ℓ}, \underline{M}} = Fr_{q^{ℓ}, \underline{M}} ∘ σ^{ℓ∗}τ_{M}. Moreover, it is injective and its cokernel is a successive extension of the σ^{i∗} coker τ_{M} for i = 0, …, ℓ − 1, whence a finitely presented R-module. Therefore Fr_{q^{ℓ}, \underline{M}} ∈ Hom_{R}(σ^{ℓ∗}\underline{M}, \underline{M}) is an isogeny, called the q^{ℓ}-Frobenius isogeny of \underline{M}. It is always inseparable, because the ℓ-th power of τ_{M}, which equals Fr_{q^{ℓ}, \underline{M}} annihilates the cokernel of Fr_{q^{ℓ}, \underline{M}}.

If \underline{M} is not effective, let n ∈ ℕ_{>0} be such that 𝔭^{n} = (a) is principal. Then (a ⊗ 1) ⊂ J and (a ⊗ 1) ⊂ σ^{i∗}J for all i. This shows that

Fr_{q^{ℓ}, \underline{M}} := τ_{M}^{ℓ} := τ_{M} ∘ σ^{∗}τ_{M} ∘ … ∘ σ^{(ℓ−1)∗}τ_{M}: σ^{ℓ∗}\underline{M}[\frac{1}{a}] ^{∼} ⟶ \underline{M}[\frac{1}{a}]  (5.9)

is a quasi-isogeny in QHom_{R}(σ^{ℓ∗}\underline{M}, \underline{M}) by Theorem 5.12, called the q^{ℓ}-Frobenius quasi-isogeny of \underline{M}.

Finally if R = k is a field contained in 𝔽_{q^{ℓ}} then σ^{ℓ∗}\underline{M} = \underline{M} and Fr_{q^{ℓ}, \underline{M}} ∈ QEnd_{k}(\underline{M}), respectively Fr_{q^{ℓ}, \underline{M}} ∈ End_{k}(\underline{M}) if \underline{M} is effective. In this case, A[π] lies in the center of End_{k}(\underline{M}) and Q[π] lies in the center of QEnd_{k}(\underline{M}), because every f ∈ End_{k}(\underline{M}) satisfies f ∘ τ_{M} = τ_{M} ∘ σ^{∗}f and σ^{ℓ∗}f = f. If k = 𝔽_{q^{ℓ}}, the center equals A[π], respectively Q[π], 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) ≠ 𝔞 = (a_{1}, …, a_{n}) ⊂ A be an ideal and let \underline{E} = (E, φ) be an abelian Anderson A-module over R. Then

\underline{E}[𝔞] := ker(φ_{a_{1},...,a_{n}} := (φ_{a_{1}}, …, φ_{a_{n}}): E ⟶ E^{n})

is called the 𝔞-torsion submodule of \underline{E}.

This definition is independent of the generators (a_{1}, …, a_{n}) of 𝔞 by the following

Lemma 6.2. (a) If (a_{1}, …, a_{n}) ⊂ (b_{1}, …, b_{m}) ⊂ A are ideals then ker(φ_{b_{1},...,b_{m}}) ↪ ker(φ_{a_{1},...,a_{n}}) is a closed immersion.

(b) If (a_{1}, …, a_{n}) = (b_{1}, …, b_{m}) then ker(φ_{b_{1},...,b_{m}}) = ker(φ_{a_{1},...,a_{n}}).

(c) For any R-algebra S we have \underline{E}[𝔞](S) = \{ P ∈ E(S): φ_{a}(P) = 0 for all a ∈ 𝔞 \}.

(d) \underline{E}[𝔞] is an A/𝔞-module via A/𝔞 → End_{R}(\underline{E}[𝔞]), b̄ ↦ φ_{b}.

(e) \underline{E}[𝔞] is a finite R-group scheme of finite presentation.

Proof. (a) By assumption there are elements c_{ij} ∈ A with a_{i} = ∑_{j} c_{ij}b_{j}. Therefore φ_{a_{i}} = ∑_{j} φ_{c_{ij}}φ_{b_{j}} and the composition of φ_{b_{1},...,b_{m}}: E → E^{m} followed by (φ_{c_{ij}})_{i,j}: E^{m} → E^{n} equals φ_{a_{1},...,a_{n}}: E → E^{n}. This proves (a) and clearly (a) implies (b). To prove (c) let P: Spec S → \underline{E} be an S-valued point in \underline{E}(S) with 0 = φ_{a}(P) := φ_{a} ∘ P for all a ∈ 𝔞. If 𝔞 = (a_{1}, …, a_{n}) then in particular φ_{a_{i}} ∘ P = 0 for i = 1, …, n. Therefore P factors through ker φ_{a_{1},...,a_{n}} = \underline{E}[𝔞].

Conversely let P: Spec S → \underline{E}[𝔞] be an S-valued point in \underline{E}[𝔞](S) and let a ∈ 𝔞. By (b) we may write 𝔞 = (a_{1}, …, a_{n}) with a_{1} = a to have \underline{E}[𝔞] = ker φ_{a_{1},...,a_{n}}. Therefore φ_{a}(P) := φ_{a} ∘ P = 0. This proves (c).

(d) The relation ab = ba in A implies φ_{a} ∘ φ_{b} = φ_{b} ∘ φ_{a}. Using that the closed subscheme \underline{E}[𝔞] is uniquely determined by (c) it follows that the ring homomorphism A → End_{R}(\underline{E}[𝔞]), b ↦ φ_{b}|_{\underline{E}[𝔞]} is well defined. If b ∈ 𝔞 then clearly φ_{b}|_{\underline{E}[𝔞]} = 0 and so this ring homomorphism factors through A/𝔞.

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

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

Lemma 6.3. Let (0) ≠ 𝔞, 𝔟 ⊂ A be two ideals with 𝔞 + 𝔟 = A.

(a) For an abelian Anderson A-module \underline{E} there is a canonical isomorphism \underline{E}[𝔞] ×_{R} \underline{E}[𝔟] ^{∼} ⟶ \underline{E}[𝔞𝔟].

(b) For an effective A-motive \underline{M} there is a canonical isomorphism \underline{M}/𝔞𝔟\underline{M} ^{∼} ⟶ \underline{M}/𝔞\underline{M} ⊕ \underline{M}/𝔟\underline{M} of finite 𝔽_{q}-shtukas.

Proof. By the Chinese remainder theorem there is an isomorphism A/𝔞𝔟 ^{∼} ⟶ A/𝔞×A/𝔟 whose inverse is given by (x_{𝔞}, x_{𝔟}) ↦ bx_{𝔞} + ax_{𝔟} for certain elements a ∈ 𝔞 and b ∈ 𝔟 which satisfy a ≡ 1 mod 𝔟 and b ≡ 1 mod 𝔞, and hence a + b ≡ 1 mod 𝔞𝔟.

(b) follows directly from this, because \underline{M}/𝔞\underline{M} = \underline{M} ⊗_{A} A/𝔞.

(a) By Lemma 6.2(a) the addition Δ on \underline{E}[𝔞𝔟] defines a canonical morphism \underline{E}[𝔞] ×_{R} \underline{E}[𝔟] ↪ \underline{E}[𝔞𝔟] ×_{R} \underline{E}[𝔞𝔟] ⟶^{Δ} \underline{E}[𝔞𝔟]. Its inverse is described as follows. The elements a, b ∈ A from above satisfy a𝔟 ⊂ 𝔞𝔟 and b𝔞 ⊂ 𝔞𝔟. By Lemma 6.2(c) the endomorphism φ_{a} of \underline{E}[𝔞𝔟] factors through \underline{E}[𝔟] and φ_{b} factors through \underline{E}[𝔞]. So the inverse is the morphism (φ_{b}, φ_{a}): \underline{E}[𝔞𝔟] → \underline{E}[𝔞] ×_{R} \underline{E}[𝔟]. Indeed, for x ∈ \underline{E}[𝔞𝔟], we compute φ_{b}(x) + φ_{a}(x) = φ_{a+b}(x) = φ_{1}(x) = x, because a + b ≡ 1 mod 𝔞𝔟. On the other hand, for x ∈ \underline{E}[𝔞] and y ∈ \underline{E}[𝔟], we compute φ_{b}(x + y) = φ_{b}(x) = x and φ_{a}(x + y) = φ_{a}(y) = y, because b ≡ 1 mod 𝔞 and a ≡ 1 mod 𝔟.

Theorem 6.4. Let \underline{E} be an abelian Anderson A-module and let (0) ≠ 𝔞 ⊂ A be an ideal.

(a) Then \underline{E}[𝔞] is a finite locally free group scheme over Spec R and a strict 𝔽_{q}-module scheme.

(b) \underline{E}[𝔞] is étale over R if and only if R · γ(𝔞) = R, that is if and only if 𝔞 + J = A_{R}.

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

\underline{M}/𝔞\underline{M} ^{∼} ⟶ \underline{M}_{q}(\underline{E}[𝔞])  of finite 𝔽_{q}-shtukas and
Dr_{q}(\underline{M}/𝔞\underline{M}) ^{∼} ⟶ \underline{E}[𝔞]  of finite locally free R-group schemes.

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

[figure: 0; \underline{E}[𝔭^{e−n}]; i_{e−n,e}; \underline{E}[𝔭^{e}]; j_{e,n}; \underline{E}[𝔭^{n}]; 0 .]  (6.1)

To prove the claim let S be an R-algebra and let P: Spec S → \underline{E}[𝔭^{n}] be an S-valued point in \underline{E}[𝔭^{n}](S). Since φ_{z^{e−n}}: \underline{E} → \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′ and a point P′ ∈ E(S′) with φ_{z^{e−n}}(P′) = P. We have to show that P′ ∈ \underline{E}[𝔭^{e}](S′). For this purpose let a ∈ 𝔭^{e}. Then \frac{a}{1} = \frac{c}{s}(\frac{z}{1})^{e} in A_{𝔭} for c ∈ A, s ∈ A ∖ 𝔭. We compute

φ_{a}(P′) = φ_{s}^{−1} ∘ φ_{c} ∘ φ_{z^{n}} ∘ φ_{z^{e−n}}(P′) = φ_{s}^{−1} ∘ φ_{c} ∘ φ_{z^{n}}(P) = 0 ,

because z^{n} ∈ 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 𝔭 ⊂ A there are (arbitrarily large) integers e such that 𝔭^{e} = (a) is principal. Then \underline{E}[𝔭^{e}] = ker φ_{a} is a finite locally free R-group scheme by Corollaries 5.11 and 5.3. If 0 ≤ n ≤ e then we show that \underline{E}[𝔭^{n}] is flat over R. Namely, using the epimorphism j_{e,n}: \underline{E}[𝔭^{e}] → \underline{E}[𝔭^{n}] from (6.1) and the flatness of \underline{E}[𝔭^{e}] over R, the flatness of \underline{E}[𝔭^{n}] will follow from [EGA, IV_{3}, Théorème 11.3.10] once we show that j_{e,n} is flat in each fiber over a point of Spec R. This follows from [DG70, § III.3, Corollaire 7.4] and so \underline{E}[𝔭^{n}] is flat over R for all n. By Lemma 6.2(e) this proves that \underline{E}[𝔭^{n}] is a finite locally free group scheme over Spec R. Moreover, it is a strict 𝔽_{q}-module scheme by [Fal02, Proposition 2], because for 𝔭^{n} = (a_{1}, …, a_{n}) the morphism φ_{a_{1},...,a_{n}} is strict 𝔽_{q}-linear by Example 4.3. So (a) is established.

If 𝔞 = 𝔭^{e} = (a) we know from Theorem 5.9(c) applied to the isogeny φ_{a} and coker \underline{M}(φ_{a}) = \underline{M}/a\underline{M} that (c) holds. If 0 ≤ n ≤ e we use the exact sequence (6.1) and the fact that the functors 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}/𝔭^{e−n}\underline{M} and image isomorphic to \underline{M}/𝔭^{n}\underline{M}. We obtain an exact sequence of finite 𝔽_{q}-shtukas

[figure: 0; \underline{M}/𝔭^{n}\underline{M}; β_{n,e}; \underline{M}/a\underline{M}; α_{e,e−n}; \underline{M}/𝔭^{e−n}\underline{M}; 0]  (6.2)

with β_{n,e} ∘ α_{e,n} = z^{e−n} on \underline{M}/a\underline{M}. Applying Dr_{q} to (6.2), using the exactness of Dr_{q}, and that Dr_{q}(\underline{M}/a\underline{M}) = \underline{E}[𝔭^{e}] and Dr_{q}(z^{e−n}) = φ_{z}^{e−n}, proves Dr_{q}(\underline{M}/𝔭^{n}\underline{M}) = \underline{E}[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}[𝔭^{e}]) and z^{e−n} = \underline{M}_{q}(φ_{z}^{e−n}), proves \underline{M}/𝔭^{n}\underline{M} = \underline{M}_{q}(\underline{E}[𝔭^{n}]). This establishes (c) in general.

(b) Let R·γ(𝔞) = R, that is there are elements a_{1}, …, a_{n} ∈ 𝔞 and b_{1}, …, b_{n} ∈ R with ∑_{i=1}^{n} b_{i}γ(a_{i}) = 1. Then the open subschemes Spec R[\frac{1}{γ(a_{i})}] ⊂ Spec R cover Spec R and it suffices to check that \underline{E}[𝔞] is étale over Spec R[\frac{1}{γ(a_{i})}] for each i. But there \underline{E}[𝔞] is a closed subscheme of \underline{E}[a_{i}] which is étale by Corollary 5.11. This shows that \underline{E}[𝔞] is unramified over R. Since it is flat by (a), it is étale as desired.

Conversely assume that R · γ(𝔞) ⊂ 𝔪 for a maximal ideal 𝔪 ⊂ R and set k = R/𝔪. Over a field extension k′ of k we have E ×_{R} k = 𝔾_{a,k′}^{d} = Spec k′[x_{1}, …, x_{d}]. We will show that \underline{E}[𝔞] ×_{R} k′ is not étale over k′ by applying the Jacobi criterion [BLR90, §2.2, Proposition 7]. Let 𝔞 = (a_{1}, …, a_{n}). Then \underline{E}[𝔞] = Spec k′[x_{1}, …, x_{d}]/(φ_{a_{1}}^{∗}(x_{1}, …, x_{d}): j = 1, …, n). The Jacobi matrix is

\frac{∂φ_{a_{j}}^{∗}}{∂x_{i}} = ( Lie φ_{a_{1}} ; ⋮ ; Lie φ_{a_{n}} ) ∈ (k′)^{nd×d}.

Since γ(a_{i}) = 0 in k′ each Lie φ_{a_{i}} is a nilpotent d × d matrix. Since φ_{a_{i}} ∘ φ_{a_{j}} = φ_{a_{i}a_{j}} = φ_{a_{j}} ∘ φ_{a_{i}} we have Lie φ_{a_{i}}(ker Lie φ_{a_{j}}) ⊂ ker Lie φ_{a_{j}}. Therefore all ker Lie φ_{a_{i}} have a non-trivial intersection. This shows that the rank of the Jacobi matrix is less than d and \underline{E}[𝔞] ×_{R} k′ is not étale over k′.

Proposition 6.5. Let \underline{M} = (M, τ_{M}) be an A-motive over R of rank r and let (0) ≠ 𝔞 ⊂ A be an ideal with R · γ(𝔞) = R, that is 𝔞 + J = A_{R}. Let s̄ = Spec Ω be a geometric base point of Spec R. Then \underline{M}/𝔞\underline{M} is an étale finite 𝔽_{q}-shtuka whose τ-invariants (\underline{M}/𝔞\underline{M})^{τ} (Ω), see (4.1), form a free A/𝔞-module of rank r which carries a continuous action of the étale fundamental group π_{1}^{ét}(Spec R, s̄).

Proof. This result and its proof are due to Anderson [And86, Lemma 1.8.2] for R a field. We let G := Res_{A/𝔞|𝔽_{q}} GL_{r,A/𝔞} be the Weil restriction with G(R′) = GL_{r}(A/𝔞 ⊗_{𝔽_{q}} R′) for all 𝔽_{q}-algebras R′. Then G is a smooth connected affine group scheme over 𝔽_{q} by [CGP10, Proposition A.5.9]. Thus by Lang’s theorem [Lan56, Corollary on p. 557] the Lang map L: G → G, g ↦ g · σ^{∗}g^{−1} is finite étale and surjective (although not a group homomorphism if r > 1 and 𝔞 ≠ A).

Since 𝔞 + J = A_{R} the isomorphism τ_{M}: σ^{∗}M|_{Spec A_{R}∖V(J)} ^{∼} ⟶ M|_{Spec A_{R}∖V(J)} of \underline{M} induces an isomorphism τ_{M/𝔞M}: σ^{∗}M/𝔞M ^{∼} ⟶ M/𝔞M and makes \underline{M}/𝔞\underline{M} into a finite 𝔽_{q}-shtuka, which is étale. After passing to a covering of Spec R by open affine subschemes, we may assume that there is an isomorphism α: (A/𝔞)^{r} ⊗_{𝔽_{q}} R ^{∼} ⟶ M/𝔞M and then α^{−1} ∘ τ_{M/𝔞M} ∘ σ^{∗}α is an element b ∈ G(R) and corresponds to a morphism b: Spec R → G. The fiber product Spec R ×_{b,G,L}G is finite étale over Spec R

and of the form Spec R′. The projection onto the second factor G corresponds to an element c ∈ G(R′) with c · σ^{∗}c^{−1} = b, that is c = b · σ^{∗}c. This implies α ∘ c = τ_{M/𝔞M} ∘ σ^{∗}(α ∘ c), and thus α ∘ c is an isomorphism (A/𝔞)^{r} ^{∼} ⟶ (\underline{M}/𝔞\underline{M})^{τ} (R′) := \{ m ⊗ M/𝔞M ⊗_{R} R′: m = τ_{M}(σ_{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) ≠ 𝔞 ⊂ A be an ideal with R · γ(𝔞) = R, that is 𝔞 + J = A_{R}. Then for every R-algebra R′ such that Spec R′ is connected, there is an isomorphism of A/𝔞-modules

\underline{E}[𝔞](R′) ^{∼} ⟶ Hom_{A/𝔞}((\underline{M}/𝔞\underline{M})^{τ} (R′) , Hom_{𝔽_{q}}(A/𝔞, 𝔽_{q})) ,
P ⟼  [ \overline{m}  ⟼  [ā ↦ m ∘ φ_{a}(P)] ] .

In particular, if s̄ = Spec Ω is a geometric base point of Spec R, then \underline{E}[𝔞](Ω) is a free A/𝔞-module of rank r which carries a continuous action of the étale fundamental group π_{1}^{ét}(Spec R, s̄).

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 𝔭 ⊂ A is a maximal ideal and the elements of γ(𝔭) ⊂ R are nilpotent. Let q̂ be the cardinality of the residue field 𝔽_{𝔭} = A/𝔭 and f = [𝔽_{𝔭} : 𝔽_{q}], that is q̂ = q^{f}. We fix a uniformizing parameter z ∈ Frac(A) at 𝔭. It defines an isomorphism 𝔽_{𝔭}[[z]] ^{∼} ⟶ Â_{𝔭} := lim_{⟵} A/𝔭^{n}. We consider the 𝔭-adic completion Â_{𝔭,R} := lim_{⟵} A_{R}/𝔭^{n} = (𝔽_{𝔭} ⊗_{𝔽_{q}} R)[[z]]. By continuity the

map γ extends to a ring homomorphism γ: Â_{𝔭} → R. We consider the ideals 𝔞_{i} = (a⊗1−1⊗γ(a)^{q^{i}}: a ∈ 𝔽_{𝔭}) ⊂ Â_{𝔭,R} for i ∈ ℤ/fℤ. By the Chinese remainder theorem Â_{𝔭,R} decomposes

Â_{𝔭,R} = (𝔽_{𝔭} ⊗_{𝔽_{q}} R)[[z]] = ∏_{i∈ℤ/fℤ} Â_{𝔭,R}/𝔞_{i} ,

and Â_{𝔭,R}/𝔞_{i} is the subset of Â_{𝔭,R} on which a ⊗ 1 acts as 1 ⊗ γ(a)^{q^{i}} for all a ∈ 𝔽_{𝔭}. Each factor is canonically isomorphic to R[[z]]. The factors are cyclically permuted by σ because σ(𝔞_{i}) = 𝔞_{i+1}. In particular σ̂ := σ^{f} stabilizes each factor and acts on it via σ̂(z) = z and σ̂(b) = bq̂ for b ∈ R. The ideal J := (a ⊗ 1 − 1 ⊗ γ(a): a ∈ A) ⊂ A_{R} decomposes as follows J · Â_{𝔭,R}/𝔞_{0} = (z − γ(z)) and J ·Â_{𝔭,R}/𝔞_{i} = (1) for i ≠ 0. In particular, Â_{𝔭,R}/𝔞_{0} equals the J -adic completion of A_{R}, as γ(z) is nilpotent in R; compare also [AH14, Lemma 5.3]. We also set R((z)) := R[[z]][\frac{1}{z}].

Definition 7.1. A local σ̂-shtuka (or local shtuka) of rank r over R is a pair \underline{M̂} = (M̂, τM̂) consisting of a locally free R[[z]]-module M̂ of rank r, and an isomorphism τM̂: σ̂^{∗}M̂[\frac{1}{z−γ(z)}] ^{∼} ⟶ M̂[\frac{1}{z−γ(z)}]. If τM̂(σ̂^{∗}M̂) ⊂ M̂ then \underline{M̂} is called effective, and if τM̂(σ̂^{∗}M̂) = M̂ then \underline{M̂} is called étale.

A morphism of local shtukas f: (M̂, τM̂) → (M̂′, τM̂′) over R is a morphism of R[[z]]-modules f: M̂ → M̂′ which satisfies τM̂′ ∘ σ̂^{∗}f = f ∘ τM̂.

Example 7.2. Let \underline{M} = (M, τ_{M}) be an A-motive over R. We consider the 𝔭-adic completion \underline{M} ⊗_{A_{R}} Â_{𝔭,R} := (M ⊗_{A_{R}} Â_{𝔭,R} , τ_{M} ⊗ 1) = lim_{⟵} \underline{M}/𝔭^{n}\underline{M}. We define the local σ̂-shtuka at 𝔭 associated with \underline{M} as

\underline{M̂}_{𝔭}(\underline{M}) := (M ⊗_{A_{R}} Â_{𝔭,R}/𝔞_{0} , (τ_{M} ⊗1)^{f}), where τ_{M}^{f} := τ_{M} ∘σ^{∗}τ_{M} ∘…∘σ^{(f−1)∗}τ_{M}. It equals the 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 𝔽_{𝔭} = 𝔽_{q}, and hence q̂ = q and σ̂ = σ, we have Â_{𝔭,R} = R[[z]] and \underline{M̂}_{𝔭}(\underline{M}) = \underline{M} ⊗_{A_{R}} Â_{𝔭,R}.

Also for f > 1 the local shtuka \underline{M̂}_{𝔭}(\underline{M}) allows to recover \underline{M} ⊗_{A_{R}} Â_{𝔭,R} via the isomorphism

⨁_{i=0}^{f−1}(τ_{M} ⊗ 1)^{i} mod 𝔞_{i}: (⨁_{i=0}^{f−1} σ^{i∗}(M ⊗_{A_{R}} Â_{𝔭,R}/𝔞_{0}), (τ_{M} ⊗ 1)^{f} ⊕ ⨁_{i≠0} id) ^{∼} ⟶ \underline{M} ⊗_{A_{R}} Â_{𝔭,R} ,

because for i ≠ 0 the equality J ·Â_{𝔭,R}/𝔞_{i} = (1) implies that τ_{M} ⊗ 1 is an isomorphism modulo 𝔞_{i}; see [HK16, Example 2.2] or [BH11, Propositions 8.8 and 8.5] for more details.

Let \underline{M̂} = (M̂, τM̂) be an effective local shtuka over R. Set \underline{M̂}_{n} := (M̂_{n}, τM̂_{n}) := (M̂/z^{n}M̂, τM̂ mod z^{n}) and G_{n} := Drq̂(\underline{M̂}_{n}). Then G_{n} is a finite locally free strict 𝔽_{𝔭}-module scheme over R and \underline{M̂}_{n} = \underline{M} q̂(G_{n}) by Theorem 4.7. Moreover, G_{n} inherits from \underline{M̂}_{n} an action of 𝔽_{𝔭}[z]/(z^{n}). The canonical epimorphisms \underline{M̂}_{n+1} ↠ \underline{M̂}_{n} induce closed immersions i_{n}: G_{n} ↪ G_{n+1}. The inductive limit Drq̂(\underline{M̂}) := lim_{⟶} G_{n} in the category of sheaves on the big fppf-site of Spec R is a sheaf of 𝔽_{𝔭}[[z]]-modules that satisfies the following

Definition 7.3. A 𝔭-divisible local Anderson module over R is a sheaf of 𝔽_{𝔭}[[z]]-modules G on the big fppf-site of Spec R such that

(a) G is 𝔭-torsion, that is G = lim_{⟶} G[z^{n}], where G[z^{n}] := ker(z^{n}: G → G), (b) G is 𝔭-divisible, that is z: G → G is an epimorphism,

(c) For every n the 𝔽_{𝔭}-module G[z^{n}] is representable by a finite locally free strict 𝔽_{𝔭}-module scheme over R (Definition 4.2), and

(d) there exist an integer d ∈ ℤ_{≥0}, such that (z − γ(z))^{d} = 0 on ω_{G} where ω_{G} := lim_{⟵} ω_{G[z^{n}]} and

ω_{G[z^{n}]} = e^{∗}Ω_{G[z^{n}]/ Spec R}^{1} is the pullback under the zero section e: Spec R → G[z^{n}].

A morphism of 𝔭-divisible local Anderson modules over R is a morphism of fppf-sheaves of 𝔽_{𝔭}[[z]]- modules. The category of divisible local Anderson modules is 𝔽_{𝔭}[[z]]-linear. It is shown in [HS15, Lemma 8.2] that ω_{G} is a finite locally free R-module and we define the dimension of G as rk ω_{G}. A 𝔭-divisible local Anderson module is called étale if ω_{G} = 0. Since ω_{G} surjects onto each ω_{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 𝔭-divisible local Anderson module G over R one associates the local shtuka \underline{M} q̂(G) := lim_{⟵} \underline{M} q̂(G[z^{n}]). Multiplication with z on G gives Mq̂(G) the structure of an R[[z]]-module. In [HS15, Theorem 8.3] we proved the following

Theorem 7.4. (a) The two contravariant functors Drq̂ and \underline{M} q̂ are mutually quasi-inverse antiequivalences between the category of effective local shtukas over R and the category of 𝔭-divisible local Anderson modules over R.

(b) Both functors are 𝔽_{𝔭}[[z]]-linear and map short exact sequences to short exact sequences. They preserve étale objects.

Let \underline{M̂} = (M̂, τM̂) be an effective local shtuka over S and let G = Drq̂(\underline{M̂}) be its associated 𝔭-divisible local Anderson module. Then

(c) G is a formal Lie group if and only if τM̂ is topologically nilpotent, that is im(τ n ; M̂) ⊂ zM̂ for an integer n.

(d) the R[[z]]-modules ω_{Drq̂(\underline{M̂})} and coker τM̂ are canonically isomorphic.

We now want to show that for an abelian Anderson A-module \underline{E} over R the local shtuka \underline{M̂}_{𝔭}(\underline{M}(\underline{E})) corresponds to the 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 γ(𝔭) ⊂ R are nilpotent. We define \underline{E}[𝔭^{∞}] := lim_{⟶} \underline{E}[𝔭^{n}] and call it the 𝔭-divisible local Anderson module associated with \underline{E}.

This definition is justified by the following

Theorem 7.6. Let \underline{E} = (E, φ) be an abelian Anderson A-module over R and assume that the elements of γ(𝔭) ⊂ R are nilpotent. Then

(a) all \underline{E}[𝔭^{n}] are finite locally free strict 𝔽_{𝔭}-module schemes,

(b) \underline{E}[𝔭^{∞}] is a 𝔭-divisible local Anderson module over R,

(c) If \underline{M} = \underline{M}(\underline{E}) is the associated effective A-motive of \underline{E} and \underline{M̂} := \underline{M̂}_{𝔭}(\underline{M}) = \underline{M} ⊗_{A_{R}} Â_{𝔭,R}/𝔞_{0} is the local σ̂-shtuka at 𝔭 associated with \underline{M}, then there are canonical isomorphisms

\underline{M} q̂(\underline{E}[𝔭^{∞}]) ≅ \underline{M̂}_{𝔭}(\underline{M})  and \underline{E}[𝔭^{∞}] ≅ Drq̂(\underline{M̂}_{𝔭}(\underline{M})) ,
\underline{M}_{q}(\underline{E}[𝔭^{∞}]) ≅ \underline{M} ⊗_{A_{R}} Â_{𝔭,R} and \underline{E}[𝔭^{∞}] ≅ Dr_{q}(\underline{M} ⊗_{A_{R}} Â_{𝔭,R}) ,
\underline{M} q̂(\underline{E}[𝔭^{n}]) ≅ \underline{M̂}/𝔭^{n}\underline{M̂}  and  \underline{E}[𝔭^{n}] ≅ Drq̂(\underline{M̂}/𝔭^{n}\underline{M̂}) .

Proof. (a) By Lemma 4.4 we may test strictness after applying a faithfully flat base change to R and assume that E = 𝔾_{a,R}^{d} = Spec R[x_{1}, …, x_{d}] = Spec R[\underline{X}] and \underline{M}(\underline{E}) = R\{τ\}^{1×d}. We set B := Γ(\underline{E}[𝔭^{n}], O_{\underline{E}[𝔭^{n}]}) and I = ker(R[\underline{X}] ↠ B) and I_{0} = (x_{1}, …, x_{d}), and consider the deformation B^{♭} = R[\underline{X}]/I · I_{0}. The endomorphisms φ_{a} of E for a ∈ A satisfy φ_{a}^{∗}(I) ⊂ I and φ_{a}^{∗}(I_{0}) ⊂ I_{0}. This defines a lift A → End_{R-algebras}(B^{♭}), a ↦ [a]^{♭} := φ_{a}^{∗} compatible with addition and multiplication as in Definition 4.2.

Let N ≥ dim \underline{E} be a positive integer which is a power of q̂ such that γ(a)^{N} = 0 for every a ∈ 𝔭^{n}. Choose λ ∈ 𝔽_{𝔭} with 𝔽_{𝔭} = 𝔽_{q}(λ) and let g be the minimal polynomial of λ over 𝔽_{q}. Choose an element t ∈ A with t mod 𝔭^{n} = λ in A/𝔭^{n} = 𝔽_{𝔭}[[z]]/(z^{n}). Then g(t) ∈ 𝔭^{n}, and hence γ(g(t))^{N} = 0. On Lie E the equation g(t^{N}) = g(t)^{N} implies Lie φ_{g(t^{N})} = Lie φ_{g(t)}^{N} − γ(g(t))^{N} = (Lie φ_{g(t)} − γ(g(t)))^{N} = 0. So φ_{g(t^{N})} ∈ End_{R-groups,𝔽_{q}-lin}(𝔾_{a,R}^{d}) = R{τ}^{d×d} as a polynomial in τ has no constant term. This means that φ_{g(t^{N})}^{∗}(x_{i}) ∈ I_{0}^{q}. Moreover, since g(t) ∈ 𝔭^{n} we have φ_{g(t)} = 0 on \underline{E}[𝔭^{n}] and hence φ_{g(t)}^{∗}(x_{i}) ∈ I. Therefore φ_{g(t^{q̂N})}^{∗}(I_{0}) = φ_{g(t)}^{∗} ∘ φ_{g(t^{q̂N−N−1})}^{∗} ∘ φ_{g(t^{N})}^{∗}(I_{0}) ⊂ φ_{g(t)}^{∗}(I_{0}^{q}) ⊂ φ_{g(t)}^{∗}(I_{0})^{2} ⊂ I ·I_{0}. In other words [g(t^{q̂N})]^{♭} = [0]^{♭} on B^{♭}. This shows that the map 𝔽_{𝔭} = 𝔽_{q}[t^{q̂N}]/(g(t^{q̂N})) → End_{R-algebras}(B^{♭}) lifts the action of 𝔽_{𝔭} ⊂ 𝔽_{𝔭}[[z]]/(z^{n}) on \underline{E}[𝔭^{n}] and is compatible with addition and multiplication.

We compute the induced action on the co-Lie complex ℓG/ Spec R^{•} of G = (Spec B, Spec B^{♭}). In degree 0 we have ℓ_{G/ Spec R}^{0} = Ω_{R[\underline{X}]/R}^{1} ⊗_{R[\underline{X}], e_{R[\underline{X}]}} R = ⨁_{i=1}^{d} R·x_{i} = I_{0}/I_{0}^{2}. From t − λ ∈ 𝔭^{n} we obtain γ(t^{q̂N}) − γ(λ) = γ(t − λ)^{q̂N} = 0 in R. On Lie E this implies Lie φ_{t^{q̂N}} − γ(λ) = (Lie φ_{t} − γ(t))^{q̂N} = 0 and therefore φ_{t^{q̂N}} − γ(λ) ∈ End_{R-groups,𝔽_{q}-lin}(𝔾_{a,R}^{d}) = R{τ}^{d×d} as a polynomial in τ has no constant term. This implies that (φ_{t^{q̂N}}^{∗} − γ(λ))(I_{0}) ⊂ I_{0}^{q} ⊂ I_{0}^{2}. We conclude that t^{q̂N} acts as the scalar γ(λ) on I_{0}/I_{0}^{2}.

To compute the action of t^{q̂N} on ℓ_{G/ Spec R}^{−1} we use that by Theorem 4.7(d), ℓG/ Spec R^{•} is homotopically

equivalent to the complex 0 → σ^{∗}M/𝔭^{n}σ^{∗}M ⟶^{τ_{M}} M/𝔭^{n}M → 0 where \underline{M}_{q}(\underline{E}[𝔭^{n}]) = \underline{M}/𝔭^{n}\underline{M} and \underline{M} = \underline{M}(\underline{E}) = (M, τ_{M}); see Theorem 6.4(c). Since t^{q̂N}−γ(λ) = (t⊗1−1⊗γ(t))^{q̂N} = 0 on coker τ_{M} there is an A_{R}-homomorphism h: M → σ^{∗}M with h τ_{M} = (t^{q̂N} −γ(λ))·id_{σ^{∗}M} and τ_{M} h = (t^{q̂N} −γ(λ))·id_{M}. This means that t^{q̂N} is homotopic to the scalar multiplication with γ(λ) on 0 → σ^{∗}M/𝔭^{n}σ^{∗}M ⟶^{τ_{M}} M/𝔭^{n}M → 0, and therefore also on ℓG/ Spec R^{•}. Let h′: I_{0}/I_{0}^{2} → ℓ_{G/ Spec R}^{−1} =: ℓ^{−1} be this homotopy, that is (t^{q̂N} − γ(λ))|_{ℓ^{−1}} = h′d and (t^{q̂N} − γ(λ))|_{I_{0}/I_{0}^{2}} = dh′. But we must show that t^{q̂N} and γ(λ) are not only homotopic on ℓG/ Spec R^{•}, but equal.

Since 0 = g(t^{q̂N}) = ∏_{i∈ℤ/fℤ}(t^{q̂N} − γ(λ)^{q^{i}}) on ℓG/ Spec R^{•}, we can decompose ℓ^{−1} = ∏_{i∈ℤ/fℤ}(ℓ^{−1})_{i}

where (ℓ^{−1})_{i} := ker(t^{q̂N} − γ(λ)^{q^{i}}: ℓ^{−1} → ℓ^{−1}). Since the differential d of ℓG/ Spec R^{•} is an R-homo-

morphism and equivariant for the action of t^{q̂N}, it maps (ℓ^{−1})_{i} into ker(t^{q̂N} − γ(λ)^{q^{i}}: I_{0}/I_{0}^{2} → I_{0}/I_{0}^{2}) which is trivial for i ≠ 0. This shows that 0 = h′d = t^{q̂N} − γ(λ) = γ(λ^{q^{i}} − λ) on (ℓ^{−1})_{i}, whence (ℓ^{−1})_{i} = (0) for i ≠ 0, because γ(λ^{q^{i}} − λ) ∈ R^{×}. We conclude that ℓ^{−1} = (ℓ^{−1})_{0} and t^{q̂N} acts as the scalar γ(λ) on ℓ^{−1}. This proves that \underline{E}[𝔭^{n}] is a finite locally free strict 𝔽_{𝔭}-module scheme over R.

(b) By construction ker(z^{n}: \underline{E}[𝔭^{∞}] → \underline{E}[𝔭^{∞}]) = \underline{E}[𝔭^{n}] and \underline{E}[𝔭^{∞}] is 𝔭-torsion. Using the epimorphism j_{n+1,n}: \underline{E}[𝔭^{n+1}] ↠ \underline{E}[𝔭^{n}] from (6.1) with i_{n,n+1} ∘ j_{n+1,n} = φ_{z} we see that \underline{E}[𝔭^{∞}] is 𝔭-divisible. In (a) we saw that \underline{E}[𝔭^{n}] is representable by a finite locally free strict 𝔽_{𝔭}-module scheme over R. It remains to verify condition (d) of Definition 7.3. Since \underline{E}[𝔭^{n}] ↪ \underline{E} is a closed immersion, ω_{\underline{E}[𝔭^{n}]} is a quotient of ω_{\underline{E}} = Hom_{R}(Lie \underline{E}, R). Since A/𝔭^{n} = 𝔽_{𝔭}[[z]]/(z^{n}), there is an element a ∈ A with z − a ∈ 𝔭^{n}, whence φ_{a} = φ_{z} on \underline{E}[𝔭^{n}]. Therefore (Lie φ_{a} − γ(a))^{d} = 0 on Lie \underline{E} implies (φ_{z} − γ(z))^{N} = (φ_{a} − γ(a))^{N} + γ(a − z)^{N} = 0 on ω_{\underline{E}[𝔭^{n}]}. It follows that (φ_{z} − γ(z))^{N} = 0 on ω_{\underline{E}[𝔭^{∞}]} := lim_{⟵} ω_{\underline{E}[𝔭^{n}]}, and that \underline{E}[𝔭^{∞}] is a 𝔭-divisible local Anderson module over R.

(c) We have \underline{M}_{q}(\underline{E}[𝔭^{n}]) = Hom_{R-groups,𝔽_{q}-lin}(\underline{E}[𝔭^{n}], 𝔾_{a,R}) = \underline{M}/𝔭^{n}\underline{M} and \underline{E}[𝔭^{n}] = Dr_{q}(\underline{M}/𝔭^{n}\underline{M}) by Theorem 6.4(c). This implies

\underline{M}_{q}(\underline{E}[𝔭^{∞}]) = lim_{⟵} \underline{M}_{q}(\underline{E}[𝔭^{n}]) = lim_{⟵} \underline{M}/𝔭^{n}\underline{M} = \underline{M} ⊗_{A_{R}} Â_{𝔭,R}

and \underline{E}[𝔭^{∞}] = lim_{⟶} Dr_{q}(\underline{M}/𝔭^{n}\underline{M}) = Dr_{q}(lim_{⟵} \underline{M}/𝔭^{n}\underline{M}) = Dr_{q}(\underline{M} ⊗_{A_{R}} Â_{𝔭,R}).

On \underline{E}[𝔭^{n}] every λ ∈ 𝔽_{𝔭} acts as φ_{λ} and on 𝔾_{a,R} as γ(λ). Therefore

\underline{M} q̂(\underline{E}[𝔭^{n}]) = Hom_{R-groups,𝔽_{𝔭}-lin}(\underline{E}[𝔭^{n}], 𝔾_{a,R})
= \underline{M}_{q}(\underline{E}[𝔭^{n}])/𝔞_{0}\underline{M}_{q}(\underline{E}[𝔭^{n}])
= \underline{M}/𝔭^{n}\underline{M} ⊗Â_{𝔭,R} Â_{𝔭,R}/𝔞_{0}
= \underline{M̂}/𝔭^{n}\underline{M̂} ,

where the second equality is due to the fact that Â_{𝔭,R}/𝔞_{0} is the summand of Â_{𝔭,R} on which λ ⊗ 1 acts as 1 ⊗ γ(λ) for all λ ∈ 𝔽_{𝔭}. This implies

\underline{M} q̂(\underline{E}[𝔭^{∞}]) = lim_{⟵} \underline{M}/𝔭^{n}\underline{M} ⊗Â_{𝔭,R} Â_{𝔭,R}/𝔞_{0} = \underline{M} ⊗_{A_{R}} Â_{𝔭,R}/𝔞_{0} = \underline{M̂}_{𝔭}(\underline{M}) = \underline{M̂} .

On the other hand, since \underline{E}[𝔭^{n}] is a finite locally free strict 𝔽_{𝔭}-module by (a), \underline{E}[𝔭^{n}] = Drq̂(\underline{M} q̂(\underline{E}[𝔭^{n}])) = Drq̂(\underline{M̂}/𝔭^{n}\underline{M̂}) by Theorem 4.7(c), and so \underline{E}[𝔭^{∞}] = lim_{⟶} Drq̂(\underline{M̂}/𝔭^{n}\underline{M̂}) = Drq̂(\underline{M̂}_{𝔭}(\underline{M})).

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Urs Hartl Universität Münster Mathematisches Institut Einsteinstr. 62 D – 48149 Münster Germany www.math.uni-muenster.de/u/urs.hartl/