Change of Coefficients for Drinfeld Modules, Shtuka, and Abelian Sheaves

Urs Hartl, Markus Hendler ^{∗}

August 10, 2006

Abstract

We study the passage from Drinfeld-A′-modules to Drinfeld-A-modules for a given finite flat inclusion A ⊂ A′. We show that this defines a morphism from the moduli space of Drinfeld-A′-modules to the moduli space of Drinfeld-A-modules which is proper but in general not representable. For Drinfeld-Anderson shtuka and abelian sheaves instead of Drinfeld modules we obtain the same results. Mathematics Subject Classification (2000): 11G09, (14G35)

Introduction

Throughout this article let 𝔽_{q} be a finite field with q elements and characteristic p and let C and C′ be two smooth projective geometrically irreducible curves over 𝔽_{q}. Let π : C′ → C be a fixed finite morphism of degree n. Let ∞ ∈ C be a closed point which does not split in C′, that is, there is exactly one point ∞′ ∈ C′ above ∞. Set A := Γ(C ∖ {∞}, O_{C}) and A′ := Γ(C′ ∖ {∞′}, O_{C′}), then A′ is a flat A-algebra via π^{∗} : A → A′.

In this situation π defines a restriction of coefficients functor from Drinfeld-A′-modules over S to Drinfeld-A-modules over S. This functor induces a morphism between the moduli spaces (moduli functors, or more sophisticated, moduli stacks) classifying Drinfeld-A′-modules, respectively Drinfeld-A-modules. We show in this article that this morphism is proper but not necessarily representable. Likewise we study the effect of π on Drinfeld-Anderson shtuka, see Definition 1.7, and on abelian sheaves, a notion introduced by the first author [9] as a higher dimensional generalization of Drinfeld modules, see Definition 1.5. For the case of Drinfeld-Anderson shtuka we may even relax the condition on π and drop the assumption on the ramification of ∞. The pushforward of sheaves along π × id_{S} : C′_{S} → C_{S} defines a restriction of coefficients functor from Drinfeld-Anderson shtuka on C′ over S to Drinfeld-Anderson shtuka on C over S, respectively from abelian sheaves on C′ over S to abelian sheaves on C over S. Again this yields proper but in general not representable morphisms between the moduli spaces classifying Drinfeld-Anderson shtuka on C′, respectively Drinfeld-Anderson shtuka on C and similarly for abelian sheaves.

Of course the results for Drinfeld modules, Drinfeld-Anderson shtuka, and abelian sheaves are strongly related by the fact that the category of Drinfeld-A-modules over S is antiequivalent to a full subcategory of the category of Drinfeld-Anderson shtuka on C over S and anti-equivalent to a full subcategory of the category of abelian sheaves on C over S. Nevertheless we give proofs also for the case of Drinfeld modules since these are particularly simple. After recalling the definitions and some basic properties in Section 1 we prove in Sections 2, 3, and 4 the properness and non-representability results for Drinfeld modules, abelian sheaves, respectively Drinfeld-Anderson shtuka.

∗Both authors acknowledge support of the Deutsche Forschungsgemeinschaft in form of DFG-grant HA3006/2-1

This article has its origin in a conversation with F. Breuer who mentioned to us a special case of the proof for properness in the case of Drinfeld modules. Our proof of Proposition 2.3 below is a generalization of his. We like to express our gratitude to him.

1 Drinfeld Modules, Shtuka, and Abelian Sheaves

We retain the notation from the introduction. In addition, we set deg(∞) := [κ(∞) : 𝔽_{q}] and we denote by ord_{∞} the normalized valuation on the fraction field of A associated with the place ∞. For an 𝔽_{q}-scheme S we set C_{S} := C ×_{𝔽_{q}} S. Unless mentioned explicitly we make no noetherian assumption on S.

For an 𝔽_{q}-algebra B we denote by B{τ} the non-commutative polynomial ring in the variable τ over B with the commutation rule τ b = b^{q} τ for all b ∈ B. As in [14, §1] one sees

Proposition 1.1. There is an isomorphism of rings between B{τ} and End_{B,𝔽_{q}}(𝔾_{a,B}) the ring of 𝔽_{q}-linear endomorphisms of the additive group scheme over Spec B given by mapping τ to the q-th power Frobenius of 𝔾_{a,B}.

Definition 1.2. (Drinfeld [5, §5.B]) Let S be an 𝔽_{q}-scheme and assume there is a morphism c : S → Spec A. Let r be a positive integer. A Drinfeld-A-module of rank r and characteristic c over S is a pair (E, φ) where E is a commutative group scheme over S and

φ : A ⟶ End_{S}(E)

is a ring homomorphism from A to the ring End_{S}(E) of endomorphisms of the S-group scheme E such that

1. E is Zariski locally on S isomorphic to the additive group scheme 𝔾_{a,S},

2. if U = Spec B is an affine open subset of S and ψ : E_{U} ⟶^{∼} 𝔾_{a,U} is an isomorphism of S-group schemes then for each a ∈ A ∖ {0}

ψ ∘ φ(a) ∘ ψ^{−1} = ∑_{i=0}^{<∞} δ_{i}(a)τ^{i} ∈ B{τ}

with δ_{0}(a) = c^{∗}(a), δ_{i}(a) ∈ B^{×} for i = d(a) := −r ord_{∞}(a) deg(∞), and δ_{i}(a) nilpotent

for i > d(a).

A morphism of Drinfeld-A-modules ε : (E, φ) → (Ẽ, φ̃) is a morphism of S-group schemes ε : E → Ẽ which satisfies φ̃(a) ∘ ε = ε ∘ φ(a) for all a ∈ A.

If f : S′ → S is a morphism of 𝔽_{q}-schemes we can pull back Drinfeld-A-modules (E, φ) over S to Drinfeld-A-modules (f^{∗}E, f^{∗}φ) over S′.

The following proposition is due to Drinfeld [5, Propositions 5.1 and 5.2] Proposition 1.3. Let (E, φ) be a Drinfeld-A-module of rank r over S. Then Zariski locally on S there exists an isomorphism ε : (E, φ) ⟶^{∼} (𝔾_{a,S}, ψ) of Drinfeld-A-modules where ψ is of the standard form

ψ : A ⟶ O_{S}{τ} , ψ(a) = ∑_{i=0}^{d(a)} δ_{i}(a)τ^{i}

with d(a) := −r ord_{∞}(a) deg(∞) and δ_{d(a)} ∈ OS^{×}. Moreover if ψ(a) is of the described form for one a ∈ A ∖ 𝔽_{q} then it already is for any a ∈ A.

Proposition 1.4. The morphism π : C′ → C defines a restriction of coefficients functor π_{∗} : (E′, φ′) ↦ (E′, φ′ ∘ π^{∗}) from Drinfeld-A′-modules of rank r′ over S to Drinfeld-A-modules of rank nr′ over S, where n is the degree of π.

Proof. The change of rank results from the fact that n ord_{∞}(a) deg(∞) = ord_{∞′}(a) deg(∞′) for all a ∈ A since π^{−1}(∞) = {∞′}. The rest is clear from the definition.

Remark. Consider the moduli problem, that is, the contravariant functor

\underline{Dr-A-Mod}^{r} : Sch_{/ Spec A} ⟶ Sets
(c : S → Spec A) ↦ { Isomorphism classes of Drinfeld-A-modules
of rank r and characteristic c over S }

from the category of schemes over Spec A to the category of sets. This functor is not representable (without adding level structures). Nevertheless the restriction of coefficients functor defines a restriction of coefficients morphism

π_{∗} : \underline{Dr-A′-Mod}^{r′} ⟶ \underline{Dr-A-Mod}^{nr′} , (E′, φ′) ↦ π_{∗}(E′, φ′) .

Remark. If we let S vary, the category of Drinfeld-A-modules of rank r becomes a stack Dr-A-Mod^{r} for the fppf topology on the category of 𝔽_{q}-schemes. It is an algebraic stack in the sense of Deligne-Mumford [4], see Laumon [12, Corollary 1.4.3]. The restriction of coefficients functor defines a restriction of coefficients 1-morphism π_{∗} : Dr-A′-Mod^{r′} → Dr-A-Mod^{nr′}.

Next we study the analogous situation for abelian sheaves. This notion was introduced in [9]. While Drinfeld modules are analogues for elliptic curves in the arithmetic of function fields, abelian sheaves are the appropriate analogues for abelian varieties as the results of [9, 2] amply demonstrate.

Let r and d be positive integers and write \frac{d}{r deg(∞)} = \frac{k}{ℓ} with relatively prime positive integers k and ℓ. Let S be an 𝔽_{q}-scheme and fix a morphism c : S → C. Let J be the ideal sheaf on C_{S} of the graph of c. We let σ := id_{C} × Frob_{q} be the endomorphism of C_{S} that acts as the identity on the underlying topological space and on the coordinates of C and as b ↦ b^{q} on the elements b ∈ O_{S}. Let pr : C_{S} → S be the projection onto the second factor. For an integer m denote by O_{C_{S}}(m · ∞) the invertible sheaf on C_{S} associated with the divisor m · ∞ and set F(m · ∞) := F ⊗_{O_{C_{S}}} O_{C_{S}}(m · ∞) for any sheaf of O_{C_{S}}-modules on C_{S}.

Definition 1.5. An abelian sheaf \underline{F} = (F_{i}, Π_{i}, τ_{i}) on C of rank r, dimension d, and characteristic c over S is a ladder of locally free sheaves F_{i} on C_{S} of rank r and injective homomorphisms Π_{i}, τ_{i} of O_{C_{S}}-modules (i ∈ ℤ) of the form

⋯ ⟶ F_{i−1} ⟶^{Π_{i−1}} F_{i}  ⟶^{Π_{i}} F_{i+1} ⟶^{Π_{i+1}} ⋯
↑^{τ_{i−2}}  ↑^{τ_{i−1}}  ↑^{τ_{i}}
⋯ ⟶ σ^{∗}F_{i−2} ⟶^{σ^{∗}Π_{i−2}} σ^{∗}F_{i−1} ⟶^{σ^{∗}Π_{i−1}} σ^{∗}F_{i} ⟶^{σ^{∗}Π_{i}} ⋯

subject to the following conditions (for all i ∈ ℤ):

1. the above diagram is commutative,

2. the morphism Π_{i+ℓ−1} ∘ … ∘ Π_{i} identifies F_{i} with the subsheaf F_{i+ℓ}(−k · ∞) of F_{i+ℓ},

3. pr_{∗} coker Π_{i} is a locally free O_{S}-module of rank d,

4. coker τ_{i} is annihilated by J^{d} and pr_{∗} coker τ_{i} is a locally free O_{S}-module of rank d.

A morphism between two abelian sheaves (F_{i}, Π_{i}, τ_{i}) and (F′_{i}, Π′_{i}, τ′_{i}) is a collection of morphisms F_{i} → F′_{i} which commute with the Π’s and the τ’s.

Remark. Abelian sheaves of dimension d = 1 are called elliptic sheaves and were studied by Drinfeld [6] and Blum-Stuhler [1]. The category of Drinfeld-A-modules of rank r over S is anti-equivalent to the category of elliptic sheaves of rank r over S which satisfy deg F_{0} = 1−r, see [1, Theorem 3.2.1].

Proposition 1.6. The push forward along π : C′_{S} → C_{S} defines a restriction of coefficients functor

π_{∗} : \underline{F}′ = (F′_{i}, Π′_{i}, τ′_{i}) ⟼ π_{∗}\underline{F}′ := (π_{∗}F′_{i}, π_{∗}Π′_{i}, π_{∗}τ′_{i})

from abelian sheaves on C′ of rank r′, dimension d′ and characteristic c′ : S → C′ over S to abelian sheaves on C of rank nr′, dimension d′ and characteristic π ∘ c′ : S → C over S. Here n is the degree of π.

Proof. Since π is finite and flat the sheaves π_{∗}F_{i} are locally free of rank nr′ by [3, Corollary 2 to Proposition II.3.2.5]. Let k and ℓ be relatively prime positive integers with \frac{k}{ℓ} = \frac{d′}{nr′ deg(∞)}. Let e be the ramification index of π at ∞′. Then n = e deg(∞′)/ deg(∞) and hence k = k′/ gcd(k′, e) and ℓ = ℓ′e/ gcd(k′, e). From axiom 2 of Definition 1.5 we obtain an isomorphism

Π′_{i+ℓ−1} ∘ … ∘ Π′_{i} : F′_{i} ⟶^{∼} F′_{i+ℓ} ⊗_{O_{C′_{S}}} O_{C′_{S}}(−ke · ∞′) .

Since π^{∗}O_{C_{S}}(∞) = O_{C′_{S}}(e · ∞′) the projection formula

π_{∗}(F′_{i+ℓ} ⊗_{O_{C′_{S}}} O_{C′_{S}}(−ke · ∞′)) = (π_{∗}F′_{i+ℓ}) ⊗_{O_{C_{S}}} O_{C_{S}}(−k · ∞)

yields

π_{∗}Π′_{i+ℓ−1} ∘ … ∘ π_{∗}Π′_{i} : π_{∗}F′_{i} ⟶^{∼} (π_{∗}F′_{i+ℓ}) ⊗_{O_{C_{S}}} O_{C_{S}}(−k · ∞)

from which the proposition is evident.

Remark. Consider the contravariant moduli functor

\underline{C-Ab-Sh}^{r,d} : Sch_{/C} ⟶ Sets
(c : S → C) ↦ { Isomorphism classes of abelian sheaves on C of
rank r, dimension d, and characteristic c over S }

Also this functor is not representable (not even after adding level structures, see [9, Remark 4.2]). Again the restriction of coefficients functor defines a restriction of coefficients morphism

π_{∗} : \underline{C′-Ab-Sh}^{r′,d′} ⟶ \underline{C-Ab-Sh}^{nr′,d′} , \underline{F}′ ↦ π_{∗}\underline{F}′ .

Remark. If we let S vary, the category of abelian sheaves on C of rank r and dimension d becomes a stack C-Ab-Sh^{r,d} for the fppf topology on the category of 𝔽_{q}-schemes. It is an algebraic stack in the sense of Deligne-Mumford [4] by [9, Theorem 3.1]. The restriction of coefficients functor defines a restriction of coefficients 1-morphism π_{∗} : C′-Ab-Sh^{r′,d′} → C-Ab-Sh^{nr′,d′}.

The construction of [1, Theorem 3.2.1] yields a 1-isomorphism of Dr-A-Mod^{r} with an open and closed substack of C-Ab-Sh^{r,1}, see [9, Example 1.8] such that the following diagram is 2-commutative

\xymatrix{Dr-A′-Mod^{r′} \ar@{^{(}->}[r] \ar[d]^{π_{∗}} & C′-Ab-Sh^{r′,1} \ar[d]^{π_{∗}} \\ Dr-A-Mod^{nr′} \ar@{^{(}->}[r] & C-Ab-Sh^{nr′,1} .}

Finally let us turn to Drinfeld-Anderson shtuka.

Definition 1.7. A right (left) Drinfeld-Anderson shtuka \underline{E} = (E, Ẽ, j, τ, b, c) on C of rank r and dimension d over S consists of two 𝔽_{q}-morphisms b, c : S → C and a diagram

[figure: E; j; E′; E; σ^{∗}E; τ; _{(resp.}; E′; τ; j; σ^{∗}E )]

of locally free sheaves E and Ẽ of rank r on C_{S} such that coker j, respectively coker τ, are locally free of rank d as O_{S}-modules and supported on the graphs of b, respectively c. The morphism b is called the pole of \underline{E} and c is called the zero of \underline{E}.

Remark. Every abelian sheaf (F_{i}, Π_{i}, τ_{i}) on C of rank r, dimension d, and characteristic c over S gives rise to a right Drinfeld-Anderson shtuka on C over S by setting for any i ∈ ℤ

E := F_{i} , Ẽ:= F_{i+1} , j := Π_{i} , τ := τ_{i} .

This defines a faithful functor from abelian sheaves to Drinfeld-Anderson shtuka on C over S. Together with the functor from Drinfeld-A-modules to elliptic sheaves on C one obtains a fully faithful functor from Drinfeld-A-modules of rank r over S to Drinfeld-Anderson shtuka on C of rank r and dimension 1 over S, see Drinfeld [7, §1]

The argument of Proposition 1.6 also shows Proposition 1.8. Relaxing the conditions on π : C′ → C assume only that π is finite of degree n. Then the push forward along π defines a restriction of coefficients functor

π_{∗} : (E , Ẽ, j , τ , b , c) ⟼ (π_{∗}E , π_{∗}Ẽ, π_{∗}j , π_{∗}τ , π ∘ b , π ∘ c)

from Drinfeld-Anderson shtuka on C′ of rank r′ and dimension d′ to Drinfeld-Anderson shtuka on C of rank nr′ and dimension d′ over S.

Remark. Consider the contravariant moduli functor

\underline{C-DA-Sht}^{r,d} : Sch_{/C×C} ⟶ Sets
((b, c) : S → C ×_{𝔽_{q}} C) ↦ { Isomorphism classes of Drinfeld-Anderson shtuka on
C of rank r, dimension d, pole b, and zero c over S }

Also this functor is not representable but the restriction of coefficients functor defines a restriction of coefficients morphism

π_{∗} : \underline{C′-DA-Sht}^{r′,d′} ⟶ \underline{C-DA-Sht}^{nr′,d′} , \underline{F}′ ↦ π_{∗}\underline{F}′ .

Here again the category of Drinfeld-Anderson shtuka of rank r and dimension d over varying 𝔽_{q}-schemes S is an algebraic stack C-DA-Sht^{r,d} for the fppf topology in the sense of Deligne-Mumford [4] and the restriction of coefficients functor defines a restriction of coefficients 1-morphism π_{∗} : C′-DA-Sht^{r′,d′} → C-DA-Sht^{nr′,d′}.

2 Restriction of Coefficients for Drinfeld Modules

Theorem 2.1. The restriction of coefficient morphism π_{∗} : \underline{Dr-A′-Mod}^{r′} → \underline{Dr-A-Mod}^{nr′} for Drinfeld modules is in general not relatively representable.

Proof. We give a counterexample to relative representability. Let q = 3, A = 𝔽_{3}[x], A′ = 𝔽_{3}[y] and π^{∗}: A → A′, x ↦ y^{2}. Let S = Spec 𝔽_{3} and c^{∗}: A → 𝔽_{3}, x ↦ 0. Consider the Drinfeld-A-module (E, φ) of rank 2 over S given by E = 𝔾_{a,S} and

φ : A ⟶ 𝔽_{3}{τ} , φ(x) = τ^{2} .

Let \underline{T} := \underline{Dr-A′-Mod}^{1} ×\underline{Dr-A-Mod}_{2} S be the fiber product of functors. Then \underline{T} is the contravariant functor

\underline{T} : Sch_{/ Spec A′×_{Spec A}S} ⟶ Sets
(S′, c′ : S′ → Spec A′ ↦ { Isomorphism classes of Drinfeld-A′-modules (E′, φ′)
f : S′ → S  )  of rank 1 over S′, such that f^{∗}(E, φ) ≅ π_{∗}(E′, φ′) } .

We show that \underline{T} is not representable. For this purpose make S into a Spec A′-scheme by (c′)^{∗} : A′ → 𝔽_{3}, y ↦ 0. Then \underline{T} (S) contains two isomorphism classes given by E′_{1} = E′_{2} = 𝔾_{a,S} and

φ′_{1} : y ↦ τ  and  φ′_{2} : y ↦ −τ .

These two isomorphism classes are different because otherwise there were an isomorphism

ε ∈ Isom((E′_{1}, φ′_{1}), (E′_{2}, φ′_{2})) = { ε ∈ OS^{×} : −τ ∘ ε = ε ∘ τ } .

That is, ε ∈ 𝔽3^{×} must satisfy −ε^{3}τ = ετ, whence ε^{2} = −1. This is impossible for ε ∈ 𝔽3^{×}.

On the other hand such an element exists in 𝔽9^{×}. So if S′ = Spec 𝔽_{9} the two isomorphism classes become equal in \underline{T} (S′). But this implies that \underline{T} is not representable. Since if it were representable by a scheme T we had two different morphisms from S to T which yield the same morphism from S′ to T

Spec 𝔽_{9} ⟶ Spec 𝔽_{3} \overrightarrow{⟶} T .

As Spec 𝔽_{9} → Spec 𝔽_{3} is a homeomorphism and 𝔽_{3} ⊂ 𝔽_{9} this is impossible.

Remark. The reason why \underline{T} is not representable is that the isomorphism α : f^{∗}(E, φ) ⟶^{∼} π_{∗}(E′, φ′) in the definition of \underline{T} (S) is only supposed to exist but is not added to the data. More precisely we have

Theorem 2.2. Let c : S → Spec A be a morphism of 𝔽_{q}-schemes and let (E, φ) be a Drinfeld-A-module of rank nr′ and characteristic c over S. Then the contravariant functor

\underline{T} : Sch_{/ Spec A′×_{Spec A}S} ⟶ Sets
(S′, c′ : S′ → Spec A′ ↦ { Isomorphism classes of tripples (E′, φ′, α) where
f : S′ → S  )  • (E′, φ′) is a Drinfeld-A′-module of rank r′

and characteristic c′ over S′ and

• α : f^{∗}(E, φ) ⟶^{∼} π_{∗}(E′, φ′) is a fixed isomorphism }

is representable by an affine S-scheme of finite presentation.

Proof. Since the question is local on S we may by Proposition 1.3 assume that S = Spec B, E = 𝔾_{a,B} and φ is given by φ : A → B{τ} such that the highest coefficient of every φ(a) is a unit in B.

Let the A-algebra A′ be generated by a′_{1}, …, a′_{N}. In order to extend φ to φ′ : A′ → B{τ} we must define φ′(a′_{1}), …, φ′(a′_{N}). Set d_{ν} := −r′ ord_{∞′}(a′_{ν}) deg(∞′) for all ν. Define

B′ := B ⊗_{A} A′ [ δ_{i,ν}, δ_{d_{ν},ν}^{−1} : ν = 1, …, N , i = 0, …, d_{ν} ]

and the morphism c′ : Spec B′ → Spec A′ by the natural map A′ → B′. Define

φ′(a′_{ν}) := ∑_{i=0}^{d_{ν}} δ_{i,ν}τ^{i} ∈ B′{τ} ,

and φ′|_{A} := φ, and let α = id_{𝔾_{a,B′}}. In order that the so defined φ′ is a Drinfeld-A′-module of rank r′ and characteristic c′ over Spec B′ we must require several conditions which are all represented by finitely presented closed subschemes of Spec B′. Namely consider successively for ν = 1, …, N the minimal polynomial of a′_{ν} over A(a′_{1}, …, a′_{ν−1})

(a′_{ν})^{m} + b_{ν,m−1}(a′_{ν})^{m−1} + … + b_{ν,1}a′_{ν} + b_{ν,0} = 0

with b_{ν,k} ∈ A(a′_{1}, …, a′_{ν−1}). The fact that φ′ : A′ → B′{τ} is a ring homomorphism is now expressed by the vanishing of

φ′(a′_{ν})^{m} + φ′(b_{ν,m−1})φ′(a′_{ν})^{m−1} + … + φ′(b_{ν,1})φ′(a′_{ν}) + φ′(b_{ν,0}) = 0

in B′{τ}. Looking at the coefficients of this τ-polynomial we get a finitely generated ideal of B′ which we must require to vanish, that is, must divide out. Likewise the commutation of φ′(a′_{ν}) with a (finite) generating system of the 𝔽_{q}-algebra A(a′_{1}, …, a′_{ν−1}) yields a finitely generated ideal of B′. Finally the condition on the characteristic means that (c′)^{∗}(a′_{ν}) = δ_{0,ν}. Putting everything together the sum of these ideals defines a closed subscheme T ⊂ Spec B′ which is of finite presentation and affine over S.

We claim that T represents \underline{T}. So let (E′, φ′, α) be an element of \underline{T} (S′). The isomorphism α : f^{∗}𝔾_{a,S} ⟶^{∼} E′ yields an isomorphism α : (𝔾_{a,S′}, ψ′) ⟶^{∼} (E′, φ′) of Drinfeld-A′-modules over S′ where ψ′(a) := α^{−1} ∘ φ′(a) ∘ α for all a ∈ A′. Since ψ′(a) = f^{∗}φ(a) for a ∈ A, ψ′ is of the form described in Proposition 1.3. In particular

ψ′(a′_{ν}) = ∑_{i=0}^{d_{ν}} δ_{i}(a′_{ν})τ^{i} ∈ Γ(S′, O_{S′}){τ} .

Mapping δ_{i,ν} to δ_{i}(a′_{ν}) defines the desired uniquely determined morphism B′ → Γ(S′, O_{S′}), whence S′ → T.

Proposition 2.3. In the situation of Theorem 2.2 the scheme T representing \underline{T} is finite over S.

Proof. We already know that T is separated and of finite presentation over S. We use the valuative criterion of properness to show that it is proper. Since it is also affine over S it must be finite.

So let R be a valuation ring with fraction field K and consider the diagram

\xymatrix{Spec K \ar[r]^{(E′,φ′,α)} \ar[d]_{g} & T \ar[d]^{π_{∗}} \\ Spec R \ar[r]_{f} \ar@{-->}[ur]_{(Ẽ,φ̃,α̃)} & S}

where the horizontal arrow on top is given by a Drinfeld-A′-module (E′, φ′) of rank r′ and characteristic c′ : Spec K → Spec A′ together with an isomorphism α : (fg)^{∗}(E, φ) ⟶^{∼} π_{∗}(E′, φ′) over Spec K. We must exhibit the dashed arrow which corresponds to a Drinfeld-A′-module (Ẽ, φ̃) of rank r′ and characteristic c̃ : Spec R → Spec A′ (note that c′ factors through a unique morphism c̃ satisfying π ∘ c̃ = c ∘ f : Spec R → Spec A because Spec A′ is proper over Spec A) together with an isomorphism α̃ : f^{∗}(E, φ) ⟶^{∼} π_{∗}(Ẽ, φ̃) over Spec R. The commutativity of the diagram means that there exists an isomorphism β : g^{∗}(Ẽ, φ̃) ⟶^{∼} (E′, φ′) with π_{∗}β ∘ g^{∗}α̃ = α.

Since R is a local ring f^{∗}E = 𝔾_{a,R} without loss of generality and f^{∗}φ : A → R{τ}. We use the isomorphism α to replace (E′, φ′) by (𝔾_{a,K}, ψ′) with ψ′(a) := α^{−1} ∘ φ′(a) ∘ α ∈ K{τ} for all a ∈ A′. Thus α is replaced by id_{𝔾_{a,K}} and ψ′(a) = (fg)^{∗}φ(a) for all a ∈ A. If we show that ψ′(a) belongs to R{τ} for all a ∈ A′, then we may take Ẽ = 𝔾_{a,R} and φ̃ = ψ′ : A′ → R{τ}, as well as α̃ = id_{𝔾_{a,R}} and β = id_{𝔾_{a,K}}, and we are done.

So let a ∈ A′ and let a^{m} + b_{m−1}a^{m−1} + … + b_{1}a + b_{0} = 0 be an equation of integral dependence of a over A. In particular

(2.1)  ψ′(a)^{m} + ψ′(b_{m−1})ψ′(a)^{m−1} + … + ψ′(b_{1})ψ′(a) + ψ′(b_{0}) = 0 .

Over an algebraic closure of K the polynomial ψ′(a)(x), where we use τ(x) = x^{q}, splits as ψ′(a)(x) = ∏_{i}(x − λ_{i}) with λ_{i} ∈ K^{alg}. From equation (2.1) we see that each λ_{i} is a root of ψ′(b_{0}) = (fg)^{∗}φ(b_{0}). Since f^{∗}φ(b_{0}) has coefficients in R with the highest coefficient in R^{×}, all λ_{i} must be integral over R. Therefore the coefficients of ψ′(a) which are symmetric polynomials in the λ_{i} are integral over R and belong to K, hence they lie in R as desired. This proves the proposition.

Theorem 2.4. The restriction of coefficients morphism π_{∗} : \underline{Dr-A′-Mod}^{r′} → \underline{Dr-A-Mod}^{nr′} satisfies the valuative criterion for properness.

Proof. Let R be a valuation ring with fraction field K and consider the diagram

[figure: \underline{T}; Spec K; (E′,φ′,α); (E′,φ′); f; \underline{Dr-A′-Mod}^{r′} ×\underline{Dr-A-Mod}_{nr′} Spec R; □; _{\underline{Dr-A′-Mod}_{r′}}; π_{∗}; Spec R; Spec R; (E,φ); _{\underline{Dr-A-Mod}_{nr′}}]

where the horizontal morphisms are induced by a Drinfeld-A′-module (E′, φ′) of rank r′ and characteristic c′ : Spec K → Spec A′ over Spec K and a Drinfeld-A-module (E, φ) of rank nr′ and characteristic c : Spec R → Spec A over Spec R and where \underline{T} is the representable functor from Theorem 2.2 for S = Spec R. The commutativity of the square on the left means that f^{∗}(E, φ) ≅ π_{∗}(E′, φ′). The choice of any such isomorphism α defines a morphism Spec K → \underline{T}. By Proposition 2.3 we find a unique morphism Spec R → \underline{T} fitting into the diagram which induces the dashed morphism. It remains to show that the dashed morphism is uniquely determined (independent of the choice of α) and this is proved in the following lemma.

Lemma 2.5. Let R be a valuation ring with fraction field K and let f : Spec K → Spec R be the induced morphism. Let (E′_{1}, φ′_{1}) and (E′_{2}, φ′_{2}) be two Drinfeld-A′-modules of rank r′ and characteristic c′ : Spec R → Spec A′ over Spec R and let α : f^{∗}(E′_{1}, φ′_{1}) ⟶^{∼} f^{∗}(E′_{2}, φ′_{2}) be an isomorphism over Spec K. Then α = f^{∗}β for a unique isomorphism β : (E′_{1}, φ′_{1}) ⟶^{∼} (E′_{2}, φ′_{2}) over Spec R.

Proof. Since R is a local ring we have without loss of generality E′_{1} = E′_{2} = 𝔾_{a,R}. Let a ∈ A′ ∖ 𝔽_{q} and write for j = 1, 2

φ′_{j}(a) = ∑_{i=0}^{m} δ_{i,j}τ^{i} .

The isomorphism over Spec K is given by an element α ∈ K^{×} which satisfies φ′_{2}(a) ∘ α = α ∘ φ′_{1}(a), whence δ_{2,m} α^{q^{m}} = α δ_{1,m}. Since δ_{1,m} and δ_{2,m} are units in R the same is true for α. So the isomorphism α is already defined over Spec R.

Remark. Phrased in the language of stacks [13], Theorems 2.1 and 2.4 say that the restriction of coefficients 1-morphism π_{∗} : Dr-A′-Mod^{r′} → Dr-A-Mod^{nr′} is proper but in general not representable. Namely by the arguments of Theorem 2.2 the stack T classifying data ((E, φ), (E′, φ,), α) where (E, φ), respectively (E′, φ′), is a Drinfeld-A-module of rank nr′, respectively a Drinfeld-A′-module of rank r′ over the same scheme S together with a fixed isomorphism α : (E, φ) ⟶^{∼} π_{∗}(E′, φ′) over S is relatively representable over Dr-A-Mod^{nr′} by a finite and finitely presented morphism of schemes. The projection T → Dr-A′-Mod^{r′} onto (E′, φ′) is an étale epimorphism and makes T into a torsor under the finite relative group scheme Aut(π_{∗}(E′, φ′)) over Dr-A′-Mod^{r′}. In particular Dr-A′-Mod^{r′} is of finite presentation over Dr-A-Mod^{nr′} since T is and it satisfies the valuative criterion for properness by the arguments of Theorem 2.4.

3 Restriction of Coefficients for Abelian Sheaves

Theorem 3.1. The restriction of coefficients morphism π_{∗} : \underline{C′-Ab-Sh}^{r′,d′} → \underline{C-Ab-Sh}^{nr′,d′} for abelian sheaves is in general not relatively representable.

Proof. This follows directly from Theorem 2.1 and the remark after Definition 1.5. The example from Theorem 2.1 yields the following abelian sheaf on C = ℙ_{𝔽_{3}}^{1} over S = Spec 𝔽_{3}. Let F_{i} = O_{C_{S}}(⌊\frac{i+1}{2}⌋ · ∞) ⊕ O_{C_{S}}(⌊\frac{i}{2}⌋ · ∞) where ⌊\frac{i}{2}⌋ is the largest integer ≤ \frac{i}{2}. Let Π_{i} be the

natural inclusion F_{i} ⊂ F_{i+1} and let τ_{i} : σ^{∗}F_{i} → F_{i+1} be given by the matrix ( \begin{matrix}0 & x \\ 1 & 0 \end{matrix} ) where

ℙ_{𝔽_{3}}^{1} ∖ {∞} = Spec 𝔽_{3}[x].

Let π : C′ = ℙ_{𝔽_{3}}^{1} → C be given by A → A′ = 𝔽_{3}[y], x ↦ y^{2}. Then (F_{i}, Π_{i}, τ_{i}) is isomorphic to π_{∗}(F′_{i}, Π′_{i}, τ′_{i}) for F′_{i} = O_{C′_{S}}(i · ∞′), Π′_{i} the natural inclusion, and τ′_{i} = ±y. The two abelian sheaves for τ′_{i} = +y and τ′_{i} = −y are not isomorphic over Spec 𝔽_{3} but become isomorphic over Spec 𝔽_{9}.

Theorem 3.2. Let S be a locally noetherian 𝔽_{q}-scheme and let c : S → C be an 𝔽_{q}-morphism. Let \underline{F} be an abelian sheaf on C of rank nr′, dimension d′ and characteristic c over S. Then the contravariant functor

\underline{T} : Sch_{/C′×_{C}S} ⟶ Sets
(S′, c′ : S′ → C′ ↦ \{ Isomorphism classes of pairs (\underline{F}′, α) where
f : S′ → S )  • \underline{F}′ is an abelian sheaf of rank r′, dimension d′,

and characteristic c′ over S′ and

• α : f^{∗}\underline{F} ⟶^{∼} π_{∗}\underline{F}′ is a fixed isomorphism \}

is representable by a (quasi-affine) S-scheme of finite type.

For the proof we need the following Lemma 3.3. Let S be a locally noetherian scheme, let ρ : Y → S be a flat projective morphism, and let π : X → Y be a finite faithfully flat morphism of degree n. For an S-scheme S′ set Y′ := Y ×_{S} S′ and X′ := X ×_{S} S′. Let F be a locally free sheaf on Y of rank rn. Then the contravariant functor

\underline{U} : Sch_{/S} ⟶ Sets
(f : S′ → S) ↦ { Isomorphism classes of pairs (F′, α) where

• F′ is a locally free sheaf of rank r on X′ and

• α : f^{∗}F ⟶^{∼} π_{∗}F′ is a fixed isomorphism }

is representable by a (quasi-affine) S-scheme of finite type.

Proof. Since the question is local on S we may assume that S is affine. By [EGA, II, Proposition 1.4.3] the functor \underline{U} is isomorphic to the functor

\underline{U}′ : (f : S′ → S) ↦ Hom_{O_{Y′}-algebras}(π_{∗}O_{X′} , End_{O_{Y′}} (f^{∗}F))

the set of O_{Y′}-algebra homomorphisms π_{∗}O_{X′} → End_{O_{Y′}} (f^{∗}F). Fix an ample invertible sheaf L on Y. For any integer N define H_{N} := Hom_{O_{Y}} (F, F ⊗_{O_{Y}} L^{⊗N}) = End_{O_{Y}} (f^{∗}F)⊗_{O_{Y}} L^{⊗N}. Then for the homomorphisms of O_{Y′}-modules we obtain

Hom_{O_{Y′}-modules}(π_{∗}O_{X′} , End_{O_{Y′}} (f^{∗}F)) = Hom_{O_{Y′}-mod}(π_{∗}O_{X′} ⊗_{O_{Y′}} L^{⊗N} , f^{∗}H_{N}) .

There is an integer N such that

• π_{∗}O_{X} ⊗_{O_{Y}} L^{⊗N} is generated by global sections and

• ρ_{∗}(π_{∗}O_{X} ⊗_{O_{Y}} L^{⊗N}) and ρ_{∗}H_{N} are locally free on S

since these conditions are achieved for N ≫ 0 by the Theorem on Cohomology and Base Change [10, Theorem III.12.11].

Shrinking S we let x_{1}, …, x_{m} be an O_{S}-basis of ρ_{∗}(π_{∗}O_{X} ⊗_{O_{Y}} L^{⊗N}). We must specify their images in ρ_{∗}H_{N}. Let U_{1} := \underline{Spec}_{S} Sym_{O_{S}}(ρ_{∗}H_{N})^{∨} and U_{2} := U_{1} ×_{S} … ×_{S} U_{1} the m-fold fiber product. Let f : U_{2} → S be the induced morphism and set Y_{2} := Y ×_{S} U_{2} and X_{2} := X ×_{S} U_{2}. Then for any S-scheme S′

Hom_{S}(S′, U_{1}) = Hom_{O_{S}-algebras}(Sym_{O_{S}}(ρ_{∗}H_{N})^{∨} , O_{S′})
= Hom_{O_{S}-modules}((ρ_{∗}H_{N})^{∨} , O_{S′})
= Γ(S′ , O_{S′} ⊗_{O_{S}} ρ_{∗}H_{N}) .

So on U_{2} there exist m universal global sections of ρ_{∗}H_{N} which we use as the images of our x_{1}, …, x_{m} to obtain a universal homomorphism of O_{U_{2}}-modules

(3.2)  ρ_{∗}(π_{∗}O_{X_{2}} ⊗_{O_{Y_{2}}} f^{∗}L^{⊗N}) ⟶ ρ_{∗}f^{∗}H_{N} .

Next we take care of the O_{Y} -algebra structures. Every x_{i} has a minimal polynomial over Γ(Y, L^{⊗N})[x_{1}, …, x_{i−1}] of the form

P (x_{i}) := x_{i}^{k} + a_{k−1}x_{i}^{k−1} + … + a_{0} = 0

inside Γ(Y, π_{∗}O_{X} ⊗_{O_{Y}} L^{⊗Nk}). Using our homomorphism (3.2) and the O_{U_{2}}-module structure of f^{∗}F we view P (x_{i}) as an element of Γ(U_{2}, f^{∗}ρ_{∗}H_{Nk}). The requirement that this element vanishes defines a closed subscheme of U_{2} by Lemma 3.4 below. Let U_{3} be the closed subscheme of U_{2} obtained in this way for i = 1, …, m. This yields a π_{∗}O_{X_{3}}-module structure on f^{∗}F, whence (an isomorphism class of) a coherent sheaf F_{3} on X_{3} := X ⊗_{S} U_{3} together with an isomorphism α : f^{∗}F ⟶^{∼} π_{∗}F_{3}.

It remains to represent the condition that F_{3} is locally free. Let V ⊂ X_{3} be the open subscheme on which F_{3} is flat, see [EGA, IV_{3}, Theorem 11.1.1]. Define U := U_{3} ∖ π(X_{3} ∖ V). Since ρπ : X_{3} → U_{3} is proper U ⊂ U_{3} is open. Since (ρπ)^{−1}U ⊂ V the coherent sheaf F_{3} is locally free on (ρπ)^{−1}U of rank r. We claim that U represents the functor \underline{U}. Indeed, let S′ be an S-scheme and (F′, α) ∈ \underline{U}(S′). Then the π_{∗}O_{X′}-module structure on π_{∗}F′ defines a uniquely determined morphism S′ → U_{3}. Since above every point s ∈ S′ the fiber F′_{s} is flat on X ×_{S} s, the image of s in U_{3} lands in U by [EGA, IV_{3}, Theorem 11.3.10]. (This is the only place where we use the assumption that π is flat.) This proves the lemma.

Lemma 3.4. Let S be a scheme and let H be a locally free sheaf on S. Let I be a set and let h_{i} ∈ Γ(S, H) for all i ∈ I. Then the condition h_{i} = 0 for all i ∈ I is represented by a closed subscheme of S.

Proof. This is [EGA, 0_{new}, Proposition 5.5.1] taking into account that on a locally noetherian topological space the set of global sections of an arbitrary direct sum equals the direct sum of the global sections.

Proof of Theorem 3.2. Let \underline{F} = (F_{i}, Π_{i}, τ_{i}) and let ℓ′ and k′ be relatively prime positive integers with \frac{k′}{ℓ′} = \frac{d′}{r′ deg(∞′)}. For i = 0, …, ℓ′ let U_{i} be the scheme from Lemma 3.3 classifying the pairs (F′_{i}, α_{i}) of locally free sheaves F′_{i} on X = C′_{S} and isomorphisms α_{i} : F_{i} ⟶^{∼} π_{∗}F′_{i}. Over T := U_{0} ×_{S} … ×_{S} U_{ℓ′} we have the universal sheaves F′_{0}, …, F′_{ℓ′} on C′_{T}. We need that the morphisms of O_{C_{T}} -modules

Π′_{i} := α_{i+1} ∘ Π_{i} ∘ α_{i}^{−1} : π_{∗}F′_{i} ⟶ π_{∗}F′_{i+1}  and
τ′_{i} := α_{i+1} ∘ τ_{i} ∘ σ^{∗}α_{i}^{−1} : π_{∗}σ^{∗}F′_{i} ⟶ π_{∗}F′_{i+1}

are actually morphisms of π_{∗}O_{C′_{T}} -modules and thus by [EGA, II, Proposition 1.4.3] morphisms Π′ : F′_{i} → F′_{i+1} and τ′_{i} : σ^{∗}F′_{i} → F′_{i+1}.

It suffices to work on an affine covering of T. Let pr : C_{T} → T be the projection onto the second factor. Let L be an ample invertible sheaf on C and let N be an integer such that for i = 0, …, ℓ′ − 1

• π_{∗}O_{C′_{T}} ⊗_{O_{C_{T}}} L^{⊗N} is generated by global sections x_{1}, …, x_{m},

• π_{∗}F′_{i} ⊗_{O_{C_{T}}} L^{⊗N} is generated by global sections y_{1}, …, y_{n}, and

• H_{i+1} := pr_{∗}(π_{∗}F′_{i+1} ⊗_{O_{C_{T}}} L^{⊗2N}) is locally free on T.

Then G_{i} := π_{∗}O_{C′_{T}} ⊗_{O_{C_{T}}} π_{∗}F′_{i}⊗_{O_{C_{T}}} L^{⊗2N} is generated by the x_{μ}⊗y_{ν}. There are two morphisms of O_{C_{T}} -modules

G_{i} \overrightarrow{⟶} π_{∗}F′_{i+1} ⊗_{O_{C_{T}}} L^{⊗2N}

depending on the order in which Π′_{i} is composed with the contraction π_{∗}O_{C′_{T}} ⊗_{O_{C_{T}}} π_{∗}F′_{i} → π_{∗}F′_{i} (coming from the O_{C′_{T}} -module structure on F′_{i}). Whether the difference of these two morphisms is the zero morphism can be tested on the images of the global sections x_{μ} ⊗ y_{ν} inside H_{i+1}. By Lemma 3.4 this condition is represented by a closed subscheme of T.

We proceed analogously for the τ_{i} and obtain a closed subscheme T_{1} ⊂ T and for i = 0, …, ℓ′ − 1 universal morphisms Π′_{i} : F′_{i} → F′_{i+1} and τ′_{i} : σ^{∗}F′_{i} → F′_{i+1} on C′_{T_{1}} which satisfy axiom 1 of Definition 1.5. Since pr_{∗}π_{∗} coker Π′_{i} = pr_{∗} coker Π_{i}, and the same for τ_{i}, also axioms 3 and 4 hold except for the condition on the support. For this condition let T_{2} := C′ ×_{C} T_{1}, let c′ : T_{2} → C′ be the projection and let J′ be the ideal defining the graph of c′. Similarly to the above argument let L and N be such that (J′)^{⊗d′} ⊗_{O_{C{′ \atop T_{2}}}} L^{⊗N} is generated

by global sections. Again by Lemma 3.4 the condition that the multiplication morphism

pr_{∗}((J′)^{⊗d′} ⊗_{O_{C{′ \atop T_{2}}}} L^{⊗N} ⊗_{O_{C{′ \atop T_{2}}}} coker τ′_{i}) ⟶ pr_{∗}(L^{⊗N} ⊗_{O_{C{′ \atop T_{2}}}} coker τ′_{i})

is zero is represented by a closed subscheme T_{3} of T_{2}.

Finally for axiom 2 consider the morphism

(3.3)  Π′_{ℓ′−1} ∘ … ∘ Π′_{0} : F′_{0} ⟶ F′_{ℓ′} ⟶ F′_{ℓ′} ⊗_{O_{C{′ \atop T_{3}}}} O_{C′_{T_{3}}} /O_{C′_{T_{3}}} (−k′ · ∞′) .

Since coker Π′_{i} has rank d′ axiom 2 is satisfied if and only if the morphism (3.3) is the zero morphism. Using that the target is locally free on T_{3} and reasoning as above the later condition is represented by a closed subscheme T_{4} of T_{3}. Over T_{4} we define F′_{i+mℓ′} := F′_{i}(k′m · ∞′) for all i = 0, …, ℓ′ − 1 and all m ∈ ℤ. Then T_{4} represents the functor \underline{T}.

Proposition 3.5. In the situation of Theorem 3.2 the scheme T representing \underline{T} is finite over S.

Proof. By Theorem 3.2 it is separated, of finite type, and quasi-affine over S. It remains to show that T is proper over S. So let R be a discrete valuation ring with fraction field K and consider the diagram

\xymatrix{Spec K \ar[r]^{(\underline{F}′,α)} \ar[d]_{g} & T \ar[d]^{π_{∗}} \\ Spec R \ar[r]_{f} \ar@{-->}[ur]_{(\underline{F̃},α̃)} & S}

where the horizontal arrow is given by an abelian sheaf \underline{F}′ on C′ over Spec K of rank r′, dimension d′ and characteristic c′ : Spec K → C′ together with an isomorphism α : (fg)^{∗}\underline{F} ⟶^{∼} π_{∗}\underline{F}′ on C_{K}. We need to construct an abelian sheaf \underline{F̃} on C′ over Spec R of rank r′, dimension d′, and characteristic c̃ : Spec R → C′ (again the properness of π implies that c′ factors through a unique morphism c̃ with π ∘ c̃ = c ∘ f : Spec R → C) together with an isomorphism α̃ : f^{∗}\underline{F} ⟶^{∼} π_{∗}\underline{F̃} on C_{R} and an isomorphism β : g^{∗}\underline{F̃} ⟶^{∼} \underline{F}′ on C′_{K} satisfying π_{∗}β ∘g^{∗}α̃ = α.

We begin by constructing for all i ∈ ℤ the locally free sheaf F̃_{i} on C′_{R} and the isomorphism α̃_{i} : f^{∗}F_{i} ⟶^{∼} π_{∗}F̃_{i}. Let L be an ample invertible sheaf on C such that π_{∗}O_{C′} ⊗_{O_{C}} L is generated by global sections x_{1}, …, x_{m}.

For the next step in the proof we need to introduce some notation. Let ϖ be the generic point of the special fiber of C_{R} over the residue field of R and let O_{ϖ} := O_{C_{R},ϖ} be the local ring at ϖ. It is a discrete valuation ring and every uniformizing parameter of R is a uniformizing parameter of O_{ϖ}. Let further K(C) be the fraction field of O_{ϖ}. It equals the function field of C_{K}. Similarly let O_{ϖ′} and K(C′) be the rings associated with the curve C′. Since the f^{∗}Π_{i} are invertible over O_{ϖ} we get τ-modules (f^{∗}F_{i} ⊗_{O_{C_{R}}} O_{ϖ}, f^{∗}(Π_{i}^{−1} ∘ τ_{i})) over O_{ϖ} with f^{∗}(Π_{i}^{−1} ∘τ_{i}) being isomorphisms. Now the argument of Gardeyn [8, Proposition 2.13(i )] shows that (f^{∗}F_{i} ⊗_{O_{C_{R}}} O_{ϖ}, f^{∗}(Π_{i}^{−1} ∘ τ_{i})) is the unique maximal f^{∗}(Π_{i}^{−1} ∘ τ_{i})-invariant O_{ϖ}-lattice in (f^{∗}F_{i} ⊗_{O_{C_{R}}} K(C), f^{∗}(Π_{i}^{−1} ∘ τ_{i})). Since every x_{μ} is an endomorphism of

(f^{∗}F_{i} ⊗_{O_{C_{R}}} K(C) , f^{∗}(Π_{i}^{−1} ∘ τ_{i}))

it must map f^{∗}F_{i} ⊗_{O_{C_{R}}} O_{ϖ} into itself. This makes f^{∗}F_{i} ⊗_{O_{C_{R}}} O_{ϖ} into a free O_{ϖ′}-module. Now we can apply Lafforgue’s [11, Lemme 2.7] which says that to give a locally free sheaf F̃_{i} on C′_{R} is equivalent to giving its restrictions F̃_{i} ⊗_{O_{C′_{R}}} O_{C′_{K}} and F̃_{i} ⊗_{O_{C′_{R}}} O_{ϖ′}. Thus out of F′_{i}

and the O_{ϖ′}-module f^{∗}F_{i} ⊗_{O_{C_{R}}} O_{ϖ} we may construct the locally free sheaf F̃_{i} together with the isomorphism β_{i} : g^{∗}F̃_{i} ⟶^{∼} F′_{i}. Since by construction

α_{i} : ((fg)^{∗}F_{i} , f^{∗}F_{i} ⊗_{O_{C_{R}}} O_{ϖ}) ⟶^{∼} (π_{∗}(F̃_{i} ⊗_{O_{C′_{R}}} O_{C′_{K}}) , π_{∗}(F̃_{i} ⊗_{O_{C′_{R}}} O_{ϖ′}))

is an isomorphism on the two restrictions we obtain the isomorphism α̃_{i} : f^{∗}F_{i} ⟶^{∼} π_{∗}F̃_{i} from Lafforgue’s lemma.

Since the Π′_{i} and the τ′_{i} are commuting homomorphisms of O_{C′_{K}}-modules they restrict to commuting homomorphisms Π̃_{i} and τ̃_{i} of O_{C′_{R}}-modules. Altogether we have shown that \underline{F̃} = (F̃_{i}, Π̃_{i}, τ̃_{i}) satisfies axioms 1, 3, and 4 from Definition 1.5 except for the condition on the support of coker τ̃_{i}. Let J̃ be the ideal sheaf on C′_{R} defining the graph of c̃. Then J̃^{d′} annihilates the generic fiber of the free R-module coker τ̃_{i}, so it annihilates all of coker τ̃_{i}. Likewise if z′ is a uniformizing parameter on C′ at ∞′ then (z′)^{k′} annihilates the generic fiber of the free R-module coker(Π̃_{i+ℓ′−1} ∘ … ∘ Π̃_{i}), so it annihilates this whole cokernel. Now all axioms are verified and \underline{F̃} is the desired abelian sheaf on C′ over Spec R.

Theorem 3.6. The restriction of coefficients morphism π_{∗} : \underline{C′-Ab-Sh}^{r′,d′} → \underline{C-Ab-Sh}^{nr′,d′} satisfies the valuative criterion for properness.

Proof. Since the stacks C-Ab-Sh^{r,d} are locally noetherian by [9, Theorem 3.1] it suffices to test the valuative criterion only for discrete valuation rings. For those the argument proceeds as in Theorem 2.4 using Lemma 3.7 below instead of Lemma 2.5.

Lemma 3.7. Let R be a valuation ring with fraction field K and let f : Spec K → Spec R be the induced morphism. Let \underline{F} and \underline{F}′ be two abelian sheaves on C over Spec R of rank r, dimension d, and characteristic c : Spec R → C. Let α : f^{∗}\underline{F} → f^{∗}\underline{F}′ be an isomorphism over Spec K. Then α = f^{∗}β for a unique isomorphism β : \underline{F} ⟶^{∼} \underline{F}′ over Spec R.

Proof. Recall the rings O_{ϖ} and K(C) introduced in the proof of Proposition 3.5 and consider the τ-modules (F_{i}⊗_{O_{C_{R}}} O_{ϖ}, Π_{i}^{−1}∘τ_{i}) and (F′_{i}⊗_{O_{C_{R}}} O_{ϖ}, Π′_{i}^{−1}∘τ′_{i}) over O_{ϖ}. By the arguments of Gardeyn [8, Proposition 2.13(i )] these are the unique maximal Π_{i}^{−1} ∘ τ_{i}-invariant O_{ϖ}- modules in F_{i} ⊗_{O_{C_{R}}} K(C), respectively F′_{i}⊗_{O_{C_{R}}} K(C). Hence they are mapped isomorphically into each other under the isomorphism α. Now the lemma follows from [11, Lemme 2.7].

Remark. Like for Drinfeld modules these results say in the language of stacks that the restriction of coefficients 1-morphism π_{∗} : C′-Ab-Sh^{r′,d′} → C-Ab-Sh^{nr′,d′} is proper but in general not representable.

4 Restriction of Coefficients for Drinfeld-Anderson Shtuka

Theorem 4.1. The restriction of coefficients morphism π_{∗} : \underline{C′-DA-Sht}^{r′,d′} → \underline{C-DA-Sht}^{nr′,d′} for Drinfeld-Anderson shtuka is in general not relatively representable.

Proof. The abelian sheaf from Theorem 3.1 provides the counter example also for Drinfeld-Anderson shtuka.

The same reasoning as in Theorem 3.2 and Proposition 3.5 yields the following

Theorem 4.2. Let S be a locally noetherian 𝔽_{q}-scheme and let b, c : S → C be two 𝔽_{q}- morphisms. Let \underline{E} = (E, Ẽ, j, τ, b, c) be a Drinfeld-Anderson shtuka on C of rank nr′ and dimension d′ over S. Then the contravariant functor

\underline{T} : Sch_{/(C′×C′)×_{(C×C)}S} ⟶ Sets
(S′, f : S′ → S  ↦ \{ Isomorphism classes of pairs (\underline{E}′, α) where
(b′, c′) : S′ → C′ ×_{𝔽_{q}} C′ )  • \underline{E}′ is a Drinfeld-Anderson shtuka of rank r′,

dimension d′, pole b′, and zero c′ over S′ and

• α : f^{∗}\underline{E} ⟶^{∼} π_{∗}\underline{E}′ is a fixed isomorphism \}

is representable by a finite S-scheme.

The following results are proved analogously to Theorem 3.6 and Lemma 3.7.

Theorem 4.3. The restriction of coefficients morphism π_{∗} : \underline{C′-DA-Sht}^{r′,d′} → \underline{C-DA-Sht}^{nr′,d′} satisfies the valuative criterion for properness.

Lemma 4.4. Let R be a valuation ring with fraction field K and let f : Spec K → Spec R be the induced morphism. Let \underline{E} and \underline{E}′ be two Drinfeld-Anderson shtuka on C over Spec R of rank r, dimension d, pole b : Spec R → C, and zero c : Spec R → C. Let α : f^{∗}\underline{E} → f^{∗}\underline{E}′ be an isomorphism over Spec K. Then α = f^{∗}β for a unique isomorphism β : \underline{E} ⟶^{∼} \underline{E}′ over Spec R.

Remark. Again these results say in the language of stacks that the restriction of coefficients 1-morphism π_{∗} : C′-DA-Sht^{r′,d′} → C-DA-Sht^{nr′,d′} is proper but in general not representable.

References

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[7] V.G. Drinfeld: Moduli variety of F -sheaves, Funct. Anal. Appl. 21 (1987), no. 2, 107–122.

[8] F. Gardeyn: The structure of analytic τ-sheaves, J. Number Th. 100 (2003), 332–362.

[9] U. Hartl: Uniformizing the Stacks of Abelian Sheaves, in Number Fields and Function fields - Two Parallel Worlds, Papers from the 4th Conference held on Texel Island, April 2004, G. van der Geer, B. Moonen, R. Schoof, Editors, pp. 167–222, Progress in Math. 239, Birkhäuser-Verlag, Basel 2005. See also arXiv:math.NT/0409341.

[10] R. Hartshorne: Algebraic Geometry, GTM 52, Springer-Verlag, Berlin etc. 1977.

[11] L. Lafforgue: Une compactification des champs classifiant les chtoucas de Drinfeld, J. Amer. Math. Soc. 11 (1998), no. 4, 1001–1036.

[12] G. Laumon: Cohomology of Drinfeld Modular Varieties I , Cambridge Studies in Advanced Mathematics 41, Cambridge University Press, Cambridge, 1996.

[13] G. Laumon, L. Moret-Bailly: Champs algébriques, Ergebnisse 39, Springer-Verlag, Berlin etc. 2000.

[14] H. Matzat: Introduction to Drinfeld modules, in Drinfeld modules, modular schemes and applications (Alden-Biesen, 1996), pp. 3–16, World Sci. Publishing, River Edge, NJ, 1997.

Urs Hartl Mathematisches Institut Albert-Ludwigs-Universität Freiburg Eckerstr. 1 D – 79104 Freiburg Germany e-mail: urs.hartl@math.uni-freiburg.de Markus Hendler Mathematisches Institut Albert-Ludwigs-Universität Freiburg Eckerstr. 1 D – 79104 Freiburg Germany