Local Shtukas and Divisible Local Anderson Modules

We develop the analog of crystalline Dieudonn\'e theory for p-divisible groups in the arithmetic of function fields. In our theory p-divisible groups are replaced by divisible local Anderson modules, and Dieudonn\'e modules are replaced by local shtukas. We show that the categories of divisible local Anderson modules and of effective local shtukas are anti-equivalent over arbitrary base schemes. We also clarify their relation with formal Lie groups and with global objects like Drinfeld modules, Anderson's abelian t-modules and t-motives, and Drinfeld shtukas. Moreover, we discuss the existence of a Verschiebung map and apply it to deformations of local shtukas and divisible local Anderson modules. As a tool we use Faltings's and Abrashkin's theory of strict modules, which we review to some extent.


Introduction
In the arithmetic of number elds, elliptic curves and abelian varieties are important objects. eir theory has been extensively developed in the last two centuries and their moduli spaces have played a major role in Faltings's proof of the Mordell conjecture [Fal ,CS ], the proof of Fermat's Last eorem by Wiles and Taylor [Wil , TW , CSS ], and the proof of the Langlands correspondence for GL n over nonarchimedean local elds of characteristic zero by Harris and Taylor [HT ]. A useful tool for studying abelian varieties and their moduli spaces are p-divisible groups. More precisely, for an elliptic curve or an abelian variety E over a Z p -algebra R the p-divisible group E[p ∞ ] = lim → E[p n ], also called the Barsotti-Tate group, captures the local p-adic information of E. One reason why E[p ∞ ] is a useful tool for the study of E is that the complicated arithmetic data of a p-divisible group over a Z p -algebra R in which p is nilpotent can be faithfully encoded by an object of semi-linear algebra, its Dieudonné module.
Elliptic curves and abelian varieties have analogs in the arithmetic of function elds. Namely, Drinfeld [Dri , Dri ] invented the notions of elliptic modules (today called Drinfeld modules) and the dual notion of F-sheaves (today called Drinfeld shtukas). ese structures are function eld analogs of elliptic curves in the following sense. eir endomorphism rings are rings of integers in global function elds of positive characteristic or orders in central division algebras over the later. On the other hand, their moduli spaces are varieties over smooth curves over a nite eld. rough these two aspects in which global function elds of positive characteristic come into play, Drinfeld shtukas and variants of them proved to be fruitful for establishing large parts of the Langlands program over local and global function elds of positive characteristic in works by Drinfeld [Dri ,Dri ,Dri ], Laumon, Rapoport, and Stuhler [LRS ], L. La orgue [Laf ], and V. La orgue [Laf ]. Beyond this the analogy between Drinfeld modules and elliptic curves is abundant.
In this spirit, Anderson [And ] introduced higher dimensional generalizations of Drinfeld modules, called abelian t-modules. ese are group schemes which carry an action of the polynomial ring F r [t] over a nite eld F r with r elements subject to certain conditions. Abelian t-modules are the function eld analogs of abelian varieties [BH ]. Although Anderson worked over a eld, abelian t-modules also exist naturally over arbitrary F r [t]-algebras R as base rings (De nition . ). ey possess an (anti-)equivalent description by semi-linear algebra objects called t-motives, which are R[t]-modules together with a Frobenius semi-linear endomorphism (Definition . and eorem . ) and are a variant and generalization of Drinfeld shtukas. rough the work of Drinfeld and Anderson it was realized very early on that a Drinfeld module or abelian t-module over a eld is completely described by its t-motive. e same is true over an arbitrary F r [t]-algebra R, as is shown for example in [Har ]. So in a way the situation in function eld arithmetic is much better than in the arithmetic of abelian varieties; the t-motive is a "global" Dieudonné module that integrates the "local" Dieudonné modules for every prime in a single object.
Correspondingly it is not di cult to come up with a de nition of a Dieudonné module at a prime p ⊂ F r [t] of an abelian t-module. It should arise as the p-adic completion of its t-motive; see Example . (ii) for details. e object one ends up with is an e ective local shtuka. To de ne these, let p = (z) for a monic irreducible polynomial z ∈ F r [t] and let F q = F r [t] p be the residue eld. en lim ← F r [t] p n = F q [[z]]. Let R be an F q [[z]]-algebra in which the image ζ of z is nilpotent. An e ective local shtuka over R is a pair M = (M, F M ) consisting of a locally free R [[z]]-module M of nite rank and an isomorphism Here σ * q is the endomorphism of R [[z]] that extends the q-Frobenius endomorphism σ * q ∶= Frob q,R ∶ b ↦ b q for b ∈ R by σ * q (z) = z, and σ * q M ∶= M⊗ R [[z]],σ * q R [[z]]. Now the goal of crystalline Dieudonné theory in the arithmetic of function elds is to describe the analogs of p-divisible groups that correspond to e ective local shtukas. In the present article we call them z-divisible local Anderson modules as in the following de nition, and we develop this theory under the technical assumption that ζ ∈ R is nilpotent. is theory was already announced in [Har ,Har ,Har ,HK ] and was used in [Har ].

De nition .
A z-divisible local Anderson module over R is a sheaf of F q [[z]]modules G on the big fppf-site of Spec R such that (i) G is z-torsion, that is, G = lim → G[z n ], where G[z n ] ∶= ker(z n ∶ G → G). (ii) G is z-divisible, that is, z∶ G → G is an epimorphism. (iii) For every n the F q -module G[z n ] is representable by a nite locally free strict F q -module scheme over R in the sense of Faltings (De nition . ).
(ii) Both functors are F q [[z]]-linear, map short exact sequences to short exact sequences, and preserve (ind-)étale objects.
(iii) G is a formal F q [[z]]-module if and only if F M is topologically nilpotent, that is im(F n M ) ⊂ zM for an integer n. (iv) e R [[z]]-modules ω Drq (M,FM ) and coker F M are canonically isomorphic.
In Section we explain the relation of z-divisible local Anderson modules and local shtukas to global objects like Drinfeld modules [Dri ], Anderson's [And ] abelian t-modules and t-motives, and Drinfeld shtukas [Dri ]. In particular, if E is a Drinfeld-F r [t]-module or an abelian t-module over R, then the z n -torsion points E[z n ] of E form a nite locally free F r [t] (z n )-module scheme over R. By Example . (ii), the limit G ∶= E[z ∞ ] ∶= lim → E[z n ] in the category of fppf-sheaves of F q [[z]]-modules on Spec R satis es G[z n ] ∶= ker(z n ∶ G → G) = E[z n ] and is a z-divisible local Anderson module over R. Moreover, the associated e ective local shtuka M q (G) from eorem . arises as the z-adic completion of the t-motive associated with E; see Example . (ii). In Section we present the above de nition of z-divisible local Anderson modules G and give equivalent de nitions. We also introduce truncated z-divisible local Anderson modules such as G[z n ] (Proposition . ). In Section we investigate, for ζ = in R, the existence of a z d -Verschiebung V z d ,G for (truncated) z-divisible local Anderson modules G, respectively for local shtukas, with V z d ,G ○ F q,G = z d ⋅ id G and F q,G ○ V z d ,G = z d ⋅ id σ * q G , where F q,G is the relative q-Frobenius of G over R. We use the z d -Verschiebung in eorem . to prove that li ing a z-divisible local Anderson module from R I to R, when I q = ( ), is equivalent to li ing the Hodge ltration on its de Rham cohomology. In Section we use the z d -Verschiebung to clarify the relation between z-divisible local Anderson modules G and formal F q [[z]]modules. Following the approach of Messing [Mes ], who treated the analogous situation of p-divisible groups and formal Lie groups, we show that a z-divisible local Anderson module is formally smooth ( eorem . ) and how to associate a formal F q [[z]]-module with it ( eorem . ). We also discuss conditions under which it is an extension of an (ind-)étale z-divisible local Anderson module by a z-divisible formal F q [[z]]-module (Proposition . ) and we prove the following corollary.

Corollary .
ere is an equivalence of categories between that of z-divisible local Anderson modules over R with G[z] radicial, and the category of z-divisible formal F q [[z]]-modules G with G[z] representable by a nite locally free group scheme, such that locally on Spec R there is an integer d with (z − ζ) d = on ω G .
In Section we explain Faltings's notion of strict F q -module schemes and give details additional to the treatments of Faltings [Fal ] and Abrashkin [Abr ]. is notion is based on certain deformations of nite locally free group schemes and the associated cotangent complex, which we review in Section . ere is an equivalent description of nite locally free strict F q -module schemes by Poguntke [Pog ] (Remark . ).

Notation .
Let F q be a nite eld with q elements and characteristic p. For a scheme S over Spec F q and a positive integer n ∈ N > , we denote by σ q n ∶= Frob q n ,S ∶ S → S its absolute q n -Frobenius endomorphism which acts as the identity on points and as the q n -power map b ↦ b q n on the structure sheaf. For an S-scheme X, respectively, ] for open U ⊂ S with the obvious restriction maps.
is is indeed a sheaf being the countable direct product of O S . Let ζ be an indeterminant over F q and let F q [ [ζ]] be the ring of formal power and O S ((z)) that acts as the identity on z and as be the tensor product sheaves. Also for a section m ∈ M we write σ * q m ∶= m ⊗ ∈ σ * q M. Note that by [HV , Proposition . ]

Lemma .
Let R be an F q [ [ζ]]-algebra in which ζ is nilpotent. en the sequence of Since ζ is nilpotent, all b i are zero. Also due to the nilpotency of ζ, the second map is well de ned and surjective. For exactness in the middle note that For We de ne the tensor product of two local shtukas (M, F M ) and (N , is a unit object for the tensor product. e dual (M Also there is a natural de nition of internal Hom, given by is makes the category of local shtukas over S into an F q [[z]]-linear, additive, rigid tensor category. It is an exact category in the sense of Quillen [Qui , § ], provided one calls a short sequence of local shtukas exact when the underlying sequence of sheaves of O S [[z]]-modules is exact.

Lemma .
Let (M, F M ) be a local shtuka over S. en locally on S there are e, Proof We work locally on Spec R ⊂ S and assume that σ * q M and M are free O S [[z]]-modules. Applying F M to a basis of σ * q M, respectively, F − M to a basis of M, proves the existence of e, respectively, e ′ . If N ≥ e ′ is an integer that is a power of p such that ζ N = in R, then We prove that the quotient K ∶= (z − ζ) −e M F M (σ * q M) is a locally free R-module of nite rank. is was already proved in [HV , Lemma . ], but the argument given there only works if R is noetherian, because it uses the fact that R [[z]] is at over R. We now give a proof in the non-noetherian case. Since we use the change of rings spectral sequence [Rot , eorem . ] and the induced epimorphism (from its associated -term sequence of low degrees, see [Rot , eorem . ]) It follows that Tor R (K, k) = and from Nakayama's lemma we conclude that K is locally free over R of nite rank; compare [Eis , Exercise . ].

Example .
We de ne the Tate objects in the category of local shtukas over S as By Lemma . every local shtuka over a quasi-compact scheme S becomes e ective a er tensoring with a suitable Tate object.
More generally, now let S be an arbitrary F q -scheme.

De nition .
A nite F q -shtuka over S is a pair M = (M, F M ) consisting of a locally free O S -module M on S of nite rank denoted rk M , and an O S -module homo- Finite F q -shtukas were studied at various places in the literature. ey were called ( nite) φ-sheaves by Drinfeld [Dri , § ], Taguchi and Wan [Tag ,TW ] and Dieudonné F q -modules by Laumon [Lau ]. Finite F q -shtukas over a eld admit a canonical decomposition.
Proposition . ( [Lau , Lemma B. . ]) If S is the spectrum of a eld L, every nite F q -shtuka M = (M, F M ) is canonically an extension of nite F q -shtukas where Fé t is an isomorphism and F nil is nilpotent, and M´e t = (Mé t , Fé t ) is the largest étale nite F q -sub-shtuka of M and equals im(F rk M M ). If L is perfect, this extension splits canonically.

Proof
is was proved by Laumon [Lau , Lemma B. . ] for perfect L. In general one considers the descending sequence is surjective, hence bijective, and therefore im(F n ′ M ) = im(F n M ) for all n ′ ≥ n. So the sequence stabilizes already for some n ≤ rk M and Mé t = im(F rk M M ). If L is perfect, M nil is isomorphic to the submodule ⋃ n≥ ker(F n M ○ σ * q n ∶ M → M) of M; see [Lau , Lemma B. . ].

Example .
Every e ective local shtuka (M, F M ) of rank r over S yields for every n ∈ N a nite F q -shtuka (M z n M, F M mod z n ) of rank rn, and (M, F M ) equals the projective limit of these nite F q -shtukas. us from Proposition . we obtain the following result.

Proposition .
If S is the spectrum of a eld L in Nilp Fq [ [ζ]] , every e ective local shtuka (M, F M ) is canonically an extension of e ective local shtukas where Fé t is an isomorphism, F nil is topologically nilpotent, and (Mé t , Fé t ) is the largest étale e ective local sub-shtuka of (M, F M ). If L is perfect, this extension splits canonically.

Review of Deformations of Finite Locally Free Group Schemes
For a commutative group scheme G over S, we denote by ε G ∶ S → G its unit section and by ω G ∶= ε * G Ω G S its co-Lie module. It is a sheaf of O S -modules. In order to describe which group objects are classi ed by nite F q -shtukas, we need to review the de nition of a strict F q -module scheme in the next two sections. We follow Faltings [Fal ] and Abrashkin [Abr ]. We begin this section with a review of deformations of nite locally free group schemes. Recall that a group scheme G over S is called nite locally free over S if on every open a ne Spec R ⊂ S the scheme G is of the form Spec A for a nite locally free R-module A. By [EGA, I new , Proposition . . ] this is equivalent to G being nite at and of nite presentation over S. e rank of the R-module A is called the order of G and is denoted ord G. It is a locally constant function on S. e following facts will be used throughout.

Remark .
(a) A morphism G ′ → G of nite locally free group schemes is a monomorphism (of schemes, or equivalently of fppf-sheaves on S) if and only if it is a closed immersion by [EGA, IV , Corollaire . . ], because it is proper.
(b) Let G and G ′′ be group schemes over S that are nite and of nite presentation, and assume that G is at over S. en a morphism G → G ′′ is an epimorphism of fppfsheaves on S if and only if it is faithfully at; compare the proof of [Mes , Chapter I, Lemma . (b)].
(c) A sequence → G ′ → G → G ′′ → of nite locally free group schemes over S is called exact if it is exact when viewed as a sequence of fppf-sheaves on S. By the above this is equivalent to the conditions that G → G ′′ is faithfully at, and that G ′ → G is a closed immersion that equals the kernel of G → G ′′ .
(d) If G ′ ↪ G is a closed immersion of nite locally free group schemes over S, then the quotient G G ′ exists as a nitely presented group scheme over S by [SGA , éorème V. . and Proposition V. . ], which is at by [EGA, IV , Corollaire . . ]. It is integral over S and hence nite, because O G G ′ ⊂ O G . In particular, G G ′ is nite locally free over S.
In the following we will work locally on S and assume that S = Spec R is a ne. Let G = Spec A be a nite locally free group scheme over S.
en G is a relative complete intersection by [SGA , Proposition III. . ]. is means that locally on S we can take A = R[X , . . . , X n ] I where the ideal I is generated by a regular sequence ( f , . . . , f n ) of length n, cf. [EGA, IV , Proposition . . ]. e unit section ε G ∶ S → G de nes an augmentation ε A ∶= ε * G ∶ A ↠ R of the R-algebra A, that is, ε A is a section of the structure morphism ι A ∶ R ↪ A. Faltings [Fal ] and Abrashkin [Abr ] de ned deformations of augmented R-algebras as follows. For every augmented R-algebra [Abr , § § . , . ] made the following de nition.

De nition .
e category DSch S has as objects all triples H = (H, H ♭ , i H ), where H = Spec A for an augmented R-algebra A that is nite locally free as an R-module, where H ♭ = Spec A ♭ for an augmented R-algebra A ♭ , and where i H ∶ H ↪ H ♭ is a closed immersion given by an epimorphism i A ∶ A ♭ ↠ A of augmented R-algebras, such that locally on Spec R there is a polynomial ring R[X ] = R[X , . . . , X n ] and an epimorphism of augmented R-algebras j∶ R[X ] ↠ A ♭ satisfying the properties that (i) the ideal I ∶= ker(i A ○ j) is generated by elements of a regular sequence of length In particular, H is a relative complete intersection. We write A = (A, A ♭ , i A ) and . Both are nite locally free R-modules of the same rank. is is obvious for t * H , and for N H a proof can be found in [HS, Lemma . ]. Also note that [Fal , § ] noted that the set is non-empty and is a principal homogeneous space under Hom R (t * H , NH). at is, for any f ♭ ∈ L the map Hom For a proof, see [HS, Lemma . ] in the expanded version of this article on arXiv [HS]. e category DSch S possesses direct products.

De nition .
Let DGr S be the category of group objects in DSch S . If G = Spec A ∈ DGr S , then its group structure is given via the comultiplication ∆∶ which satisfy the usual axioms. In particular, we require the and that ε and ε ♭ are the augmentation maps. e morphisms in DGr S are morphisms of group objects.
If G = (G, G ♭ ) ∈ DGr S , note that G = Spec A is a nite locally free group scheme over R with the comultiplication ∆, the counit ε, and the coinversion [− ]. But, in general, G ♭ is not a group scheme over S when the comultiplication ∆ Faltings and Abrashkin [Abr , § . ] made the following remarks.
(a) If G = Spec(A, A ♭ , i A ) ∈ DSch S and G = Spec A is a nite locally free group scheme over R, then there exists a unique structure of a group object on G that is compatible with that of G. It satis es Faltings de ned the co-Lie complex of G over S = Spec R (that is, the ber at the unit section of G of the cotangent complex) as the complex of nite locally free R-modules Recall that the co-Lie complex of G S and more generally the cotangent complex of a morphism were de ned by Illusie [Ill ,Ill ] generalizing earlier work of Lichtenbaum and Schlessinger [LS ]. If G = Spec A for A = R[X ] I, where I is generated by a regular sequence, then the cotangent complex of Illusie [Ill , II. . . ] is quasi-isomorphic to the complex of nite locally free A-modules A → concentrated in degrees − and with d being the di erential map [Ill , Corollaire III. . . ]. e co-Lie complex of G over S was de ned by Illusie [Ill , §VII. . ] as ℓ To see that this is equal to Faltings's de nition note that and that the di erential of both co-Lie complexes sends an element x ∈ I to the linear term in its expansion as a polynomial in X , because all terms of higher degree are sent to zero under ε * G . Up to homotopy equivalence both L are quasi-isomorphic by [Mes , Chapter II, Proposition . . ], where ι∶ G → S is the structure map.

De nition .
We (re-)de ne the co-Lie module of G over S as G Ω G S , which is also canonically isomorphic to the R-module of invariant di erentials on G.
We record the following lemmas.

Lemma .
If G ∈ DGr S , the following are equivalent: Conversely, since Ω G S is a nitely generated O G -module, by Nakayama ω G = implies that G is étale along the zero section. Being a group scheme, it is étale everywhere.
be an exact sequence of nite locally free group schemes over S. en there is an exact sequence of R-modules

Strict F q -module Schemes
We keep the notation of the previous section. Let O be a commutative unitary ring.

De nition .
In this article an O-module scheme over S is a nite locally free commutative group scheme G over S together with a ring homomorphism O → End S (G). We denote the category of O-module schemes over S by Gr(O) S .

Proposition .
If S is the spectrum of a eld L, every O-module scheme G over S is canonically an extension → G → G → G´e t → of an étale O-module scheme G´e t by a connected O-module scheme G . e O-module scheme G´e t is the largest étale quotient of G. If L is perfect, G´e t is canonically isomorphic to the reduced closed O-module subscheme G red of G and the extension splits canonically, G = G × S G red .

Proof
e constituents of the canonical decomposition of the nite S-group scheme G are O-invariant.

De nition .
Let We let DGr(O) S be the category whose objects are pairs (G, is a strict O-action, and whose morphisms We let DGr * (O) S be the quotient category of DGr(O) S having the same objects, whose morphisms are the equivalence classes of morphisms (G,

So by de nition the forgetful functor DGr
Faltings [Fal , Remark b)

Remark .
Note that there can be di erent non-isomorphic strict O-actions on a deformation G. For example, let be the polynomial ring in the variable a, and let R be an O-algebra by sending a to in R. For every u ∈ R the endomorphism [a] = ∶ α α α p → α α α p , X ↦ li s to [a]∶ A ♭ → A ♭ , X ↦ uX p . All these li s de ne strict O-actions on (G, Spec A ♭ ) that are non-isomorphic in DGr * (F p [a]) S . In particular, the forgetful functor is not fully faithful.
In contrast, for O = F q we have the following lemma.

Lemma .
e forgetful functor DGr * (F q ) S → Gr(F q ) S is fully faithful. In particular, if G ∈ Gr(F q ) S and G = (G, G ♭ ) ∈ DGr S is a deformation of G, then there is at most one strict F q -action on G that li s the action on G.
We claim that it also satis es h a+b = h a + h b . Namely, denoted by the same symbol. We evaluate this expression on X ν , where is means that ( f , f ♭ ) de nes a morphism in DGr * (F q ) S that maps to f under the forgetful functor. So this functor is fully faithful. e remaining assertion follows by takingÃ ♭ = A ♭ ,Ã = A and f ♭ = id.

De nition .
A nite locally free F q -module scheme G over R is called a strict F q -module scheme if it lies in the essential image of the forgetful functor DGr * (F q ) S → Gr(F q ) S , that is, if it has a deformation G carrying a strict F q -action that li s the F q -action on G. We identify DGr * (F q ) S with the category of nite locally free strict F q -module schemes over S.

Lemma .
For a nite locally free F q -module scheme G over R, the property of being a strict F q -module scheme is local on Spec R.
Proof LetG be a deformation of G over Spec R. Let Spec R i ⊂ Spec R be an open covering and let G i be deformations of G × R Spec R i carrying a strict F q -action which li s the F q -action on G. is action induces by [Fal , Remark b) a er De nition ] a strict F q -action onG × R Spec R i for all i. Above Spec R i ∩ Spec R j these actions coincide by Lemma . , and hence they glue to a strict F q -action onG as desired.

Example .
We give examples for nite locally free strict F q -module schemes. Let R be an F q -algebra.
with d = and a ∈ F q acts on it as scalar multiplication by a because N [a] (X q ) = (aX) q = aX q and t * [a] (X) = aX. erefore α α α q is a nite locally free strict F q -module scheme. ( and so a ∈ F q acts on N G by a p which is not scalar multiplication by a when a p ≠ a. is action does not li to µ µ µ ♭ p = Spec R[X] (X − ) p+ , because on µ µ µ ♭ p we have ∆(X) = X ⊗ X and hence [a](X) = X a , which satis es [p](X) = X p ≠ . erefore no deformation of µ µ µ p can carry a strict F p -action and µ µ µ p is not a strict F p -module scheme. Note that nevertheless F p acts through scalar multiplication on the co-Lie complex ℓ • µ µ µ p S . Part (c) generalizes to the following lemma.

Lemma .
Any nite étale F q -module scheme is a nite locally free strict F q -module. In particular, if → G ′ → G → G ′′ → is an exact sequence of nite locally free F q -module schemes with G a strict F q -module and G ′′ étale, then both G ′ and G ′′ are strict F q -modules.

Proof
e rst assertion was remarked by Faltings [Fal , § , p. ] more generally for nite étale O-module schemes, and also follows from [Dri , Proposition . ( )] and eorem . below. (For a direct proof, see [HS, Lemma . ].) e last assertion on the strictness of G ′ can be proved on a ne open subsets of S. ere Lemma . implies that the morphism G → G ′′ is F q -strict in the sense of Faltings [Fal , Denition ], and by [Fal , Proposition ] its kernel G ′ is a strict F q -module.

Equivalence Between Finite F q -shtukas and Strict F q -modules
Let S be a scheme over Spec F q . Recall that a nite locally free commutative group scheme G over S is equipped with a relative p-Frobenius F p,G ∶ G → σ * p G and a p-Ver- For more details see [SGA , Exposé VII A , § . ]. Example . is generalized by the following results of Abrashkin. e rst is concerned with nite locally free strict F p -module schemes.
eorem . ( [Abr , eorem ]) Let G be a nite locally free group scheme equipped with an F p -action over an F p -scheme S. en this action li s (uniquely) to a strict F p -action on some (any) deformation of G if and only if the p-Verschiebung of G is zero. In particular, the forgetful functor induces an equivalence between DGr * (F p ) S and the category of those group schemes in Gr(F p ) S that have p-Verschiebung zero.
To explain Abrashkin's classi cation of nite locally free strict F q -module schemes we recall that Drinfeld [Dri , § ] de ned a functor from nite F q -shtukas over S to nite locally free F q -module schemes over S. Abrashkin [Abr ] proved that the essential image of Drinfeld's functor consists of nite locally free strict F q -module schemes. Other descriptions of the essential image were given by Taguchi [Tag , § ] and Laumon [Lau , §B. ]. (But note that [Lau , Propositions . . , B. . , and Lemma B. . ] are incorrect as the F q -module scheme G = α α α p = Spec R[x] (x p ) shows when p ≠ q.) Drinfeld's functor is de ned as follows. Let M = (M, F M ) be a nite F q -shtuka over S. Let be the geometric vector bundle corresponding to M, and let F q,E ∶ E → σ * q E be its relative q-Frobenius morphism over S. On the other hand, the map F M induces another . Also we equip every R-algebra T with the Frob q,R -semi-linear R-module endomorphism F semi T ∶= Frob q,T ∶ T → T.
en Dr q (M ) is the group scheme over S that is given on R-algebras T as where I = ⊕ n≥ Sym n O S M is the ideal generated by all m ∈ M. is deformation is equipped with both the comultiplication ∆ ♭ ∶ m ↦ m ⊗ + ⊗ m and the F q -action [a] ♭ ∶ m ↦ am. We set Dr q (M ) ∶= (Dr q (M ), Dr q (M ) ♭ ). Its co-Lie complex is . is de nes the functor Dr q ∶ F q -Sht S → DGr(F q ) S . We also compose Dr q with the projection to is makes G a,S into an R-module scheme, and in particular, into an F q -module scheme via F q ⊂ R. We associate with G the R-module of F q -equivariant homomorphisms on S M q (G) ∶= Hom R-groups,Fq -lin (G, G a,S ) with its action of R via R → End R-groups,Fq -lin (G a,S ). It is a nite locally free R-module by [Pog , Proposition . , Remark . ]; see also [SGA , VII A , . . ] in the reedited version of SGA by P. Gille and P. Polo. e composition on the le with the relative q-Frobenius endomorphism F q,G a,S of G a,S = Spec R[x] given by x ↦ We thank L. Taelman for mentioning this to us.
It factors through the category DGr * (F q ) S and further over the forgetful functor through the category of nite locally free strict F q -module schemes over S.
ere is a natural morphism ere is also a natural morphism of group schemes G → Dr q (M q (G)) given on the structure sheaves by which is well de ned because A large part of the following theorem was already proved by Drinfeld [Dri , Proposition . ] without using the notion of strict F q -modules.

eorem .
(i) e contravariant functors Dr q and M q are mutually quasi-inverse anti-equivalences between the category of nite F q -shtukas over S and the category of nite locally free strict F q -module schemes over S.
(ii) Both functors are F q -linear and map short exact sequences to short exact sequences. ey preserve étale objects and map the canonical decompositions from Propositions . and . to each other. Conversely let → G ′ → G → G ′′ → be a short exact sequence of nite locally free strict F q -module schemes. en the exactness of → M q (G ′′ ) → M q (G) → M q (G ′ ) is obvious. Applying Dr q , whose exactness we just established, to the injec- Consider the exact sequences from Propositions . and . . en Dr q (M´e t ) is an étale quotient of Dr q (M ).
is yields a morphism Dr q (M )´e t → Dr q (M´e t ).
is yields a morphism M q (G´e t ) → M q (G)é t . e equivalence of (i) shows that both morphisms are isomorphisms. is proves the compatibility of Dr q and M q with the canonical decompositions.
(iii) (b) By de nition G ∶= Dr q (M ) is radicial over S if G(K) → S(K) is injective for all elds K. is can be tested by applying the base change Spec K → S. By (ii) and Propositions . and . the base change G × S Spec K is connected if and only if F M ⊗ id K is nilpotent. is implies (iii)(b) over Spec K. It remains to show that F M is nilpotent locally on S if G is radicial. Locally on an a ne open Spec R ⊂ S we may choose an R-basis of M and write F M as an r × r-matrix where r = rk M. For every point s ∈ S, Proposition . implies that F r M = in κ(s) r×r . erefore the entries of the matrix F r M lie in the nil-radical of R. If n is an integer such that their q n -th powers are zero, then F r(n+ ) M = F r M ⋅ ⋅ ⋅ σ n * q (F r M ) = . is establishes (iii)(b). (iii) (c) If locally on S we choose an isomorphism M ≅ ⊕ n ν= O S ⋅ X ν and let (t i j ) be the matrix of the morphism F M ∶ σ * q M → M with respect to the basis (X , . . . , X n ), then Dr q (M ) is the subscheme of G n a,S , given by the system of equations Note that the latter is surjective by de nition and injective because both σ * q M and I (II ) are locally free O S -modules of the same rank.

Remark .
Finite locally free strict F q -module schemes over S = Spec R were equivalently described by Poguntke [Pog ]. He de ned the category F q -gr +,b S of nite locally free F q -module schemes G = Spec A that locally on S can be embedded into G N a,S for some set N and are balanced in the following sense. e R-module of morphisms of group schemes over R decomposes under the action of F q on G into eigenspaces for i ∈ Z eZ, where q = p e . Now G is balanced if the composition on the right with the relative p-Frobenius F p,G a,S of the additive group scheme G a,S induces isomorphisms e latter holds if and only if G is étale by eorem . (ii).
Abrashkin [Abr , . . ] showed that every nite locally free strict F q -module scheme over S belongs to F q -gr +,b S . And Poguntke [Pog , eorem . ] conversely showed that Dr q and M q provide an anti-equivalence between the category of nite F q -shtukas over S and the category F q -gr +,b S .

Relation to Global Objects
Without giving proofs, in this section we want to relate local shtukas and divisible local Anderson modules (de ned in the next section), as well as nite F q -shtukas and nite locally free strict F q -module schemes to global objects like A-motives, global shtukas, Drinfeld modules, Anderson A-modules, etc. which are de ned as follows. Let C be a smooth, projective, geometrically irreducible curve over F q . For an F q -scheme S we set C S ∶= C × Fq S and we consider the endomorphism

De nition .
(i) Let n and r be positive integers. A global shtuka of rank r with n legs over an F q -scheme S is a tuple N = (N, c , . . . , c n , τ N ) consisting of • a locally free sheaf N of rank r on C S , Γc i outside the graphs Γ ci of the c i . In this article we will only consider the case where Γ ci ∩ Γ c j = ∅ for i ≠ j.
(ii) A global shtuka over S is a Drinfeld shtuka if n = , Γ c ∩ Γ c = ∅, and τ N satis es τ N (σ * q N) ⊂ N on C S ∖ Γ c with cokernel locally free of rank as O S -module, and τ − N (N) ⊂ σ * q N on C S ∖ Γ c with cokernel locally free of rank as O S -module. Drinfeld shtukas were introduced by Drinfeld [Dri ] under the name F-sheaves.
An important class of special examples is de ned as follows. Let ∞ ∈ C be a closed point and put A ∶= Γ(C ∖ {∞}, O C ). en Spec A = C ∖ {∞}. We will consider a ne A-schemes c∶ S = Spec R → Spec A and the ideal J ∶= (a ⊗ − ⊗ c * (a) ∶ a ∈ A) ⊂ A R ∶= A ⊗ Fq R whose vanishing locus V(J) is the graph Γ c of the morphism c. e endomorphism σ q ∶= id C ⊗ Frob q,S ∶ C S → C S induces the ring endomorphism for a ∈ A and b ∈ R. e following de nition generalizes Anderson's [And ] notion of t-motives, which is obtained as the special case, where C = P , A = F q [t] and R is a eld.

De nition .
Let d and r be positive integers and let S = Spec R be an a ne A-scheme. An e ective A-motive of rank r and dimension d over S is a pair N = (N , τ N ) consisting of a locally free A R -module N of rank r and a morphism τ N ∶ σ * q N → N of A R -modules, such that coker τ N is a locally free R-module of rank d and J d ⋅ coker τ N = . More generally, an A-motive of rank r over S is a pair N = (N , τ N ) consisting of a locally free A R -module N of rank r and an isomorphism outside the vanishing locus V(J) = Γ c of J. N = (N, c , c , τ N ) is a global shtuka of rank r over S = Spec R with two legs such that c = c and c ∶ S → {∞} ⊂ C, then

Example . (a) If
is an A-motive of rank r over S.
(b) Conversely, if ∞ ∈ C(F q ), every A-motive N = (N , τ N ) over an a ne A-scheme c∶ S = Spec R → Spec A can be obtained from a global shtuka by taking c = c and c ∶ S → {∞} ⊂ C, and taking N as an extension to C S of the sheaf associated with N on Spec A R , and τ N = τ N . ese global objects give rise to nite and local shtukas, and that motivates the names for the latter. is example gave rise to the name nite F q -shtuka.
(b) Let v ∈ C be a closed point de ned by a sheaf of ideals p ⊂ O C , letq be the cardinality of the residue eld F v of v, let f ∶= [F v ∶F q ], and let z ∈ F q (C) be a uniformizing parameter at v. Let N = (N, c , . . . , c n , τ N ) be a global shtuka of rank r over S = Spec R such that for some i the elements of c * i (p) are nilpotent in R. Set ζ ∶= c * i (z) ∈ R. en the formal completion of C S along the graph Γ ci of c i is canonically isomorphic to Spf R[[z]] by [AH , Lemma . ]. e formal completion M of (N, τ N ) So far we have discussed the semi-linear algebra side given by shtukas. On the side of group schemes, an important source from which the corresponding strict F qmodule schemes arise are Drinfeld A-modules, or more generally abelian Anderson A-modules. To de ne them, let c∶ S = Spec R → Spec A be an a ne A-scheme. Recall that for a smooth commutative group scheme E over Spec R the co-Lie module ω E ∶= ε * E Ω E R is a locally free R-module of rank equal to the relative dimension of E over R. Moreover, on the additive group scheme G a,R = Spec R[x] the elements b ∈ R, and in particular c * (a) ∈ R for a ∈ F q ⊂ A, act via endomorphisms is makes G a,R into an F q -module scheme. In addition, let τ ∶= F q,Ga,R be the relative q-Frobenius endomorphism of G a,R = Spec R[x] given by x ↦ x q . It satis es τ ○ ψ b = ψ b q ○ τ. We let with τb = b q τ be the non-commutative polynomial ring in the variable τ over R.
ere is an isomorphism of rings R{τ} ∼ → End R-groups,Fq -lin (G a,R ) sending an ele-

De nition .
Let d and r be positive integers. An abelian Anderson A-module of rank r and dimension d over an a ne A-scheme c∶ Spec R → Spec A is a pair E = (E, φ) consisting of a smooth a ne 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 (i) there is a faithfully at ring homomorphism R → R ′ for which E × R Spec R ′ ≅ G d a,R ′ as F q -module schemes, where F q acts on E via φ and F q ⊂ A; (ii) (a ⊗ − ⊗ c * a) d ⋅ ω E = for all a ∈ A under the action of a ⊗ induced from φ a and the natural action of ⊗ b for b ∈ R on the R-module ω E ; (iii) the set N ∶= M q (E ) ∶= Hom R-groups,Fq -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 ∈ N by If d = , this is called a Drinfeld A-module over S; cf. [Har , eorem . ].
e case in which C = P , A = F q [t], and R is a eld was considered by Anderson [And ] under the name abelian t-module. In [Har , eorem . ] we gave a proof of the following relative version of Anderson's theorem [And , eorem ].

eorem .
If E = (E, φ) is an abelian Anderson A-module of rank r and dimension d, we consider, in addition, on N ∶= M q (E ), the map τ semi is an e ective A-motive of rank r and dimension d. ere is a canonical isomorphism of R-modules coker τ N e contravariant functor E ↦ M q (E ) is fully faithful. Its essential image consists of all e ective A-motives N = (N , τ N ) over R for which there exists a faithfully at ring

Example .
Let E = (E, φ) be an abelian Anderson A-module over an a ne A-scheme c∶ Spec R → Spec A, and let N ∶= M q (E ) be its associated e ective Amotive.
(a) Let a ⊂ A be a non-zero ideal. By [Har , eorem . ] the a-torsion submodule of E, de ned as the scheme-theoretic intersection E [a] ∶= ⋂ a∈a ker(φ a ∶ E → E), is a nite locally free A a-module scheme and a strict F q -module scheme over S, which satis es M q (E [a]) = N aN and E [a] = Dr q (N aN ).
(b) Let p ⊂ A be a maximal ideal and assume that the elements of c * (p) ⊂ R are nilpotent. Letq be the cardinality of the residue eld F p ∶= A p and let f ∶= [F p ∶F q ]. We x a uniformizing parameter z ∈ F q (C) = Frac(A) at p and set ζ ∶= c * (z) ∈ R. We obtain an isomorphism

Divisible Local Anderson Modules
e name divisible local Anderson module is motivated by Example . (b). ese are the function eld analogs of p-divisible groups. ey were introduced in [Har ], but their de nition in [Har , § . ] and the claimed equivalence in [Har , § . ] is false. We give the correct de nition below analogously to Messing [Mes , Chapter I, Denition . ]. We x the following notation. For an fppf-sheaf of F q [z]-modules G over a scheme S, we denote the kernel of z n ∶ G → G by G[z n ]. Clearly (G[z n+m ])[z n ] = G[z n ] for all n, m ∈ N.

De nition .
A z-divisible local Anderson module over a scheme S ∈ Nilp Fq[ [ζ]] is a sheaf of F q [[z]]-modules G on the big fppf-site of S such that (ii) G is z-divisible, that is, z∶ G → G is an epimorphism.
(iii) For every n the F q -module G[z n ] is representable by a nite locally free strict F q -module scheme over S (De nition . ). (iv) Locally on S there exists an integer We de ne the co-Lie module of a z-divisible local Anderson module G over S as ω G ∶= lim ← ω G[z n ] . We will see in Lemma . and eorem . that ω G is a nite locally free O S -module and we de ne the dimension of G as rk ω G . It is locally constant on S.
A z-divisible local Anderson module is called (ind-)étale if ω G = .
Since ω G surjects onto each ω G[z n ] because ω in ∶ ω G[z n+ ] ↠ ω G[z n ] is an epimorphism, ω G = if and only if all G[z n ] are étale; see Lemma . .
A morphism of z-divisible local Anderson modules over S is a morphism of fppf-sheaves of F q [[z]]-modules. e category of z-divisible Anderson modules over S is F q [[z]]-linear and an exact category in the sense of Quillen [Qui , § ].

Remark .
We will frequently use that for a quasi-compact S-scheme X, any S-morphism f ∶ X → lim → G[z n ] factors through f ∶ X → G[z m ] for some m; see for example [HV , Lemma . ].
Remark . (On axiom (iv) in De nition . ) Note the following di erence to the theory of p-divisible groups. On a commutative group scheme, multiplication by p always induces multiplication with the scalar p on its co-Lie module. In the case of F q [[z]]-module schemes, axiom (iv) is the appropriate substitute for this fact, taking into account Example . . It allows that z−ζ is nilpotent on ω G[z n ] . Without axiom (iv) the O S -module ω G is not necessarily nite; see Example . .

Notation .
Let G be a z-divisible local Anderson module. We denote by i n the inclusion map G[z n ] ↪ G[z n+ ] and by i n,m ∶ G[z n ] → G[z m+n ] the composite of the inclusions i n+m− ○⋅ ⋅ ⋅○ i n . We denote by j n,m the unique homomorphism G[z m+n ] → G[z m ] that is induced by multiplication with z n on G[z m+n ] such that i m,n ○ j n,m = z n id G[z m+n ] . Observe that also j n,m ○ i m,n = z n id G[z m ] for all m, n ∈ N, as can be seen by composing with the F q [z]-equivariant monomorphism i m,n ∶ G[z m ] ↪ G[z m+n ]. e following two propositions give an alternative characterization of divisible local Anderson modules, which is analogous to Tate's de nition [Tat ] of p-divisible groups.

Proposition .
Let G be a z-divisible local Anderson module.
(i) For any ≤ m, n the following sequence of group schemes over S is exact (ii) ere is a locally constant function h∶ S → N , s ↦ h(s) such that the order of G[z n ] equals q nh . We call h the height of the z-divisible local Anderson module G.
Proof (i) Since z∶ G → G is an epimorphism, j n,m is also. e rest of (i) is clear. Let . en ord G[z] = q h by eorem . (iii)(c). Now (ii) follows from (i) and the multiplicativity of the order.

Proposition .
Let (G n , i n ∶ G n ↪ G n+ ) n∈N be an inductive system of F q [z]-module schemes that are nite locally free strict F q -module schemes over S satisfying the following conditions.
ere is a locally constant function h∶ S → N such that ord G n = q nh for all n. (iii) Locally on S there exists an integer d ∈ Z ≥ , such that (z − ζ) d = on ω G , where ω G = lim ← ω Gn . en G = lim → G n is a z-divisible local Anderson module.
Proof From (i) it follows that G n = G m [z n ] ⊂ G[z n ] for all m ≥ n. Conversely let x ∈ G[z n ](T) for an S-scheme T. On each quasi-compact open subscheme U ⊂ T we can nd an m such that x U ∈ G m (U) by Remark . . Now z n x = implies x U ∈ G m [z n ](U) = G n (U). In total, x ∈ G n (T). is shows that G n = G[z n ] and G = lim → G[z n ] is z-torsion. e quotient G n G is a nite locally free group scheme over S by Remark . (d). Its order is q (n− )h , by (ii) and the multiplicativity of the order. e natural map z∶ G n G ↪ G n [z n− ] ≅ G n− is a monomorphism and hence a closed immersion by Remark . (a). It must be an isomorphism because ord(G n G ) = ord(G n− ) by (ii). is proves that z∶ G n → G n− is an epimorphism of fppf-sheaves. Let x ∈ G(T) for an S-scheme T. Choose a quasi-compact open covering {U i } i of T. For each i we nd by Remark . an integer n i such that x Ui ∈ G ni (U i ). By the above, there is a Note that we require De nition . (iv) and Proposition . (iii) due to the following example that we do not want to consider a z-divisible local Anderson module.

Example .
Let S be the spectrum of a ring R in which ζ is zero, and let G n be the subgroup of G n a,S = Spec R[x , . . . , x n ] de ned by the ideal (x q , . . . , x q n ). Make G n into an F q [[z]]-module scheme by letting z act through z * (x ) = and z * (x ν ) = x ν− , for < ν ≤ n. De ne i n ∶ G n → G n+ as the inclusion of the closed subgroup scheme de ned by the ideal (x n+ ).
As in Proposition . one proves that G ∶= lim → G n satis es axioms (i)-(iii) of Definition . , but not (iv). Here ω Gn = ⊕ n i= R ⋅ dx i ≅ R n , and so ω G is not a nite R-module. erefore we cannot drop the conditions (iv) in De nition . and (iii) in Proposition . .
In the remainder of this section we introduce truncated z-divisible local Anderson modules.

Lemma .
Let n ∈ N and let G be an fppf-sheaf of F q [z]-modules over S, such that G = G[z n ]. en the following conditions are equivalent.
Proof (i) ⇒ (ii) Because of (i), the multiplication with z i induces isomorphisms for i ≤ n − . is gives us ker(z n− ) ⊂ im(z), and the opposite inclusion ker(z n− ) ⊃ im(z) follows from G = G[z n ]. Now ker(z n−i ) ⊂ ker(z n− ) ⊂ im(z) implies that ker(z n−i ) = z ker(z n−i+ ) = z ⋅ z i− G = z i G, by induction on i.
(ii) ⇒ (i) Taking i = implies im(z) = ker(z n− ), and hence multiplication with z n− induces an isomorphism G zG ∼ → z n− G. Since this factors through the epimorphisms G zG → zG z G → ⋅ ⋅ ⋅ → z n− G, we see that each of these maps is an isomorphism. us we have gr [Bou , Chapter III, § . , eorem ] implies that G is a at F q [z] (z n )-module.

De nition .
Let d, n ∈ N > . A truncated z-divisible local Anderson module with order of nilpotence d and level n is an fppf-sheaf of F q [z]-modules over S, such that (i) if n ≥ d, it is an F q [z] (z n )-module scheme G that is nite locally free and strict as an F q -module scheme, such that (z − ζ) d is homotopic to on ℓ • G S and G satis es the equivalent conditions of Lemma . ; (ii) if n < d, it is of the form ker(z n ∶ G → G) for some truncated z-divisible local Anderson module G with order of nilpotence d and level d.
If G is a z-divisible local Anderson module over S ∈ Nilp Fq [ [ζ]] with (z − ζ) d = on ω G , we will see in Proposition . that G[z n ] is a truncated z-divisible local Anderson module with order of nilpotence d and level n. is justi es the name.

The Local Equivalence
e category of z-divisible local Anderson modules over S and the category of local shtukas over S are both F q [[z]]-linear. Our next aim is to extend Drinfeld's construction and the equivalence from Section to an equivalence between the category of e ective local shtukas over S and the category of z-divisible local Anderson modules over S. For We deduce from [Bou , §III. . , Proposition and Corollaire ] that M q (G) is a nitely generated O S [[z]]-module and the canonical map We claim that multiplication with z on M q (G) is injective. So let lim ← ( f n ) n ∈ M q (G), f n ∈ M q (G[z n ]) with z ⋅ f n = in M q (G[z n ]) for all n. To prove the claim consider the factorization obtained from Notation . . eorem . (ii) implies that M q ( j ,n ) is injective, and hence f n = M q (i n, )( f n+ ) is zero for all n as desired.
Locally on Spec R ⊂ S the R-module M q (G[z]) is free. By eorem . (iii)(c) its rank is r. Let m , . . . , m r be representatives in M q (G) of an R-basis of M q (G[z]) and consider the presentation Note that α is surjective by Nakayama's Lemma [Eis , Corollary . ] because z is contained in the radical of R [[z]]. e snake lemma applied to multiplication with z on the sequence ( . ) yields the exact sequence in which the right map is an isomorphism. is implies that multiplication with z n is surjective on ker α for all n, and hence ker α ⊂ ⋂ n z n ⋅ In particular, n G[z n ] ≅ ker F Mq (G[z n ]) and ω G[z n ] ≅ coker F Mq (G[z n ]) for the O S -modules from De nition . .

Lemma .
Let S ∈ Nilp Fq [ [ζ]] and let G = lim → G[z n ] be a z-divisible local Anderson module over S.
e projective system (n G[z n ] ) n satis es the Mittag-Le er condition. (iii) M q (G) is an e ective local shtuka over S and coker(F Mq (G) ) is canonically isomorphic to ω G . In particular, ω G is a nite locally free O S -module.
Proof Working locally on S we may assume that ζ N ′ = in O S and that (z−ζ) d ω G = for some integers N ′ and d. Let N ≥ max{N ′ , d} be an integer which is a power of (i) e closed immersion i n ∶ G[z n ] ↪ G[z n+ ] induces an epimorphism and therefore ω G surjects onto each ω G[z n ] . is implies that z N ω G[z n ] = for all n. Applying Lemma . to the exact sequence ( . ) for m = , and using i ,n ○ j n, = z n id G[z n+ ] , we obtain that ker(ω is an isomorphism for all n ≥ N. To prove (ii) we x an n ≥ N. We abbreviate the O S -modules M q (G[z k ]) by M k and the map F Mq(G[z k ]) by F k . From Proposition . and eorem . (ii) we have an exact sequence , which is nite locally free over O S by Lemma . . Furthermore, condition (iv) of De nition . implies that (z − ζ) d annihilates coker F Mq (G) . is proves that the map of the same rank, the map is an isomorphism. us M q (G) is an e ective local shtuka.
We can now prove the following theorem. It generalizes [And , § . ], which treated the case of formal F q [[z]]-modules and which we state in (i).
(i) e two contravariant functors Dr q and M q are mutually quasi-inverse antiequivalences between the category of e ective local shtukas over S and the category of z-divisible local Anderson modules over S.
(ii) Both functors are F q [[z]]-linear, map short exact sequences to short exact sequences, and preserve (ind-)étale objects.
Suppose furthermore that M = (M, F M ) is an e ective local shtuka over S and let G = Dr q (M ) be its associated z-divisible local Anderson module. en the following hold. the G n are nite locally free strict F q -module schemes over S of order q nh , and the exact sequence of nite F q -shtukas yields an exact sequence of group schemes → G n → G n+ z n → G n+ . is implies that G n = ker(z n ∶ G n+ → G n+ ) =∶ G n+ [z n ]. By Lemma . we know that locally on S there exist positive integers e ′ , N such that (z − ζ) e ′ = on coker F M and z N = on coker F M . Applying the snake lemma to the diagram the columns are exact by de nition of G ′′ n and the two upper rows are exact by Proposition . . By the snake lemma, this de nes the exact sequence in the bottom row. By eorem . , this implies that is surjective for all n. In particular, the projective system M q (G ′′ n ) satis es the Mittag-Le er condition, and the morphism M q ( f )∶ M ∶= M q (G) → M ′ ∶= M q (G ′ ) of e ective local shtukas corresponding to f by eorem . is surjective by [Har , Proposition II , because this is true for M and M ′ .
us M ′′ is an e ective local shtuka over S. Applying the snake lemma to the (injective) multiplication with z n on the sequence erefore, eorem . implies that G G ′ = Dr q (M ′′ ) = lim → G ′′ n is a z-divisible local Anderson module over S.

De nition .
Let G be an fppf-sheaf of groups over an F q -scheme S. For n ∈ N we let G[F n q ] be the kernel of the relative q n -Frobenius F q n ,G ∶ G → σ * q n G of G over S.
. Later we will assume that ζ = in O S . Let G be a z-divisible local Anderson module over S and let M = (M, F M ) = M q (G) be its associated local shtuka from eorem . . en the q-Frobenius morphism F q,G ∶= lim . In addition to the q-Frobenius, G carries a q-Verschiebung that is identically zero by eorem . . erefore, if ζ = in O S , we will introduce a "z d -Verschiebung" in Remark . and Corollary . , which is more useful for z-divisible local Anderson modules. We begin with the following lemma.

Lemma .
Let M be an e ective local shtuka with (z − ζ) d = on coker F M . en there exists a uniquely determined homomorphism of Proof Since F M is injective by Lemma . and (z − ζ) d = on coker F M , the lemma follows from diagram ( . ).
and hence is a quasi-isogeny between the local shtukas σ

Corollary .
Assume that ζ = in O S . Let G be a z-divisible local Anderson module over S with (z − ζ) d = on ω G . en there is a uniquely determined morphism Proof Let M = M q (G) be the e ective local shtuka associated with G. Since is proves the corollary.

Proposition .
Let G be a z-divisible local Anderson module with (z − ζ) d = on ω G , and let n ∈ N. en G[z n ] ∶= ker(z n ∶ G → G) is a truncated z-divisible local Anderson module with order of nilpotence d and level n; see De nition . .
By eorem . (iii), (z − ζ) d = on coker F Mq(G) . We reduce the map V Mq (G) from Lemma . modulo z n to obtain a homomorphism . Under the identi cation of the co-Lie complex ℓ

Proposition .
Assume that ζ = in O S . Let G = G[z l ] be a truncated z-divisible local Anderson module over S with order of nilpotence d and level l.
for all i. Now let n ∈ N > and l = nd. In particular, if n = , there is a truncated divisible local Anderson moduleG of level d with G =G[z d ] and we assume that V z d ,G = V z d ,G σ * q G . en the following hold.

Remark .
We only treated the case where S ′ ⊂ S is de ned by an ideal I with I q = . e general case for d = is treated by Genestier and La orgue [GL , Proposition . ] using ζ-divided powers in the style of Grothendieck and Berthelot.

Divisible Local Anderson Modules and Formal Lie Groups
In this section we clarify the relation between z-divisible local Anderson modules and formal F q [[z]]-modules; see De nition . . We follow the approach of Messing [Mes ] who treated the analogous situation of p-divisible groups and formal Lie groups.

De nition .
Let G be an fppf-sheaf of abelian groups over S ∈ Nilp Fq [ [ζ]] . We say that G is F-torsion if G = lim → G[F n q ], and that G is F-divisible if F q,G ∶ G → σ * q G is an epimorphism.
Recall that Messing [Mes , Chapter II, eorem . . ] proved that a sheaf of groups G on S is a formal Lie group [Mes , Chapter II, De nitions . . , . . ], if and only if G is F-torsion, F-divisible, and the G[F n q ] are nite locally free S-group schemes.

eorem .
When ζ = in O S and G is a z-divisible local Anderson module over S, then lim = ( )such that the pull-back x ∈ G(Spec R i I i ) is zero}.
Proof By [Mes , Chapter II, eorem . . ] it su ces to show that lim → G[F n q ] is F-torsion, F-divisible, and that the G[F n q ] are nite locally free. By construction lim → G[F n q ] is F-torsion. By De nition . (iv) there is locally on S an integer d with (z − ζ) d ⋅ ω G = ( ), and then G[F n q ] ⊂ G[z nd ] by Corollary . and G[z nd ] is a truncated z-divisible local Anderson module with order of nilpotence d and level nd by Proposition . . erefore Proposition . shows that G[F n q ] is nite locally free, and that F q,G ∶ G[F n q ] → σ * q G[F n− q ] is an epimorphism. Consequently, is an epimorphism and so lim → G[F n q ] is F-divisible, and hence a formal Lie group. e action of F q [[z]] makes it into a formal F q [[z]]-module.
To prove the last statement of the theorem, observe that for any S-scheme T, the homomorphism F q n ,G ∶ G(T) → (σ * q n G)(T) is simply the map sending x to x ○ Frob q n ,T as can be seen from the following diagram: / / S erefore, the monomorphism G[F n q ] ↪ G de nes an inclusion G[F n q ] ⊂ Inf q n − G, the ideals I i in ( . ) being the augmentation ideal in O G[F n q ] de ning the zero section. We claim that this inclusion is an equality. So let x ∈ (Inf q n − G)(T) and let R i and I i be as in ( . ). en I q n i = ( ) implies that Frob q n ,Ri factors through R i → R i I i j → R i . So F q n ,G (x) Spec Ri = x ○ Frob q n ,Ri = j * (x Spec Ri Ii ) = , that is, x ∈ G[F n q ]. us we have G[F n q ] = Inf q n − G and lim → G[F n q ] = lim → Inf k (G) ⊂ G which completes the proof.
Our next aim is to extend the theorem to all S ∈ Nilp Fq [ [ζ]] . For that purpose we start with the following lemma.

Lemma
. Let S be a scheme with ζ N+ = in O S , and let G = G[z nd ] be a truncated z-divisible local Anderson module over S with order of nilpotence d and level nd with n ≥ N + . en for any a ne open subset U of S and any quasicoherent sheaf F of O U -modules, the natural homomorphism for the co-Lie complexes Ext OU (ℓ and there is nothing to prove. If n ≥ , we use [Mes , Chapter II, Corollary . . ] for the sequence → G[z (n− )d ] → G[z nd ] z (n− )d → G[z d ] → . So we must show that ω G[z nd ] ↠ ω G[z (n− )d ] is an isomorphism, that ω G[z nd ] and ω G[z d ] are locally free O S -modules, and that rk ω s * G[z (n− )d ] ≤ rk ω s * G[z d ] for all points s ∈ S. All three statements follow from Proposition . (vi). is concludes the proof when N = .
Since f ∈ G[z n ](X), we have z n f = , and so z n f ′ ∈ G[z n ′ ](X ′ ) has the property that its restriction to G[z n ′ ](X) is zero. Since I = , the group of sections of G[z n ′ ] over X ′ whose restriction to X is zero, is, by [SGA , III, éorème . . (a)], isomorphic to the group Hom OX (ω G[z n ′ ] ⊗ O S O X , I) under an isomorphism that sends the zero morphism X ′ → G[z n ′ ] to the zero element, and the morphism z n f ′ to an element that we denote by h ∈ Hom OX (ω G ⊗ O S O X , I). Since ζ N kills I and N ′ ≥ N, we obtain ζ N ′ ⋅ h = . On ω G the assumption (z − ζ) d = implies z N ′ = ζ N ′ , and so the section

Corollary .
Let ζ N = in O S and let G and d be as in Lemma . . Let N ′ be the smallest integer that is a power of p and greater or equal to N and d. en the k-th in nitesimal neighborhood of G[z n ] in G is the same as that of G[z n ] in G[z n+k N ′ ]. In particular, Inf k (G) = Inf k (G[z k N ′ ]) and this is therefore representable.
Proof By de nition [Mes , Chapter II, De nition ( . )], an S-morphism f ∶ T ′ → G belongs to the k-th in nitesimal neighborhood of G[z n ] in G, if and only if there is an fppf-covering {Spec R i → T ′ } i and ideals I i ⊂ R i with I k+ i = ( ) such that f Spec Ri Ii ∈ G[z n ](Spec R i I i ). But then f ∈ G[z n+k N ′ ](T ′ ) by Lemma . . e last statement is the special case with n = .
Proof As G clearly is an F q [[z]]-submodule of G, we must show that it is a formal Lie variety; see [Mes , Chapter II, De nition . . ]. By construction it is indin nitesimal. Since the question is local on S, we may assume that there are integers N and d as in Corollary . . en the sheaf Inf k (G) is representable for all k. By eorem . we know that G is formally smooth and by de nition ( . ) of Inf k (G) this implies that G is formally smooth. Let N ′ be the smallest integer that is a power of p and greater or equal to N and d. en G[z k N ′ ] satis es the li ing condition ) of [Mes , Chapter II, Proposition . . ] by eorem . and Lemma . . erefore, by [Mes , loc. cit] G[z k N ′ ] satis es condition ) and ) of [Mes , Chapter II, De nition . . ], and hence is a formal F q [[z]]-module.

Remark .
We already know from eorem . and Lemma . that ω G is locally free of nite rank. is now follows again from the theorem because ω G = ω G .
Next we pursue the question when a z-divisible local Anderson module is a formal F q [[z]]-module and vice versa.

Lemma .
Let B be a ring in which ζ is nilpotent, and let I be a nilpotent ideal of B. De ne a sequence of ideals I ∶= ζI + I , . . . , I n+ ∶= ζI n + (I n ) . en for n su ciently large I n = ( ).