Local Shtukas and Divisible Local Anderson Modules

Urs Hartl, Rajneesh Kumar Singh January 24, 2019

Abstract

We develop the analog of crystalline Dieudonné 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 Dieudonné 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. Mathematics Subject Classification (2010): 11G09, (13A35, 14L05)

Contents

1 Introduction 2 2 Local and finite shtukas 5 3 Review of deformations of finite locally free group schemes 8 4 Strict 𝔽_{q}-module schemes 12 5 Equivalence between finite 𝔽_{q}-shtukas and strict 𝔽_{q}-modules 17 6 Relation to global objects 20 7 Divisible local Anderson modules 23 8 The local equivalence 26 9 Frobenius, Verschiebung and deformations of local shtukas 30 10 Divisible local Anderson modules and formal Lie groups 34 A Review of the cotangent complex 39 References 42

This article was published as [HS19]. The present arXiv version of this article contains a few more details most notably Lemmas 3.3, 3.4, 4.4 and 4.11, Remark 3.6, Example 8.4, Corollary 10.17, and Appendix A.

1 Introduction

In the arithmetic of number fields elliptic curves and abelian varieties are important objects. Their theory has been vastly developed in the last two centuries and their moduli spaces have played a major role in Faltings’s proof of the Mordell conjecture [Fal83, CS86], the proof of Fermat’s Last Theorem by Wiles and Taylor [Wil95, TW95, CSS97], and the proof of the Langlands correspondence for GL_{n} over non-archimedean local fields of characteristic zero by Harris and Taylor [HT01]. A useful tool to study 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 Barsotti-Tate group, captures the local p-adic information of E. One reason why E[p^{∞}] is a useful tool to study 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 fields. Namely, Drinfeld [Dri74, Dri87] invented the notions of elliptic modules (today called Drinfeld modules) and the dual notion of F -sheaves (today called Drinfeld shtukas). These structures are function field analogs of elliptic curves in the following sense. Their endomorphism rings are rings of integers in global function fields 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 finite field. Through these two aspects in which global function fields 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 fields of positive characteristic in works by Drinfeld [Dri74, Dri77, Dri87], Laumon, Rapoport, and Stuhler [LRS93], L. Lafforgue [Laf02] and V. Lafforgue [Laf18]. Beyond this the analogy between Drinfeld modules and elliptic curves is abundant.

In this spirit, Anderson [And86] introduced higher dimensional generalizations of Drinfeld modules, called abelian t-modules. These are group schemes which carry an action of the polynomial ring 𝔽_{r}[t] over a finite field 𝔽_{r} with r elements subject to certain conditions. Abelian t-modules are the function field analogs of abelian varieties; see for example [BH09]. Although Anderson worked over a field, abelian t-modules also exist naturally over arbitrary 𝔽_{r}[t]-algebras R as base rings; see Definition 6.5. They 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, see Definition 6.2 and Theorem 6.6, and are a variant and generalization of Drinfeld shtukas. Through the work of Drinfeld and Anderson it was realized very early on that a Drinfeld module or abelian t-module over a field is completely described by its t-motive. The same is true over an arbitrary 𝔽_{r}[t]-algebra R, as is shown for example in [Har19]. So in a way the situation in function field arithmetic is much better than in the arithmetic of abelian varieties: the t-motive is a “global” Dieudonné module which integrates the “local” Dieudonné modules for every prime in a single object.

Correspondingly it is not difficult to come up with a definition 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 6.7(b) for details. The object one ends up with is an effective local shtuka. To define these let 𝔭 = (z) for a monic irreducible polynomial z ∈ 𝔽_{r}[t] and let 𝔽_{q} = 𝔽_{r}[t]/𝔭 be the residue field. Then 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 effective local shtuka over R is a pair \underline{M} = (M, F_{M}) consisting of a locally free R[[z]]-module M of finite rank, and an isomorphism F_{M}: σ_{q}^{∗}M[\frac{1}{z−ζ}] ^{∼} ⟶ M[\frac{1}{z−ζ}] with F_{M}(σ_{q}^{∗}M) ⊂ M. Here σ_{q}^{∗} is the endomorphism of R[[z]] which 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 fields is to describe the analogs of p-divisible groups which correspond to effective local shtukas. In the present article we call them z-divisible local Anderson modules as in the following definition, and we develop this theory under the technical assumption that ζ ∈ R is nilpotent. This theory was already announced in [Har05, Har09, Har11, HK19] and is used in [Har19].

Definition 7.1. 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

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

(b) G is z-divisible, that is z: G → G is an epimorphism,

(c) For every n the 𝔽_{q}-module G[z^{n}] is representable by a finite locally free strict 𝔽_{q}-module scheme over R in the sense of Faltings (Definition 4.8), and

(d) locally on Spec R there exists an integer d ∈ ℤ_{≥0}, such that (z − ζ)^{d} = 0 on ω_{G} where ω_{G} := lim_{⟵} ω_{G[z^{n}]} and ω_{G[z^{n}]} := ε^{∗}Ω_{G[z^{n}]/ Spec R}^{1} for the unit section ε of G[z^{n}] over R.

Such objects were studied in the special case with d = 1 in work of Drinfeld [Dri76], Genestier [Gen96], Laumon [Lau96], Taguchi [Tag93] and Rosen [Ros03]. Generalizations for d > 1 and their semi-linear algebra description by the analog of Dieudonné theory were attempted by the first author in [Har05, Definition 6.2] and by W. Kim [Kim09, Definition 7.3.1]. But unfortunately both definitions and the statements about the analog of Dieudonné theory [Har05, Theorem 7.2] and [Kim09, Theorem 7.3.2] are wrong. The problem lies in the fact that the strictness assumption from (c) is missing. Our above definition corrects this error. It generalizes Anderson’s [And93, § 3.4] definition of formal t-modules who considered the case where the G[z^{n}] are radicial and G is a formal 𝔽_{q}[[z]]-module in the following sense.

Definition 1.1. In this article we define a formal 𝔽_{q}[[z]]-module over an 𝔽_{q}-scheme S to be a formal Lie group G equipped with an action of 𝔽_{q}[[z]]. In particular, we do not impose a condition for the 𝔽_{q}[[z]]-action on ω_{G}.

The description of z-divisible local Anderson modules by effective local shtukas is deduced from Abrashkin’s [Abr06] anti-equivalence between finite locally free strict 𝔽_{q}-module schemes over Spec R and finite 𝔽_{q}-shtukas. The latter are pairs (M, F_{M}) consisting of a locally free R-module M of finite rank and an R-module homomorphism F_{M}: σ_{q}^{∗}M → M. We define finite and local shtukas in Section 2 and we recall Abrashkin’s results in Section 5. His equivalence is given by Drinfeld’s functor

(M, F_{M}) ⟼ Dr_{q}(M, F_{M}) := Spec (⨁_{n≥0} Sym_{R}^{n} M)/(m^{⊗q} − F_{M}(σ_{q}^{∗}m): m ∈ M) ,

and its quasi-inverse defined on a finite locally free strict 𝔽_{q}-module scheme G as

G ⟼ \underline{M}_{q}(G) := Hom_{R-groups,𝔽_{q}-lin}(G, 𝔾_{a,R}),

by which we mean the R-module of 𝔽_{q}-equivariant morphisms of group schemes over R on which the Frobenius F_{M_{q}(G)} is provided by the relative q-Frobenius of the additive group scheme 𝔾_{a,R} over R. Various properties of \underline{M} are reflected in properties of Dr_{q}(\underline{M}); see Theorem 5.2 for details. The functors Dr_{q} and \underline{M}_{q} are extended to effective local shtukas \underline{M} and z-divisible local Anderson modules G by

\underline{M} ⟼ Dr_{q}(\underline{M}) := lim_{⟶_{n}} Dr_{q}(\underline{M}/z^{n}\underline{M})  and
G ⟼ \underline{M}_{q}(G) := lim_{⟵_{n}} \underline{M}_{q}(G[z^{n}]) .

Generalizing [And93, § 3.4], who treated the case of formal 𝔽_{q}[[z]]-modules, we prove the following

Theorem 8.3.

(a) The two contravariant functors Dr_{q} and \underline{M}_{q} are mutually quasi-inverse anti-equivalences between the category of effective local shtukas over R and the category of z-divisible local Anderson modules over R.

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

(c) G is a formal 𝔽_{q}[[z]]-module if and only if F_{M} is topologically nilpotent, that is im(F_{M}^{n}) ⊂ zM for an integer n.

(e) the R[[z]]-modules ω_{Dr_{q}(M,F_{M})} and coker F_{M} are canonically isomorphic.

In Section 6 we explain the relation of z-divisible local Anderson modules and local shtukas to global objects like Drinfeld modules [Dri74], Anderson’s [And86] abelian t-modules and t-motives, and Drinfeld shtukas [Dri87]. 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 finite locally free 𝔽_{r}[t]/(z^{n})-module scheme over R. By Example 6.7(b), the limit G := E[z^{∞}] := lim_{⟶} E[z^{n}] in the category of fppf -sheaves of 𝔽_{q}[[z]]-modules on Spec R satisfies G[z^{n}] := ker(z^{n}: G → G) = E[z^{n}] and is a z-divisible local Anderson module over R. Moreover, the associated effective local shtuka \underline{M}_{q}(G) from Theorem 8.3 arises as the z-adic completion of the t-motive associated with E; see Example 6.7(b).

In Section 7 we present the above definition of z-divisible local Anderson modules G and give equivalent definitions. We also introduce truncated z-divisible local Anderson modules, like for example G[z^{n}]; see Proposition 9.5. In Section 9 we investigate, for ζ = 0 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 Theorem 9.8 to prove that lifting a z-divisible local Anderson module from R/I to R when I^{q} = (0) is equivalent to lifting the Hodge filtration on its de Rham cohomology. In Section 10 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 [Mes72] who treated the analogous situation of p-divisible groups and formal Lie groups, we show that a zdivisible local Anderson module is formally smooth (Theorem 10.4), and how to associate a formal 𝔽_{q}[[z]]-module with it (Theorem 10.7). 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 10.16) and we prove the following

Corollary 10.12. There 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 finite locally free group scheme, such that locally on Spec R there is an integer d with (z − ζ)^{d} = 0 on ω_{G}.

In Section 4 we explain Faltings’s notion of strict 𝔽_{q}-module schemes and give details additional to the treatments of Faltings [Fal02] and Abrashkin [Abr06]. This notion is based on certain deformations of finite locally free group schemes and the associated cotangent complex, which we review in Section 3, respectively in Appendix A. There is an equivalent description of finite locally free strict 𝔽_{q}-module schemes by Poguntke [Pog17]; see Remark 5.3.

Acknowledgements. We are grateful to Alain Genestier and Vincent Lafforgue for many valuable discussions on the topics of the present article. We also thank the referees for allowing us to include formulations from their reports to improve our introduction. Both authors were supported by the Deutsche Forschungsgemeinschaft (DFG) in form of the research grant HA3002/2-1 and the SFB’s 478 and 878.

Notation

Let 𝔽_{q} be a finite field with q elements and characteristic p. For a scheme S over Spec 𝔽_{q} and a positive integer n ∈ ℕ_{>0} 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 an O_{S}-module M we write σ_{q^{n}}^{∗}X := X ×_{S,σ_{q^{n}}} S, respectively σ_{q^{n}}^{∗}M := M ⊗_{O_{S},σ_{q^{n}}^{∗}} O_{S} for the pullback under σ_{q^{n}}. For m ∈ M we also write σ_{q^{n}}^{∗}(m) := m ⊗ 1 ∈ σ_{q^{n}}^{∗}M and note that σ_{q^{n}}^{∗}(bm) = bm ⊗ 1 = m ⊗ b^{q^{n}} = b^{q^{n}} · σ_{q}^{∗}m for b ∈ O_{S} and m ∈ M.

Let z be an indeterminant over 𝔽_{q}. Let O_{S}[[z]] be the sheaf on S of formal power series in z. That is Γ(U, O_{S}[[z]]) = Γ(U, O_{S})[[z]] for open U ⊂ S with the obvious restriction maps. This 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 series in ζ over 𝔽_{q}. Let Nilp_{𝔽_{q}[[ζ]]} be the category of 𝔽_{q}[[ζ]]-schemes on which ζ is locally nilpotent. For S ∈ Nilp_{𝔽_{q}[[ζ]]} let O_{S}((z)) be the sheaf of O_{S}-algebras on S associated with the presheaf U ↦ Γ(U, O_{U})[[z]][\frac{1}{z}]. If U is quasi-compact then O_{S}((z))(U) = Γ(U, O_{S}[[z]])[\frac{1}{z}]. Since ζ is locally nilpotent on S, the sheaf O_{S}((z)) equals the sheaf associated with the presheaf U ↦ Γ(U, O_{S}[[z]])[\frac{1}{z−ζ}]. We denote by σ_{q}^{∗} the endomorphism of O_{S}[[z]] and O_{S}((z)) that acts as the identity on z and as b ↦ b^{q} on local sections b ∈ O_{S}. For a sheaf M of O_{S}[[z]]-modules on S we let σ_{q}^{∗}M := M ⊗_{O_{S}[[z]],σ_{q}^{∗}} O_{S}[[z]] and M[\frac{1}{z−ζ}] := M ⊗_{O_{S}[[z]]} O_{S}[[z]][\frac{1}{z−ζ}] = M ⊗_{O_{S}[[z]]} O_{S}((z)) be the tensor product sheaves. Also for a section m ∈ M we write σ_{q}^{∗}m := m ⊗ 1 ∈ σ_{q}^{∗}M. Note that a sheaf M of O_{S}[[z]]-modules which fpqc-locally on S is isomorphic to O_{S}[[z]]^{⊕r} is already Zariski-locally on S isomorphic to O_{S}[[z]]^{⊕r} by [HV11, Proposition 2.3]. We therefore call such a sheaf simply a locally free sheaf of O_{S}[[z]]-modules of rank r.

2 Local and finite shtukas

Let S be a scheme in Nilp_{𝔽_{q}[[ζ]]}.

Definition 2.1. A local shtuka of rank (or height) r over S is a pair \underline{M} = (M, F_{M}) consisting of a locally free sheaf M of O_{S}[[z]]-modules of rank r, and an isomorphism F_{M}: σ_{q}^{∗}M[\frac{1}{z−ζ}] ^{∼} ⟶ M[\frac{1}{z−ζ}].

A morphism of local shtukas f: (M, F_{M}) → (M′, F_{M′}) over S is a morphism of the underlying sheaves f: M → M′ which satisfies F_{M′} ∘ σ_{q}^{∗}f = f ∘ F_{M}.

A quasi-isogeny between local shtukas f: (M, F_{M}) → (M′, F_{M′}) over S is an isomorphism of O_{S}((z))-modules f: M ⊗_{O_{S}[[z]]} O_{S}((z)) ^{∼} ⟶ M′ ⊗_{O_{S}[[z]]} O_{S}((z)) with F_{M′} ∘ σ_{q}^{∗}(f) = f ∘ F_{M}. A morphism which is a quasi-isogeny is called an isogeny.

For any local shtuka (M, F_{M}) over S ∈ Nilp_{𝔽_{q}[[ζ]]} the homomorphism M → M[\frac{1}{z−ζ}] is injective by the flatness of M and the following

Lemma 2.2. Let R be an 𝔽_{q}[[ζ]]-algebra in which ζ is nilpotent. Then the sequence of R[[z]]-modules

\xymatrix{0 \ar[r] & R[[z]] \ar[r] & R[[z]] \ar[r] & R \ar[r] & 0 \\  & 1 \ar@{|->}[r] & z − ζ , z \ar@{|->}[r] & ζ}

is exact. In particular R[[z]] ⊂ R[[z]][\frac{1}{z−ζ}].

Proof. If ∑_{i} b_{i}z^{i} lies in the kernel of the first map, that is, 0 = (z − ζ)(∑_{i} b_{i}z^{i}) = ∑_{i}(b_{i−1} − ζb_{i})z^{i}, then b_{i} = ζb_{i+1} = ζ^{n}b_{i+n} for all n. Since ζ is nilpotent, all b_{i} are zero. Also due to the nilpotency of ζ the second map is well defined and surjective. For exactness in the middle note that ∑_{i} b_{i}ζ^{i} = 0 implies ∑_{i} b_{i}z^{i} = ∑_{i} b_{i}(z^{i} − ζ^{i}) which is a multiple of z − ζ.

For a morphism f: S′ → S in Nilp_{𝔽_{q}[[ζ]]} we can pull back a local shtuka (M, F_{M}) over S to the local shtuka (M ⊗_{O_{S}[[z]]} O_{S′}[[z]], F_{M} ⊗ id) over S′.

We define the tensor product of two local shtukas (M, F_{M}) and (N, F_{N}) over S as the local shtuka (M ⊗_{O_{S}[[z]]} N , F_{M} ⊗ F_{N}). The local shtuka \underline{1l}(0) := (O_{S}[[z]], F_{1l(0)} = id_{O_{S}[[z]]} : σ_{q}^{∗}O_{S}[[z]] = O_{S}[[z]] ^{∼} ⟶ O_{S}[[z]]) is a unit object for the tensor product. The dual (M^{∨}, F_{M^{∨}}) of a local shtuka (M, F_{M}) over S is defined as the sheaf M^{∨} = Hom_{O_{S}[[z]]}(M, O_{S}[[z]]) together with

F_{M^{∨}}: σ_{q}^{∗}M^{∨}[\frac{1}{z−ζ}] ^{∼} ⟶ M^{∨}[\frac{1}{z−ζ}] , f ↦ f ∘ F_{M}^{−1}.

Also there is a natural definition of internal Hom’s with Hom(\underline{M}, \underline{N}) = \underline{M}^{∨} ⊗ \underline{N}. This 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 [Qui73, §2] if one calls a short sequence of local shtukas exact when the underlying sequence of sheaves of O_{S}[[z]]-modules is exact.

Lemma 2.3. Let (M, F_{M}) be a local shtuka over S. Then locally on S there are e, e′, N ∈ ℤ such that (z − ζ)^{e′}M ⊂ F_{M}(σ_{q}^{∗}M) ⊂ (z − ζ)^{−e}M and z^{N}M ⊂ F_{M}(σ_{q}^{∗}M). For any such e the map F_{M}: σ_{q}^{∗}M → (z − ζ)^{−e}M is injective, and the quotient (z − ζ)^{−e}M/F_{M}(σ_{q}^{∗}M) is a locally free O_{S}- module of finite rank.

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}^{−1} to a basis of M, proves the existence of e, respectively e′. If N ≥ e′ is an integer which is a power of p such that ζ^{N} = 0 in R, then z^{N}M = (z^{N} − ζ^{N})M = (z − ζ)^{N}M ⊂ F_{M}(σ_{q}^{∗}M).

We prove that the quotient K := (z − ζ)^{−e}M/F_{M}(σ_{q}^{∗}M) is a locally free R-module of finite rank. This was already proved in [HV11, Lemma 4.3], but the argument given there only works if R is noetherian, because it uses that R[[z]] is flat over R. We now give a proof also in the non-noetherian case. Since K = coker(F_{M} mod (z − ζ)^{e+e′}: σ_{q}^{∗}M/(z − ζ)^{e+e′}σ_{q}^{∗}M → (z − ζ)^{−e}M/(z − ζ)^{e′}M), it is of finite presentation over R. Since R[[z]] ⊂ R[[z]][\frac{1}{z−ζ}] is a subring by Lemma 2.2 and M is locally free, the map F_{M}: σ_{q}^{∗}M → (z − ζ)^{−e}M is injective. Let 𝔪 ⊂ R be a maximal ideal and set k = R/𝔪. In the exact sequence

0 → Tor_{1}^{R[[z]]}(K, k[[z]]) → σ_{q}^{∗}M ⊗_{R[[z]]} k[[z]] → (z − ζ)^{−e}M ⊗_{R[[z]]} k[[z]] → K ⊗_{R[[z]]} k[[z]] → 0,

we have isomorphisms σ_{q}^{∗}M ⊗_{R[[z]]} k[[z]] ≅ k[[z]]^{⊕ rk M} ≅ (z −ζ)^{−e}M ⊗_{R[[z]]} k[[z]]. Moreover, ζ = 0 in k and hence z^{e+e′}K ⊗_{R[[z]]} k[[z]] = 0. Since k[[z]] is a PID, the map σ_{q}^{∗}M ⊗_{R[[z]]} k[[z]] → (z − ζ)^{−e}M ⊗_{R[[z]]} k[[z]] is injective by the elementary divisor theorem, and hence 0 = Tor_{1}^{R[[z]]}(K, k[[z]]). To relate this to

Tor_{1}^{R}(K, k) = Tor_{1}^{R[[z]]/(z−ζ)^{e+e′}} (K, k[[z]]/(z^{e+e′})) we use the change of rings spectral sequence [Rot09, Theorem 10.71] and the induced epimorphism (from its associated 5-term sequence of low degrees, see [Rot09, Theorem 10.31])

… → Tor_{1}^{R[[z]]}(K, k[[z]]) → Tor_{1}^{R[[z]]/(z−ζ)^{e+e′}} (K, k[[z]]/(z^{e+e′})) → 0 .

It follows that Tor_{1}^{R}(K, k) = 0 and from Nakayama’s lemma we conclude that K is locally free over R of finite rank; compare [Eis95, Exercise 6.2].

Definition 2.4. A local shtuka \underline{M} = (M, F_{M}) over S is called effective if F_{M} is actually a morphism F_{M}: σ_{q}^{∗}M ↪ M. Let (M, F_{M}) be effective of rank r = rk \underline{M}. We say that

(a) (M, F_{M}) has dimension d if coker F_{M} is locally free of rank d as an O_{S}-module.

(b) (M, F_{M}) is étale if F_{M}: σ_{q}^{∗}M ^{∼} ⟶ M is an isomorphism.

(c) F_{M} is topologically nilpotent if locally on S there is an integer n such that im(F_{M}^{n}) ⊂ zM, where F_{M}^{n} := F_{M} ∘ σ_{q}^{∗}F_{M} ∘ … ∘ σ_{q^{n−1}}^{∗}F_{M}: σ_{q^{n}}^{∗}M → M.

(d) \underline{M} is bounded by (d, 0, …, 0) ∈ ℤ^{r} for an integer d ≥ 0, if \underline{M} satisfies

⋀_{O_{S}[[z]]}^{r} F_{M}(σ_{q}^{∗}M) = (z − ζ)^{d} · ⋀_{O_{S}[[z]]}^{r} M .

Example 2.5. We define the Tate objects in the category of local shtukas over S as

\underline{1l}(n) := (O_{S}[[z]] , F_{M} : 1 ↦ (z − ζ)^{n}) .

By Lemma 2.3 every local shtuka over a quasi-compact scheme S becomes effective after tensoring with a suitable Tate object.

More generally, let now S be an arbitrary 𝔽_{q}-scheme.

Definition 2.6. A finite 𝔽_{q}-shtuka over S is a pair \underline{M} = (M, F_{M}) consisting of a locally free O_{S}- module M on S of finite rank denoted rk \underline{M}, and an O_{S}-module homomorphism F_{M}: σ_{q}^{∗}M → M. A morphism f: (M, F_{M}) → (M′, F_{M′}) of finite 𝔽_{q}-shtukas is an O_{S}-module homomorphism f: M → M′ which makes the following diagram commutative

\xymatrix{σ_{q}^{∗}M \ar[r]^{σ_{q}^{∗}f} \ar[d]^{F_{M}} & σ_{q}^{∗}M′ \ar[d]^{F_{M′}} \\ M \ar[r]^{f} & M′ .}

We denote the category of finite 𝔽_{q}-shtukas over S by 𝔽_{q}-Sht_{S}.

A finite 𝔽_{q}-shtuka over S is called étale if F_{M} is an isomorphism. We say that F_{M} is nilpotent if there is an integer n such that F_{M}^{n} := F_{M} ∘ σ_{q}^{∗}F_{M} ∘ … ∘ σ_{q^{n−1}}^{∗}F_{M} = 0.

Finite 𝔽_{q}-shtukas were studied at various places in the literature. They were called “(finite) φsheaves” by Drinfeld [Dri87, § 2], Taguchi and Wan [Tag95, TW96] and “Dieudonné 𝔽_{q}-modules” by Laumon [Lau96]. Finite 𝔽_{q}-shtukas over a field admit a canonical decomposition.

Proposition 2.7. ([Lau96, Lemma B.3.10]) If S is the spectrum of a field L every finite 𝔽_{q}-shtuka \underline{M} = (M, F_{M}) is canonically an extension of finite 𝔽_{q}-shtukas

0 ⟶ (M_{ét}, F_{ét}) ⟶ (M, F_{M}) ⟶ (M_{nil}, F_{nil}) ⟶ 0

where F_{ét} is an isomorphism and F_{nil} is nilpotent. \underline{M}_{ét} = (M_{ét}, F_{ét}) is the largest étale finite 𝔽_{q}-sub-shtuka of \underline{M} and equals im(F_{M}^{rk \underline{M}}). If L is perfect this extension splits canonically.

Proof. This was proved by Laumon [Lau96, Lemma B.3.10] for perfect L. In general one considers the descending sequence of L-subspaces … ⊃ im(F_{M}^{n}) ⊃ im(F_{M}^{n+1}) ⊃ … of M which stabilizes at some finite n. If im(F_{M}^{n+1}) = im(F_{M}^{n}) then F_{M}: σ_{q}^{∗}(im F_{M}^{n}) ↠ im F_{M}^{n+1} = im F_{M}^{n} is surjective, hence bijective, and therefore im(F_{M}^{n′}) = im(F_{M}^{n}) for all n′ ≥ n. So the sequence stabilizes already for some n ≤ rk \underline{M} and M_{ét} = im(F_{M}^{rk \underline{M}}). If L is perfect, M_{nil} is isomorphic to the submodule ⋃_{n≥0} ker(F_{M}^{n} ∘ σ_{q^{n}}^{∗}: M → M) of M; see [Lau96, Lemma B.3.10].

Example 2.8. Every effective local shtuka (M, F_{M}) of rank r over S yields for every n ∈ ℕ a finite 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 finite 𝔽_{q}-shtukas.

Thus from Proposition 2.7 we obtain

Proposition 2.9. If S is the spectrum of a field L in Nilp_{𝔽_{q}[[ζ]]} every effective local shtuka (M, F_{M}) is canonically an extension of effective local shtukas

0 ⟶ (M_{ét}, F_{ét}) ⟶ (M, F_{M}) ⟶ (M_{nil}, F_{nil}) ⟶ 0

where F_{ét} is an isomorphism and F_{nil} is topologically nilpotent. (M_{ét}, F_{ét}) is the largest étale effective local sub-shtuka of (M, F_{M}). If L is perfect this extension splits canonically.

3 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}^{1} its co-Lie module. It is a sheaf of O_{S}-modules. In order to describe which group objects are classified by finite 𝔽_{q}-shtukas we need to review the definition of strict 𝔽_{q}-module schemes in the next two sections. We follow Faltings [Fal02] and Abrashkin [Abr06]. We begin in this section with a review of deformations of finite locally free group schemes. Recall that a group scheme G over S is called finite locally free over S if on every open affine Spec R ⊂ S the scheme G is of the form Spec A for a finite locally free R-module A. By [EGA, I_{new}, Proposition 6.2.10] this is equivalent to G being finite flat and of finite presentation over S. The 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. The following facts will be used throughout.

Remark 3.1. (a) A morphism G′ → G of finite 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_{4}, Corollaire 18.12.6], because it is proper.

(b) Let G and G′′ be group schemes over S which are finite and of finite presentation and assume that G is flat over S. Then a morphism G → G′′ of is an epimorphism of fppf -sheaves on S if and only if it is faithfully flat; compare the proof of [Mes72, Chapter I, Lemma 1.5(b)].

(c) A sequence 0 → G′ → G → G′′ → 0 of finite 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 flat, and that G′ → G is a closed immersion which equals the kernel of G → G′′. In this case ord(G) = ord(G′) · ord(G′′) as can be seen from the isomorphism O_{G′} ≅ O_{G} ⊗_{O_{G′′},ε_{G′′}^{∗}} O_{S} where ε_{G′′}: S → G′′ is the unit section, and from the multiplicativity of ranks rk_{O_{S}} O_{G} = (rk_{O_{G′′}} O_{G}) · (rk_{O_{S}} O_{G′′}) = (rk_{O_{S}} O_{G′}) · (rk_{O_{S}} O_{G′′}).

(d) If G′ ↪ G is a closed immersion of finite locally free group schemes over S, then the quotient G/G′ exists as a finitely presented group scheme over S by [SGA 3, Théorème V.4.1 and Proposition V.9.1], which is flat by [EGA, IV_{3}, Corollaire 11.3.11]. It is integral over S and hence finite, because O_{G/G′} ⊂ O_{G}. In particular, G/G′ is finite locally free over S.

In the following we will work locally on S and assume that S = Spec R is affine. Let G = Spec A be a finite locally free group scheme over S. Then G is a relative complete intersection by [SGA 3, Proposition III.4.15]. This means that locally on S we can take A = R[X_{1}, …, X_{n}]/I where the ideal I is generated by a regular sequence (f_{1}, …, f_{n}) of length n; compare [EGA, IV_{4}, Proposition 19.3.7]. The unit section ε_{G}: S → G defines 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 [Fal02] and Abrashkin [Abr06] define deformations of augmented R-algebras as follows. For every augmented R-algebra (A, ε_{A}: A ↠ R) set I_{A} := ker ε_{A}. For the polynomial ring R[\underline{X}] = R[X_{1}, …, X_{n}] set I_{R[\underline{X}]} = (X_{1}, …, X_{n}) and ε_{R[\underline{X}]} : R[\underline{X}] ↠ R, X_{ν} ↦ 0. Abrashkin [Abr06, §§ 1.1 and 1.2] makes the following

Definition 3.2. The category DSch_{S} has as objects all triples H = (H, H^{♭}, i_{H}), where H = Spec A for an augmented R-algebra A which is finite 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[\underline{X}] = R[X_{1}, …, X_{n}] and an epimorphism of augmented R-algebras j: R[\underline{X}] ↠ A^{♭} satisfying the properties that

• the ideal I := ker(i_{A} ∘ j) is generated by elements of a regular sequence of length n in R[\underline{X}],

• ker j = I · I_{R[\underline{X}]}, and hence A = R[\underline{X}]/I and A^{♭} = R[\underline{X}]/(I · I_{R[\underline{X}]}).

In particular, H is a relative complete intersection. We write A = (A, A^{♭}, i_{A}) and H = Spec A. A morphism Spec(Ã, Ã^{♭}, iÃ) → Spec(A, A^{♭}, i_{A}) in DSch_{S} is given by morphisms f: A → Ã and f^{♭}: A^{♭} → Ã^{♭} of augmented R-algebras such that f ∘ i_{A} = iÃ ∘ f^{♭}. Sometimes i_{H} and i_{A} are ommited.

For an object H = Spec(A, A^{♭}, i_{A}) of DSch_{S} define the two R-modules N_{H} = ker i_{A} and t_{H}^{∗} = I_{A^{♭}}/I_{A^{♭}}^{2}, where I_{A^{♭}} is the kernel of the augmentation ε_{A^{♭}}: A^{♭} → R. After choosing locally on Spec R an epimorphism j: R[\underline{X}] ↠ A^{♭} we have I_{A^{♭}} = I_{R[\underline{X}]}/(I ·I_{R[\underline{X}]}), which implies N_{H} = I/(I · I_{R[\underline{X}]}) and t_{H}^{∗} = I_{R[\underline{X}]}/I_{R[\underline{X}]}^{2}. Both are finite locally free R-modules of the same rank. This is obvious for t_{H}^{∗}, and for N_{H} we give a proof in Lemma 3.3 below. Also note that I_{A^{♭}} ·ker i_{A} = 0, because ker i_{A} = I/(I ·I_{R[\underline{X}]}). We write n = n_{H}: N_{H} ↪ A^{♭} for the natural inclusion and π = π_{H} := ( id −ι_{A^{♭}}ε_{A^{♭}}) mod I_{A^{♭}}^{2} : A^{♭} ↠ t_{H}^{∗}. If H = (A, A^{♭}) and H̃ = Spec(Ã, Ã^{♭}) every morphism (f, f^{♭}): (A, A^{♭}) → (Ã, Ã^{♭}) in Hom_{DSch}(H̃, H) induces morphisms of R-modules N_{f}: N_{H} → NH̃ and t_{f}^{∗}: t_{H}^{∗} → t∗ ; H̃ with f^{♭} ∘ n_{H} = nH̃ ∘ N_{f} and πH̃ ∘ f^{♭} = t_{f}^{∗} ∘ π_{H}.

Lemma 3.3. If H = Spec(A, A^{♭}, i_{A}) ∈ DSch_{S} then A^{♭} and N_{H} are finite locally free R-modules with rk_{R} N_{H} = rk_{R} t_{H}^{∗}.

Proof. Considering the exact sequence 0 → ker i_{A} → A^{♭} → A → 0 of R-modules it suffices to prove that N_{H} = ker i_{A} is finite locally free. Working locally on Spec R we assume that there is an epimorphism j: R[\underline{X}] ↠ A^{♭} as in Definition 3.2 such that I := ker(i_{A} ∘ j) is generated by a regular sequence (f_{1}, …, f_{n}). Then ker i_{A} = I/(I · I_{R[\underline{X}]}) = I ⊗_{R[\underline{X}]} R. From Lemma A.1 we get an exact sequence of R[\underline{X}]-modules

\xymatrix{⨁_{1≤μ<ν≤n} R[\underline{X}] · h_{μν} \ar[r] & ⨁_{ν=1}^{n} R[\underline{X}] · g_{ν} \ar[r] & I \ar[r] & 0 \\ h_{μν} \ar@{|->}[r] & f_{ν}g_{μ} − f_{μ}g_{ν} , g_{ν} \ar@{|->}[r] & f_{ν}}

Applying . ⊗_{R[\underline{X}]} R, the first homomorphism becomes zero because f_{ν} ∈ I_{R[\underline{X}]}, whence f_{ν} = 0 in R. So ker i_{A} ≅ R^{⊕n}. The equality of ranks follows from t_{H}^{∗} = I_{R[\underline{X}]}/I_{R[\underline{X}]}^{2} = ⨁_{ν=1}^{n} R · X_{ν}.

Faltings [Fal02, § 2] notes the following

Lemma 3.4. Let H = Spec(A, A^{♭}, i_{A}) and H̃ = Spec(Ã, Ã^{♭}, iÃ) be objects in DSch_{S} and let f: A → Ã be a morphism of augmented R-algebras. Then the set

L := { f^{♭}: A^{♭} → Ã^{♭} morphisms of augmented R-algebras for which (f, f^{♭}) ∈ Hom_{DSch_{S}}(H̃, H) }

is non-empty and is a principal homogeneous space under Hom_{R}(t_{H}^{∗}, NH̃). That is, for any f^{♭} ∈ L the map Hom_{R}(t_{H}^{∗}, NH̃) → L, h ↦ f^{♭} + nH̃ ∘ h ∘ π_{H} is a bijection.

For the convenience of the reader we include a

Proof. We first show that for every f^{♭} ∈ L the map f̃^{♭} := f^{♭} + nH̃hπ_{H}: A^{♭} → Ã^{♭} is a morphism of augmented R-algebras. Clearly it is a map of R-modules with f̃^{♭}(I_{A^{♭}}) ⊂ IÃ^{♭}. We must show that f̃^{♭}(xy) = f̃^{♭}(x)f̃^{♭}(y). We write x = x′ + x′′ and y = y′ + y′′ with x′, y′ ∈ ι_{A^{♭}}(R) and x′′, y′′ ∈ I_{A^{♭}}. Since nH̃hπ_{H}(x) · nH̃hπ_{H}(y) ∈ ker(iÃ)^{2} = 0 and f^{♭}(x′′) · nH̃hπ_{H}(y) ∈ IÃ^{♭} · ker(iÃ) = 0, as well as nH̃hπ_{H}(x′y′) = nH̃hπ_{H}(x′′y′′) = 0 we compute

f̃^{♭}(x)f̃^{♭}(y) = f^{♭}(x)f^{♭}(y) + (f^{♭}(x′) + f^{♭}(x′′)) · nH̃hπ_{H}(y) + nH̃hπ_{H}(x) · (f^{♭}(y′) + f^{♭}(y′′))
= f^{♭}(xy) + x′ · nH̃hπ_{H}(y) + y′ · nH̃hπ_{H}(x)
= f^{♭}(xy) + nH̃hπ_{H}(x′y + y′x − x′y′ + x′′y′′)
= f̃^{♭}(xy) .

Since im(nH̃hπ_{H}) ⊂ ker(iÃ^{♭}) we have iÃ^{♭} ∘ f̃^{♭} = iÃ^{♭} ∘ f^{♭} = f ∘ i_{A^{♭}} and so f̃^{♭} ∈ L.

Next if f^{♭}, f̃^{♭} ∈ L then im(f^{♭} − f̃^{♭}) ⊂ ker(iÃ^{♭}). If further x = x′ + x′′ ∈ A^{♭} with x′ ∈ ι_{A^{♭}}(R) and x′′ ∈ I_{A^{♭}}, then f^{♭}(x) − f̃^{♭}(x) = x′ + f^{♭}(x′′) − x′ − f̃^{♭}(x′′) is independent of x′. And for x′′, y′′ ∈ I_{A^{♭}} the equation f^{♭}(x′′y′′) − f̃^{♭}(x′′y′′) = f^{♭}(x′′)(f^{♭}(y′′) − f̃^{♭}(y′′)) + (f^{♭}(x′′) − f̃^{♭}(x′′))f̃^{♭}(y′′) ∈ IÃ^{♭} · ker(iÃ) = 0 shows that f^{♭}−f̃^{♭} factors through a unique R-homomorphism h ∈ Hom_{R}(t_{H}^{∗}, NH̃) as f^{♭}−f̃^{♭} = nH̃hπ_{H}.

It remains to show that L ≠ ∅. Locally on open affine subsets U_{i} := Spec R_{i} ⊂ S we consider presentations A = R_{i}[\underline{X}]/I, A^{♭} = R_{i}[\underline{X}]/(I I_{R_{i}[\underline{X}]}) and Ã = R_{i}[ ˜\underline{X}]/Ĩ, Ã^{♭} = R_{i}[ ˜\underline{X}]/(ĨI_{R_{i}[ ˜\underline{X}]}).

We choose an R_{i}-homomorphism F_{i}: R_{i}[\underline{X}] → R_{i}[ ˜\underline{X}] which lifts f. In particular, F_{i}(I) ⊂ Ĩ and F_{i}(I_{R_{i}[\underline{X}]}) ⊂ I_{R_{i}[ ˜\underline{X}]}. Therefore F_{i} induces a homomorphism f_{i}^{♭}: A^{♭} → Ã^{♭} which lifts f. In order to glue the f_{i}^{♭} we consider the quasi-coherent sheaf H := Hom_{S}(t_{H}^{∗}, NH̃) on S. Over U_{ij} := U_{i} ∩ U_{j} both f_{i}^{♭} and f_{j}^{♭} lift f. By the above there is a section h_{ij} ∈ Γ(U_{ij}, H) with f_{i}^{♭} − f_{j}^{♭} = nH̃h_{ij}π_{H}. The h_{ij} form a Čech cocycle. Since Ȟ^{1}({U_{i}}, H) = 0 we find elements h_{i} ∈ Γ(U_{i}, H) with h_{ij} = h_{i} − h_{j}. This means that the f̃_{i}^{♭} := f_{i}^{♭} − nH̃h_{i}π_{H} coincide over U_{ij} and glue to a morphism f̃^{♭}: A^{♭} → Ã^{♭} which lies in L.

The category DSch_{S} possesses direct products. If H = Spec(A, A^{♭}, i_{A}) and H̃ = Spec(Ã, Ã^{♭}, iÃ), then the product H ×_{S} H̃ is given by Spec(A ⊗_{R} Ã, (A ⊗_{R} Ã)^{♭}, κ), where

(A ⊗_{R} Ã)^{♭} := (A^{♭} ⊗_{R} Ã^{♭})/(ker i_{A} ⊗ Ã^{♭} + A^{♭} ⊗ ker iÃ) · (I_{A^{♭}} ⊗ Ã^{♭} + A^{♭} ⊗ IÃ^{♭})

and κ is the natural epimorphism (A⊗_{R} A)^{♭} ↠ A⊗_{R} A. After choosing locally on Spec R presentations A = R[\underline{X}]/I, A^{♭} = R[\underline{X}]/(I · I_{R[\underline{X}]}) and Ã = R[ ˜\underline{X}]/Ĩ, Ã^{♭} = R[ ˜\underline{X}]/(Ĩ· I_{R[ ˜\underline{X}]}) we can write

(A ⊗_{R} Ã)^{♭} = R[\underline{X} ⊗ 1, 1 ⊗ ˜ \underline{X}] / (I ⊗ R[ ˜\underline{X}] + R[\underline{X}] ⊗ Ĩ) · (I_{R[\underline{X}]} ⊗ R[ ˜\underline{X}] + R[\underline{X}] ⊗ I_{R[ ˜\underline{X}]}).

The projections pr_{1} : H ×_{S} H̃ → H and pr_{2} : H ×_{S} H̃ → H̃ come from the natural embeddings of R[\underline{X}] and R[ ˜\underline{X}] into R[\underline{X} ⊗ 1, 1 ⊗ ˜ \underline{X}].

Definition 3.5. 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 Δ: A → A⊗_{R} A and Δ^{♭}: A^{♭} → (A⊗_{R} A)^{♭}, the counit ε: A → R and ε^{♭}: A^{♭} → R, and the coinversion [−1]: A → A and [−1]^{♭}: A^{♭} → A^{♭}, which satisfy the usual axioms. In particular, we require the counit axiom ( id_{A^{♭}} ⊗ε^{♭})∘Δ^{♭} = id_{A^{♭}} = (ε^{♭} ⊗ id_{A^{♭}})∘Δ^{♭}, and that ε and ε^{♭} are the augmentation maps. The morphisms in DGr_{S} are morphisms of group objects.

If G = (G, G^{♭}) ∈ DGr_{S}, note that G = Spec A is a finite locally free group scheme over R with the comultiplication Δ, the counit ε and the coinversion [−1]. But in general G^{♭} is not a group scheme over S when the comultiplication Δ^{♭}: A^{♭} → (A ⊗_{R} A)^{♭} does not lift to A^{♭} ⊗_{R} A^{♭}. Faltings and Abrashkin [Abr06, § 1.2] make the following

Remark 3.6. (a) If G = Spec(A, A^{♭}, i_{A}) ∈ DSch_{S} and G = Spec A is a finite locally free group scheme over R, then there exists a unique structure of a group object on G, which is compatible with that of G. It satisfies Δ^{♭}(x) − x ⊗ 1 − 1 ⊗ x ∈ I_{A^{♭}} ⊗ I_{A^{♭}} for all x ∈ I_{A^{♭}}.

(b) If G, H ∈ DGr_{S} are group objects and (f, f^{♭}) ∈ Hom_{DSch_{S}}(G, H) such that f: G → H is a morphism of group schemes, then (f, f^{♭}) ∈ Hom_{DGr_{S}}(G, H).

For the convenience of the reader we give a

Proof. (a) By Lemma 3.4 we may choose a homomorphism Δ̃^{♭}: A^{♭} → (A ⊗_{R} A)^{♭} which lifts the comultiplication map Δ: A → A ⊗_{R} A. We want to modify Δ̃^{♭} to Δ^{♭} := Δ̃^{♭} + n_{G×G} ∘ h ∘ π_{G} for an R-homomorphism h ∈ Hom_{R}(t_{G}^{∗}, N_{G×G}) such that ( id_{A^{♭}} ⊗ε^{♭}) ∘ Δ^{♭} = id_{A^{♭}} = (ε^{♭} ⊗ id_{A^{♭}}) ∘ Δ^{♭} holds. Thus we can take n_{G×G} ∘h∘π_{G}(x) = (x−( id_{A^{♭}} ⊗ε^{♭})∘Δ̃^{♭}(x))⊗1+1⊗(x−(ε^{♭}⊗ id_{A^{♭}})∘Δ̃^{♭}(x)). Note that this lies in N_{G×G}, because ( id_{A} ⊗ε) ∘ Δ = id_{A} = (ε ⊗ id_{A}) ∘ Δ implies x − ( id_{A^{♭}} ⊗ε^{♭}) ∘ Δ̃^{♭}(x) ∈ ker i_{A} and x − (ε^{♭} ⊗ id_{A^{♭}}) ∘ Δ̃^{♭}(x) ∈ ker i_{A}. This also shows that n_{G×G} ∘ h ∘ π_{G}: A^{♭} → (A ⊗_{R} A)^{♭} factors through t_{G}^{∗} and therefore h exists.

To prove uniqueness of Δ^{♭} we work locally on S and choose a presentation A = R[\underline{X}]/I and A^{♭} = R[\underline{X}]/(I · I_{R[\underline{X}]}). Then we have A ⊗_{R} A = R[\underline{X} ⊗ 1, 1 ⊗ \underline{X}]/(I ⊗ R[\underline{X}] + R[\underline{X}] ⊗ I) and

(A ⊗_{R} A)^{♭} = R[\underline{X} ⊗ 1, 1 ⊗ \underline{X}] / (I ⊗ R[\underline{X}] + R[\underline{X}] ⊗ I) · (I_{R[\underline{X}]} ⊗ R[\underline{X}] + R[\underline{X}] ⊗ I_{R[\underline{X}]}) .

Note that every element u ⊗ ũ ∈ I ⊗ R[\underline{X}] with u ∈ I and ũ = ũ′ + ũ′′ ∈ R[\underline{X}], where ũ′ ∈ R and ũ′′ ∈ I_{R[\underline{X}]}, satisfies u ⊗ ũ = u ⊗ ũ′ = (ũ′u) ⊗ 1 in

ker(i_{A⊗A}) = (I ⊗ R[\underline{X}] + R[\underline{X}] ⊗ I) / (I ⊗ R[\underline{X}] + R[\underline{X}] ⊗ I) · (I_{R[\underline{X}]} ⊗ R[\underline{X}] + R[\underline{X}] ⊗ I_{R[\underline{X}]}) .

Now assume that Δ̃^{♭} and Δ^{♭} both satisfy the counit axiom and lift Δ. Then for every x ∈ A^{♭} there are u, v ∈ I such that Δ^{♭}(x)−Δ̃^{♭}(x) = u⊗1+1⊗v in (A⊗_{R}A)^{♭}. We obtain u = ( id_{A^{♭}} ⊗ε^{♭})(Δ^{♭}(x)−Δ̃^{♭}(x)) = x − x = 0 and v = (ε^{♭} ⊗ id_{A^{♭}})(Δ^{♭}(x) − Δ̃^{♭}(x)) = x − x = 0. This proves that Δ^{♭} = Δ̃^{♭}.

The last assertion is standard. Namely, write Δ^{♭}(x) − x ⊗ 1 − 1 ⊗ x = ∑_{i} u_{i} ⊗ v_{i} for u_{i} = u′_{i} + u′′_{i}, v_{i} = v′_{i} + v′′_{i} with u′_{i}, v′_{i} ∈ ι_{A^{♭}}(R) and u′′_{i}, v′′_{i} ∈ I_{A^{♭}}. Then ∑_{i} u′_{i}v_{i} = (ε^{♭} ⊗ id_{A^{♭}})(∑_{i} u_{i} ⊗ v_{i}) = x − ε^{♭}(x) − x = 0 implies ∑_{i} u_{i} ⊗ v_{i} = ∑_{i} u′′_{i} ⊗ v_{i}. And ∑_{i} u′′_{i}v′_{i} = ( id_{A^{♭}} ⊗ε^{♭})(∑_{i} u′′_{i} ⊗ v_{i}) = ( id_{A^{♭}} ⊗ε^{♭})(Δ^{♭}(x) − x ⊗ 1 − 1 ⊗ x) = x − x − ε^{♭}(x) = 0 implies ∑_{i} u′′_{i} ⊗ v_{i} = ∑_{i} u′′_{i} ⊗ v′′_{i}.

(b) We write G = Spec(A, A^{♭}) with comultiplication (Δ, Δ^{♭}) and counit (ε, ε^{♭}), and H = Spec(Ã, Ã^{♭}) with comultiplication (Δ̃, Δ̃^{♭}) and counit (ε̃, ε̃^{♭}). We also write A = R[\underline{X}]/I locally on S. We have to show that F := (Δ^{♭} ∘ f^{♭} − (f^{♭} ⊗ f^{♭}) ∘ Δ̃^{♭}): Ã^{♭} → (A ⊗_{R} A)^{♭} is zero. From Δ ∘ f = (f ⊗ f) ∘ Δ̃ we see as in (a) that for every x ∈ Ã^{♭} there are u, v ∈ I with F (x) = u ⊗ 1 + 1 ⊗ v in (A ⊗_{R} A)^{♭}. Now

u = ( id_{A^{♭}} ⊗ε^{♭}) ∘ F (x)
= ( id_{A^{♭}} ⊗ε^{♭}) ∘ Δ^{♭} ∘ f^{♭}(x) − ( id_{A^{♭}} ⊗ε^{♭}) ∘ (f^{♭} ⊗ f^{♭}) ∘ Δ̃^{♭}(x)
= id_{A^{♭}} ∘f^{♭}(x) − f^{♭} ∘ ( id_{A^{♭}} ⊗ε̃^{♭}) ∘ Δ̃^{♭}(x)
= f^{♭}(x) − f^{♭}(x)
= 0 ,

and likewise v = 0. This shows that F = 0 and (f, f^{♭}) ∈ Hom_{DGr_{S}}(G, H) as claimed.

Let G = (G, G^{♭}, i_{G}) ∈ DGr_{S}. Faltings defines the co-Lie complex of G over S = Spec R (that is, the fiber at the unit section of G of the cotangent complex) as the complex of finite locally free R-modules

ℓG/S^{•}:  0 ⟶ N_{G} ⟶^{d} t_{G}^{∗} ⟶ 0  (3.1)

concentrated in degrees −1 and 0 with differential d := π_{G} ∘ n_{G}. Recall that the co-Lie complex of G/S and more generally the cotangent complex of a morphism was defined by Illusie [Ill71, Ill72] generalizing earlier work of Lichtenbaum and Schlessinger [LS67]; cf. Appendix A. If G = Spec A for A = R[\underline{X}]/I where I is generated by a regular sequence then the cotangent complex of Illusie [Ill71, II.1.2.3] is quasi-isomorphic to the complex of finite locally free A-modules

LG/S^{•}: 0 ⟶ I/I^{2} ⟶^{d} Ω_{R[\underline{X}]/R}^{1} ⊗_{R[\underline{X}]} A ⟶ 0

concentrated in degrees −1 and 0 with d being the differential map; see [Ill71, Corollaire III.3.2.7]. The co-Lie complex of G over S is defined by Illusie [Ill72, § VII.3.1] as ℓG/S^{•} := ε_{G}^{∗}LG/S^{•} where ε_{G}: S → G is the unit section. To see that this is equal to Faltings’s definition note that

ε_{G}^{∗}(I/I^{2}) = I/I^{2} ⊗_{A} R = I ⊗_{R[\underline{X}]} R = I/(I · I_{R[\underline{X}]}) = N_{G}  and
ε_{G}^{∗}(Ω_{R[\underline{X}]/R}^{1} ⊗_{R[\underline{X}]} A) = Ω_{R[\underline{X}]/R}^{1} ⊗_{R[\underline{X}]} R = ⊕_{ν=1}^{n}R · dX_{ν} = I_{R[\underline{X}]}/I_{R[\underline{X}]}^{2} = t_{G}^{∗},

4 STRICT 𝔽_{Q}-MODULE SCHEMES 12

and that the differential of both co-Lie complexes sends an element x ∈ I to the linear term in its expansion as a polynomial in \underline{X}, because all terms of higher degree are sent to zero under ε_{G}^{∗}.

Up to homotopy equivalence both LG/S^{•} and ℓG/S^{•} only depend on G, and not on the presentation A = R[\underline{X}]/I nor on the deformation G of G. Note that LG/S^{•} and ι^{∗}ℓG/S^{•} are quasi-isomorphic by [Mes72, Chapter II, Proposition 3.2.9] where ι: G → S is the structure map.

Definition 3.7. We (re-)define the co-Lie module of G over S as ω_{G} := H^{0}(ℓG/S^{•}) := coker d and set n_{G} := H^{−1}(ℓG/S^{•}) := ker d. These R-modules only depend on G and not on G. Since H^{0}(LG/S^{•}) = Ω_{G/S}^{1} we have ω_{G} = ε_{G}^{∗}Ω_{G/S}^{1} which is also canonically isomorphic to the R-module of invariant differentials on G.

We record the following lemmas.

Lemma 3.8. If G ∈ DGr_{S} the following are equivalent:

(a) G is étale over S,

(b) ω_{G} = 0,

(c) the differential of ℓG/S^{•} is an isomorphism.

Proof. If G is étale then Ω_{G/S}^{1} = 0. Conversely, since Ω_{G/S}^{1} is a finitely generated O_{G}-module, ω_{G} = 0 implies by Nakayama that G is étale along the zero section. Being a group scheme it is étale everywhere.

Clearly (c) implies (b). Conversely if ω_{G} = 0, that is, if d is surjective, then d is also injective, because both t_{G}^{∗} and N_{G} are finite locally free of the same rank by Lemma 3.3.

Lemma 3.9 ([Mes72, Chapter II, Proposition 3.3.4]). Let 0 → G′ → G → G′′ → 0 be an exact sequence of finite locally free group schemes over S. Then there is an exact sequence of R-modules

0 ⟶ n_{G′′} ⟶ n_{G} ⟶ n_{G′} ⟶ ω_{G′′} ⟶ ω_{G} ⟶ ω_{G′} ⟶ 0 .

In particular, if G′ ↪ G is a closed immersion then ω_{G} ↠ ω_{G′} is surjective.

4 Strict 𝔽_{q}-module schemes

We keep the notation of the previous section. Let O be a commutative unitary ring.

Definition 4.1. In this article an O-module scheme over S is a finite 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 4.2. If S is the spectrum of a field L every O-module scheme G over S is canonically an extension 0 → G^{0} → G → G^{ét} → 0 of an étale O-module scheme G^{ét} by a connected O-module scheme G^{0}. The O-module scheme G^{ét} is the largest étale quotient of G. If L is perfect, G^{ét} is canonically isomorphic to the reduced closed O-module subscheme G^{red} of G and the extension splits canonically, G = G^{0} ×_{S} G^{red}.

Proof. The constituents of the canonical decomposition of the finite S-group scheme G are O-invariant.

Definition 4.3. Let S = Spec R be a scheme over O and let G ∈ DGr_{S}. A strict O-action on G is a homomorphism O → End_{DGr_{S}}(G) such that the induced action on ℓG/S^{•} is equal to the scalar multiplication via O → R; compare Remark 4.5.

We let DGr(O)_{S} be the category whose objects are pairs (G, [ .]) where G ∈ DGr_{S} and [ .]: O → End_{DGr_{S}}(G), a ↦ [a] is a strict O-action, and whose morphisms f: (G, [ .]) → (G′, [ .]′) are those morphisms f: G → G′ in DGr_{S} which are compatible with the O-actions, that is, which satisfy f ∘ [a] = [a]′ ∘ f for all a ∈ O.

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, G^{♭}) → (H, H^{♭}) in DGr(O)_{S} which induce the same morphism G → H.

So by definition the forgetful functor DGr^{∗}(O)_{S} → Gr(O)_{S}, which sends (G, G^{♭}) to G and morphisms (G, G^{♭}) → (H, H^{♭}) to their restriction to G → H, is faithful.

Faltings [Fal02, Remark b) after Definition 1] notes the following property of strict O-actions for which we include a proof.

Lemma 4.4. A strict O-action [ .] on G induces on every deformation G̃ of G a unique strict O-action ˜[ .] which is compatible with all lifts G̃ → G and G → G̃ of the identity on G. In particular, the pairs (G, [ .]) and (G̃, ˜[ .]) are isomorphic in DGr^{∗}(O)_{S}.

Proof. Let G = Spec(A, A^{♭}) and G̃ = Spec(A, Ã^{♭}). By Lemma 3.4 we may choose lifts f: A^{♭} → Ã^{♭} and g: Ã^{♭} → A^{♭} of the identity on A, and there are homotopies h: t_{G}^{∗} → N_{G} and h̃: t∗ ; G̃ → NG̃ satisfying

gf = id +nhπ and fg = id +ñh̃π̃ where n := n_{G}, π := π_{G} and ñ := nG̃, π̃ := πG̃. If we have a strict O-action O → End_{DGr_{S}}(G̃), a ↦ ˜[a] on G̃ satisfying f ∘ [a] = ˜[a] ∘ f and g ∘ ˜[a] = [a] ∘ g then we necessarily must have ˜[a] = ˜[a](fg − ñh̃π̃) = f ∘ [a] ∘ g − ñN˜[a]h̃π̃ = f[a]g − a ñh̃π̃. This proves uniqueness.

So to define a strict O-action on G̃ we choose f and g as in the previous paragraph and we set ˜[a] := f[a]g − a ñh̃π̃: Ã^{♭} → Ã^{♭}. To show that this is a ring homomorphism we first note that it is R-linear. To prove compatibility with multiplication let b, c ∈ Ã^{♭}. We write b = b′ + b′′ and c = c′ + c′′

with b′, c′ ∈ ιÃ^{♭}(R) and b′′, c′′ ∈ IÃ^{♭}. Then π̃(b) = (b′′ mod I2 ; Ã^{♭}) =: \overline{b′′} and ˜[a](b) = f[a]g(b)−a ñh̃(\overline{b′′}) =

b′ + f[a]g(b′′) − a ñh̃(\overline{b′′}) with f[a]g(b′′) ∈ IÃ^{♭}, because f, [a] and g are homomorphisms of augmented R-algebras. Since ñ(NG̃)^{2} = ñ(NG̃) · IÃ^{♭} = (0) and \overline{b′′c′′} = 0 we can compute

˜[a](b) · ˜[a](c) = f[a]g(bc) − b′ · a ñh̃(\overline{c′′}) − c′ · a ñh̃(\overline{b′′})
= f[a]g(bc) − a ñh̃(b′ \overline{c′′} + c′ \overline{b′′} + \overline{b′′c′′})
= f[a]g(bc) − a ñh̃π̃(bc)
= ˜[a](bc) .

We next claim that the map O → End_{DGr_{S}}(G̃), a ↦ ˜[a] is a ring homomorphism. First of all ˜[1] = fg − ñh̃π̃ = id. Next note that

˜[a] + ˜[b]:= m̃ ∘ ((f[a]g − a ñh̃π̃) ⊗ (f[b]g − b ñh̃π̃)) ∘ Δ̃^{♭}
= m̃∘((f ⊗ f)([a] ⊗ [b])(g ⊗ g) − (a ñh̃π̃) ⊗ f[b]g − f[a]g ⊗ (b ñh̃π̃) + (a ñh̃π̃) ⊗ (b ñh̃π̃))∘Δ̃^{♭},

where m̃: (Ã ⊗_{R} Ã)^{♭} → Ã^{♭} is induced from the multiplication in the ring Ã^{♭} and the homomorphism (f[a]g−a ñh̃π̃)⊗(f[b]g−b ñh̃π̃): Ã^{♭}⊗_{R}Ã^{♭} → Ã^{♭}⊗_{R}Ã^{♭} induces a homomorphism (Ã⊗_{R}Ã)^{♭} → (Ã⊗_{R}Ã)^{♭} denoted by the same symbol. To prove ˜[a] + ˜[b] =^{!} { \atop [a + b]} := f[a + b]g − (a + b)ñh̃π̃ we observe

m̃ ∘ (f ⊗ f)([a] ⊗ [b])(g ⊗ g) ∘ Δ̃^{♭} = f ∘ m ∘ ([a] ⊗ [b]) ∘ Δ^{♭} ∘ g =: f ∘ [a + b] ∘ g ,

and we evaluate the claimed equality on X̃_{ν} where Ã^{♭} = R[ ˜\underline{X}]/Ĩ · I_{R[ ˜\underline{X}]}. For every ν there are

u_{i}, v_{i} ∈ I_{R[ ˜\underline{X}]} such that Δ̃^{♭}(X̃_{ν}) = X̃_{ν} ⊗ 1 + 1 ⊗ X̃_{ν} + ∑_{i} u_{i} ⊗ v_{i}; see (a) after Definition 3.5. Now π̃(1) = 0, as well as (ñh̃π̃)(I_{R[ ˜\underline{X}]}) ⊂ Ĩ/Ĩ· I_{R[ ˜\underline{X}]} ⊂ Ã^{♭} and f[b]g(I_{R[ ˜\underline{X}]}) ⊂ I_{R[ ˜\underline{X}]} imply

m̃ ∘ ((a ñh̃π̃) ⊗ f[b]g) ∘ Δ^{♭}(X̃_{ν}) = a ñh̃π̃(X̃_{ν})  and
m̃ ∘ (f[a]g ⊗ (b ñh̃π̃)) ∘ Δ^{♭}(X̃_{ν}) = b ñh̃π̃(X̃_{ν})  and  ((a ñh̃π̃) ⊗ (bñh̃π̃)) ∘ Δ^{♭}(X̃_{ν}) = 0 .

From this ˜[a] + ˜[b] = { \atop [a + b]} follows. To prove that ˜[a] ∘ ˜[b] = ˜[ab], we use that N_{[a]} = a and t_{[b]}^{∗} = b implies [a]n = nN_{[a]} = a · n and π[b] = t_{[b]}^{∗}π = b · π, as well as [a]gñ = [a]nN_{g} = a · gñ and π̃f[b] = t_{f}^{∗}π[b] = b · π̃f. We compute

˜[a] ∘ ˜[b] = (f[a]g − a ñh̃π̃) ∘ (f[b]g − b ñh̃π̃)
= f[a]( id +nhπ)[b]g − a ñh̃π̃f[b]g − b · f[a]g ñh̃π̃ + ab ñh̃π̃ñh̃π̃
= f[a][b]g + ab fnhπg − ab ñh̃π̃fg − ab fgñh̃π̃ + ab ñh̃π̃ñh̃π̃
= f[ab]g + ab f(gf − id)g − ab (fg − id)fg − ab (fg − ñh̃π̃)ñh̃π̃
= f[ab]g − ab ñh̃π̃
= ˜[ab] .

This proves that O → End_{DGr_{S}}(G̃), a ↦ ˜[a] is a ring homomorphism. From

\begin{aligned}t∗ ; ˜[a]π̃ & = & π̃˜[a] & = & π̃f[a]g − a π̃ñh̃π̃ & = & a π̃(fg − ñh̃π̃) & = & a π̃ and \\ ñN˜[a] & = & ˜[a]ñ & = & f[a]gñ − a ñh̃π̃ñ & = & a(fg − ñh̃π̃)ñ & = & a ñ \end{aligned}

we conclude that O → End_{DGr_{S}}(G̃), a ↦ ˜[a] is a strict O-action on G̃.

To prove that ˜[a] is compatible with f and g, we compute ˜[a]f = f[a]gf −a ñh̃π̃f = f[a]+f[a]nhπ− a(fgf −f) = f[a] and g˜[a] = gf[a]g −a gñh̃π̃ = [a]g +nhπ[a]g −a(gfg −g) = [a]g. If f′: A^{♭} → Ã^{♭} and g′: Ã^{♭} → A^{♭} are other lifts of the identity on A then f′ = f+ñℓ̃π and g′ = g+nℓπ̃ for R-homomorphisms

ℓ̃: t_{G}^{∗} → NG̃ and ℓ: t∗ ; G̃ → N_{G}. Then f′[a] = f[a] + ñℓ̃π[a] = ˜[a]f + a ñℓ̃π = ˜[a]f + ˜[a]ñℓ̃π = ˜[a]f′ and

[a]g′ = [a]g + [a]nℓπ̃ = g˜[a] + a nℓπ̃ = g˜[a] + nℓπ̃˜[a] = g′˜[a]. This proves the first part of the lemma. Finally f and g are mutually inverse isomorphisms between (G, [ .]) and (G̃, ˜[ .]) in DGr^{∗}(O)_{S}.

Remark 4.5. The co-Lie complex ℓG/S^{•} depends on the deformation G of G. For another deformation G̃ the complex ℓG̃/S^{•} is homotopically equivalent to ℓG/S^{•}. Therefore one might try to weaken Definition 4.3

and only require that the action of a ∈ O on ℓG/S^{•} is homotopic to the scalar multiplication with a. We do not know whether this is equivalent to Definition 4.3 and whether Lemma 4.4 remains valid for general O. Both is true for the polynomial ring O = 𝔽_{p}[a].

Remark 4.6. Note that there can be different non-isomorphic strict O-actions on a deformation G. For example let G = ∣α_{p} = Spec R[X]/(X^{p}) and A^{♭} = R[X]/(X^{p+1}). Let O = 𝔽_{p}[a] be the polynomial ring in the variable a, and let R be an O-algebra by sending a to 0 in R. For every u ∈ R the endomorphism [a] = 0: ∣α_{p} → ∣α_{p}, X ↦ 0 lifts to [a]: A^{♭} → A^{♭}, X ↦ uX^{p}. All these lifts define strict O-actions on (G, Spec A^{♭}) which are non-isomorphic in DGr^{∗}(𝔽_{p}[a])_{S}. In particular, the forgetful functor DGr^{∗}(𝔽_{p}[a])_{S} → Gr(𝔽_{p}[a])_{S} is not fully faithful.

In contrast, for O = 𝔽_{q} we have the following

Lemma 4.7. The 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 which lifts the action on G.

Proof. Let (G, [ .]) and (G̃, ˜[ .]) be in DGr^{∗}(𝔽_{q})_{S} with G = Spec(A, A^{♭}) and G̃ = Spec(Ã, Ã^{♭}). Let f: A → Ã be a morphism in Gr(𝔽_{q})_{S}, that is ˜[a]f = f[a]. Take any lift f^{♭}: A^{♭} → Ã^{♭} of f. Then for each a ∈ 𝔽_{q} there is an R-homomorphism h_{a}: t_{G}^{∗} → NG̃ with ˜[a]f^{♭} − f^{♭}[a] = ñh_{a}π. It satisfies h_{ab} = ah_{b} + bh_{a} because π[b] = t_{[b]}^{∗}π = b π and ˜[a]ñ = ñN˜[a] = a ñ, and hence

ñh_{ab}π = ˜[ab]f^{♭} − f^{♭}[ab] = ˜[a]( ˜[b] f^{♭} − f^{♭}[b]) + (˜[a]f^{♭} − f^{♭}[a])[b] = ˜[a]ñh_{b}π + ñh_{a}π[b] = ñ(ah_{b} + bh_{a})π .

We claim that it also satisfies h_{a+b} = h_{a} + h_{b}. Namely

ñh_{a+b}π = { \atop [a + b]}f^{♭} − f^{♭}[a + b]
= m̃ ∘ (˜[a] ⊗ ˜[b] ) ∘ Δ̃^{♭} ∘ f^{♭} − f^{♭} ∘ m ∘ ([a] ⊗ [b]) ∘ Δ^{♭}
= m̃ ∘ (˜[a]f^{♭} ⊗ ˜[b] f^{♭} − f^{♭}[a] ⊗ f^{♭}[b]) ∘ Δ^{♭}
= m̃ ∘ ((˜[a]f^{♭} − f^{♭}[a]) ⊗ ˜[b] f^{♭} + f^{♭}[a] ⊗ ( ˜[b] f^{♭} − f^{♭}[b]))
= m̃ ∘ (ñh_{a}π ⊗ ˜[b] f^{♭} + f^{♭}[a] ⊗ ñh_{b}π) ∘ Δ^{♭}

where m̃: (Ã⊗_{R} Ã)^{♭} → Ã^{♭} is induced from the multiplication in the ring Ã^{♭} and (˜[a]f^{♭} ⊗ ˜[b] f^{♭}): A^{♭} ⊗_{R} A^{♭} → Ã^{♭} ⊗_{R} Ã^{♭} induces a homomorphism (A ⊗_{R} A)^{♭} → (Ã ⊗_{R} Ã)^{♭} denoted by the same symbol. We evaluate this expression on X_{ν} where A^{♭} = R[\underline{X}]/I · I_{R[\underline{X}]}. For every ν there are u_{i}, v_{i} ∈ I_{R[\underline{X}]} such that Δ^{♭}(X_{ν}) = X_{ν} ⊗ 1 + 1 ⊗ X_{ν} + ∑_{i} u_{i} ⊗ v_{i}; see (a) after Definition 3.5. Now π(1) = 0, as well as (ñh̃_{a}π)(I_{R[\underline{X}]}) ⊂ Ĩ/Ĩ·I_{R[ ˜\underline{X}]} ⊂ Ã^{♭} and ˜[b] f^{♭}(I_{R[\underline{X}]}) ⊂ I_{R[ ˜\underline{X}]} imply ñh_{a+b}π(X_{ν}) = ñh_{a}π(X_{ν})+ñh_{b}π(X_{ν}) as desired. This proves h_{a+b} = h_{a} + h_{b}. If a lies in the image 𝔽_{p} of ℤ in 𝔽_{q} then h_{a} = a · h_{1} = 0. In other words a ↦ h_{a}, 𝔽_{q} → Hom_{R}(t_{G}^{∗}, NG̃) is an 𝔽_{p}-derivation. Since Ω_{𝔽_{q}/𝔽_{p}}^{1} = (0) we must have

h_{a} = 0 and ˜[a]f^{♭} = f^{♭}[a] for all a ∈ 𝔽_{q}. This means that (f, f^{♭}) defines a morphism in DGr^{∗}(𝔽_{q})_{S} which maps to f under the forgetful functor. So this functor is fully faithful. The remaining assertion follows by taking Ã^{♭} = A^{♭}, Ã = A and f^{♭} = id.

Definition 4.8. A finite 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 which lifts the 𝔽_{q}-action on G. We identify DGr^{∗}(𝔽_{q})_{S} with the category of finite locally free strict 𝔽_{q}-module schemes over S.

Lemma 4.9. For a finite 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. Let G̃ 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 lifts the 𝔽_{q}-action on G. This action induces by Lemma 4.4 a strict 𝔽_{q}-action on G̃ ×_{R} Spec R_{i} for all i. Above Spec R_{i} ∩ Spec R_{j} these actions coincide by Lemma 4.7, and hence they glue to a strict 𝔽_{q}-action on G̃ as desired.

Example 4.10. We give some examples for finite locally free strict 𝔽_{q}-module schemes. Let R be an 𝔽_{q}-algebra.

(a) Let ∣α_{q} = Spec R[X]/(X^{q}) and ∣α_{q}^{♭} = Spec R[X]/(X^{q+1}). Then [a](X) = aX for a ∈ 𝔽_{q} defines a strict 𝔽_{q}-action on G = (∣α_{q}, ∣α_{q}^{♭}). Indeed, the co-Lie complex is

ℓG/S^{•}: 0 ⟶ X^{q} · R ⟶ X · R ⟶ 0

with d = 0 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. Therefore ∣α_{q} is a finite locally free strict 𝔽_{q}-module scheme.

(b) On ∣α_{p} = Spec R[X]/(X^{p}) there is an 𝔽_{q}-action given by [a](X) = aX. If q ≠ p it does not lift to a strict 𝔽_{q}-action on ∣α_{p}^{♭} = Spec R[X]/(X^{p+1}). Although we may lift the action to G = (∣α_{p}, ∣α_{p}^{♭}) via [a](X) = aX, the co-Lie complex is

ℓG/S^{•}: 0 ⟶ X^{p} · R ⟶ X · R ⟶ 0

and so a ∈ 𝔽_{q} acts on N_{G} by a^{p} which is not scalar multiplication by a when a^{p} ≠ a. Any other lift ˜[a] of the 𝔽_{q}-action on ∣α_{p} to G satisfies ˜[a] = [a] + nh_{a}π for an R-homomorphism h_{a}: t_{G}^{∗} → N_{G} and yields nN˜[a] = ˜[a]n = [a]n + nh_{a}πn = [a]n = nN_{[a]} because πn = d = 0 on ℓG/S^{•}. So no such action is strict and ∣α_{p} is not a strict 𝔽_{q}-module scheme.

(c) The constant étale group scheme \underline{𝔽_{q}} = Spec R[X]/(X^{q} − X) over Spec R and its deformation \underline{𝔽_{q}}^{♭} = Spec R[X]/(X^{q+1} − X^{2}) carry a strict 𝔽_{q}-action via [a](X) = aX. Indeed, the co-Lie complex is

ℓG/S^{•}: 0 ⟶ (X − X^{q}) · R ⟶ X · R ⟶ 0

with d: X − X^{q} ↦ X and a ∈ 𝔽_{q} acts on it by N_{[a]}(X − X^{q}) = aX − (aX)^{q} = a(X − X^{q}) and t_{[a]}^{∗}(X) = aX. Therefore \underline{𝔽_{q}} is a finite locally free strict 𝔽_{q}-module scheme.

(d) The multiplicative group lμ .. _{p} = Spec R[X]/(X^{p} − 1) has an 𝔽_{p}-action via [a](X) = X^{a}. This action does not lift to lμ .. _{p}^{♭} = Spec R[X]/(X − 1)^{p+1}, because on lμ .. _{p}^{♭} we have Δ(X) = X ⊗ X and hence [a](X) = X^{a}, which satisfies [p](X) = X^{p} ≠ 1. Therefore no deformation of lμ .. _{p} can carry a strict 𝔽_{p}-action and lμ .. _{p} is not a strict 𝔽_{p}-module scheme. Note that nevertheless 𝔽_{p} acts through scalar multiplication on the co-Lie complex ℓlμ .. _{p}/S^{•}.

Part (c) generalizes to the following

Lemma 4.11. Any finite étale 𝔽_{q}-module scheme is a finite locally free strict 𝔽_{q}-module. In particular, if 0 → G′ → G → G′′ → 0 is an exact sequence of finite 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. The first assertion was remarked by Faltings [Fal02, § 3, p. 252] more generally for finite étale O-module schemes, and also follows from [Dri87, Proposition 2.1(6)] and Theorem 5.2 below. For the convenience of the reader we include a direct proof which also works for general O. Let G be a finite étale 𝔽_{q}-module scheme. Since it is clearly locally free we must prove its strictness. Locally we choose a presentation G = Spec A with A = R[\underline{X}]/I and A^{♭} = R[\underline{X}]/I ·I_{R[\underline{X}]}. Since G is étale, its zero section is open and A = R × A_{1} is a product of rings, where R = R[\underline{X}]/I_{R[\underline{X}]} corresponds to the zero section and A_{1} to its complement. If I_{1} := ker(R[\underline{X}] ↠ A_{1}) then I = I_{R[\underline{X}]} · I_{1} and I_{R[\underline{X}]} + I_{1} = (1). We fix elements u_{0} ∈ I_{R[\underline{X}]} and u_{1} ∈ I_{1} with u_{0} + u_{1} = 1. For a ∈ 𝔽_{q} let [a]: A → A denote its action on A and lift it arbitrarily to an R-homomorphism [a]: R[\underline{X}] → R[\underline{X}]. This lift satisfies [a](I_{R[\underline{X}]}) ⊂ I_{R[\underline{X}]} and [a](I_{1}) ⊂ I_{1}, because [a] fixes the zero section of G and stabilizes its complement. We define the R-homomorphism [a]^{♭}: R[\underline{X}] → A^{♭} by

[a]^{♭}(X_{i}) := aX_{i}u_{1} + [a](X_{i})(1 − u_{1}) = aX_{i}(1 − u_{0}) + [a](X_{i})u_{0} ∈ I_{R[\underline{X}]} .

5 EQUIVALENCE BETWEEN FINITE 𝔽_{Q}-SHTUKAS AND STRICT 𝔽_{Q}-MODULES 17

Since u_{1} ∈ I_{1} and X_{i}, [a](X_{i}), u_{0} ∈ I_{R[\underline{X}]}, it follows that [a]^{♭}(X_{i}) ≡ [a](X_{i}) mod I = I_{R[\underline{X}]}I_{1} and [a]^{♭}(X_{i}) ≡ aX_{i} mod I_{R[\underline{X}]}^{2}. Therefore [a]^{♭}(I) ⊂ I, whence [a]^{♭}(I · I_{R[\underline{X}]}) ⊂ I · I_{R[\underline{X}]}, and so this defines an R-homomorphism [a]^{♭}: A^{♭} → A^{♭} which lifts the action of [a] on A. Since for every x ∈ I ⊂ I_{R[\underline{X}]} we have [a]^{♭}(x) − ax ∈ I_{R[\underline{X}]}^{2} and x − xu_{1} = xu_{0} ∈ II_{R[\underline{X}]}, we compute for [a]^{♭}(x) in I/II_{R[\underline{X}]} = I_{R[\underline{X}]}I_{1}/I_{R[\underline{X}]}^{2}I_{1} that [a]^{♭}(x) = [a]^{♭}(x) · u_{1} = axu_{1} = ax. In particular, [a] acts as scalar multiplication by a on t_{G}^{∗} = I_{R[\underline{X}]}/I_{R[\underline{X}]}^{2} and N_{G} = I/II_{R[\underline{X}]}. Moreover, this shows that the map 𝔽_{q} → End_{R-Alg}(R[\underline{X}]/I_{R[\underline{X}]}^{2}), a ↦ [a]^{♭} is a ring homomorphism. Likewise 𝔽_{q} → End_{R-Alg}(A_{1}), a ↦ [a]^{♭} = [a] is a ring homomorphism. Since I ·I_{R[\underline{X}]} = I_{R[\underline{X}]}^{2}I_{1} and I_{R[\underline{X}]}^{2} +I_{1} = (1) imply A^{♭} = R[\underline{X}]/I_{R[\underline{X}]}^{2} ×A_{1}, it follows that the map 𝔽_{q} → End_{DGr_{S}}(G), a ↦ ([a], [a]^{♭}) is a ring homomorphism. This defines a strict 𝔽_{q}-action on G = (Spec A, Spec A^{♭}).

The last assertion on the strictness of G′ can be proved on affine open subsets of S. There Lemma 4.7 implies that the morphism G → G′′ is 𝔽_{q}-strict in the sense of Faltings [Fal02, Definition 1], and by [Fal02, Proposition 2] its kernel G′ is a strict 𝔽_{q}-module.

5 Equivalence between finite 𝔽_{q}-shtukas and strict 𝔽_{q}-modules

Let S be a scheme over Spec 𝔽_{q}. Recall that a finite 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-Verschiebung morphism V_{p,G}: σ_{p}^{∗}G → G which satisfy F_{p,G} ∘V_{p,G} = p id_{σ_{p}^{∗}G} and V_{p,G} ∘F_{p,G} = p id_{G}. For more details see [SGA 3, Exposé VII_{A}, § 4.3]. Example 4.10 is generalized by the following results of Abrashkin. The first is concerned with finite locally free strict 𝔽_{p}-module schemes.

Theorem 5.1 ([Abr06, Theorem 1]). Let G be a finite locally free group scheme equipped with an 𝔽_{p}-action over an 𝔽_{p}-scheme S. Then this action lifts (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} which have p-Verschiebung zero.

To explain Abrashkin’s classification of finite locally free strict 𝔽_{q}-module schemes we recall that Drinfeld [Dri87, § 2] defined a functor from finite 𝔽_{q}-shtukas over S to finite locally free 𝔽_{q}-module schemes over S. Abrashkin [Abr06] proved that the essential image of Drinfeld’s functor consists of finite locally free strict 𝔽_{q}-module schemes. Other descriptions of the essential image were given by Taguchi [Tag95, § 1] and Laumon [Lau96, § B.3]. (But note that [Lau96, Propositions 2.4.11, B.3.13 and Lemma B.3.16] are incorrect as the 𝔽_{q}-module scheme G = ∣α_{p} = Spec R[x]/(x^{p}) shows when p ≠ q.) Drinfeld’s functor is defined as follows. Let \underline{M} = (M, F_{M}) be a finite 𝔽_{q}-shtuka over S. Let

E = \underline{Spec}_{S} ⨁_{n≥0} Sym_{O_{S}}^{n} M

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 S-morphism Spec(Sym^{•} F_{M}): E → σ_{q}^{∗}E. Drinfeld defines

Dr_{q}(\underline{M}) := ker(Spec(Sym^{•} F_{M}) − F_{q,E}: E → σ_{q}^{∗}E) = \underline{Spec}_{S} (⨁_{n≥0} Sym_{O_{S}}^{n} M)/I

where the ideal I is generated by the elements m^{⊗q} − F_{M}(σ_{q}^{∗}m) for all local sections m of M. (Here m^{⊗q} lives in Sym_{O_{S}}^{q} M and F_{M}(σ_{q}^{∗}m) in Sym_{O_{S}}^{1} M.)

There is an equivalent description of Dr_{q}(\underline{M}) as follows. Let S = Spec R be affine and denote the R-module Γ(S, M) again by M. Let Frob_{q,R}: R → R be the q-Frobenius on R with x ↦ x^{q}. We equip M with the Frob_{q,R}-semi-linear endomorphism F_{M}^{semi} : M → M, m ↦ F_{M}(σ_{q}^{∗}m), which satisfies F_{M}^{semi}(bm) = F_{M}(σ_{q}^{∗}(bm)) = F_{M}(b^{q}σ_{q}^{∗}m) = b^{q}F_{M}^{semi}(m). Also we equip every R-algebra T with the Frob_{q,R}-semi-linear R-module endomorphism F_{T}^{semi} := Frob_{q,T} : T → T. Then Dr_{q}(\underline{M}) is the group scheme over S which is given on R-algebras T as

Dr_{q}(\underline{M})(T) = Hom_{F^{semi}}(\underline{M}, T) := \{ h ∈ Hom_{R-Mod}(M, T): h(m)^{q} = h(F_{M}(σ_{q}^{∗}m)) ∀ m ∈ M \} ,

because Hom_{R-Mod}(M, T) = Hom_{R-Alg}(Sym_{R}^{•} M, T) = E(Spec T). We thank L. Taelman for mentioning this to us.

Dr_{q}(\underline{M}) is an 𝔽_{q}-module scheme over S via the comultiplication Δ: m ↦ m ⊗ 1 + 1 ⊗ m and the 𝔽_{q}-action [a]: m ↦ am which it inherits from E. It has a canonical deformation

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

where I_{0} = ⨁_{n≥1} Sym_{O_{S}}^{n} M is the ideal generated by all m ∈ M. This deformation is equipped with the comultiplication Δ^{♭}: m ↦ m ⊗ 1 + 1 ⊗ m and the 𝔽_{q}-action [a]^{♭}: m ↦ am. We set Dr_{q}(\underline{M}) := (Dr_{q}(\underline{M}), Dr_{q}(\underline{M})^{♭}). Its co-Lie complex is

0 ⟶ I/(I · I_{0}) ⟶ I_{0}/I_{0}^{2} ⟶ 0  (5.2)

with differential d: m^{⊗q} − F_{M}(σ_{q}^{∗}m) ↦ −F_{M}(σ_{q}^{∗}m). On it [a] acts by scalar multiplication with a because (am)^{q} − F_{M}(σ_{q}^{∗}(am)) = a^{q}(m^{⊗q} − F_{M}(σ_{q}^{∗}m)). This defines the functor Dr_{q}: 𝔽_{q}-Sht_{S} → DGr(𝔽_{q})_{S}. We also compose Dr_{q} with the projection to DGr^{∗}(𝔽_{q})_{S}.

Conversely, let G = (G, G^{♭}) = Spec(A, A^{♭}) ∈ DGr(𝔽_{q})_{S} in the affine situation S = Spec R. Note that on the additive group scheme 𝔾_{a,S} = Spec R[x] the elements b ∈ R act via endomorphisms ψ_{b}: 𝔾_{a,S} → 𝔾_{a,S} given by ψ_{b}^{∗}: R[x] → R[x], x ↦ bx. This 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,𝔽_{q}-lin}(G, 𝔾_{a,S}) = {x ∈ A: Δ(x) = x ⊗ 1 + 1 ⊗ x, [a](x) = ax, ∀a ∈ 𝔽_{q}} ,

with its action of R via R → End_{R-groups,𝔽_{q}-lin}(𝔾_{a,S}). It is a finite locally free R-module by [Pog17, Proposition 3.6 and Remark 5.5]; see also [SGA 3, VII_{A}, 7.4.3] in the reedited version of SGA 3 by P. Gille and P. Polo. The composition on the left with the relative q-Frobenius endomorphism F_{q,𝔾_{a,S}} of 𝔾_{a,S} = Spec R[x] given by x ↦ x^{q} defines a map M_{q}(G) → M_{q}(G), m ↦ F_{q,𝔾_{a,S}} ∘ m which is not R-linear, but σ_{q}^{∗}-linear, because F_{q,𝔾_{a,S}} ∘ ψ_{b} = ψ_{b^{q}} ∘ F_{q,𝔾_{a,S}}. Therefore, F_{q,𝔾_{a,S}} induces an Rhomomorphism F_{M_{q}(G)} : σ_{q}^{∗}M_{q}(G) → M_{q}(G). Then \underline{M}_{q}(G) := (M_{q}(G), F_{M_{q}(G)}) is a finite shtuka over S. Note that for m ∈ M_{q}(G) the commutative diagram

\xymatrix{G \ar[r]^{F_{q,G}} \ar[d]_{m} & σ_{q}^{∗}G \ar[d]^{σ_{q}^{∗}m} \\ 𝔾_{a,S} \ar[r]^{F_{q,𝔾_{a,S}}} & 𝔾_{a,S}}  (5.3)

implies that F_{M_{q}(G)}(σ_{q}^{∗}m) := F_{q,𝔾_{a,S}} ∘ m = σ_{q}^{∗}m ∘ F_{q,G}. If H ∈ DGr(𝔽_{q})_{S} and (f, f^{♭}): G → H is a morphism in the category DGr(𝔽_{q})_{S}, then \underline{M}_{q}(f): \underline{M}_{q}(H) → \underline{M}_{q}(G), m ↦ m ∘ f. This defines the functor \underline{M}_{q}: DGr(𝔽_{q})_{S} → 𝔽_{q}-Sht_{S}. It factors through the category DGr^{∗}(𝔽_{q})_{S} and further over the forgetful functor through the category of finite locally free strict 𝔽_{q}-module schemes over S.

There is a natural morphism \underline{M} → \underline{M}_{q}(Dr_{q}(\underline{M})), m ↦ f_{m}, where f_{m}: Dr_{q}(\underline{M}) → 𝔾_{a,S} = Spec R[x] is given by f_{m}^{∗}(x) = m. There is also a natural morphism of group schemes G → Dr_{q}(\underline{M}_{q}(G)) given on the structure sheaves by ⨁_{n≥0} Sym_{O_{S}}^{n} M_{q}(G)/I → O_{G}, m ↦ m^{∗}(x), which is well defined be-

cause

F_{M_{q}(G)}(σ_{q}^{∗}m)^{∗}(x) = (F_{q,𝔾_{a,S}} ∘ m)^{∗}(x) = m^{∗}(x^{q}) = (m^{∗}(x))^{q}.

A large part of the following theorem was already proved by Drinfeld [Dri87, Proposition 2.1] without using the notion of strict 𝔽_{q}-modules.

Theorem 5.2. (a) The contravariant functors Dr_{q} and \underline{M}_{q} are mutually quasi-inverse anti-equiva-lences between the category of finite 𝔽_{q}-shtukas over S and the category of finite locally free strict 𝔽_{q}-module schemes over S.

(b) Both functors are 𝔽_{q}-linear and map short exact sequences to short exact sequences. They preserve étale objects and map the canonical decompositions from Propositions 4.2 and 2.7 to each other.

Let \underline{M} = (M, F_{M}) be a finite 𝔽_{q}-shtuka over S and let G = Dr_{q}(\underline{M}). Then

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

(d) the 𝔽_{q}-module scheme Dr_{q}(\underline{M}) is radicial over S if and only if F_{M} is nilpotent locally on S.

(e) the order of the S-group scheme Dr_{q}(\underline{M}) is q^{rk M}.

(f) the co-Lie complex ℓDr_{q}(\underline{M})/S^{•} is canonically isomorphic to the complex 0 → σ_{q}^{∗}M ⟶^{F_{M}} M → 0. In particular, ω_{Dr_{q}(\underline{M})} = coker F_{M} and n_{Dr_{q}(\underline{M})} = ker F_{M}.

Proof. Assertions (a) and (c) were proved by Abrashkin [Abr06, Theorem 2] in terms of the category DGr^{∗}(𝔽_{q})_{S}.

(b) The 𝔽_{q}-linearity is clear from the definitions and the compatibility with étale objects follows from (f) and Lemma 3.8. Let 0 → \underline{M}′′ → \underline{M} → \underline{M}′ → 0 be a short exact sequence of finite 𝔽_{q}- shtukas. Then by construction Dr_{q}(\underline{M}′) → Dr_{q}(\underline{M}) is a closed immersion. Using (a) we consider the local sections of M′′ = Hom_{S-groups,𝔽_{q}-lin}(Dr_{q}(\underline{M}′′), 𝔾_{a,S}) which are obtained by the closed immersion Dr_{q}(\underline{M}′′) ↪ \underline{Spec}_{S}(Sym_{O_{S}}^{•} M′′) composed with local coordinate functions on \underline{Spec}_{S}(Sym_{O_{S}}^{•} M′′). These local sections go to zero in M′ and this yields a morphism Dr_{q}(\underline{M})/ Dr_{q}(\underline{M}′) → Dr_{q}(\underline{M}′′). The latter must be an isomorphism by (a) due to the identification

\underline{M}_{q}(Dr_{q}(\underline{M})/ Dr_{q}(\underline{M}′)) = ker(\underline{M}_{q}(Dr_{q}(\underline{M})) → \underline{M}_{q}(Dr_{q}(\underline{M}′))) = \underline{M}′′ = \underline{M}_{q}(Dr_{q}(\underline{M}′′)).

Conversely let 0 → G′ → G → G′′ → 0 be a short exact sequence of finite locally free strict 𝔽_{q}- module schemes. Then the exactness of 0 → \underline{M}_{q}(G′′) → \underline{M}_{q}(G) → \underline{M}_{q}(G′) is obvious. Applying Dr_{q}, whose exactness we just established, to the injection \underline{M}_{q}(G)/\underline{M}_{q}(G′′) → \underline{M}(G′) yields an isomorphism Dr_{q}(\underline{M}_{q}(G)/\underline{M}_{q}(G′′)) = ker(G → G′′) = G′. From (a) it follows that \underline{M}_{q}(G)/\underline{M}_{q}(G′′) → \underline{M}(G′) is an isomorphism.

Finally consider the exact sequences from Propositions 4.2 and 2.7. Then Dr_{q}(\underline{M}_{ét}) is an étale quotient of Dr_{q}(\underline{M}). This yields a morphism Dr_{q}(\underline{M})^{ét} → Dr_{q}(\underline{M}_{ét}). Conversely \underline{M}_{q}(G^{ét}) is an étale 𝔽_{q}-subshtuka of \underline{M}_{q}(G). This yields a morphism \underline{M}_{q}(G^{ét}) → \underline{M}_{q}(G)_{ét}. The equivalence of (a) shows that both morphisms are isomorphisms. This proves the compatibility of Dr_{q} and \underline{M}_{q} with the canonical decompositions.

(d) By definition G := Dr_{q}(\underline{M}) is radicial over S if G(K) → S(K) is injective for all fields K. This can be tested by applying the base change Spec K → S. By (b) and Propositions 4.2 and 2.7 the base change G ×_{S} Spec K is connected if and only if F_{M} ⊗ id_{K} is nilpotent. This implies (d) over Spec K. It remains to show that F_{M} is nilpotent locally on S if G is radicial. Locally on an affine 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 2.7 implies that F_{M}^{r} = 0 in κ(s)^{r×r}. Therefore the entries of the matrix F_{M}^{r} lie in the nil-radical of R. If n is an integer such that their q^{n}-th powers are zero then F_{M}^{r(n+1)} = F_{M}^{r} · . . . · σ_{q}^{n∗}(F_{M}^{r}) = 0. This establishes (d).

(e) If locally on S we choose an isomorphism M ≅ ⨁_{ν=1}^{n} O_{S} · X_{ν} and let (t_{ij}) be the matrix of the morphism F_{M}: σ_{q}^{∗}M → M with respect to the basis (X_{1}, …, X_{n}), then Dr_{q}(\underline{M}) is the subscheme of 𝔾_{a,S}^{n}, given by the system of equations

X_{j}^{q} = ∑_{i=1}^{n} t_{ij}X_{i} for j = 1, …, n.

Therefore O_{Dr_{q}(\underline{M})} is a free O_{S}-module with basis X_{1}^{m_{1}} · . . . · X_{n}^{m_{n}}, 0 ≤ m_{i} < q. Thus ord Dr_{q}(\underline{M}) := rk_{O_{S}} O_{Dr_{q}(\underline{M})} = q^{rk M}.

(f) In the presentation of ℓDr_{q}(\underline{M})/S^{•} given in (5.2) with I = (m^{⊗q} − F_{M}(σ_{q}^{∗}m): m ∈ M) and I_{0} = ⨁_{n≥1} Sym_{O_{S}}^{n} M we use the isomorphisms of O_{S}-modules M ^{∼} ⟶ I_{0}/I_{0}^{2}, m ↦ m and σ_{q}^{∗}M ^{∼} ⟶ I/(I I_{0}), b σ_{q}^{∗}m = m ⊗ b ↦ b F_{M}(σ_{q}^{∗}m) − b m^{⊗q}. Note that the latter is surjective by definition and injective because both σ_{q}^{∗}M and I/(I I_{0}) are locally free O_{S}-modules of the same rank.

Remark 5.3. Finite locally free strict 𝔽_{q}-module schemes over S = Spec R were equivalently described by Poguntke [Pog17] as follows. He defines the category 𝔽_{q}-gr_{S}^{+,b} of finite locally free 𝔽_{q}-module schemes G = Spec A which locally on S can be embedded into 𝔾_{a,S}^{N} for some set N and are balanced in the following sense. The R-module

M_{p}(G) := Hom_{S-groups}(G, 𝔾_{a,S}) = { x ∈ A: Δ(x) = x ⊗ 1 + 1 ⊗ x }

of morphisms of group schemes over R decomposes under the action of 𝔽_{q} on G into eigenspaces

M_{p}(G)_{p^{i}} := { m ∈ M_{p}(G): [α](m) = α^{p^{i}} · m for all α ∈ 𝔽_{q} }

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 M_{p}(G)_{p^{i}} ^{∼} ⟶ M_{p}(G)_{p^{i+1}} for all i = 0, …, e − 2. Note that it is neither required nor implied that also M_{p}(G)_{p^{e−1}} → M_{p}(G)_{1} =: M_{q}(G) is an isomorphism. The latter holds if and only if G is étale by Theorem 5.2(b).

Abrashkin [Abr06, 2.3.2] already showed that every finite locally free strict 𝔽_{q}-module scheme over S belongs to 𝔽_{q}-gr_{S}^{+,b}. And Poguntke [Pog17, Theorem 1.4] conversely shows that Dr_{q} and \underline{M}_{q} provide an anti-equivalence between the category of finite 𝔽_{q}-shtukas over S and the category 𝔽_{q}-gr_{S}^{+,b}.

6 Relation to global objects

Without giving proofs, we want in this section to relate local shtukas and divisible local Anderson modules (defined in the next section), as well as finite 𝔽_{q}-shtukas and finite locally free strict 𝔽_{q}- module schemes to global objects like A-motives, global shtukas, Drinfeld modules, Anderson Amodules, etc. which are defined 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 ×_{𝔽_{q}} S and we consider the endomorphism σ_{q} := id_{C} ⊗ Frob_{q,S}: C_{S} → C_{S}.

Definition 6.1. (a) 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 \underline{N} = (N , c_{1}, …, c_{n}, τ_{N}) consisting of

• a locally free sheaf N of rank r on C_{S},

• 𝔽_{q}-morphisms c_{i}: S → C called the legs of \underline{N} and

• an isomorphism τ_{N} : σ_{q}^{∗}N |_{C_{S}∖⋃_{i} Γ_{c_{i}}} ^{∼} ⟶ N |_{C_{S}∖⋃_{i} Γ_{c_{i}}} outside the graphs Γ_{c_{i}} of the c_{i}.

In this article we will only consider the case where Γ_{c_{i}} ∩ Γ_{c_{j}} = ∅ for i ≠ j.

(b) A global shtuka over S is a Drinfeld shtuka if n = 2, Γ_{c_{1}} ∩Γ_{c_{2}} = ∅, and τ_{N} satisfies τ_{N} (σ_{q}^{∗}N) ⊂ N on C_{S} ∖ Γ_{c_{2}} with cokernel locally free of rank 1 as O_{S}-module, and τ_{N}^{−1}(N) ⊂ σ_{q}^{∗}N on C_{S} ∖ Γ_{c_{1}} with cokernel locally free of rank 1 as O_{S}-module.

Drinfeld shtukas were introduced by Drinfeld [Dri87] under the name F -sheaves.

An important class of special examples is defined as follows. Let ∞ ∈ C be a closed point and put A := Γ(C ∖ {∞}, O_{C}). Then Spec A = C ∖ {∞}. We will consider affine A-schemes c: S = Spec R → Spec A and the ideal J := (a⊗1−1⊗c^{∗}(a): a ∈ A) ⊂ A_{R} := A⊗_{𝔽_{q}} R whose vanishing locus V(J) is the graph Γ_{c} of the morphism c. The endomorphism σ_{q} := id_{C} ⊗ Frob_{q,S}: C_{S} → C_{S} induces the ring endomorphism σ_{q}^{∗} := id_{A} ⊗ Frob_{q,R}: A_{R} → A_{R}, a ⊗ b ↦ a ⊗ b^{q} of A_{R} for a ∈ A and b ∈ R. The following definition generalizes Anderson’s [And86] notion of t-motives, which is obtained as the special case, where C = ℙ^{1}, A = 𝔽_{q}[t] and R is a field.

Definition 6.2. Let d and r be positive integers and let S = Spec R be an affine A-scheme. An effective A-motive of rank r and dimension d over S is a pair \underline{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} = 0. More generally, an A-motive of rank r over S is a pair \underline{N} = (N, τ_{N}) consisting of a locally free A_{R}-module N of rank r and an isomorphism τ_{N}: σ_{q}^{∗}N|_{Spec A_{R}∖V(J)} ^{∼} ⟶ N|_{Spec A_{R}∖V(J)} outside the vanishing locus V(J) = Γ_{c} of J.

Example 6.3. (a) If \underline{N} = (N , c_{1}, c_{2}, τ_{N}) is a global shtuka of rank r over S = Spec R with two legs such that c_{1} = c and c_{2} : S → \{∞\} ⊂ C, then \underline{N}(\underline{N}) := (N, τ_{N}) := (Γ(Spec A_{R}, N), τ_{N} ) is an A-motive of rank r over S.

(b) Conversely, if ∞ ∈ C(𝔽_{q}), every A-motive \underline{N} = (N, τ_{N}) over an affine A-scheme c: S = Spec R → Spec A can be obtained from a global shtuka \underline{N} = (N , c_{1}, c_{2}, τ_{N}) by taking c_{1} = c and c_{2} : 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}.

These global objects give rise to finite and local shtukas, and that motivates the names for the latter.

Example 6.4. (a) Let i: D ↪ C be a finite closed subscheme and let \underline{N} = (N , c_{1}, …, c_{n}, τ_{N}) be a global shtuka of rank r over S such that τ_{N} (σ_{q}^{∗}N) ⊂ N in a neighborhood of D_{S} := D ×_{𝔽_{q}} S. (For example this is satisfied if \underline{N} is a Drinfeld-shtuka and D_{S} ∩ Γ_{c_{2}} = ∅ or if \underline{N} is as in Example 6.3 with \underline{N}(\underline{N}) an effective A-motive and D ⊂ Spec A.) Then

(M, F_{M}) := (i^{∗}N , i^{∗}τ_{N})

is a finite 𝔽_{q}-shtuka over S, because M is locally free over S of rank r · dim_{𝔽_{q}} O_{D}. The sense in which \underline{N} is global and (M, F_{M}) is finite, is with respect to the coefficients: \underline{N} lives over all of C and \underline{M} lives over the finite scheme D. This example gave rise to the name “finite 𝔽_{q}-shtuka”.

(b) Let v ∈ C be a closed point defined by a sheaf of ideals 𝔭 ⊂ O_{C}, let q̂ be the cardinality of the residue field 𝔽_{v} of v, let f := [𝔽_{v} : 𝔽_{q}], and let z ∈ 𝔽_{q}(C) be a uniformizing parameter at v. Let \underline{N} = (N , c_{1}, …, 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. Then the formal completion of C_{S} along the graph Γ_{c_{i}} of c_{i} is canonically isomorphic to Spf R[[z]] by [AH14, Lemma 5.3]. The formal completion M of (N , τ_{N}) along Γ_{c_{i}} together with τ_{M} := τ_{N}^{f} : σ_{q}^{f∗}M[\frac{1}{z−ζ}] ^{∼} ⟶ M[\frac{1}{z−ζ}] is a local shtuka over S of rank r (as in Definition 2.1 with q and 𝔽_{q}[[z]] and σ_{q}^{∗} replaced by q̂ and 𝔽_{v}[[z]] and σ_{q}^{f∗}). See [Har19, § 6] for more details. Again \underline{M} is local with respect to the coefficients as it lives over the complete local ring Ô_{C,v} = 𝔽_{v}[[z]] of C at v. This gave rise to the name “local shtuka”.

So far we discussed the semi-linear algebra side given by shtukas. On the side of group schemes, an important source from which the corresponding strict 𝔽_{q}-module schemes arise are Drinfeld A-modules, or more generally abelian Anderson A-modules. To define them, let c: S = Spec R → Spec A be an affine 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}^{1} 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 ψ_{b}: 𝔾_{a,R} → 𝔾_{a,R} given by ψ_{b}^{∗}: R[x] → R[x], x ↦ bx. This makes 𝔾_{a,R} into an 𝔽_{q}-module scheme. In addition, let τ := F_{q,𝔾_{a,R}} be the relative q-Frobenius endomorphism of 𝔾_{a,R} = Spec R[x] given by x ↦ x^{q}. It satisfies τ ∘ ψ_{b} = ψ_{b^{q}} ∘ τ. We let R{τ} := {∑_{i=0}^{n} b_{i}τ^{i}: n ∈ ℕ_{0}, b_{i} ∈ R} with τb = b^{q}τ be the non-commutative polynomial ring in the variable τ over R. There is an isomorphism of rings R{τ} ^{∼} ⟶ End_{R-groups,𝔽_{q}-lin}(𝔾_{a,R}) sending an element f = ∑_{i} b_{i}τ^{i} ∈ R{τ} to the 𝔽_{q}-equivariant endomorphism f: 𝔾_{a,R} → 𝔾_{a,R} given by f^{∗}(x) := ∑_{i} b_{i}x^{q^{i}}.

Definition 6.5. Let d and r be positive integers. An abelian Anderson A-module of rank r and dimension d over an affine A-scheme c: Spec R → Spec A is a pair \underline{E} = (E, φ) consisting of a smooth affine group scheme E over Spec R of relative dimension d, and a ring homomorphism φ: A → End_{R-groups}(E), a ↦ φ_{a} such that

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

(b) (a ⊗ 1 − 1 ⊗ c^{∗}a)^{d} · ω_{E} = 0 for all a ∈ A under the action of a ⊗ 1 induced from φ_{a} and the natural action of 1 ⊗ b for b ∈ R on the R-module ω_{E},

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

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

If d = 1 this is called a Drinfeld A-module over S; compare [Har19, Theorem 2.13].

The case in which C = ℙ^{1}, A = 𝔽_{q}[t], and R is a field was considered by Anderson [And86] under the name abelian t-module. In [Har19, Theorem 2.10] we give a proof the following relative version of Anderson’s theorem [And86, Theorem 1].

Theorem 6.6. If \underline{E} = (E, φ) is an abelian Anderson A-module of rank r and dimension d, we consider in addition on N := M_{q}(\underline{E}) the map τ_{N}^{semi} : m ↦ F_{q,𝔾_{a,R}}∘ m. Since τ_{N}^{semi}(bm) = b^{q}τ_{N}^{semi}(m) for b ∈ R, the map τ_{N}^{semi} is σ_{q}-semilinear and induces an A_{R}-linear map τ_{N}: σ_{q}^{∗}N → N with τ_{N}^{semi}(m) = τ_{N}(σ_{q}^{∗}m). Then \underline{M}_{q}(\underline{E}) := (N, τ_{N}) is an effective A-motive of rank r and dimension d. There is a canonical isomorphism of R-modules

coker τ_{N} ^{∼} ⟶ ω_{E} , m mod τ_{N}(σ_{q}^{∗}N) ⟼ m^{∗}(1) ,  (6.4)

where m^{∗}(1) means the image of 1 ∈ ω_{𝔾_{a,R}} = R under the induced R-homomorphism m^{∗}: ω_{𝔾_{a,R}} → ω_{E}.

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

Example 6.7. Let \underline{E} = (E, φ) be an abelian Anderson A-module over an affine A-scheme c: Spec R → Spec A, and let \underline{N} := \underline{M}_{q}(\underline{E}) be its associated effective A-motive.

(a) Let 𝔞 ⊂ A be a non-zero ideal. By [Har19, Theorem 5.4] the 𝔞-torsion submodule of E, defined as the scheme-theoretic intersection

\underline{E}[𝔞] := ⋂_{a∈𝔞} ker(φ_{a}: E → E) ,

is a finite locally free A/𝔞-module scheme and a strict 𝔽_{q}-module scheme over S, which satisfies \underline{M}_{q}(\underline{E}[𝔞]) = \underline{N}/𝔞\underline{N} and \underline{E}[𝔞] = Dr_{q}(\underline{N}/𝔞\underline{N}).

(b) Let 𝔭 ⊂ A be a maximal ideal and assume that the elements of c^{∗}(𝔭) ⊂ R are nilpotent. Let q̂ be the cardinality of the residue field 𝔽_{𝔭} := A/𝔭 and let f := [𝔽_{𝔭} : 𝔽_{q}]. We fix a uniformizing parameter z ∈ 𝔽_{q}(C) = Frac(A) at 𝔭 and set ζ := c^{∗}(z) ∈ R. We obtain an isomorphism 𝔽_{𝔭}[[z]] ^{∼} ⟶ A_{𝔭} := lim_{⟵} A/𝔭^{n}. As in Example 6.4 the J-adic completion M of \underline{N} together with τ_{M} := τ_{N}^{f}: σ_{q}^{f∗}M → M is an effective local shtuka \underline{M} = (M, τ_{M}) over R of rank r (as in Definition 2.1 with q and 𝔽_{q}[[z]] and σ_{q}^{∗} replaced by q̂ and 𝔽_{𝔭}[[z]] and σ_{q}^{f∗}). By [Har19, Theorem 6.6] the torsion module \underline{E}[𝔭^{n}] is a finite locally free strict 𝔽_{𝔭}-module scheme which satisfies Drq̂(\underline{M}/𝔭^{n}\underline{M}) = \underline{E}[𝔭^{n}] and \underline{M}/𝔭^{n}\underline{M} = \underline{M} q̂(\underline{E}[𝔭^{n}]). Moreover, in the sense of Section 7 below, the inductive limit \underline{E}[𝔭^{∞}] := lim_{⟶} \underline{E}[𝔭^{n}] is a 𝔭-divisible local Anderson module over R which satisfies Drq̂(\underline{M}) = \underline{E}[𝔭^{∞}] and \underline{M} = \underline{M} q̂(\underline{E}[𝔭^{∞}]) under the functors from Theorem 8.3. Note that condition (b) of Definition 6.5 implies that (z − ζ)^{d} = 0 on ω_{\underline{E}[𝔭^{n}]} for every n and on ω_{\underline{E}[𝔭^{∞}]} := lim_{⟵} ω_{\underline{E}[𝔭^{n}]}.

7 Divisible local Anderson modules

The name “divisible local Anderson module” is motivated by Example 6.7(b). These are the function field analogs of p-divisible groups. They were introduced in [Har09], but their definition in [Har09, § 3.1] and the claimed equivalence in [Har09, § 3.2] is false. We give the correct definition below analogously to Messing [Mes72, Chapter I, Definition 2.1]. We fix 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.

Definition 7.1. A z-divisible local Anderson module over a scheme S ∈ Nilp_{𝔽_{q}[[ζ]]} is a sheaf of 𝔽_{q}[[z]]- modules G on the big fppf -site of S such that

(a) G is z-torsion, that is G = lim_{⟶_{n}} G[z^{n}],

(b) G is z-divisible, that is z: G → G is an epimorphism,

(c) For every n the 𝔽_{q}-module G[z^{n}] is representable by a finite locally free strict 𝔽_{q}-module scheme over S (Definition 4.8), and

(d) locally on S there exists an integer d ∈ ℤ_{≥0}, such that (z −ζ)^{d} = 0 on ω_{G} where ω_{G} := lim_{⟵_{n}} ω_{G[z^{n}]}.

We define 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 later in Lemma 8.2 and Theorem 10.7 that ω_{G} is a finite locally free O_{S}-module and we define 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} = 0. Since ω_{G} surjects onto each ω_{G[z^{n}]} because ω_{i_{n}}: ω_{G[z^{n+1}]} ↠ ω_{G[z^{n}]} is an epimorphism, ω_{G} = 0 if and only if all G[z^{n}] are étale, see Lemma 3.8.

A morphism of z-divisible local Anderson modules over S is a morphism of fppf -sheaves of 𝔽_{q}[[z]]- modules.

The category of z-divisible Anderson modules over S is 𝔽_{q}[[z]]-linear and an exact category in the sense of Quillen [Qui73, §2].

Remark 7.2. 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 [HV11, Lemma 5.4].

Remark 7.3 (on axiom (d) in Definition 7.1). Note the following difference 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 (d) is the appropriate substitute for this fact, taking into account Example 6.7. It allows that z − ζ is nilpotent on ω_{G[z^{n}]}. Without axiom (d) the O_{S}-module ω_{G} is not necessarily finite; see Example 7.7 below.

Notation 7.4. Let G be a z-divisible local Anderson module. We denote by i_{n} the inclusion map G[z^{n}] ↪ G[z^{n+1}] and by i_{n,m}: G[z^{n}] → G[z^{m+n}] the composite of the inclusions i_{n+m−1} ∘ … ∘ i_{n}. We denote by j_{n,m} the unique homomorphism G[z^{m+n}] → G[z^{m}] which 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}].

The following two propositions give an alternative characterization of divisible local Anderson modules, which is analogous to Tate’s definition [Tat66] of p-divisible groups.

Proposition 7.5. Let G be a z-divisible local Anderson module.

(a) For any 0 ≤ m, n the following sequence of group schemes over S is exact

0 → G[z^{n}] ⟶^{i_{n,m}} G[z^{m+n}] ⟶^{j_{n,m}} G[z^{m}] → 0.  (7.5)

(b) There is a locally constant function h: S → ℕ_{0}, 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. (a) Since z: G → G is an epimorphism, also j_{n,m} is. The rest of (a) is clear. Let h := rk_{O_{S}} \underline{M}_{q}(G[z]). Then ord G[z] = q^{h} by Theorem 5.2(e). Now (b) follows from (a) and the multiplicativity of the order; see Remark 3.1(c).

Proposition 7.6. Let (G_{n}, i_{n}: G_{n} ↪ G_{n+1})_{n∈ℕ} be an inductive system of 𝔽_{q}[z]-module schemes which are finite locally free strict 𝔽_{q}-module schemes over S such that

(a) i_{n} induces an isomorphism i_{n}: G_{n} ^{∼} ⟶ G_{n+1}[z^{n}],

(b) there is a locally constant function h: S → ℕ_{0} such that ord G_{n} = q^{nh} for all n,

(c) locally on S there exist an integer d ∈ ℤ_{≥0}, such that (z − ζ)^{d} = 0 on ω_{G} where ω_{G} = lim_{⟵} ω_{G_{n}}.

Then G = lim_{⟶} G_{n} is a z-divisible local Anderson module.

Proof. Since i_{n}: G_{n} ↪ G_{n+1} is a monomorphism the maps G_{n}(T) ↪ G_{n+1}(T) are injective for all Sschemes T and we may identify G_{n}(T) with a subset of G(T). From (a) 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 find an m such that x|_{U} ∈ G_{m}(U) by Remark 7.2. Now z^{n}x = 0 implies x|_{U} ∈ G_{m}[z^{n}](U) = G_{n}(U). In total x ∈ G_{n}(T). This shows that G_{n} = G[z^{n}] and G = lim_{⟶} G[z^{n}] is z-torsion.

The quotient G_{n}/G_{1} is a finite locally free group scheme over S by Remark 3.1(d). Its order is q^{(n−1)h} by (b) and the multiplicativity of the order; see Remark 3.1(c). The natural map z: G_{n}/G_{1} ↪ G_{n}[z^{n−1}] ≅ G_{n−1} is a monomorphism and hence a closed immersion by Remark 3.1(a). Thus O_{G_{n−1}} ↠ O_{G_{n}/G_{1}} is an epimorphism of finite locally free O_{S}-modules. It must be an isomorphism because rk_{O_{S}} O_{G_{n}/G_{1}} = ord(G_{n}/G_{1}) = ord(G_{n−1}) = rk_{O_{S}} O_{G_{n−1}} by (b). This proves that z: G_{n} → G_{n−1} 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 find by Remark 7.2 an integer n_{i} such that x|_{U_{i}} ∈ G_{n_{i}}(U_{i}). By the above, there is a y_{i} ∈ G_{n_{i}+1}(U_{i}) ⊂ G(U_{i}) with z · y_{i} = x|_{U_{i}}. This shows that G is z-divisible. By (c) it is a z-divisible local Anderson module.

Note that we require the conditions (d) in Definition 7.1 and (c) in Proposition 7.6 due to the following example which we do not want to consider a z-divisible local Anderson module.

Example 7.7. Let S be the spectrum of a ring R in which ζ is zero, and let G_{n} be the subgroup of 𝔾_{a,S}^{n} = Spec R[x_{1}, …, x_{n}] defined by the ideal (x_{1}^{q}, …, x_{n}^{q}). Make G_{n} into an 𝔽_{q}[[z]]-module scheme by letting z act through

z^{∗}(x_{1}) = 0  and  z^{∗}(x_{ν}) = x_{ν−1} for 1 < ν ≤ n .

Define i_{n}: G_{n} → G_{n+1} as the inclusion of the closed subgroup scheme defined by the ideal (x_{n+1}).

As in Proposition 7.6 one proves that G := lim_{⟶} G_{n} satisfies axioms (a) to (c) of Definition 7.1, but not (d). Here ω_{G_{n}} = ⨁_{i=1}^{n} R · dx_{i} ≅ R^{n}, and so ω_{G} is not a finite R-module. Therefore we cannot drop the conditions (d) in Definition 7.1 and (c) in Proposition 7.6.

In the remainder of this section we introduce truncated z-divisible local Anderson modules.

Lemma 7.8. Let n ∈ ℕ and let G be an fppf -sheaf of 𝔽_{q}[z]-modules over S, such that G = G[z^{n}]. Then the following conditions are equivalent

(a) G is a flat 𝔽_{q}[z]/(z^{n})-module,

(b) ker(z^{n−i}) = im(z^{i}) for i = 0, …, n, that is the morphism z^{i}: G → G[z^{n−i}] is an epimorphism.

Proof. (a)⟹(b). Because of (a), the multiplication with z^{i} induces isomorphisms

𝔽_{q}[z]/(z) ^{∼} ⟶ z^{i}𝔽_{q}[z]/z^{i+1}𝔽_{q}[z]  and  G/zG ^{∼} ⟶ z^{i}G/z^{i+1}G

for i ≤ n − 1. This gives us ker(z^{n−1}) ⊂ im(z), and the opposite inclusion ker(z^{n−1}) ⊃ im(z) follows from G = G[z^{n}]. Now ker(z^{n−i}) ⊂ ker(z^{n−1}) ⊂ im(z) implies that ker(z^{n−i}) = z ker(z^{n−i+1}) = z · z^{i−1}G = z^{i}G by induction on i.

(b)⟹(a). Taking i = 1 implies im(z) = ker(z^{n−1}), and hence multiplication with z^{n−1} induces an isomorphism G/zG ^{∼} ⟶ z^{n−1}G. Since this factors through the epimorphisms G/zG → zG/z^{2}G → ⋯ → z^{n−1}G we see that each of these maps is an isomorphism. Thus we have

gr^{•}(𝔽_{q}[z]/(z^{n})) ⊗_{𝔽_{q}} gr^{0}(G) ⟶^{∼} gr^{•}(G).  (7.6)

Note that the ideal (z) ⊂ 𝔽_{q}[z]/(z^{n}) is nilpotent. Since G/zG is flat over 𝔽_{q}[z]/(z) = 𝔽_{q} [Bou61, Chapter III, § 5.2, Theorem 1] implies that G is a flat 𝔽_{q}[z]/(z^{n})-module.

Definition 7.9. Let d, n ∈ ℕ_{>0}. 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:

(a) If n ≥ 2d it is an 𝔽_{q}[z]/(z^{n})-module scheme G which is finite locally free and strict as 𝔽_{q}-module scheme, such that (z − ζ)^{d} is homotopic to 0 on ℓG/S^{•} and G satisfies the equivalent conditions of Lemma 7.8.

(b) If n < 2d 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 2d.

If G is a z-divisible local Anderson module over S ∈ Nilp_{𝔽_{q}[[ζ]]} with (z − ζ)^{d} = 0 on ω_{G}, we will see in Proposition 9.5 below that G[z^{n}] is a truncated z-divisible local Anderson module with order of nilpotence d and level n. This justifies the name.

8 The local equivalence

The 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 5 to an equivalence between the category of effective local shtukas over S and the category of z-divisible local Anderson modules over S.

For every effective local shtuka \underline{M} = (M, F_{M}) over S we observe \underline{M} = lim_{⟵}(M/z^{n}M, F_{M} mod z^{n}M) and we set

Dr_{q}(\underline{M}) := lim_{⟶_{n}} Dr_{q}(M/z^{n}M, F_{M} mod z^{n}M) .

The action of 𝔽_{q}[[z]] on M makes Dr_{q}(\underline{M}) into an fppf -sheaf of 𝔽_{q}[[z]]-modules on S. Conversely, for every z-divisible local Anderson module G = lim_{⟶} G[z^{n}] over S we set

\underline{M}_{q}(G) = (M_{q}(G), F_{M_{q}(G)}) := lim_{⟵_{n}} (M_{q}(G[z^{n}]), F_{M_{q}(G[z^{n}])}) .

Multiplication with z on G gives M_{q}(G) the structure of an O_{S}[[z]]-module.

Lemma 8.1. Let G = lim_{⟶} G[z^{n}] be a z-divisible local Anderson module of height r over S, see Proposition 7.5, then M_{q}(G) is a locally free sheaf of O_{S}[[z]]-modules of rank r.

Proof. Applying \underline{M}_{q} to the exact sequence 0 → G[z^{n}] ⟶^{i_{n}} G[z^{n+1}] ⟶^{z^{n}} G[z^{n+1}] yields an exact sequence of O_{S}[[z]]-modules

M_{q}(G[z^{n+1}]) ⟶^{z^{n}} M_{q}(G[z^{n+1}]) ⟶^{M_{q}(i_{n})} M_{q}(G[z^{n}]) → 0 .

We deduce from [Bou61, § III.2.11, Proposition 14 and Corollaire 1] that M_{q}(G) is a finitely generated O_{S}[[z]]-module and the canonical map M_{q}(G) → M_{q}(G[z^{n}]) identifies M_{q}(G[z^{n}]) with M_{q}(G)/z^{n}M_{q}(G).

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} = 0 in M_{q}(G[z^{n}]) for all n. To prove the claim consider the factorization

z · id_{M_{q}(G[z^{n+1}])} = M_{q}(j_{1,n}) ∘ M_{q}(i_{n,1}): M_{q}(G[z^{n+1}]) ⟶ M_{q}(G[z^{n+1}])

obtained from Notation 7.4. Theorem 5.2(b) implies that M_{q}(j_{1,n}) is injective, and hence f_{n} = M_{q}(i_{n,1})(f_{n+1}) is zero for all n as desired.

Locally on Spec R ⊂ S the R-module M_{q}(G[z]) is free. By Theorem 5.2(e) its rank is r. Let m_{1}, …, m_{r} be representatives in M_{q}(G) of an R-basis of M_{q}(G[z]) and consider the presentation

0 ⟶ ker α ⟶ ⨁_{i=1}^{r} R[[z]] m_{i} ⟶^{α} M_{q}(G) ⟶ 0 .  (8.7)

Note that α is surjective by Nakayama’s Lemma [Eis95, Corollary 4.8] because z is contained in the radical of R[[z]]. The snake lemma applied to multiplication with z on the sequence (8.7) yields the exact sequence

0 ⟶ coker(z: ker α → ker α) ⟶ ⨁_{i=1}^{r} R m_{i} ^{∼} ⟶ M_{q}(G[z]) → 0

in which the right map is an isomorphism. This implies that multiplication with z^{n} is surjective on ker α for all n, and hence ker α ⊂ ⋂_{n} z^{n} · (⨁_{i=1}^{r} R[[z]] m_{i}) = 0 because R[[z]] is z-adically separated. Therefore M_{q}(G) is locally on S a free O_{S}[[z]]-module of rank r.

Recall from Theorem 5.2(f) that the co-Lie complex ℓG[z^{n}]/S^{•} of G[z^{n}] is canonically isomorphic

to the complex of O_{S}-modules 0 → σ_{q}^{∗}M_{q}(G[z^{n}]) ⟶^{F_{Mq(G[z^{n}])}} M_{q}(G[z^{n}]) → 0. In particular, n_{G[z^{n}]} ≅ ker F_{M_{q}(G[z^{n}])} and ω_{G[z^{n}]} ≅ coker F_{M_{q}(G[z^{n}])} for the O_{S}-modules from Definition 3.7.

Lemma 8.2. Let S ∈ Nilp_{𝔽_{q}[[ζ]]} and let G = lim_{⟶} G[z^{n}] be a z-divisible local Anderson module over S. Then

(a) locally on S there is an N ∈ ℕ such that the morphism i_{n}: G[z^{n}] ↪ G[z^{n+1}] induces an isomorphism ω_{G[z^{n+1}]} ^{∼} ⟶ ω_{G[z^{n}]} for all n ≥ N.

(b) The projective system (n_{G[z^{n}]})_{n} satisfies the Mittag-Leffler condition.

(c) \underline{M}_{q}(G) is an effective local shtuka over S and coker(F_{M_{q}(G)}) is canonically isomorphic to ω_{G}. In particular, ω_{G} is a finite locally free O_{S}-module.

Proof. Working locally on S we may assume that ζ^{N′} = 0 in O_{S} and that (z − ζ)^{d}ω_{G} = 0 for some integers N′ and d. Let N ≥ max{N′, d} be an integer which is a power of p. Then z^{N}ω_{G} = (z^{N} − ζ^{N})ω_{G} = (z − ζ)^{N}ω_{G} = 0.

(a) The closed immersion i_{n}: G[z^{n}] ↪ G[z^{n+1}] induces an epimorphism ω_{i_{n}}: ω_{G[z^{n+1}]} ↠ ω_{G[z^{n}]} and therefore ω_{G} surjects onto each ω_{G[z^{n}]}. This implies that z^{N}ω_{G[z^{n}]} = 0 for all n. Applying Lemma 3.9 to the exact sequence (7.5) for m = 1,

0 ⟶ G[z^{n}] ⟶^{i_{n,1}} G[z^{n+1}] ⟶^{j_{n,1}} G[z] ⟶ 0,

and using i_{1,n} ∘ j_{n,1} = z^{n} id_{G[z^{n+1}]} in

\xymatrix{ω_{G[z]} \ar[r]^{j_{n,1}^{∗}} & ω_{G[z^{n+1}]} \ar[r]^{i_{n,1}^{∗}} & ω_{G[z^{n}]} \ar[r] & 0 \\ ω_{G[z^{n+1}]} \ar@{->>}[u]^{i_{1,n}^{∗}} \ar[ur]_{z^{n}}}

we obtain that ker(ω_{G[z^{n+1}]} ↠ ω_{G[z^{n}]}) = z^{n}ω_{G[z^{n+1}]}. Therefore ω_{G[z^{n+1}]} ^{∼} ⟶ ω_{G[z^{n}]} is an isomorphism for all n ≥ N.

To prove (b) we fix an n ≥ N. We abbreviate the O_{S}-modules M_{q}(G[z^{k}]) by M_{k} and the map F_{M_{q}(G[z^{k}])} by F_{k}. From Proposition 7.5 and Theorem 5.2(b) we have an exact sequence

0 → M_{k} ⟶^{M_{q}(j_{n,k})} M_{n+k} ⟶^{M_{q}(i_{n,k})} M_{n} → 0 .

It remains exact after applying σ_{q}^{∗} because M_{n} is locally free. For all k, we consider the commutative diagrams

\xymatrix{0 \ar[r] & ker F_{n+k} \ar[r] \ar[d]^{ρ_{k}} & σ_{q}^{∗}M_{n+k} \ar[r]^{F′_{n+k}} \ar@{->>}[d]^{σ_{q}^{∗}M_{q}(i_{n,k})} & im F_{n+k} \ar[r] \ar@{->>}[d] & 0 & 0 \ar[r] & im F_{n+k} \ar[r]^{F′′_{n+k}} \ar@{->>}[d] & M_{n+k} \ar[r] \ar@{->>}[d]^{M_{q}(i_{n,k})} & coker F_{n+k} \ar[r] \ar[d]^{≅} & 0 \\ 0 \ar[r] & ker F_{n} \ar[r] & σ_{q}^{∗}M_{n} \ar[r]_{F′_{n}} & im F_{n} \ar[r] & 0 & 0 \ar[r] & im F_{n} \ar[r]_{F′′_{n}} & M_{n} \ar[r] & coker F_{n} \ar[r] & 0}

where we have split F_{n} = F′′_{n} ∘ F′_{n} with F′_{n} surjective and F′′_{n} injective, and where the vertical map on the right in the second diagram is an isomorphism by the identification coker F_{n} = ω_{G[z^{n}]} from Theorem 5.2(f) and by what we proved in (a) above. We denote the vertical map on the left in the first diagram by ρ_{k}. The snake lemma applied to both diagrams yields the following exact sequence

σ_{q}^{∗}M_{k} ⟶^{F_{k}} M_{k} ⟶ coker ρ_{k} ⟶ 0.

Therefore coker ρ_{k} ≅ coker F_{k} = ω_{G[z^{k}]}. In the diagram

[figure: 0; im ρ_{k+1}; ker F_{n}; coker ρ_{k+1}; ≅; 0; 0; im ρ_{k}; ker F_{n}; coker ρ_{k}; 0]

the vertical map on the right is an isomorphism for k ≥ N by what we have proved in (a) above. Therefore the image of ρ_{k} stabilizes for k ≥ N, that is n_{G[z^{n}]} = ker F_{n} satisfies the Mittag-Leffler condition. Note that also (im F_{n})_{n} satisfies the Mittag-Leffler condition. We will use this for proving (c).

(c) We still abbreviate M_{q}(G[z^{n}]) by M_{n} and F_{M_{q}(G[z^{n}])} by F_{n}. The maps F_{n}: σ_{q}^{∗}M_{n} → M_{n} give us two short exact sequences of projective systems

0 ⟶ ker F_{n} ⟶ σ_{q}^{∗}M_{n} ⟶ im F_{n} ⟶ 0  and  0 ⟶ im F_{n} ⟶ M_{n} ⟶ coker F_{n} ⟶ 0 .

Taking the projective limit, using the Mittag-Leffler conditions via [Har77, Proposition II.9.1(b)], the isomorphism σ_{q}^{∗}(M_{q}(G)) ≅ lim_{⟵} σ_{q}^{∗}(M_{n}) which is due to the flatness of M_{q}(G) over O_{S}, and combining both exact sequences we obtain an exact sequence

0 ⟶ lim_{⟵_{n}} ker F_{n} ⟶ σ_{q}^{∗}(M_{q}(G)) ⟶^{F_{Mq(G)}} M_{q}(G) ⟶ lim_{⟵_{n}} coker F_{n} ⟶ 0 .

This shows that ω_{G} := lim_{⟵} ω_{G[z^{n}]} = lim_{⟵} coker F_{n} = coker F_{M_{q}(G)}, which is finite locally free over O_{S} by

Lemma 2.3. Furthermore, condition (d) of Definition 7.1 implies that (z−ζ)^{d} annihilates coker F_{M_{q}(G)}. This proves that the map F_{M_{q}(G)} : σ_{q}^{∗}(M_{q}(G))[\frac{1}{z−ζ}] → M_{q}(G)[\frac{1}{z−ζ}] is surjective. As both modules are locally free over O_{S}[[z]][\frac{1}{z−ζ}] of the same rank, the map is an isomorphism. Thus \underline{M}_{q}(G) is an effective local shtuka.

We can now prove the following theorem. It generalizes [And93, § 3.4], who treated the case of formal 𝔽_{q}[[z]]-modules, which we state in (c).

Theorem 8.3. Let S ∈ Nilp_{𝔽_{q}[[ζ]]}.

(a) The two contravariant functors Dr_{q} and \underline{M}_{q} are mutually quasi-inverse anti-equivalences between the category of effective local shtukas over S and the category of z-divisible local Anderson modules over S.

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

Let \underline{M} = (M, F_{M}) be an effective local shtuka over S and let G = Dr_{q}(\underline{M}) be its associated z-divisible local Anderson module. Then

(c) G is a formal 𝔽_{q}[[z]]-module (Definition 1.1) if and only if F_{M} is topologically nilpotent.

(d) the height (see Proposition 7.5) and dimension of G are equal to the rank and dimension of \underline{M}.

(e) the O_{S}[[z]]-modules ω_{Dr_{q}(\underline{M})} and coker F_{M} are canonically isomorphic.

(f) if \underline{M} is bounded by (d, 0, …, 0) for an integer d ≥ 0, then dim G = d is constant and (z−ζ)^{d}·ω_{G} = (0) globally on S, but the converse is false in general.

Proof. (a) We already saw in Lemma 8.2(c) that \underline{M}_{q} sends z-divisible local Anderson modules to effective local shtukas. To prove the converse we use Proposition 7.6. Let \underline{M} = (M, F_{M}) be an effective local shtuka over S and abbreviate \underline{M}/z^{n}\underline{M} =: \underline{M}_{n} = (M_{n}, F_{M_{n}}) and G_{n} := Dr_{q}(\underline{M}_{n}). Then G := Dr_{q}(\underline{M}) = lim_{⟶} G_{n}. Consider the locally constant function h := rk_{O_{S}[[z]]} M on S. It satisfies rk_{O_{S}}(M_{n}) = nh. By Theorem 5.2 the G_{n} are finite locally free strict 𝔽_{q}-module schemes over S of order q^{nh}, and the exact sequence of finite 𝔽_{q}-shtukas \underline{M}_{n+1} ⟶^{z^{n}} \underline{M}_{n+1} ⟶ \underline{M}_{n} ⟶ 0 yields an exact sequence of group schemes 0 ⟶ G_{n} ⟶ G_{n+1} ⟶^{z^{n}} G_{n+1}. This implies that G_{n} = ker(z^{n}: G_{n+1} → G_{n+1}) =: G_{n+1}[z^{n}]. By Lemma 2.3 we know that locally on S there exist positive integers e′, N such that (z − ζ)^{e′} = 0 on coker F_{M} and z^{N} = 0 on coker F_{M}. Applying the snake lemma to the diagram

\xymatrix{0 \ar[r] & σ_{q}^{∗}M \ar[r]^{F_{M}} \ar[d]^{z^{n}} & M \ar[r] \ar[d]^{z^{n}} & coker F_{M} \ar[r] \ar[d]^{z^{n}} & 0 \\ 0 \ar[r] & σ_{q}^{∗}M \ar[r]^{F_{M}} & M \ar[r] & coker F_{M} \ar[r] & 0}

shows that coker F_{M} → coker F_{M_{n}} is an isomorphism for n ≥ N. Therefore by Theorem 5.2(f)

ω_{G} := lim_{⟵} ω_{G_{n}} = lim_{⟵} coker(F_{M_{n}}) = coker F_{M} .

This establishes (e) and implies (z − ζ)^{e′} = 0 on ω_{G}. Therefore G = lim_{⟶} G_{n} is a z-divisible local Anderson module by Proposition 7.6. By Theorem 5.2 the functors Dr_{q} and \underline{M}_{q} are quasi-inverse to each other. This proves (a).

(d) From our proof above, the height of Dr_{q}(\underline{M}) equals the rank of \underline{M}. The equality of dimensions follows from (e).

(b) The 𝔽_{q}[[z]]-linearity of the functors is clear by construction. From (e) it follows that both functors Dr_{q} and \underline{M}_{q} preserve (ind-)étale objects. To prove the exactness of Dr_{q} let 0 → \underline{M}′′ → \underline{M} → \underline{M}′ → 0 be a short exact sequence of effective local shtukas. Modulo z^{n} it yields a short exact sequence of finite 𝔽_{q}-shtukas 0 → \underline{M}′′_{n} → \underline{M}_{n} → \underline{M}′_{n} → 0, where \underline{M}′′_{n} := \underline{M}′′/z^{n}\underline{M}′′, etc. Theorem 5.2 produces the exact sequence 0 → G′[z^{n}] → G[z^{n}] → G′′[z^{n}] → 0, where G = Dr_{q}(\underline{M}), G′ = Dr_{q}(\underline{M}′), G′′ = Dr_{q}(\underline{M}′′). This implies that 0 → G′ → G → G′′ → 0 is exact, because taking direct limits in the category of sheaves is an exact functor.

Conversely let 0 → G′ → G → G′′ → 0 be a short exact sequence of z-divisible local Anderson modules. Since multiplication with z^{n} is an epimorphism on G′′, the snake lemma yields the exact sequence of finite locally free strict 𝔽_{q}-module schemes 0 → G′[z^{n}] → G[z^{n}] → G′′[z^{n}] → 0. Theorem 5.2 implies that the sequence 0 → \underline{M}′′_{n} → \underline{M}_{n} → \underline{M}′_{n} → 0 is exact, where \underline{M} = \underline{M}_{q}(G), \underline{M}′ = \underline{M}_{q}(G′), \underline{M}′′ = \underline{M}_{q}(G′′). Since \{\underline{M}′′_{n}\} satisfies the Mittag-Leffler condition we obtain the exactness of 0 → \underline{M}′′ → \underline{M} → \underline{M}′ → 0.

(c) Let G = Dr_{q}(\underline{M}). In Proposition 10.11 below we will see that G is a formal 𝔽_{q}[[z]]-module if and only if G[z] = ker(z: G → G) is radicial, which by Theorem 5.2(d) is equivalent to F_{M_{1}} = (F_{M} mod z) being nilpotent locally on S. The latter is the case if and only if locally on S there is an integer n such that (F_{M})^{n} ≡ 0 mod z, that is, if and only if F_{M} is topologically nilpotent.

(f) If \underline{M} is bounded by (d, 0, …, 0) then (z − ζ)^{d} annihilates coker F_{M} = ω_{G}. The dimension of G can be computed by pullback to the closed points s: Spec k → S. There M ⊗_{O_{S}} k ≅ k[[z]]^{rk \underline{M}} ≅ σ_{q}^{∗}M ⊗_{O_{S}} k and ζ = 0 in k. The elementary divisor theorem implies (dim G)(s) := dim_{k} s^{∗}ω_{G} = dim_{k}(coker F_{M}) ⊗_{O_{S}} k = ord_{z} det(F_{M} ⊗_{O_{S}} k) = d by definition of boundedness of \underline{M} by (d, 0, …, 0). That the converse fails can be seen from the following Example 8.4. Let R = k[ε]/(ε^{2}) for a field k. Then the local shtuka \underline{M} = (R[[z]]^{2}, F_{M} = ( \begin{matrix}z−ζ & 0 \\ 0 & z−ζ−ε \end{matrix} )) satisfies (z − ζ)^{2} = 0 on coker F_{M} = R^{2}. So its associated z-divisible local Anderson module G = Dr_{q}(\underline{M}) satisfies the conclusion of Theorem 8.3(f). But (∧^{2}F_{M})(1) = (z−ζ)^{2}−ε(z−ζ) ∉ (z−ζ)^{2}∧^{2}M, and hence \underline{M} is not bounded by (2, 0). This also shows that [HV11, Example 4.5] is false.

Corollary 8.5. Let S ∈ Nilp_{𝔽_{q}[[ζ]]} and let f: G′ → G be a monomorphism of z-divisible local Anderson modules over S. Then the quotient sheaf G/G′ is a z-divisible local Anderson module over S.

Proof. Since the question is local on S we may assume that S = Spec R is affine. For all n the induced map G′[z^{n}] → G[z^{n}] is a monomorphism, hence a closed immersion by Remark 3.1(a). By Lemma 4.7 it is strict 𝔽_{q}-linear in the sense of Faltings [Fal02, Definition 1], and by [Fal02, Proposition 2] the cokernel G′′_{n} := G[z^{n}]/G′[z^{n}] is a strict 𝔽_{q}-module scheme which is finite locally free by Remark 3.1(d). By Theorem 5.2 this induces the exact sequence of finite 𝔽_{q}-shtukas 0 → \underline{M}_{q}(G′′_{n}) → \underline{M}_{q}(G[z^{n}]) → \underline{M}_{q}(G′[z^{n}]) → 0. In the following diagram

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & G′[z^{n}] \ar[r]^{i′_{n,m}} \ar[d] & G′[z^{n+m}] \ar[r]^{j′_{n,m}} \ar[d] & G′[z^{m}] \ar[r] \ar[d] & 0 \\ 0 \ar[r] & G[z^{n}] \ar[r]^{i_{n,m}} \ar[d] & G[z^{n+m}] \ar[r]^{j_{n,m}} \ar[d] & G[z^{m}] \ar[r] \ar[d] & 0 \\ 0 \ar[r] & G′′_{n} \ar[r]^{i′′_{n,m}} \ar[d] & G′′_{n+m} \ar[r]^{j′′_{n,m}} \ar[d] & G′′_{m} \ar[r] \ar[d] & 0 \\  & 0 & 0 & 0}

the columns are exact by definition of G′′_{n}, and the two upper rows are exact by Proposition 7.5. By the snake lemma this defines the exact sequence in the bottom row. By Theorem 5.2 this implies that \underline{M}_{q}(i′′_{n,1}): \underline{M}_{q}(G′′_{n+1}) → \underline{M}_{q}(G′′_{n}) is surjective for all n. In particular, the projective system \underline{M}_{q}(G′′_{n}) satisfies the Mittag-Leffler condition, and the morphism \underline{M}_{q}(f): \underline{M} := \underline{M}_{q}(G) → \underline{M}′ := \underline{M}_{q}(G′) of effective local shtukas corresponding to f by Theorem 8.3 is surjective by [Har77, Proposition II.9.1(b)]. The kernel \underline{M}′′ := ker \underline{M}_{q}(f) = lim_{⟵} \underline{M}_{q}(G′′_{n}) is a locally free R[[z]]-module with a

morphism F_{M′′}: σ_{q}^{∗}M′′ → M′′ inducing an isomorphism F_{M′′}: σ_{q}^{∗}M′′[\frac{1}{z−ζ}] → M′′[\frac{1}{z−ζ}], because this is true for \underline{M} and \underline{M}′. Thus \underline{M}′′ is an effective local shtuka over S. Applying the snake lemma to the (injective) multiplication with z^{n} on the sequence 0 → \underline{M}′′ → \underline{M} → \underline{M}′ → 0 shows that \underline{M}′′/z^{n}\underline{M}′′ = \underline{M}_{q}(G′′_{n}). Therefore, Theorem 8.3 implies that G/G′ = Dr_{q}(\underline{M}′′) = lim_{⟶} G′′_{n} is a zdivisible local Anderson module over S.

9 Frobenius, Verschiebung and deformations of local shtukas

Definition 9.1. Let G be an fppf -sheaf of groups over an 𝔽_{q}-scheme S. For n ∈ ℕ_{0} we let G[F_{q}^{n}] be the kernel of the relative q^{n}-Frobenius F_{q^{n},G}: G → σ_{q^{n}}^{∗}G of G over S. In particular, G[F_{q}^{0}] := ker( id_{G}) = (0).

Let S ∈ Nilp_{𝔽_{q}[[ζ]]}. Later we will assume that ζ = 0 in O_{S}. Let G be a z-divisible local Anderson module over S and let \underline{M} = (M, F_{M}) = \underline{M}_{q}(G) be its associated local shtuka from Theorem 8.3. Then the q-Frobenius morphism F_{q,G} := lim_{⟶} F_{q,G[z^{n}]} : G → σ_{q}^{∗}G corresponds by diagram (5.3) to the morphism

\underline{M}_{q}(F_{q,G}) = F_{M}: M_{q}(σ_{q}^{∗}G) = σ_{q}^{∗}M_{q}(G) ⟶ M_{q}(G), m ↦ m ∘ F_{q,G} = F_{M}(m) .

In addition to the q-Frobenius, G carries a q-Verschiebung which is identically zero by Theorem 5.1. Therefore, if ζ = 0 in O_{S} we will introduce a “z^{d}-Verschiebung” in Remark 9.3 and Corollary 9.4 below, which is more useful for z-divisible local Anderson modules. We begin with the following Lemma 9.2. Let M be an effective local shtuka with (z − ζ)^{d} = 0 on coker F_{M}. Then there exists a uniquely determined homomorphism of O_{S}[[z]]-modules V_{M}: M → σ_{q}^{∗}M with F_{M} ∘V_{M} = (z −ζ)^{d} · id_{M} and V_{M} ∘ F_{M} = (z − ζ)^{d} · id_{σ_{q}^{∗}M}.

Proof. Since F_{M} is injective by Lemma 2.3 and (z − ζ)^{d} = 0 on coker F_{M}, the lemma follows from the following diagram

\xymatrix{0 \ar[r] & σ_{q}^{∗}M \ar[r]^{F_{M}} \ar[d]_{(z−ζ)^{d}} & M \ar[r] \ar[d]^{(z−ζ)^{d}} \ar@{-->}[dl]_{V_{M}} & coker F_{M} \ar[r] \ar[d]^{(z−ζ)^{d} = 0} & 0 \\ 0 \ar[r] & σ_{q}^{∗}M \ar[r]_{F_{M}} & M \ar[r] & coker F_{M} \ar[r] & 0 .}  (9.8)

Remark 9.3. If ζ = 0 in O_{S}, the Frobenius f := F_{M}: σ_{q}^{∗}\underline{M}[\frac{1}{z}] ^{∼} ⟶ \underline{M}[\frac{1}{z}] satisfies F_{M} ∘ σ_{q}^{∗}f = F_{M} ∘ σ_{q}^{∗}F_{M} = f ∘ σ_{q}^{∗}F_{M}, and hence is a quasi-isogeny between the local shtukas σ_{q}^{∗}\underline{M} = (σ_{q}^{∗}M, σ_{q}^{∗}F_{M}) and \underline{M}. Likewise, if \underline{M} is effective with (z − ζ)^{d} = 0 on coker F_{M}, the homomorphism V_{M} from Lemma 9.2 is an isogeny V_{z^{d},\underline{M}} := V_{M}: \underline{M}[\frac{1}{z}] ^{∼} ⟶ σ_{q}^{∗}\underline{M}[\frac{1}{z}], called the z^{d}-Verschiebung of \underline{M}. It satisfies F_{M} ∘ V_{z^{d},\underline{M}} = z^{d} · id_{\underline{M}} and V_{z^{d},\underline{M}} ∘ F_{M} = z^{d} · id_{σ_{q}^{∗}\underline{M}}. Indeed, ζ = 0 = ζ^{q} implies that the following diagram is commutative

\xymatrix{σ_{q}^{∗}M \ar[r]^{σ_{q}^{∗}V_{z^{d},\underline{M}}} \ar[d]^{F_{M}} & σ_{q}^{∗}(σ_{q}^{∗}M) \ar[d]^{σ_{q}^{∗}F_{M} = F_{σ^{∗}_{q}M}} \\ M \ar[r]^{V_{z^{d},\underline{M}}} & σ_{q}^{∗}M ,}

as F_{σ_{q}^{∗}M} ∘ σ_{q}^{∗}V_{z^{d},\underline{M}} = σ_{q}^{∗}F_{M} ∘ σ_{q}^{∗}V_{z^{d},\underline{M}} = σ_{q}^{∗}((z − ζ)^{d} · id_{M}) = (z − ζ^{q})^{d} · id_{σ_{q}^{∗}M} = V_{z^{d},\underline{M}} ∘ F_{M}.

Corollary 9.4. Assume that ζ = 0 in O_{S}. Let G be a z-divisible local Anderson module over S with (z − ζ)^{d} = 0 on ω_{G}. Then there is a uniquely determined morphism V_{z^{d},G}: σ_{q}^{∗}G → G with F_{q,G}∘V_{z^{d},G} = z^{d}· id_{σ_{q}^{∗}G} and V_{z^{d},G}∘F_{q,G} = z^{d}· id_{G}. It is called the z^{d}-Verschiebung of G. In particular, G[F_{q}^{n}] := ker(F_{q^{n},G}: G → σ_{q^{n}}^{∗}G) is contained in G[z^{nd}] and ker(V_{z^{d},G}^{n}: σ_{q^{n}}^{∗}G → G) ⊂ σ_{q^{n}}^{∗}G[z^{nd}] for all n.

Proof. Let \underline{M} = \underline{M}_{q}(G) be the effective local shtuka associated with G. Since (z − ζ)^{d} = 0 on ω_{G} = coker F_{M}, the z^{d}-Verschiebung V_{z^{d},\underline{M}} of \underline{M} from Remark 9.3 corresponds by Theorem 8.3 to a morphism V_{z^{d},G} := Dr_{q}(V_{z^{d},\underline{M}}): σ_{q}^{∗}G → G with F_{q,G} ∘ V_{z^{d},G} = z^{d} · id_{σ_{q}^{∗}G} and V_{z^{d},G} ∘ F_{q,G} = z^{d} · id_{G}, and hence V_{z^{d},G}^{n} ∘ F_{q,G}^{n} = z^{nd} · id_{G} and F_{q,G}^{n} ∘ V_{z^{d},G}^{n} = z^{nd} · id_{σ_{q^{n}}^{∗}G}. This proves the corollary.

Proposition 9.5. Let G be a z-divisible local Anderson module with (z − ζ)^{d} = 0 on ω_{G}, and let n ∈ ℕ. Then 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 Definition 7.9.

Proof. The equivalent conditions of Lemma 7.8 for the 𝔽_{q}[z]/(z^{n})-module scheme G[z^{n}] follow from Proposition 7.5 by considering for all ν = 0, …, n the commutative diagram

[figure: 0; G[z^{n−ν}]; i_{n−ν,ν}; G[z^{n}]; j_{n−ν,ν}; z^{n−ν}; G[z^{ν}]; i_{ν,n−ν}; 0; G[z^{n}]; j_{ν,n−ν}; z^{ν}; G[z^{n}]]

in which i_{ν,n−ν} is a monomorphism and j_{ν,n−ν} an epimorphism, and hence ker(z^{n−ν}) = ker(j_{n−ν,ν}) = im(i_{n−ν,ν}) = im(z^{ν}). By Theorem 8.3(e), (z − ζ)^{d} = 0 on coker F_{M_{q}(G)}. We reduce the map V_{M_{q}(G)} from Lemma 9.2 modulo z^{n} to obtain a homomorphism V_{M_{q}(G[z^{n}])} : M_{q}(G[z^{n}]) → σ_{q}^{∗}M_{q}(G[z^{n}]) with F_{M_{q}(G[z^{n}])}∘V_{M_{q}(G[z^{n}])} = (z−ζ)^{d}· id_{M_{q}(G[z^{n}])} and V_{M_{q}(G[z^{n}])}∘F_{M_{q}(G[z^{n}])} = (z−ζ)^{d}· id_{σ_{q}^{∗}M_{q}(G[z^{n}])}. Under

the identification of the co-Lie complex ℓG[z^{n}]/S^{•} with 0 → σ_{q}^{∗}M_{q}(G[z^{n}]) ⟶^{F_{Mq(G[z^{n}])}} M_{q}(G[z^{n}]) → 0 from Theorem 5.2(f) the map V_{M_{q}(G[z^{n}])} corresponds to a homotopy h: t_{G[z^{n}]}^{∗} → N_{G[z^{n}]} with dh = (z − ζ)^{d} on t_{G[z^{n}]}^{∗} and hd = (z − ζ)^{d} on N_{G[z^{n}]}. This means that (z − ζ)^{d} is homotopic to zero on ℓG[z^{n}]/S^{•}.

Proposition 9.6. Assume that ζ = 0 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. Then

(a) there exists a morphism V_{z^{d},G}: σ_{q}^{∗}G → G with F_{q,G}∘V_{z^{d},G} = z^{d}· id_{σ_{q}^{∗}G} and V_{z^{d},G}∘F_{q,G} = z^{d}· id_{G}. It is not uniquely determined, unless G is étale.

(b) G[F_{q}^{i}] ⊂ G[z^{id}] and ker V_{z^{d},G}^{i} ⊂ σ_{q^{i}}^{∗}G[z^{id}] for all i.

Now let n ∈ ℕ_{>0} and l = nd. In particular, if n = 1 there is a truncated divisible local Anderson module G̃ of level 2d with G = G̃[z^{d}] and we assume that V_{z^{d},G} = V_{z^{d},G̃}|_{σ_{q}^{∗}G}. Then

(c) for all i with 0 ≤ i ≤ n the morphism F_{q,G}^{i}: G[F_{q}^{n}] → σ_{q^{i}}^{∗}G[F_{q}^{n−i}] is an epimorphism,

(d) the morphisms V_{z^{d},G}^{n}: σ_{q^{n}}^{∗}G → ker F_{q,G}^{n} and F_{q,G}^{n}: G → ker V_{z^{d},G}^{n} are epimorphisms,

(e) G[F_{q}^{i}] and ker V_{z^{d},G}^{i} are finite locally free strict 𝔽_{q}-module schemes over S for all 0 ≤ i ≤ n,

(f) for all 0 ≤ i ≤ n we have ω_{G} = ω_{G[z^{id}]} = ω_{G[F_{q}]} and this is a finite locally free O_{S}-module.

Proof. (a) Let h: t_{G}^{∗} → N_{G} be a homotopy with dh = (z − ζ)^{d} on t_{G}^{∗} and hd = (z − ζ)^{d} on N_{G}. Note that h is determined only up to adding a homomorphism t_{G}^{∗} ↠ coker d = ω_{G} → n_{G} = ker d ↪ N_{G}, and in particular, is not unique unless G is étale. Let V : M_{q}(G) → σ_{q}^{∗}M_{q}(G) be the homomorphism

which corresponds to h under the identification of the co-Lie complex ℓG/S^{•} with 0 → σ_{q}^{∗}M_{q}(G) ⟶^{F_{Mq(G)}} M_{q}(G) → 0 from Theorem 5.2(f). Then V ∘ F_{M_{q}(G)} = z^{d} · id_{σ_{q}^{∗}M_{q}(G)} = σ_{q}^{∗}(z^{d} · id_{M_{q}(G)}) = σ_{q}^{∗}(F_{M_{q}(G)} ∘ V) = F_{σ_{q}^{∗}M_{q}(G)} ∘ σ_{q}^{∗}V implies that V : \underline{M}_{q}(G) → σ_{q}^{∗}\underline{M}_{q}(G) = \underline{M}_{q}(σ_{q}^{∗}G) is a morphism of finite 𝔽_{q}- shtukas. It induces the desired morphism V_{z^{d},G} := Dr_{q}(V): σ_{q}^{∗}G → G with F_{q,G} ∘ V_{z^{d},G} = z^{d} · id_{σ_{q}^{∗}G} and V_{z^{d},G} ∘ F_{q,G} = z^{d} · id_{G}.

(b) follows from V_{z^{d},G}^{i} ∘ F_{q,G}^{i} = z^{id} · id_{G} and F_{q,G}^{i} ∘ V_{z^{d},G}^{i} = z^{id} · id_{σ_{q^{i}}^{∗}G} which are consequences of (a).

(c) is trivial if n = 1 and i = 0 or 1. If n ≥ 2 there is by (a) a factorization F_{q,G}^{i} ∘ V_{z^{d},G}^{i} = z^{id}: σ_{q^{i}}^{∗}G → σ_{q^{i}}^{∗}G. Since the morphism z^{id}: σ_{q^{i}}^{∗}G → σ_{q^{i}}^{∗}G[z^{(n−i)d}] is an epimorphism by Lemma 7.8, and since σ_{q^{i}}^{∗}G[F_{q}^{n−i}] ⊂ σ_{q^{i}}^{∗}G[z^{(n−i)d}] by (b), we obtain (c).

(d) is proved by induction on n. For n = 1 we use G = G̃[z^{d}]. By Lemma 7.8 there is an exact sequence

0 ⟶ G̃[z^{d}] ⟶ G̃ ⟶^{z^{d}} G̃[z^{d}] ⟶ 0 .

Since G[F_{q}] ⊂ G̃[F_{q}] ⊂ G̃[z^{d}] by (b), the map V_{z^{d},G̃} ∘ F_{q,G̃} = z^{d}: (z^{d})^{−1}(G[F_{q}]) → G[F_{q}] is an epimorphism. From F_{q,G̃}: (z^{d})^{−1}(G[F_{q}]) → σ_{q}^{∗}G̃[z^{d}] = σ_{q}^{∗}G we see that V_{z^{d},G} = V_{z^{d},G̃}|_{σ_{q}^{∗}G}: σ_{q}^{∗}G → G[F_{q}] is an epimorphism. The statement for F_{q,G} is proved in the analogous way using ker V_{z^{d},G} ⊂ ker V_{z^{d},G̃} ⊂ σ_{q}^{∗}G̃[z^{d}]. Thus we have proved (d) for the case n = 1.

To prove it in general by induction on n, consider the diagram

[figure: σ_{q^{n}}^{∗}G[z^{nd}]; V_{z^{d},G}^{n}; z^{d}; G[F_{q}^{n}]; F_{q,G}; σ_{q^{n}}^{∗}G[z^{(n−1)d}]; σ_{q}^{∗}V_{z^{d},G}^{n−1}; (σ_{q}^{∗}G[z^{(n−1)d}])[F_{q}^{n−1}]; σ_{q}^{∗}G[F_{q}^{n−1}] .]

In the bottom row, σ_{q}^{∗}V_{z^{d},G}^{n−1} is an epimorphism by the induction hypothesis, and the equality comes from (b). The vertical map on the left is an epimorphism by Lemma 7.8, and therefore F_{q,G} ∘ V_{z^{d},G}^{n} is an epimorphism. Thus if we can show that ker(F_{q,G}) = G[F_{q}] is contained in the image of V_{z^{d},G}^{n}, it will follow that V_{z^{d},G}^{n} is an epimorphism. But by the case n = 1 settled above

G[F_{q}] = V_{z^{d},G}(σ_{q}^{∗}G[z^{d}]) = V_{z^{d},G} ∘ z^{(n−1)d}(σ_{q}^{∗}G[z^{nd}]) = V_{z^{d},G}^{n} ∘ F_{q,G}^{n−1}(σ_{q}^{∗}G[z^{nd}]) ⊂ V_{z^{d},G}^{n}(σ_{q^{n}}^{∗}G[z^{nd}]).

This proves that V_{z^{d},G}^{n} is an epimorphism. The statement for F_{q,G}^{n} is proved in the same way.

(e) The morphisms F_{q,G}^{i}: G → σ_{q^{i}}^{∗}G and V_{z^{d},G}^{i}: σ_{q^{i}}^{∗}G → G between group schemes of finite presentation over S are themselves of finite presentation by [EGA, IV_{1}, Proposition 1.6.2(v)]. Therefore G[F_{q}^{i}] := ker F_{q,G}^{i} and ker V_{z^{d},G}^{i} are of finite presentation over S by [EGA, IV_{1}, Proposition 1.6.2(iii)]. As closed subschemes of G, respectively σ_{q^{i}}^{∗}G, they are also finite over S. Since in (d) we proved that V_{z^{d},G}^{i}: σ_{q^{i}}^{∗}G[z^{id}] ↠ (G[z^{id}])[F_{q}^{i}] = G[F_{q}^{i}] and F_{q,G}^{i}: G[z^{id}] ↠ ker V_{z^{d},G[z^{id}]}^{i} = ker V_{z^{d},G}^{i} are epimorphisms, they are faithfully flat by Remark 3.1(b). Therefore G[F_{q}^{i}] and ker V_{z^{d},G}^{i} are flat over S by [EGA, IV_{3}, Corollaire 11.3.11], and hence finite locally free. Over any affine open U ⊂ S the 𝔽_{q}-equivariant morphisms F_{q,G}^{i} and V_{z^{d},G}^{i} lift by Lemma 4.7 to morphisms in DGr(𝔽_{q})_{U}. Thus they are 𝔽_{q}-strict morphisms in the sense of Faltings [Fal02, Definition 1]. By [Fal02, Proposition 2] their kernels G[F_{q}^{i}] ×_{S} U and ker(V_{z^{d},G}^{i}) ×_{S} U are strict 𝔽_{q}-module schemes over U. So the 𝔽_{q}-strictness of G[F_{q}^{i}] and ker V_{z^{d},G}^{i} over all of S follows from Lemma 4.9.

(f) For any group scheme G = Spec R[X_{1}, …, X_{r}]/I of finite type over Spec R, we compute G[F_{q}] = Spec R[X_{1}, …, X_{r}]/(I, X_{1}^{q}, …, X_{r}^{q}). By the conormal sequence [Har77, Proposition II.8.12] for the closed immersion G[F_{q}] ⊂ G this implies ω_{G} = ω_{G[F_{q}]}. The inclusion G[F_{q}] ⊂ G[z^{d}] from (b) therefore implies G[F_{q}] = (G[z^{id}])[F_{q}], and hence ω_{G} = ω_{G[z^{id}]} = ω_{G[F_{q}]} for all i. Moreover, since G[F_{q}] is a finite locally free strict 𝔽_{q}-module scheme over S by (e), we can compute ω_{G[F_{q}]} as coker F_{M_{q}(G[F_{q}])} where (M_{q}(G[F_{q}]), F_{M_{q}(G[F_{q}])}) is the associated finite 𝔽_{q}-shtuka from Theorem 5.2. In particular, F_{M_{q}(G[F_{q}])} = \underline{M}_{q}(F_{q,G[F_{q}]}) = 0 and this implies that coker F_{M_{q}(G[F_{q}])} = M_{q}(G[F_{q}]) is a finite locally free O_{S}-module.

In the remainder of this section we will show that to lift a z-divisible local Anderson-module is equivalent to lifting its “Hodge filtration”. Let S ∈ Nilp_{𝔽_{q}[[ζ]]} and let G be a z-divisible local Anderson-module over S satisfying (z − ζ)^{d} · ω_{G} = 0. Let (M, F_{M}) be its effective local shtuka. Then (z − ζ)^{d} · coker F_{M} = 0 and we consider the map V_{M} from Lemma 9.2. The injective morphism F_{M} induces by diagram (9.8) an exact sequence of O_{S}[[z]]-modules

0 ⟶ coker V_{M} ⟶^{F_{M}} M/(z − ζ)^{d}M ⟶ coker F_{M} ⟶ 0 .

In particular coker V_{M} is a locally free O_{S}-module of finite rank. Conversely, V_{M} induces the exact sequence of O_{S}[[z]]-modules

0 ⟶ coker F_{M} ⟶^{V_{M}} σ_{q}^{∗}M/(z − ζ)^{d}σ_{q}^{∗}M ⟶ coker V_{M} ⟶ 0 .  (9.9)

Definition 9.7 (compare [HJ19, § 5.7]). We call H(G) := H_{dR}^{1}(G, O_{S}[z]/(z−ζ)^{d}) := σ_{q}^{∗}M/(z−ζ)^{d}σ_{q}^{∗}M the de Rham cohomology of G with coefficients in O_{S}[z]/(z − ζ)^{d}. It is a locally free O_{S}[z]/(z − ζ)^{d}- module of rank equal to rk \underline{M} = height G. The O_{S}[z]-submodule V_{M}(coker F_{M}) ⊂ H(G) is called the Hodge filtration of the z-divisible local Anderson-module G.

Now let i: S′ ↪ S be a closed subscheme defined by an ideal I with I^{q} = 0. Then the morphisms Frob_{q,S} and Frob_{q,S′} factor through i

Frob_{q,S} = i ∘ j: S → S′ → S  and  Frob_{q,S′} = j ∘ i: S′ → S → S′ .

where j: S → S′ is the identity on the underlying topological space |S′| = |S| and on the structure sheaf this factorization is given by

O_{S} ⟶^{i^{∗}} O_{S′}  ⟶^{j^{∗}} O_{S}
b ↦ b mod I ↦ b^{q} .

Let G′ be a divisible local Anderson-module over S′ with (z−ζ)^{d}·ω_{G′} = 0, and denote by (M′, F_{M′}) its local shtuka. We set H(G′)_{S} := j^{∗}M′/(z − ζ)^{d} j^{∗}M′. This is a locally free module over O_{S}[z]/(z − ζ)^{d} and satisfies i^{∗}H(G′)_{S} = H(G′).

Theorem 9.8. The functor G ↦ ( i^{∗}G, V_{M}(coker F_{M}) ⊂ H(G)) defines an equivalence between

(a) the category of z-divisible local Anderson-modules G over S with (z − ζ)^{d} · ω_{G} = 0, and

(b) the category of pairs ( G′, F il ⊂ H(G′)_{S} ) where G′ is a z-divisible local Anderson-module over S′ and F il ⊂ H(G′)_{S} is an O_{S}[[z]]-submodule whose quotient is a flat O_{S}-module, and which specializes to the O′_{S}[[z]]-submodule V_{M′}(coker F_{M′}) ⊂ H(G′) under i.

Proof. We describe the quasi-inverse functor. Let ( G′, F il ⊂ H(G′)_{S} ) be given and let (M′, F_{M′}) be the local shtuka of G′. We define V_{M}: M ↪ j^{∗}M′ as the kernel of the morphism j^{∗}M′ ↠ H(G′)_{S}/F il. Since F il ⊂ H(G′)_{S} specializes to V_{M}(coker F_{M′}) ⊂ H(G′) we obtain i^{∗}(H(G′)_{S}/F il) = H(G′)/V_{M′}(coker F_{M′}) = coker V_{M′}. This implies i^{∗}M ≅ M′ and σ_{q}^{∗}M = j^{∗}i^{∗}M ≅ j^{∗}M′. Moreover coker V_{M} is annihilated by (z−ζ)^{d}. Thus there is an injective morphism of O_{S}[[z]]-modules F_{M}: σ_{q}^{∗}M → M with F_{M}V_{M} = (z − ζ)^{d} id_{M} and V_{M}F_{M} = (z − ζ)^{d} id_{σ_{q}^{∗}M}. From sequence (9.9) we see that the cokernel of F_{M} is a locally free O_{S}-module. Clearly the z-divisible local Anderson-module G over S associated with the local shtuka (M, F_{M}) specializes to G′ and has F il ⊂ H(G′)_{S} = H(G) as its Hodge filtration.

Remark 9.9. We only treated the case where S′ ⊂ S is defined by an ideal I with I^{q} = 0. The general case for d = 1 is treated by Genestier and Lafforgue [GL11, Proposition 6.3] using ζ-divided powers in the style of Grothendieck and Berthelot.

10 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 Definition 1.1. We follow the approach of Messing [Mes72] who treated the analogous situation of p-divisible groups and formal Lie groups.

Definition 10.1. Let G be an fppf -sheaf of abelian groups over S ∈ Nilp_{𝔽_{q}[[ζ]]}. We say that G is

• F -torsion if G = lim_{⟶_{n}} G[F_{q}^{n}], see Definition 9.1,

• F -divisible if F_{q,G}: G → σ_{q}^{∗}G is an epimorphism.

Recall that Messing [Mes72, Chapter II, Theorem 2.1.7] proved that a sheaf of groups G on S is a formal Lie group [Mes72, Chapter II, Definitions 1.1.4 and 1.1.5], if and only if G is F -torsion, F -divisible, and the G[F_{q}^{n}] are finite locally free S-group schemes.

Theorem 10.2. When ζ = 0 in O_{S} and G is a z-divisible local Anderson module over S, then lim_{⟶_{n}} G[F_{q}^{n}] is a formal 𝔽_{q}[[z]]-module. It is equal to \overline{G} := lim_{⟶_{k}} Inf^{k}(G), where for any S-scheme T,

Messing [Mes72, Chapter II, (1.1)] defines

(Inf^{k} G)(T) := { x ∈ G(T): there is an fppf -covering {Spec R_{i} → T }_{i}  (10.10)
and for every i an ideal I_{i} ⊂ R_{i} with I_{i}^{k+1} = (0)
such that the pull-back x ∈ G(Spec R_{i}/I_{i}) is zero } .

Proof. By [Mes72, Chapter II, Theorem 2.1.7] it suffices to show that lim_{⟶} G[F_{q}^{n}] is F -torsion, F - divisible and that the G[F_{q}^{n}] are finite locally free. By construction lim_{⟶} G[F_{q}^{n}] is F -torsion. By

Definition 7.1(d) there is locally on S an integer d with (z − ζ)^{d} · ω_{G} = (0), and then G[F_{q}^{n}] ⊂ G[z^{nd}] by Corollary 9.4 and G[z^{nd}] is a truncated z-divisible local Anderson module with order of nilpotence d and level nd by Proposition 9.5. Therefore Proposition 9.6 shows that G[F_{q}^{n}] is finite locally free, and that F_{q,G}: G[F_{q}^{n}] → σ_{q}^{∗}G[F_{q}^{n−1}] is an epimorphism. Consequently, F_{q,G}: lim_{⟶} G[F_{q}^{n}] → σ_{q}^{∗}(lim_{⟶} G[F_{q}^{n}]) =

lim_{⟶} σ_{q}^{∗}G[F_{q}^{n−1}] is an epimorphism and so lim_{⟶} G[F_{q}^{n}] is F -divisible, and hence a formal Lie group. The 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

[figure: T; ^{Frob^{q^{n},T}}; x; F_{q^{n},G}(x); T; x; G; F_{q^{n},G}; _{Frob_{q^{n},G}}; σ_{q^{n}}^{∗}G; G; S; S; _{Frob_{q^{n},S}}; S .]

Therefore, the monomorphism G[F_{q}^{n}] ↪ G defines an inclusion G[F_{q}^{n}] ⊂ Inf^{q^{n}−1} G, the ideals I_{i} in (10.10) being the augmentation ideal in O_{G[F_{q}^{n}]} defining the zero section. We claim that this inclusion is an equality. So let x ∈ (Inf^{q^{n}−1} G)(T) and let R_{i} and I_{i} be as in (10.10). Then I_{i}^{q^{n}} = (0) implies that

Frob_{q^{n},R_{i}} factors through R_{i} ⟶ R_{i}/I_{i} ⟶^{j} R_{i}. So F_{q^{n},G}(x)|_{Spec R_{i}} = x ∘ Frob_{q^{n},R_{i}} = j^{∗}(x|_{Spec R_{i}/I_{i}}) =

0, that is x ∈ G[F_{q}^{n}]. Thus we have G[F_{q}^{n}] = Inf^{q^{n}−1} G and lim_{⟶} G[F_{q}^{n}] = lim_{⟶} Inf^{k}(G) ⊂ G which completes the proof.

Our next aim is to extend the theorem to all S ∈ Nilp_{𝔽_{q}[[ζ]]}. For that purpose we start with the following

Lemma 10.3. Let S be a scheme with ζ^{N+1} = 0 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 + 1. Then for any affine open subset U of S and any quasi-coherent sheaf F of O_{U}-modules the natural homomorphism for the co-Lie complexes Ext_{O_{U}}^{1} (ℓG[z^{(n−N−1)d}]/U^{•}, F) ⟶ Ext_{O_{U}}^{1} (ℓG[z^{nd}]/U^{•}, F) is zero.

Proof. We proceed by induction on N and begin with N = 0. If n = 1, then G[z^{(n−N−1)d}] = (0) and there is nothing to prove. If n ≥ 2 we use [Mes72, Chapter II, Corollary 3.3.9] for the sequence

0 ⟶ G[z^{(n−1)d}] ⟶ G[z^{nd}] ⟶^{z^{(n−1)d}} G[z^{d}] ⟶ 0 .

So we have to show that

(a) ω_{G[z^{nd}]} ↠ ω_{G[z^{(n−1)d}]} is an isomorphism,

(b) ω_{G[z^{nd}]} and ω_{G[z^{d}]} are locally free O_{S}-modules, and

(c) rk ω_{s^{∗}G[z^{(n−1)d}]} ≤ rk ω_{s^{∗}G[z^{d}]} for all points s ∈ S.

All three statements follow from Proposition 9.6(f). This concludes the proof when N = 0. For general N we take the exact sequence

0 → ζF → F → F/ζF → 0.

and consider the commutative diagram with exact rows

\xymatrix{Ext_{O_{U}}^{1} (ℓG[z^{(n−N−1)d}]/U^{•}, ζF) \ar[r] \ar[d] & Ext_{O_{U}}^{1} (ℓG[z^{(n−N−1)d}]/U^{•}, F) \ar[r] \ar[d] & Ext_{O_{U}}^{1} (ℓG[z^{(n−N−1)d}]/U^{•}, F/ζF) \ar[d] \\ Ext_{O_{U}}^{1} (ℓG[z^{(n−N)d}]/U^{•}, ζF) \ar[r] \ar[d] & Ext_{O_{U}}^{1} (ℓG[z^{(n−N)d}]/U^{•}, F) \ar[r] \ar[d] & Ext_{O_{U}}^{1} (ℓG[z^{(n−N)d}]/U^{•}, F/ζF) \\ Ext_{O_{U}}^{1} (ℓG[z^{nd}]/U^{•}, ζF) \ar[r] & Ext_{O_{U}}^{1} (ℓG[z^{nd}]/U^{•}, F) .}

Since ζ · (F/ζF) = (0), the right vertical arrow can be computed by base change to the zero locus V(ζ) ⊂ S of ζ. So it is the zero map by what we have proved above, and hence the image of

Ext_{O_{U}}^{1} (ℓG[z^{(n−N−1)d}]/U^{•}, F) in Ext_{O_{U}}^{1} (ℓG[z^{(n−N)d}]/U^{•}, F) lies inside the image of Ext_{O_{U}}^{1} (ℓG[z^{(n−N)d}]/U^{•}, ζF).

Since ζ^{N} · (ζF) = (0), the lower left vertical arrow can similarly be computed by base change to the zero locus V(ζ^{N}) ⊂ S, and hence it is the zero map by our induction hypothesis. This proves the lemma.

Theorem 10.4. If S ∈ Nilp_{𝔽_{q}[[ζ]]} and G is a z-divisible local Anderson module over S, then G is formally smooth.

Proof. Let X′ be an affine scheme over S and let X be a closed subscheme defined by an ideal of square zero. Let f: X → G be an S-morphism. We must show that f can be lifted to an S-morphism f′: X′ → G.

\xymatrix{X \ar[d]_{f} \ar@{^{(}->}[r] & X′ \ar@{-->}[dl]^{f′} \\ G}

As X is quasi-compact we have G(X) = lim_{⟶_{n}} G[z^{n}](X) = lim_{⟶_{n}} G[z^{nd}](X), and hence f: X → G[z^{nd}] for

some n by Remark 7.2. We cover X by a finite number of affine opens U_{i}, i = 1, …, m such that the image of U_{i} in S is contained in an affine open V_{i}. Since ζ is nilpotent on each V_{i} there is an integer N such that ζ^{N+1} is zero on ⋃ V_{i}. Replacing S by S′ = ⋃ V_{i} and G by G_{S′} we are led to the case where ζ^{N+1} = 0 in O_{S}. But now Lemma 10.3 and [Mes72, Chapter II, Proposition 3.3.1] show that f can be lifted to an f′: X′ → G[z^{(n+N+1)d}] and the theorem is proved.

Lemma 10.5. Let G be a z-divisible local Anderson module over S with (z − ζ)^{d} = 0 on ω_{G} for some d ∈ ℕ. Assume we are given an S-scheme X′ and a subscheme X defined by a sheaf of ideals I such that I^{k+1} = (0) and ζ^{N} · I/I^{2} = 0 for some integer N. Let N′ be the smallest integer which is a power of p and greater or equal to N and d. If an S-morphism f′: X′ → G satisfies f = f′|_{X}: X → G[z^{n}], then f′ factors through f′: X′ → G[z^{n+kN′}] ⊂ G.

Proof. The problem is local on X′ and hence we can assume that X′ is affine and thus quasi-compact. But then f′ ∈ G(X′) = lim_{⟶} G[z^{m}](X′) and hence we can assume that f′: X′ → G[z^{n′}] for some n′

by Remark 7.2. We now use induction on k and the sequence of closed subschemes V(I^{l}) ⊂ X′ for l = 1, …, k + 1. Thus we can assume that I^{2} = 0 and k = 1.

Since f ∈ G[z^{n}](X) we have z^{n}f = 0, and so z^{n}f′ ∈ G[z^{n′}](X′) has the property that its restriction to G[z^{n′}](X) is zero. Since I^{2} = 0, the group of sections of G[z^{n′}] over X′ whose restriction to X is zero, is by [SGA 3, III, Théorème 0.1.8(a)] isomorphic to the group Hom_{O_{X}} (ω_{G[z^{n′}]} ⊗_{O_{S}} O_{X}, I) under an isomorphism which sends the zero morphism X′ → G[z^{n′}] to the zero element, and the morphism z^{n}f′ to an element which we denote by h ∈ Hom_{O_{X}} (ω_{G} ⊗_{O_{S}} O_{X}, I). Since ζ^{N} kills I and N′ ≥ N we obtain ζ^{N′} · h = 0. On ω_{G} the assumption (z − ζ)^{d} = 0 implies z^{N′} = ζ^{N′}, and so the section z^{N′}(z^{n}f′) is sent to z^{N′} · h = ζ^{N′} · h = 0. This implies z^{n+N′}f′ = 0, that is, f′ ∈ G[z^{n+N′}](X′).

Corollary 10.6. Let ζ^{N} = 0 in O_{S} and let G and d be as in Lemma 10.5. Let N′ be the smallest integer which is power of p and greater or equal to N and d. Then the k-th infinitesimal neighborhood of G[z^{n}] in G is the same as that of G[z^{n}] in G[z^{n+kN′}]. In particular, Inf^{k}(G) = Inf^{k}(G[z^{kN′}]) and this is therefore representable.

Proof. By definition [Mes72, Chapter II, Definition (1.01)], an S-morphism f: T′ → G belongs to the k-th infinitesimal 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_{i}^{k+1} = (0) such that f|_{Spec R_{i}/I_{i}} ∈ G[z^{n}](Spec R_{i}/I_{i}). But then f ∈ G[z^{n+kN′}](T′) by Lemma 10.5. The last statement is the special case with n = 0.

Theorem 10.7. Let G be a z-divisible local Anderson module over S ∈ Nilp_{𝔽_{q}[[ζ]]}. Then \overline{G} = lim_{⟶} Inf^{k}(G) is a formal 𝔽_{q}[[z]]-module.

Proof. As \overline{G} clearly is an 𝔽_{q}[[z]]-submodule of G, we must show that it is a formal Lie variety; see [Mes72, Chapter II, Definition 1.1.4]. By construction it is ind-infinitesimal. Since the question is local on S we may assume that there are integers N and d as in Corollary 10.6. Then the sheaf Inf^{k}(G) is representable for all k. By Theorem 10.4 we know that G is formally smooth and by definition (10.10) of Inf^{k}(G) this implies that \overline{G} is formally smooth. Let N′ be the smallest integer which is power of p and greater or equal to N and d. Then G[z^{kN′}] satisfies the lifting condition 2) of [Mes72, Chapter II, Proposition 3.1.1] by Theorem 10.4 and Lemma 10.5. Therefore, by [Mes72, loc. cit.] G[z^{kN′}] satisfies condition 2) and 3) of [Mes72, Chapter II, Definition 1.1.4], and hence is a formal 𝔽_{q}[[z]]-module.

Remark 10.8. We already know from Theorem 8.3 and Lemma 2.3 that ω_{G} is locally free of finite rank. This now follows again from the theorem, because ω_{G} = ω_{\overline{G}}.

Next we pursue the question when a z-divisible local Anderson module is a formal 𝔽_{q}[[z]]-module and vice versa.

Lemma 10.9. Let B be a ring in which ζ is nilpotent, and let I be a nilpotent ideal of B. Define a sequence of ideals I_{1} := ζI + I^{2}, …, I_{n+1} := ζI_{n} + (I_{n})^{2}. Then for n sufficiently large I_{n} = (0).

Proof. Let J = ζB + I. Then it is easy to check that I_{n} ⊂ J^{n+1}. Since ζ and I both are nilpotent, so is the ideal J. This implies I_{n} = 0 for n sufficiently large.

Lemma 10.10. If S ∈ Nilp_{𝔽_{q}[[ζ]]} and G is a formal 𝔽_{q}[[z]]-module over S such that locally on S there is an integer d with (z − ζ)^{d} = 0 on ω_{G}, then G is z-torsion.

Proof. We must show G = lim_{⟶} G[z^{n}] and since this is a statement about sheaves, it suffices to check it locally on S. Thus we can assume S = Spec R with ζ ∈ R nilpotent and G is given by a power series ring R[[X_{1}, …, X_{d}]]; see [Mes72, p. 26]. If T is any affine S-scheme, say T = Spec B, then an element of G(T) will be an N-tuple (b_{1}, …, b_{d}) with each b_{i} nilpotent. Let I be the ideal generated by {b_{1}, …, b_{d}}. Let N′ be a power of p with N′ ≥ d. Then multiplication with z^{N′} on G is given by power series (z^{N′})^{∗}(X_{i}) ∈ R[[X_{1}, …, X_{d}]] with linear term ζ^{N′}X_{i} and without constant term, because ω_{G} = (X_{1}, …, X_{d})/(X_{1} …, X_{d})^{2}. Therefore each component of z^{N′} · (b_{1}, …, b_{d}) belongs to ζ^{N′}I + I^{2} ⊂ ζI + I^{2} =: I_{1}. Then each component of z^{nN′} · (b_{1}, …, b_{d}) belongs to the ideal I_{n} from Lemma 10.9, and hence the lemma shows that (b_{1}, …, b_{d}) is z-torsion.

The next result is analogous to Messing’s characterization [Mes72, Chapter II, Proposition 4.4] for a p-divisible group to be a formal Lie group, and also its proof follows similarly using Theorem 10.7.

Proposition 10.11. Let S ∈ Nilp_{𝔽_{q}[[ζ]]} and let G be a z-divisible local Anderson module over S. Then the following conditions are equivalent:

(a) G = \overline{G}.

(b) G is a formal 𝔽_{q}[[z]]-module.

(c) G[z^{n}] is radicial for all n.

(d) G[z] is radicial.

Corollary 10.12. For S ∈ Nilp_{𝔽_{q}[[ζ]]}, there is an equivalence of categories between that of z-divisible local Anderson modules over S with G[z] radicial, and the category of z-divisible formal 𝔽_{q}[[z]]-modules G with G[z] representable by a finite locally free group scheme, such that locally on S there is an integer d for which (z − ζ)^{d} = 0 on ω_{G}.

Proof. By Lemma 10.10 and Proposition 10.11 both categories are identified with the same full subcategory of fppf -sheaves of 𝔽_{q}[z]-modules on S, once we observe that G[z^{n}] := ker(z^{n}: G → G) is a strict 𝔽_{q}-module as the kernel of an 𝔽_{q}-linear homomorphism of formal Lie groups which are 𝔽_{q}-modules.

Corollary 10.13. Let S ∈ Nilp_{𝔽_{q}[[ζ]]} be the spectrum of an Artinian local ring. Then a z-divisible formal 𝔽_{q}[[z]]-module, such that locally on S there is an integer d for which (z − ζ)^{d} = 0 on ω_{G}, is a z-divisible local Anderson module with G[z] radicial and conversely.

Proof. This follows from Corollary 10.12, because the G[z^{n}] are automatically representable by finite locally free group schemes by [Mes72, Chapter II, Proposition 4.3].

The next result is analogous to Messing’s characterization [Mes72, Chapter II, Proposition 4.7] for a p-divisible group to be ind-étale, and also its proof follows verbatim.

Proposition 10.14. Let S ∈ Nilp_{𝔽_{q}[[ζ]]} and let G be a z-divisible local Anderson module over S. In order that \overline{G} = 0 it is necessary and sufficient that G is (ind-)étale.

We have the following lemma for z-divisible local Anderson modules over S similarly to and with the same proof as [Mes72, Chapter II, Proposition 4.11].

Lemma 10.15. Let S ∈ Nilp_{𝔽_{q}[[ζ]]} and let 0 → G_{1} → G_{2} → G_{3} → 0 be an exact sequence of z-divisible local Anderson modules over S. Then 0 → \overline{G}_{1} → \overline{G}_{2} → \overline{G}_{3} → 0 is also exact.

Finally there is a criterion when \overline{G} is itself a z-divisible local Anderson module in analogy to Messing’s criterion [Mes72, Chapter II, Proposition 4.9].

Proposition 10.16. Let S ∈ Nilp_{𝔽_{q}[[ζ]]} and let G be a z-divisible local Anderson module over S. Then the following conditions are equivalent.

(a) \overline{G} is a z-divisible local Anderson module.

(b) G is an extension of an (ind-)étale z-divisible local Anderson module G′′ by an ind-infinitesimal z-divisible local Anderson module G′.

(c) G is an extension of an (ind-)étale z-divisible local Anderson module G′′ by a z-divisible formal 𝔽_{q}[[z]]-module G′.

(d) For all n, G[z^{n}] is an extension of a finite étale group by a finite locally-free radicial group.

(e) G[z] is an extension of a finite étale group by a finite locally-free radicial group.

(f) the map S → ℤ, s ↦ ord(G[z]_{s})_{ét} =: separable rank (G[z]_{s}) is a locally constant function on S.

Proof. The proof proceeds in the same way as [Mes72, Chapter II, Proposition 4.9] using Corollary 8.5 and Lemma 10.15 in (a)⟺(b), Corollary 10.12 and ω_{G} = ω_{G′} in (b)⟺(c), and Lemma 4.11 in (d)⟹(c).

Corollary 10.17. If S is the spectrum of a field L every z-divisible local Anderson-module G = lim_{⟶} G[z^{n}] over S is canonically an extension of an (ind-)étale divisible local Anderson-module G^{ét} by

a z-divisible formal 𝔽_{q}[[z]]-module \overline{G}

0 ⟶ \overline{G} ⟶ G ⟶ G^{ét} ⟶ 0 .

G^{ét} is the largest (ind-)étale quotient of G. With notation as in Proposition 4.2 we have \overline{G} = lim_{⟶} G[z^{n}]^{0}

and G^{ét} = lim_{⟶} G[z^{n}]^{ét}. If L is perfect the extension splits canonically.

This decomposition is compatible with the decomposition of the local shtuka \underline{M}_{q}(G) from Proposition 2.9 under the functors \underline{M}_{q} and Dr_{q} from Theorem 8.3.

Proof. Proposition 10.16, whose condition (f) is trivially satisfied, provides the extension and the equalities \overline{G} = lim_{⟶} G[z^{n}]^{0} and G^{ét} = lim_{⟶} G[z^{n}]^{ét}. From this the characterization of G^{ét} and the canonical splitting for perfect L follows; see Proposition 4.2. Finally the compatibility with the decomposition of the local shtuka \underline{M}_{q}(G) from Proposition 2.9 follows from the characterization of G^{ét} being (ind-)étale, respectively \overline{G} being a formal 𝔽_{q}[[z]]-module in terms of their associated local shtukas proved in Theorem 8.3.

A Review of the cotangent complex

In this appendix we carry out the elementary exercise to compare the definitions of the cotangent complex given by Illusie [Ill72, § VII.3.1], Lichtenbaum and Schlessinger [LS67, § 2.1], and Messing [Mes72, Chapter II, § 3.2] for a finite locally free group scheme G = Spec A over S = Spec R. Recall that G is a relative complete intersection by [SGA 3, Proposition III.4.15]. This means that locally on S we can take A = R[X_{1}, …, X_{n}]/I where the ideal I is generated by a regular sequence (f_{1}, …, f_{n}) of length n; compare [EGA, IV_{4}, Proposition 19.3.7].

The cotangent complex in the sense of Lichtenbaum and Schlessinger

We follow the notation of Lichtenbaum and Schlessinger [LS67, § 2.1] and take the free R[\underline{X}]-module F = R[\underline{X}] · g_{1} ⊕ … ⊕ R[\underline{X}] · g_{n}. We set U := ker(j: F ↠ I, g_{ν} ↦ f_{ν}) and let U_{0} be the image of the R[\underline{X}]-linear map F ⊗_{R[\underline{X}]} F → F, x ⊗ y ↦ j(x)y − j(y)x.

Lemma A.1. There is an exact sequence of R[\underline{X}]-modules

\xymatrix{⨁_{1≤μ<ν≤n} R[\underline{X}] · h_{μν} \ar[r]^{i} & ⨁_{ν=1}^{n} R[\underline{X}] · g_{ν} \ar[r]^{j} & I \ar[r] & 0 . \\ h_{μν} \ar@{|->}[r] & f_{ν}g_{μ} − f_{μ}g_{ν} , g_{ν} \ar@{|->}[r] & f_{ν}}  (A.11)

In particular the ideal I is finitely presented and U = U_{0}.

Proof. By definition the map j in (A.11) is surjective. To prove exactness in the middle let ∑_{ν=1}^{n} a_{ν}g_{ν} ∈ ker j, that is ∑_{ν=1}^{n} a_{ν}f_{ν} = 0 in R[\underline{X}]. This implies a_{n}f_{n} = 0 in R[\underline{X}]/(f_{1}, …, f_{n−1}). Since f_{n} ∈ I is a non-zero-divisor in R[\underline{X}]/(f_{1}, …, f_{n−1}) we have a_{n} = 0 in R[\underline{X}]/(f_{1}, …, f_{n−1}). Thus there exist b_{nμ} ∈ R[\underline{X}] for 1 ≤ μ ≤ n − 1 such that a_{n} = ∑_{μ=1}^{n−1} b_{nμ}f_{μ}. It follows that

∑_{ν=1}^{n} a_{ν}g_{ν} ≡ ∑_{ν=1}^{n} a_{ν}g_{ν} − i(∑_{μ=1}^{n−1} b_{nμ}h_{nμ}) ≡ (a_{1} + b_{n1}f_{n})g_{1} + … + (a_{n−1} + b_{n,n−1}f_{n})g_{n−1} mod im(i) .

Continuing in this way we get

∑_{ν=1}^{n} a_{ν}g_{ν} ≡ (a_{1} + b_{n1}f_{n} + b_{n−1,1}f_{n−1} + … + b_{2,1}f_{2})g_{1} mod im(i),

and hence (a_{1} + b_{n1}f_{n} + b_{n−1,1}f_{n−1} + … + b_{2,1}f_{2})f_{1} = 0 in R[\underline{X}]. Since f_{1} is a non-zero-divisor in R[\underline{X}] we conclude ∑_{m=1}^{n} a_{ν}g_{ν} ∈ im(i). This proves that I is finitely presented over R[\underline{X}]. Moreover, U = ker(j) = im(i) = U_{0}.

Consequently the cotangent complex of Lichtenbaum and Schlessinger [LS67, Definition 2.1.3] is the complex of O_{G}-modules concentrated in degrees −1 and 0 given by

LLS^{•}(G/S): 0 ⟶ I/I^{2} ⟶ Ω_{R[\underline{X}]/R} ⊗_{R[\underline{X}]} A ⟶ 0 .

By [Ill71, Corollaire III.3.2.7] this complex is quasi-isomorphic to the cotangent complex LG/S^{•} defined by Illusie [Ill71, II.1.2.3], which we considered in Section 3 before Definition 3.7.

The cotangent complex in the sense of Messing

We next recall the definition of the cotangent complex of G/S given by Messing [Mes72, Chapter II, § 3.2]. Since G is a group scheme, A is a bi-algebra with comultiplication Δ: A → A ⊗_{R} A and counit ε_{A}: A → R. Then Ǎ = Hom_{R-Mod}(A, R) carries an R-algebra structure via the dual morphisms Δ̌ of Δ and ε̌_{A} of ε_{A}.

Definition A.2. We let U(G) := Spec(Sym_{R}^{•} A). It represents the contravariant functor from Sschemes to rings, sending an S-scheme T to the ring Γ(T, Ǎ ⊗_{R} O_{T}); see [EGA, II, 1.7.9].

We let U(G)^{×} be the contravariant functor from S-schemes to abelian groups whose points with values in an S-scheme T are the invertible elements in the ring U(G)(T) = Γ(T, Ǎ ⊗_{R} O_{T}).

So U(G)^{×} is defined by the fiber product diagram

[figure: U(G)^{×} = S ×_{U(G)} (U(G) × U(G)); U(G) × U(G); Δ̌; S; ε̌_{A}; U(G)]

Since U(G) is affine and of finite presentation over S, its unit section ε̌_{A} is a closed immersion of finite presentation by [EGA, IV_{1}, Proposition 1.6.2], and the same is true for U(G)^{×} ↪ U(G) × U(G). Therefore U(G)^{×} is an affine group scheme of finite presentation over S. By [EGA, IV_{4}, Proposition 19.3.7] the unit section of the smooth S-scheme U(G) is a regular immersion. Therefore the immersion U(G)^{×} ↪ U(G) × U(G) is also regular by [EGA, IV_{4}, Proposition 19.1.5].

U(G)^{×} is smooth over S because U(G) is and the inclusion U(G)^{×} → U(G) is a smooth monomorphism (and hence an open immersion by [EGA, IV_{4}, Théorème 17.9.1]). Indeed, smoothness can be tested by the infinitesimal lifting criterion [BLR90, § 2.2, Proposition 6] as follows. Let I ⊂ B be an ideal in a ring B with I^{2} = 0 and let b ∈ Ǎ⊗_{R} B = U(G)(B) be a point with (b mod I) ∈ U(G)^{×}(B/I). Then any lift b′ ∈ Ǎ ⊗_{R} B of (b mod I)^{−1} ∈ U(G)^{×}(B/I) ⊂ Ǎ ⊗_{R} B/I satisfies bb′ − 1 ∈ Ǎ ⊗_{R} I and 0 = (bb′ − 1)^{2} = 1 − bb′(2 − bb′). It follows that b ∈ U(G)^{×}(B).

Messing considers the natural monomorphism i: G ↪ U(G)^{×} which is defined by viewing a T - valued point of G as a homomorphism of O_{T} -algebras A ⊗_{R} O_{T} → O_{T} and hence as an element of Γ(T, Ǎ ⊗_{R} O_{T}). The fact that such a homomorphism when viewed as an element of U(G)(T) is invertible, follows from the commutativity of the following diagram

[figure: G(T) × G(T); i × i; Spec Δ; Γ(T, Ǎ ⊗_{O_{S}} O_{T}) × Γ(T, Ǎ ⊗_{O_{S}} O_{T}); Δ̌; G(T); i; Γ(T, Ǎ ⊗_{O_{S}} O_{T})]

This diagram is commutative because both the left and right vertical arrow come from Δ. For every f ∈ G(T) there exists a g ∈ G(T) such that (Spec Δ)(f, g) = 1. Since i(f) · i(g) = Δ̌ ∘ (i × i)(f, g) = i ∘ (Spec Δ)(f, g) = i(1), it is enough to prove that i(1) = 1. Now 1 ∈ G(T) is equivalent to ε_{A}: A ⊗_{O_{S}} O_{T} → O_{T} which in turn is equivalent to (ε̌_{A}: O_{T} → Ǎ ⊗_{O_{S}} O_{T}) ≡ 1 ∈ Γ(T, Ǎ ⊗_{O_{S}} O_{T}). This shows i(1) = 1. Also it shows that the morphism i: G ↪ U(G)^{×} is a homomorphism of group schemes. Since G is a relative complete intersection and finite over S, it follows that the monomorphism i: G ↪ U(G)^{×} is a regular closed immersion; see [Mes72, Chapter II, Lemmas 3.2.5 and 3.2.6]. Let J be the ideal defining G in U(G)^{×}. Then Messing [Mes72, Chapter II, Definition 3.2.8] defines the cotangent complex of G over S as the complex of O_{G}-modules concentrated in degrees −1 and 0

LME^{•}(G/S): 0 ⟶ J/J^{2} ⟶ i^{∗}(Ω_{U(G)^{×}/S}) ⟶ 0 .

Proposition A.3. The cotangent complexes LLS^{•}(G/S) and LME^{•}(G/S) are homotopically equivalent.

Proof. The scheme U(G)×U(G) is locally on S of the form U(G)×U(G) = Spec R[\underline{X}] for a polynomial algebra and we can form the cotangent complex LLS^{•}(G/S) using R[\underline{X}]. We set U(G)^{×} = Spec \overline{R}. Let I be the ideal defining G in U(G) × U(G) and let K be the ideal defining U(G)^{×} in U(G) × U(G). Then J = I/K. By [EGA, IV_{4}, Proposition 19.1.5] the composition G ⟶^{i} U(G)^{×} ⟶ U(G) × U(G) is a regular closed immersion and the canonical sequence

0 ⟶ i^{∗}(K/K^{2}) ⟶ I/I^{2} ⟶^{g^{(−1)}} J/J^{2} ⟶ 0

is also exact on the left. We have denoted the third map by g^{(−1)}. Since U(G)^{×} is smooth over S, [EGA, IV_{4}, Proposition 17.2.5] yields an exact sequence of finite locally free \overline{R}-modules

0 ⟶ K/K^{2} ⟶ Ω_{R[\underline{X}]/R}^{1} ⊗_{R[\underline{X}]} \overline{R} ⟶ Ω_{\overline{R}/R}^{1} ⟶ 0

which we tensor with A to get an exact sequence of A-modules

0 ⟶ i^{∗}(K/K^{2}) ⟶ Ω_{R[\underline{X}]/R}^{1} ⊗_{R[\underline{X}]} A ⟶^{g^{(0)}} Ω_{\overline{R}/R}^{1} ⊗_{\overline{R}} A ⟶ 0.

We have denoted the third map by g^{(0)}. Since Ω_{\overline{R}/R}^{1} ⊗_{\overline{R}} A is a finite locally free A-module we may

choose a section f^{(0)} of g^{(0)}. In the diagram with commuting solid arrows

[figure: 0; 0; i^{∗}(K/K^{2}); α^{(−1)}; i^{∗}(K/K^{2}); α^{(0)}; 0; I/I^{2}; d^{(−1)}; g^{(−1)}; s^{(−1)}; Ω_{R[\underline{X}]/R}^{1} ⊗_{R[\underline{X}]} A; s^{(0)}; g^{(0)}; h^{(−1)}; 0; 0; J/J^{2}; d̃^{(−1)}; f^{(−1)}; Ω_{\overline{R}/R}^{1} ⊗_{\overline{R}} A; f^{(0)}; 0; 0; 0]

we define the section s^{(0)} of α^{(0)} by id −f^{(0)}g^{(0)} = α^{(0)}s^{(0)}. Then s^{(−1)} := s^{(0)}d^{(−1)} satisfies s^{(−1)}α^{(−1)} = s^{(0)}d^{(−1)}α^{(−1)} = s^{(0)}α^{(0)} = id_{i^{∗}(K/K^{2})}. We define the section f^{(−1)} of g^{(−1)} by id −α^{(−1)}s^{(−1)} = f^{(−1)}g^{(−1)}. Then d^{(−1)}f^{(−1)}g^{(−1)} = d^{(−1)}( id −α^{(−1)}s^{(−1)}) = d^{(−1)} − α^{(0)}s^{(0)}d^{(−1)} = f^{(0)}g^{(0)}d^{(−1)} = f^{(0)}d̃^{(−1)}g^{(−1)} and hence d^{(−1)}f^{(−1)} = f^{(0)}d̃^{(−1)}. This means that we obtain homomorphisms of complexes

\xymatrix{LLS^{•}(G/S): \ar[d]^{g} & 0 \ar[r] & I/I^{2} \ar[r] \ar[d]^{g^{(−1)}} & Ω_{R[\underline{X}]/R}^{1} ⊗_{R[\underline{X}]} A \ar[r] \ar[d]^{g^{(0)}} & 0 \\ LME^{•}(G/S): \ar[u]^{f} & 0 \ar[r] & J/J^{2} \ar[r] \ar[u]^{f^{(−1)}} & Ω_{\overline{R}/R}^{1} ⊗_{\overline{R}} A \ar[r] \ar[u]^{f^{(0)}} & 0}

with gf = id. We define the homotopy h^{(−1)} := α^{(−1)}s^{(0)}. Then

\begin{aligned}id −f^{(0)}g^{(0)} & = & α^{(0)}s^{(0)} & = & d^{(−1)}α^{(−1)}s^{(0)} & = & d^{(−1)}h^{(−1)} and \\ id −f^{(−1)}g^{(−1)} & = & α^{(−1)}s^{(−1)} & = & α^{(−1)}s^{(0)}d^{(−1)} & = & h^{(−1)}d^{(−1)} . \end{aligned}

This proves that f and g form a homotopy equivalence between LLS^{•}(G/S) and LME^{•}(G/S).

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Urs Hartl Universität Münster Mathematisches Institut Einsteinstr. 62 D – 48149 Münster Germany www.math.uni-muenster.de/u/urs.hartl/ Rajneesh Kumar Singh Ramakrishna Vivekananda University PO Belur Math, Dist Howrah 711202, West Bengal India