The Newton stratification on deformations of local G-shtukas

Urs Hartl and Eva Viehmann

February 12, 2010

Abstract

Bounded local G-shtukas are function field analogs for p-divisible groups with extra structure. We describe their deformations and moduli spaces. The latter are analogous to Rapoport-Zink spaces for p-divisible groups. The underlying schemes of these moduli spaces are affine Deligne-Lusztig varieties. For basic Newton polygons the closed Newton stratum in the universal deformation of a local G-shtuka is isomorphic to the completion of a corresponding affine Deligne-Lusztig variety in that point. This yields bounds on the dimension and proves equidimensionality of the basic affine Deligne-Lusztig varieties. Mathematics Subject Classification (2000): 20G25 (11G09, 14L05, 14M15)

1 Introduction

Deformations and moduli spaces of p-divisible groups play an important role for the local theory and the reduction modulo p of Shimura varieties. A first case was studied by Drinfeld [Dr] who used such a moduli space to uniformize certain Shimura curves. Generalizing Drinfeld’s result, Rapoport and Zink [RZ] constructed formal schemes over ℤ_{p} parametrizing p-divisible groups together with a quasi-isogeny to a fixed p-divisible group, and also variants including extra structure such as a polarization, endomorphisms, or a level structure. These spaces are used to uniformize Shimura varieties (for a corresponding group, i.e. a restriction of scalars of some general linear or symplectic group) along Newton strata.

In this article we consider an analog over the local function field 𝔽_{q}((z)). Here we are not restricted to the general linear or symplectic groups. We replace p-divisible groups by so-called local G-shtukas for any split connected reductive group G over the finite field 𝔽_{q}. To define them, let LG be the loop group of G, that is, the ind-scheme over 𝔽_{q} representing the sheaf of groups for the fpqc-topology whose sections for an 𝔽_{q}-algebra R are given by LG(Spec R) = G(R[[z]][\frac{1}{z}]). Let K be the infinite dimensional affine group scheme over 𝔽_{q} with K(Spec R) = G(R[[z]]). Every K-torsor over an 𝔽_{q}- scheme S for the fpqc-topology is already a K-torsor for the étale topology (Proposition 2.2). For such a K-torsor G on S let LG be the associated LG-torsor and σ^{∗}LG the pullback of LG under the q-Frobenius morphism Frob_{q} : S → S. As our base scheme for a local G-shtuka let S be an 𝔽_{q}[[z]]-scheme on which z is locally nilpotent and denote the image of z in O_{S} by ζ. Then a local Gshtuka over S is a pair \underline{G} = (G, φ) consisting of a K-torsor G on S and an isomorphism of LG-torsors φ : σ^{∗}LG ^{∼} ⟶ LG. Let k be an algebraically closed field extension of 𝔽_{q}. Local G-shtukas over k can also be described as follows. There exists a trivialization G ≅ K_{k} and with respect to such a trivialization φ corresponds to an element b ∈ LG(k). A change of the trivialization replaces b by g^{−1}bσ^{∗}(g) for g ∈ K(k), where σ^{∗} is the endomorphism of K and of LG induced by Frob_{q} : S → S.

An important invariant of \underline{G} over k is its Hodge polygon which controls the relative position of G and the image of σ^{∗}G under φ. We fix a Borel subgroup B of G containing a split maximal torus T. Let b ∈ LG(k) be as above. Then the Hodge polygon of \underline{G} is defined (using the Cartan decomposition) to be the unique dominant cocharacter μ\underline{G} ∈ X_{∗}(T) with b ∈ K(k)z^{μ\underline{G}}K(k). Clearly μ\underline{G} does not depend on the chosen trivialization G ≅ K_{k}. If μ is any dominant cocharacter of G we say that \underline{G} is bounded by μ if μ\underline{G} ⪯ μ in the Bruhat ordering of X_{∗}(T). In Definition 3.5 we also extend this ad hoc definition of boundedness to not necessarily reduced base schemes S.

The affine Grassmannian is the quotient sheaf Gr = LG/K for the fppf -topology. It is an indscheme over 𝔽_{q} which is of ind-finite type; see [BD, §4.5], [BL], [LS], who comprehensively develop the theory of the affine Grassmannian over the field of complex numbers. Most of their results, and in particular all we use here, also hold with the same proofs over 𝔽_{q}. For instance [NP], [Fa] reprove some statements. Moreover [PR2] present a generalization of results and proofs to twisted affine flag varieties over 𝔽_{q}.

One main result of our present article, Theorem 5.6, identifies the universal deformation space Spf D for deformations bounded by μ of a local G-shtuka \underline{G} over a field k′ with the completion of a closed subscheme of Gr ×_{Spec 𝔽_{q}} Spec k′[[ζ]] determined by μ. It is noetherian of relative dimension ⟨2ρ, μ⟩ over k′[[ζ]], and \overline{D} := (D/ζD)_{red} is normal and Cohen-Macaulay. Here ρ is the half-sum of the positive roots of G. We call Spf \overline{D} the universal (ζ = 0)-deformation space of \underline{G}.

We construct moduli spaces of local G-shtukas \underline{G} bounded by μ together with a quasi-isogeny δ to a fixed trivializable local G-shtuka \underline{𝔾} over k′, which are the analogs of the moduli spaces of p-divisible groups defined by Rapoport and Zink, [RZ]. They are formal schemes locally formally of finite type over k′[[ζ]] (Theorem 6.3) whose underlying topological spaces can be described as follows. Inside the affine Grassmannian we consider the affine Deligne-Lusztig variety which is defined to be the locally closed reduced ind-subscheme of Gr whose points over an algebraically closed extension k of k′ are

X_{μ}(b)(k) := { g ∈ Gr(k) = LG(k)/K(k) : g^{−1}bσ^{∗}(g) ∈ K(k)z^{μ}K(k) }

and the closed affine Deligne-Lusztig variety X_{⪯μ}(b) = ⋃_{μ′⪯μ} X_{μ′}(b). They are schemes locally of finite type over k′ (Corollary 6.5). The underlying topological space of the Rapoport-Zink space of local G-shtukas bounded by μ is isomorphic to X_{⪯μ}(b) where b ∈ LG(k′) describes the Frobenius φ of \underline{𝔾}.

We consider a second invariant of a local G-shtuka \underline{G} over k, its Newton polygon. It is the quasi-cocharacter ν\underline{G} ∈ X_{∗}(T)_{ℚ} associated by Kottwitz [Ko1, Ko2] to the σ-conjugacy class of b ∈ LG(k). Kottwitz’s articles only consider the analogous case of σ-conjugacy classes of elements b ∈ G(Frac(W (k)) where W (k) is the ring of Witt vectors over k. The arguments carry over literally to the equal characteristic case. By definition ν\underline{G} is invariant under isogeny. Under specialization Hodge and Newton polygon behave like those of p-divisible groups: there is a generalization of Grothendieck’s specialization theorem by Rapoport and Richartz [RR]. By results of Vasiu [Vas] the Newton stratification is pure, that is the Newton polygon of a local G-shtuka over a connected scheme jumps in codimension one or is constant (Theorem 7.4). Using these two properties we study the Newton stratification on the universal (ζ = 0)-deformation space of an arbitrary local G-shtuka \underline{G} over k and give lower bounds for the dimensions of the strata (Proposition 7.8).

We show that a local G-shtuka over a reduced complete local ring whose Newton polygon ν = ν\underline{G} is basic, that is central in G, is isogenous to a constant local G-shtuka (Proposition 8.1), which is an analog of a result of Oort and Zink, [OZ, Proposition 3.3]. It allows to compare the Newton stratum N_{ν} in the universal (ζ = 0)-deformation to the Rapoport-Zink space. Here we use a bijection between bounded local shtukas over Spec R and Spf R for any complete local ring R of characteristic p (Proposition 3.16) to pass from the universal deformation (which is a formal scheme) to the associated scheme. We obtain

Theorem 1.1. Let b ∈ LG(k) be basic and let μ ∈ X_{∗}(T) be dominant and such that X_{⪯μ}(b) ≠ ∅. Let g ∈ X_{⪯μ}(b) be a k-valued point and let (X_{⪯μ}(b))g^{∧} be the completion of X_{⪯μ}(b) in this point. Let N_{ν} be the basic Newton stratum in the universal (ζ = 0)-deformation of \underline{G} = (K_{k}, g^{−1}bσ^{∗}(g)σ^{∗}) bounded by μ. Then (X_{⪯μ}(b))g^{∧} is canonically isomorphic to N_{ν}.

Using the dimension formula for affine Deligne-Lusztig varieties in [GHKR1], [Vi1], and Theorem 1.1, we show that these varieties are also equidimensional. A similar strategy is used by de Jong and Oort in [dJO, Proposition 5.19 and Corollary 5.20] to show equidimensionality of a moduli space of p-divisible groups.

Theorem 1.2. Let b ∈ LG(k) be basic and let μ ∈ X_{∗}(T) be dominant and such that X_{⪯μ}(b) ≠ ∅. Then X_{μ}(b) and X_{⪯μ}(b) are equidimensional of dimension ⟨ρ, μ⟩ − \frac{1}{2}def(b). Here ρ is the half-sum of the positive roots of G and def(b) = rk(G) − rk(J) is the defect of b (see 7.9).

For X_{μ}(b), this finishes the proof of a conjecture by Rapoport [Ra, Conjecture 5.10]. For X_{μ}(b) with b ∈ T (k((z))) (not necessarily basic) Theorem 1.2 has been shown in [GHKR1, Proposition 2.17.1]. For X_{⪯μ}(b), this theorem immediately implies

Corollary 1.3. Let b ∈ LG(k) be basic and μ ∈ X_{∗}(T) be dominant. Then X_{⪯μ}(b) is the closure of X_{μ}(b) in Gr.

Proof. We have ⟨ρ, μ′⟩ < ⟨ρ, μ⟩ for every μ′ ⪯ μ with μ′ ≠ μ. Thus dim X_{μ′}(b) < dim X_{μ}(b) for each such μ′. Hence equidimensionality of X_{⪯μ}(b) implies that X_{μ}(b) is dense in X_{⪯μ}(b).

A more careful analysis of the proof of Theorem 1.2 yields

Corollary 1.4. Let \underline{G} be a local G-shtuka over k which is basic and bounded by some μ ∈ X_{∗}(T). Let \overline{D} be the universal (ζ = 0)-deformation ring for deformations of \underline{G} that are bounded by μ. Then the generic fibre of the local G-shtuka over Spec \overline{D} corresponding to the universal family over Spf \overline{D} has Newton polygon μ.

We prove Theorem 1.1, Theorem 1.2 and Corollary 1.4 in Section 8.

In the non-basic case the dimensions of the affine Deligne-Lusztig varieties and the lower bound on the dimension of the corresponding (non-basic) Newton stratum differ by ⟨2ρ, ν⟩. Especially, Theorem 1.1 does not hold in this general situation. The same difference between the dimensions already occurs for p-divisible groups. There, work of Oort [O1] shows that up to a finite morphism the Newton stratum N_{ν} is a product of the underlying scheme of a Rapoport-Zink space with a so-called central leaf, and by [O2], or [Ch2] the difference ⟨2ρ, ν⟩ between the dimensions is the dimension of this central leaf. In a sequel to this article we show that a similar structure exists in the non-basic case for our deformations of local G-shtukas.

In the last section we define local G-shtukas with Iwahori-level structure. Using a variant of Theorem 1.1 we can compare the Newton stratum of an associated universal (ζ = 0)-deformation with an affine Deligne-Lusztig variety inside the affine flag manifold in the basic case. This proves the basic case of a conjecture by Beazley [Be, Conjecture 1]. The strategy used to prove Theorem 1.2 also gives a lower bound on the dimension of the affine Deligne-Lusztig variety in the Iwahori setting. However, in this case the lower bound depends on the maximal length of a chain of Newton polygons in the universal deformation, for which a general formula is not known to us.

Acknowledgement. We are grateful to O. Gabber, R. Kottwitz, and W. Soergel for helpful explanations and to A. Genestier for many stimulating discussions. We also thank W. Soergel for providing the proof of Proposition 3.14 and M. E. Arasteh Rad for telling us about an inaccuracy in a previous version of this article. The first author acknowledges support of the DFG (German Research Foundation) in form of DFG-grant HA3006/2-1 and SFB 478, project C5. This work was also supported by the SFB/TR 45 “Periods, Moduli Spaces and Arithmetic of Algebraic Varieties” of the DFG.

Contents

1 Introduction 1

2 Notation and complements on torsors for loop groups 4

3 Local G-shtukas 8

4 The general linear group 15

5 Deformation theory of local G-shtukas 17

6 Rapoport-Zink spaces for local G-shtukas and affine Deligne-Lusztig varieties 22

7 The Newton stratification 27

8 Basic Newton strata 30

9 The Iwahori case 32

Appendix: Admissibility 36

References 38

2 Notation and complements on torsors for loop groups

We denote by 𝔽_{q} the finite field with q elements and by k an algebraically closed field which is also an 𝔽_{q}-algebra. Let 𝔽_{q}[[z]] be the ring of power series in the indeterminate z and let Nilp_{𝔽_{q}[[z]]} be the category of 𝔽_{q}[[z]]-schemes on which z is locally nilpotent. For a scheme S in Nilp_{𝔽_{q}[[z]]} we denote by ζ ∈ O_{S} the image of z and we write from now on S ∈ Nilp_{𝔽_{q}[[ζ]]}. We view z and ζ as algebraically independent indeterminates.

A reference for the various topologies (fpqc, fppf , ét in addition to the Zariski topology) that we consider on the category of all 𝔽_{q}-schemes is [SGA 3, Exposé IV, §6.3]. Let O_{S}[[z]] be the sheaf of O_{S}-algebras on S for the fpqc-topology whose ring of sections on an S-scheme Y is the ring of power series O_{S}[[z]](Y) := Γ(Y, O_{Y})[[z]]. This is indeed a sheaf being the countable direct product of O_{S}. Let O_{S}((z)) be the fpqc-sheaf of O_{S}-algebras on S associated with the presheaf Y ↦ Γ(Y, O_{Y})[[z]][\frac{1}{z}]. If Y is quasi-compact then O_{S}((z))(Y) = Γ(Y, O_{Y})[[z]][\frac{1}{z}] by [Ha, Exercise II.1.11]. We denote by σ^{∗} the endomorphism of O_{S}[[z]] and O_{S}((z)) that acts as the identity on the variable z, and as b ↦ b^{q} on local sections b ∈ O_{S}. For a sheaf M of O_{S}[[z]]-modules on S we set σ^{∗}M := M ⊗_{O_{S}[[z]],σ^{∗}} O_{S}[[z]].

Let G be a split connected reductive group over 𝔽_{q}. Let B ⊃ T be a Borel subgroup and a maximal split torus of G. Recall that a weight λ ∈ X^{∗}(T) is called dominant if ⟨λ, α^{∨}⟩ ≥ 0 for every positive coroot α^{∨} of G, and similarly for coweights. We consider the ordering ⪯ on the group of coweights X_{∗}(T) of G, which is defined as μ_{1} ⪯ μ_{2} if and only if the difference μ_{2} − μ_{1} is a non-negative integral linear combination of simple coroots. If μ_{1}, μ_{2} are dominant it coincides with the Bruhat ordering. On X^{∗}(T) we consider the analogous ordering. On X_{∗}(T)_{ℚ} = X_{∗}(T) ⊗_{ℤ} ℚ we use the ordering with μ_{1} ⪯ μ_{2} if and only if the difference μ_{2} − μ_{1} is a non-negative rational linear combination of simple coroots. For μ ∈ X_{∗}(T) we denote by z^{μ} the image of z ∈ 𝔾_{m}(𝔽_{q}((z))) under the morphism μ : 𝔾_{m} → T.

Let LG be the loop group of G, see [Fa, Definition 1]. That is, LG is the ind-scheme of ind-finite type over 𝔽_{q} representing the sheaf of groups for the fpqc-topology on 𝔽_{q}-schemes S given by

LG(S) = G(O_{S}((z))(S)) .

On Nilp_{𝔽_{q}[[ζ]]} the local nilpotency of ζ on S implies that the sheaf LG is canonically isomorphic to the sheaf associated with the presheaf S ↦ G(Γ(S, O_{S})[[z]][\frac{1}{z−ζ}]). We denote by K the infinite dimensional affine group scheme over 𝔽_{q} whose S-valued points for an 𝔽_{q}-scheme S are K(S) = G(O_{S}[[z]](S)) = G(Γ(S, O_{S})[[z]]). In the natural way K can be viewed as a subsheaf of LG. For any scheme S ∈ Nilp_{𝔽_{q}[[ζ]]} we denote by K_{S} the induced sheaf of groups for the fpqc-topology on S. The endomorphism σ^{∗} induces an endomorphism of LG and K.

Let ∗ ∈ {fpqc, fppf , ét}. Let S ∈ Nilp_{𝔽_{q}[[ζ]]} and let H be a sheaf of groups on S for the ∗-topology. In this article a (right) H-torsor for the ∗-topology on S is a sheaf G for the ∗-topology on S together with a (right) action of the sheaf H such that G is isomorphic to H on a ∗-covering of S. Here H is viewed as an H-torsor by right multiplication. The H-torsors are classified up to isomorphism by Čech cohomology Ȟ^{1}(S_{∗}, H), which is a pointed set. For H = K the categories of torsors for the three different topologies fpqc, fppf , ét are equivalent by Proposition 2.2 below. Therefore we simply speak of K-torsors on S.

Let K′ be equal to K or to the Iwahori subgroup

I(S) := { g ∈ K(S) = G(Γ(S, O_{S})[[z]]) : g mod z ∈ B(S) } .

Let G be a K′-torsor on S for the ∗-topology. Then G is trivialized by a ∗-covering S′ → S and G_{S′} ≅ K′_{S′} is an affine scheme. The covering S′ → S is in particular an fpqc-covering, and the fpqcdescent is effective by [BLR, §6.1, Theorem 6]. Hence G is an (infinite dimensional) affine scheme over S. With G we associate an LG-torsor LG for the ∗-topology via the map Ȟ^{1}(S_{∗}, K′) → Ȟ^{1}(S_{∗}, LG). Also for an LG-torsor LG on S we denote by σ^{∗}LG the pullback of LG under the q-Frobenius morphism Frob_{q} : S → S.

The pointed set Ȟ^{1}(S_{∗}, K′) also classifies another kind of torsors which we introduce now. Throughout this article (i.e. for the definition of local G-shtukas) we only use the first kind of torsors. However, the second interpretation of Ȟ^{1}(S_{∗}, K′) turns out to be useful to compare these sets for the various topologies in Proposition 2.2. Besides, it relates the torsors underlying local G-shtukas to global versions as studied by Varshavsky [Var]. To explain it let S be an 𝔽_{q}-scheme and let S[[z]] be the formal scheme over Spf 𝔽_{q}[[z]] in the sense of [EGA, I_{new}, Section 10] consisting of the topological space S endowed with the structure sheaf O_{S}[[z]]. Equivalently S[[z]] is the formal completion of 𝔸_{S}^{1} = S ×_{𝔽_{q}} Spec 𝔽_{q}[z] along the closed subscheme S ×_{𝔽_{q}} V(z). Let G′ be the smooth affine parahoric group scheme over 𝔽_{q}[[z]] with K′(𝔽_{q}) = G′(𝔽_{q}[[z]]), with generic fiber G ×_{𝔽_{q}} 𝔽_{q}((z)) and connected special fiber; see [HR] or [BT, 4.6.2, 4.6.26, and 5.2.6]. We write Ĝ′ = G′ ×_{Spec 𝔽_{q}[[z]]} Spf 𝔽_{q}[[z]] for the z-adic formal completion of G′. It is an affine formal scheme over Spf 𝔽_{q}[[z]]. There is an equality

K′(S) = G′(Γ(S, O_{S})[[z]])
= Hom_{𝔽_{q}[[z]]}(Γ(G′, O′_{G}) , Γ(S, O_{S})[[z]])
= Hom_{𝔽_{q}[[z]]}^{cont}(Γ(Ĝ′, OĜ′) , Γ(S, O_{S})[[z]])
= Hom_{Spf 𝔽_{q}[[z]]}(S[[z]] , Ĝ′)

where the latter Hom group denotes morphisms of formal schemes over Spf 𝔽_{q}[[z]], and the last equation holds by [EGA, I_{new}, Proposition 10.4.6] since Ĝ′ is affine.

Definition 2.1. Let ∗ ∈ {fpqc, fppf , ét}. We denote by PHS_{∗}(S[[z]], Ĝ′) (for principal homogenous space) the set of isomorphism classes of z-adic formal schemes Ĝ over S[[z]] together with an action Ĝ′ ˆ×_{Spf 𝔽_{q}[[z]]}Ĝ → Ĝ of Ĝ′ on Ĝ such that there is a covering S′[[z]] → S[[z]] where S′ → S is a ∗-covering and a Ĝ′-equivariant isomorphism α : Ĝˆ×_{S[[z]]}S′[[z]] ^{∼} ⟶ Ĝ′ ˆ×_{Spf 𝔽_{q}[[z]]}S′[[z]]. Here Ĝ′ acts on itself by right multiplication. The set PHS_{∗}(S[[z]], Ĝ′) is a pointed set, the distinguished point being the trivial torsor.

Proposition 2.2. (a) There is a natural bijection of pointed sets Ȟ^{1}(S_{∗}, K′) = PHS_{∗}(S[[z]], Ĝ′).

(b) Ȟ^{1}(S_{ét}, K′) = Ȟ^{1}(S_{fppf} , K′) = Ȟ^{1}(S_{fpqc}, K′).

(c) Let S = Spec A be an affine 𝔽_{q}-scheme, let 𝔞 ⊂ A be a nilpotent ideal and set \overline{A} = A/𝔞 and \overline{S} = Spec \overline{A}. Then for any K′-torsor G over S whose pull back to \overline{S} admits a trivialization ᾱ : G\overline{S} ^{∼} ⟶ K′ ; \overline{S}, there is a trivialization α : G ^{∼} ⟶ K′_{S} lifting ᾱ. Likewise for any formal Ĝ′-torsor Ĝ over S[[z]] as in Definition 2.1 whose pull back to \overline{S}[[z]] admits a trivialization ᾱ, there is a trivialization α of Ĝ lifting ᾱ.

We are mainly interested in statement (c) for K′ and in statement (b). However, we prove them using Ĝ′.

Proof. (a) Let Ĝ be a formal Ĝ′-torsor representing a class in the set PHS_{∗}(S[[z]], Ĝ′), and let α : Ĝˆ×_{S[[z]]}S′[[z]] ^{∼} ⟶ Ĝ′ ˆ×_{Spf 𝔽_{q}[[z]]}S′[[z]] be a trivialization over S′[[z]] for a ∗-covering S′ → S. We can pull back α to S′[[z]]ˆ×_{S[[z]]}S′[[z]] = S′′[[z]] for S′′ = S′ ×_{S} S′ under the two projections p_{i} : S′′[[z]] → S′[[z]]. Then h := p_{2}^{∗}α ∘ p_{1}^{∗}α^{−1} is a Ĝ′-equivariant automorphism of Ĝ′ ˆ×_{Spf 𝔽_{q}[[z]]}S′′[[z]], hence given by an element h ∈ Hom_{Spf 𝔽_{q}[[z]]}(S′′[[z]] , Ĝ′) = K′(S′′). Clearly h is a 1-cocycle for the covering S′ → S and induces a cohomology class in Ȟ^{1}(S_{∗}, K′).

Conversely any such cohomology class is represented by a ∗-covering S′ → S and a 1-cocycle h ∈ K′(S′′). The latter defines a descent datum on Ĝ′ ˆ×_{Spf 𝔽_{q}[[z]]}S′[[z]] which we must prove to be effective. Let n be a positive integer and consider the situation modulo z^{n}. Then h (mod z^{n}) is a descent datum on

Ĝ′ ˆ×_{Spf 𝔽_{q}[[z]]}S′[[z]]ˆ×_{Spf 𝔽_{q}[[z]]} Spec 𝔽_{q}[[z]]/(z^{n}) = G′ ×_{Spec 𝔽_{q}[[z]]} (S′ ×_{𝔽_{q}} Spec 𝔽_{q}[z]/(z^{n})) .

Since G′ is affine and S′×_{𝔽_{q}} Spec 𝔽_{q}[z]/(z^{n}) → S×_{𝔽_{q}} Spec 𝔽_{q}[z]/(z^{n}) is an fpqc-covering by base change from S′ → S, the descent is effective by [BLR, §6.1, Theorem 6a] and we obtain an affine scheme Ĝ_{n} over S ×_{𝔽_{q}} Spec 𝔽_{q}[z]/(z^{n}). It is of finite presentation and smooth by [EGA, IV_{2}, Proposition 2.7.1 and IV_{4}, Corollaire 17.7.3] since the same holds for G′ over 𝔽_{q}[[z]]. For varying n we obtain an inductive system Ĝ_{n} with Ĝ_{n+1} ×_{𝔽_{q}[[z]]/(z^{n+1})} 𝔽_{q}[[z]]/(z^{n}) = Ĝ_{n} whose limit exists as a z-adic affine formal scheme Ĝ over S[[z]] with underlying topological space Ĝ_{1} by [EGA, I_{new}, Corollary 10.6.4].

(b) Due to (a) and the injectivity of the natural maps PHS_{ét}(S[[z]], Ĝ′) ↪ PHS_{fppf} (S[[z]], Ĝ′) ↪ PHS_{fpqc}(S[[z]], Ĝ′) we only need to prove surjectivity of these maps. So let Ĝ be a formal Ĝ′-torsor over S[[z]] representing an element of PHS_{fpqc}(S[[z]], Ĝ′). We must show that Ĝ is trivial over S′[[z]] for an étale covering S′ → S. As in (a), Ĝ = lim_{⟶} Ĝ_{n} is the limit of smooth affine schemes Ĝ_{n} over

S ×_{𝔽_{q}} Spec 𝔽_{q}[z]/(z^{n}). Since Ĝ_{1} is smooth over S it has a section s_{1} : S → Ĝ_{1} over an étale covering S′ → S. We may further assume that S′ is (the disjoint union of a family of) affine (schemes). Then the section s_{1} lifts inductively for all n to a section s_{n} : S′ ×_{𝔽_{q}} Spec 𝔽_{q}[z]/(z^{n}) → Ĝ_{n} by the smoothness of Ĝ_{n}. In the limit this gives a section s : S′[[z]] → Ĝ which induces a trivialization of Ĝˆ×_{S[[z]]}S′[[z]] over S′[[z]] as desired.

(c) We first consider a cohomology class in Ȟ^{1}(S_{ét}, K′) whose image under Ȟ^{1}(S_{ét}, K′) → Ȟ^{1}(\overline{S}_{ét}, K′) is trivial. By (a) this class corresponds to a formal Ĝ′-torsor Ĝ over Spf A[[z]] whose pull back to Spf \overline{A}[[z]] is trivial. As above Ĝ is the limit of Spec A[z]/(z^{n})-schemes Ĝ_{n} which are of finite presentation and smooth. By the triviality of Ĝˆ×_{Spf A[[z]]} Spf \overline{A}[[z]] there exists a section ᾱ : Spf \overline{A}[[z]] → Ĝ. Modulo z it induces a section Spec \overline{A} → Ĝ_{1} which lifts by the smoothness of Ĝ_{1} to a section Spec A → Ĝ_{1}. It inductively lifts by the smoothness of Ĝ_{n} to a section Spec A[z]/(z^{n}) → Ĝ_{n}. In the limit we obtain a section β : Spf A[[z]] → Ĝ (which in general does not lift the initial section ᾱ : Spf \overline{A}[[z]] → Ĝ). Nevertheless, the section yields a trivialization of Ĝ over Spf A[[z]] and this proves that the only cohomology class which becomes trivial under Ȟ^{1}(S_{ét}, K′) → Ȟ^{1}(\overline{S}_{ét}, K′) is the trivial one.

We have to prove the existence of a trivialization α which lifts ᾱ both in the situation for formal Ĝ′-torsors Ĝ and for K′-torsors G. Let the torsor be represented by S′ → S and h ∈ K′(S′′). The trivialization ᾱ after pull back to \overline{S} of the formal Ĝ′-torsor Ĝ over \overline{S}[[z]], respectively the K′-torsor G over \overline{S}, is given by an element ḡ ∈ K′(\overline{S}′) with p_{2}^{∗}(ḡ) p_{1}^{∗}(ḡ)^{−1} = h (mod 𝔞) in K′(\overline{S}′′). Similarly the trivialization β of Ĝ over S[[z]], respectively G over S, is given by an element f ∈ K′(S′) with p_{2}^{∗}(f) p_{1}^{∗}(f)^{−1} = h in K′(S′′). Let f̄ be the pullback of f to \overline{S}. Then p_{2}^{∗}(f̄^{−1}ḡ) = p_{1}^{∗}(f̄^{−1}ḡ) in K′(\overline{S}′′) implies that f̄^{−1}ḡ ∈ K′(\overline{S}) = G′(\overline{A}[[z]]). Since ker(A[[z]] → \overline{A}[[z]]) is nilpotent and G′ is smooth over 𝔽_{q}[[z]] we can lift f̄^{−1}ḡ to an element f^{−1}g ∈ G′(\overline{A}[[z]]) with which we multiply f to obtain an element g ∈ K′(S′) satisfying p_{2}^{∗}(g) p_{1}^{∗}(g)^{−1} = h and lifting ḡ ∈ K′(\overline{S}′). Then g induces a trivialization α of Ĝ over S[[z]], respectively G over S, which lifts the trivialization ᾱ as desired.

Proposition 2.3 (Hilbert 90 for loop groups). Let G = GL_{r}. As before let K be the associated fpqc-sheaf over 𝔽_{q} whose Y -valued points are K(Y) = GL_{r}(Γ(Y, O_{Y})[[z]]). Then for any 𝔽_{q}-scheme S

Ȟ^{1}(S_{Zar}, K) = Ȟ^{1}(S_{ét}, K) = Ȟ^{1}(S_{fppf} , K) = Ȟ^{1}(S_{fpqc}, K)

and this pointed set parametrizes isomorphism classes of locally free sheaves of rank r on the formal scheme S[[z]], or equivalently sheaves of O_{S}[[z]]-modules on S which locally for the Zariski-topology on S are isomorphic to O_{S}[[z]]^{⊕r}. In particular, any fpqc-sheaf of O_{S}[[z]]-modules on S which is isomorphic to O_{S}[[z]]^{⊕r} fpqc-locally on S is already isomorphic to O_{S}[[z]]^{⊕r} Zariski-locally on S.

Proof. The second and third equality were proved in the preceding proposition. Consider a cohomology class in Ȟ^{1}(S_{fpqc}, K) and represent it by an fpqc-covering S′ → S and a 1-cocycle h ∈ K(S′′) for S′′ = S′ ×_{S} S′. Let p_{i} : S′′ → S′ be the projection onto the i-th factor. On the sheaf F′ = O_{S′}[[z]]^{⊕r} consider the descent datum given by the isomorphism h : p_{1}^{∗}O_{S′}[[z]]^{⊕r} ^{∼} ⟶ p_{2}^{∗}O_{S′}[[z]]^{⊕r}, v′′ ↦ h v′′ over S′′. In this way Ȟ^{1}(S_{fpqc}, K) parametrizes isomorphism classes of pairs (S′ → S , h) where S′ → S is an fpqc-covering and h is a descent datum on F′ = O_{S′}[[z]]^{⊕r}.

Modulo z^{n} such a descent datum induces a descent datum on the trivial sheaf F′_{n} of rank r on S′ ×_{𝔽_{q}} Spec 𝔽_{q}[z]/(z^{n}). By fpqc-descent [BLR, §6.1, Theorem 4] and [EGA, IV_{2}, Proposition 2.5.2] the sheaf F′_{n} descends to a locally free sheaf F_{n} of rank r on S ×_{𝔽_{q}} Spec 𝔽_{q}[z]/(z^{n}). By [EGA, I_{new}, Proposition 10.10.8.6] the projective limit F = lim_{⟵} F_{n} is a locally (in the Zariski-topology) free sheaf of rank r on S[[z]]. Since S is the underlying topological space and O_{S}[[z]] is the structure sheaf of the formal scheme S[[z]] the sheaf F is nothing else than a sheaf of O_{S}[[z]]-modules on S. This proves the proposition.

3 Local G-shtukas

Definition 3.1. A local G-shtuka over S is a pair \underline{G} = (G, φ) consisting of a K-torsor G on S together with an isomorphism φ : σ^{∗}LG ^{∼} ⟶ LG of the associated LG-torsors.

If f : S′ → S is a morphism in Nilp_{𝔽_{q}[[ζ]]} there is a natural way to pull back a local G-shtuka \underline{G} from S to S′. We denote the pullback by \underline{G}_{S′} or f^{∗}\underline{G}.

Remark 3.2. In particular, if \underline{G} = (G, φ) is a local G-shtuka over a noetherian complete local ring R with algebraically closed residue field, then G is isomorphic to the trivial torsor, and the morphism φ can be written as bσ^{∗} for some b ∈ LG(R). Indeed, G is trivialized by an étale R-algebra by Proposition 2.2, but every étale R-algebra decomposes into a direct sum of copies of R.

For G = GL_{r} local G-shtukas previously appeared in the literature as pairs (M, φ) where M is a locally free sheaf of O_{S}[[z]]-modules of rank n, and where φ : σ^{∗}M[\frac{1}{z−ζ}] ^{∼} ⟶ M[\frac{1}{z−ζ}] is an isomorphism. They were first introduced by Anderson [An] in the case when S is the spectrum of a complete discrete valuation ring. Genestier [Ge] constructed Rapoport-Zink spaces for them in the Drinfeld case and used these to uniformize Drinfeld modular varieties. Local GL_{r}-shtukas were further used and studied, among others, in [Ha1, Ha2]. (To be precise, in all these references they have to satisfy an additional boundedness condition in the style of Definition 3.5 and Example 4.5 below.) The two definitions of local GL_{r}-shtukas given above are equivalent by Proposition 2.3. For more details see Section 4. Local GL_{r}-shtukas are analogs in the arithmetic of local function fields of p-divisible groups, or more precisely of F -crystals. To a large extent they behave similarly (see also [Ha3] and [Ha1] where they were called Dieudonné 𝔽_{q}[[z]]-modules). Similarly to p-divisible groups which describe the local behaviour of abelian varieties at the prime p, there is also a global version of local G-shtukas. In [Var] Varshavsky describes G-shtukas which are defined using a smooth projective curve instead of Spf 𝔽_{q}[[z]].

We define the Hodge polygon of a local G-shtuka. Let S ∈ Nilp_{𝔽_{q}[[ζ]]}. Let G and H be K-torsors on S and let δ : LH ^{∼} ⟶ LG be an isomorphism of the associated LG-torsors. For a geometric point s : Spec k → S, we can choose trivializations of s^{∗}G and s^{∗}H. Then δ corresponds to an element g ∈ LG(k). Changing the trivializations changes g within its K(k)-double coset. The Cartan decomposition shows that LG(k) is the disjoint union of the sets K(k)z^{μ}K(k) where μ ∈ X_{∗}(T) is a dominant coweight. Obviously the double coset of g does not depend on the chosen algebraic closure k of κ(s).

Definition 3.3. The dominant element μ_{δ}(s) ∈ X_{∗}(T) with g ∈ K(k)z^{μ_{δ}(s)}K(k) is called the Hodge polygon of δ at s. If \underline{G} = (G, φ) is a local G-shtuka over S, we write μ\underline{G}(s) := μ_{φ}(s).

Let π_{1}(G) be the quotient of X_{∗}(T) by the lattice generated by the coroots. For any coweight μ ∈ X_{∗}(T) let [μ] be its image in π_{1}(G).

Proposition 3.4. Let G and H be K-torsors on S and let δ : LH ^{∼} ⟶ LG be an isomorphism of the associated LG-torsors. Then the function S → π_{1}(G), s ↦ [μ_{δ}(s)] is locally constant on S.

Proof. By Proposition 2.2 we can choose trivializations for the torsors G and H over an étale covering f : S′ → S. Thus δ corresponds to an element of LG(S′). This induces a morphism from S′ to the affine Grassmannian Gr. By [PR2, Theorem 0.1] the group π_{1}(G) equals the set of connected components π_{0}(Gr) and the induced morphism S′ → π_{0}(Gr) = π_{1}(G) does not depend on the chosen trivializations. Hence it descends to a morphism S → π_{0}(Gr) which coincides with [μ_{δ}( .)]. Therefore [μ_{δ}( .)] is locally constant.

Let \overline{B} ⊂ G be the Borel subgroup opposite to our fixed B. For a dominant weight λ of G we let V (λ) := (IndG ; \overline{B}(−λ)_{dom})^{∨} be the Weyl module of G with highest weight λ. It is a cyclic G-module generated by a B-stable line on which B acts through λ. Any other such G-module is a quotient of V (λ), see for example [Ja, II.2.13]. For a K-torsor G on a scheme S we denote by G_{λ} the fpqc-sheaf of O_{S}[[z]]-modules on S associated with the presheaf

Y ⟼ (G(Y) × (V (λ) ⊗_{𝔽_{q}} O_{S}[[z]](Y)))/K(Y) .

This means in particular that if S′ → S is an étale covering trivializing G (see Proposition 2.2) and if α : G_{S′} ^{∼} ⟶ K_{S′} is an isomorphism of K-torsors with p_{2}^{∗}α ∘ (p_{1}^{∗}α)^{−1} = g ∈ K(S′′) on S′′ = S′ ×_{S} S′ (with p_{i} the projection onto the i-th factor) then

G_{λ}(Y) ≅ { v′ ∈ V (λ) ⊗_{𝔽_{q}} O_{S}[[z]](Y ×_{S} S′) : p_{2}^{∗}v′ = g · p_{1}^{∗}v′ on Y ×_{S} S′′ } .

By Proposition 2.3 the sheaf G_{λ} is locally free in the Zariski-topology on S.

Furthermore, if G and H are K-torsors on S and δ : LH ^{∼} ⟶ LG is an isomorphism of the associated LG-torsors then δ induces an isomorphism of sheaves of O_{S}((z))-modules

δ : H_{λ} ⊗_{O_{S}[[z]]} O_{S}((z)) ^{∼} ⟶ G_{λ} ⊗_{O_{S}[[z]]} O_{S}((z)) .

Indeed, if over an étale covering S′ → S there are trivializations α : G_{S′} ^{∼} ⟶ K_{S′} and β : H_{S′} ^{∼} ⟶ K_{S′} with p_{2}^{∗}α ∘ (p_{1}^{∗}α)^{−1} = g ∈ K(S′′) and p_{2}^{∗}β ∘ (p_{1}^{∗}β)^{−1} = h ∈ K(S′′) and αδβ^{−1} = Δ ∈ LG(S′), then δ sends v′ ∈ H_{λ}(Y) ⊗_{O_{S}[[z]](Y)} O_{S}((z))(Y) to Δ · v′.

Definition 3.5. Let S be a connected scheme in Nilp_{𝔽_{q}[[ζ]]} and let μ be a dominant coweight of G. Let either z̃ = z − ζ or z̃ = z.

(a) Let G and H be K-torsors on S and let δ : LH ^{∼} ⟶ LG be an isomorphism of the associated LG-torsors. The isomorphism δ is bounded by (μ, z̃) if for each dominant weight λ of G

δ(H_{λ}) ⊂ z̃^{−⟨ (−λ)_{dom},μ⟩} G_{λ} ⊂ G_{λ} ⊗_{O_{S}[[z]]} O_{S}((z)) and  (3.1)
[μ] = [μ_{δ}(s)] in π_{1}(G) for all s ∈ S.  (3.2)

(b) A local G-shtuka (G, φ) over S is bounded by μ if the isomorphism

φ : σ^{∗}LG ^{∼} ⟶ LG

is bounded by (μ, z − ζ).

Remark 3.6. Bounds using z̃ = z are only used in some of the proofs to bound quasi-isogenies between local G-shtukas (see for example Definition 3.8 and the proof of Theorem 6.3).

Instead of using Weyl modules for the definition of boundedness one could also use a different class of representations such as for example the class of all induced representations V (λ)^{∨} = IndG ; \overline{B}(−λ)_{dom}, or the class of all representations with lowest weight (−λ)_{dom}, or the class of all tilting modules [Ja, Chapter E]. It follows from Lemma 3.10(b) below that on reduced schemes S in Nilp_{𝔽_{q}[[ζ]]} the boundedness definitions for all these classes are equivalent. However, they may differ in their nilpotent structure.

Lemma 3.7. The monoid X^{∗}(T)_{+} of dominant weights is finitely generated. Condition (3.1) holds for all dominant weights (and fixed z̃ = z − ζ or z̃ = z) if and only if it holds for a finite generating system.

Proof. Every dominant weight is a non-negative integral linear combination of the dominant weights in a generating system. So it suffices to show that if condition (3.1) holds for two dominant weights λ and λ′ then it also holds for λ + λ′. Now V (λ + λ′) is a G-submodule of V (λ) ⊗ V (λ′). Indeed, as V (λ) and V (λ′) are Weyl modules, they trivially admit a Weyl-filtration, i.e. a filtration with Weylmodules as factors. By [Ja, II.4.19, II.4.21] the same is true for V (λ) ⊗ V (λ′). By Weyl’s character formula [Ja, Corollary II.5.11] the character of V (λ) ⊗ V (λ′) contains λ + λ′ as highest weight. Thus we can find V (λ + λ′) as a submodule of V (λ) ⊗ V (λ′) by [Ja, II.4.19]. Consequently also H_{λ+λ′} is a submodule of H_{λ} ⊗ H_{λ′} and the same holds for CG_{λ+λ′}. Now the linearity of λ ↦ ⟨ (−λ)_{dom}, μ⟩ for dominant λ proves the lemma.

Definition 3.8. A quasi-isogeny between local G-shtukas (G, φ) → (G′, φ′) over S is an isomorphism of the associated LG-torsors f : LG ^{∼} ⟶ LG′ with φ′σ^{∗}(f) = fφ. The set of quasi-isogenies between \underline{G} and \underline{G}′ over S is denoted QIsog_{S}(\underline{G}, \underline{G}′).

As one might expect from the analogy with p-divisible groups, quasi-isogenies are rigid in the following sense.

Proposition 3.9. Let \underline{G} = (G, φ) and \underline{G}′ = (G′, φ′) be two local G-shtukas over S ∈ Nilp_{𝔽_{q}[[ζ]]} and let i : \overline{S} ↪ S be a closed immersion defined by a sheaf of ideals I which is locally nilpotent. Then

i^{∗} : QIsog_{S}(\underline{G}, \underline{G}′) ^{∼} ⟶ QIsog\overline{S}(i^{∗}\underline{G}, i^{∗}\underline{G}′) , f ↦ i^{∗}f

is a bijection of sets. Let now S be quasi-compact and let \underline{G}, \underline{G}′ both be bounded. Then an element of the right hand side is bounded by (μ, z) for some μ if and only if the corresponding element over S is bounded by (μ̃, z) for some μ̃.

Proof. Arguing by induction over O_{S}/I^{q^{n}} it suffices to treat the case where I^{q} = (0). In this case the q-Frobenius Frob_{p} factors as S ⟶^{j} \overline{S} ⟶^{i} S where j is the identity on the underlying topological space |\overline{S}| = |S| and on the structure sheaf this factorization is given by

O_{S} ⟶^{i^{∗}} O\overline{S}  ⟶^{j^{∗}} O_{S}
b ↦ b mod I ↦ b^{q} .

Therefore σ^{∗}f = j^{∗}(i^{∗}f) for any f ∈ QIsog_{S}(\underline{G}, \underline{G}′). We obtain the diagram

[figure: LG; ≅; f; LG′; σ^{∗}LG; ≅; φ; ≅; j^{∗}(i^{∗}f); σ^{∗}LG′; ≅; φ′]  (3.3)

from which the bijectivity is obvious.

For the assertion on boundedness we may again assume that I^{q} = (0). One direction is obvious, thus we now consider the case that i^{∗}f is bounded by μ. We use Lemma 3.7. Let Λ be a finite generating system of the monoid of dominant weights. The λ ∈ Λ fall into two classes according to whether ⟨λ, 2ρ^{∨}⟩ = 0 (i.e. λ is central) or ⟨λ, 2ρ^{∨}⟩ > 0. Since (−2ρ^{∨})_{dom} = 2ρ^{∨}, we have ⟨(−λ)_{dom}, 2ρ^{∨}⟩ > 0 for each λ in the second class. Since S is quasi-compact, we can find an m ≥ 0 such that for all those λ, (3.1) is satisfied for the coweight 2mρ^{∨} + μ. For the λ with ⟨λ, 2ρ^{∨}⟩ = 0 note that as λ is central, −λ is also dominant. If \underline{G} is bounded by some ω, applying (3.1) to λ and −λ implies that φ induces an isomorphism

φ : σ^{∗}G_{λ} ^{∼} ⟶ (\frac{1}{z−ζ} )^{⟨−λ,ω⟩}G_{λ} .

We get a similar isomorphism for \underline{G}′ with some ω′. Observe that ⟨λ, ω⟩ = ⟨λ, ω′⟩ since λ is central, [ω] = [ω′] in π_{1}(G) by (3.2), and f ∘ φ = φ′ ∘ σ^{∗}f. Also the boundedness of i^{∗}f over \overline{S} by μ yields (by applying j^{∗}) an isomorphism

f_{λ} : G_{λ} ≅ (\frac{1}{z−ζ} )^{⟨λ,ω⟩}σ^{∗}G_{λ} ^{∼} ⟶ (\frac{1}{z−ζ} )^{⟨λ,ω⟩}(\frac{1}{z})^{⟨−λ,μ⟩}σ^{∗}G′_{λ} ≅ (\frac{1}{z})^{⟨−λ,μ⟩}G′_{λ} = (\frac{1}{z})^{⟨−λ,2mρ^{∨}+μ⟩}G′_{λ}

over S, proving that f is bounded by 2mρ^{∨} + μ.

Lemma 3.10. Let G and H be K-torsors on S for a connected scheme S ∈ Nilp_{𝔽_{q}[[ζ]]}. Let either z̃ = z − ζ or z̃ = z and let δ : LH ^{∼} ⟶ LG be an isomorphism of the associated LG-torsors. Let μ be a dominant coweight of G satisfying (3.2).

(a) Then the condition that δ is bounded by (μ, z̃) is representable by a finitely presented closed immersion into S.

(b) If S is reduced then δ is bounded by (μ, z̃) if and only if this holds for the pullback to every geometric point of S. By (a) it is even enough to consider the pullback to the generic points of S.

Proof. We prove the lemma for z̃ = z − ζ, the other case is completely analogous. Fix a generating system Λ of the monoid X^{∗}(T)_{+} of dominant weights and consider for each λ ∈ Λ the isomorphism

δ : H_{λ} ⊗_{O_{S}[[z]]} O_{S}[[z]][\frac{1}{z−ζ}] ^{∼} ⟶ G_{λ} ⊗_{O_{S}[[z]]} O_{S}[[z]][\frac{1}{z−ζ}].

Since both questions are local on S we can assume that S = Spec R and G_{λ} and H_{λ} are free R[[z]]- modules. Then δ(H_{λ}) is contained in (z − ζ)^{−N_{λ}}G_{λ} ⊂ G_{λ} ⊗_{R[[z]]} R[[z]][\frac{1}{z−ζ}] for some N_{λ} ≫ 0. So δ is bounded by μ if and only if δ maps all generators of H_{λ} to zero in (z−ζ)^{−N_{λ}}G_{λ}/(z−ζ)^{−⟨(−λ)_{dom},μ⟩}G_{λ} =: M_{λ} for all λ ∈ Λ (by Lemma 3.7). Let M := ⨁_{λ∈Λ} M_{λ}. Since M is a free R-module of finite rank this condition is represented by a finitely presented closed immersion. This proves (a).

The condition in (b) is clearly necessary. Since M ↪ M ⊗_{R} ∏_{𝔭⊂R} κ(𝔭)^{alg} where the product is taken over all prime ideals 𝔭 of R, the condition is also sufficient.

Lemma 3.11. Let k be an algebraically closed field in Nilp_{𝔽_{q}[[ζ]]} (hence ζ = 0 in k) and let G and H be K-torsors over k. Let δ : LH ^{∼} ⟶ LG be an isomorphism of the associated LG-torsors with Hodge polygon μ_{δ}(Spec k) and let μ be a dominant coweight of G. Then δ is bounded by (μ, z) if and only if μ_{δ}(Spec k) ⪯ μ.

Proof. After choosing trivializations of G and H the isomorphism δ is represented by left multiplication with an element sz^{μ′}t ∈ LG(k) for μ′ := μ_{δ}(Spec k) and s, t ∈ K(k). Let λ be a dominant weight of G and consider the Weyl module V (λ). By Weyl’s character formula [Ja, Corollary II.5.11] its weights λ′ all satisfy −(−λ)_{dom} ⪯ λ′ ⪯ λ and the weight −(−λ)_{dom} occurs. Let v_{λ′} ∈ V (λ) be an element of the weight space of λ′. Then

δ(t^{−1}v_{λ′}) = z^{⟨λ′,μ′⟩} · s(v_{λ′}) ∈ z^{⟨λ′,μ′⟩} G_{λ} .

In particular δ(H_{λ}) ⊂ z^{−⟨(−λ)_{dom},μ′⟩}G_{λ} ∖ z^{−n}G_{λ} for all n < ⟨(−λ)_{dom}, μ′⟩. So δ is bounded by μ if and only if μ − μ′ is a linear combination of coroots (by (3.2)) with ⟨(−λ)_{dom}, μ − μ′⟩ ≥ 0 for every dominant λ. This is equivalent to μ′ ⪯ μ.

Example 3.12. For G = 𝔾_{m} we give an example of a local G-shtuka over R = k[ε]/(ε^{2}) which is not bounded by any coweight μ. Let \underline{G} = (K_{R}, bσ^{∗}) with b = 1 + \frac{ε}{z}. The Hodge polygon in the special point is equal to μ = (0) ∈ ℤ. Thus in the special point it is bounded by μ = (0) and by no other μ′. But as b ∉ 𝔾_{m}(R[[z]]), the boundedness condition does not hold on all of Spec R.

On the other hand we have Lemma 3.13. Let G and H be K-torsors on a quasi-compact connected scheme S ∈ Nilp_{𝔽_{q}[[ζ]]}. Let either z̃ = z − ζ or z̃ = z and let δ : LH ^{∼} ⟶ LG be an isomorphism of the associated LG-torsors. Assume that G is semi-simple or that S is reduced. Then there is a dominant μ ∈ X_{∗}(T) such that δ is bounded by (μ, z̃).

Proof. Let {λ_{1}, …, λ_{n}} with λ_{i} ≠ 0 be a finite generating set of the monoid of dominant weights of G. Let {λ_{n′}, …, λ_{n}} be the subset of weights that are central in G, that is orthogonal to the sum 2ρ^{∨} of the positive coroots. If G is semi-simple, the latter set is empty. Since S is quasi-compact, there are constants c_{i} for i = 1, …, n with δ(H_{λ_{i}}) ⊂ z̃^{−c_{i}}G_{λ_{i}}. We have to show that there is a dominant μ with [μ] = [μ_{δ}(s)] in π_{1}(G) for all s ∈ S and ⟨(−λ_{i})_{dom}, μ⟩ ≥ c_{i} for all i. As S is connected, [μ_{δ}(s)] ∈ π_{1}(G) is constant. Let μ be dominant with [μ] = [μ_{δ}(s)]. Replacing μ by μ + 2cρ^{∨} for c ∈ ℕ leaves the image in π_{1}(G) invariant and replaces ⟨(−λ_{i})_{dom}, μ⟩ by ⟨(−λ_{i})_{dom}, μ⟩ + c⟨(−λ_{i})_{dom}, 2ρ^{∨}⟩. For non-central λ_{i}, the last bracket is strictly positive. Thus for i = 1, …, n′ − 1 we can choose c large enough to ensure that ⟨(−λ_{i})_{dom}, μ + 2cρ^{∨}⟩ ≥ c_{i}. For G semi-simple this proves the lemma. Consider now the case that S is reduced. We want to show that μ + 2cρ^{∨} also satisfies the analogous condition for i ≥ n′. As S is reduced, it is by Lemma 3.10 enough to check the condition in each geometric point separately. Thus we have to show that ⟨(−λ_{i})_{dom}, μ_{δ}(s)⟩ ≤ ⟨(−λ_{i})_{dom}, μ + 2cρ^{∨}⟩ for i ≥ n′. As λ_{i} is central, both sides are determined by the images of μ_{δ}(s) resp. μ + 2cρ^{∨} in π_{1}(G) = X_{∗}(T)/(coroot lattice). As those images agree, the two sides are equal.

Proposition 3.14 (Soergel). Let V = ⨁_{λ∈Λ} V (λ) be the direct sum of Weyl modules over a finite generating system Λ of the monoid X^{∗}(T)_{+} of dominant weights for G. Then the natural map η : G → GL(V) is a closed immersion.

Proof. We first prove injectivity for points with values in an algebraic closure of 𝔽_{q}. Every such point lies in a Borel subgroup and is thus conjugate to an element g ∈ B(𝔽_{q}^{alg}). Assume that g ∈ ker(η)(𝔽_{q}^{alg}). Let g = g_{s}g_{u} be its decomposition into semi-simple and unipotent part. For each λ ∈ Λ consider a highest weight vector v_{λ} ∈ V (λ). That is, v_{λ} generates the G-module V (λ), and B stabilizes 𝔽_{q} · v_{λ} and operates through the quotient T and the weight λ on v_{λ}. In particular λ(g_{s}) · v_{λ} = λ(g) · v_{λ} = η(g)(v_{λ}) = v_{λ}, and hence λ(g_{s}) = 1. Since this holds for all λ in a generating system of X^{∗}(T), we must have g_{s} = 1. So ker(η) is a closed normal subgroup consisting solely of unipotent elements. Since G is reductive we conclude ker(η)(𝔽_{q}^{alg}) = (1).

For η to be a closed immersion it remains to prove that the induced map on Lie algebras is injective. Since ker(η) is normal in G, the torus T acts on it and the Lie algebra of ker(η) decomposes into weight spaces under T. So it suffices to show that η is injective on Lie T and on Lie U_{α} for each root subgroup U_{α}. Our argument for injectivity on points also shows that η is injective on Lie T.

Now we consider U_{α}. By conjugation with the longest element of the Weyl group it suffices to treat the case where α is a negative root. For this consider a dominant weight λ (not necessarily in Λ) and the B-representation W_{λ} := (−λ) ⊗ Res_{B}^{G} IndG ; \overline{B} λ where

IndG ; \overline{B} λ := \{ f ∈ 𝔽_{q}[G] : f(xb) = λ(b)^{−1}f(x) for all x ∈ G, b ∈ \overline{B} \} .

Let U be the unipotent radical of B. Via restriction to U we get a map W_{λ} → 𝔽_{q}[U]. This is an inclusion, since every f ∈ W_{λ} which is zero on U is also zero on the big cell U\overline{B}, hence zero in 𝔽_{q}[G]. For t ∈ T we have (tf)(x) = f(t^{−1}x) = f(t^{−1}xt t^{−1}) = λ(t) Int_{t}^{∗}(f)(x) with Int_{t}^{∗}(f)(x) := f(t^{−1}xt). The inclusion W_{λ} ⊂ 𝔽_{q}[U] is equivariant under T, where T acts on 𝔽_{q}[U] through Int_{t}^{∗}. Note that the weights of 𝔽_{q}[U] are in ℚ_{≤0} · R^{+} := {∑_{β∈R^{+}} n_{β} · β : n_{β} ∈ ℚ, n_{β} ≤ 0}, where R^{+} is the set of positive roots. The inclusion is also equivariant under U, where U operates on 𝔽_{q}[U] through left translation.

We choose λ such that w(λ + ρ) − λ − α − ρ ∉ ℚ_{≥0} · R^{+} for all w ≠ 1 in the Weyl group W, where ρ is the halfsum of all positive roots. To see that such a λ exists, let α_{1}, …, α_{r} be the simple roots of G and choose μ_{j} ∈ X_{∗}(T)_{ℚ} for j = 1, …, r with ⟨α_{i}, μ_{j}⟩ = δ_{ij}. Let

N = max{ ⟨−α , μ_{j}⟩ : j = 1, …, r } ∈ ℕ_{0}

and let λ = 2Nρ. For any 1 ≠ w ∈ W there exists a μ_{j} with ⟨ρ − w(ρ), μ_{j}⟩ ≠ 0, i. e. ⟨ρ − w(ρ), μ_{j}⟩ ≥ 1/2. Then

⟨w(λ + ρ) − λ − α − ρ , μ_{j}⟩ ≤ (2N + 1) ⟨w(ρ) − ρ , μ_{j}⟩ + ⟨−α , μ_{j}⟩ < 0 ,

and this implies that w(λ + ρ) − λ − α − ρ ∉ ℚ_{≥0} · R^{+} for all w ≠ 1. Note that for this choice of λ we have λ = (−λ)_{dom}.

We next use Kostant’s character formula [Kn, Corollary 5.83]. Note that although it is stated in loc. cit. for Lie algebra representations in characteristic zero, the formula also holds in our situation because it is a formal consequence (see [Kn, proof of Corollary 5.83]) of Weyl’s character formula [Ja, Corollary II.5.10]. Let μ be a weight and

P(μ) := #{ (n_{β})_{β∈R^{+}} : n_{β} ∈ ℕ_{>0}, μ = ∑_{β∈R^{+}} n_{β} · β } = dim_{𝔽_{q}}(𝔽_{q}[U])^{−μ} ,

the dimension of the (−μ)-weight space of 𝔽_{q}[U]. If μ ∉ ℚ_{≥0} · R^{+} then P(μ) = 0. Kostant’s formula says that the α-weight space of W_{λ} has dimension

dim_{𝔽_{q}}(W_{λ})^{α} = dim_{𝔽_{q}}(IndG ; \overline{B} λ)^{λ+α} = ∑_{w∈W}(−1)^{ℓ(w)}P(w(λ + ρ) − λ − α − ρ)

which is for our choice of λ

= P(λ + ρ − λ − α − ρ) = dim_{𝔽_{q}}(𝔽_{q}[U])^{α} .

In particular, W_{λ} generates the 𝔽_{q}-algebra 𝔽_{q}[U_{α}]. Assume that u ∈ Lie U_{α} lies in ker(η). Since V (λ) is a G-submodule of a tensor power of V (see the argument of Lemma 3.7), u acts trivially on V (λ). As λ = (−λ)_{dom}, it also acts trivially on W_{λ}. Therefore u acts trivially on 𝔽_{q}[U_{α}]. Since this operation is via left translation we must have u = 0. This proves that η is injective on Lie U_{α} and finishes the proof.

Remark 3.15. The corresponding map η′ : G → GL(V′) where V′ is a sum over Λ of corresponding irreducible highest weight representations is in general not an immersion. This already occurs for G = PGL_{2} in characteristic 2, where one can choose Λ to consist of the only positive root α. The corresponding Weyl module V (α) is the dual of the adjoint representation. With respect to the decomposition V (α) = (Lie T ⊕ Lie U_{α} ⊕ Lie U_{−α})^{∨} it is given by

η : PGL_{2} → GL(V (α)) , g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ⟼ ( \begin{matrix}1 & \frac{ac}{det g} & \frac{bd}{det g} \\ 0 & \frac{a^{2}}{det g} & \frac{b^{2}}{det g} \\ 0 & \frac{c^{2}}{det g} & \frac{d^{2}}{det g} \end{matrix} ) .

It is an extension of a two-dimensional irreducible representation V′ by the trivial representation, and the corresponding map η′ : PGL_{2} → GL(V′) is not injective on tangent spaces.

For the next proposition recall that by [EGA, 0_{I}, Lemma 7.7.2], a linearly topologized 𝔽_{q}[[ζ]]- algebra R is admissible if R = lim_{⟵} R_{α} for a projective system (R_{α}, u_{αβ}) of discrete rings such that the filtered index-poset has a smallest element 0, all maps R → R_{α} are surjective, and the kernels I_{α} := ker u_{α,0} ⊂ R_{α} are nilpotent.

Proposition 3.16. Let R be an admissible 𝔽_{q}[[ζ]]-algebra as above with filtered index-poset ℕ_{0}. Then the pullback under the natural morphism Spf R → Spec R defines a bijection between local G-shtukas bounded by μ over Spec R and over Spf R.

Remark 3.17. Without the boundedness by μ the pullback map is in general only injective. The corresponding result for p-divisible groups is shown by Messing in [Me, Lemma II.4.16] and by de Jong in [dJ, Lemma 2.4.4].

Proof of Proposition 3.16. A local G-shtuka over Spf R is by definition a projective system (\underline{G}_{n})_{n∈ℕ_{0}} of local G-shtukas \underline{G}_{n} over R_{n} with \underline{G}_{n−1} ≅ \underline{G}_{n} ⊗_{R_{n}} R_{n−1}. The pullback under Spf R → Spec R sends a local G-shtuka \underline{G} to the projective system (\underline{G} ⊗_{R} R_{n})_{n}. We describe the inverse map. By Proposition 2.2 there is an étale R_{0}-algebra A_{0} which trivializes G_{0}. By [SGA 1, Théorème I.5.5] there is a unique étale R-algebra A with A ⊗_{R} R_{0} ≅ A_{0}. Set A_{n} := A ⊗_{R} R_{n}. By Proposition 2.2 we can simultaneously trivialize all G_{n} over A_{n}. Thus \underline{G}_{n} ⊗_{R_{n}} A_{n} ≅ (K_{A_{n}}, b_{n}σ^{∗}) for b_{n} ∈ LG(A_{n}) = G(A_{n}[[z]][\frac{1}{z−ζ}]) with b_{n} ⊗_{A_{n}} A_{n−1} = b_{n−1}. We must show that b := lim_{⟵} b_{n} exists as an element of

G(A[[z]][\frac{1}{z−ζ}]). Note that without the boundedness by μ this is in general false.

Consider the representation η : G → GL(V) with V being the 𝔽_{q}-vector space ⨁_{λ∈Λ} V (λ) from Proposition 3.14. Let N be a positive integer with N ≥ ⟨(−λ)_{dom}, μ⟩ for all λ ∈ Λ. Then the boundedness by μ implies that (z − ζ)^{N} · η(b_{n}) ∈ M_{r}(A_{n}[[z]]), that is the denominator of b_{n} is bounded by (z − ζ)^{N} independently of n. Clearly the projective system ((z − ζ)^{N} · η(b_{n}))_{n} has a limit in M_{r}(A[[z]]). It is of the form (z − ζ)^{N} · η(b) for a b ∈ G(A[[z]][\frac{1}{z−ζ}]) by construction. The local G-shtuka (K_{Spec A}, bσ^{∗}) over Spec A inherits a descent datum from the \underline{G}_{n}. This gives us the desired local G-shtuka over Spec R.

4 The general linear group

In this section we consider the case G = GL_{r} and give the translation between GL_{r}-torsors and locally free sheaves of finite rank. This special case has the advantage that the similarity to the theory of crystals is visible more clearly.

Let B ⊂ GL_{r} be the Borel subgroup of upper triangular matrices and let T be the torus of diagonal matrices. Then X_{∗}(T) ≅ ℤ^{r} with simple coroots e_{i} − e_{i+1} for i = 1, …, r − 1. Also X^{∗}(T) ≅ ℤ^{r}. Let λ_{i} = (1, …, 1, 0, …, 0) with multiplicities i and r−i. The Weyl module V (λ_{1}) = (IndGL_{r} ; \overline{B}(−λ_{1})_{dom})^{∨} of highest weight λ_{1} is simply the standard representation of GL_{r} on the space of column vectors with r rows, and V (λ_{i}) = ∧^{i}V (λ_{1}). For an 𝔽_{q}-scheme S we have K(S) = GL_{r}(Γ(S, O_{S})[[z]]). By Proposition 2.3 there is an equivalence between the category of K-torsors on S and the category of sheaves of O_{S}[[z]]-modules which Zariski-locally on S are free of rank r with isomorphisms as the only morphisms. This equivalence sends G to the sheaf G_{λ_{1}} associated with the presheaf

Y ⟼ (G(Y) × (V (λ_{1}) ⊗_{𝔽_{q}} O_{S}[[z]](Y)))/ GL_{r}(Y [[z]]);

compare the discussion before Definition 3.5. On the sheaf G_{λ_{1}} the group GL_{r} acts via the standard representation (which corresponds to λ_{1} as above).

Definition 4.1. A local shtuka over S ∈ Nilp_{𝔽_{q}[[ζ]]} (of rank r) is a pair (M, φ) where M is a sheaf of O_{S}[[z]]-modules on S which Zariski-locally is free of rank r, together with an isomorphism of O_{S}((z))- modules φ : σ^{∗}M ⊗_{O_{S}[[z]]} O_{S}((z)) ^{∼} ⟶ M ⊗_{O_{S}[[z]]} O_{S}((z)). A quasi-isogeny between local shtukas (M, φ) → (M′, φ′) is an isomorphism of O_{S}((z))-modules f : M ⊗_{O_{S}[[z]]} O_{S}((z)) ^{∼} ⟶ M′ ⊗_{O_{S}[[z]]} O_{S}((z)) with φ′σ^{∗}(f) = fφ.

Lemma 4.2. Under the functor G ↦ G_{λ_{1}} =: M the category of local GL_{r}-shtukas over S with quasiisogenies as morphisms is equivalent to the category of local shtukas over S with quasi-isogenies as morphisms.

Let \underline{M} = (M, φ) be a local shtuka over S ∈ Nilp_{𝔽_{q}[[ζ]]} and let s : Spec L → S be a point with L a field. By the elementary divisor theorem (i.e. the Cartan decomposition for G = GL_{r}) there are L[[z]]-bases m_{1}, …, m_{r} of M_{s} and n_{1}, …, n_{r} of σ^{∗}M_{s} with φ(n_{i}) = z^{−e_{i}} ·m_{i} and e_{i} = e_{i}(\underline{M}; s). The elementary divisors e_{1} ≥ … ≥ e_{r} are called the Hodge weights of \underline{M}_{s}. The decreasing ordering e_{1} ≥ … ≥ e_{r} corresponds to the choice that the Borel subgroup B is the group of upper triangular matrices and that we want (e_{1}, …, e_{r}) to be dominant. The vector μ = (e_{1}, …, e_{r}) is also called the Hodge polygon of \underline{M}_{s}. This name comes from the fact that one often considers the polygon associated to μ which is the graph of the piecewise linear continuous function [0, r] → ℝ with 0 ↦ 0 and slope e_{i} on [i − 1, i]. Note that we chose the opposite ordering e_{1} ≥ … ≥ e_{r} than Katz [Ka] does and therefore the shape of our polygons is opposite to the one of Katz.

The second coordinate e_{1} + … + e_{r} of the endpoint of the polygon associated to μ equals the valuation ord_{z}(det φ) of the determinant of φ with respect to any L[[z]]-bases of M_{s} and σ^{∗}M_{s}. One easily checks that it is a locally constant function on S. In view of π_{1}(GL_{r}) ≅ ℤ and [μ_{\underline{M}}(s)] = ord_{z}(det s^{∗}φ) this gives a simple proof of Proposition 3.4 for G = GL_{r}.

The ordering on X_{∗}(T) can also be visualized using the associated polygons: μ′ ⪯ μ if and only if the polygon associated to μ′ lies below the polygon associated to μ and both polygons have the same endpoint. Writing μ′ = (d_{i}) and μ = (e_{i}) this is equivalent to e_{1} + … + e_{i} ≥ d_{1} + … + d_{i} for all 1 ≤ i ≤ r with equality for i = r.

Lemma 4.3. Let d_{1} ≥ … ≥ d_{r} be integers. Then a local GL_{r}-shtuka \underline{G} is bounded by μ = (d_{1}, …, d_{r}) ∈ ℤ^{r} = X_{∗}(T) if and only if its associated local shtuka (M, φ) satisfies

φ(∧^{i}σ^{∗}M) ⊂ (z − ζ)^{d_{r−i+1}+...+d_{r}} ∧^{i} M for 1 ≤ i ≤ r with equality for i = r .

In this case coker(φ : σ^{∗}M → (z − ζ)^{d_{r}}M) is a locally free sheaf of O_{S}-modules of finite rank on S.

Proof. To prove the first assertion we use Lemma 3.7. The dominant weights λ_{i} = (1, …, 1, 0, …, 0) with 1 repeated i times for 1 ≤ i ≤ r, together with −λ_{r} generate the monoid X^{∗}(T)_{+} of dominant weights of GL_{r}. The Weyl module V (λ_{i}) associated with λ_{i} is the i-th exterior power of the standard representation V (λ_{1}). The Weyl module V (λ_{r}) is one dimensional and therefore equal to V (−λ_{r})^{∨}; cf. [Ja, Remark 3, p. 177]. This is responsible for the equality for i = r.

To prove that coker φ is locally free note that φ : σ^{∗}M → (z − ζ)^{d_{r}}M is injective. Let s : Spec κ(s) → S be a point of S and consider the sequence

0 ⟶ Tor_{1}^{O_{S}}(κ(s), coker φ) ⟶ s^{∗}σ^{∗}M ⟶^{s^{∗}φ} s^{∗}(z − ζ)^{d_{r}}M ⟶ s^{∗} coker φ ⟶ 0 .

As M is locally free we have s^{∗}σ^{∗}M ≅ κ(s)[[z]]^{⊕r} ≅ s^{∗}(z − ζ)^{d_{r}}M. The boundedness condition says for i = r that det(s^{∗}φ) equals z^{d_{1}+...+d_{r}} times a unit. Therefore the map s^{∗}φ is injective. So coker φ is finitely presented by construction and flat over S by Nakayama’s Lemma; e.g. [Ei, Exercise 6.2], hence locally free of finite rank.

Remark 4.4. Over a field S = s = Spec k the elementary divisor theorem (or Lemma 3.11) tells us that \underline{M} is bounded by μ = (d_{i}) if and only if e_{1}(\underline{M}; s) + … + e_{i}(\underline{M}; s) ≤ d_{1} + … + d_{i} for all 1 ≤ i ≤ r with equality for i = r, that is the Hodge polygon μ_{\underline{M}}(s) := (e_{i}(\underline{M}; s)) satisfies μ_{\underline{M}}(s) ⪯ μ.

Example 4.5. Consider μ = (d, 0, …, 0) for some d ≥ 0. Then a local GL_{r}-shtuka \underline{G} over S is bounded by μ if and only if its associated local shtuka (M, φ) satisfies that φ : σ^{∗}M → M is a true morphism of O_{S}[[z]]-modules and coker φ is locally free of rank d on S.

5 Deformation theory of local G-shtukas

Before we treat deformations let us return to the affine Grassmannian Gr. Consider the formal scheme Spf 𝔽_{q}[[ζ]] as the ind-scheme lim_{⟶} Spec 𝔽_{q}[ζ]/(ζ^{n}) and let Ĝr be the fiber product Gr ×_{Spec 𝔽_{q}} Spf 𝔽_{q}[[ζ]]

in the category of ind-schemes ([BD, 7.11.1]). Thus Ĝr is the restriction of the sheaf Gr to the fppf -site of schemes in Nilp_{𝔽_{q}[[ζ]]}. We can also think of Ĝr as the formal completion of Gr ×_{Spec 𝔽_{q}} Spec 𝔽_{q}[[ζ]] along the special fiber V(ζ) ⊂ Gr ×_{Spec 𝔽_{q}} Spec 𝔽_{q}[[ζ]].

The following proposition is a variant of [LS, Proposition 3.10]. Namely loc. cit. treats the case where \underline{G} is a G-torsor over X ×_{𝔽_{q}} S for a smooth projective connected curve, and δ is a trivialization outside a single point p ∈ X. Our proposition is the infinitesimal variant and the argument of loc. cit. carries over literally. We will give a detailed proof of this fact in a slightly modified situation in Theorem 6.2.

Proposition 5.1. The ind-scheme Ĝr pro-represents the functor (Nilp_{𝔽_{q}[[ζ]]})^{o} ⟶ Sets

S ⟼ { Isomorphism classes of pairs (G, δ) where

G is a K-torsor on S and

δ : LG ^{∼} ⟶ LG_{S} is an isomorphism of the associated LG-torsors } .

Here (G, δ) and (G′, δ) are isomorphic if δ^{−1} ∘ δ′ is an isomorphism G′ → G.

Definition 5.2. We write Ĝr = lim_{⟶} X_{n} for some quasi-compact schemes X_{n} ∈ Nilp_{𝔽_{q}[[ζ]]}. Let

either z̃ = z − ζ or z̃ = z. Then the condition that the inverse γ := δ^{−1} of the universal isomorphism δ : LG_{X_{n}} ^{∼} ⟶ LG_{X_{n}} from Proposition 5.1 is bounded by (μ, z̃) is represented by a closed subscheme of X_{n} by Lemma 3.10. The inductive limit of these closed subschemes defines a closed ind-subscheme of Ĝr, which we call Ĝr^{⪯(μ,z̃)}. The fibers over ζ = 0 of Ĝr^{⪯(μ,z−ζ)} and Ĝr^{⪯(μ,z)} coincide as indsubschemes of Gr.

To describe Ĝr^{⪯(μ,z−ζ)} and Ĝr^{⪯(μ,z)} recall that the Cartan decomposition defines a stratification of Gr by the Schubert cells Gr^{μ} := Kz^{(−μ)_{dom}}K/K for dominant coweights μ. Note that our unusual definition of Gr^{μ} is motivated by Proposition 5.5 and Remark 5.8 below. The following result is well-known.

Proposition 5.3. (a) The Schubert cell Gr^{μ} is a locally closed subscheme of Gr which is a quasiprojective scheme over 𝔽_{q}.

(b) The closure of Gr^{μ} in Gr equals the union Gr^{⪯μ} := ⋃_{μ′⪯μ} Gr^{μ′}. The latter is a projective scheme over 𝔽_{q} and Gr^{μ} is open in Gr^{⪯μ}.

(c) Both Gr^{μ} and Gr^{⪯μ} are irreducible and of dimension ⟨2ρ, μ⟩ where 2ρ is the sum of all positive roots of G.

Proof. Note that (−μ)_{dom} = w_{0}(−μ) where w_{0} is the longest element of the Weyl group. Thus μ′ ⪯ μ if and only if (−μ′)_{dom} ⪯ (−μ)_{dom}. Besides, ⟨2ρ, (−μ)_{dom}⟩ = ⟨2ρ, μ⟩. Now the assertion follows from [NP, Lemma 2.2 and the remarks thereafter] or [BD, 4.5.8, 4.5.12], except for the projectivity of Gr^{⪯μ} which we now prove. Since K is an affine scheme, it and Gr^{μ} are quasi-compact. Since there are only finitely many μ′ ⪯ μ, also Gr^{⪯μ} is quasi-compact and then its projectivity follows from the facts that Gr is ind-projective (that is Gr = lim_{⟶} X_{n} for projective 𝔽_{q}-schemes X_{n}; see [BD, Proof of

4.5.1(iv) and 7.11.2(iii)]), that Gr^{⪯μ} is closed in Gr, and the following Lemma 5.4. Note that strictly speaking the results in [BD, §4.5] are proved only over the base field ℂ but their proofs also work over 𝔽_{q}.

Lemma 5.4. Let X be an ind-scheme over a base scheme S, cf. [BD, 7.11.1], that is X is a sheaf of sets for the fppf -topology on S, which can be represented as the limit lim_{⟶} X_{α} of an inductive system of quasi-compact S-schemes X_{α} and closed immersions X_{α} ↪ X_{β} for α ≤ β in a directed set. If U is a quasi-compact S-scheme, then any S-morphism f : U → X, that is section f ∈ X(U), factors through some X_{α_{0}}.

Note that since the presheaf U ↦ lim_{⟶} X_{α}(U) is not a sheaf, an argument for this is required.

Proof. Since X is the sheaf associated with the presheaf U ↦ lim_{⟶} X_{α}(U), the morphism f is given by

a covering U′ → U for the fppf -topology and an element f′ ∈ lim_{⟶} X_{α}(U′). This means that there is an fppf -morphism U′′ → U and a Zariski covering U′′ = ⋃_{i} U′′_{i} such that U′ = ∐_{i} U′′_{i}. In particular f′ ∈ ∏_{i} lim_{⟶} X_{α}(U′′_{i}) and for each i there is an α_{i} with f′|_{U′′_{i}} ∈ X_{α_{i}}(U′′_{i}). Since U is quasi-compact, already finitely many of the U′′_{i} form an fppf -covering of U. Replacing U′ by their union we can assume that U′ is also quasi-compact. Then taking α_{0} as an upper bound of the finitely many α_{i} involved yields f′ ∈ X_{α_{0}}(U′). By fppf -descent [BLR, §6.1, Theorem 6a] the morphism f factors through X_{α_{0}}.

Proposition 5.5. Let either z̃ = z−ζ or z̃ = z. Then the ind-scheme Ĝr^{⪯(μ,z̃)} is a ζ-adic noetherian formal scheme over 𝔽_{q}[[ζ]] whose underlying topological space is Gr^{⪯μ}.

Proof. We claim that Ĝr^{⪯(μ,z̃)} ×_{Spf 𝔽_{q}[[ζ]]} Spec 𝔽_{q}[[ζ]]/(ζ^{n}) =: Y_{n} is a scheme locally of finite type over 𝔽_{q}[[ζ]]/(ζ^{n}) with underlying topological space Gr^{⪯μ}. From the claim the proposition follows by [EGA, I_{new}, Corollary 10.6.4].

To prove the claim let us first recall the structure of Gr as an ind-scheme. Choosing an embedding G ⊂ GL_{r} induces an injection of Gr into the affine Grassmannian for GL_{r} which we denote by G̃r. The ind-scheme structure on G̃r can be described as follows. The valuation ord_{z} det(A) for A ∈ GL_{r}(k((z))) yields a locally constant function on G̃r; see Proposition 3.4. So G̃r is the disjoint union of the G̃r(h) := { A ∈ G̃r : ord_{z} det(A) = h }. Let N be a positive integer and let X̃(h)^{N} be the reduced subscheme of G̃r(h) defined by

X̃(h)^{N}(k) := { A ∈ G̃r(h)(k) : A ∈ M_{r}(z^{−N}k[[z]])/ GL_{r}(k[[z]]) } .

Then the X̃(h)^{N} are projective 𝔽_{q}-schemes, X̃^{N} := ∐_{h} X̃(h)^{N} is a scheme locally of finite type over 𝔽_{q}, and G̃r = lim_{⟶} X̃^{N}. The inclusion Gr ↪ G̃r realizes Gr as a closed ind-subscheme and the induced ind-scheme structure of Gr does not depend on the chosen embedding; see [BD, Theorem 4.5.1] in characteristic zero, but note that the proof also works over 𝔽_{q}.

We consider the G-module V and the induced embedding η : G → GL(V^{∨}) of Proposition 3.14. Let N = N(n, μ) ∈ ℕ be a power of q such that N ≥ n and N ≥ ⟨(−λ)_{dom}, μ⟩ for all λ ∈ Λ. On

Y_{n} we have (z − ζ)^{N} = z^{N} − ζ^{N} = z^{N}. By our definition of Ĝr^{⪯(μ,z̃)}, the universal isomorphism γ := δ^{−1} : LG_{Y_{n}} ^{∼} ⟶ LG|_{Y_{n}} on Y_{n} is bounded by (μ, z̃), hence satisfies

γ(K_{Y_{n}})_{λ} = γ(V (λ) ⊗_{𝔽_{q}} O_{Y_{n}}[[z]]) ⊂ z̃^{−⟨(−λ)_{dom},μ⟩} G_{λ} ⊂ z^{−N}G_{λ}

for all λ ∈ Λ and both choices of z̃. By Proposition 2.3 we may choose bases of the G_{λ} Zariski-locally on Y_{n}. Then γ is represented on V = ⨁_{λ∈Λ} V (λ) as a matrix A in the right coset

A ∈ GL_{dim V} (Y_{n}[[z]])\M_{dim V} (z^{−N}Y_{n}[[z]]) .

On V^{∨} = (⨁_{λ∈Λ} V (λ))^{∨} its inverse δ = γ^{−1} is represented by its transpose

A^{T} ∈ M_{dim V} (z^{−N}Y_{n}[[z]])/ GL_{dim V} (Y_{n}[[z]]) .

Therefore Y_{n} is a closed subscheme of X̃^{N} ×_{Spec 𝔽_{q}} Spec 𝔽_{q}[[ζ]]/(ζ^{n}), hence locally of finite type over 𝔽_{q}[[ζ]]/(ζ^{n}). The reduced closed subscheme underlying Y_{n} can be computed by looking at geometric points x ∈ Gr. Over x the isomorphism γ_{x} = δ_{x}^{−1} is given by an element g ∈ K(κ(x))z^{μ_{γ}(x)}K(κ(x)) after choosing a trivialization of G_{x}. By definition of Ĝr this means that the point x is given by g^{−1} ∈ LG(κ(x)) with g^{−1} ∈ Gr^{μ_{γ}(x)}. By Lemma 3.11 the point x belongs to Y_{n} if and only if μ_{γ}(x) ⪯ μ, which is equivalent to x lying in Gr^{⪯μ}.

Let μ ∈ X_{∗}(T) be dominant and let \underline{𝔾} = (𝔾, φ_{𝔾}) be a local G-shtuka bounded by μ over a field k′ ∈ Nilp_{𝔽_{q}[[ζ]]}. Assume that there is a trivialization α : 𝔾 ^{∼} ⟶ K_{k′} such that α : \underline{𝔾} ^{∼} ⟶ (K_{k′}, b_{0}σ^{∗}) for some b_{0} ∈ LG(k′). Note that if k′ is algebraically closed, a trivialization always exists by Remark 3.2. The inverse b_{0}^{−1} of b_{0} defines a point x ∈ Gr(k′) (which depends on the chosen trivialization). Since \underline{𝔾} is bounded by μ, (b_{0}^{−1})^{−1} = b_{0} is bounded by μ and we have x ∈ Gr^{⪯μ}(k′) by Proposition 5.5. Let

D be the complete local ring of Ĝr^{⪯(μ,z−ζ)} at the point x. It is a complete noetherian local ring over k′[[ζ]].

Theorem 5.6. D pro-represents the formal deformation functor of \underline{𝔾}

F : (Artinian local k′[[ζ]]-algebras with residue field k′) ⟶ Sets ,
A ⟼ \{ Isomorphism classes of pairs (\underline{G}, β) where
\underline{G} is a local G-shtuka over Spec A bounded by μ and
β : \underline{𝔾} ^{∼} ⟶ \underline{G} ⊗_{A} k′ is an isomorphism of local G-shtukas \}

where (\underline{G}, β) and (\underline{G}′, β′) are isomorphic if there exists an isomorphism η : \underline{G} → \underline{G}′ with β′ = (η ⊗_{A} k′) ∘ β.

Proof. We first construct the universal pro-object over D. Let D_{n} = D/𝔪_{D}^{n} where 𝔪_{D} is the maximal ideal of D. By [NP, Lemma 2.1] the projection morphism LG → Gr has a section in an open neighborhood of the image of 1 ∈ LG(k′). Translating by b_{0}^{−1} ∈ LG(k′) we obtain a section in an open

neighborhood of x ∈ Gr(k′). Since 𝔪_{D} is nilpotent on D_{n}, the natural map Spec D_{n} → Ĝr^{⪯(μ,z−ζ)} → Ĝr can be given by some element b̃_{n}^{−1} ∈ LG(D_{n}) = G(D_{n}[[z]][\frac{1}{z−ζ}]) with b̃_{n}^{−1} ⊗_{D_{n}} k′ = b_{0}^{−1}. Note that b̃_{n}^{−1} is only determined up to multiplication on the right with elements g ∈ K(D_{n}) satisfying g ⊗_{D_{n}} k′ = 1. By construction γ_{n} = (b̃_{n}^{−1})^{−1} = b̃_{n} is bounded by (μ, z − ζ). We choose the b̃_{n} in a compatible way and let b̃ ∈ G(D[[z]][\frac{1}{z−ζ}]) be the limit. The boundedness by μ of all b̃_{n} implies the existence and boundedness by μ of b̃ by the same argument as in Proposition 3.16. Although b̃^{−1} is still only determined up to right multiplication with elements g ∈ K(D) as above, we take \underline{G}^{univ} := (K_{D}, b̃σ^{∗}) and α : \underline{𝔾} ^{∼} ⟶ \underline{G}^{univ} ⊗_{D} k′ so that (\underline{G}^{univ}, α) ∈ F (D) because b̃ and b̃σ^{∗} are bounded by μ. In particular we obtain a morphism of functors Hom_{k′[[ζ]]}(D, .) → F by sending u : D → A to the deformation (K_{A}, u(b̃)σ^{∗}, α) ∈ F (A).

We will show that this morphism of functors is an isomorphism. Let (\underline{G}, β) ∈ F (A). Let 𝔪 ⊂ A be the maximal ideal. We use induction on A_{n} := A/𝔪^{q^{n}} to show the following Claim. For each n there is a uniquely determined k′[[ζ]]-homomorphism u_{n} : D → A_{n} with

(K_{A_{n}}, u_{n}(b̃)σ^{∗}, α) = (\underline{G}, β) ⊗_{A} A_{n} in F (A_{n}) .

For n = 0 we have A_{0} = k′ and there is only one k′[[ζ]]-homomorphism, namely u_{0} : D → D/𝔪_{D} = k′ = A_{0}. This proves the claim for n = 0 since u_{0}(b̃) = b_{0} and (K_{k′}, b_{0}σ^{∗}, α) is isomorphic to (\underline{G}, β) ⊗_{A} k′ via β ∘ α^{−1}.

Now let n ≥ 1 and assume that we have already constructed the uniquely determined u_{n−1}. In particular there is an isomorphism η : (K_{A_{n−1}}, u_{n−1}(b̃)σ^{∗}) ^{∼} ⟶ \underline{G} ⊗_{A} A_{n−1} with (η ⊗_{A_{n−1}} k′) ∘ α = β. By Proposition 2.2 the trivialization η lifts to a trivialization η′ : K_{A_{n}} ^{∼} ⟶ G ⊗_{A} A_{n}. We let b_{n} := (η′)^{−1} φ_{G} σ^{∗}(η′) ∈ G(A_{n}[[z]][\frac{1}{z−ζ}]) and replace (\underline{G}, β) ⊗_{A} A_{n} by (K_{A_{n}}, b_{n}σ^{∗}, α). Then b_{n} ⊗_{A_{n}} A_{n−1} = u_{n−1}(b̃). Consider the diagram

[figure: Spec A_{n}; b_{n}^{−1}; Spec u_{n}; LG; Ĝr; Spec A_{n−1}; Spec u_{n−1}; Spec D; b̃^{−1}; _{Ĝr_{⪯(μ,z−ζ)}}]

Here the right square commutes by the definition of b̃. The element b_{n}^{−1} yields a morphism Spec A_{n} → Ĝr which factors through Ĝr^{⪯(μ,z−ζ)}, since \underline{G} and hence (b_{n}^{−1})^{−1} are bounded by μ. As b_{n}⊗_{A_{n}} k′ = b_{0},

the maximal ideal of A_{n} is sent to x ∈ Ĝr^{⪯(μ,z−ζ)}. Since A_{n} is complete, the morphism even factors through Spec D. We obtain a homomorphism u_{n} : D → A_{n} with u_{n}(b̃) ⊗_{A_{n}} A_{n−1} = u_{n−1}(b̃) and such that g := b_{n}u_{n}(b̃^{−1}) ∈ K(A_{n}) by definition of Ĝr. In particular g ⊗_{A_{n}} A_{n−1} = 1, hence σ^{∗}(g) = 1. We now replace (K_{A_{n}}, b_{n}σ^{∗}, α) by (K_{A_{n}}, g^{−1}b_{n}σ^{∗}(g) · σ^{∗}, α), that is, we change η′ to η′g. Then u_{n}(b̃) = g^{−1}b_{n} = g^{−1}b_{n}σ^{∗}(g) as desired.

It remains to show that u_{n} is uniquely determined. Let u_{n}, u′_{n} : D → A_{n} be two homomorphisms as in our claim. Applying ⊗_{A_{n}}A_{n−1} and exploiting the uniqueness for n − 1 we see that u_{n}(b̃) ⊗_{A_{n}} A_{n−1} = u′_{n}(b̃) ⊗_{A_{n}} A_{n−1}. So there is a g ∈ K(A_{n}) with g^{−1}u_{n}(b̃)σ^{∗}(g) = u′_{n}(b̃) and g ⊗_{A_{n}} A_{n−1} = 1. In particular σ^{∗}(g) = 1 and hence u′_{n}(b̃^{−1}) = u_{n}(b̃^{−1})g. This implies that the compositions of u_{n} and u′_{n} with Spec D → Ĝr coincide. By the definition of D this implies u_{n} = u′_{n} as desired.

Remark 5.7. From now on we always consider the deformation ring D with a fixed isomorphism of functors Hom_{k′[[ζ]]}(D, .) ≅ F (and thus a fixed universal local G-shtuka over D) as in the proof.

Remark 5.8. One could avoid the inversion b_{0}^{−1} and the definition of Gr^{μ} as Kz^{(−μ)^{dom}}K/K by

instead considering the affine Grassmannian (K\LG) ×_{Spec 𝔽_{q}} Spf 𝔽_{q}[[ζ]] and defining Ĝr^{⪯(μ,z−ζ)} as a closed subscheme of the latter. On the other hand, it is not clear to us whether the two conditions that δ^{−1} is bounded by μ and that δ is bounded by (−μ)_{dom} are equivalent over a non-reduced scheme.

Therefore we do not know whether one could also define Ĝr^{⪯(μ,z−ζ)} as the subscheme on which δ is bounded by (−μ)_{dom}. As mentioned in Remark 3.6, we could in the definition of boundedness also take the class of all G-modules of lowest weight (−λ)_{dom} or the class of all tilting modules. The latter are self-dual by [Ja, E.6]. Then we would have that δ^{−1} is bounded by μ if and only if δ is bounded by (−μ)_{dom}.

Proposition 5.9. Let D be the deformation ring from Theorem 5.6. Then the reduced quotient \overline{D} = (D/ζD)_{red} is a complete normal noetherian integral domain of dimension ⟨2ρ, μ⟩ and Cohen-Macaulay.

Proof. The ring \overline{D} equals the completion of the local ring of x in Gr^{⪯μ}. Since Gr^{⪯μ} is irreducible, reduced, normal, and Cohen-Macaulay (by [PR1, Theorem 6.1] for SL_{n} or [Fa, Theorem 8 and Remark before Corollary 11] in general) this local ring is a normal integral domain and Cohen-Macaulay. By Zariski’s Main Theorem [ZS, Theorem VIII.32] and [Ei, Proposition 18.8] the same is then true for its completion \overline{D}. By Proposition 5.3, Gr^{⪯μ} has dimension ⟨2ρ, μ⟩. Thus the same holds for \overline{D}.

Note that if one is only interested in the dimension (as we are in Proposition 7.8), one does not need the normality. Namely, Gr^{⪯μ} is equidimensional of dimension ⟨2ρ, μ⟩. Thus the same holds for every local ring. By [HIO, Theorem 18.17] the completion of a local equidimensional and universally catenary ring is equidimensional.

Example 5.10. Contrary to p-divisible groups the deformation space of a local G-shtuka is in general not smooth. Consider for example the ring A = 𝔽_{q}[[ζ]]/(ζ^{2}) and the local shtuka

(M, φ) = (A[[z]]^{⊕2}, ( \begin{matrix}z & 0 \\ 0 & z − 2ζ \end{matrix} ) σ^{∗}).

According to Lemma 4.3 and since the determinant of φ equals (z − ζ)^{2}, it is bounded by (2, 0) . We claim that it cannot be deformed to a local shtuka over Ã = 𝔽_{q}[[ζ]]/(ζ^{3}) bounded by any μ (which is then of the form (μ_{1}, 2 − μ_{1})_{dom}). Indeed assume there is a deformation (M̃, φ̃) = (Ã[[z]]^{⊕2}, b̃σ^{∗}) with

b̃ − ( \begin{matrix}z & 0 \\ 0 & z − 2ζ \end{matrix} ) = ζ^{2} ( \begin{matrix}c_{11} & c_{12} \\ c_{21} & c_{22} \end{matrix} ) ∈ ζ^{2}M_{2}(Ã[[z]]).

Then det b̃ = z^{2} − 2ζz + ζ^{2}z(c_{11} + c_{22}) and the boundedness by (μ_{1}, 2 − μ_{1}) requires that this determinant differs from (z − ζ)^{2} by a unit in Ã[[z]]^{×}. However this is not the case.

Nevertheless deformation from k′ to k′[[ζ]] is always possible:

Lemma 5.11. Let \underline{𝔾} be a local G-shtuka over a field k′ ∈ Nilp_{𝔽_{q}[[ζ]]} of the form \underline{𝔾} = (K_{k′}, bσ^{∗}) for some b ∈ LG(k′). Then there exists a local G-shtuka \underline{𝔾̂} over k′[[ζ]] with \underline{𝔾̂} ⊗_{k′[[ζ]]} k′ ≅ \underline{𝔾}. If \underline{𝔾} is bounded by some dominant coweight μ ∈ X_{∗}(T) then we can find a \underline{𝔾̂} which is also bounded by μ.

Proof. Write b = s_{0}z^{μ′}t_{0} for some dominant μ′ ∈ X_{∗}(T) and s_{0}, t_{0} ∈ K(k′). Choose lifts s, t ∈ K(k′[[ζ]]) of s_{0} and t_{0} and set \underline{𝔾̂} := (K_{k′[[ζ]]}, s(z − ζ)^{μ′}t · σ^{∗}). Then \underline{𝔾̂} is a local G-shtuka over k′[[ζ]]. Moreover, \underline{𝔾} is bounded by μ if and only if μ′ ⪯ μ by Lemma 3.11. Then ⟨(−λ)_{dom}, μ′⟩ ≤ ⟨(−λ)_{dom}, μ⟩ for all dominant weights λ. This shows that \underline{𝔾̂} is bounded by μ.

6 Rapoport-Zink spaces for local G-shtukas and affine Deligne-Lusztig varieties

We explain the relation between local G-shtukas and affine Deligne-Lusztig varieties. Let k′ be a field containing 𝔽_{q}. Recall that LG(k′) = ∐ K(k′)z^{μ}K(k′) where the union is over all dominant coweights μ. We recall the definition of affine Deligne-Lusztig varieties from [GHKR1].

Definition 6.1. For an element b ∈ LG(k′) and a dominant μ ∈ X_{∗}(T) the affine Deligne-Lusztig variety X_{μ}(b) is the locally closed reduced ind-subscheme over k′ of the affine Grassmannian Gr_{k′} := Gr ⊗_{𝔽_{q}}k′ defined by

X_{μ}(b)(\overline{k′}) = \{ g ∈ Gr_{k′}(\overline{k′}) : g^{−1}bσ^{∗}(g) ∈ K(\overline{k′})z^{μ}K(\overline{k′}) \} ,

where \overline{k′} is an algebraic closure of k′. The closed affine Deligne-Lusztig variety X_{⪯μ}(b) is the closed reduced ind-subscheme of Gr_{k′} defined by

X_{⪯μ}(b)(\overline{k′}) = ⋃_{μ′⪯μ} X_{μ′}(b)(\overline{k′}).

Left multiplication by g ∈ LG(k′) induces an isomorphism between X_{μ}(g^{−1}bσ^{∗}(g)) and X_{μ}(b). There is a criterion for X_{μ}(b) to be non-empty, see [KR] and [Ga]. In Corollary 6.5 we give a proof for the well known fact that both X_{μ}(b) and X_{⪯μ}(b) are schemes locally of finite type over k′.

We fix a local G-shtuka \underline{𝔾} over k′ of the form \underline{𝔾} = (K_{k′}, bσ^{∗}) for some b ∈ LG(k′). For a scheme S ∈ Nilp_{k′[[ζ]]} we denote by S̄ the closed subscheme V(ζ) ⊂ S. Recall the ind-scheme Ĝr from Section 5 and set Ĝr_{k′} := Ĝr ⊗_{𝔽_{q}} k′.

Theorem 6.2. The ind-scheme Ĝr_{k′} pro-represents the functor (Nilp_{k′[[ζ]]})^{o} ⟶ Sets

S ⟼ \{ Isomorphism classes of pairs (\underline{G}, δ̄) where
\underline{G} is a local G-shtuka over S and
δ̄: \underline{G}S̄ → \underline{𝔾}S̄ is a quasi-isogeny \}

Here (\underline{G}, δ̄) and (\underline{G}′, δ̄′) are called isomorphic if δ̄^{−1} ∘ δ̄′ lifts to an isomorphism \underline{G}′ → \underline{G}.

Proof. Let us first reformulate the theorem by lifting the statement to k′[[ζ]]. By Lemma 5.11 we may choose a local G-shtuka \underline{𝔾̂} over k′[[ζ]] which lifts \underline{𝔾}. Note that 𝔾̂ has a trivialization 𝔾̂ = (K_{k[[ζ]]}, b̂σ^{∗}); see the proof of Lemma 5.11. By rigidity of quasi-isogenies (Proposition 3.9) the functor in question is isomorphic to the functor (Nilp_{k′[[ζ]]})^{o} ⟶ Sets

S ⟼ \{ Isomorphism classes of pairs (\underline{G}, δ) where \underline{G} is a local
G-shtuka over S and δ : \underline{G} → \underline{𝔾̂}_{S} is a quasi-isogeny \}

Here (\underline{G}, δ) and (\underline{G}′, δ′) are called isomorphic if δ^{−1} ∘ δ′ is an isomorphism \underline{G}′ → \underline{G}.

Let x ∈ Ĝr_{k′}(S) for a scheme S ∈ Nilp_{k′[[ζ]]}. The projection morphism LG → Gr admits local sections for the étale topology by [BD, Theorem 4.5.1]. Note that in [BD] this is proved to hold even Zariski locally over an algebraically closed field of characteristic zero. Using [NP, Lemma 2.1] the proof carries over for our base field 𝔽_{q} after allowing a finite separable extension of 𝔽_{q}. Consequently there is an étale covering S′ → S such that x is represented by an element g′ ∈ LG(S′). Define (G′, φ′, δ′) over S′ as follows. Let G′ = K_{S′}, let the quasi-isogeny δ′ : (G′, φ′) → \underline{𝔾̂}_{S′} be given by y ↦ g′y, and the Frobenius by φ′ = (g′)^{−1}b̂σ^{∗}(g′)σ^{∗}. We descend (G′, φ′, δ′) to S. For an S-scheme Y let Y′ = Y ×_{S} S′ and Y′′ = Y′ ×_{Y} Y′, and let p_{i} : Y′′ → Y′ be the projection onto the i-th factor. Since g′ comes from an element x ∈ Ĝr_{k′}(S) there is an h ∈ K(S′′) with p_{1}^{∗}(g′) = p_{2}^{∗}(g′) · h. Consider the fpqc-sheaf G on S whose sections over an S-scheme Y are given by

G(Y) := { y′ ∈ K(Y′) : p_{1}^{∗}(y′) = h^{−1} · p_{2}^{∗}(y′) in K(Y′′) }

on which K(Y) acts by right multiplication. Then \underline{G} is a K-torsor on S because over Y = S′ there is a trivialization

K_{S′} ^{∼} ⟶ G_{S′} , f ↦ h p_{1}^{∗}(f) ∈ K(S′′)

due to the cocycle condition on h. Moreover, φ′ descends to an isomorphism φ : σ^{∗}LG(Y) ^{∼} ⟶ LG(Y) , σ^{∗}(y′) ↦ (g′)^{−1}b̂σ^{∗}(g′)σ^{∗}(y′) making (G, φ) a local G-shtuka over S. Also δ′ descends to a quasi-isogeny of local G-shtukas

δ : LG(Y) ^{∼} ⟶ LG(Y) = { f′ ∈ LG(Y′) : p_{1}^{∗}(f′) = p_{2}^{∗}(f′) in LG(Y′′) } , y′ ↦ g′y′ .

Note that this is well defined. Namely, if g′ is replaced by g̃′ with u′ = (g̃′)^{−1}g′ ∈ K(S′) then left multiplication with u′ defines an isomorphism (K_{S′}, (g′)^{−1}b̂σ^{∗}(g′)σ^{∗}, g′) ^{∼} ⟶ (K_{S′}, (g̃′)^{−1}b̂σ^{∗}(g̃′)σ^{∗}, g̃′). Also h̃ = p_{2}^{∗}(u′) h p_{1}^{∗}(u′)^{−1} and hence left multiplication with u′ descends to an isomorphism \underline{G} ^{∼} ⟶ \underline{G̃} over S.

Conversely let \underline{G} = (G, φ) be a local G-shtuka and let δ : \underline{G} → \underline{𝔾}_{S} be a quasi-isogeny over S. By Proposition 2.2 there is an étale covering S′ → S such that the pull back of G to S′ is trivial. After choosing a trivialization, the map δ is given by an element g′ ∈ LG(S′) whose image in Ĝr_{k′}(S′) is independent of the chosen trivialization. The quasi-isogeny δ also determines φ = (g′)^{−1}b̂σ^{∗}(g′)σ^{∗}. Since \underline{G} was defined over S the element g′ descends to a point x ∈ Ĝr_{k′}(S). Clearly these two constructions are inverse to each other.

Next we want to show that X_{⪯μ}(b) is the underlying reduced subscheme of an equal characteristic Rapoport-Zink space for local G-shtukas. More precisely, consider again the local G-shtuka \underline{𝔾} = (K_{k′}, bσ^{∗}) from above. Note that each isogeny class of local G-shtukas has a representative over a finite field, so we may assume that k′ is finite. Besides we may assume that bσ^{∗} is decent, i.e. there are an integer s > 0 and a cocharacter ω ∈ X_{∗}(T), such that

(bσ^{∗})^{s} := b σ^{∗}(b) · . . . · (σ^{∗})^{s−1}(b) (σ^{∗})^{s} = z^{ω}(σ^{∗})^{s} .  (6.1)

Let μ ∈ X_{∗}(T) be a dominant coweight such that \underline{𝔾} is bounded by μ, and consider the functor M : (Nilp_{k′[[ζ]]})^{o} → Sets,

S ⟼ \{ Isomorphism classes of pairs (\underline{G}, δ̄) where
\underline{G} is a local G-shtuka over S bounded by μ and
δ̄: \underline{G}S̄ → \underline{𝔾}S̄ is a quasi-isogeny \} .

Again S̄ denotes the closed subscheme V(ζ) ⊂ S.

Theorem 6.3. Let X̂_{⪯μ}(b) be the closed ind-subscheme of Ĝr_{k′} defined by the condition that the universal G-shtuka \underline{G} over Ĝr_{k′} from Theorem 6.2 is bounded by μ. Then X̂_{⪯μ}(b) is a formal scheme over Spf k′[[ζ]] which is locally formally of finite type. It pro-represents the functor M : (Nilp_{k′[[ζ]]})^{o} → Sets. Its underlying reduced subscheme equals X_{⪯μ}(b).

Here by the underlying reduced subscheme we mean the subscheme corresponding to the largest ideal of definition. Also recall that a formal scheme over k′[[ζ]] in the sense of [EGA, I_{new}, 10] is called locally formally of finite type if it is locally noetherian and adic and its reduced subscheme is locally of finite type over k′. It is called formally of finite type if in addition it is quasi-compact. For the proof of the theorem we need the following proposition in which 2ρ^{∨} is the sum of all positive coroots of G.

Proposition 6.4. Let b ∈ LG(k′) satisfy a decency condition with the integer s as in (6.1). Assume k′ ⊂ 𝔽_{q^{s}}. Then there is a natural number d_{0} such that for every algebraically closed field k and any point (\underline{G}, δ̄) ∈ M(k) there is a point (\underline{G}′, δ̄′) ∈ M(𝔽_{q^{s}}) such that δ̄^{−1} ∘ δ̄′ is bounded by 2d_{0}ρ^{∨}.

Proof. By trivializing \underline{G} the assertion is equivalent to the statement that for any g ∈ LG(k) with g ∈ X_{⪯μ}(b)(k) there is a g′ ∈ LG(𝔽_{q^{s}}) with g′ ∈ X_{⪯μ}(b)(𝔽_{q^{s}}) and such that g^{−1}g′ ∈ Kz^{μ′}K for some μ′ ⪯ 2d_{0}ρ^{∨}. This last condition is especially satisfied if the distance of g and g′ in the Bruhat-Tits building of G is less than d_{0}. Hence the proposition follows from [RZ2, Theorem 1.4 and Subsection 2.1].

Proof of Theorem 6.3. Note that by construction X̂_{⪯μ}(b) represents M. Also X̂_{⪯μ}(b) is a closed ind-subscheme of Ĝr_{k′} by Lemma 3.10. We show that the underlying topological space of X̂_{⪯μ}(b) is the ind-scheme X_{⪯μ}(b). As X_{⪯μ}(b) and the underlying reduced ind-subscheme of X̂_{⪯μ}(b) are both reduced ind-subschemes of Gr, it then follows that they are equal. Let x ∈ Ĝr be a point with values in an algebraically closed field κ(x). Put (\underline{Ḡ}, δ̄) := (\underline{G}, δ)_{Spec κ(x)}, and let δ̄ be given by g ∈ LG(κ(x)) with respect to some trivialization of \underline{Ḡ} = (Ḡ, φ̄). Then φ̄ = g^{−1}bσ^{∗}(g) · σ^{∗} and g ∈ X_{μ′}(b) for the Hodge polygon μ′ = μ\underline{Ḡ}(x) of (Ḡ, φ̄) (Definition 3.3). We have that g ∈ X̂_{⪯μ}(b) if and only if (Ḡ, φ̄) is bounded by μ. By Lemma 3.11 this is equivalent to μ′ ⪯ μ, or to g ∈ X_{⪯μ}(b) as desired.

It remains to show that X̂_{⪯μ}(b) is a formal scheme locally formally of finite type. We follow the proof of [RZ, Theorem 2.16]. By Lemma 5.11 we can choose a lift \underline{𝔾̂} of \underline{𝔾} to k′[[ζ]] which is bounded by μ, and using rigidity (Proposition 3.9) we replace M by the isomorphic functor (Nilp_{k′[[ζ]]})^{o} ⟶ Sets

S ⟼ \{ Isomorphism classes of pairs (\underline{G}, δ) where
\underline{G} is a local G-shtuka over S bounded by μ and
δ : \underline{G} → \underline{𝔾̂}_{S} is a quasi-isogeny \} .

For v ∈ π_{1}(G) let M(v) be the open and closed ind-subscheme of M = X̂_{⪯μ}(b) where the image of δ in π_{1}(G) is v (using Proposition 3.4). By [Vi2, Proof of Lemma 1] there is a self-quasi-isogeny j_{v} of \underline{𝔾̂} whose image in π_{1}(G) is the given v. Composition of the quasi-isogenies δ with j_{v} yields an isomorphism between the ind-subschemes M(0) and M(v). Hence it is enough to prove the theorem for \underline{M} := M(0) instead of X̂_{⪯μ}(b).

We consider two kinds of ind-schemes related to \underline{M}. Let n ∈ ℕ. By Lemma 3.10, the condition that the universal δ, δ^{−1}, and σ^{∗}δ over X̂_{⪯μ}(b) are all bounded by (2nρ^{∨}, z) is represented by a closed ind-subscheme M^{n} of X̂_{⪯μ}(b). Note that (3.2) in Definition 3.5 implies that M^{n} is contained in \underline{M}. We show that M^{n} is a ζ-adic noetherian formal scheme over k′[[ζ]]. The condition that the universal δ on Ĝr from Proposition 5.1, as well as δ^{−1} and σ^{∗}(δ) are all bounded by (2nρ^{∨}, z) is represented by a closed ind-subscheme X̂^{n} of Ĝr due to Lemma 3.10. By an argument analogous to Proposition 5.5, X̂^{n} is a ζ-adic noetherian formal scheme over k′[[ζ]] whose underlying topological space is the projective scheme Gr^{⪯2nρ^{∨}} from Proposition 5.3. The Frobenius φ_{G} of the universal Gshtuka \underline{G} = (G, φ_{G}) over X̂^{n} from Theorem 6.2 satisfies φ_{G} = δ^{−1} ∘ φ𝔾̂ ∘ σ^{∗}(δ) and hence φ_{G}(σ^{∗}G_{λ}) ⊆ z^{−⟨(−λ)_{dom},4nρ^{∨}⟩}(z − ζ)^{−⟨(−λ)_{dom},μ⟩}G_{λ} due to the definition of X̂^{n} and the boundedness of \underline{𝔾̂} by μ. Analogous to Lemma 3.10, the condition that φ_{G} as above is bounded by (μ, z − ζ) is equivalent to the following condition. For each λ, the images of the generators of σ^{∗}G_{λ} in the locally free sheaf z^{−⟨(−λ)_{dom},4nρ^{∨}⟩}(z −ζ)^{−⟨(−λ)_{dom},μ⟩}G_{λ}/(z −ζ)^{−⟨(−λ)_{dom},μ⟩}G_{λ} of finite rank on X̂^{n} have to vanish. Hence M^{n} is a ζ-adic noetherian closed formal subscheme of X̂^{n} over k′[[ζ]] whose reduced subscheme is projective.

For n ∈ ℕ let M_{n} be the ind-scheme which is the formal completion of X̂_{⪯μ}(b) along the closed subset (M^{n})_{red}, where (M^{n})_{red} denotes the topological space underlying M^{n}. In particular (M^{n})_{red} = (M_{n})_{red}. Over a field in Nilp_{k′[[ζ]]}, Lemma 3.11 implies that δ_{s} is bounded by 2nρ^{∨} if and only if δ^{−1} or σ^{∗}δ are bounded by 2nρ^{∨}. Hence M_{n} represents the subfunctor of M

S ⟼ \{(\underline{G}, δ) ∈ M(S) : for all closed points s ∈ S the map δ_{s} is bounded by 2nρ^{∨}\}.

Claim. M_{n} is representable by a formal scheme which is formally of finite type over Spf k′[[ζ]].

Fix n and for each m ≥ n let M_{n}^{m} be the formal completion of M^{m} along (M_{n})_{red}, i.e. the indscheme on which all the above boundedness conditions are satisfied. It is an adic noetherian formal scheme over k′[[ζ]]. We fix an affine open subscheme U of (M_{n})_{red}. For m ≥ n we have (M_{n})_{red} = (M_{n}^{m})_{red}, and thus we get an affine open formal subscheme Spf R_{m} of M_{n}^{m} whose underlying set is U. Since by construction M^{m} → M^{m+1} is a closed immersion, we have a projective system of surjective maps of adic rings R_{m+1} → R_{m}. Let R be its limit. We write R_{m} = R/𝔞_{m} for ideals 𝔞_{m} ⊂ R. Let J be the inverse image in R of the largest ideal of definition of R_{n}. We have to prove that R is a J-adic ring. Since R_{m} is J-adic for all m we may write R = lim_{⟵} R/(𝔞_{m} + J^{c}). The limit is taken independently over m and c. By [RZ, 2.5] it is enough to show that for each c the descending sequence 𝔞_{m} + J^{c} ⊇ 𝔞_{m+1} + J^{c} ⊇ ⋯ stabilizes. Let \underline{G}_{m} be the universal local G-shtuka on Spf R_{m}. Since R/J^{c} = lim_{⟵} R_{m}/J^{c} is an admissible ring, \underline{G} = lim_{⟵} \underline{G}_{m} defines a local G-shtuka on Spec R/J^{c} for every c by Proposition 3.16. By rigidity (Proposition 3.9), we may also lift the quasi-isogeny δ from R/J = R_{n}/J to R/J^{c}. We want to see that there is an integer m_{0} ≥ n, such that δ, δ^{−1}, and σ^{∗}δ are bounded by 2m_{0}ρ^{∨} over R/J^{c}. By the second assertion of Proposition 3.9, they are all bounded by some coweight ω whose image in π_{1}(G) is trivial. Choosing m_{0} such that ω ⪯ 2m_{0}ρ^{∨} we obtain a bound of the desired form. Hence by the universal property of M_{n}^{m_{0}} there is a unique map R_{m_{0}} → R/J^{c} inducing the given point (\underline{G}, δ). For any m ≥ m_{0} the composite R_{m} → R_{m_{0}} → R/J^{c} → R_{m}/J^{c}R_{m} is the projection, since both maps induce (\underline{G}_{m}, δ) over R_{m}/J^{c}R_{m}. Thus the first map yields an isomorphism R_{m}/J^{c}R_{m} → R_{m_{0}}/J^{c}R_{m_{0}}. This implies that the sequence above stabilizes and proves our claim.

Let d_{0} be as in Proposition 6.4. The assertion of the proposition obviously remains valid for \underline{M}. For two closed points x = (\underline{G}, δ), x′ = (\underline{G}′, δ′) ∈ \underline{M} we define d(x, x′) to be the smallest n ∈ ℕ such that δ^{−1} ∘ δ′ is bounded by 2nρ^{∨}. For a point y ∈ \underline{M}(𝔽_{q^{s}}) we denote by M_{n}(y) the closed subset of points x of M_{n} with d(x, y) ≤ d_{0}. By the triangular inequality for d we obtain that M_{n}(y) = ∅ if d((\underline{𝔾}, id), y) > n + d_{0}, that is if y ∉ M_{n+d_{0}}. For each positive integer f let U_{n}^{f} be the open formal subscheme of M_{n} whose underlying set is the complement of

⋃_{y∈\underline{M}(𝔽_{q^{s}}) , d((\underline{𝔾}, id),y)≥f} M_{n}(y) = ⋃_{y∈M_{n+d_{0}}(𝔽_{q^{s}}) , d((\underline{𝔾}, id),y)≥f} M_{n}(y) .

Note that this union is finite because the underlying reduced subscheme of M_{n+d_{0}} is the projective scheme (M^{n+d_{0}})_{red}.

Claim 2. If n ≥ f + d_{0} then U_{n}^{f} = U_{n+1}^{f}.

We first show this for the underlying sets. Let x ∈ U_{n+1}^{f}(k) be a point with values in an algebraically closed field k. We have to show that d((\underline{𝔾}, id), x) ≤ n. By Proposition 6.4 there is a y ∈ \underline{M}(𝔽_{q^{s}}) such that d(x, y) ≤ d_{0} and x ∈ M_{n+1}(y). By the definition of U_{n+1}^{f}, we have d((\underline{𝔾}, id), y) < f, hence d((\underline{𝔾}, id), x) < f + d_{0} ≤ n. The equality of formal schemes follows because M_{n} is the completion of M_{n+1} along the (closed) image of (M_{n})_{red}. Indeed, this implies that U_{n}^{f} is the completion of U_{n+1}^{f} along U_{n+1}^{f}. Hence the claim follows.

Let U^{f} = U_{n}^{f} for some n ≥ f + d_{0}. Then U^{f} → U^{f+1} is an open immersion of formal schemes formally of finite type. Moreover, \underline{M} has the limit topology of the limit over its intersections with the subschemes of Ĝr_{k′} where the universal δ is bounded by 2nρ^{∨}. The underlying topological space of such an intersection is equal to that of M_{n}. Since the topological space |U^{f}| underlying U^{f} is open in each M_{n}, it is also open in \underline{M}. This shows that the formal scheme U^{f} equals the formal completion of the open ind-scheme \underline{M}|_{|U^{f} |} of \underline{M} supported on |U^{f}| along the whole set |U^{f}|, whence \underline{M}|_{|U^{f} |} = U^{f}. Since U^{f} is formally of finite type by construction, also \underline{M}|_{|U^{f} |} is. In order to prove the theorem it remains to show that \underline{M} = ⋃_{f} U^{f}. We prove that every point x ∈ \underline{M} with d((\underline{𝔾}, id), x) < f − d_{0} is contained in the open set U^{f}. Indeed, if x is in the complement of U^{f}, there is a y ∈ \underline{M}(𝔽_{q^{s}}) with d(x, y) ≤ d_{0} and d((\underline{𝔾}, id), y) ≥ f, a contradiction. The theorem follows.

Corollary 6.5. Let b ∈ LG(k′) be decent. The affine Deligne-Lusztig varieties X_{⪯μ}(b) and X_{μ}(b) are schemes locally of finite type over k′. The irreducible components of X_{⪯μ}(b) (respectively of X_{μ}(b)) are projective (respectively quasi-projective).

Proof. Let V ⊂ X_{⪯μ}(b) be an irreducible component and let η be its generic point. At η the universal δ, δ^{−1}, and σ^{∗}δ are bounded by 2nρ^{∨} for some n. Due to Lemma 3.10 they are then bounded by 2nρ^{∨} on all of V. Hence V is contained in the projective scheme (M^{n})_{red} as a closed subscheme, and therefore V is projective. Since X_{μ}(b) is open in X_{⪯μ}(b), each irreducible component of X_{μ}(b) is quasi-projective.

Note that if one only wants to prove this corollary, one can use a significantly simplified version of the proof of Theorem 6.3. Namely one does not need the distinction between M^{n} and M_{n}, and the fact that M_{n} is a formal scheme formally of finite type.

Remark 6.6. If one chooses a different definition for boundedness as mentioned in Remark 3.6, one possibly obtains a different Rapoport-Zink space X̂_{⪯μ}(b), whose underlying topological space still coincides with X_{⪯μ}(b). If one enlarges the class of G-modules which are used to define boundedness, one obtains the new Rapoport-Zink space as a closed subspace of the former one.

7 The Newton stratification

Let \underline{G} be a local G-shtuka over an algebraically closed field k. By Remark 3.2, G is isomorphic to the trivial K-torsor on k and the Frobenius is given by some element b ∈ LG(k). Changing the trivialization corresponds to σ-conjugation of b by elements of K(k). On the other hand, considering b up to σ-conjugation by LG(k) corresponds to considering the local G-isoshtuka \underline{LG} ≅ (LG_{k}, bσ^{∗}). We briefly review Kottwitz’s classification of the latter kind of σ-conjugacy classes. To be precise, we use the analog for the equal characteristic case of his classification.

Remark 7.1. Let G be a connected reductive group over an algebraically closed field k of characteristic p. Let B(G) be the set of σ-conjugacy classes of elements b ∈ LG(k). We denote the class of b (which contains all g^{−1}bσ^{∗}(g) for g ∈ LG(k)) by [b]. In [Ko1], [Ko2], Kottwitz classifies the classes [b] ∈ B(G) by two invariants. For split groups the first invariant, the Kottwitz point κ(b) has the following explicit definition. Let μ\underline{G} ∈ X_{∗}(T) be dominant with b ∈ K(k)z^{μ\underline{G}}K(k). Then κ(b) is the image of μ\underline{G} under the projection X_{∗}(T) → π_{1}(G). Indeed, the maps κ_{G} considered by Kottwitz are invariant under σ-conjugation of b, they are group homomorphisms, and are natural transformations in G. For G = 𝔾_{m} we have κ_{G}(b) = v_{z}(b). See [RR, Section 1] for the reformulation as maps to the fundamental group. Let now G be split. The properties of κ_{G} mentioned above show that κ_{G} is trivial on K(k) as each such element is σ-conjugate to 1 (by a version of Lang’s theorem), and that the torus element z^{μ} is mapped to its image in π_{1}(G). Together we obtain the explicit description of κ_{G} given above. Hence κ_{G} : LG(k) → π_{1}(G) coincides with the map in Proposition 3.4. The second invariant is the Newton point or Newton polygon ν_{b} ∈ X_{∗}(T)_{ℚ}. For G = GL_{n} this is the usual Newton polygon of the σ-linear map bσ^{∗}, for general G it is defined by requiring that it be functorial in G. The two invariants have the same image in π_{1}(G)_{ℚ}, which is the quotient of X_{∗}(T)_{ℚ} by the sub-vector space generated by the coroots. Note that not all elements of X_{∗}(T)_{ℚ} occur as Newton points. For a description of the set of Newton polygons in X_{∗}(T)_{ℚ} see also [Ch1, 4].

If \underline{G} is a local G-shtuka on S ∈ Nilp_{𝔽_{q}[[ζ]]}, we consider the function [b\underline{G}] : S → B(G) with s ↦ [b(s)] where [b(s)] is the element associated to the reduction of \underline{G} in the geometric point s. The Newton polygon of [b(s)] is called the Newton polygon of \underline{G} in s.

We use the following analogy to isocrystals with G-structure defined by Rapoport and Richartz, [RR]. Let S be a connected 𝔽_{q}-scheme. Let Isosh_{S} be the category of pairs (N, F) consisting of an S((z))-module N which is étale-locally on S free of finite rank and of an isomorphism F : σ^{∗}(N) → N. The elements of Isosh_{S} are called local isoshtukas, and are a function field analog of isocrystals. Let G be as above and \underline{G} = (G, φ) a local G-shtuka on S. Let V be an 𝔽_{q}((z))-representation of G. Then let G_{V} be the sheaf associated with the presheaf

Y ⟼ (G(Y) × (V ⊗_{𝔽_{q}((z))} O_{S}((z))(Y)))/K(Y) .

Together with the σ-linear isomorphism induced by (φ, id_{V}) it is an element of Isosh_{S}, since G is trivialized by an étale covering of S according to Proposition 2.2. This defines an exact faithful tensor functor \underline{G} : Rep_{𝔽_{q}((z))} G → Isosh_{S}. These functors, called isoshtukas with G-structure, are the function field analog of isocrystals with G-structure as defined by Rapoport and Richartz in [RR, Definition 3.3]. Note that the rational invariant [b] of \underline{G} as defined above is the same as the σ-conjugacy class of the Frobenius of the local isoshtuka with G-structure associated to \underline{G} in [RR, 3.4(i)].

Proposition 7.2. Let \underline{G} be a local G-shtuka over an algebraically closed field in Nilp_{𝔽_{q}[[ζ]]}. Let ν be its Newton polygon and let μ be its Hodge polygon (as in Definition 3.3). Then ν ⪯ μ as elements of X_{∗}(T)_{ℚ}.

Proof. This result is first shown by Katz in [Ka, 1.4.1] (Mazur’s Theorem) for σ^{a}-F -crystals. It is generalized in [RR, Theorem 4.2 (ii)] to elements b ∈ G(L) where G is an unramified reductive group over k and where L is the quotient field of the Witt ring of k. Proposition 7.2, which is the function field analog of those results, can be shown by the same argument.

The behaviour of [b] under specialization is described by the following theorem. Its proof is completely analogous to the corresponding proof in [RR], using again the above analogy between local G-shtukas and isocrystals with G-structure. Note that the additional statement in [RR, Theorem 3.6(i)] that the image of ν in π_{1}(G)_{ℚ} is locally constant is in our context a consequence of Proposition 3.4.

Theorem 7.3 ([RR, Theorem 3.6]). Let S ∈ Nilp_{𝔽_{q}[[ζ]]} and let \underline{G} be a local G-shtuka on S. For each b_{0} ∈ B(G), the subset {s ∈ S : ν_{b(s)} ⪯ ν_{b_{0}}} is Zariski-closed on S and locally on S the zero set of a finitely generated ideal.

Let S and \underline{G} be as in the theorem and let ν ∈ X_{∗}(T)_{ℚ} be a Newton polygon. Then by Theorem 7.3 the reduced subscheme N_{⪯ν} of S with

N_{⪯ν} = {s ∈ S : ν_{b(s)} ⪯ ν}

is a closed subscheme of S and similarly

N_{ν} = {s ∈ S : ν_{b(s)} = ν}

defines a locally closed subscheme of S, called the Newton stratum associated to \underline{G} and ν.

Theorem 7.4. Let S be an integral and locally noetherian 𝔽_{q}-scheme and let \underline{G} be a local G-shtuka on S. Let ν be the Newton polygon in the generic point of S. Then the Newton stratification on S defined by \underline{G} satisfies the purity property, that is the inclusion of the stratum N_{ν} in S is an affine morphism.

Proof. For G = GL_{r} this can be shown in mostly the same way as Vasiu’s corresponding result for F -crystals, [Vas, Theorem 6.1]. The only place where one needs more than the obvious translation is that the results from his Section 5.1 (whose proof does not generalize directly) have to be replaced by our Corollary 10.3. As the composition of affine morphisms is affine, this also implies the theorem for all G which are a product of finitely many factors GL_{r_{i}}. For general G one considers a finite number of representations λ_{i} of G distinguishing the different Newton polygons on S (for example one for each simple factor of the adjoint group G_{ad}). Then the Newton stratifications of S corresponding to \underline{G} and \underline{G}_{λ_{1}} × … × \underline{G}_{λ_{l}} coincide.

Corollary 7.5. Let S be an integral, locally noetherian 𝔽_{q}-scheme of dimension d and let \underline{G} be a local G-shtuka on S. Let S′ be the complement of the generic Newton stratum. Then S′ is empty or pure of dimension d − 1.

Proof. By replacing S by suitable open subschemes, we may assume that S is affine and that S′ ≠ ∅. Similarly it is enough to show that dim(S′) = d − 1. By Theorem 7.4, S ∖ S′ is affine. By [EGA, IV Corollaire 21.12.7] this implies that dim(S) − dim(S′) ≤ 1.

See [Vas, Remark 6.3(a)] for another proof of this corollary.

Definition 7.6. Let S be a scheme and let \underline{G} be a local G-shtuka on S. Let Y be an irreducible component of the Newton stratum N_{ν} ⊆ S for some ν. We call a chain of Newton polygons ν = ν_{n} ≺ ν_{n−1} ≺ … ≺ ν_{0} with ν_{i} ≠ ν_{i−1} for all i realizable in S at Y if for each i there is an irreducible subscheme S_{i} of the corresponding Newton stratum N_{ν_{i}} such that S_{i} ⊆ \overline{S_{i−1}} ∖ S_{i−1} for all i > 0 and such that S_{n} = Y. We call n the length of the chain.

Let α1^{∨}, …, αr^{∨} be the simple coroots of G. Then we choose ω_{i} ∈ X^{∗}(T)_{ℚ} for i = 1, …, r with ⟨ω_{i}, αj^{∨}⟩ = δ_{ij}. Note that these elements are in general not unique. However, we will only use them in expressions of the form ⟨ω_{i}, μ⟩ for μ in the sub-vector space of X_{∗}(T)_{ℚ} generated by the coroots. These values do not depend on the particular choice of the ω_{i}.

Corollary 7.7. Let S be an irreducible 𝔽_{q}-scheme and let \underline{G} be a local G-shtuka on S. Let ν_{0} be the Newton polygon of \underline{G} at the generic point of S. Let ν ⪯ ν_{0} with N_{ν} ≠ ∅ and let Y be an irreducible component of N_{ν}.

(a) The difference dim S − dim Y equals the maximal length n of a realizable chain of Newton polygons in S at Y.

(b) dim(Y) ≥ dim(S) − ∑_{i}⌈⟨ω_{i}, ν_{0} − ν⟩⌉.

Proof. Let ν_{n} ≺ … ≺ ν_{0} be a maximal realizable chain at Y and let S_{i} be corresponding irreducible subschemes of N_{ν_{i}}. As the chain is maximal, ν_{0} is equal to the Newton polygon at the generic point of S, so we may assume that S_{0} is dense in S. We have to show that we can modify the S_{i} such that they satisfy in addition that dim S_{i} − dim S_{i+1} = 1 for all i. As the S_{i} are irreducible and not equal, the difference is at least one. As S_{i+1} is irreducible, it is contained in an irreducible component of \overline{S_{i}} ∖ (N_{ν_{i}} ∩ \overline{S_{i}}), which by Corollary 7.5 has dimension dim(S_{i}) − 1. As the chain is maximal, the generic Newton polygon of this component is ν_{i+1}. For i + 1 < n this implies that we may replace S_{i+1} by that component (using increasing induction on i) and obtain dim S_{i+1} = dim S_{i} − 1. For i + 1 = n this implies that S_{i+1} is already equal to that component, and therefore has dimension dim S_{n−1} − 1.

(b) is an immediate consequence of [Ch1, Theorem 7.4 (iv)] on the maximal length of chains of comparable Newton polygons between ν_{0} and ν (that are not necessarily realizable).

Note that a consequence of the proof is that already the S_{i} we started with have dimension dim S − i.

Proposition 7.8. Let D be the coordinate ring of the universal deformation bounded by μ of a local G-shtuka over k with Newton polygon ν and let \overline{D} = (D/ζD)_{red}. Let \underline{G} be the local G-shtuka over S = Spec \overline{D} corresponding to the universal family over Spf \overline{D} as in Proposition 3.16. Let N_{ν} be the closed Newton stratum inside S and let Y be any irreducible component of N_{ν}. Then

dim(Y) ≥ ⟨2ρ, μ⟩ − ∑_{i}⌈⟨ω_{i}, μ − ν⟩⌉.

Proof. By Proposition 5.9, S = Spec \overline{D} is equidimensional of dimension ⟨2ρ, μ⟩. We apply Corollary 7.7 (b) to the reduced subscheme S′ of an irreducible component of S. By the corollary it is enough to show that the Newton polygon in the generic point of S′ is ⪯ μ. This follows from Proposition 7.2, because the same inequality holds for the Hodge polygon by Lemma 3.11.

Remark 7.9. By [Ko3], the estimate can be rewritten as

⟨2ρ, μ⟩ − ∑_{i}⌈⟨ω_{i}, μ − ν⟩⌉ = ⟨ρ, μ + ν⟩ − \frac{1}{2}def(b).  (7.2)

Here b ∈ LG(k) is any element with Newton polygon ν and κ(b) = [μ]. Furthermore, def(b) = rk(G) − rk_{𝔽_{q}((z))}(J) where J := QIsog(\underline{G}) is the group of quasi-isogenies of \underline{G} = (K_{k}, bσ^{∗}) and where rk_{𝔽_{q}((z))} J is the rank of a maximal 𝔽_{q}((z))-split subtorus of J. The group J is the set of 𝔽_{q}((z))-valued points of a reductive algebraic group over 𝔽_{q}((z)) which is an inner form of a Levi subgroup of G. This can be shown using the same argument as for the analogous statement for p-divisible groups, compare [Ko1], [RZ, Corollary 1.14] or [Ko2]. Note that using 1.1 of [Ko3], one sees that (7.2) also holds without the additional assumption that the derived group of G is simply connected (which is a general assumption in Kottwitz’s paper).

8 Basic Newton strata

An element b ∈ LG(k) is called basic if its Newton point ν ∈ X_{∗}(T)_{ℚ} is central in G (compare [Ko1, §5.1]). Note that this property only depends on the σ-conjugacy class of b.

Proposition 8.1. Let \underline{G} = (G, φ) be a local G-shtuka over a reduced noetherian complete local ring R over 𝔽_{q} with algebraically closed residue field. Let its Newton polygon be constant on Spec R and basic. Then there exists a local G-Shtuka \underline{F} over 𝔽_{q} and a quasi-isogeny \underline{G} → \underline{F} ×_{Spec 𝔽_{q}} Spec R.

Proof. By Remark 3.2 the K-torsor G over R is isomorphic to the trivial torsor K_{R}, and choosing a trivialization φ can be written as bσ^{∗} for some b ∈ LG(R). Let 𝔪 be the maximal ideal of R and let k = R/𝔪. There is an h > 0 and a lift of hν ∈ X_{∗}(T)_{ℚ} to X_{∗}(T) which maps to h[μ\underline{G}] in π_{1}(G). Indeed, after multiplying ν by the common denominator h_{1} we obtain an element of X_{∗}(T) whose class in π_{1}(G) coincides with h_{1}[μ\underline{G}] up to torsion. By multiplying with a second integer, we may assume that the two elements are equal. As k is algebraically closed, [Ko1, §4.3] shows that there is a y ∈ LG(k) such that the reduction \overline{φ}^{h} of φ^{h} modulo 𝔪 maps y to cy for the central element c := z^{hν} ∈ LG(k) (where hν denotes the lifted element of X_{∗}(T)). Since k ⊂ R we may replace \underline{G} ≅ (K_{R}, bσ^{∗}) by the quasi-isogenous local G-shtuka (K_{R}, y^{−1}bσ^{∗}(y) · σ^{∗}) and we may assume that \overline{φ}^{h} = c (σ^{h})^{∗}.

Claim. There is a quasi-isogeny between \underline{G} and a local G-shtuka (G′, φ′) with G′ = K_{R} and (φ′)^{h} = c (σ^{h})^{∗}.

Identifying G with the trivial torsor, the claim is equivalent to the existence of an element x ∈ LG(R) with φ^{h}(x) = cx. Replacing φ by c^{−1}φ^{h} and σ by σ^{h} we may assume that h = 1, and c = 1. As c is central and commutes with σ^{∗}, the σ-conjugacy class of this renormalized φ is [1] in each point of Spec R. Let x̃ ∈ LG(R) = G(R((z))) be a lift of 1 ∈ LG(k). Then φ(x̃) ≡ x̃ (mod 𝔪). As φ is σ-linear, φ^{n}(x̃) ≡ φ^{n−1}(x̃) (mod (σ^{∗})^{n−1}(𝔪)). Thus the sequence φ^{n}(x̃) has a limit x ∈ G({ \atop R((z))}). The congruences further show that φ(x) = x. To prove the claim it remains to show that x ∈ LG(R). As in the proof of Proposition 3.16 we have to show that the denominators occurring in φ^{n}(x̃) are bounded independently of n. As R is a reduced noetherian ring, it is enough to check this in each generic point η of Spec R separately. Let \overline{k(η)} be an algebraic closure of k(η) and let x̃_{η} be the image of x̃ in LG(\overline{k(η)}). Let φ_{η} be the induced map on LG(\overline{k(η)}). As \overline{k(η)} is algebraically closed and as φ_{η} is in the class [1] ∈ B(G), there is a y ∈ LG(\overline{k(η)}) with φ_{η}(y) = y. Then

φ_{η}^{n}(x̃_{η}) = φ_{η}^{n}(yy^{−1}x̃_{η}) = φ_{η}^{n}(y) · (σ^{n})^{∗}(y^{−1}x̃_{η}) = y · (σ^{n})^{∗}(y^{−1}x̃_{η}).

This shows that the z-powers occurring in φ_{η}^{n}(x̃_{η}) are bounded independently of n. Especially, x ∈ LG(R). This proves the claim.

It remains to show that if \underline{G} is a local G-shtuka with φ^{h} = c (σ^{h})^{∗}, then \underline{G} is isogenous to a constant local G-shtuka \underline{F}_{R} as in the assertion. Over the residue field k this follows again from the classification of σ-conjugacy classes, see Remark 7.1. As G is trivial we may thus assume that φ = bσ^{∗} for some b whose reduction \overline{b} modulo 𝔪 is defined over 𝔽_{q}. Let b̃ be the image of \overline{b} ∈ LG(𝔽_{q}) in LG(R). It suffices to show that b = b̃. We use induction on n to show that b ≡ b̃ (mod (σ^{∗})^{n}(𝔪)). For n = 0 this follows from the definition of b̃. Assume that b ≡ b̃ (mod (σ^{∗})^{n}(𝔪)). Then

bσ^{∗}(bσ^{∗})^{h−1} = c = b̃^{h} = b̃σ^{∗}(b̃^{h−1}) ≡ b̃σ^{∗}(bσ^{∗})^{h−1} (mod (σ^{∗})^{n+1}(𝔪)).

Thus b ≡ b̃ (mod (σ^{∗})^{n+1}(𝔪)).

We are now able to give the proofs of Theorem 1.1, Theorem 1.2 and Corollary 1.4.

Proof of Theorem 1.1. By Theorem 6.3 we obtain a local G-shtuka \underline{G}′ over (X_{⪯μ}(b))g^{∧} together with a quasi-isogeny to (K_{(X_{⪯μ}(b))_{g}^{∧}}, bσ^{∗}). Then \underline{G}′ is a deformation of its special fibre, which is isomorphic to (K_{k}, g^{−1}bσ^{∗}(g)σ^{∗}). By definition it is isogenous to a constant local G-shtuka, and especially its Newton polygon is constant. Also, its Hodge polygon is ⪯ μ in every point of (X_{⪯μ}(b))g^{∧}. Since ζ = 0 on (X_{⪯μ}(b))g^{∧} and (X_{⪯μ}(b))g^{∧} is reduced (being the completion of a reduced local ring) \underline{G}′ defines a morphism f : (X_{⪯μ}(b))g^{∧} → N_{ν}.

To construct a morphism in the opposite direction, we use Proposition 8.1. Let \underline{G} be the universal object over N_{ν}. By the proposition there is a quasi-isogeny δ from \underline{G} to a constant local G-shtuka. Changing this quasi-isogeny by a quasi-isogeny between constant local G-shtukas, we may assume that the constant local G-shtuka is (K_{k}, bσ^{∗}) and that the quasi-isogeny at the special fibre is given by g. Since ζ = 0 on N_{ν} and N_{ν} is reduced this yields a morphism h : N_{ν} → (X_{⪯μ}(b))g^{∧}. The composition f ∘ h first constructs the additional quasi-isogeny δ, and then forgets it. Thus it is the identity. The composition h ∘ f is the identity by rigidity of quasi-isogenies of local G-shtukas (Proposition 3.9).

Proof of Theorem 1.2. Note that as b is basic, ⟨ρ, ν⟩ = 0. By [GHKR1], and [Vi1] we already know that X_{μ}(b) and X_{⪯μ}(b) have dimension ⟨ρ, μ⟩ − \frac{1}{2}def(b). It remains to show that each irreducible component has at least this dimension. This is an immediate consequence of Theorem 1.1 together with the lower bound on the dimension of each irreducible component of N_{ν} in Proposition 7.8 and Remark 7.9.

Proof of Corollary 1.4. From Theorem 1.2 and Theorem 1.1 we obtain that N_{ν} is equidimensional of dimension ⟨ρ, μ⟩ − \frac{1}{2}def(b) = ⟨2ρ, μ⟩ − ∑_{i}⌈⟨ω_{i}, μ − ν⟩⌉. Let ν_{0} be the Newton point at the generic point of Spec \overline{D}. By Corollary 7.7(b) we obtain

∑_{i}⌈⟨ω_{i}, μ − ν⟩⌉ ≤ ∑_{i}⌈⟨ω_{i}, ν_{0} − ν⟩⌉.

The two sides of this inequality are the maximal lengths of chains of comparable Newton polygons between ν and μ, respectively ν_{0} (see [Ch1, Theorem 7.4 (iv)]). As ν_{0} ⪯ μ, the right hand side is strictly smaller than the left hand side unless μ = ν_{0}.

9 The Iwahori case

In this section we replace the maximal bounded open subgroup K by an Iwahori subgroup I of G and explain how some of the preceding results can be generalized to this situation. Again we can relate the dimension of the basic Newton stratum of some universal (ζ = 0)-deformation to the dimension of an affine Deligne-Lusztig variety. However, in this case one does not have a closed expression for the dimension of the basic Newton stratum yet.

As before let G be a split connected reductive group over 𝔽_{q}. Let B be a Borel subgroup and let T ⊆ B be a split maximal torus. Let π : R[[z]] → R be the projection. Let I be the Iwahori subgroup scheme of K over 𝔽_{q} defined as

I(R) := { g ∈ G(R[[z]]) : π(g) ∈ B(R) }

on any 𝔽_{q}-algebra R. Let W̃ ≅ W ⋉ X_{∗}(T) denote the extended affine Weyl group of G. Then LG(k) = ∐_{x∈W̃} I(k)xI(k) by the Bruhat-Tits decomposition.

We now define the analog of local G-shtukas, with K replaced by I. We restrict our attention to base schemes S with ζ = 0.

Definition 9.1. (a) A local G-shtuka with I-structure over an 𝔽_{q}-scheme S is a pair \underline{I} = (I, φ) consisting of an I-torsor I on S (for the étale topology) together with an isomorphism φ : σ^{∗}Ĩ ^{∼} ⟶ Ĩ. Here Ĩ denotes the LG-torsor associated to I.

(b) We fix an element x ∈ W̃. A local G-shtuka with I-structure \underline{I} over S is of affine Weyl type x if étale-locally on S, the shtuka is isomorphic to a trivial I-torsor I_{S} with φ = bσ^{∗} for some b ∈ I(S)xI(S).

Remark 9.2. (a) Lemma 9.3 shows that this rather ad hoc definition is analogous to the notion of a local G-shtuka with fixed Hodge polygon μ.

(b) Let R be a complete local noetherian ring with algebraically closed residue field. In the same way as Proposition 3.16 one can show that there is an equivalence of categories between local G-shtukas with I-structure of affine Weyl type x over Spec R and over Spf R.

Lemma 9.3. Let (G, φ) be a local G-shtuka which has constant Hodge polygon μ and is bounded by μ over an 𝔽_{q}[[ζ]]/(ζ)-scheme S. Then étale-locally on S the K-torsor G is trivial and φ is given by an element of K(S)z^{μ}K(S).

Proof. By Proposition 2.2 there is an étale covering S′ → S such that G_{S′} is trivial. Choosing a trivialization we can write φ = bσ^{∗} for some b ∈ LG(S′). Then b^{−1} induces a morphism S′ → Gr which factors through Gr^{⪯μ} by Proposition 5.5 since φ is bounded by μ. Since the Hodge polygon is equal to μ it factors through the open subscheme Gr^{μ} which is the homogeneous space Kz^{(−μ)_{dom}}K/K. This proves our assertion.

Like Gr also the affine flag variety F := LG/I is an ind-scheme of ind-finite type. Analogously to Proposition 5.3 we have for any x ∈ W̃:

Remark 9.4. (a) The set I(k)xI(k)/I(k) is the set of k-valued points of a locally closed reduced subscheme F_{x} of F which is of finite type over k.

(b) F_{x} is irreducible and of dimension ℓ(x), the length of x ∈ W̃.

Let \underline{𝔾} = (𝔾, φ_{𝔾}) be a local G-shtuka with I-structure of affine Weyl type x over an algebraically closed field k ∈ Nilp_{𝔽_{q}[[ζ]]}. Choosing a trivialization of 𝔾, we can write φ_{𝔾} = b_{0}σ^{∗} for some b_{0} ∈ I(k)xI(k). Then b_{0}^{−1} defines a point in F_{x^{−1}}(k). Let \overline{D}_{I} be the complete local ring of F_{x^{−1}} at this point. It is a complete noetherian local ring over k. Note that ℓ(x^{−1}) = ℓ(x), so F_{x^{−1}} has dimension ℓ(x).

Theorem 9.5. \overline{D}_{I} pro-represents the formal (ζ = 0)-deformation functor of \underline{𝔾}

F : (Artinian local k[[ζ]]/(ζ)-algebras with residue field k) ⟶ Sets

A ⟼ \{ Isomorphism classes of pairs (\underline{G}, β) where
\underline{G} is a local G-shtuka with I-structure of affine Weyl type x over Spec A,

β : \underline{𝔾} ^{∼} ⟶ \underline{G} ⊗_{A} k is an isomorphism of local G-shtukas with I-structure \}

where (\underline{G}, β) and (\underline{G}′, β′) are isomorphic if there exists an isomorphism γ : \underline{G} → \underline{G}′ with β′ = (γ ⊗_{A} k) ∘ β.

Proof. The proof of this theorem is the same as for Theorem 5.6, always replacing K by I.

Let \underline{G} be the local G-shtuka with I-structure over Spec \overline{D}_{I} which corresponds via Remark 9.2 (b) to the chosen universal local G-shtuka with I-structure over Spf \overline{D}_{I}. It consists of the trivial I-torsor I_{\overline{D}_{I}} together with φ = b_{\overline{D}_{I}} σ^{∗} for some element b_{\overline{D}_{I}} ∈ LG(\overline{D}_{I}). As in Section 7 we can consider the function assigning to each geometric point of Spec \overline{D}_{I} the σ-conjugacy class of b_{\overline{D}_{I}} in that point. This induces a Newton stratification on Spec \overline{D}_{I}. We denote by N_{ν}^{I} the locally closed reduced subscheme of Spec \overline{D}_{I} where the Newton polygon of b_{\overline{D}_{I}} is equal to ν.

We fix an element b ∈ LG(k) and some x ∈ W̃. The affine Deligne-Lusztig variety X_{x}^{I}(b) associated to G, I, b, and x is defined as the locally closed reduced subscheme of the affine flag variety F whose k-valued points for an algebraically closed field k are given by

X_{x}^{I}(b)(k) = { g ∈ LG(k)/I(k) : g^{−1}bσ^{∗}(g) ∈ I(k)xI(k) }.

Theorem 9.6. Let g ∈ LG(k) be a representative of an element of X_{x}^{I}(b) and assume that the Newton polygon ν of b is basic. Let N_{ν}^{I} be the closed Newton stratum in the universal (ζ = 0)- deformation of the local G-shtuka with I-structure (I_{k}, g^{−1}bσ^{∗}(g)σ^{∗}). Then N_{ν}^{I} is isomorphic to the completion of X_{x}^{I}(b) at g.

Proof. The proof of this theorem is the same as for Theorem 1.1, always replacing K by I. The main ingredient is in both cases Proposition 8.1, which considers elements of LG(\overline{D}_{I}) up to σ-conjugation. This result does not depend on the subgroups K or I.

Corollary 9.7. Let b be basic. Let S be an irreducible component of X_{x}^{I}(b) and let g ∈ S be a k-valued point not contained in any other irreducible component of X_{x}^{I}(b). Then dim S = ℓ(x) − d(g, x) where d(g, x) is the maximal length of a realizable chain of Newton polygons in the universal deformation of (I_{k}, g^{−1}bσ^{∗}(g)σ^{∗}) at its closed Newton stratum (which is irreducible).

Proof. This is an immediate consequence of Theorem 9.6 and Corollary 7.7(a).

Remark 9.8. For deformations of local G-shtukas (without Iwahori-structure) we saw that such maximal lengths of realizable chains in the universal deformation of g bounded by some μ are equal to the maximal length of chains of Newton polygons between μ and the basic Newton polygon. These lengths are given by Chai’s formula, compare Corollary 7.7(b). For deformations of local G-shtukas with I-structure, the situation is more difficult. Let b ∈ I(k)xI(k) and let μ be the dominant element in the Weyl group orbit of the translation part of x ∈ W̃. Let ν be the Newton polygon of b. Then in general not every Newton polygon ν′ with ν ⪯ ν′ ⪯ μ arises as the Newton polygon of some point in the universal deformation of b of affine Weyl type x. Let N(b, x) be the set of Newton polygons that occur in the universal deformation of b. Then the maximal chain lengths in N(b, x) are in general shorter than those in the set of all Newton polygons between ν and μ. Up to now there is no closed expression for N(b, x). A second question is whether there always is a chain in N(b, x) of maximal length that is realizable in the universal deformation. However this last assertion holds in all known examples (for example for G = SL_{3} and all b, x by [Be]).

We spend the rest of this section using Theorem 9.6 to prove the basic case of a conjecture of Beazley [Be, Conjecture 1] comparing dimensions of affine Deligne-Lusztig varieties inside the affine flag variety with codimensions of Newton strata in I(k)xI(k). Let b ∈ LG(k) be basic and x ∈ W̃ with X_{x}^{I}(b) ≠ ∅. We consider the subset IxI := I(k)xI(k) ⊂ LG(k). Let ν be the Newton polygon of b and let (IxI)_{ν} be the subset of IxI of elements having Newton polygon ν. Theorem 10.1 for B = IxI implies that there is a bounded open subgroup I_{0} of I(k) such that the Newton polygon of an element of IxI only depends on its image in IxI/I_{0}. This last set is naturally the set of k-valued points of a scheme. The specialization property of Newton polygons then shows that (IxI)_{ν}/I_{0} is the set of k-valued points of a closed subscheme of IxI/I_{0}. We define codim((IxI)_{ν} ⊆ IxI) := dim IxI/I_{0} − dim(IxI)_{ν}/I_{0}. By our choice of I_{0} it is independent of I_{0}, provided I_{0} is small enough. A subset S of IxI that is right invariant under a (sufficiently small) group I_{0} is called irreducible if S/I_{0} is irreducible. Again this is independent of the choice of I_{0}. The notions of irreducible component or smoothness for such subsets are defined similarly.

Proposition 9.9. Let b ∈ LG(k) be basic and x ∈ W̃ such that X_{x}^{I}(b) ≠ ∅. Then

dim X_{x}^{I}(b) = ℓ(x) − codim((IxI)_{ν} ⊆ IxI).

This proposition proves the basic case of [Be, Conjecture 1].

Proof. By Theorem 9.6 and Remark 9.4 it is enough to show that codim((IxI)_{ν} ⊆ IxI) is equal to dim Spec \overline{D}_{I} −dim N_{ν}^{I} where N_{ν}^{I} is the Newton stratum in the universal (ζ = 0)-deformation Spec \overline{D}_{I} associated to some g ∈ X_{x}^{I}(b) such that X_{x}^{I}(b) is smooth and of maximal dimension in g. As in the proof of Proposition 7.8 one sees that Spec \overline{D}_{I} and the completion of IxI/I_{0} in b are equidimensional. The assertion is then an immediate consequence of Proposition 9.10 and Corollary 7.7 (a).

Proposition 9.10. Let b ∈ I(k)xI(k) with Newton polygon ν, let Y be an irreducible component of (IxI)_{ν}. Assume that Y is the only such component containing b, and that Y is normal in b. Then the closed Newton stratum N_{ν}^{I} in the universal (ζ = 0)-deformation of the local G-shtuka with I-structure (I_{k}, bσ^{∗}) is irreducible. Let I_{0} be a subgroup of I such that the Newton polygon of an element of IxI only depends on its I_{0}-coset. Then a chain of Newton polygons is realizable in the universal (ζ = 0)-deformation at its closed Newton stratum if and only if it is realizable in IxI/I_{0} at Y/I_{0}.

Proof. If Spf R_{1} and Spf R_{2} are the completions of IxI/I_{0} and (IxI)_{ν}/I_{0} in b, let Y = Spec R_{1} and Y_{ν} = Spec R_{2}. We choose a representative f ∈ I(Y)xI(Y) of the universal element of IxI/I_{0} over Y with f ⊗_{R_{1}} k = b. We consider the local G-shtuka with I-structure \underline{I} = (I_{Y} , fσ^{∗}) over Y. Since \underline{I} is a deformation of its special fiber (I_{k}, bσ^{∗}) this induces a morphism π : Y → Spec \overline{D}_{I} mapping Y_{ν} to N_{ν}^{I} and also each other Newton stratum of Y to the corresponding Newton stratum in Spec \overline{D}_{I}. Let f′ be such that (I_{\overline{D}_{I}} , f′σ^{∗}) is the universal object over Spec \overline{D}_{I} and that f′ ⊗_{\overline{D}_{I}} k = b. Then f′ ∈ I(\overline{D}_{I})xI(\overline{D}_{I}). This provides a section of π. Especially π is surjective. Our assumptions on Y imply that the closed Newton stratum in Y is irreducible by Zariski’s Main Theorem [ZS, Theorem VIII.32]. Therefore the closed Newton stratum N_{ν}^{I} in Spec \overline{D}_{I} is also irreducible. Thus the chains realizable in Y at Y_{ν} are also realizable in Spec \overline{D}_{I} at N_{ν}^{I}.

For the other direction let ν = ν_{n} ≺ … ≺ ν_{0} be a realizable chain of Newton polygons in Spec \overline{D}_{I} at S_{n} = N_{ν}^{I} and let S_{i} be corresponding irreducible subschemes of Spec \overline{D}_{I}. Let Y_{i} ⊆ Y be the inverse image of S_{i} under π. Then Y_{n} = Y_{ν}. We want to show that ν_{n} ≺ … ≺ ν_{0} is also realizable in Y. To do that we use decreasing induction on i to show that there is an irreducible component Y′_{i} of Y_{i} with Y′_{i+1} ⊆ \overline{Y′_{i}}. As Y′_{i+1} is (by induction) irreducible, it is enough to show that Y′_{i+1} ⊆ \overline{Y_{i}}, or that Y_{i+1} ⊆ \overline{Y_{i}}. Let y be a geometric point of Y_{i+1} with values in some field k̃. As S_{i+1} ⊆ \overline{S_{i}} there is a k̃[[t]]-valued point of S_{i} ∪ S_{i+1} with special point π(y) and general point in S_{i}. Let f′_{k̃[[t]]} be the reduction of f′ at this point and let f ′ ; k̃ be the reduction in its special point. As π(f ′ ; k̃) = π(y) there is an i ∈ I(k̃) with i^{−1}f ′ ; k̃σ^{∗}(i) = y. Then i^{−1}f′_{k̃[[t]]}σ^{∗}(i) is a k̃[[t]]-valued point with special point y

and general point in Y_{i}. This finishes the proof that chains realizable in Spec \overline{D}_{I} are also realizable in Y.

Appendix: Admissibility

In this section we provide the results needed to generalize Vasiu’s proof of purity for F -crystals to local G-shtukas. Theorem 10.1 also implies admissibility of the Newton stratification of I-double cosets needed in Section 9.

For n ≥ 0 let I_{n} = {g ∈ K(k) : g ≡ 1 (mod z^{n}), g ∈ B (mod z^{n+1})}. Here B is the chosen Borel subgroup. Recall that a subset of LG(k) is bounded if and only if it is contained in a finite union of K-double cosets Kz^{μ}K.

Theorem 10.1. Let B be a bounded subset of LG(k). Then there is a c ∈ ℕ such that for each d ∈ ℕ, each g ∈ B and each h ∈ I_{d+c} there is a k ∈ I_{d} with gh = k^{−1}gσ^{∗}(k).

Proof. Note that the set of σ-conjugacy classes of elements of B is finite. We write B as a disjoint union of its intersections with the different σ-conjugacy classes and consider each subset separately. Thus from now on we assume that all elements of B are σ-conjugate under LG(k). Let ν be their Newton polygon. Then B lies in a finite union of double cosets Kz^{μ′}K with ν ⪯ μ′ and all occurring μ′ have the same image in π_{1}(G). Hence B ⊂ ∐_{μ′⪯μ} Kz^{μ′}K for some μ.

We first show that there is a bounded subset C of LG(k) and an element b ∈ LG(k) such that each element of B is σ-conjugate via an element of C to b. There is a standard parabolic subgroup P = MN of G with Levi component M and a b ∈ M in the given σ-conjugacy class with M-dominant Newton polygon ν such that b is superbasic in M. Indeed, M is simply the smallest standard Levi subgroup of the centralizer of ν in G such that the given σ-conjugacy class contains an element of M. Each g ∈ B is of the form g = f^{−1}bσ^{∗}(f) for some f ∈ LG(k). We have to show that we may take all f in some bounded subset of LG(k). Using the Iwasawa decomposition we write f = mnk with m ∈ M(k((z))), n ∈ N(k((z))) and k ∈ K. Thus all elements of B are σ-conjugate via K to elements of the form m^{−1}bσ^{∗}(m)((m^{−1}bσ^{∗}(m))^{−1}n^{−1}(m^{−1}bσ^{∗}(m))σ^{∗}n) ∈ MN. These elements are still all in ∐_{μ′⪯μ} Kz^{μ′}K and in the same σ-conjugacy class. We claim that the Levi parts l := m^{−1}bσ^{∗}(m) of these elements are then again in a finite union of M(k[[z]])-double cosets, namely those contained in ∐_{μ′⪯μ} Kz^{μ′}K. To see this, consider for a λ ∈ X_{∗}(T) which is central in M and with ⟨α, λ⟩ > 0 for each root of T in N the element g_{x} ∈ LG(k[x]) given by g_{0} = l and g_{x} = λ(x)gλ(x)^{−1} for x ≠ 0. An easy calculation shows that this defines an element of LG(k[x]). As λ(x) ∈ K, all g_{x} for x ≠ 0 are in ∐_{μ′⪯μ} Kz^{μ′}K. As this union is closed, it also contains g_{0} = l. Reformulating the boundedness for the elements l in terms of m, we obtain m ∈ X_{μ′}^{M}(b), where μ′ is one of these finitely many dominant coweights of M. By [Vi2, Proposition 1], m is of the form m = jm_{0} for some j ∈ J_{M} = {x ∈ M(k((z))) : x^{−1}bσ^{∗}(x) = b} and m_{0} in some fixed connected component of X_{⪯μ′}^{M}(b). As the element g we started with only determines m up to left multiplication by J_{M}, we may assume that m = m_{0}. As b is superbasic in M, each connected component of X_{⪯μ′}^{M}(b) is a projective scheme of finite type (see for example [Vi1]). Especially, all elements m (for one μ′, but then also for all of the finitely many occurring μ′) are in some bounded subset of LG(k). By σ-conjugating g with m^{−1} = m_{0}^{−1} we obtain an element g′ of the form ñ^{−1}bσ^{∗}(ñ) for ñ = mnm^{−1} ∈ N(k((z))). Thus g′ = bn′ with n′ = b^{−1}ñ^{−1}bσ^{∗}(ñ) ∈ N. So far we only conjugated g by elements in a bounded subset of LG(k), thus the n′ still are contained in a subset of N(k((z))) which can be bounded by a bound only depending on the original B. Let N_{0} be a given bounded open subgroup of N(k((z))). By σ-conjugating by a sufficiently dominant element of T (𝔽_{q}((z))) which is central in M we may map all n′ in this bounded subset to N_{0}. Note that again the element we conjugate by is uniformly bounded (by a bound depending on B and the chosen N_{0}). Using Lemma 10.2 for b and N_{0} = I_{c_{b}} ∩ N (for d = 0), we obtain that each element of B is σ-conjugate by an element of a bounded set C to b.

Let c_{1} be such that fI_{d+c_{1}}f^{−1} ∈ I_{d} for all f ∈ C and all d. Let further c_{2} be such that σ^{∗}(f)^{−1}I_{d+c_{2}}σ^{∗}(f) ∈ I_{d} for all f ∈ C and all d. Let c = c_{b} + c_{1} + c_{2}. Let g ∈ B and h ∈ I_{d+c} for some d > 0. Let f ∈ C with f^{−1}gσ^{∗}(f) = b. Then

gh = fb(σ^{∗}(f^{−1})hσ^{∗}(f))σ^{∗}(f^{−1})

As the expression in the bracket is in I_{d+c_{b}+c_{1}} by our choice of c_{2}, Lemma 10.2 shows that there is an k′ ∈ I_{d+c_{1}} such that this equals

= fk^{′−1}bσ^{∗}(k′)σ^{∗}(f^{−1})
= (fk′f^{−1})^{−1}gσ^{∗}(fk′f^{−1}).

As fk′f^{−1} ∈ I_{d} by the choice of c_{1}, this proves the theorem.

Lemma 10.2. Let b ∈ LG(k). Then there is a c_{b} ∈ ℕ such that for each d ∈ ℕ and each g ∈ I_{d+c_{b}} there is a k ∈ I_{d} with bg = k^{−1}bσ^{∗}(k).

Proof. By σ-conjugating b we may assume that b is equal to the standard representative of its σconjugacy class as defined in [GHKR2, 7.2]. More precisely, this implies that b has the following properties. Let M be the centralizer of the G-dominant Newton polygon of b, it is the Levi component of a standard parabolic subgroup P of G. Then the standard representative is an element b of the extended affine Weyl group W̃_{M} of M with bI_{M}b^{−1} = I_{M} where I_{M} = I ∩ M. Note that this M is not the same as in the proof of Theorem 10.1. There we considered a Levi such that no σ-conjugate of b is contained in an even smaller one. Here, the Levi subgroup M is the largest standard Levi subgroup such that b is basic in M. As b is in W̃_{M}, there is an r > 0 such that b^{r} = z^{rν}.

It is enough to prove the lemma for d > 0. Let c_{b} be so large that b^{i}I_{d+c_{b}}b^{−i} ⊆ I_{d} for all i ∈ [−r + 1, …, r − 1]. (Using that I_{d} is generated by the corresponding affine root subgroups, one sees that if this condition holds for one d then it holds for all.) For the groups I_{n} we have an Iwahori decomposition I_{n} = I_{N,n}I_{M,n}I_{\overline{N},n} where the factors are the intersections of I_{n} with N, M and \overline{N} and where N and \overline{N} are the unipotent radicals of P and its opposite parabolic. Let g ∈ I_{d+c_{b}} and let g = g_{N}g_{M}g\overline{N} be its decomposition. We begin by showing that there is an f\overline{N} ∈ I_{\overline{N},d} with

f −1 ; \overline{N}bgσ^{∗}(f\overline{N}) ∈ bI_{M,d+c_{b}}I_{N,d+c_{b}}. Recall that b^{r} = z^{rν} for a ν with ⟨α, ν⟩ < 0 for every root of T in \overline{N}.

Thus g ↦ b^{−1}gb is an elementwise topologically nilpotent map on \overline{N}. The definition of c_{b} together with b^{r} = z^{rν} also implies that b^{−i}gb^{i} ∈ I_{d} for all i ≥ 0 and g ∈ I_{\overline{N},d+c_{b}}. Let f_{1} = (σ^{∗})^{−1}(g\overline{N}). Then

f_{1}^{−1}bg_{N}g_{M}g\overline{N}σ^{∗}(f_{1}) = f_{1}^{−1}bg_{N}g_{M}
= b(b^{−1}f_{1}^{−1}b)g_{N}g_{M}.

Note that [I_{a}, I_{a′}] ⊂ I_{a+a′} for all a, a′. As b^{−1}f_{1}^{−1}b ∈ I_{d}, the above product lies in bg_{N}g_{M}(b^{−1}f_{1}^{−1}b)I_{2d+c_{b}}. Repeating this construction for g′ ∈ g_{N}g_{M}(b^{−1}f_{1}^{−1}b)I_{2d+c_{b}} and its Iwahori decomposition g′ =

g′_{N}g′_{M}g′ ; \overline{N} yields an f_{2} = (σ^{∗})^{−1}(g′ ; \overline{N}). We have g′ ; \overline{N} ∈ (b^{−1}f_{1}^{−1}b)I_{\overline{N},2d+c_{b}}, and by the choice of c_{b},

this is in I_{d}. We iterate this process and let f\overline{N} = f_{1} ∘ f_{2} ∘ · · ·. As conjugation by b is elementwise topologically nilpotent on \overline{N}, the product converges. We obtain that bg is σ-conjugate via an element of I_{\overline{N},d} to an element of the form bg̃ with g̃ = g̃_{N} g̃_{M} ∈ I_{N,d+c_{b}}I_{M,d+c_{b}}. A similar argument (using f_{1} = bg̃_{N}b^{−1} and that g ↦ bgb^{−1} is elementwise topologically nilpotent on N) shows that we may also achieve that g̃_{N} = 1. For g ∈ I_{M,d+c_{b}} we use that b^{−1}I_{M,a}b = I_{M,a} for all a. In this situation the argument in [GHKR2, 6.3] also shows that bg is σ-conjugate via I_{M,d} (or even I_{M,d+c_{b}}) to b.

Corollary 10.3. Let (M, bσ^{∗}) and (M′, b′σ^{∗}) be two effective local shtukas over k of rank r bounded by μ and μ′. Then there is a c ∈ ℕ only depending on μ and μ′ with the following property. Let g : M → M′ be a linear map with (b′σ^{∗}(g) − gb)(x) ∈ z^{c+d}M′ for some d and every x ∈ σ^{∗}M. Then there exists a homomorphism of local shtukas f : (M, bσ^{∗}) → (M′, b′σ^{∗}) with (f − g)(x) ∈ z^{d}M′ for every x ∈ M.

Recall that a local shtuka over k is effective if and only if its Hodge polygon has only non-negative slopes.

Proof. Let M̃ = M ⊕ M′ and b̃: σ^{∗}M̃ → M̃ with b̃ = (b, b′). Consider g̃ : M̃ → M̃ with g̃|_{M′} = id_{M′} and g̃(m) = m + g(m) for m ∈ M. In matrix form we obtain

g̃^{−1}b̃σ^{∗}(g̃) = ( \begin{matrix}b & 0 \\ −gb + b′σ^{∗}(g) & b′ \end{matrix} ) ,

so the entries in the lower left block are congruent to zero modulo z^{c+d}. We use Theorem 10.1 for G = GL(M̃) and B = Kb̃K. We obtain that there is a c (only depending on B, i. e. on the Hodge polygon of b̃ or equivalently on those of b and b′), such that such a matrix is σ-conjugate via a k ∈ I_{d}

to b̃. An easy calculation shows that if we write k = ( \begin{matrix}k_{1} & k_{2} \\ k_{3} & k_{4} \end{matrix} ), then f = gk_{1} + k_{3} has all claimed

properties.

The last corollary was also proved by Fessler [Fe] using a different method.

References

[EGA] A. Grothendieck: Élements de Géométrie Algébrique, Publ. Math. IHES 4, 8, 11, 17, 20, 24, 28, 32, Bures-Sur-Yvette, 1960–1967; see also Grundlehren 166, Springer-Verlag, Berlin etc. 1971.

[SGA 1] A. Grothendieck: Revêtements étales et groupe fondamental, LNM 224, Springer-Verlag, Berlin-Heidelberg 1971.

[SGA 3] M. Demazure, A. Grothendieck: Schémas en Groupes I, II, III, LNM 151, 152, 153, Springer-Verlag, Berlin etc. 1970.

[An] G. Anderson: On Tate Modules of Formal t-Modules, Internat. Math. Res. Notices 2 (1993), 41–52.

[BL] A. Beauville, Y. Laszlo: Conformal blocks and generalized theta functions, Comm. Math. Phys. 164 (1994), no. 2, 385–419.

[Be] E. T. Beazley: Codimensions of Newton strata for SL_{3}(F) in the Iwahori case, preprint on

         arXiv:0711.3820.

[BD] A. Beilinson, V. Drinfeld: Quantization of Hitchin’s integrable system and Hecke eigensheaves,

         preprint on http://www.math.uchicago.edu/∼mitya/langlands.html.

[BLR] S. Bosch, W. Lütkebohmert, M. Raynaud: Néron models, Ergebnisse (3) 21, Springer-Verlag, Berlin, 1990.

[BT] F. Bruhat, J. Tits, Groupes réductifs sur un corps local II, Schémas en groupes. Existence d’une donnée radicielle valuée, Inst. Hautes Études Sci. Publ. Math. 60 (1984) 197–376.

[Ch1] C.-L. Chai: Newton polygons as lattice points, Amer. J. Math. 122 (2000), 967–990.

[Ch2] C.-L. Chai: Canonical coordinates on leaves of p-divisible groups: The two-slope case, preprint,

         http://www.math.upenn.edu/∼chai/papers pdf/2slope v14.pdf.

[Dr] V. G. Drinfeld: Coverings of p-adic symmetric domains, Funct. Anal. Appl. 10 (1976), 107–115.

[Ei] D. Eisenbud: Commutative Algebra with a View towards Algebraic Geometry, Springer-Verlag, New York 1994.

[Fa] G. Faltings: Algebraic loop groups and moduli spaces of bundles, J. Eur. Math. Soc. 5 (2003), no. 1, 41–68.

[Fe] T. Fessler: The Semilinear Theorem and its consequences for isomorphisms of F -crystals and local shtuka, Diploma thesis, University of Freiburg 2008.

[Ga] Q. R. Gashi: The conjecture of Kottwitz and Rapoport in the case of split groups, preprint on

         arXiv:0805.4575.

[Ge] A. Genestier: Espaces symétriques de Drinfeld, Astérisque 234, Soc. Math. France, Paris 1996.

[GHKR1] U. Görtz, T. Haines, R. Kottwitz, D. Reuman: Dimensions of some affine Deligne-Lusztig varieties, Ann. Sci. École Norm. Sup. (4) 39 (2006), no. 3, 467–511.

[GHKR2] U. Görtz, T. Haines, R. Kottwitz, D. Reuman: Affine Deligne-Lusztig varieties in affine flag varieties,

         preprint on arXiv:0805.0045.

[HR] T. Haines, M. Rapoport: On parahoric subgroups, Appendix to [PR2], Advances in Mathematics 219 (2008), 188–198.

[Ha1] U. Hartl: Uniformizing the Stacks of Abelian Sheaves, in Number Fields and Function fields - Two Parallel Worlds, Papers from the 4th Conference held on Texel Island, April 2004, G. van der Geer, B. Moonen, R. Schoof, Editors, 167–222, Progress in Math. 239, Birkhäuser-Verlag, Basel 2005.

[Ha2] U. Hartl: Period Spaces for Hodge Structures in Equal Characteristic, to appear in Ann. of Math.

         (2010); see also arXiv:math.NT/0511686.

[Ha3] U. Hartl: A Dictionary between Fontaine-Theory and its Analogue in Equal Characteristic, J. Number Th. 129 (2009), 1734–1757; see also arXiv:math.NT/0607182.

[Ha] R. Hartshorne: Algebraic Geometry, Springer-Verlag, New-York, Heidelberg (1977).

[HIO] M. Herrmann, S. Ikeda, U. Orbanz: Equimultiplicity and blowing up. An algebraic study. With an appendix by B. Moonen, Springer-Verlag, Berlin, 1988.

[Ja] J. C. Jantzen: Representations of algebraic groups, Mathematical Surveys and Monographs 107, AMS, Providence, RI, 2003.

[dJ] A. J. de Jong: Crystalline Dieudonné module theory via formal and rigid geometry, Publ. Math. IHES 82 (1995), 5–96.

[dJO] A. J. de Jong, F. Oort: Purity of the stratification by Newton polygons, J. Amer. Math. Soc. 13 (2000), 209–241.

[Ka] N. Katz: Slope filtration of F -crystals. Journées de Géométrie Algébrique de Rennes (Rennes, 1978), Vol. I, 113–163, Astérisque 63, Soc. Math. France, Paris 1979.

[Kn] A. Knapp: Lie groups beyond an introduction, Progress in Mathematics 140, Birkhäuser Boston, Boston, MA, 2002.

[Ko1] R. E. Kottwitz: Isocrystals with additional structure, Compositio Math. 56 (1985), no. 2, 201–220.

[Ko2] R. E. Kottwitz: Isocrystals with additional structure II, Compositio Math. 109 (1997), no. 3, 255– 339.

[Ko3] R. E. Kottwitz: Dimensions of Newton strata in the adjoint quotient of reductive groups, Pure Appl. Math. Q. 2 (2006), 817–836.

[KR] R. E. Kottwitz, M. Rapoport: On the existence of F -crystals, Comment. Math. Helv. 78 (2003), 153–184.

[LS] Y. Laszlo, C. Sorger: The line bundles on the moduli of parabolic G-bundles over curves and their sections, Ann. Sci. École Norm. Sup. (4) 30 (1997), no. 4, 499–525.

[Me] W. Messing, The crystals associated to Barsotti-Tate groups: with applications to abelian schemes, Lecture Notes in Math. 264, Springer, 1972.

[NP] B.C. Ngô, P. Polo: Résolutions de Demazure affines et formule de Casselman-Shalika géométrique, J. Algebraic Geom. 10 (2001), no. 3, 515–547.

[O1] F. Oort: Foliations in moduli spaces of abelian varieties, J. Amer. Math. Soc. 17 (2004), no. 2, 267–296

[O2] F. Oort: Foliations in moduli spaces of abelian varieties and dimension of leaves. To appear in Algebra, Arithmetic and Geometry, Volume I. In Honor of Y.I. Manin, Progress in Mathematics 269, Birkhäuser, 2009.

[OZ] F. Oort, Th. Zink: Families of p-divisible groups with constant Newton polygon, Documenta Math. 7 (2002), 183–201.

[PR1] G. Pappas, M. Rapoport: Local models in the ramified case I, The EL-case, J. Algebraic Geom. 12 (2003), no. 1, 107–145.

[PR2] G. Pappas, M. Rapoport: Twisted loop groups and their affine flag varieties, Advances in Mathematics 219 (2008), 118–198.

[Ra] M. Rapoport: A guide to the reduction modulo p of Shimura varieties, Astérisque 298 (2005), 271–318.

[RR] M. Rapoport, M. Richartz: On the classification and specialization of F -isocrystals with additional structure, Compositio Math. 103 (1996), 153–181.

[RZ] M. Rapoport, T. Zink: Period Spaces for p-divisible Groups, Ann. Math. Stud. 141, Princeton University Press, Princeton 1996.

[RZ2] M. Rapoport, T. Zink: A finiteness theorem in the Bruhat-Tits building: an application of Landvogt’s embedding theorem, Indag. Mathem., N.S. 10, 449–458.

[Var] Y. Varshavsky: Moduli spaces of principal F -bundles, Selecta Math. (N.S.) 10 (2004), no. 1, 131– 166.

[Vas] A. Vasiu: Crystalline boundedness principle, Ann. Sci. École Norm. Sup. 39 (2006), 245–300.

[Vi1] E. Viehmann: The dimension of some affine Deligne-Lusztig varieties, Ann. Sci. École Norm. Sup. (4) 39 (2006), 513–526.

[Vi2] E. Viehmann: Connected components of closed affine Deligne-Lusztig varieties, Math. Ann. 340 (2008), 315–333.

[ZS] O. Zariski, P. Samuel: Commutative algebra, Vol. II, Van Nostrand Co., Princeton-Toronto-London-New York 1960.

Urs Hartl

Universität Münster

Mathematisches Institut

Einsteinstr. 1

D – 48149 Münster

Germany

www.math.uni-muenster.de/u/urs.hartl/

Eva Viehmann Universität Bonn Mathematisches Institut Endenicher Allee 60 D – 53115 Bonn Germany