Purity of the stratification by Newton Polygons

by A.J. de Jong, Massachusetts Institute of Technology and F. Oort, Universiteit Utrecht

CONTENTS

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Results on F -crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3. Cohomology of the link of a surface singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4. Stratification by Newton polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5. Irreducibility of a catalogue of Barsotti-Tate groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1 Introduction

In this paper we prove four theorems: one on surface singularities, two on F -crystals, and one on moduli of p-divisible groups. The reason to put together these results in one paper is that the proofs, as given here, show how they are related. Let us first describe our results.

Let (S, 0) be a normal surface singularity over an algebraically closed field of characteristic p. Let S̃ → S be a resolution of singularities. Theorem 3.2 states that

(1)  H_{ét}^{1}(S \ {0}, ℚ_{p}) ≅ H_{ét}^{1}(S̃, ℚ_{p}).

The interest in this lies in the fact that we can work with ℚ_{p}-coefficients in characteristic p. The formula can be seen as a weak form of purity with ℚ_{p}-coefficients; when we consider ℚ_{ℓ}-coefficients the same result holds and follows in a straightforward manner from purity for étale cohomology. The result also holds for singularities of mixed characteristic, see Remark 3.13.

For the following two statements, let S be a scheme of characteristic p > 0 and let (E, F) be a nondegenerate F -crystal over S.

For any geometric point s̄ of S we can define the Newton polygon of (E, F) at s̄, see 2.2. We say that (E, F) is isoclinic if all slopes of the Newton polygon are equal, in all points of S. Our investigations show that the following working hypothesis is reasonable: if π_{1}(S) = {1}, then an isoclinic F -crystal E is isogenous to a constant F -crystal. We prove a result in this direction when S = Spec A, and A is a complete local Noetherian ring with algebraically closed residue field k. See Proposition 2.15 and Remark 2.18. This result implies the Isogeny Theorem 2.17 for p-divisible groups:

(2) If G over A is a p-divisible group, G is isoclinic, and A is normal, then G is isogenous to a constant p-divisible group G_{0} × Spec A.

Grothendieck showed that under a specialization s → s_{0} the Newton polygon of E at s_{0} lies on or above the Newton polygon at s. Katz analyzed and completed this result by proving that the locus where the Newton polygon is bounded from below is closed in S. Our central result, which we call the Purity Theorem 4.1, says:

(3) If the Newton polygon jumps somewhere, then it jumps already in codimension one.

As an application of (2) and (3) we study a problem on local moduli of simple pdivisible groups: describe all deformations which do not change the Newton polygon. By the Isogeny Theorem (2) we know that local deformations are obtained by isogenies. Because of this, and because a global moduli space does not exist, we look at the moduli space T of isogenies φ : H → X, where H is a fixed simple p-divisible group and where φ has a fixed degree, see Subsection 5.9. We prove that T is a catalogue: a classifying space which contains all objects that we want to consider over an algebraically closed field (Proposition 5.10). Our main theorem in Section 4 is that T is irreducible. We obtain two corollaries:

(4a) A simple local-local p-divisible group can be deformed into a p-divisible group with the same Newton polygon and with a = 1 (Corollary 5.12).

(4b) Grothendieck’s conjecture holds for a simple local-local p-divisible group (Corollary 5.13).

We quickly recall Grothendieck’s conjecture alluded to above, see [12, page 150]. Let X be a p-divisible group and let D denote its formal deformation space. Let δ be the Newton polygon of the covariant Dieudonné module of X. Let β be a Newton polygon, having the same endpoints as δ. If β occurs in the family over D, then β lies on or below δ, see discussion preceding (3). Grothendieck’s conjecture is that the converse holds: if β lies on or below δ, then U_{β} = {d ∈ D where the Newton polygon is β} is not empty. We would like to understand better the stratification D = ∐ U_{β} of D. For example: what is the dimension of U_{β}? Our result (3) gives an estimate. If U_{β} ≠ ∅ then

dim(U_{β}) ≥ dim(D) − c,

where

c = maximum length k of a chain γ = β_{0} ≻ β_{1} ≻ ⋯ ≻ β_{k} = β, β_{i} ≠ β_{i+1}

The symbol γ is the generic Newton polygon (see Lemma 5.15) and the relation β_{i} ≻ β_{i+1} means β_{i+1} lies on or above β_{i}.

Generally speaking the Purity Theorem can be used to bound codimensions of Newton polygon strata. This can be applied to the F -crystal defined by the ith crystalline cohomology of a family of varieties over S. For example the universal family of hypersurfaces of a given degree in ℙ^{n}.

The Purity Theorem also gives the correct lower bound for the dimension of the Newton polygon strata in the moduli spaces of principally polarized abelian varieties A_{g,1,n}⊗𝔽_{p} (n ≥ 3, p |̸n). Li and Oort [18] have proven that the supersingular stratum S_{g} ⊂ A_{g,1,n}⊗𝔽_{p} has the correct dimension [g^{2}/4]. It follows that any irreducible component of a Newton polygon stratum U_{β} ⊂ A_{g,1,n}⊗𝔽_{p} whose closure meets S_{g} has the correct dimension (apply the Purity Theorem to bound the codimension of S_{g} in \overline{U_{β}}). For more information on this, compare [29].

We turn to a discussion of the proof of the Purity Theorem. Let us analyze the situation where S is normal, local, two dimensional with closed point 0, and where E has constant Newton polygon over S \ {0}. We consider a slope λ which occurs in the generic fiber. This gives us an isoclinic F -crystal over S \ {0}, which is determined by a certain monodromy representation of π_{1}(S \ {0}). By taking exterior powers we can make this abelian. Here the result (1) applies. Finally, we apply the extension theorem on homomorphisms of F -crystals [8]. From this we conclude that λ occurs in the Newton polygon of E over 0 as desired.

The proof of (1) has four ingredients. Algebraization [2], to make the singularity algebraic. Alteration [7], to view the situation as a fibering by curves. Patching à la Harbater and Raynaud to produce a global class from a local one. Extension in equal characteristic [8] to extend a class in the 1st étale cohomology group of the generic fiber of a family of curves to a class lying on the whole family. In the mixed characteristic case, we use the result of Tate [35] on extensions of homomorphisms of Barsotti-Tate groups; see Remark 3.13.

Acknowledgements. The first author would like to thank the second author.

2 Results on F-crystals

2.1 Conventions. In this section S denotes a connected scheme of characteristic p. We use the term crystal to mean a crystal of finite locally free O_{cris}-modules. See [3]. Here O_{cris} denotes the structure sheaf on the category CRIS(S/ Spec ℤ_{p}) (big crystalline site of S). If T → S is a morphism, then we use E|_{T} to denote the pullback of E to CRIS(T/ Spec ℤ_{p}). For a crystal E, we denote by E^{(n)} the pullback of E by the nth iterate of the Frobenius endomorphism of S. An F -crystal over S is a pair (E, F), where E is a crystal over S and F : E^{(1)} → E is a morphism of crystals. We usually denote an F -crystal by E, the map F being understood. Recall that E is a nondegenerate F -crystal if the kernel and cokernel of F are annihilated by some power of p. See [32]. All F -crystals in this paper will be nondegenerate.

Let λ ∈ ℚ_{≥0} be a nonnegative rational integer. Write λ = b/a in lowest terms. We will say that an F -crystal E is divisible by λ, if there exists a morphism of F -crystals φ : E^{(a)} → E such that F^{a} = p^{aλ}φ.

A perfect scheme S in characteristic p is a scheme such that the Frobenius map (−)^{p} : O_{S} → O_{S} is an isomorphism. A crystal over a perfect scheme S is simply given by a finite locally free sheaf of W (O_{S})-modules.

2.2 Definition of isoclinic F -crystals. Let k be an algebraically closed field of characteristic p. Let W be the Witt ring of k, with Frobenius map σ. Recall that a nondegenerate F - crystal over Spec k corresponds to a finite free W -module M with a σ-linear endomorphism F : M → M such that M/F (M) has finite length. We briefly describe the Newton polygon attached to (M, F), see [16, (1.3)]. Let L be the fraction field of W. The vector space M ⊗ L has a slope decomposition M ⊗ L = ⊕_{λ∈ℚ}V_{λ}. This decomposition is characterized by the following properties: (1) it is F -stable, (2) each V_{λ} has a basis of elements e such that F^{n}(e) = p^{nλ}e for n sufficiently divisible. The slope λ occurs in the Newton polygon of (M, F) with multiplicity dimV_{λ}, and they are organized in increasing order. We say that (M, F) is isoclinic of slope λ if V_{λ′} = 0 for λ′ ≠ λ.

Let S be a scheme of characteristic p. Let E be a nondegenerate F -crystal on S. We say that E is isoclinic of slope λ over S if the pullback of E to every geometric point of S is isoclinic of slope λ.

2.3 The standard F -crystal of slope λ. Let λ ∈ ℚ_{≥0} be a nonnegative rational number. Write λ = b/a in lowest terms (note 0 = 0/1). We define the standard F -isocrystal E_{λ} of slope λ over Spec 𝔽_{p} to be the crystal O_{cris}^{a} with basis {e_{1}, …, e_{a}}, and with F acting by the formulae: F (e_{i}) = e_{i+1}, for i = 1, …, a − 1 and F (e_{a}) = p^{aλ}e_{1}. Note that E_{λ} is divisible by λ, with φ_{λ} : E_{λ}^{(a)} → E_{λ} given by φ_{λ}(e_{i}) = e_{i}. We recall that

D_{λ} := End(E_{λ}|_{Spec} _{ 𝔽̄_{p}}) ⊗ ℚ

is the division algebra over ℚ_{p} whose invariant λ is the class of λ in ℚ/ℤ. We remark that all endomorphisms of E_{λ} are realized over 𝔽_{p^{a}}.

2.4 Divisible isoclinic F -crystals. Let S be a scheme of characteristic p, and let E be an isoclinic F -crystal of slope λ over S which is λ-divisible. Denote φ : E^{(a)} → E the map such that F^{a} = p^{aλ}φ which is supposed to exist by our definition of divisibility. Since (E, F) is isoclinic of slope λ, we see that φ is an isomorphism, i.e., the pair (E, φ) is a ”unit-root F^{a}-crystal”.

Let us consider the sheaves H_{n} on the étale site of S given by

Γ(T, H_{n}) = Γ(CRIS(T/ Spec ℤ_{p}), (E/p^{n}E))^{φ=1}.

This is a locally constant sheaf of finite free ℤ/p^{n}ℤ-modules of rank equal to a(rk E). This follows from the results of [4, Section 2.4] in the following way. Let E(φ) denote the crystal E ⊕ E^{(1)} ⊕ … ⊕ E^{(a−1)}, with F given by the matrix

( \begin{matrix}0 & 0 & … & φ \\ id & 0 & … & 0 \\ … \\ … & … & 0 & 0 \\ … & … & id & 0 \end{matrix} ) .

Then E(φ) is a unit root F -crystal, and the sheaf H_{n} is clearly canonically isomorphic to the sheaf H(E(φ)/p^{n}) defined in [4, page 205]. The desired result then follows from the results of [4], in particular the remarks at the bottom of page 205 and the discussion directly preceding Proposition 2.4.4.

We can identify H_{n} with the sheaf Hom((E_{λ}, F, φ_{λ}), (E/p^{n}, F, φ)) of homomorphisms α : E_{λ} → E/p^{n} of truncated F -crystals which are compatible with φ_{λ} and φ, i.e., α^{(a)}∘φ_{λ} = φ ∘ α. Indeed, given such an α we have α(e_{1}) ∈ Γ(E/p^{n}E)^{φ=1}. The reverse construction is left to the reader. Thus H_{n} is canonically a sheaf of right modules over the sheaf of endomorphisms of (E_{λ}/p^{n}, F) compatible with φ_{λ}:

D_{λ,n} := End ((E_{λ}/p^{n}, F, φ_{λ})).

For the same reasons as above, the sheaves D_{λ,n} are finite locally constant sheaves of ℤ/p^{n}ℤ-algebras over S_{ét}. In fact, each H_{n} is finite locally free over D_{λ,n}. Thus the system {H_{n}} = {Hom((E_{λ}, F, φ_{λ}), (E/p^{n}, F, φ))} is a finite locally free sheaf of right {D_{λ,n}}- modules over S_{ét}, in other words it is a lisse sheaf. We abbreviate: Hom(E_{λ}, E) := {H_{n}} and D_{λ} = {D_{λ,n}}.

Note that Γ( Spec 𝔽̄_{p}, D_{λ,n}) = O_{λ}/p^{n}O_{λ}, where O_{λ} ⊂ D_{λ} is the maximal order. Hence Γ( Spec 𝔽̄_{p}, D_{λ}) = O_{λ}. We also note that if S → Spec 𝔽_{p} factors through Spec 𝔽_{p^{a}}, then D_{λ} is in fact a constant sheaf (i.e., all D_{λ,n} are constant).

We leave it to the reader to establish the following result. It generalizes the case of slope 0 done in [4], and follows from this case by the discussion above.

2.5 Proposition. The functor

E ⟼ Hom(E_{λ}, E)

establishes an equivalence between the category of λ-divisible F -crystals isoclinic of slope λ endowed with a choice of φ and the category of lisse right D_{λ}-modules on S_{ét}.

2.6 There is a small snag to this proposition. The map φ may not be uniquely determined by E and its Frobenius map F. However, any two choices φ, φ′ differ by a map which is annihilated by p^{aλ}, namely p^{aλ}(φ − φ′) = p^{aλ}φ − p^{aλ}φ′ = F^{a} − F^{a} = 0. For good schemes (good means no p-torsion in the universal PD-envelopes, e.g., regular schemes and perfect schemes) there is at most one φ. In general however, the categories of λ-divisible λ-isoclinic F -crystals and of lisse right D_{λ}-modules over S will still be equivalent up to isogeny by the above.

In particular, such F -crystals up to isogeny correspond bijectively to continuous twisted Galois representations

ρ : π_{1}(S, s̄) ⟶ GL_{d}(D_{λ}).

Here twisted means that π_{1}(S) acts nontrivially on the coefficients D_{λ} as in 2.3. The integer d equals (rk E)/a. The idea to consider these representations is not new, and we could have used other references above. See for example [11] and [15, pp. 142–146]; as we saw above, in [4] we find the case of slope zero over any base.

2.7 Study of F -crystals over a field. Suppose that S is the spectrum of a field K. Choose a Cohen ring Λ for K, and let σ : Λ → Λ be a lift of Frobenius on K. Let K^{pf} be a perfect closure of K. We remark that

W (K^{pf}) = lim_{⟶} (Λ⟶^{σ}Λ⟶^{σ} …).

Thus we get an injection Λ → W (K^{pf}) compatible with σ.

We recall, see for example [4, Proposition 1.3.3], that an F -crystal E over K is given by a triple (M, ∇, F) over Λ. (Thus M is a finite free Λ-module, ∇ is a topologically quasi-nilpotent continuous connection, and F is a horizontal σ-linear selfmap of M.) The pullback E|_{Spec K^{pf}} of E to K^{pf} corresponds to the pair (M ⊗_{Λ} W (K^{pf}), F ⊗ σ). The slopes of the F -crystal E are simply defined as the slopes of the pullback of E to Spec K^{pf}, i.e., the slopes of (M ⊗_{Λ} W (K^{pf}), F ⊗ σ).

Suppose these slopes are λ_{1} ≤ λ_{2} ≤ … ≤ λ_{s} with multiplicities A_{1}, …, A_{s}. We claim that the result of [16, Corollary 2.6.3] holds in this situation.

2.8 Claim. (M, ∇, F) is isogenous to a F -crystal (M′, ∇′, F′) which admits a filtration

0 ⊂ (M′_{1}, ∇′, F′) ⊂ (M′_{2}, ∇′, F′) ⊂ … ⊂ (M′_{s}, ∇′, F′) = (M′, ∇′, F′)

in which M′_{i}/M′_{i−1} is a free Λ-module such that the resulting F -crystal (M′_{i}/M′_{i−1}, ∇′, F′) is divisible by λ_{i}, has rank A_{i} and is isoclinic of slope λ_{i}.

Proof. We claim the results 2.4.2 (Newton-Hodge filtration), 2.6.1 (Isogeny theorem) and 2.6.2 of [16] in our situation. For the proof of 2.4.2, just set A_{0} = K and A_{∞} = Λ, and the proof given in [16] goes through with only minor modifications. For 2.6.1 the proof is exactly the same as for the case of a perfect field given in [16], i.e., the module M′ works. The proofs of 2.6.2 and 2.6.3 go through without changes.

2.9 Note that the F -crystals E_{i} corresponding to (M′_{i}/M′_{i−1}, ∇′, F′) are well defined up to isogeny by E. We define the Galois representation associated to E in slope λ_{i} to be the representation

ρ : Gal(K^{sep}/K) ⟶ GL_{d_{i}}(D_{λ_{i}})

associated to the λ_{i}-divisible λ_{i}-isoclinic F -crystal E_{i} over Spec K, see 2.6.

2.10 Proposition. Let R be a discrete valuation ring of characteristic p with fraction field K. Let E be a F -crystal over R. Suppose that at the two points of Spec R the Newton polygons of E coincide. Then the Galois representations in every slope λ associated to E over K are unramified (i.e., they come from representations of π_{1}(Spec R)).

Proof. We replace R by its strict henselization. We have to show that the representations are trivial.

Let us denote R^{∧} the completion of R, and let K^{∧} be the fraction field of R^{∧}. Since [33, Ch II, section 3, Cor 4] the map of absolute Galois groups

Gal_{K^{∧}} ⟶ Gal_{K}

is surjective, it suffices to handle the case where R is complete.

In this case we have R = k[[t]] with k separably closed. Let k̄ be an algebraic closure of k. Consider the ring extension k[[t]] ⊂ k̄[[t]]. It induces an extension of fraction fields K ⊂ L. Again the map

Gal_{L} ⟶ Gal_{K}

is surjective. See [5, Lemma 3.2.5]. We sketch the proof. The union R′ of the henselian local rings k^{p^{−n}}[[t]] is a henselian discrete valuation ring whose completion is k̄[[t]]. Hence, by the first part of the proof of the proposition we have that Gal_{L} → Gal_{K′} is surjective. After this, we simply observe that K ⊂ K′ is purely inseparable. Thus we may assume that R = k[[t]] with k algebraically closed.

In this case the result follows from [16, Theorem 2.7.4], which states that over k((t))^{pf} the F -crystal E becomes isogenous to a constant F -crystal, and therefore has trivial associated Galois representations.

2.11 Remark. In fact, we can be a little more precise. If R is henselian with residue field k, then we have π_{1}(Spec R) ≅ Gal(k^{sep}/k). The resulting representations

Gal(k^{sep}/k) ⟶ GL_{d_{i}}(D_{λ_{i}})

are in fact the Galois representations associated to the F -crystal E|_{Spec k} in slope λ_{i}. To prove this we may replace R by k^{pf}[[t]], where k^{pf} is the perfect closure of k. From [16] we see that the slope filtration over the fraction field comes from a slope filtration over k^{pf}[[t]], with ith piece a λ_{i}-divisible F -crystal. From this the result follows. We will not use this result.

2.12 Before we can state the next proposition we introduce some notations. We remark that the result we have in mind can be proved in much greater generality. However, since the authors do not see a use for this at present, we have decided to avoid the added technicalities by formulating a less general result.

Let A be an excellent domain, with fraction field K. We write the perfect closure K^{pf} of K as the union of its finite extensions K_{α} of K. Let A_{α} denote the integral closure of A in K_{α}. The union A^{pf} = ⋃ A_{α} is a perfect ring. Write S = Spec A, S_{α} = Spec A_{α} and S^{pf} = Spec A^{pf}. Thus S^{pf} = lim_{⟵}S_{α}. All the S_{α} are excellent integral schemes, and S^{pf} is integral as well.

A nice scheme T over S^{pf}, is any scheme obtained as follows: For some α_{∘} we are given a modification T_{α_{∘}} → S_{α_{∘}}, and for α > α_{∘} the scheme T_{α} is the normalization of T_{α_{∘}} in the field K_{α}. All the transition morphisms T_{α} → T_{α′} are affine, hence the scheme T = lim_{⟵}T_{α} exists (and it dominates S^{pf}). We remark that the schemes T_{α} are modifications of S_{α}, they are proper over S = Spec A, integral, excellent and have function field K_{α}. Thus any nice scheme T is integral, perfect and has function field K^{pf}.

Note that if X → T is a morphism of finite presentation, then there exists a β and a morphism of finite presentation X_{β} → T_{β} such that X ≅ X_{β} ×_{T_{β}} T. In case X → T is (finite) étale, we can choose X_{β} → T_{β} to be (finite) étale. Similarly for proper. See EGA IV 8.8 & 8.10.5.

Suppose that φ : X → T is proper, of finite presentation, and that η ∈ X is a point mapping to the generic point of T inducing an isomorphism κ(η) = R(T) = K^{pf}. Then there exists a nice scheme T′ → S and a morphism T′ → X over S such that the generic point of T′ maps to η.

To prove this, write X = X_{β} ×_{T_{β}} T as above, with X_{β} proper over T_{β}. The point η maps to a point η_{β} ∈ X_{β}. Note that K ⊂ κ(η_{β}) ⊂ κ(η) = K^{pf}. Thus there exists a γ such that κ(η_{β}) ⊂ K_{γ} (use that X_{β} is of finite type over T_{β}). In other words, the point η_{γ} in X_{γ} = X_{β} ×_{T_{β}} T_{γ} will satisfy κ(η_{γ}) = K_{γ} = R(T_{γ}). Let T′_{γ} ⊂ X_{γ} be the schematic closure of {η_{γ}}. Then the composition T′_{γ} → T_{γ} → S_{γ} is a modification. Let T′_{δ} be the normalization of T′_{γ} in K_{δ} for δ > γ. Then there are maps T′_{δ} → T_{δ} by the universal properties of normalizations, and we obtain the desired morphism

T′ = lim_{⟵} T′_{δ} ⟶ lim_{⟵} X_{γ} ×_{T_{γ}} T_{δ} = X.

2.13 Proposition. Let A ⊂ K, S, S_{α}, etc be as above. Let E be a nondegenerate F -crystal over S^{pf}. Assume that all the slopes of E|_{Spec K^{pf}} are ≥ λ. Then there exists a nice morphism π : T → S^{pf} such that E|_{T} is isogenous to an F -crystal which is divisible by λ.

Proof. Since S^{pf} is perfect, the F -crystal E is simply a finite locally free sheaf of W (O_{S^{pf}} )- modules endowed with a Frobenius linear endomorphism F : E → E. Let us use the suggestive notation E ⊗ W (K^{pf}) to denote the W (K^{pf})-module which corresponds to the pull back E|_{Spec K^{pf}} . By the isogeny theorem [16, 2.6.1], there exists an isogeny

N ⟶ E ⊗ W (K^{pf})

such that N is divisible by λ. Note that for some r ∈ ℕ:

(∗)  p^{r}E ⊗ W (K^{pf}) ⊂ N ⊂ E ⊗ W (K^{pf}).

We claim that, for any submodule N as in (∗), there exist (1) a nice morphism π : T → S^{pf}, (2) a finite locally free W (O_{T})-module E′, and (3) a W (O_{T})-linear injection

E′ ⟶ π^{∗}E := π^{−1}E ⊗_{W (π^{−1}O)} W (O_{T})

such that E′ ⊗ W (K^{pf}) = N ⊂ π^{∗}E ⊗ W (K^{pf}) = E ⊗ W (K^{pf}) and such that π^{∗}E/E′ has a finite filtration whose graded is a locally free O_{T} -module.

We prove this by induction on r. Let N′ := N + p^{r−1}E ⊗ W (K^{pf}). By induction hypothesis we obtain π : T → S^{pf}, E′ → π^{∗}E such that E′ ⊗ W (K^{pf}) = N′. Note that pN′ ⊂ N ⊂ N′. Hence, N is the inverse image of a subvector space V ⊂ E′ ⊗ K^{pf} (under the map E′ ⊗ W (K^{pf}) → E′ ⊗ K^{pf}). Say dimV = m. The sheaf of O_{T} -modules E′/pE′ = E′ ⊗_{W (O_{T})} O_{T} is finite locally free. Hence the Grassmanian X = 𝔾r(m, E′/pE′) parameterizing rank m locally direct summands is of finite presentation over T. The subvector space V ⊂ E′ ⊗ K^{pf} determines a point η ∈ X with κ(η) = K^{pf}. Therefore, by the discussion preceding the proposition, we can find a nice scheme T′ → S^{pf} and a morphism T′ → X mapping the generic point of T′ to η. Letting τ : T′ → T be the induced map, this means that V extends to a locally direct summand V ⊂ τ^{∗}(E′/pE′). Set E′′ equal to the inverse image of V under the map τ^{∗}E′ → τ^{∗}(E′/pE′). Then E′′ is finite locally free over W (O_{T′}), and E′′ ⊗ W (K^{pf}) = N. The quotient (π ∘ τ)^{∗}E/E′′ has a filtration 0 ⊂ τ^{∗}E′/E′′ ⊂ τ^{∗}π^{∗}E/E′′ with quotients τ^{∗}E′/E′′, locally free by construction, and τ^{∗}(π^{∗}E/E′), having a suitable filtration by induction hypothesis. This proves the claim.

We apply the claim to the λ-divisible subcrystal N of E ⊗W (K^{pf}). We obtain π : T → S^{pf} and E′ ⊂ π^{∗}E. Since F (N) ⊂ N, the condition on π^{∗}E/E′ assures that F (E′) ⊂ E′ (use that T is integral and perfect). The fact that N is divisible by λ means that F^{a}(N) ⊂ p^{aλ}N. Again the condition on π^{∗}E/E′ insures that this implies F^{a}(E′) ⊂ p^{aλ}E′. Since W (O_{T}) does not have p-torsion, we can form the map φ = p^{−aλ}F^{a} on E′ as desired.

2.14 Let us consider a Noetherian complete local ring A of characteristic p with algebraically closed residue field k. Let E be an isoclinic F -crystal over A of slope λ. According to 2.9 we obtain a Galois representation (associated to E|_{Spec κ(𝔭)} in slope λ)

ρ_{𝔭} : Gal(κ(𝔭)^{sep}/κ(𝔭)) ⟶ GL_{d}(D_{λ}),

where 𝔭 ⊂ A is any prime ideal of A. (Note that our assumptions on A imply that 𝔽̄_{p} ⊂ A so that we get actual representations, not twisted ones.)

2.15 Proposition. The Galois representation ρ_{𝔭} is trivial.

Proof. We may replace A by A/𝔭. Then 𝔭 = (0) and K = κ(𝔭) is the fraction field of the excellent domain A. We use the notation introduced in Subsection 2.12. According to Proposition 2.13 we can find a nice morphism π : T → Spec A^{pf} such that π^{∗}E becomes isogenous to a λ-divisible F -crystal E′ over T. By 2.6 this isocrystal determines a continuous representation

ρ : π_{1}(T) ⟶ GL_{d}(D_{λ}).

When we restrict ρ to π_{1}(Spec K^{pf}), we recover the representation ρ_{𝔭}.

Let us write 0 ∈ Spec A for the closed point of Spec A. Let T_{0} denote the fiber over 0 of the morphism T → Spec A. Obviously, E′|_{T_{0}} is isogenous to the constant F -crystal with fiber E|_{0}. We conclude that, for any geometric point t̄ of T_{0}, the composition

π_{1}(T_{0}, t̄) ⟶ π_{1}(T, t̄) ⟶^{ρ} GL_{d}(D_{λ})

is trivial. We claim that this implies that ρ (and so ρ_{𝔭}) is trivial.

We have to prove the following statement. Any finite étale covering X → T such that X ×_{T} T_{0} is a trivial covering of T_{0} is itself trivial. By the discussion in 2.12 there exists a diagram

X ⟶ X_{α}
↓  ↓
T ⟶ T_{α}
↓  ↓
Spec A^{pf} ⟶ Spec A_{α}
↓  ↓
Spec A ⟶ Spec A

such that X = X_{α} ×_{T_{α}} T. We may assume that X_{α} → T_{α} is finite étale. Since T_{0} = lim_{⟵}(T_{γ})_{0}, we may assume, after increasing α, that X_{α} ×_{T_{α}} (T_{α})_{0} is a trivial covering of (T_{α})_{0}.

At this point we can use Zariski’s theorem on formal functions. The morphism X_{α} → Spec A is proper being the composition of the finite morphism X_{α} → T_{α} and the proper morphism T_{α} → Spec A, see 2.12. Thus the map on connected components π_{0}(X_{α}) → π_{0}((X_{α})_{0}) is bijective. The triviality of (X_{α})_{0} over (T_{α})_{0} implies that π_{0}(X_{α}) has at least d elements, if the degree of X → T is d. Thus X → T is trivial.

2.16 Remark. We mention a generalization of this proposition that can be proved with the same methods. Assume that E is a nondegenerate F -crystal over Spec A whose Newton polygon is constant (i.e., the Newton polygon is the same in all points of Spec A). Then the Galois representations in all slopes associated to the fibers of E in the points of Spec A are trivial. We remark that the case dimA = 1 of this generalization follows from Proposition 2.10. To prove this more general statement one has to strengthen Proposition 2.13 to a theorem like the filtration theorem of [16].

2.17 Corollary. (Isogeny Theorem.) Let A be as in Subsection 2.14. Let G be a pdivisible group over Spec A. Assume that A is normal, and that the covariant crystalline Dieudonné module 𝔻(G) (see [22]) is isoclinic over Spec A. Then there exists a p-divisible group G_{0} over Spec k and an isogeny G_{0} ×_{Spec k} Spec A → G over A.

Proof. By the proposition above, the Galois representation associated to 𝔻(G ⊗ K) = 𝔻(G)|_{Spec K} is trivial. This means that there is an isogeny E_{λ}^{n}|_{Spec K} → 𝔻(G), where λ = a/b is the slope of 𝔻(G), see Proposition 2.5 and Subsection 2.6. Since λ comes from a p-divisible group, we have λ ∈ [0, 1] ∩ ℚ. This means that (E_{λ})^{n} is isogenous to the Dieudonné module of a p-divisible group G_{0} over Spec 𝔽_{p}. (For example the p-divisible group (G_{b−a,a})^{n} of [20].) Therefore we have an isogeny G_{0} ⊗ K → G ⊗ K over the field of fractions K of A. This extends to a morphism of p-divisible groups over A by [8, Introduction].

2.18 Remark. One would like to have the corollary also for F -crystals. Proposition 2.15 implies that there exists an isogeny (E_{λ})^{n}|_{Spec K} → E|_{Spec K} (for some λ and n) over the generic point Spec K ↪ Spec A. The extension theorem of [8] holds for F -crystals over discrete valuation rings, but we do not know whether an extension theorem over Noetherian normal schemes holds. The reason is that the universal PD-envelopes may behave badly, even for excellent normal rings. However, the corollary should hold in a suitable setting of convergent F -isocrystals.

3 Cohomology of the link of a surface singularity

3.1 Let A be a local complete Noetherian ring, normal of dimension 2 with algebraically closed residue field k. Let S = Spec A, let 0 ∈ Spec A be the closed point, and let U = S \ {0}. Let us choose a resolution of singularities π : S̃ → S (see for example [19]), and let E = π^{−1}(0) be the exceptional fiber. We identify π^{−1}(U) with U:

U ⟶ S̃ ⟵ E
↓ ↓ ↓
U ⟶ S ⟵ 0

3.2 Theorem. The natural map

H_{ét}^{1}(S̃, ℚ_{p}) ⟶ H_{ét}^{1}(U, ℚ_{p})

is an isomorphism.

Informally speaking this means that any étale ℤ_{p}-covering of U extends to an étale covering of S̃. The map of the theorem is easily seen to be injective. Note that H_{ét}^{1}(S̃, ℚ_{p}) = H_{ét}^{1}(E, ℚ_{p}).

If k is the complex numbers and the singularity is algebraic, one can find this result in Mumford’s paper [26, pp. 234-235]. More generally, when k is an algebraically closed field of characteristic different from p, the theorem follows easily using standard methods of étale cohomology and the nondegeneracy of the intersection matrix of the components of E. However, the reader can verify that our methods lead to a proof of the theorem in these cases also.

In the rest of this section we will discuss the case where A has characteristic p, i.e., where k ⊂ A. At the end of the section we will indicate how to change the arguments to prove the mixed characteristic (0, p) case.

We fix a nonzero element α of H_{ét}^{1}(U, ℚ_{p}). It will be viewed as a continuous homomorphism π_{1}(U) → ℚ_{p}. We may assume (by replacing α by a multiple) that the image of α lies in ℤ_{p} and that α : π_{1}(U) → ℤ_{p} is surjective.

Let η ∈ S̃ be the generic point of a component of E. Consider the complete local ring O = Ô_{_{S},η} of S̃ at η. Let K be the field of fractions of O. There is a map Gal_{K} → π_{1}(U) induced by functoriality of the fundamental group from the morphism Spec K → U.

3.3 Lemma. If the composition Gal_{K} → π_{1}(U) ⟶^{α} ℚ is unramified for every η as above, then α extends.

Proof. The assumption implies that each of the finite étale coverings V_{n} → U associated to α mod p^{n} : π_{1}(U) → ℤ_{p}/p^{n}ℤ_{p} extends to a finite étale covering outside codimension 2. By Zariski–Nagata on purity of branch locus [SGA2, X Thm 3.4], we see they extend to the whole of S̃.

3.4 Lemma. Let A → B be a finite ring homomorphism such that B is a normal domain also. If the element α ∈ H_{ét}^{1}(U, ℚ_{p}) extends to an étale cohomology class on the desingularization of Spec B, then it extends to S̃.

Proof. We may dominate the desingularization of Spec B by a desingularization T̃ which dominates S̃. Let η ∈ S̃ be as above, and let ξ ∈ T̃ be a point of T̃ mapping to η. Such a point exists as T̃ → S̃ is proper and dominant, hence surjective.

Let O′ be the complete local ring of T̃ at ξ, and let K′ be its field of fractions. We remark that O ⊂ O′ is a finite extension of complete discrete valuation rings. The inertia group I′ ⊂ Gal_{K′} therefore maps onto an open subgroup of the inertia subgroup I ⊂ Gal_{K}.

By assumption we know that the composition

I′ → I → Gal_{K} → π_{1}(U) ⟶^{α} ℚ_{p}

is trivial. Since ℚ_{p} is torsion free this implies that α|_{I} is trivial. Thus by Lemma 3.3 we are done.

We are going to use that the singularity S is algebraizable, see [Artin, Theorem 3.8]. This means that there exists a projective surface X over k, and a point 0 ∈ X such that the complete local ring of X at 0 is isomorphic to A. (Recall that we are treating the case k ⊂ A.) After blowing up X, we may assume in addition that there exists a flat morphism X → ℙ_{k}^{1}. By [7], we may perform an alteration:

\begin{matrix}Y & ⟶^{ψ} & X \\ ↓ & & ↓ \\ C & ⟶ & ℙ_{k}^{1}. \end{matrix}

Here Y is a projective surface smooth over k, C is a projective curve smooth over k, and Y → C is a strict semi-stable curve over C smooth over an open part of C. The map ψ : Y → X is an alteration.

Let E′ be a connected component of ψ^{−1}(0); let P ∈ C be the image of E′ in C. Note that E′ is a union of irreducible components of Y_{P}. Also, E′ is a normal crossings divisor in Y. We remark that E′ ≠ Y_{P}. (For example because the class of Y_{P} maps to the class of a fiber of X → ℙ^{1}, but the class of E′ maps to zero. Classes are taken in Chow groups modulo algebraic equivalence.)

Let Y → Y′ → X be the Stein factorization of ψ. Let 0′ ∈ Y′(k) be the image of E′ in Y′. The extension

A ≅ Ô_{X,0} ⊂ Ô_{Y′,0′}

is a finite extension of normal local rings. Thus by Lemma 3.4 we have reduced our problem to the case where A is as in Situation 3.5 below.

3.5 Situation. Here X → Spec k[[t]] is a flat projective family of curves. The scheme X is regular. The special fiber X_{0} is a strict normal crossings divisor in X, and the generic fiber X_{K} is a smooth curve. Furthermore, E ⊂ X_{0} is a connected union of irreducible components, and E ≠ X_{0}.

Finally, the ring A is the complete local ring of the scheme obtained from X by collapsing E. More precisely, there exists a modification X → X′, with the following properties: (1) It is an isomorphism outside E ⊂ X. (2) The image of E is a single point P ∈ X′. (3) The complete local ring of X′ at P is isomorphic to A.

3.6 Let us interpret the data α in a slightly different manner in this situation. The map α mod p^{n} : π_{1}(U) → ℤ_{p}/p^{n}ℤ_{p} corresponds to a finite morphism T_{n} → Spec A with T_{n} a normal 2-dimensional scheme; T_{n} is endowed with an action of ℤ/p^{n}ℤ, such that over U it is a ℤ/p^{n}ℤ-principal homogeneous space.

Let X_{/E} denote the formal scheme obtained by formally completing X along E. There is a morphism of formal schemes

X_{/E} ⟶ Spf A

which induces an isomorphism A → Γ(X_{/E}, O_{X_{/E}}). Let 𝔗_{n} denote the formal scheme obtained from T_{n} under base change with Spf A → Spec A. We can form the fiber products

X_{/E} ×_{Spf A} 𝔗_{n} = X_{/E} ×_{Spec A} T_{n}.

Finally, we can “formally normalize” these formal schemes to obtain formal schemes ℨ_{n} finite over X_{/E}. (“Formally normalize” means to take the integral closure of the formal affine rings.)

In more down to earth terms one can describe ℨ_{n} as follows. Let 𝔘 = Spf C be a formal affine open of X_{/E}. There is a ring map A → C (see above), such that 𝔪_{A}C is an ideal of definition of C. Write T_{n} = Spec B_{n}. Then the inverse image of 𝔘 in ℨ_{n} is the formal spectrum of the following algebra:

C_{n} := normalization of B_{n} ⊗_{A} C.

The ideal of definition is 𝔪_{A}C_{n}; we do not have to complete as A → B_{n} is finite and C is excellent, so that C_{n} is finite over C, hence complete.

A few more remarks. The complete local rings of C at its maximal ideals are regular local rings of dimension two, as X is regular. Hence by [21, (18.H)], the maps C → C_{n} are flat. Further, C → C_{n} is finite étale outside of Spec C/𝔪_{A}C. Finally, there is a canonical action of ℤ/p^{n}ℤ on ℨ_{n}, coming from its action on T_{n}.

3.7 Let D be the union of the components of X_{0} not contained in E, considered as a reduced scheme over Spec k. Let E ∩ D = {x_{1}, …, x_{r}}. For each i, the morphism D → Spec k is smooth at x_{i}. Let K_{i} be the fraction field of the complete local ring O_{i} of D at x_{i}. Note that A maps to O_{i}, for example since A maps to Γ(X_{/E}, O_{X_{/E}}). Also, A → O_{i} is local. Thus we obtain a map Spec K_{i} → U and hence we get a Galois representation

α_{i} : Gal_{K_{i}} ⟶ ℤ_{p}. Here is another way to obtain these representations. Clearly, there is a morphism

Spf O_{i} → X_{/E}. The pullback of the ideal of definition of X_{/E} to O_{i} is an ideal of definition of O_{i}. Let us form the fiber products

Spf O_{i} ×_{X_{/E}} ℨ_{n}.

These will correspond to generically étale, finite flat ring extensions O_{i} ⊂ O_{i,n}, endowed with an action of ℤ/p^{n}ℤ. The corresponding system of separable ring extensions K_{i} ⊂ K_{i,n} (with ℤ/p^{n}ℤ actions) is another avatar of α_{i}.

3.8 Lemma. Let D be a projective 1 dimensional scheme over k, let {x_{1}, …, x_{r}} be a finite set of smooth points of D → Spec k, and let K_{i} be the fraction field of O_{i} = Ô_{D,x_{i}}. Furthermore, suppose we are given maps α_{i} : Gal_{K_{i}} ⟶ ℤ_{p}. Then there exists an element

α_{D} ∈ H_{ét}^{1}(D \ {x_{1}, …, x_{r}}, ℤ_{p})

such that we recover each α_{i} from this by pulling back via Spec K_{i} → D \ {x_{1}, …, x_{r}}.

Proof. (Similar results occurs in the work of Harbater, see e.g. [13].) Let D′ be the union of those irreducible components of D which contain a point x_{i}; so {x_{1}, …, x_{n}} ⊂ D′. The map

H_{ét}^{1}(D \ {x_{1}, …, x_{r}}, ℤ_{p}) ⟶ H_{ét}^{1}(D′ \ {x_{1}, …, x_{r}}, ℤ_{p})

is surjective; we leave this to the reader as an exercise in étale cohomology. Thus it is sufficient to prove the lemma in case every component of D contains at least one x_{i}.

Let us write W = D \ {x_{1}, …, x_{r}}, and j : W → D for the inclusion morphism. By the above, we may assume that W is affine. Note that Rj_{∗}O_{W} = j_{∗}O_{W}, and that there exists an exact sequence

0 → O_{D} → j_{∗}O_{W} → ⨁K_{i}/O_{i} → 0.

Therefore we obtain the exact sequence

H^{0}(W, O) → ⨁K_{i}/O_{i} → H_{ét}^{1}(D, O_{D}) → 0.

On each of these we have the map “Frobenius − identity”. The cokernel of this map on H^{0}(W, O) is equal to H_{ét}^{1}(W, ℤ/pℤ) and on ⨁ K_{i}/O_{i} it is ⨁ Hom(Gal_{K_{i}}, ℤ/pℤ). The map on cokernels is surjective as the cokernel of the map “Frobenius − identity” on the finite dimensional k vector space H_{ét}^{1}(D, O_{D}) is zero, see [SGA7, XXII 1]. Thus the result holds with coefficients ℤ/pℤ.

Finally, note that the map H_{ét}^{1}(W, ℤ_{p}) → H_{ét}^{1}(W, ℤ/pℤ) is surjective. [As the pcohomological dimension of W is ≤ 1.] From this, and the mod p case, the result follows formally. Indeed, suppose that we already have α_{n} ∈ H_{ét}^{1}(W, ℤ_{p}) such that α_{n}|_{Gal_{K_{i}}} −α_{i} = p^{n}β_{i} for certain β_{i}. Then we can find a β ∈ H_{ét}^{1}(W, ℤ_{p}) which gives back β_{i} modulo p when we restrict it to Gal_{K_{i}}. As the next element we take α_{n+1} = α_{n} + p^{n}β. In the end we put α_{D} = lim α_{n}.

3.9 We return to the situation at the end of Subsection 3.7. Choose an element α_{D} ∈ H_{ét}^{1}(D \ {x_{1}, …, x_{r}}, ℤ_{p}) as in the lemma.

For each m ∈ ℕ we consider the scheme X_{m} = X ⊗ k[t]/(t^{m}). The formal scheme X_{/E}⊗k[t]/(t^{m}) is simply the formal completion of X_{m} in E. Let U_{m} be the open subscheme of X_{m} whose underlying open set is X_{0} \ E = D \ {x_{1}, …, x_{r}}. Thus we have (U_{m})_{red} ≅ D\{x_{1}, …, x_{r}}. By the topological invariance of étale cohomology, the class of α_{D} modulo p^{n} gives rise to finite étale coverings V_{n,m} → U_{m}.

For each x_{i} we choose an affine neighborhood Spec A_{i} of x_{i} in X, containing no other x_{j}. We may assume, by shrinking if necessary, that the ideal of E in A_{i} is generated by a single element, say f_{i} ∈ A_{i}. Then, locally around x_{i}, we can describe our spaces as follows:

X_{m}  corresponds to the ring  A_{i}/t^{m}A_{i}
U_{m}  − // −  A_{i}/t^{m}A_{i}[f_{i}^{−1}]
X_{/E}  − // −  (A_{i})_{(f_{i})}^{∧}
X_{/E} ⊗ k[t]/(t^{m})  − // −  (A_{i}/t^{m}A_{i})_{(f_{i})}^{∧}

We are going to glue the finite flat coverings ℨ_{n} ⊗ k[t]/(t^{m}) → X_{/E} ⊗ k[t]/(t^{m}) to the coverings V_{n,m} → U_{m}. For this we use the formulation of [6, 4.6], see [25], [1], [10] for proofs. Thus we need to prove that these coverings agree over the rings

R_{i,m} = (A_{i}/t^{m}A_{i})_{(f_{i})}^{∧}[f_{i}^{−1}].

Both ℨ_{n} ⊗ k[t]/(t^{m}) and V_{n,m} give rise to a finite étale ring extension of R_{i,m}. We leave it to the reader to see that R_{i,m} is an Artinian local ring with residue field K_{i}. Thus we have to see that these finite étale extensions restrict to isomorphic étale ring extensions of K_{i}. This follows from our choice of α_{D}, see end of Subsection 3.7.

By the references given above, we deduce the existence of finite flat coverings X_{n,m} → X_{m}, agreeing with ℨ_{n} ⊗ k[t]/(t^{m}) over X_{/E} ⊗ k[t]/(t^{m}) and with V_{n,m} over U_{m}.

Everything is compatible with varying m. Thus we obtain a formal scheme 𝔛_{n} and a finite flat morphism 𝔛_{n} → 𝔛; here 𝔛 denotes the formal completion of X along X_{0}. By Grothendieck’s theorem on algebraization of formal schemes [EGA, III 5.1.4] we conclude that 𝔛_{n} is the completion of a scheme Y_{n} finite flat over X. It is clear from the construction that the morphism Y_{n} → X ramifies at most along E ⊂ X.

3.10 Conclusion. For every element α as above there exists an element β ∈ H_{ét}^{1}(X_{η}, ℤ_{p}) which extends to an étale cohomology class over X \ E = X′ \ {P } (see Situation 3.5) and which pulls back to α under the morphism U → X′ \ {P }.

3.11 Let J_{η} be the Jacobian of the projective curve X_{η} smooth over κ(η); we think of J_{η} as Pic^{o}. Let J be the Néron model of J_{η} over Spec k[[t]]. Since the curve X_{η} has a semistable model over k[[t]], the group scheme J is semi-abelian over k[[t]]. See [9, Theorem 2.4 and Proposition 2.3]. Choose a section σ : Spec k[[t]] → X of the structural morphism X → Spec k[[t]]; the image of σ lies in the smooth locus Sm(X/k[[t]]) of X → Spec k[[t]]. We may and do assume that σ(Spec k) lies in X_{0} \ E. We can use the rational point σ( Spec k((t))) to obtain the morphism X_{η} → J_{η}. By the universal property of Néron models this extends to a morphism

Sm(X/k[[t]]) ⟶ J

which maps σ to the zero section.

3.12 Lemma. Let C be a smooth projective irreducible curve over a field K. Let c : Spec K → C be a rational point. Let C → J be the map of C into its Jacobian, mapping c to zero. Suppose β ∈ H_{ét}^{1}(C, ℤ_{p}) is a nonzero étale cohomology class such that c^{∗}β is zero in H_{ét}^{1}(Spec K, ℤ_{p}). In this situation, there exists a surjective homomorphism of p-divisible groups φ : J[p^{∞}] → ℚ_{p}/ℤ_{p} such that the following system of finite étale coverings {J_{n} → J} corresponds to an étale cohomology class on J that pulls back to β:

0 → J[p^{n}] → J ⟶^{p^{n}} J → 0
↓^{φ[p^{n}]} ↓ ↓
0 → ℤ/p^{n}ℤ → J_{n} ⟶ J → 0

Proof. This is geometric class field theory. See for example [34, VII §2, Proposition 9].

We apply this lemma to the situation described just before the lemma. Note that σ^{∗}β is trivial as Spec k[[t]] has trivial fundamental group and β extends to X \ E. We get φ : J_{η}[p^{∞}] → ℚ_{p}/ℤ_{p}, and the sequence of abelian varieties J_{n,η}. Let J be the Néron model of J_{η} as above, and let J_{n} be the Néron model of J_{n,η}. We remark that J_{n} is a semi-abelian scheme, as J_{n,η} is isogenous to J_{η}. Our goal is to show that the map J_{n} → J is étale.

Recall that the p-divisible group J_{η}[p^{∞}] has a natural filtration 0 ⊂ G_{η}^{f} ⊂ J_{η}[p^{∞}]. See [24, Chapter IV, Section 2] and compare [8, Section 2]. In fact, G_{η}^{f} is the generic fiber of the p-divisible group G^{f} of the Raynaud extension associated to the semi-abelian scheme J over k[[t]]. Further, the quotient J_{η}[p^{∞}]/G_{η}^{f} is a p-divisible group which extends to an étale p-divisible group over k[[t]].

By [8], the restriction of the map φ to G_{η}^{f} extends to a homomorphism of p-divisible groups ψ : G^{f} → ℚ_{p}/ℤ_{p} over Spec k[[t]].

There are two cases. Case 1: ψ is not trivial. Let s be the largest integer such that ψ is divisible by p^{s}. Set ψ′ = p^{−s}ψ. Then the restriction of ψ′ : G^{f}[p] → \underline{ℤ/pℤ} is a surjection of finite flat groups schemes over k[[t]]. In view of this, the pushout H_{n} in the diagram

0 → G^{f}[p^{n}] → G^{f} ⟶^{p^{n}} G^{f} → 0
↓^{ψ′[p^{n}]}  ↓  ↓
0 → ℤ/p^{n}ℤ → H_{n} ⟶ G^{f} → 0

is a p-divisible group over Spec k[[t]]. We remark that, for n ≥ s, there is a homomorphism H_{n−s,η} → J_{n,η}[p^{∞}]. In terms of elements this is given by class of (a, g) ↦ class of (p^{s}a, g′), where a ∈ ℤ/p^{n−s}ℤ, g ∈ G_{η}^{f} and g′ ∈ G_{η}^{f} is an element such that p^{s}g′ = g. It is clear that H_{n−s,η} → J_{n,η}[p^{∞}] is an injection; in fact it can easily be seen that H_{n,η} is the p-divisible part of the inverse image of G_{η}^{f} under the map J_{n,η}[p^{∞}] → J_{η}[p^{∞}].

On the other hand, let G_{n}^{f} denote the p-divisible group of the Raynaud extension associated to the Néron model J_{n}. Clearly, we also have that G_{n,η}^{f} is the p-divisible part of the inverse image of G_{η}^{f} in J_{n,η}[p^{∞}]. We conclude that G_{n,η}^{f} ≅ H_{n,η}, and hence by [8] we obtain G_{n}^{f} ≅ H_{n}. We conclude that the map G_{n}^{f} → G^{f} has kernel group scheme \underline{ℤ/p^{n−s}ℤ} over k[[t]]. Thus the induced map on Raynaud extensions is étale and hence the morphism of Néron models J_{n} → J is étale.

Case 2: the map ψ is trivial. In this case one sees easily that G_{n}^{f} → G^{f} is an isomorphism. Thus the same conclusion holds. (In this case we can think of the integer s as being infinite.)

Thus the morphism J_{n} → J is étale, and the group ℤ/p^{n}ℤ operates on J_{n}. This means that J_{n} → J is finite étale over its image. Let J^{∘} ⊂ J denote the connected Néron model. The image of J_{n} → J certainly contains J^{∘}. We conclude that there exists a class γ ∈ H_{ét}^{1}(J^{∘}, ℤ_{p}) which pulls back to β under the map X_{η} → J_{η}.

There exists a natural number N ∈ ℕ such that the composition

Sm(X/k[[t]]) ⟶ J ⟶^{[N]} J

has image contained in J^{∘}. The pullback of γ via this map agrees with Nβ on the generic fiber X_{η}. We conclude that the étale cohomology class α extends to the generic points of E, and hence by Lemma 3.3 we are done.

3.13 Remark. In the case that A has mixed characteristic we can argue in exactly the same fashion. Algebraizable now means that S is isomorphic to the formal completion in a point of an algebraic family of projective curves over W (k). One can perform stable reduction for these and one reduces to a situation as in Situation 3.5, with base not k[[t]], but a finite extension R ⊃ W (k). In the Subsections 3.6, 3.7 and 3.9 the reader should think of t as a uniformizer of R; then these go through with only minor modifications. Finally, in the last part of the argument, we use the result of Tate [35] on extensions of homomorphisms of p-divisible groups, in stead of that of [8].

However, there is another way to prove the result in the mixed characteristic case, namely by using a result of Miki [23]. This result says that ℤ_{p}-extensions of complete discretely valued fields of characteristic (0, p) are “elementary”. Let S → Spec W (k) be the morphism given by the canonical map W (k) → A; we assume for simplicity that W (k) is algebraically closed in A. For any finite extension of dvr W (k) ⊂ R, we let S̃_{R} denote the resolution of singularities of Spec A ⊗_{W (k)} R. The result of Miki implies that the ℤ/p^{n}ℤ-coverings that define α extend to finite étale coverings of the scheme S̃_{R} for some R = R_{n}. We leave it to the reader to see that π_{1}(S̃_{R}) ≅ π_{1}(S̃) for every R. The desired result follows.

4 Stratification by Newton polygons

Let S be a locally Noetherian scheme of characteristic p, and let E be a nondegenerate F -crystal of rank r over S. Let B denote the set of Newton polygons on [0, r] as described in [16, page 122]. By [16], we obtain a stratification {U_{β}}, indexed by polygons β ∈ B, of S characterized by the following properties: (1) The points s ∈ U_{β} are the points s ∈ S, where the Newton polygon of E is β. (2) Each stratum U_{β} is a reduced locally closed subscheme of S. Of course S = ∐ U_{β}. Warning: It is not true in general that the closure \overline{U}_{β} of a stratum is the union of irreducible components of strata U_{γ}.

Our goal is to show that this stratification has all its “jumps” in codimension 1. This can be formulated in the following way: Suppose that η is a generic point of the scheme \overline{U}_{β} \ U_{β}. Then

dim O_{\overline{U}_{β},η} = 1.

This follows immediately from the following result.

4.1 Theorem. (Purity Theorem.) Let A be a Noetherian local ring of characteristic p. Let E be an F -crystal over Spec A. Assume that the Newton polygon of E is constant over Spec A\{𝔪_{A}}. Then either dimA ≤ 1 or the Newton polygon of E is constant over Spec A.

Proof. Let us assume that dimA > 1. We will prove that E has constant Newton polygon over A.

Suppose that A → B is a local homomorphism of Noetherian local rings such that 𝔪_{B} = \sqrt{𝔪_{A}B}, and dimB ≥ 2. If we prove the result for E|_{Spec B}, then the result for E follows. Thus we may perform the following reductions We may replace A by the completion of A. If dimA > 2, then we choose some nonzero divisor x ∈ 𝔪_{A}, and we see that it suffices to prove the theorem for A/xA. Thus we reduce to the case where A is a two dimensional Noetherian complete local ring. We can find a homomorphism A → B as above, where B is a Noetherian complete local normal domain of dimension 2 with algebraically closed residue field. (E.g., if A ≅ k[[x_{i}]]/(f_{j}), then take B to be the normalization of a quotient of k̄[[x_{i}]]/(f_{j}).)

Thus we may assume that A is as in Section 3. We write U = Spec A \ {0}. We remark that U is a 1 dimensional regular scheme.

Let K be the field of fractions of A. Let

0 ⊂ E_{1} ⊂ E_{2} ⊂ … ⊂ E_{s} ⟶ E|_{Spec K}

be the slope filtration as in Claim 2.8. Let r_{i} denote the rank of E_{i}. Then the slope filtration of the F -crystal ⋀^{r_{i}} E over Spec K starts with ⋀^{r_{i}} E_{i} ⊂ ⋀^{r_{i}} E|_{Spec K}. Of course ⋀^{r_{i}} E_{i} has rank one, and an integral slope, say n_{i} ∈ ℤ_{≥0}. We remark that the Newton polygon of E|_{U} is determined by the point (r_{i}, n_{i}); these are the break points of the Newton polygon in question. Consider the Galois representation

ρ_{i} : Gal_{K} ⟶ GL_{1}(ℚ_{p})

associated to ⋀^{r_{i}} E_{i} over Spec K. See Subsection 2.9. By assumption the Newton slopes of ⋀^{r_{i}} E are constant over U, and according to Proposition 2.10 this implies that ρ_{i} is unramified over U. Thus we can write ρ_{i} as a continuous representation

ρ_{i} : π_{1}(U) ⟶ GL_{1}(ℚ_{p}) ≅ ℤ_{p} × 𝔽_{p}^{∗} × ℤ × (ℤ/2ℤ if p = 2).

We may replace A by the integral closure of A in a finite extension of K. Thus we may assume that the image of ρ_{i} lies in the factor ℤ_{p} for all i.

At this point we apply the main result of Section 3. Thus we see that on a resolution of singularities S̃ → Spec A, the representations ρ_{i}, extend to representations ρ_{i} : π_{1}(S̃) → GL_{1}(ℚ_{p}). By the equivalence of categories 2.5, this means that ⋀^{r_{i}} E_{i} extends to a rank 1 F -crystal L_{i} of slope n_{i} over S̃. Let us write φ_{i} : (L_{i})_{Spec K} → (⋀^{r_{i}} E)_{Spec K} for the nontrivial map we have by the above.

Let η denote the generic point of a component of the exceptional fiber of S̃ → Spec A. (We remark that if A happens to be regular, then we let S̃ be the blowing up of the special point of Spec A.) Let O = O_{_{S},η}^{∧} be the completed local ring of S̃ at η. Since κ(η) is the function field of a curve over k, it has a p-basis consisting of 1 element. Therefore O has a p-basis, see [6, Lemma 1.1.3] for example. Thus we may apply [8, Main Theorem] to the two nondegenerate F -crystals ⋀^{r_{i}} E|_{Spec O} and L_{i}|_{Spec O} and the map φ_{i} (which lives over K and by pullback over the fraction field of O). We obtain a nontrivial map L_{i}|_{Spec O} → ⋀^{r_{i}} E|_{Spec O}. Restricting to η, we see that (⋀^{r_{i}} E)|_{η} contains a 1 dimensional subcrystal of slope n_{i}. Thus the Newton polygon of E|_{η} goes through or below the point (r_{i}, n_{i}), see [16, page 122]. On the other hand, by Grothendieck’s specialization theorem [16, 2.3.1] it lies above the unique polygon P with break points {(r_{i}, n_{i})}, since P is the Newton polygon of E over U. Thus the Newton polygon of E|_{η} equals P. Since η maps to 0, we derive that E|_{0} has Newton polygon P as desired.

4.2 Remark. Let G be the universal deformation of a p-divisible group G_{0} over R = k[[t_{1}, …, t_{dd^{∗}}]]. We obtain a stratification S = Spec A = ⋃ U_{β} as above. Are the closed subschemes \overline{U}_{β} set theoretic complete intersections in S?

5 Irreducibility of a catalogue of Barsotti-Tate groups

Let P be a moduli problem, for example moduli of elliptic curves or moduli of pdivisible groups, or moduli of finite group schemes. A catalogue for P is a family X → T in P such that every object of P over an algebraically closed field k appears as the fibre of the family X over a point Spec k → T. Of course such a catalogue, if it exists, is not unique. If the moduli problem admits a fine moduli scheme (i.e. if it is representable), the pull back by any surjective morphism gives a catalogue. For elliptic curves over ℚ one can take either the universal Weierstrass elliptic curve, or the family defined by Y^{2} = X^{3} + AX + B over ℚ[A, B, 1/(4A^{3} + 27B^{2})]. The advantage of studying catalogues is that they often exist where moduli spaces do not, for example in the case of p-divisible groups. In addition we can ask intelligible questions about them, e.g.: Does there exist an irreducible catalogue for P?

In this section we show that all p-divisible groups in characteristic p isogenous with a given iso-simple p-divisible group fit into an irreducible catalogue T. From this we derive the corollaries (4a) and (4b) mentioned in the introduction.

Let G_{m,n} be the iso-simple p-divisible group (it is a formal group of dimension m and with dual of dimension n). Our choice of T is described in Subsection 5.9. Let X denote some p-divisible group isogenous to G_{m,n}. For the proof of the irreducibility we show on the one hand that the Purity Theorem 4.1 implies that any component of the Newton polygon stratum in the local deformation space of X has dimension at least r := (m − 1)(n − 1)/2, see 5.17. By the Isogeny Theorem 2.17 the same holds for every component of the catalogue, see 5.20. On the other hand we can construct a stratification of T by types A of X, see 5.27. The strata with A ≠ ⟨0⟩ have dimension < r and the stratum with A = ⟨0⟩ is irreducible. Hence T is irreducible.

The types involved are semi-modules; we need some elementary combinatorial facts to describe them. Description of the strata is elementary. However, the fact that every stratum is in the Zariski-closure of the stratum with A = ⟨0⟩ seems a non-trivial fact. (And we have used the results of the previous sections to prove this.)

Acknowledgements. The combinatorial facts needed are recorded in the appendix of this section. We thank Don Zagier and Daan Krammer for helpful discussions on these combinatorial facts; especially for the suggestion of introducing Young diagrams in this situation.

5.1 We fix in this section positive integers m, n ∈ ℤ_{>0} which are coprime. We write

r := (m − 1)(n − 1)/2.

The symbol δ will denote the Newton polygon consisting of m+n times the slope n/(m+n).

5.2 The p-divisible group G_{m,n} over Spec 𝔽_{p}. This is introduced in [20, page 35]. We describe G_{m,n} in terms of its covariant Dieudonné module (in [20] contravariant Dieudonné module theory is used). This module can be given as

M_{m,n} = ℤ_{p}[F, V]/ℤ_{p}[F, V](F^{m} − V^{n}),

where ℤ_{p}[F, V] is the usual ring. We recall that G_{m,n} is a formal group of dimension m and has dual (G_{m,n})^{t} = G_{n,m} of dimension n. It has height h = m + n.

5.3 The p-divisible group H = H_{m,n} over Spec 𝔽_{p}. In the following we will use more often the p-divisible group H which is isogenous to G_{m,n}. Again we describe H by its covariant Dieudonné module M(H). This is a free ℤ_{p}-module with basis e_{0}, e_{1}, …, e_{m+n−1}. We will write e_{j} = p^{a}e_{i}, for j ∈ ℕ such that j = i + a(m + n). The actions of F and V are given by F (e_{i}) = e_{i+n}, and V (e_{i}) = e_{i+m}. Note that M(H) (and hence H) has an endomorphism π given by π(e_{i}) = e_{i+1}. Clearly, the isogeny π : H → H has a kernel of order p, which is isomorphic to α_{p}.

The reader checks immediately that this Dieudonné module is isoclinic of slope n/(m+ n). Since (F^{m} − V^{n})e_{0} = 0, the Dieudonné module generated by e_{0} is isomorphic with M(G_{m,n}), and we have an inclusion M(G_{m,n}) → M(H). Thus H is isogenous to G_{m,n} over 𝔽_{p}. For coprime integers m and n we choose a, b ∈ ℤ such that am + bn = 1. In the ring End(M(H_{𝔽_{p}})) we have the following relations: F = π^{m}, and V = π^{n}, hence π^{m+n} = p. We could have defined π by π = F^{b}V^{a} in End(M(H_{𝔽_{p}})).

5.4 Lemma. For every perfect field K containing 𝔽_{p^{m+n}} we have

End(H_{K}) = W (𝔽_{p^{m+n}})[π].

The ring End(H_{K}) is a (noncommutative) discrete valuation ring with uniformizer π and valuation given by log_{p} deg(−). Here λ ∈ W (𝔽_{p^{m+n}}) acts on e_{0} via multiplication by λ. For λ ∈ W (𝔽_{p^{m+n}}) we have π · λ = σ^{b−a}(λ) · π. Consider the filtration N^{•} of M(H_{k}) = M(H) ⊗ W (k) given by

M(H_{k}) = N^{0} ⊃ … ⊃ N^{j} ⊃ …

with

N^{j} = ⟨e_{j}, e_{j+1}, …⟩ = Σ_{t≥j}W ·e_{t} = π^{j}(M(H_{K})).

For every element τ of End(H_{k}) and every j we have τ(N^{j}) = N^{j+v}, where v = v(τ) is the valuation of τ.

The proof of this lemma is left to the reader. Note that for K ⊃ 𝔽_{p^{m+n}} the algebra End(H_{K}) ⊗_{W (K)} W (K)[1/p] is a central simple algebra over ℚ_{p} of rank (m + n)^{2}. Its invariant can be computed as follows. Cearly the field of fractions of W (𝔽_{p^{m+n}}) is cyclic Galois with Galois group generated by σ, the automorphism of W (𝔽_{p^{m+n}}) lifting (x ↦ x^{p}) ∈ Aut(𝔽_{p^{m+n}}). Choose an integer s ∈ ℤ such that s(a−b)−1 is divisible by m+n. In the notation of [31, pp. 252, 277] this algebra can be given as (E, σ^{a−b}, π) = (E, σ^{(a−b)s}, π^{s}); hence by [31, 17.10, Theorem on page 338] we see that its invariant equals s/(m + n).

5.5 Lemma. Let k be an algebraically closed field of characteristic p. Let φ : H_{k} → X and ψ : H_{k} → X be isogenies of p-divisible groups. Then either φ = ψ ∘ τ or ψ = φ ∘ τ for some τ ∈ End(H_{k}). If deg(φ) = deg(ψ), then τ is an automorphism of H_{k}. A similar result holds for isogenies X → H_{k}.

Proof. Let β : X → H_{k} be any isogeny. Then both β ∘ φ and β ∘ ψ are endomorphisms of H_{k}. Since End(H_{k}) is a discrete valuation ring, either τ = (β ∘ ψ)^{−1} ∘ (β ∘ φ) ∈ End(H_{k}) or τ = (β ∘ φ)^{−1} ∘ (β ∘ ψ) ∈ End(H_{k}). The result follows.

5.6 Canonical filtrations and the semi-module attached to a formal group. This can be found in [20, page 47].

Let k be an algebraically closed field of charceteristic p. Let X be a formal group over k isogenous to G_{m,n}. Then there exists an isogeny ψ : X → H_{k}. This induces an inclusion of Dieudonné modules M(X) ⊂ M(H_{k}). The filtration N^{•} on M(H_{k}) (see Lemma 5.4) induces a filtration on M(X). This filtration has jumps, and we obtain a set A_{ψ} ⊂ ℤ such that

a ∈ A_{ψ} ⟺ M(X) ∩ N^{a} ≠ M(X) ∩ N^{a+1}.

If ψ′ : X → H_{k} is isogeny, then either ψ′ = τ ∘ ψ or ψ = τ ∘ ψ′ for some τ ∈ End(H_{k}), see Lemma 5.5. Thus, by Lemma 5.4, we see that the filtration of M(X) is well determined up to a shift in numbering, and that A_{ψ} ⊂ ℤ is well determined up to translation.

Note that F : M(H_{k}) → M(H_{k}) and V : M(H_{k}) → M(H_{k})) are strict for the filtration N^{•}, but with a shift of n, resp. m. See Subsection 5.3. This implies that A_{ψ} is a semi-module (see Subsection 5.29), i.e., a ∈ A_{ψ} ⇒ a + n, a + m ∈ A_{ψ}.

We conclude that the semi-module A_{ψ} obtained is unique up to equivalence (translation inside ℤ) once X is given. By Lemma 5.34 we see that there is a unique admissible semi-module A attached to this filtration. This semi-module will be called the type of X,

A = Type(X).

5.7 Examples. If X = G_{m,n} we obtain the “maximal” admissible semi-module. This is the semi-module generated by 0, i.e., A = ⟨0⟩ := ℤ_{≥0}·m + ℤ_{≥0}·n ⊂ ℤ.

If X = H_{m,n} we obtain the “minimal” admissible semi-module A = [r, ∞).

5.8 We continue the discussion of the previous subsection. The fact that A is admissible means that after renumbering the filtration N^{•} on M(H_{k}) we may assume that

N^{0} ⊃ M(X) ⊃ N^{2r},

and that both inclusions have colength r over W (k). Note that any N^{j} is isomorphic (as a Dieudonné module) to the Dieudonné module of H_{k}. Therefore we conclude that, for any X as above, there exist isogenies H_{k} → X and X → H_{k} of degree p^{r}.

5.9 In this subsection we introduce the catalogue that we study in the rest of this section. It is the scheme representing the following contravariant functor:

\begin{matrix}Sch/ Spec 𝔽_{p} & ⟶ & Sets \\ S & ⟼ & \{(φ, X) | φ : H_{S} → X, deg(φ) = p^{r}\}/ ∼ . \end{matrix}

Thus this functor associates to S the set of quotients of H_{S} by finite locally free closed subgroup schemes of rank p^{r}. (Recall r = (n − 1)(m − 1)/2.) This functor is representable by a scheme T = T_{m,n} → Spec 𝔽_{p} projective over Spec 𝔽_{p}. For a proof, see [17, Lemma (2.8)].

The universal object over T will be denoted (G, Φ), so G is a p-divisible group over T and Φ : H_{T} → G is an isogeny of degree p^{r}. Note that T depends on the choice of the coprime positive integers m and n; having fixed this choice, they will be omitted form notations (otherwise we should have to write T_{m,n}, etc.).

5.10 Proposition. The family G → T of p-divisible groups over T is a catalogue for p-divisible groups isogenous to G_{m,n}: if X is a p-divisible group over an algebraically closed field k of characteristic p, and X is isogenous to G_{m,n}, then there exists a point t : Spec k → T such that X ≅ G_{t}.

Proof. This follows from Subsection 5.8.

The following is the main result of this section.

5.11 Theorem. The scheme T is geometrically irreducible of dimension r over 𝔽_{p}.

To state a consequence of the theorem, we recall the following notation. If K is a field of characteristic p, and X is a group scheme over k, then the a-number of X is a(X) := dim_{K}Hom(α_{p}, X).

5.12 Corollary. Let X be a p-divisible group over an algebraically closed field k isogenous to G_{m,n}. Then there exist a deformation X /k[[t]] of X over k[[t]] such that the Newton polygon is constant, and such that a(X_{η}) = 1.

To deduce the corollary from the theorem, consider T_{k}. There is a point t ∈ T (k) such that G_{t} ≅ X by Proposition 5.10. To find a point of T (k) with a = 1, one simply finds any p-divisible group isogenous to H with a-number 1 (this exists, as T_{k} is a catalogue). For example G_{m,n} ⊗ k will do. Since T_{k} is irreducible, we conclude that the (a = 1)- locus of the family G is an open dense subscheme U of T_{k}. Thus we can find a morphism φ : Spec k[[t]] → T_{k} over k such that φ(η) ∈ U and φ(Spec k) = t. The pullback X = φ^{∗}G gives the desired deformation.

The following was conjectured by Grothendieck, see [12, page 150].

5.13 Corollary. Let X be a p-divisible group over an algebraically closed field k isogenous to G_{m,n}. Let γ be a Newton polygon with end points (0, 0) and (m + n, n). Then there exists a deformation X /k[[t]] of X over k[[t]] such that the Newton polygon of X_{η} is γ.

Proof. Let Y/k[[t]] be the deformation of X that we produced in the previous corollary. According to [6, 3.2.2, 3.2.3 and 3.2.4] we can find a deformation Ỹ of Y over k[[t, t_{1}, …, t_{mn}]] which is versal in the directions given by t_{1}, …, t_{mn}. Thus the restriction of this family of p-divisible groups to \overline{k((t))}[[t_{1}, …, t_{mn}]] is the universal deformation of a p-divisible group with a = 1. According to [30, Corollary 3.8], the Newton polygon γ occurs in this deformation. This proves that the locally closed stratum U_{γ} ⊂ Spec k[[t, t_{1}, …, t_{mn}]] (see Section 4) is not empty. Finally, one finds a suitable map Spec k[[t′]] → Spec k[[t, t_{1}, …, t_{mn}]] mapping the generic point into U_{γ}.

5.14 Before we start the proof of the theorem we make some remarks on deformations of p-divisible groups. Let k be an algebraically closed field of characteristic p. Let X be a p-divisible group over k, of dimension d and with Serre dual X^{t} of dimension c. As is well known, the deformation problem posed by X over k is pro-representable by a formally smooth complete local k-algebra R ≅ k[[t_{1}, …, t_{dc}]], see [14] for example. Any p-divisible group over Spf R comes from a p-divisible group over Spec R, see [6, Lemma 2.4.4] for example. Let X over Spec R be the universal deformation of X. We need the following well known result for which we were unable to find a reference.

5.15 Lemma. The generic fibre of X has Newton polygon equal to the Newton polygon of (𝔾_{m})^{d} × (ℚ_{p}/ℤ_{p})^{c} (i.e., d times slope 0 and c times slope 1 on the covariant Dieudonné module).

Proof. By Grothendieck’s specialization theorem we need only find some point of Spec R where X has the desired Newton polygon. Write X = X^{μ} ⊕ X′ ⊕ X^{ét} as the direct sum of its multiplicative part, its local-local part and its étale part. Deforming X′ while keeping fixed X^{ét} and X^{μ} constant, and keeping the direct sum decompositon we obtain a closed subscheme of Spec R. Thus, if we prove the lemma for X′, then the lemma follows for X.

Hence we may assume that X is local-local. We will use the display of a deformation of X. See [27], [28] and [36]. According to these references, the display of X is given by a matrix

(∗)  ( \begin{matrix}A & B \\ C & D \end{matrix} ) ,

with A of size d × d over W = W (k) and B of size d × c, etc. The universal deformation over k[[t_{i,j}]] is given by the matrix

( \begin{matrix}A + T C & B + T C \\ C & D \end{matrix} ) ,

where T = ([t_{i,j}]). See [28, Section 1] and [36, (2.34)]. Let A_{0} and C_{0} denote the reductions of A and C modulo p. The Hasse-Witt matrix of this deformation is the matrix A_{0} + (t_{i,j})C_{0}, seen as a matrix over k[[t_{i,j}]]. We want to show that the determinant of this matrix is nonzero in k[[t_{i,j}]].

If not, then this determinant is zero in k[t_{i,j}] as well. This means that every matrix of the form A_{0} + T C_{0} over k has determinant zero. Clearly this can only be the case when Ker(A_{0}) ∩ Ker(C_{0}) is not zero. However, this contradicts the axiom of displays that the matrix (*) is invertible over W.

5.16 Assume k, X and other notations are as in Subsection 5.14, but now let X be isogenous to G_{m,n} ⊗ k. The Newton polygon of the covariant Dieudonné module of X is δ, see 5.1. According to Grothendieck’s specialization theorem (see [16, Theorem 2.3.1]), the locus of points of Spec R where the Newton polygon of X lies above δ is a closed subset W_{δ} ⊂ Spec R. Remark that, since the Newton polygon in the unique closed point of Spec R is δ, the Newton polygon of X |_{W_{δ}} is constant.

5.17 Proposition. With the notations above. The dimension of any irreducible component of W_{δ} is at least r

Proof. The fibre of X at the generic point of Spec R gives a p-divisible group with Newton polygon equal to ρ := m × (slope 0) + n × (slope 1). See Lemma 5.15. Consider Newton polygons of height m + n and dimension m, i.e., those which start at the point (0, 0) ∈ ℝ^{2} and end at the point (m + n, n). The number of points with integral coordinates in ℝ^{2} strictly below δ and on or above ρ equals mn − r, as is easy to see. Hence any chain of comparable, mutually different Newton polygons starting at δ and ending at ρ has length at most mn − r. Using the Purity Theorem 4.1, the result of the proposition follows.

5.18 Suppose G → T and Φ : H_{T} → G are the catalogue as in Subsection 5.9. Let t : Spec k → T be a geometric point, and let (X, φ) be the fibre at this point: X = G_{t} and φ = Φ_{t} : H_{k} → X. We denote O^{∧} the complete local ring of T_{k} at t. The pullback of G to O^{∧} is a deformation of X to Spec O^{∧}. By the universal property of the deformation ring R (see 5.14), we obtain a k-algebra homomorphism R → O^{∧}. Of course, the (reduced) image of the associated morphism Spec O^{∧} → Spec R is contained in W_{δ}. (We do not know whether T and hence O^{∧} is reduced.)

5.19 Proposition. The morphism Spec O^{∧} → Spec R is a closed immersion. The (reduced) image of this morphism equals W_{δ}.

Proof. To prove that it is a closed immersion we need only show that it is unramified at the special point. This is clear: if not then we would obtain a nontrivial deformation φ_{A} : H_{A} → X_{A} of the initial map φ : H_{k} → X over an Artinian k-algebra A. By rigidity of homomorphisms of p-divisible groups this is impossible.

Let S be an irreducible component of W_{δ}. Note that S = Spec A′, and that A′ is an excellent domain. Thus the normalization A of A′ is another complete local Noetherian k-algebra with residue field k. Let Y be the restriction of the p-divisible group X over Spec R to Spec A. We may apply the Isogeny Theorem 2.17 to Y over Spec A: we obtain an isogeny ψ′_{A} : H_{A} → Y. We may assume that the degree p^{t} = deg(ψ′_{A}) is ≥ p^{r}.

Let K be the field of fractions of A, and let K̄ be an algebraic closure of K. According to Subsection 5.8, we may factor ψ′ ; K̄ as ψK̄∘π^{t−r}, for some ψK̄ : HK̄ → X of degree p^{r}. This means that Ker(π^{t−r})K̄ ⊂ Ker(ψ′ ; K̄). We conclude that Ker(π^{t−r}) × Spec A is contained in Ker(ψ′_{A}), as A → K̄ is injective. In other words, there exists an isogeny ψ′_{A} : H_{A} → Y of degree p^{r}, such that ψ_{A} ∘ π^{t−r} = ψ′_{A}.

Let ψ = ψ_{A} ⊗ k be the special fibre of ψ_{A}. It may not be true that ψ = φ (see 5.18 for meaning of φ). However, according to Lemma 5.5, there exists an automorphism τ of H_{k} such that φ = ψ ∘ τ. Thus the pair (Y, ψ_{A} ∘ τ) is a deformation of (X, φ) over A. Hence, by the defining property of the moduli scheme T, we get a morphism Spec A → T, whose special point is t. By definition of O^{∧}, this induces a map O^{∧} → A. We leave it to the reader to see that the diagram

Spec A  ⟶ Spec O^{∧} ↪ Spec R
↓  ||
S = Spec A′ ⟶  W_{δ}  ⊂ Spec R

is commutative. Since S was any irreducible component of W_{δ}, the last statement of the proposition follows.

5.20 Corollary. Every component of T has dimension at least r.

Proof. This follows trivially from the previous two propositions.

5.21 Let K be a perfect field of characteristic p. Suppose φ : H_{K} → X defines a point of T over K. We are going to find standard elements in M(X). Note that since K is perfect, we can use (covariant) Dieudonné module theory over K; the reader will have to check in what follows that some of the results obtained above for algebraically closed fields K remain true for perfect fields.

First, we recall from 5.8 that there exists an isogeny β : X → H_{K} of degree p^{r}. After changing β with an automorphism of H_{K}, we may assume that β ∘ φ = π^{2r}, compare 5.5. Note that β is uniquely determined by this condition. Thus we obtain a canonical injection M(β) : M(X) ↪ M(H_{K}) determined by (X, φ). From now on we think of M(X) as a submodule of M(H_{K}).

Any element z of M(H_{K}) can be written uniquely in the form

z = [c_{0}]e_{0} + [c_{1}]e_{1} + [c_{2}]e_{2} + …

with c_{j} ∈ K. Here [c] = (c, 0, 0, 0, …) is the Teichmüller lift of c ∈ K. (Recall that e_{i} are defined for every i ≥ 0.) Suppose we have a second element z′ ∈ M(H_{K}) which has the form

z′ = [c_{t}]e_{t} + ∑_{j>t}[c′_{j}]e_{j}.

Then the difference z − z′ has the expression

z − z′ = ∑_{0≥j<t}[c_{j}]e_{j} + ∑_{j>t}[p_{j}]e_{j},

where the p_{j} ∈ K have the following shape: there exist integers n(j) and certain universal polynomial expressions P_{j} in the elements c_{i} and c′_{i} with t < i ≤ j such that

(p_{j})^{p^{n(j)}} = P_{j}(c_{t+1}, c′_{t+1}, …, c_{j}, c′_{j}).

We will call such an expression p_{j} a quasi-polynomial expression in c_{t+1}, c′_{t+1}, …, c_{j}, c′_{j}. This statement follows from the following two assertions: (1) the addition of Witt vectors is given by certain universal polynomial expressions, and (2) when we write a Witt vector c̄ = (c_{0}, c_{1}, c_{2}, …) in the form c̄ = [c_{0}]+p(γ_{0}, γ_{1}, γ_{2}, …) the elements (γ_{i})^{p} are certain universal polynomial combinations of c_{0}, …, c_{i}. In addition, one easily sees that polynomial P_{j} has the following shape:

P_{j} = (c_{j})^{p^{n(j)}} − (c′_{j})^{p^{n(j)}} + Q(c_{t+1}, c′_{t+1}, …, c_{j−1}, c′_{j−1}).

Let A be the type of X, defined as in Subsection 5.6. We say that x ∈ M(X) is a standard element of M(X) if for some a ∈ A

(∗)  x = e_{a} + ∑_{j>a}[c_{j}]e_{j},

with c_{j} ∈ K, and c_{j} = 0 if j ∈ A.

5.22 Lemma. See [20, Lemma 3.9]. Let X be as above. For every a ∈ A, there exists a standard element x_{a} ∈ M(X) as in (*), and it is unique.

Proof. By definition of A there exists an element y ∈ M ∩ N^{a}, which is non-zero modulo M ∩ N^{a+1}. Scaling gives y the form y = e_{a} + ∑_{j>a}[c_{j}]e_{j}, with c_{j} ∈ K.

Suppose that for some t, a < t we have already that c_{a′} = 0 for a < a′ < t with a′ ∈ A. Consider t and c_{t}. If t ∉ A, then we may replace t by t + 1. If t ∈ A, then we find an element y′ ∈ M(X) such that y′ ∈ M(X) ∩ N^{t}, but y′ ∉ M(X) ∩ N^{t+1}. After subtracting a suitable multiple of y′ from y, we obtain a new y as above with in addition c_{t} = 0. We can continue like this until t ≥ 2r. Then since N^{2r} ⊂ M(X), we can subtract the tail end ∑_{j≥2r}[c_{j}]e_{j} from y to get the desired result.

5.23 Let B be the cycle associated to the type A of X. We freely use the definitions introduced in the appendix, in particular Subsection 5.35. We are going to consider the standard element corresponding to t(i) ∈ A; we write them as

x_{t(i)} = e_{t(i)} + ∑_{j>t(i)}[X_{t(i),j}]e_{j}.

Here i ∈ {1, …, a}. Similarly, we also consider the standard elements

x_{c(i)} = e_{c(i)} + ∑_{j>t(i)}[Y_{c(i),j}]e_{j}.

Since ⊤(B) ∩ ⊥(B) = ∅, no confusion can result from these formulae. These formulae define elements X_{t(i),j} and Y_{c(i),j} of K. Note that since the elements above are standard only the pairs (t(i), j) ∈ P(A) occur, and similarly only (c(i), j) belonging to P(A) occur.

5.24 Lemma. Given the elements X_{t(i),j} and Y_{c(i),j} of K we can reconstruct the submodule M(X) ⊂ M(H_{K}).

Proof. This follows, since the elements x_{t(i)} generate the Dieudonné module M(X). This is proved in [20, Lemma 3.9 part b].

In particular, the isogeny H_{K} → X can be defined over the perfect closure in K of the subfield of K generated by the elements X_{t(i),j} and Y_{c(i),j}. Hence the morphism Spec K → T can be defined over this perfect closure. We will assume this is the case in the following discussions.

The strategy of the proof of Theorem 5.11 is as follows. We will show that, given A, there are certain algebraic relations between the elements X_{t(i),j} and Y_{c(i),j} of K. In fact, we will show that the maximal number of algebraically independent elements in the set Γ := {X_{t(i),j}, Y_{c(i),j}} is at most the volume of the Young diagram Y(A) associated to A. Thus the morphism Spec K → T does not map to a generic point of T, unless A = ⟨0⟩, by Proposition 5.20. Finally, we will show that T has only one generic point by considering the case A = ⟨0⟩ in more detail.

5.25 We produce relations among the elements X_{t(i),j} and Y_{c(i),j} as follows. This will be an increasing induction with respect to a variable t ∈ ℤ_{>0}.

Consider the subset Γ^{t} ⊂ Γ defined by

Γ^{t} = {X_{t(i),j} | j − t(i) < t} ∪ {Y_{c(i),j} | j − c(i) < t}.

Obviously we have ∅ = Γ^{1} ⊂ Γ^{2} ⊂ … ⊂ Γ^{2r} = Γ, as only pairs in P(A) occur and A ⊃ [2r, ∞).

Pick an equivalence class C ⊂ P(A), and let i_{0}, i_{1} be the integers associated to C by Lemma 5.38. Let t denote the integer t = j − a which is fixed for elements (a, j) ∈ C.

5.26 Lemma. The elements X_{t(i),t(i)+t}, i_{0} ≤ i ≤ i_{1} and Y_{c(i),c(i)+t}, i_{0} < i ≤ i_{1} generate a field extension of transcendence degree at most one over the subfield of K generated by Γ^{t}.

Let us assume the lemma for the moment and finish the proof of the Theorem 5.11. We bound the transcendence degree of the field 𝔽_{p}(Γ) ⊂ K as follows. By induction on t we show that the transcendence degree of 𝔽_{p}(Γ^{t}) is at most equal to the number of equivalence classes C ⊂ P(A) such that (a, j) ∈ C implies j − a < t. The lemma above, together with Lemma 5.38, furnishes the induction step. Hence the transcendence degree of 𝔽_{p}(Γ) is at most the number of equivalence classes of P(A). By Lemma 5.40 this is the number of elements of V(A); by definition of the Young diagram of A (see 5.41) this equals the volume of Y(A) and by Proposition 5.43, this is less than r, unless A = ⟨0⟩. We conclude that if t : Spec K → T maps to a generic point, then A = ⟨0⟩.

The case A = ⟨0⟩. Here we note that there are only X_{t(i),j} and no Y_{c(i),j}. Furthermore, every class C ⊂ P(A) has one element in this case, hence the number of X_{t(i),j} equals r, see Example 5.42. Thus, if Spec K → T maps to a generic point of T, then the elements X_{t(i),j} are algebraically independent. We may (see above) also assume that K is the perfect closure of 𝔽_{p}(X_{t(i),j}). Clearly, there is only one such K (up to isomorphism), and only one M(X) ⊂ M(H_{K}) with the correct parameters X_{t(i),j}. This means that T has a unique generic point, hence T is irreducible. This finishes the proof of Theorem 5.11.

Proof of Lemma 5.26. We first introduce specific elements x_{a} for all a ∈ A. When a ∈ ⊤(B), or a ∈ ⊥(B), we already have defined x_{a} above. If b ∈ B and a ∉ ⊥(B) ∪ ⊤(B), then we can write b = t(i)+s·m for some i and s, 0 < s < n_{i}, or we can write b = t(i)+s·n for some i and s, 0 < s < m_{i}, see Subsection 5.35. In the first case we put x_{b} = V^{s}(x_{t(i)}); in the second case we put x_{b} = F^{s}(x_{t(i)}). If a ∈ A and a ∉ B, then we can write a = b+s·h for a unique b ∈ B and s > 0. In this case we put x_{a} = p^{s}x_{b} (since pe_{j} = e_{j+h} this just means shifting the coefficients). The elements x_{a} so defined are in general not standard elements. However, they all have the form

x_{a} = e_{a} + ∑_{j>a}[Z_{a,j}]e_{j},

where each Z_{a,j} is either 0 or a p-power (negative or positive) of some X_{t(i),j′}, or some Y_{c(i),j′} with the property that t(i) − j′ = j − a or c(i) − j′ = a − j. In particular, Z_{a,j} is a quasi-polynomial expression in the elements of Γ^{j−a+1}. (See Subsection 5.21 for an explanation of the term “quasi-polynomial”.)

We show how to produce an algebraic relation between X_{t(i),t(i)+r} and Y_{c(i),c(i)+r} over 𝔽_{p}(Γ^{r}). We will leave it to the reader to establish the analogous relation between X_{t(i),t(i)+r} and Y_{c(i+1),c(i+1)+r}. Recall from 5.35, that c(i) = t(i)+n_{i} ·m. Thus if we apply V^{n_{i}} to x_{t(i)} we obtain

V^{n_{i}}(x_{t(i)}) = e_{c(i)} + ∑_{j>t(i)}[X_{t(i),j}^{p^{−n_{i}}}]e_{j+n_{i}·m}.

If this happens to be a standard element, then by uniqueness of standard elements we deduce that this equals x_{c(i)} and hence

Y_{c(i),c(i)+r} = X_{t(i),t(i)+r}^{p^{n_{i}}}.

In general the element V^{n_{i}}(x_{t(i)}) need not be standard. As explained in the proof of Lemma 5.22 we can modify the expression until we obtain a standard element. We claim that during this process the coefficient X_{t(i),t(i)+t}^{p^{n_{i}}} will only be modified by adding onto it a quasi-polynomial expressions in the variables in Γ^{t}. This will prove that

Y_{c(i),c(i)+t} = X_{t(i),t(i)+t}^{p^{n_{i}}} + a quasi-polynomial expression in Γ^{t},

which is a precise version of what we wanted to prove.

Let us consider any element

z = e_{a} + ∑_{j>a}[W_{j}]e_{j}

of M(X) such that W_{j} ∈ 𝔽_{p}(Γ^{j−a}). We want to bring it into standard form. Thus, let a′ ∈ A, a < a′ be the smallest element such that W_{a′} ≠ 0. We form the difference

z′ = z − [W_{a′}]x_{a′} = (e_{a} + ∑_{j>a}[W_{j}]e_{j}) − ([W_{a′}]e_{a′} + ∑_{j′>a′}[W_{a′}Z_{a′,j′}]e_{j′} ) .

Next, we write z′ in the form e_{a} + ∑_{j>a}[W′_{j}]e_{j}. The reader easily checks, using the discussion in 5.21, that (1) W′_{j} = W_{j} for j < a′, (2) W′_{a′} = 0, and (3) for j > a′, W′_{j} equals W_{j}+ a quasi-polynomial expression in W_{a′} and Z_{a′,j′}, with a′ < j′ ≤ j. Note that in case (3) the elements W_{a′} and Z_{a′,j′} have “lower level” then W_{j} (more precisely, they lie in the perfect closure of 𝔽_{p}(Γ^{j−a})). By repeatedly applying this procedure, starting with V^{n_{i}}(x_{t(i)}) we arrive at the claim that was stated in the middle of the preceding paragraph. This ends the proof of Lemma 5.26.

5.27 Remark. Possibly the proof of the theorem becomes more clear by the following explanation. For every admissible semi-module A we define the set U_{A} ⊂ T consisting of all points corresponding with a p-divisible group, isogenous with H, with semi-module equal to A. We can show that U_{A} ⊂ T is locally closed in T, irreducible and of dimension equal to the volume of Y(A). In this way we have a stratification T = ⊔ U_{A}, which is implicitly used above. The group H admits a unique H → H of degree p^{r}; this gives the stratum defined by A = [r, ∞). It is the unique zero-dimensional stratum. It is easy to see that every stratum contains this in its closure. One of the non-elementary facts seems to be to determine in which way these strata fit together. The central idea of this section is the fact that the largest straum contains all others in its closure, plus a proof of this via the Purity Theorem.

5.28 Appendix, combinatorial facts on semi-modules

In this appendix give the combinatorial definitions and facts used above. Some of these are already to be found in [20, pp. 45–46]. As above, we fix coprime positive integers m and n; we write h := n + m and r := (m − 1)(n − 1)/2.

For integers a < b we write [a, b] := {a, a + 1, ⋯, b}. We write [a, ∞) = ℤ_{≥a}. For subsets V_{1}, V_{2} ⊂ V we write V_{1} \ V_{2} for the set of elements in V_{1} not in V_{2}. For a subset A ⊂ ℤ and t ∈ ℤ we write A + t = {a + t | a ∈ A}.

5.29 A semi-module is a subset A ⊂ ℤ bounded from below such that x ∈ A ⇒ x+m, x+ n ∈ A. We say that semi-modules A, A′ are equivalent if one is obtained from the other by translation by an element of ℤ.

A numerical type is a map δ : {1, ⋯, m + n} → {+, −} such that #δ^{−1}(+) = m and #δ^{−1}(−) = n. Two numerical types are called equivalent if they can be obtained from each other by a cyclic permutation.

A cycle is a sequence B = (b_{0}, b_{1}, ⋯, b_{m+n−1}) of integers with a partition in disjoint subsets: B = B^{+} ∐ B^{−} with #B^{+} = m and #B^{−} = n such that the following conditions hold (here we set b_{m+n} := b_{0} for convenience): (1) If b_{i} + n ∈ B, then b_{i} ∈ B^{+} and b_{i+1} = b_{i} + n. (2) If b_{i} − m ∈ B, then b_{i} ∈ B^{−} and b_{i+1} = b_{i} − m. Thus the subsets B^{+} and B^{−} are well defined by B. Cycles are called equivalent if they can be obtained from each other by a combination of a translation and a cyclic permutation.

5.30 We will show the three types of objects are in one to one correspondence up to equivalence.

The correspondence between cycles and numerical types is rather obvious. If δ is a numerical type, then we choose some b_{0} ∈ ℤ and we inductively define b_{i+1} = b_{i} + n if δ(i) = + and b_{i+1} = b_{i} − n if δ(i) = −. (With this definition we indeed have b_{n+m} = b_{0}.) We leave the inverse construction to the reader.

We define the cycle B of a semi-module A by setting B^{+} = A \ (m + A), B^{−} = (m + A) \ (h + A) and B = B^{+} ∪ B^{−} as sets. It is easy to show that #B^{+} = m and #B^{−} = n. The reader shows that +n induces a map B^{+} → B and −m induces a map B^{−} → B. Starting at any b_{0} ∈ B these operations define a walk through the set B. This walk has to close; an easy number theoretic lemma shows that such a loop has length at least h = n + m. Therefore, all members of B are visited in this manner and B is a cycle. Conversely, a cycle B defines a semi-module by considering the semi-module generated by it: A = B + mℤ_{≥0} + nℤ_{≥0}.

5.31 Remark. The number of equivalence classes equals \binom{m+n}{m}/(m + n). This follows from the description in terms of numerical types.

5.32 A semi-module A is called admissible if: (1) A ⊂ ℤ_{≥0}, (2) the first r elements a_{1}, …, a_{r} of A are all smaller than 2r and (3) A = {a_{1}, ⋯, a_{r}} ∪ [2r, ∞).

For any semi-module A we define the dual semi-module to be the set A^{t} = ℤ \ (2r − 1 − A) = {b | b ∈ ℤ, 2r − 1 − b ∉ A}. Trivially A^{tt} = A. Also, if A is an admissible semi-module, then A^{t} is an admissible semi-module (proof left to the reader).

Finally, we define the symbol ⟨0⟩ to indicate the semi-module generated by 0. So ⟨0⟩ = ℤ_{≥0}n + ℤ_{≥0}m.

5.33 Lemma. (i) The semi-module ⟨0⟩ is admissible. We have 2r − 1 ∉ ⟨0⟩. (ii) For an admissible semi-module A the following are equivalent: (1) A = ⟨0⟩, (2) 0 ∈ A, and (3) 2r − 1 ∉ A.

Proof. The first statement is a combination of [20, Lemma 3.8] and its two corollaries. For the second. Clearly (1) implies (2). We have (2) ⇒ (1) as 0 ∈ A implies ⟨0⟩ ⊂ A, and the equality follows by counting. In particular, by (i), we see that ⟨0⟩^{t} = ⟨0⟩. By considering the duality A ↦ A^{t}, we see that (3) is dual to (2).

5.34 Lemma. For every semi-module there is a unique equivalent semi-module which is admissible.

Proof. Let A be a semi-module. If necessary, we translate A such that A has exactly r elements smaller than 2r, and such that 2r ∈ A. This translation is possible and it is unique. If a ∈ A with a < 0, then a + ⟨0⟩ ⊂ A and we obtain a contradiction, using # (a + ⟨0⟩ ∩ (−∞, 2r − 1]) = r − a > r. Hence A ⊂ ℤ_{≥0}.

Next, we consider A^{t}. There are exactly r elements of A^{t} which are smaller than 2r by what we just proved. Also, 2r ∈ A^{t}, as −1 ∉ A. Thus we can apply the argument above to see that A^{t} does not contain any negative integers. These two facts imply that A is admissible.

5.35 For a given semi-module A we define the a-number to be the number of generators of A, i.e., a(A) := # (A \ (n + A) ∪ (m + A)). In the language of cycles this is the number of elements b ∈ B such that b + n ∈ B and b + m ∈ B. We leave the proof of this to the reader.

More definitions. Let B = (b_{0}, b_{1}, …, b_{n+m−1}) be a cycle. Let us write

⊤(B) = {b ∈ B|b + m ∈ B and b + n ∈ B}.

Similarly,

⊥(B) = {b ∈ B|b − n ∈ B and b − m ∈ B}.

In terms of the numerical type δ associated to B, we have b_{i} ∈ ⊤(B) if δ(i) = + and δ(i − 1) = −, and analogously for elements of ⊥(B). This implies that #⊤(B) = #⊥(B). This number is the a-number a of the semi-module associated to B, see above.

We can use ⊤(B) and ⊥(B) to write B in a standard form. After a cyclic renumbering, we may assume that b_{0} ∈ B is the largest element of B, so b_{0} ∈ ⊥(B). This determines uniquely bijections

t : {1, ⋯, a} → ⊤(B)

and

c : {1, ⋯, a} → ⊥(B)

which preserve the order in which the elements are numbered in B. (In other words, if t(i) = b_{j} and t(i + 1) = b_{j′}, then j < j′. Similarly for c; in particular b_{0} = c(1).) A moments consideration shows that there are positive integers n_{1}, …, n_{a} and m_{1}, …, m_{a} such that

c(1) = n_{1} · m + t(1), t(1) + m_{1} · n = c(2), c(2) = n_{2} · m + t(2), …,

and t(a) + m_{a} · n = c(1). We remark that n_{1} + … + n_{a} = n, and m_{1} + … + m_{a} = m. Such sequences of integers give us yet another way of finding cycles. Finally, any element b of B can be written uniquely as t(i) + s · m for some i and some s, with 0 ≤ s ≤ n_{i} or as t(i) + s · n for some i and some s, 0 ≤ s ≤ m_{1}.

Here is the way we picture our cycle:

t(1)  t(2)  …  t(a)
↙  ↘  ↙  ↘ …  ↙  ↘
c(1)  c(2)  …  c(a)  c(1)

The south-west arrows indicate adding various multiples of m, and the south-east arrows indicate adding multiples of n. We think of the rest of A as lying ”below” this picture.

5.36 Lemma. Let b_{0} ∈ B be the largest element of B as above. Then [b_{0}, ∞) ⊂ A.

Proof. Let x be an integer, x ≥ b_{0}. By definition of B^{+} there is an element b ∈ B^{+} such that x−b is divisible by m. By our choice of b_{0}, we see that s = (x−b)/m is a nonnegative integer. Thus x = b + s · m ∈ A.

5.37 Suppose given a semi-module A. We denote by

P(A) := {(a, j) | a ∈ A, j ∈ ℤ \ A, a < j}.

On P(A) we introduce an equivalence relation ∼. It is the smallest generated by the following two implications: (a, j), (a + n, j + n) ∈ P(A) ⇒ (a, j) ∼ (a + n, j + n), and (a, j), (a + m, j + m) ∈ P(A) ⇒ (a, j) ∼ (a + m, j + m). We write C(a, j) ⊂ P(A) for the equivalence class containing (a, j).

An equivalence class C ⊂ P(A) is mapped to A by the map q : C → A given by q(a, j) = a. Note that q is injective as the difference k = j − a is fixed for (a, j) in C. Hence C = {(a, a + k)|a ∈ q(C)}. If a ∈ A and a + n ∈ q(C), then a ∈ q(C). Similarly, if a ∈ A and a + m ∈ q(C), then a ∈ q(C).

5.38 Lemma. Let C be an equivalence class as above. Then there exist 1 ≤ i_{0} ≤ i_{1} ≤ a such that t(i) ∈ q(C) if and only if i_{0} ≤ i ≤ i_{1}, and c(i) ∈ q(C) if and only if i_{0} < i ≤ i_{1}. Notations used are as in Subsection 5.35.

Proof. According to the remarks made before the lemma, for any element a ∈ q(C), if a − n (or a − m) is in A, then it lies in q(C). Thus we see that q(C) ∩ ⊤(B) is not empty. Let i_{0}, resp. i_{1} be the smallest, resp. the largest index in {1, …, a} such that t(i_{0}), resp. t(i_{1}) lies in q(C).

Suppose that c(i) ∈ q(C). By Lemma 5.36 c(1) ∉ C, so i ≠ 1. We have t(i − 1) = c(i) − m_{i−1} · n and t(i) = c(i) − n_{i} · m. This implies t(i − 1), t(i) ∈ q(C). Hence i_{0} < i ≤ i_{1}. We have shown the ”only if” clause.

By definition of the equivalence relation on P(A), there exists a sequence t(i_{0}) = a_{0}, a_{1}, a_{2}, …, a_{s−1}, a_{s} = t(i_{1}) of elements of C such that for each index 0 ≤ j < s we have one of the following four possibilities a_{j} = a_{j+1} + n, a_{j} = a_{j+1} + m, a_{j} = a_{j+1} − n, or a_{j} = a_{j+1} −m. We may suppose (by ”abbreviating”) that this sequence has no repetitions. Suppose that for some s > j ≥ 1 we have a_{j−1} = a_{j} − n and a_{j+1} = a_{j} − m. In this case we have a_{j} − m − n ∈ A unless a_{j+1} ∈ ⊥(B) (left to the reader). If a_{j} − n − m ∈ A, then we replace a_{j} by a_{j} − n − m which is also in q(C). A similar replacement can be performed if a_{j−1} = a_{j} − n, a_{j+1} = a_{j} − m and a_{j} − n − m ∈ A. Continuing this process, we see that after a finite number of steps, all the local maxima in the sequence a_{0}, …, a_{q} are elements of ⊥(B). This implies that the sequence is part of the picture in 5.35: it is contained in B, and the sequence of local maxima of a_{0}, …, a_{q} is either c(i_{0} + 1), c(i_{0} + 2), …, c(i_{1}) or c(i_{0}), c(i_{0} − 1), …, c(1), c(a), …, c(i_{1} + 1). The second possibility is excluded by what was said above. This proves the “if” clause.

5.39 We write

V(A) := {(v, f) | f ∈ B^{+}, v ∈ B^{−}, v < f}.

5.40 Lemma. The natural map V(A) → P(A)/∼ given by (v, f) ↦ C(v − m, f − m) is bijective.

Proof. The map is well defined: f ∈ B^{+} implies that f −m ∉ A, and v ∈ B^{−} implies that v − m ∈ A. We will construct the inverse to this map. Let C ⊂ P(A) be an equivalence class, and let i_{0} ≤ i_{1} be the integers of Lemma 5.38. Let s ≥ 0 be the largest integer such that t(i_{0}) + s · m ∈ q(C). Since c(i_{0}) ∉ q(C), we have s < n_{i_{0}}. Thus v = t(i_{0}) + (s + 1) · m is an element of B. It is trivial to check that v ∈ B^{−}. We take f = v + k, where k is the constant difference k = j − a, for (a, j) ∈ C. We leave it to the reader to check that f ∈ B^{+}. (We remark that in the picture of 5.35, the point v is the left-most-point of q(C) ∩ B.)

5.41 We think of a Young diagram as a finite subset Y ⊂ ℕ × ℕ such that (x, y) ∈ Y and 1 ≤ i ≤ x, 1 ≤ j ≤ y imply (i, j) ∈ Y. The volume of Y is the number of elements of Y.

Given a semi-module A, we may write

B^{+} = {f_{m}, ⋯, f_{1}}, f_{m} < ⋯ < f_{2} < f_{1}.

The Young diagram Y(A) associated with A is the Young diagram with columns for x = 1, …, m whose heights are given by y = #{v ∈ B^{−}|v ≤ f_{x}}. By definition the volume of Y(A) is equal to the number of elements of V(A).

5.42 Example. (max) In the notions defined above there is a “maximal” Young diagram (why this is maximal will be specified). This is the case when A = ⟨0⟩. In this case B^{+} = {0, n, 2n, …, (m − 1)n}, and B^{−} = {mn, mn − m, mn − 2m, …, m}. The Young diagram is given as follows: the height of the column x is

[(m − x)·n/m]

(where [ ] denotes the integral part). We leave it to the reader to see that the volume equals r = (n − 1)(m − 1)/2 in this case.

(min) Consider the admissible semi-module A = [r, ∞) = ℤ_{≥r}. In this case B^{+} = [r, r + n − 1], B^{−} = [r + n, r + n + m − 1]. The Young diagram given by this is empty and has volume equal to zero.

5.43 Proposition. Let A be a semi-module. Then its Young diagram is contained in the maximal one: Y(A) ⊂ Y(⟨0⟩). Moreover, Y(A) = Y(⟨0⟩) ⟺ A ∼ ⟨0⟩.

Proof. We define a function g : [0, n + m] → ℝ, where [0, n + m] ⊂ ℝ denotes the usual interval. Let B = {b_{0}, …, b_{n+m−1}} as in Subsection 5.35. We require that g is piecewise linear with breakpoints only at integers, and such that g(i) = b_{i} and g(n + m) = b_{0}. Thus the graph of g has only two slopes, namely +n and −m: if b_{i} ∈ B^{+}, then the slope is +n on the interval [i, i + 1], and if b_{i} ∈ B^{−} then the slope is −m on the interval [i, i + 1]. We may also assume that b_{0} is the largest element of B (as in 5.35). If g(i) = g(j) for some integers 0 ≤ i < j ≤ n + m, then i = 0 and j = n + m. (The reader will be able to follow the arguments much more easily after visualizing the graph of g by making some pictures.)

We choose an integer t, 1 ≤ t ≤ m. Let I_{t} ⊂ [0, n + m] be the part of the interval where the graph of g lies below f_{t}, so I_{t} = {x | g(x) ≤ f_{t}}. Let I_{t}^{+} be the part of I_{t} where the slope of g is positive (+n), and let I_{t}^{−} is the part where the slope is negative (-m). By the description of the slopes above, we see that the length of I_{t}^{+} (i.e., its measure) is at most the number of elements of b_{i} ∈ B^{+} with b_{i} < f_{t}, i.e., m(I_{t}^{+}) ≤ (m − t). We leave it to the reader to show the trivial equality:

m(I_{t}^{−}) · m = m(I_{t}^{+}) · n.

On the other hand, by the description of the slopes of g, we see that the number of i ∈ I_{t} with b_{i} ∈ B^{−} is ≤ m(I_{t}^{−}). Putting all of this together we obtain:

#\{v ∈ B^{−}|v ≤ f_{t}\} ≤ [\frac{(m − t)n}{m}] .

This proves the first assertion.

The second assertion means the following. If we have equality in the last displayed equation for all 1 ≤ t ≤ m, then the graph of g is an inverted triangle. Assume that equality holds for all t. Suppose that for some i, 2 ≤ i ≤ n + m − 2 we have b_{i−1} + n = b_{i}, and b_{i} = b_{i+1}+m. We claim that there exists a t such that b_{i−1} < f_{t} < b_{i}. To prove this look at the elements b_{i+1}, b_{i+2}, …, b_{n+m−1}}. Let j be the first index in {i+1, i+2, …, n+m−1} such that b_{j} ∈ B^{+}; note that such an index exists and that b_{j} ≤ b_{i+1} < b_{i}. There are two possibilities: either b_{i−1} < b_{j}, in which case f_{t} = b_{j} works, or b_{i−1} > b_{j}. In the last case, the series of elements b_{j}, b_{j+1}, …, b_{n+m−1}, b_{n+m} = b_{0} starts below b_{i−1} and ends in b_{0} which is above b_{i} = b_{i−1} + n. The only upward steps in the series, are steps of size +n and the interval [b_{i−1}, b_{i}] has size n also. Then the pigeon hole principle says that some j′, j < j′ < n + m exists, with b_{i−1} < b_{j′} < b_{i}, and b_{j′+1} = b_{j′} + n. Thus b_{j′} ∈ B^{+} and we have proved the claim.

Let f_{t} = b_{j} be the element of B^{+} that exists by the claim above. There is another cycle B′ which is given by B′ = {b′_{0} = b_{0}, …, b′_{i−1} = b_{i−1}, b′_{i} = b_{i} − n − m, b′_{i+1} = b_{i+1}, …, b′_{n+m−1} = b_{n+m−1}}. The only elements that changed signs are b′_{i−1}, which is now a −, and b′_{i}, which is now a +. Thus the element b_{j} will still be listed as f_{t} in the enumeration scheme of Subsection 5.41 (because the number of elements of B^{+} above it has not changed). However, the number of elements of (B′)^{−} below f_{t} = b_{j} has increased by 1, as the element b′_{i} has been added. However, this is impossible as by assumption the number of elements of B^{−} below f_{t} was already maximal. This means that an i as in the third sentence of the previous paragraph doesn’t exist, and hence the graph of g is an inverted triangle.

5.44 Remark. We have shown that the set of cycles defined by the numbers m, n maps to the set of Young diagrams contained in Y(⟨0⟩). Both set have the same cardinality. It seems plausible that this map is bijective. A “better” proof of the previous proposition would be to construct from every such Young diagram Y a cycle B such that this defines an inverse to B ↦ Y(A); we do not know whether this is possible.

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