Flatness

Chapter V: Γ( p ) -level structure

5.3 Flatness

The goal of this section is to prove that Theorem 5.3.1. M_Γ(p)new is flat overZp.

Recall from the previous chapter that the mapsM_Γ

1(p) → M_Γ

0(p)andM_Γ

0(p) →Zp

are flat. We will verify thatM_Γ(p)new is flat overM_Γ

0(p). Since the latter is flat over Zp, our result will follow.

Theorem 5.3.2. M_Γ(p)new is flat overM_Γ

0(p).

Proof. We will construct a tower of moduli spacesM_i fori =0,1, . . . ,h−1,h, so that M₀ = M_Γ

0(p),M_h = M_Γ(p)new and such that M_i is flat overM_i−1 for every i= 1, . . . ,h.

M_iwill be moduli space representing the following moduli problem:

For a base schemeS/Zp, we consider the moduli problem classifying p-divisible groups of heighthoverS, together with a filtration 0= H₀ ⊂ H₁ ⊂ . . . ⊂ H_h =G[p] together withicompatible group homomorphismsϕ₁, . . . , ϕ_isuch that the induced mapsZ/pZ→ H_j/H_j−1, 1 ≤ j ≤ iare Oort-Tate generators:

0 ^//Z/pZ ^//

ϕ₁

(Z/pZ)^{2 //}

ϕ₂

. . . ^//(Z/pZ)^{i−1 //}

ϕ_i−1

(Z/pZ)ⁱ

ϕ_i

H₀ =0 ^//H₁ ^//H₂ ^//. . . ^//Hi−1 //Hi

Note thatM₀is justM_Γ

0(p)andM_h=M_Γ(p)new. There is a natural mapM_i → M_i−1 obtained by forgettingϕ_i and the Oort-Tate generatorZ/pZ → H_i/H_i−1. We will show thatM_iis flat overM_i−1for eachi. It will follow thatM_Γ(p)new =M_h is flat overM₀= M_Γ

0(p), as desired.

The casei= 1 is slightly different from the others, and we treat it first:

Leti =1. By definition,M₁represents the functor overM_Γ

0(p)whose S-points are given by ap-divisible groupG/Stogether with a filtration(H_i)as in the definition ofΓ₀(p)-level, together with an Oort-Tate generator ofH₁. This means thatM₁is the 2-fiber product of the following diagram:

M₁

//G^×

M₀ =M_Γ

0(p) //OT where the bottom arrow extractsH₁.

SinceG^× is flat overOT (see 3.1), it follows thatM₁is flat overM_Γ

0(p). Assume now that 2 ≤i ≤ hand let’s show thatM_iis flat overM_i−1.

Note that the forgetful mapM_i → M_i−1 factors throughM_i−1×_OT G^×. Indeed, this corresponds to forgettingϕi, but not the induced Oort-Tate generatorZ/pZ→ H_i/H_i−1. By a similar argument as in thei= 1 case,M_i−1×_OT G^× → M_i−1is flat.

It thus suffices to establish flatness ofM_i → M_i−1×_OT G^×.

We show thatM_iis in fact a fppf torsor overM_i−1×_OT G^× underH_i−1and thus flat.

Consider the diagram:

0 ^//(Z/pZ)ⁱ⁻¹

ϕi−1

//(Z/pZ)ⁱ

//Z/pZ

γi

//0

0 ^//H_i−1 ^//H_i ^//H_i/H_i−1 ^//0

Observe that since ϕ_i−1 is a Drinfeld-Katz-Mazur level structure and Z/pZ → H_i/H_i−1is an Oort-Tate generator, Lemma 3.1.1 implies that any group homomor- phism, if it exists, completing the diagram as above is automatically a Drinfeld-Katz- Mazur level structure.

Group homomorphismsϕ_i :(Z/pZ)ⁱ → H_iyielding the above diagram commutative, if they exist, are classified byH :=Hom(Z/pZ,H_i−1)= H_i−1. We claim thatM_iis aH_i−1-torsor overM_i−1×_OT G^×. For that we need to verify that locally, the above diagram can be completed with a dotted arrow.

Any extension filling in the above diagram arises in the following form:

0 ^//(Z/pZ)ⁱ⁻¹

ϕ_i−1

//(Z/pZ)ⁱ

//Z/pZ ^//0

0 ^//H_i−1 //?1

//(Z/pZ) ^//0

0 ^//H_i−1 //?₂ ^//

Z/pZ

γ_i

//0

0 ^//H_i−1 ^//H_i ^//H_i/H_i−1 ^//0

where the second row is the pushout under ϕ_i :(Z/pZ)ⁱ⁻¹→ H_i−1and the third row is the pull-back underγ_i :Z/pZ → H_i/H_i−1. We thus need to verify that a dotted arrow such as above exists, locally on the base.

Note that the top row 0→ (Z/pZ)ⁱ⁻¹ → (Z/pZ)ⁱ →Z/pZ→0 is a split extension, and thus corresponds to the trivial element in the corresponding Ext group. Thus the extension

0 ^//H_i−1 ^//?₁ ^//(Z/pZ) ^//0

is trivial, as it is obtained by pushout from a trivial extension. A dotted arrow as above then exists if and only if the class of

0 ^//H_i−1 //?₂ ^//(Z/pZ) ^//0

in the Ext group is trivial, i.e. if the sequence splits locally. Note that ?₂is the fiber productH_i×_Hi/Hi−

1 (Z/pZ). SinceH_i(andZ/pZ) are killed byp(see [TO70]), ?₂is also killed byp. We then invoke following lemma:

Lemma 5.3.1. LetGbe a commutative finite flat group scheme killed byp. Every short exact sequence

0→H → G→ (Z/pZ) →0 splits locally in the fppf topology.

Proof of lemma: IfSis our base scheme, then any section in(Z/pZ)_S(S)has, fppf locally, a preimage inG. SayT → Sis such a scheme, i.e. 1∈ (Z/pZ)(T)acquires a preimage. ButG(T), by assumption is killed byp.

Note thatG(T), as an abstract group, could be infinite (for exampleG =α_p×α_pover an infinite fieldk of characteristicpandT = k[]/(^p)). However,anypre-image of 1∈ (Z/pZ)(T)gives a splitting, asG(T)is killed by p. In other words, every exact sequence of (abstract) abelian groups

0→ A→G(T) →Z/pZ→0,

where AandG(T)are killed byp, splits.

C h a p t e r 6

EPIPELAGIC LEVEL AND CERTAIN GROUP SCHEMES OF ORDER P

²

6.1 Epipelagic level structure

In this section, we discuss yet another notion of level structure onp-divisible groups.

The motivation comes from the study of epipelagic representations. Just like our notion ofΓ(p)-level, it ‘‘lives’’ overM_Γ

1(p), however it is not ‘‘comparable’’ to it.

To introduce the moduli problem onp-divisible groups, we first define:

Definition 6.1.1. LetS be a scheme overZpandGbe ap-divisible group of height hoverS. An epipelagic level structure onGis the data consisting of

1. A filtration 0 = H₀ ⊂ H₁ ⊂ . . . ⊂ H_h = G[p] of G[p] by finite flat group schemes H_i of rank pⁱ. Note that the isomorphism G[p²]/G[p] −→^w G[p] induced by multiplication by pallows us to extend this filtration to the right H_h−1⊂ H_h= G[p]= H_h+1 ⊂ . . .⊂ G[p²], such that H_h+1/G[p] w H₁. 2. Oort-Tate generatorsγ_ifor the quotientsH_i/H_i−1(which are finite flat of rank

p), fori= 1, . . . ,h+1such thatγ_h+1= γ₁.

3. For 2 ≤ i ≤ h+ 1, group homomorphisms Z/pZ → H_i/H_i−2 such that the composition Z/pZ → H_i/H_i−2 H_i/H_i−1 is the Oort-Tate generator γ_i :Z/pZ→ H_i/H_i−1.

Denote byM_I++ the set-valued functor on Sch/Zpwhose values onSare given by p-divisible groups with an epipelagic level structure. Note that we have a natural mapM_I++ → M_Γ

1(p)by forgetting the group homomorphismsZ/pZ→ H_i/H_i−2. We now show thatM_I++is flat overZpby showing that it flat overM_Γ

1(p), which is flat overZp(see chapter 4). In fact we prove that

Proposition 6.1.1. M_I++is a torsor overM_Γ

1(p) under Öh+1

i=2

H_i/H_i−1.

Proof. The data of a map Z/pZ → Hi/Hi−2 such that the composition with H_i/H_i−2 → H_i/H_i−1 is the Oort-Tate generator γ_i is the same as a dotted group

homomorphism making the following diagram commute:

0 ^//Z/pZ ^//

γ_i−1

(Z/pZ)^{2 //}

Z/pZ ^//

γ_i

0

0 ^//H_i−1/H_i−2 //H_i/H_i−2 //H_i/H_i−1 //0

If a group homomorphism completing the above exists, then all such group homo- morphisms are classified by Hom(Z/pZ,H_i−1/H_i−2)=H_i−1/H_i−2. The argument for i= 2 used in the proof of flatness forΓ(p)-level structures shows that existence is

indeed satisfied.

Corollary 6.1.1. M_I++is flat overZp.

We now take a different turn. Recall that in writing down the local models of Shimura varieties forΓ₁(p)-level structure, a crucial gadget was the classification of group schemes of orderpdue to Oort-Tate and level structures on them (see [Sha15]

and [HR12]). Similarly, when trying to write down the local model for the moduli space of p-divisible groups with epipelagic level structure, a useful gadget would be a classification of group schemesGof orderp², killed by p, equipped with an extension structure 0 →H₁→ G→ H₂→ 0, or at least of such extensions when H₂= Z/pZ.

We will give a partial result in this direction, giving an explicit classification of extensions ofZ/pZbyµ_poverZp-algebrasR. Note that one can compute (abstractly) the group Ext¹(Z/pZ, µ_p)as H¹

fppf(Spec(R), µ_p). Using Kummer theory, the latter is (locally) isomorphic to R^∗/(R^∗)^p. We will arrive at this result from a different direction, by analyzing in detail what happens on the level of Hopf algebras, using Oort-Tate theory along the way. It is our hope that such an approach can be generalized to classify other extensions.

6.2 Certain group schemes of order p²

Let Rbe any algebra and ∈ R^× a unit. Consider the following group schemes, which we denote byG, introduced in [KM85]. The affine ring ofG is

B :=

p−1

Ê

i=0

R[X_i]/(X_i^p−ⁱ).

For any R-algebraAwith connected spectrumT = SpecA, we have G(T)= {(t,i)| t^p =ⁱ, 0≤ i ≤ p−1}

Addition is defined as

(t,i) · (s,j)=











(st,i+ j), ifi+ j < p (st/,i+ j− p), ifi+ j ≥ p Comultiplication is given by

X_i =(0, . . . ,0,X_i,0, . . . ,0) 7→

i

Õ

j=0

X_j ⊗ X_i−j+

p−1

Õ

j=i+1

Xj ⊗Xp+i−j

and

e_i =(0, . . . ,0,1,0, . . . ,0) 7→ Õ

j+k=i (mod p)

e_j ⊗e_k,

as is usual for constant group schemes. Here we viewei andXias elements in the ith component R[X_i]/(X_i^p−ⁱ). The augmentation ideal has rankp²−1 and equals hX₀−1i ⊕ R[X₁]/(X^p

1 −) ⊕ · · · ⊕ R[X_p−1]/(X_p−^p

1−^p−1). Projection onto the first factorB R[X₀]/(X^p

0 −1)and the map sendingX_i →0 for everyi(on functor of points, this sends(a,i) 7→i) induces a short exact sequence of group schemes

0→ µ_p→G →Z/pZ→0

This sequence splits if and only if is a pth power of a unit inR.

6.3 Extensions ofZ/pZby µ_pin characteristicp

In this section we prove that in characteristic p, locally on the base, the group schemesG defined above are the only finite flat extensions ofZ/pZbyµ_p. LetRbe a (connected)Fp-algebra. SinceZ/pZ, as a scheme overS =Spec(R), haspdisjoint components, an extensionGof Z/pZby µ_pis of the formG₀Ý

S₁Ý. . .Ý S_p−1, whereG₀= µ_pandS_iare schemes, flat over S=Spec(R)of rankp.

For eachi = 1, . . . ,p−1 we have an actionG₀×S_i → S_i. Let us investigate this action. We need to use certain results about µ_pactions from [Tzi17]. That paper studies actions ofα_pand µ_pon schemes and the result we need is only proved for α_p, so we prove it for µ_phere:

Lemma 6.3.1. Let Xbe a scheme of finite type over a ringRof characteristicp> 0. X admits a nontrivial µ_paction if and only if X has a nontrivial global vector field Dsuch that D^p= D.

Before we begin the proof, let us summarize the results of Proposition 2.2.1 in characteristicp:

Proposition 6.3.1([TO70]). Let Rbe a ring of characteristicpand consider the Hopf algebra B = R[z]/(z^p−1)of µ_p/SpecR. Then, there is an element y such that:

1. I₁is generated by y andI_j is generated by y^j, for1 ≤ j ≤ p−1(notation from chapter II).

2. For1 ≤ j ≤ p−1, we have y^j = j!y_j. We denotew_j := j!. Moreover y^p= w_py= 0,

for somew_p ∈Zpequal to p· (unit). 3.

∆(y_i)= y_i⊗1+1⊗ y_i+

p−1

Õ

j=1

y_j ⊗ y_i−j

and in particular

∆(y)= y⊗1+1⊗ y+

p−1

Õ

j=1

yⁱ

w_i ⊗ y^p−i w_p−i

4. z=1+ y+ y²

2! +· · ·+ y^p−1 (p−1)!.

Let us go back now to the proof of the Lemma 6.3.1.

Proof. Suppose first thatX has a nontrivial vector fieldDsuch thatD^p= D. Lety be the element in the Hopf algebra of µ_pdescribed above (so the Hopf algebra of µ_p is SpecR[y]/(y^p−w_py)= SpecR[y]/(y^p); howeveryisnotz−1 in the traditional representation of µpasR[z]/(z^p−1)).

Consider the mapΦ: O_X → O_X[y]/(y^p)by setting Φ(a)=

p−1

Õ

m=0

D^(m)(a) m!

y^m

Let’s check that this defines an action of µ_ponX. Recall that the comultiplication

∆:R[y]/(y^p) → R[y]/(y^p) ⊗R[y]/(y^p)sendsy 7→ y⊗1+1⊗ y+

p−1

Õ

i=1

yⁱ

w_i ⊗ y^p−i w_p−i

.

We need to verify that the following diagram commutes:

O_X ^{Φ //}

Φ

O_X ⊗_RR[y]/(y^p)

Φ⊗1

O_X ⊗_R R[y]/(y^p) ^1⊗∆//O_X ⊗_RR[y]/(y^p) ⊗_R R[y]/(y^p)

Indeed, a sectiona ∈ O_Xgets sent to

p−1

Õ

m,n=0

D^(m+n)(a)

m!n! y^m⊗yⁿunder the above-diagonal composition of the above diagram. The below-diagonal composition sends

a 7→

p−1

Õ

m=0

D^(m)(a) m!

∆(yⁱ)

Since

∆(yⁱ)=∆(w_iy_i)= w_i∆(y_i)=w_i©

­

«

y_i ⊗1+1⊗y_i+

p−1

Õ

j=1

y_j⊗ y_i−jª

®

¬

=

=w_i©

­

« 1

w_iyⁱ⊗1+ 1

w_i1⊗yⁱ+

p−1

Õ

j=1

y^j

w_j ⊗ y^i−j w_i−j

ª

®

¬ we can see that the coefficients of yⁿ⊗ y^m is equal to

D^(m+n)(a) (m+n)!

·w_m+n· 1

w_n ⊗ 1 w_m

, ifm+n < pand

D^(m+n−p+1)(a) (m+n)!

·w_m+n· 1

w_n ⊗ 1 w_m

,

if m +n ≥ p respectively. The result then follows since w_i = i! (mod p) for 1≤ i ≤ p−1 andD^p = D.

Assume now X admits a non-trivial µ_p-action given by α : µ_p × X → X. This corresponds to a co-action map

α^∗ : O_X → O_X ⊗_RR[y]/(y^p). By definition, we then have the commutative diagram:

O_X ^{α∗ //}

α^∗

O_X ⊗_RR[y]/(y^p)

α^∗⊗1

O_X ⊗_R R[y]/(y^p) ^1⊗∆//O_X ⊗_RR[y]/(y^p) ⊗_R R[y]/(y^p)

Write

α^∗(a)= a⊗1+Φ₁(a) ⊗y+

p−1

Õ

j=2

Φ_j(a) ⊗ y^j.

It is clear that theΦ_i’s areR-linear. Since y^p= 0 andα^∗ is a ring homomorphism, the relationα^∗(ab) = α^∗(a)α^∗(b)implies thatΦ₁(ab)= aΦ₁(b)+bΦ₁(a), i.e. that Φ₁is a derivation. We now prove by induction thatΦ_i = Φ⁽ⁱ⁾

1

i! .

Indeed, the coefficient ofy⊗yⁱunder the above-diagonal compositions isΦ₁(Φ_i(a)). Under the below-diagonal composition, this is

w_i+1Φ_i+1(a) w_iw₁ .

By induction,Φ_i(a)= Φ⁽ⁱ⁾

1 (a)

i! , thusΦ_i+1(a)= Φ⁽ⁱ⁺¹⁾

1 (a)

(i+1)! sincew_i =i! for everyi. Finally, equating the coefficients ofy⊗y^p−1under the two compositions, one obtains D^p= D, proving the lemma.

We will also need the following result (mild generalization of lemma 4.1 in [Tzi17]):

Lemma 6.3.2. Let Abe an Zp-algebra and M and A-module. LetΦ : M → M be an A-module homomorphism such that Φ^p = Φ. Then M = ⊕_i=0^p−1M_i, where M_i = {m ∈ M|Φ(m) = χ(i)m}, where χ : Fp → Zpis the Teichmuller character defined in section 2.2.

We remark that this lemma is a purely algebraic result, similar to the one used in [TO70] to decompose the augmentation ideal of a group scheme of orderpinto a direct sum of p−1 modules. The only prerequisite is the presence of p−1 roots of unity. Note that the proof referenced in [Tzi17] is for the case ofFp-algebras, and is based on Lemma 1, A.V. 104 in [Bou03]. TheZpcase is nearly identical to the Fp case: while in loc. cit., overFp one has the identity 1= Õ

P_i(x), where P_i(x)= X− X^p

X − χ(i); overZpthe sum is still a unit inZp[X]/(X^p−X), so the argument of loc. cit. goes through. Indeed, one has the identity

1+ p

1−pX^p−1 Õ

P_i(x)=1.

Returning to our problem, the action of µ_ponS_i =Spec(A_i)thus givesR-derivations D_i on A_i such that D_i^p = D_i. Using the above lemma, we decompose each R_i as a direct sum of p submodules Mi,j, j = 0, . . . ,p−1. We shall analyze these derivations and decompositions. Because we repeat this process for every A_i, we

drop (temporarily) the subscriptiand thus writeA,D,M_j instead of A_i,D_iand Mi,j

respectively.

We check that each Mj is locally free of rank 1. Indeed, it suffices to prove that at every geometric pointx, the vector space(M_j)_x is 1-dimensional. But at geometric points all extensions ofZ/pZby µ_p are split and thus the schemesS_iare actually copies of µ_p = Speck[X]/(X^p−1), where k is an algebraically closed field. In this case one checks that M_iis spanned by yⁱ, wherey is, as usual, the generating Oort-Tate section in the Oort-Tate description ofµ_p(see Proposition 6.3.1).

Thus M₁is free of rank 1 and let f be a generating section ofM₁. ThenD(f)= f by definition and an easy induction shows that D(fⁱ)=i fⁱ. In fact because the image of fⁱat geometric points is nonzero, it follows that fⁱ , 0. Moreover, because the decomposition ofO_Si as a sum of eigenspaces ofDisdirect, it follows that the fⁱs are linearly independent for 0 ≤ i ≤ p−1 (compare also Lemma 2 on page 9 of [TO70]). Since howeverS_iis finite flat of rank poverS = Spec(R), the section f satisfies a monic polynomial of degreep(e.g. the characteristic polynomial), and this polynomial is killed byD. On the other hand, D(fⁱ)=i fⁱ for 0 ≤i ≤ p−1. It follows that the only possible polynomial relation must be of the form f^p= , where is some constant inR.

Thus eachS_i of the form SpecR[X]/(X^p−_i). Note that we don’t know yet_i are units inR. However, we have also not fully used the fact thatGis a group scheme of orderp²- we have only exploited the action ofµ_pon theS_i’s. We will now use the ‘‘global’’ group structure to conclude that each_iis a unit and in fact_i =ⁱ

1·u_i^p where theu_iare some units (i.e. _i = ⁱup to multiplication by pth power of units).

Denote byα_ithe action map µ_p×S_i → S_i. The group structure onGinduces further actionsα_{i j} :S_i×S_j → S_i+jifi+j < pand actionsα_{i j} :S_i×S_j → S_i+j−pifi+j ≥ p. All these actions commute with theS₀ =G₀ = µpaction. In other words, we have, fori+ j < p, a commutative diagram

G₀×S_i×S_j^{id×αi j //}

αi×id

G₀×S_i+j

S_i×S_j ^{αi j //}S_i+j

Let’s analyze the ring maps induced by the above diagram:

R[y]/(y^p) ⊗R[X_i]/(X_i^p−_i) ⊗R[X_j]/(X^p_j −^id_j^×α)^{ooi j} R[y]/(y^p) ⊗R[X_i+j]/(X_i^p_+j −_i+j)

R[Xi]/(X_i^p−i) ⊗R[Xj]/(X_j^p−j)

α_i^∗×id

OO

R[Xi+j]/(X_i+j^p −i+j)

αi j

oo

α_i+j^∗

OO

Note that we can write the mapsα_i^∗explicitly. They are determined by derivations Di on SpecR[Xi]/(X_i^p−i)such thatDi(X_i^k)= k X_i^k. An easy induction shows that D_i^m(X_i^k)= k^mX_i^k. Thus the co-actionsα_i^∗ satisfy:

α_i^∗(X_i^k)=

p−1

Õ

m=0

D_i^m(X_i^k) m!

⊗ y^m =

p−1

Õ

m=0

k^mX_i^k m!

⊗ y^m.

Fix some indicesiand j. Assumeα_{i j}(X_i+j) = Õ

kl

c_klX_i^k ⊗ X^l_j. Let’s look at what happens toX_i+j in the above commutative diagram. Its image in the top-left ring is a sum of monomialsX_i^k⊗X^l_j⊗y^mwith various coefficients. For eachk,l,m, equating these coefficients from the two compositions yields

c_kl · k^m m! = c_kl

m!.

Since this is true for anyk,l,m, we conclude thatc_kl = 0 if k , 1. By symmetry ckl = 0 ifl ,1. Thusαi j(Xi+j)= c₁₁Xi⊗ Xj. Since X_i^p=i, this implies

_i+j = c^p

11_ij.

Recall that the indices were fixed andc₁₁depends on the pair(i,j). A straightforward induction then shows that_i =ⁱ

1·u_i^p, for someu_i ∈ R.

It remains to check that_iare units. By the above, we can assume that_i =ⁱwhere =₁.

Note that for every 1 ≤ i ≤ p−1 we also have maps β_i : S_i×S_p−i → G₀ = µ_p. These commute with the actions ofG₀on theS_i’s, making the following diagram commutative:

G₀×S_i×S_p−i^{id×βi //}

αi×id

G₀×G₀

m

S_i×S_p−i ^{βi //}G₀

The induced maps on rings are

R[z]/(z^p−1) ⊗R[X_i]/(X_i^p−ⁱ) ⊗R[X_p−i]/(X_p−i^p −^p−iid×)^βooi R[z]/(z^p−1) ⊗R[z]/(z^p−1)

R[X_i]/(X_i^p−ⁱ) ⊗R[X_p−i]/(X_p−i^p −^p−i)

α_i^∗×id

OO

R[z]/(z^p−1)

βi

oo

∆

OO

Note that we have switched, for computational convenience, to the presentation R[z]/(z^p−1)ofµ_p, rather thanR[y]/(y^p−w_py). The relations betweenyand zare given in proposition 2.2.1.

Assume β_i(z)= Õ

k,l

d_klX_i^k ⊗ X_p^l_−i.

Then the image of zin the top-left ring under the above-diagonal composition is Õ

k,l

d_klz⊗ X_i^k ⊗ X_p−i^l .

Recall thatα_i^∗(X_i^k)= Õ

m≥0

y^m ⊗ D^(m)(X_i^k) m! = Õ

m≥0

k^m

m!y^m ⊗ X_i^k. Thus, the image of zunder the below-diagonal composition is

Õ

k,l

d_klα_i^∗(X_i^k) ⊗X_p−i^l = Õ

k,l,m

d_kl · k^m m!

y^m ⊗X_i^k ⊗ X_p−i^l .

Equating the coefficients ofX_i^k ⊗ X_p−i^l , we obtain d_klz= d_kl Õ

m≥0

k^m m!

y^m.

On the other hand, from 2.2.1,z = Õ

m≥0

1 m!

y^m. Thus, ifk ,1, we must haved_kl =0.

By symmetry (i.e. letting G₀ act onS_p−i instead of S_i), we obtain that d_kl = 0 if l , 1. Thusβ_i(z)= d₁₁X_i⊗ X_p−i. Sincez^p= 1, we must haved^p

11ⁱ·^p−i =1, and

thus is a unit, as wanted.

6.4 Extensions ofZ/pZby µ_pover anyZp-algebra

Most of the above results go through, with minor modifications which we explain here. We still have the commutative diagram:

O_X ^{α∗ //}

α^∗

O_X ⊗_RR[y]/(y^p)

α^∗⊗1

O_X ⊗_R R[y]/(y^p) ^1⊗∆//O_X ⊗_RR[y]/(y^p) ⊗_R R[y]/(y^p)

Write

α^∗(a)= a⊗1+Φ₁(a) ⊗y+

p−1

Õ

j=2

Φ_j(a) ⊗ y^j.

We show that

Φ_i(a)= Φ⁽ⁱ⁾

1 (a)(1−p)ⁱ⁻¹ w_i . Whenp=0, we recover the formulaΦ_i(a)= Φⁱ

1(a)

i! . The proof is nearly identical to the characteristicpcase - one looks at the coefficient ofy⊗ yⁱ⁻¹. One picks up an extra(1−p)because it is present in the comultiplication rule fory(see 2.2.1). By looking at the coefficient of y⊗ y^p−1, we deduce that

D(1−p)p

= D(1−p).

Thus in the presence ofp−1 roots of unity, we still have a nice decomposition ofO_X into eigenspaces ofD(1−p), with eigenvalues 0, χ(1)= 1, χ(2), . . . , χ(p−1). In our caseX = S₁is of rank pand looking at geometric points just as in the characteristic pcase, since every extension ofZ/pZbyµ_psplits, one sees that these are all of rank 1.

A difference from the case whenp=0 is thatα^∗(ab)= α^∗(a)α^∗(b)no longer implies that D=Φ₁is a derivation. However we see below that it still has strong properties.

The relationα^∗(ab)=α^∗(a)α^∗(b)together with y^p =w_pythen implies D(ab)= aD(b)+bD(a)+w_p

p−1

Õ

j=1

Φ_j(a)Φ_p−j(a)

= aD(b)+bD(a)+w_p

p−1

Õ

j=1

(1−p)^p−2D^(j)(a)D^(p−j)(a) w_jw_p−j

.

Let f generate the invertible module corresponding to eigenvalue 1, so(1−p)D(f)= f. Using the above relation, we show that(1−p)D(f)= f implies(1−p)D(f²)= χ(2)f and more generally(1−p)D(fⁱ)= χ(i)fⁱ.

Indeed, (1− p)D(f) = f implies (1− p)ⁱD⁽ⁱ⁾(f) = f, thus D^(j)(f)D^(p−j)(f) = 1

(1−p)^p f². Thus, plugginga =b= f in the above relation yields:

D(f²)= f²©

­

« 2

1−p + w_p (1−p)²

p−1

Õ

j=1

1 w_jw_p−j

ª

®

¬

This implies that(1−p)D(f²)= c₂f²for some constantc₂. Since(1− p)Dis fully diagonalizable with a full set of distinct eigenvalues 0, χ(1), . . . , χ(p−1), it follows thatc₂is one of those. Sincew_p = p·unit, we havec₂≡ 2 (mod p), thusc₂must be χ(2). In fact, an easy induction then shows that there are constantsc_i, congruent toimod p, such thatD(fⁱ)= cifⁱ and then a similar argument shows thatci = χ(i). Alternatively, to show directly D(f²)= χ(2)

1−pf², it suffices to verify the identity:

χ(2)= 2+ w_p 1−p

p−1

Õ

j=1

1 w_jw_p−j

.

A series of similar identities can be derived for each χ(i), 2≤ i ≤ p−1.

Computing thenD(f^p)(for example fromα^∗(f^p)=α^∗(f)α^∗(f^p−1)), one finds that it vanishes (even if we are not in characteristic pandDis not a derivation!). Thus, any polynomial relation involving f must be of the form f^p= as before.

Thus, Si = Spec(R[Xi]/(X_i^p−i)). The rest of the argument in the characteristic pcase goes through unchanged, and we deduce that _i = ⁱ

1·u_i^p for some unitu_i. Moreover, the global group structure implies, by a nearly identical argument, that₁ is a unit. The only changes are the appearance of(1−p)factors in the formulas, the replacement ofm! byw_m throughout the formulas andD^(m)(X_i^k)= k^mX_i^k is replaced by χ(k)^m

(1−p)^mX_i^k.

ThusGis isomorphic to a group scheme of the formG, as we wanted to show.

C h a p t e r 7

LEVEL STRUCTURES ON DIEUDONNÉ MODULES

In this chapter, we present an auxiliary result about interpreting Drinfeld-Katz-Mazur level structures using (contravariant) Dieudonné theory. Gabber (unpublished) and Eike Lau [Lau10] have shown the Dieudonné equivalence over perfect bases. We briefly recalled this equivalence in Section 2.4. Thus, the notion of Drinfeld-Katz- Mazur level structure should have a (semi-)linear algebraic analogue. We restrict our attention to a perfect base schemesS, locally given byS= SpecR, whereRis a perfect ring (overFp). The goal of this chapter is to give an interpretation of the notion of Drinfeld-Katz-Mazur level structure in the Dieudonnś setting.

We will use some results of [HT01] but to simplify the exposition, we will work with (usual) p-divisible groups instead of Barsotti-TateO_K-modules as is done in [HT01] (soK =QpandO_K =Zp). Let thusSbe a base scheme overZpandH/Sa p-divisible group of heighthendowed with an embeddingZp,→ End(H). We do not assumeH is 1-dimensional.

Recall that a Drinfeld-Katz-Mazur p^m-level structure onH/Sis then a morphism of Zp-modules:

α:(Z/p^mZ)^⊕h w (p^−mZp/Zp)^⊕h→ H[p^m](S)

so that the set ofα(x)for x ∈ (Z/p^mZ)^hforms a full set of sections ofH[p^m]. Note that, from the definition of endomorphisms ofp-divisible groups,Zpacts on each ‘‘level’’H[p^m], i.e. H[p^m](T)is aZp-module for any schemeT overS. By embeddingZpinto End(H), we embedZpinto End(M_H), hence the Dieudonné moduleM_HofHbecomes itself aZp-module. Then an abstract group homomorphism (Z/p^mZ)^h→ H[p^m](S)is a homomorphism ofZp-modules if and only if the induced homomorphism of Dieudonné modules M_H = W_m(R)^h → M_const = W_m(R)^h is a Zp-module homomorphism.

7.1 A commutative diagram

LetHbe a p-divisible group as above,m≥ 1 an integer andG:= H[p^m].

Consider a group homomorphism f :(p^−mZ/Z)^⊕h→ G. This must commute with

the action of Frobenius (and Verschiebung), so we have the following diagram:

(p^−mZ/Z)^{⊕h //}

w Frob

G

Frob

(p^−mZ/Z)^{⊕h(p) //}G^(p)

(7.1)

Locally on S, the Dieudonné module ofH is free of rank h overW(R) (and thus the Dieudonné module of Gis free of rank hoverW_m(R)). Choose a basis. The corresponding diagram of contravariant Dieudonné modules is

W_m(R)^⊕hoo W_m(R)^⊕h

W_m(R)^⊕h

Frob w

OO

W_m(R)^⊕h

oo A

B Frob

OO (7.2)

Note that the left side of the diagram are the Dieudonné modules of the constant group and the ones on the right those ofG.

The diagram can be interpreted and used in two (equivalent) ways: we either view the bottom two Dieudonné modules as identical to the ones above, and the vertical maps aresemi-linear maps; or the vertical maps arelinearand the modules on the bottom row are Frobenius twists of the ones above. We will mainly work in the linear setting.

HereAandBareh×hmatrix with entries inW_m(R). The condition for the map to

‘‘commute’’ with the Frobenius is

FrobA= AB,

where FrobAis obtained by raisingeachentry ofAto the pth power. To see this one can take a basise₁, . . . ,ehof the Dieudonné module corresponding toG(which is free overW_m(R)) and look at the action of each map.

Let’s make this a bit more explicit as follows: writeM_G for the Dieudonné module ofGandM_const for that of(p^−mZ/Z)^h

S. Then, in thesemi-linearpoint of view, we have:

M_const ^{oo A} M_G

M_const

Frob w

OO

M_G

oo A

B Frob

OO