An introduction to Lubin-Tate spaces and p-divisible groups

Laurent Fargues

CNRS-IHES-université Paris-Sud Orsay

4 septembre 2005

1 One dimensional formal group laws

1.1 Definitions

Let R be a ring.

Definition 1. A one dimensional formal group law over R is a F (X, Y) ∈ R[[X, Y]] s.t. – F (X, Y) = F (Y, X) (commutativity)

– F (X, 0) = X and F (0, Y) = Y (the section T = 0 is the unit section) (in particular F (X, Y) ≡ X + Y mod deg 2)

– F (F (X, Y), Z) = F (X, F (Y, Z)) (associativity)

Lemma 1. For any one dimensional formal group law F over R there exists a unique series f ∈ R[[T]] such that f(0) = 0 and F (f(T), T) = 0.

Proof. This is an easy induction/approximation argument by constructing and verifying its unicity modulo T^{k} for all k starting with k = 1. In fact thanks to the hypothesis F (X, Y) ≡ X + Y mod deg 2 one has f(T) ≡ −T [T^{2}]. And if f is such that

F (f(T), T) ≡ 0 [T^{k}]

Then ∃a ∈ R s.t.

F (f(T), T) ≡ aT^{k} [T^{k+1}]

then for b ∈ R

F (f(T) + bT^{k}, T) ≡ aT^{k} + bT^{k} \frac{∂F}{∂X} (f(T), T) [T^{k+1}]

But the power series g(T) = \frac{∂F}{∂X} (f(T), T) verifies g(0) = 1. Thus f(T)−aT^{k} is the unique solution modulo T^{k+1} to the equation F (f, T) ≡ 0 [T^{k+1}].

This series f will be denote [−1]_{F}, it is the “formal inversion” on F.

We will often denote F (X, Y) = X +_{F} Y. More generally if A is an R-algebra complete with

respect to the ideal 𝔪 and a, b ∈ 𝔪 we can define a +_{F} b and thus consider (𝔪, +_{F}) that is an abelian group.

Definition 2. Let F and G be two formal group laws. We define

Hom(F, G) = { f ∈ T R[[T]] | f(X +_{F} Y) = f(X) +_{G} f(Y) }

And thus we have a category structure on formal group laws over R. This is even a ℤ-linear category.

Remark 1. A morphism f : F ⟶ G is an isomorphism iff f′(0) ∈ R^{×} that is to say iff the associated tangent map is an isomorphism.

If R ⟶ R′ is a ring morphism there is an evident base change functor from the category of one dimensional formal groups laws over R to the one over R′.

Example 1. – The additive group law 𝔾̂_{a} associated to F (X, Y) = X + Y

– The multiplicative one 𝔾̂_{m} associated to F (X, Y) = XY + X + Y (this is (X + 1)(Y + 1) − 1, for formal group laws 0 is the neutral element thus on has to translate by 1)

– If R is a ℚ-algebra there is an isomorphism of formal group laws log(1 + T) : 𝔾̂_{m} ⟶^{∼} 𝔾̂_{a} where log(1 + T) = ∑_{k≥1}(−1)^{k−1} \frac{T^{k}}{k} For n ∈ ℤ set [n]_{F} ∈ End(F) be the multiplication by n via ℤ → End(F). Thus if n > 0

[n]_{F} = \underbrace{T +_{F} … +_{F} T}_{n−times}

and if n < 0

[n]_{F} = \underbrace{[−1]_{F} (T) +_{F} … +_{F} [−1]_{F} (T)}_{(−n) −times}

Then

[n]′_{F} (0) = n

and thus if R is a ℤ_{(p)}-algebra then End(F) is a ℤ_{(p)}-algebra (we refer to remark 2). Moreover one verifies that if R is p-adically complete then End(R) is naturally a ℤ_{p}-algebra. The series [p]_{F} will play an important role in the sequel when R will be p-adically complete.

1.2 Invariant differential forms

Definition 3. An invariant differential form on F is an expression ω = f(T)dT such that

F_{∗}ω = f(X)dX + f(Y)dY

where

F_{∗}ω = f(F (X, Y))\frac{∂F}{∂X} (X, Y)dX + f(F (X, Y))\frac{∂F}{∂Y} (X, Y)dY

We note ω_{F} for the R-module of invariant differential forms on F.

Definition 4. A continuous translation invariant derivation on F is a T -adically continuous R-derivation ∂ : R[[T]] ⟶ R[[T]] such that for any g ∈ R[[T]]

∂_{X}(g(X +_{F} Y)) = (∂g)(X +_{F} Y)

where ∂_{X} is the continuous R[[Y]]-derivation on R[[X, Y]] obtained by base change from ∂. We note Lie F for the R-module of continuous invariant derivations.

Being continuous such a derivation ∂ is completely determined by ∂T and is of the form

∂ = f(T)\frac{d}{dT}

for some f = ∂T ∈ R[[T]].

Next proposition says that such translation invariant differential forms/derivations are determined by their fiber at the origin.

Proposition 1. The morphism

ω_{F} ⟶ R
f(T)dT ⟼ f(0)

is an isomorphism and thus ω_{F} is free of rank 1. Idem for Lie F via

Lie F ⟶ R
∂ ⟼ (∂T)_{|T =0}

Moreover there is a pairing

ω_{F} × Lie F ⟶ R
(f(T)dT, g(T)\frac{d}{dT} ) ⟼ f(T)g(T)

that is perfect and identifies ω_{F} with (Lie F)^{∗}.

Proof. Consider the case of ω_{F}. One has to solve the system of equations

\{{f(X) = f(F (X, Y))\frac{∂F}{∂X} (X, Y) \atop f(Y) = f(F (X, Y))\frac{∂F}{∂Y} (X, Y)}

which by f(X, Y) = f(Y, X) is reduced to the first equation. Putting Y = 0 one finds that necessarily

f(Y)\frac{∂F}{∂X} (0, Y) = f(0)

But the series g(Y) = \frac{∂F}{∂X} (0, Y) verifies g(0) = 1 and is thus invertible. Thus any solution of the system is a multiple of

\frac{∂F}{∂X} (0, T)^{−1}

Reciprocally, deriving the equation F (Z, F (X, Y)) = F (F (Z, X), Y) with respect to Z and putting Z = 0 one verifies the preceding is a solution of the system.

The remaining part of the proposition about Lie F and the pairing is easy and left to the reader.

Of course F ⟼ ω_{F} is functorial contravariant in F and F ⟼ Lie F is functorial covariant in F. For f ∈ Hom(F, G) there is a commutative diagram

\xymatrix{Lie F \ar[d]^{≃} \ar[r]^{f_{∗}} & Lie G \ar[d]^{≃} \\ R \ar[r]^{×f′(0)} & R}

where the vertical arrows are the one defined in the preceding proposition.

Example 2. For 𝔾̂_{m} the one has ω_{F} = R.\frac{dT}{T +1} and Lie F = R.(1 + T)\frac{d}{dT}. For 𝔾̂_{a} one has ω_{F} = R.dT and Lie F = R.\frac{d}{dT}.

Remark 2. A morphism f : F ⟶ G is an isomorphism iff the associated tangent morphism Lie F ⟶ Lie G is an isomorphism that is to say iff f′(0) ∈ R^{×}.

1.3 Lazard’s key lemma

The following is the key lemma to the study of one-dimensional formal group laws.

Definition 5. Let 𝔪 = (X, Y) ⊂ R[[X, Y]] and n ∈ ℕ^{∗}. Call a truncated at the order n one dimensional formal group law over R a F ∈ R[[X, Y]]/𝔪^{n} satisfying all the axioms of a formal group law in R[[X, Y]]/𝔪^{n} that is to say modulo degree ≥ n polynomials.

Lemma 2 (Lazard). Set for all n ∈ ℕ^{∗}

C_{n}(X, Y) = \{(X + Y)^{n} − X^{n} − Y^{n} if n is not a power of a prime \frac{(X + Y)^{n} − X^{n} − Y^{n}}{p} if n = p^{α} with p prime

Let n ∈ ℕ^{∗}. Let F (X, Y) be a truncated at the order n formal group law over R that can be extended to a truncated at the order n + 1 formal group law. Then the set of such extensions G to a truncated at the order n + 1 formal group law is is a principal homogeneous space under R via

∀a ∈ R ∀G a.G = G + aC_{n}(X, Y)

Let G ∈ R[[X, Y]]/𝔪^{n+1} be an extension as in the theorem. We look for a degre n homogeneous polynomial

Γ(X, Y) = ∑_{{i+j=n \atop 0<i,j<n}} a_{ij}X^{i}Y^{n−i}

such that F = G + Γ is a truncated formal group law at the order n + 1. The first evident condition is a_{ij} = a_{ji}. Now writing F (F (X, Y), Z) = F (X, F (Y, Z)) modulo degree n + 1 we find the equivalent cocyle condition

Γ(X, Y) + Γ(X + Y, Z) = Γ(Y, Z) + Γ(X, Y + Z)

This is a linear system in the unkown (a_{ij})_{i,j}. More precisely, the equations are

∀i, j, k > 0 i + j + k = n a_{i+j,k} \binom{i + j}{j} = a_{i,j+k} \binom{j + k}{k}

The C_{n}(X, Y) are solutions of this linear system as one can see immediatly by using their definition. Reciprocally one has to see that those are the unique solutions up to a scalar. This is a linear algebra problem solved in [?] section I.4..

1.4 The characteristic zero case

Proposition 2. Let R be a ℚ-algebra. Then for a one dimensional formal group law F over R there exists a unique series log_{F} ∈ T R[[T]] called the logarithm of F s.t.

log_{F} : F ⟶^{∼} 𝔾̂_{a} and log′_{F} (0) = 1

Proof. The uniqueness assertion is clear since if f_{1}, f_{2} : F ⟶^{∼} 𝔾̂_{a} then f_{1}^{−1} ∘ f_{2} ∈ End(𝔾̂_{a}). And since R is a ℚ-algebra End(𝔾̂_{a}) = R.

Let now F be such that F (X, Y) ≡ X + Y mod 𝔪^{n} for some n ∈ ℕ^{∗}. Then from lemma 2 and since R is a ℚ-algebra, for a u ∈ R

F (X, Y) ≡ X + Y + u((X + Y)^{n} − X^{n} − Y^{n}) mod 𝔪^{n+1}

where 𝔪 = (X, Y). Now setting

h(T) = T − uT^{n}

we find

h(h^{−1}(X) +_{F} h^{−1}(Y)) ≡ X + Y mod 𝔪^{n+1}

which expresses the fact that h is a truncated isomorphism

h : F ⟶^{∼} 𝔾̂_{a} mod 𝔪^{n+1}

By induction we thus construct a sequence (h_{n})_{n≥1} such that h_{n} ≡ T [T^{n}] and h_{n} ∘ ⋯ ∘ h_{1} is a truncated isomorphism modulo 𝔪^{n+1} between F and 𝔾̂_{a}. One concludes easily that

lim_{n→+∞}(h_{n} ∘ ⋯ ∘ h_{1})

converges to the desired series log_{F}.

Remark 3. This proposition is a very particular case of the following theorem : over a ℚ-algebra R the Lie algebra functor induces a category equivalence between formal group laws (not necessarily one dimensional or commutative) and Lie algebras over R that are finite free R-modules.

If log_{F} is the logarithm of F then (log_{F})_{∗}dT is an invariant by translations differential form on F since dT is one on 𝔾̂_{a}. Reciprocally if ω = f(T)dT is an invariant by translations differential form on F such that f(0) = 1 then log_{F} = ∫_{0}^{T} ω and thus

log_{F} = ∫_{0}^{T} \frac{∂F}{∂X} (0, T)^{−1}dT

If R is p-adically complete and without p-torsion one has the following formula for the logarithm of F after inverting p still denoted log_{F} although F is defined over R

log_{F} = lim_{k→+∞} \frac{1}{p^{k}} [p^{k}]_{F} (T)

where the limit is for the p-adic topology on each of the coefficients of the power series and this limit lies in R[\frac{1}{p}][[T]]. In fact the logarithm of F satisfies

∀k ≥ 1 \frac{1}{p} log_{F} ∘[p]_{F} = log_{F}

Thus if T is the following operator on R[\frac{1}{p}][[T]] : T (f) = \frac{1}{p}f ∘ [p]_{F} then log_{F} is a fixed point of this operator. But one checks that if forall k ≥ 1 and E = R[\frac{1}{p}][T]/(T^{k})

∀ ∑_{0≤i<k} a_{i}T^{i} ∈ E ∥ ∑_{0≤i<k} a_{i}T^{i}∥ = inf{j ∈ ℤ | ∀i a_{i} ∈ p^{j}R}

then the induced operator T on E is contracting for the “additive” norm ∥.∥ (by additive norm we mean “− log_{p}” of the usual notion of norm). Thus we obtain the preceding formula for log_{F}.

Corollary 1. Let R be a ℚ-algebra. Then the application that associates to f ∈ T R[[T]] such that f′(0) = 1 the series

F (X, Y) = f^{−1}(f(X) + f(Y))

gives a bijection between T + T^{2}R[[T]] and the set of formal group laws over R.

Remark 4. The preceding bijection is compatible with base change of rings. If R ⟶ R′ is a morphism of ℚ-algebras then the base change functor F ⟼ F ⊗_{R} R′ is the evident map T + T^{2}R[[T]] ⟶ T + T^{2}R′[[T]] at the level of logarithms.

1.5 Lazard’s ring

Definition 6. Let Λ be the ring representing the functor

Rings ⟶ Sets
R ⟼ the set of one dim. formal group laws over R

In fact this ring has the following presentation :

Λ = ℤ[a_{ij}]_{i,j≥1}/I

where the a_{ij} are indeterminates, the universal group law is

F (X, Y) = X + Y + ∑_{i,j≥1} a_{ij}X^{i}Y^{j}

and the ideal I is generated by the relations a_{ij} = a_{ji} and the one given by writing F (F (X, Y), Z) = F (X, F (Y, Z)).

Lemma 3. With the preceding presentation for Λ put deg a_{ij} = i + j − 1. Then the ideal I is homogeneous.

Proof. Put deg X = deg Y = −1. Then F (X, Y) is homogeneous of degree −1 in the variables (a_{ij})_{i,j}, X and Y. Thus if we put deg Z = −1 the series F (F (X, Y), Z) and F (X, F (Y, Z)) are homogeneous of degree −1 in the variables (a_{ij})_{i,j}, X, Y and Z. Thus the equation F (F (X, Y), Z) = F (X, F (Y, Z)) gives degree 0 homogeneous equations in the (a_{ij})_{i,j}.

Thus the ring Λ is graded by

Λ = ⨁_{k≥0} Λ^{k}

with Λ^{0} = ℤ. Now put

∀k ∈ ℕ Λ̃^{k} = Λ^{k}/ < xy | x ∈ Λ^{i}, y ∈ Λ^{j} i + j = k and i, j > 0 >

where < . > means the R-submodule generated. Then a translation of lemma 2 is

Corollary 2. For all k ∈ ℕ the ℤ-module Λ̃^{k} if free of rank 1.

Proof. For all n ∈ ℕ consider the ideal ⨁_{k≥n} Λ^{k} of Λ and the quotient ring

Λ/ ⨁_{k≥n} Λ^{k} ≃ ⨁_{0≤k≤n−1} Λ^{k}

the preceding isomorphism being a ℤ-modules isomorphism. This quotient ring represents the functor

Rings ⟶ Sets
R ⟼ truncated at the order n + 1 one dim. formal group laws

Now for a truncated at the order n + 1 formal group law F corresponding to a morphism

f : ⨁_{0≤k≤n−1} Λ^{k} ⟶ R

we look for the truncated at the order n+1 formal group laws that are congruent to F mod degree ≥ n. This corresponds to ℤ-linear maps ψ : Λ^{n−1} ⟶ R such that if we set

∀k ≤ n − 1 ∀x ∈ Λ^{k} f′(x) = \{{f(x) if k < n − 1 \atop f(x) + ψ(x) if k = n − 1}

then f′ is a ring morphism. One sees easily that this is equivalent to says that ψ factorises as a morphism of abelian groups in the following way

\xymatrix{Λ^{n−1} \ar[rr]^{ψ} \ar@{->>}[dr] &  & R \\  & Λ̃^{n−1} \ar[ur]}

Thus we conclude thanks to lemma 2.

From this one deduces :

Theorem 1 (Lazard). The graded ring Λ is a polynomial algebra

Λ ≃ ℤ[t_{k}]_{k≥1}

with deg t_{k} = k.

Proof. Choose for all k a lifting to Λ^{k} of a generator of Λ̃^{k} and define a graded morphism from ℤ[t_{k}]_{k≥1} to Λ by sending t_{k} to this generator. On has to see this is an isomorphism. But this is a morphism of graded rings A = ⊕_{k≥0}A^{k} ⟶ B = ⊕_{k}B^{k} inducing an isomorphism a the level of the ℤ-modules Ã^{k} ⟶ B̃^{k} for all k where Ã^{k} and B̃^{k} are defined as Λ̃^{k}. From this one deduces that this morphism A = ℤ[t_{k}]_{k≥1} ⟶ Λ = B is surjective.

Now to prove injectivity it suffices to prove it after tensoring with ℚ, for the map

ℚ[t_{k}]_{k≥1} ⟶ Λ ⊗ ℚ

But now define the following “universal logarithm” series

h(T) = T + ∑_{k≥1} u_{k}T^{k+1} ∈ ℚ[u_{k}]_{k≥1}[[T]]

and set G(X, Y) = h^{−1}(h(X)+h(Y)) ∈ ℚ[u_{k}]_{k≥1}[[X, Y]]. By the universal property of Λ associated to G there is a unique morphism

Λ ⊗ ℚ ⟶ ℚ[u_{k}]_{k≥1}

such that the universal group law on Λ ⊗ ℚ is sent to G(X, Y) through this morphism. This morphism is an isomorphism since by proposition 1 and remark 4 it defines an isomorphism of functors of ℚ-algebras between Hom(ℚ[u_{k}]_{k≥1}, −) and Hom(Λ⊗ℚ, −). More precisely, the inverse of this morphism is constructed in the following way : by proposition 2 there exists a unique g ∈ Λ ⊗ ℚ[[T]] such that g(0) = 0, g′(0) = 1 and if F is the universal group law on Λ ⊗ ℚ then

F (X, Y) = g^{−1}(g(X) + g(Y))

Then the inverse of Λ ⊗ ℚ ⟶ ℚ[u_{k}]_{k≥1} is given by sending u_{k} to the k + 1-th coefficient of g.

Now this isomorphism

Λ ⊗ ℚ ⟶^{∼} ℚ[u_{k}]_{k≥1}

is a graded isomorphism if we set deg u_{k} = k. In fact, putting deg T = −1 we see the universal logarithm h(T) ∈ ℚ[u_{k}]_{k≥1}[[T]] is homogeneous with degree −1. Thus, with deg X = deg Y = −1, G(X, Y) = h^{−1}(h(X) + h(Y)) is homogeneous with degree −1 which implies our isomorphism respects the grading.

Now look at the composition

ℚ[t_{k}]_{k≥1} ↠ Λ ⊗ ℚ ⟶^{∼} ℚ[u_{k}]_{k≥1}

it is a surjective graded morphism

ℚ[t_{k}]_{k≥1} ↠ ℚ[u_{k}]_{k≥1}

and thus an isomorphism since it induces a surjective morphism of finite dimensional ℚ-vector spaces of the same dimension between elements of degree less than a given integer.

Remark 5. One can chose an isomorphism Λ ≃ ℤ[t_{k}]_{k≥1} s.t. if F is the universal group law on Λ then

∀k ≥ 1 F (X, Y) ≡ X + Y + t_{k}C_{k+1}(X, Y) mod (t_{1}, …, t_{k−1}, (X, Y)^{k+2})

1.6 The theory of one-dim. formal groups laws is unobstructed

Corollary 3. Let R be a ring.

– Any truncated at the order k one dimensional formal group law over R extends to a formal group law over R.

– Let I be an ideal in R. Any one dimensional formal group law over R/I lifts to a formal group law over R.

Proof. This is a consequence of Lazard’s theorem, since “morphisms of polynomials algebras can be lifted”.

1.7 Disgression about the universal group law

The reader may ask the following question : we have proved the isomorphism

Λ ⟶^{∼} ℤ[t_{k}]_{k≥1}

but our construction is not “constructive” in the sens it does not give explicitely a universal group law ; can we do this ?

The answer is yes. One can give an explicit universal logarithm f ∈ ℚ[t_{k}]_{k≥1}[[T]] such that f^{−1}(f(X) + f(Y)) lies in ℤ[t_{k}]_{k≥1}[[X, Y]] and is a universal group law. This is done in section I.5.2 of [?]. To understand the formula given in [?] one needs to know Cartier theory of formal groups. In fact in [?] first the formula for the universal group law is given (and it is proved it is a universal group law), then, further in the book, Cartier theory is treated. This seems strange since kwnowing Cartier theory one can deduce without computations the given power series gives a universal group law over ℤ[t_{k}]_{k≥1}.

1.8 Classification over car. p algebraically closed fields

1.8.1 Frobenius morphism

Let R be a ring such that pR = 0. Let G be a one dim. formal group law over R.

Definition 7. We let G^{(p)} be the formal group law obtained by raising the all the coefficients of G to the p-power.

In fact one verifies easily this is a group law. The only difficulty is to verify the associativity condition but using the fact that the morphism R[[X, Y, Z]] ⟶ R[[X, Y, Z]] that sends X to X^{p}, Y to Y^{p} and Z to Z^{p} is injective and the equality G^{(p)}(X^{p}, Y^{p}) = G(X, Y)^{p} one concludes easily.

Definition 8. We denote

F : G ⟶ G^{(p)}

the Frobenius morphism defined by the power series T^{p}.

1.8.2 Height

Let k be a characteristic p field and G defined over k.

Lemma 4. Let f ∈ End(F) be s.t. f′(0) = 0 and f ≠ 0. Then ∃!h ∈ ℕ^{∗} ∃g ∈ k[[T]] g(0) = 0, g′(0) ≠ 0 and

f = g(T^{p^{h}})

Proof. Differentiating the equality f(F (X, Y)) = F (f(X), f(Y)) with respect to X one finds

f′(F (X, Y))\frac{∂F}{∂X} (X, Y) = \frac{∂F}{∂X} (f(X), f(Y))f′(X)

with X = 0 one finds

f′(Y)\frac{∂F}{∂X} (0, Y) = 0 ⟹ f′ = 0

since \frac{∂F}{∂X} (0, Y) is invertible.

Thus f can be written as a composite

\xymatrix{G \ar[r]^{F} \ar[rr]_{f} & G^{(p)} \ar[r]^{g} & G}

where g ∈ End(F), and one can apply by induction the preceding process to g.

Definition 9. Let [p]_{G} ∈ End(G). If [p]_{G} = 0 we say G has infinite height. If not let h ≥ 1 be the integer defined in the previous lemma which is the biggest integer k such that [p]_{F} can be factorized by F^{(p^{k})}. We say G has height h.

Thus if G has height h < +∞ then [p]_{F} can be factorized as

\xymatrix{G \ar[r]^{F^{h}} \ar[rr]_{[p]_{G}} & G^{(p^{h})} \ar[r]^{g}_{≃} & G}

where g is an isomorphism. From this it is clear that the height of G is an invariant of the isomorphism class of G.

1.9 The infinite height case

We keep the hypothesis of the last section.

Proposition 3. G has infinite height ⟺ G ≃ 𝔾̂_{a}.

Proof. Suppose G ≡ X + Y mod deg ≥ n. Then ∃a ∈ k

G ≡ X + Y + aC_{n}(X, Y) mod deg ≥ n + 1

If n is not a power of p then putting h(T) = t − aT^{n} one finds

h(h^{−1}(X) +_{G} h^{−1}(Y)) ≡ X + Y mod deg ≥ n + 1

If n is a power of p then one finds [p]_{G} ≡ −aT^{n} mod deg ≥ n + 1 (cf. lemma 5) and thus from the hypothesis [p]_{G} = 0 we find a = 0. By induction we thus can construct for all n truncated isomorphisms G ⟶^{∼} 𝔾̂_{a} mod deg ≥ n which in the limit gives us an isomorphism as in the proof of proposition 2.

1.10 The finite height case

The notations are the same as in the preceding section.

1.10.1 Normalized group law

Definition 10. Let h ∈ ℕ^{∗}. The height h group law G over k is normalized if [p]_{G} = T^{p^{h}}.

Remark 6. If G is normalized then G ∈ 𝔽_{p^{h}}[[X, Y]].

As we have seen we can write

[p]_{G} = g ∘ F^{h}

where g is an isomorphism between G^{(p^{h})} and G. Now the problem to know whether or not G is isomorphic to a normalized group law is equivalent to know whether or not g can be written as

g = g_{1}g_{1}^{−p^{h}}

for some g_{1} ∈ Aut(G), that is to say if g is “a coboundary”.

Remark 7. This problem is related to the problem to know if in a height h one dimensional crystal over k one can find an non-zero element e s.t. F^{h}.e = p.e.

Proposition 4. If k is separably closed then any finite height one dimensional group law over k is isomorphic to a normalized one.

Proof. If

[p]_{G} ≡ aT^{p^{h}} [T^{p^{h}+1}]

with a ≠ 0 then if b ∈ k is such that a = b^{1−p^{h}} and g(T) = bT then g ∘ [p]_{G} ∘ g^{−1} ≡ T^{p^{h}} [T^{p^{h}+1}]. And thus we can suppose the coefficient of T^{p^{h}} is 1. Now if for some k ≥ 2

[p]_{F} ≡ T^{p^{h}} + aT^{kp^{h}} [T^{(k+1)p^{h}}]

putting g(T) = T − bT^{k} we find

g ∘ [p]_{G} ∘ g^{−1}(T) ≡ T^{p^{h}} + (a + b^{p^{h}} − b)T^{kp^{j}} [T^{(k+1)p^{h}}]

and thus using a b solution of the Artin-Schreier equation a = b − b^{p^{h}} we can suppose

[p]_{G} ≡ T^{p^{h}} [T^{(k+1)p^{h}}]

We conclude as usual by induction and passing to the limit. Lemma 5. Let G_{1}, G_{2} be two formal group laws over R s.t.

G_{1} ≡ G_{2} + aC_{p^{i}}(X, Y) mod deg p^{i} + 1

then

[p]_{G_{1}} ≡ [p]_{G_{2}} + a(−1 + p^{p^{i}−1})T^{p^{i}} mod deg p^{i} + 1

In particular if pR = 0 then

[p]_{G_{1}} ≡ [p]_{G_{2}} − aT^{p^{i}} mod deg p^{i} + 1

Proof. By an easy induction argument one finds

∀n ∈ ℕ^{∗} [n]_{G_{1}} ≡ [n]_{G_{2}} + a ∑_{k=1}^{n−1} C_{p^{i}}(T, kT) mod deg p^{i} + 1

Moreover

∑_{k=1}^{n−1} C_{p^{i}}(T, kT) = ∑_{k=1}^{n−1} \frac{(k + 1)^{p^{i}}T^{p^{i}} − k^{p^{i}}T^{p^{i}} − T^{p^{i}}}{p}
= (−1 + p^{p^{i}−1})T^{p^{i}}

Proposition 5. Let G_{1}, G_{2} be two formal group laws over k separably closed. Then G_{1} ≃ G_{2} iff ht(G_{1}) = ht(G_{2}).

Proof. Suppose ht(G_{1}) = ht(G_{2}) = h, G_{1} and G_{2} are normalized, and we have constructed g ∈ 𝔽_{p^{h}}[[T]] s.t. g(0) = 0, g′(0) ≠ 0 and

g(X +_{G_{1}} Y) ≡ g(X) +_{G_{2}} g(Y) mod deg ≥ n

Then replacing G_{1} by g(G_{1}(g^{−1}(X) + g^{−1}(Y))) we can suppose

G_{1} ≡ G_{2} mod deg ≥ n

and thus

∃a ∈ 𝔽_{p^{h}} G_{1} ≡ G_{2} + aC_{n}(X, Y) mod deg ≥ n + 1

If n is not a power of p, using g(T) = T − aT^{n} we have g(G_{1}(g^{−1}(X), g^{−1}(Y))) ≡ G_{2} mod deg ≥ n + 1. If n is a power of p then by the preceding lemma

[p]_{G_{1}} ≡ [p]_{G_{2}} − aT^{n} [T^{n+1}]
⟹ a = 0 since [p]_{G_{1}} = [p]_{G_{2}} = T^{p^{h}}

Thus starting from a truncated at the order n isomorphism between G_{1} and G_{2} we have constructed a truncated isomorphism at the order n + 1. Moreover since g ∈ 𝔽_{p^{h}}[[T]] the new formal group laws are still normalized. By induction and a limit process one then construct an isomorphism

G_{1} ⟶^{∼} G_{2}.

Corollary 4. Any finite height h one dim. formal group law is isomorphic to a normalized one F s.t.

F (X, Y) ≡ X + Y − C_{p^{h}}(X, Y) mod deg ≥ p^{h} + 1

Proof. This can be deduced from the proof of the last proposition.

Theorem 2. The isomorphism classes of one dimensional formal group laws over a separably closed field are in bijection with ℕ^{∗} ∪ {∞}, the bijection being given by the height.

Proof. It remains to prove that for any h ∈ ℕ^{∗} there exists a height h formal group law over k. But if Λ ≃ ℤ[t_{i}]_{i≥1} is Lazard’s ring then if we define the ring morphism Λ ⟶ k by sending t_{i} ↦ 0 if i < p^{h} − 1, t_{p^{h}−1} ↦ 1 and t_{i} ↦ any element of k for other i then if G(X, Y) is the corresponding formal group law over k we have

G(X, Y) ≡ X + Y + C_{p^{h}}(X, Y) mod deg ≥ p^{h} + 1

(we refer to remark 5) which implies as already seen

[p]_{G} ≡ −T^{p^{h}} [T^{p^{h}+1}]

Remark 8. The preceding construction gives a group law of any given height over 𝔽_{p}.

1.11 Explicit formula for a group law with given height

One can prove the following, again using Cartier theory.

Proposition 6. Let h ∈ ℕ^{∗},

f(T) = ∑_{k≥0} \frac{T^{p^{kh}}}{p^{k}} ∈ ℚ[[T]]

and the group law F (X, Y) = f^{−1}(f(X)+f(Y)). Then F ∈ ℤ_{(p)}[[X, Y]] and moreover its reduction modulo p is a height h group law over 𝔽_{p}.

1.12 Endomorphism algebra

One can show that for a G of height h < +∞ over k algebraicaly closed

End(G) = O_{D}

where D is a division algebra with invariant \frac{1}{n} and O_{D} is its maximal order. This is a consequence of the classification of p-divisible groups via crystals. Concretely

O_{D} = ℤ_{p^{n}}[Π]

where ℤ_{p^{n}} = W (𝔽_{p^{n}}) is the ring of integers in the degree n unramified extension of ℚ_{p} and Π is the uniformizing element in D verifying

∀x ∈ ℤ_{p^{n}} Πx = x^{σ}Π

where σ stands for the Frobenius of ℚ_{p^{n}}.

It results from the proof of theorem 2 that there exists one dim. formal group law of any height defined over 𝔽_{p}. Then such a formal group law G admits the Frobenius as an endomorphism. In the identification End(G) = O_{D} the Frobenius is sent to a uniformizing element of O_{D} and thus can be identified with Π.

For the inclusion ℤ_{p^{n}} ⊂ End(G) one can use proposition 6. In fact with the given formula in this proposition for the logarithm f(T) one verifies that if ζ ∈ μ_{p^{n}−1} is a root of unity then

f(ζT) = ζf(T)

and thus the action of ℤ_{p^{n}} over 𝔾̂_{a} ⊗ ℤ_{p^{n}} induces an action of ℤ_{p^{n}} on the group law F ⊗ ℤ_{p^{n}}. By reducing modulo p this gives an action of ℤ_{p^{n}} on our group law after extending the scalars to 𝔽_{p^{n}}.

2 Deformation theory in the unobstructed case

In this section we present a particular case of Schlessinger’s deformation theorem ([?]).

2.0.1 Definitions

Let k be a field and C be the category of local Artin rings with residue field k, morphisms being morphisms inducing the identity on k. Let

F : C ⟶ { Sets }

be a covariant functor. We will make the assumption F (k) = {∅} that is to say F is “connected”.

Remark 9. Suppose k is perfect. Let W (k) be the Witt vectors of k. Then for any A ∈ C there is a unique lift

\xymatrix{A \ar@{->>}[rr] &  & k \\  & W (k) \ar@{->>}[ur] \ar@{-->}[ul]}

Thus any A ∈ C is canonicaly a W (k)-algebra. In particular any functor F on C is equiped with a structural morphism F ⟶ Spf(W (k)) defined by sending any element of F (A) to this canonical morphism in Spf(W (k))(A) = Hom(W (k), A).

Definition 11. The functor F is formally smooth if for all A, B ∈ C and A ↠ B a surjective morphism then F (A) ⟶ F (B) is surjective.

Of course, as usual, in the preceding definition one can assume the kernel I of A ↠ B verifies I^{2} = (0) and even 𝔪.I = (0) where 𝔪 is the maximal ideal of A. This is a consequence of the fact that if A ↠ B is a surjective morphism in C then there is a sequence

A ↠ A_{1} ↠ ⋯ ↠ A_{k−1} ↠ A_{k} = B

where forall i the morphism A_{i} ↠ A_{i+1} satisfies this hypothesis “𝔪.I = (0)”.

Definition 12. The functor F satisfies the Mayer-Vietoris property if for all diagrams in C

\xymatrix{A \ar[dr] &  & B \ar[dl] \\  & C}

The natural morphism F (A ×_{C} B) ⟶ F (A) ×_{F (C)} F (B) is an isomorphism.

Example 3. Suppose k is perfect. Any formal scheme over Spf(W (k)) gives a functor on C. If F is the functor associated to a formal scheme then F satisfies the Mayer Vietoris property.

Remark 10. One has to be careful with the following. If k is perfect and 𝔛 is a formal scheme over Spf(W (k)) then let’s note h_{𝔛} for the associated functor on C. Then if k is algebraicaly closed and 𝔛_{red} its special fiber (the reduced subscheme of the reduction of 𝔛 modulo a definition ideal) is a finite type scheme one has

h_{𝔛} ≃ ∐_{x∈𝔛_{red}(k)} h𝔛̂_{/{x}}

Where 𝔛̂_{/{x}} stands for the formal completion of 𝔛 at the point x. Thus the category of test objects C is rich enough only for formal schemes 𝔛 such that dim 𝔛_{red} = 0. For other formal schemes one has to use more test schemes (rings on which p is nilpotent).

2.0.2 Tangent functor

Let’s suppose F satisfies the Mayer-Vietoris property. Let M be the category of finite dimensional k-vector spaces. There is a functor

M ⟶ C
V ⟼ k ⊕ V

where k ⊕ V is the trivial extension of k by the squared zero ideal V. One has ∀(x, y), (x′, y′) ∈ k ⊕ V (x, y).(x′, y′) = (xx′, xy′ + x′y). For example if V = k, k ⊕ V = k[ϵ] ≃ k[T]/(T^{2}) is the dual number algebra.

Definition 13. We denote T F : M ⟶ Sets to be the composite M ⟶ C ⟶ Sets.

There is a canonical factorisation

\xymatrix{M \ar[rr]^{T F} \ar[dr] &  & Sets \\  & k-vector spaces \ar[ur]}

defined in the following way : ∀V ∈ Ob M there is a morphism in M

V × V ⟶ V
(x, y) ⟼ x + y

inducing by functoriality a morphism in C

k ⊕ V ×_{k} k ⊕ V ⟶^{+} k ⊕ V

where k ⊕ V ×_{k} k ⊕ V ≃ k[X, Y]/(X^{2}, Y^{2}). In the same way for all x ∈ k there is a morphism in M

V ⟶ V
v ⟼ x.v

inducing

m_{x} : k ⊕ V ⟶ k ⊕ V

They induce

\xymatrix{T F (V) × T F (V) \ar[r]^{≃} \ar[rr] & F (k ⊕ V ×_{k} k ⊕ V) \ar[r]^{F (+)} & T F (V)}

and

∀x ∈ k T F (V) ⟶^{F (m_{x})} T F (V)

And one verifies those two operations induce a k-vector space structure on T F (V) . One verifies immediately that this vector space structure is functorial in V. Thus we have our factorization.

Remark 11. The deep reason why there exists such a factorization is the following. Let C and D be categories admiting fiber products. One can define the notion of a group, k-vector space... in such a category (for example a group in the category of schemes is a group-schemes). This consists in a set of diagrams satisfying certain properties in such cateogories (for example a morphism X ×_{S} X ⟶ X satisfying some associativity condition...). Now if F : C ⟶ D is a functor commuting with fiber products it transforms such a group, k-vector space... in C in a group, kvector space... in D. This is the case of the functor M ⟶ C that associates k ⊕ V to V. Thus if F : C ⟶ Sets satisfies the Mayer-Vietoris property the composition T F has values in k-vector spaces in the category of Sets that is to say k-vector spaces.

Lemma 6. There is a natural isomorphism of functors

T F (−) ≃ T F (k) ⊗_{k} (−)

Proof. Fix an equivalence between M and the subcategory of vector spaces (k^{n})_{n≥0}. It suffices to construct natural isomorphisms

T F (k^{n}) ≃ T F (k) ⊗_{k} k^{n}

They are deduced from the Mayer -Vietoris property : T F (k × ⋯ × k) ≃ T F (k)^{n}. The naturality is easily checked.

Proposition 7. Let A ∈ C with maximal ideal 𝔪 and let I be an ideal of A s.t. 𝔪.I = 0. Let f : F (A) ⟶ F (A/I). Then for each x ∈ F (A/I) either f^{−1}({x}) = ∅ either f^{−1}({x}) is a principal homogeneous space under T F (I).

Proof. There is an isomorphism in C

A ×_{A/I} A ⟶^{∼} A ×_{k} (k ⊕ I)
(a_{1}, a_{2}) ↦ (a_{1}, ā_{1} ⊕ (a_{1} − a_{2}))

where ā_{1} is the reduction of a_{1} modulo the maximal ideal of A. Applying the Mayer-Vietoris property we find

F (A) × T F (I) ⟶^{∼} F (A) ×_{F (A/I)} F (A)

and one verifies easily this defines an action of T F (I) on T F (A).

2.1 Formally étale morphisms

Definition 14. A morphism f : F ⟶ G of covariant functors on C is formay étale if for all A ∈ C and ideal I of A the following diagram is cartesian

\xymatrix{F (A) \ar[r] \ar[d] & G(A) \ar[d] \\ F (A/I) \ar[r] & G(A/I)}

Of course as usual, by devissage, it suffices to consider the case when I^{2} = 0 or even when 𝔪.I = (0) where 𝔪 is the maximal ideal of A.

Lemma 7. A morphism f : F ⟶ G is an isomorphism iff it is formally étale and induces a bijection F (k) ⟶ G(k).

Proof. This is clear since in a cartesian square if the bottom horizontal line is a bijection then so is the top horizontal one.

Proposition 8. Suppose F is formally smooth. Suppose F and G satisfy the Mayer Vietoris property. A morphism f : F ⟶ G is formally étale iff it induces an isomorphism T F (k) ⟶^{∼} T G(k).

Proof. If f induces an isomorphism between T F (k) and T G(k) then by lemma 6 is induces an isomorphism of the tangent functors T F (−) ⟶^{∼} T G(−). We can suppose we have a diagram

\xymatrix{F (A) \ar[r] \ar@{->>}[d] & G(A) \ar[d] \\ F (A/I) \ar[r] & G(A/I)}

where 𝔪.I = 0. The left vertical map is surjective since F is formally smooth. Then one verifies to prove this diagram is cartesian it suffices to prove the morphism F (A) ⟶ G(A) induces an isomorphism between the fibers of the vertical maps. By proposition 7 and the isomorphism T F ⟶^{∼} T G this is the case since a morphism of principal homogeneous spaces is an isomorphism.

2.1.1 Main theorem

If k is a perfect field any complete local ring with residue field k is naturally a W (k)-algebra. Thus the category C is the category of local artinian W (k)-algebras with residue field k and F can be considered as a functor over Spf(W (k)).

Theorem 3. Suppose k is perfect. Suppose F satisfies the Mayer-Vietoris property, is formally smooth verifies F (k) = {∅} and satisfies

dim_{k} T F (k) = n < +∞

Then

F ≃ Spf(W (k)[[T_{1}, …, T_{n}]])

Proof. Let (e_{1}, …, e_{n}) be a basis of T F (k). It gives an element of

F (k[T_{1}, …, T_{n}]/(T_{1}^{2}, …, T_{n}^{2})) ≃ ∏_{i=1}^{n} T F (k)

By the formal smoothness property this can be lifted to an element of

F (W (k)[[T_{1}, …, T_{n}]]) := lim_{⟵}_{k} F (W (k)[[T_{1}, …, T_{n}]]/I^{k})

where I = (p, T_{1}, …, T_{n}). This gives a morphism

f : G = Spf(W (k)[[T_{1}, …, T_{n}]]) ⟶ F

Now because of our choices

T G(k) ⟶^{∼} T F (k)

and thus by proposition 8 f is an isomorphism.

3 Lubin-Tate spaces : first approach via formal group laws

3.1 Definition

Let F be a one dimensional formal group law of finite height n over \overline{𝔽}_{p}.

Definition 15. Let A be a local artinian algebra with maximal ideal 𝔪 and residue field \overline{𝔽}_{p}. – A formal group law G over A is a deformation of G if G ≡ F modulo 𝔪

– Two deformations G_{1}, G_{2} are isomorphic if there is an isomorphism

f : G_{1} ⟶^{∼} G_{2}

such that f ≡ Id mod 𝔪 (that is ot say f(T) ≡ T). Let Def be the associated functor of isomorphism classes of deformations of F on artin local rings with residue field \overline{𝔽}_{p}.

3.2 Computation of the tangent space

We note k = \overline{𝔽}_{p}.

Lemma 8. The tangent space to Def is the k-vector space H^{1}(C^{•}) where

C^{•} = [T k[[T]] ⟶^{∂} XY k[[X, Y]]^{𝔖_{2}} ⟶^{∂} k[[X, Y, Z]]]_{0  +1  +2}

where k[[X, Y]]^{𝔖_{2}} stands for the symmetric formal power series and

∂φ(T) = φ(X +_{F} Y) − φ(X) − φ(Y)
∂ψ(X, Y) = ψ(Y, Z) − ψ(X +_{F} Y, Z) + ψ(X, Y +_{F} Z) − ψ(X, Y)

Proof. Let h(T)\frac{d}{dT} be a base of Lie F. Take for example h(T) = \frac{∂F}{∂X} (0, T). The series h(T) lies in \overline{𝔽}_{p}[[T]]^{×}. Writte F′(X, Y) = F (X, Y) + ϵψ(X, Y).h(F (X, Y)) with ϵ^{2} = 0. Then

∂ψ = 0 ⟺ F′(F′(X, Y), Z) = F′(X, F′(Y, Z))

And if h(T) = T + ϵφ(T).h(T),

F′ = h(F (h^{−1}(X), h^{−1}(Y))) ⟺ ψ = ∂φ

Suppose now F satisfies

F (X, Y) ≡ X + Y + C_{p^{n}}(X, Y) mod deg ≥ p^{n} + 1

We will suppose more explicitly that F is associated to the morphism from Lazard’s ring ℤ[t_{i}]_{i≥1} to k defined by t_{i} ↦ 0 if i < p^{n} − 1, t_{p^{n}−1} ↦ 1 and for other i t_{i} maps to anything.

Proposition 9. The tangent space to Def is n − 1 dimensional generated by cocyles ψ_{1}, …, ψ_{n−1} that verify ψ_{i}(X, Y) ≡ C_{p^{i}}(X, Y) mod deg ≥ p^{i} + 1.

Proof. We have

∂T^{k} ≡ (X + Y)^{k} − X^{k} − Y^{k} mod deg ≥ k + 1
∂T^{p^{i}} ≡ C_{p^{i+n}}(X, Y) mod deg ≥ p^{in} + 1

Moreover if a cocyle ψ verifies

ψ ≡ 0 mod deg ≥ k

then thanks to lemma 2

ψ ≡ aC_{k}(X, Y) mod deg ≥ k + 1

for some a ∈ \overline{𝔽}_{p}. With this one can prove that if k ≥ p^{h} and ψ is a cocyle that is a coboundary mod deg≥ k then ψ is a couboundary mod deg≥ k + 1.

Now for each 1 ≤ i ≤ n − 1 let ψ_{i} consider the map from Lazard’s ring ℤ[t_{j}]_{j≥1} to k[ϵ] defined by t_{j} ↦ 0 if j < p^{i} −1, t_{p^{i}−1} ↦ ϵ, t_{j} ↦ 0 if p^{i} −1 < j < p^{n} −1, t_{p^{n}−1} ↦ 1 and t_{j} maps to anything for other j (the same anything as the one we took for defining F). This defines a deformation of F and thus a cocyle ψ_{i}. It follows by computing modulo T^{p^{n}} and the formulas given for ∂T^{k} with k not a power of p that those cocyles form a basis of the tangent space to Def.

3.3 Representability

Proposition 10. There is an isomorphism

Def ≃ Spf(W (k)[[T_{1}, …, T_{n−1}]])

Proof. We apply theorem 3. The functor Def clearly satisfies Mayer-Vietoris property. Formal smoothness follows from corollary 3 and the assertion about the tangent space is proposition 9.

3.4 Good coordinates on Lubin-Tate spaces

Proposition 11. There is a system of formal coordinates (x_{1}, …, x_{n−1}) on Def that is to say an isomorphism Def ≃ Spf((W (k)[[x_{1}, …, x_{n−1}]]) such that if G is the associated universal formal group law then

∀1 ≤ i ≤ n − 1 G(X, Y) ≡ x_{i}C_{p^{i}}(X, Y) mod (deg p^{i} + 1, x_{1}, …, x_{i−1})

Proof. Let G be the group law over W (k)[[x_{1}, …, x_{n−1}]] associated to the map from Lazard’s ring

Λ = ℤ[t_{k}]_{k≥1}

to W (k)[[x_{1}, …, x_{n−1}]] that maps t_{p^{i}−1} to x_{i} for 1 ≤ i ≤ n − 1, t_{p^{n}−1} to 1 and other t_{i} to 0. It gives a morphism from Spf(W (k)[[x_{1}, …, x_{n−1}]]) to Def. To see this is an isomorphism we have to prove it is one at the level of the tangent spaces, that is to say look at the reduction of G to k[x_{1}, …, x_{n−1}]/(x_{1}^{2}, …, x_{n−1}^{2}). But this is clear from the computation of the tangent space of Def. Now the remaining assertion follows from remark 5.

Corollary 5. There is a system of formal coordinates (x_{1}, …, x_{n−1}) on Def such that if G is the associated universal formal group law then

[p]_{G} = pu_{0}T + x_{1}u_{1}T^{p} + ⋯ + x_{n−1}u_{n−1}T^{p^{n−1}} + u_{n}T^{p^{n}}

where u_{0}, …, u_{n} ∈ W (k)[[x_{1}, …, x_{n−1}]][[T]]^{×} are units.

Proof. This follows from the last proposition coupled with lemma 5.

Remark 12. In the following sections we will explain the link with the Newton stratification and this type of coordinates.

3.5 The action of O_{D}^{×}

Let F and Def be as in the preceding section. Let O_{D} = End(F) and O_{D}^{×} = Aut(F).

Definition 16. For g ∈ Aut(F), g ∈ \overline{𝔽}_{p}[[T]], we define the following automorphism of Def : for each artinian algebra A and deformation G associated to an isomorphism class of deformation ∈ Def(A) let g̃ ∈ A[[T]] be a lift of g and consider the isomorphism class of the deformation g(G(g^{−1}(X), g^{−1}(Y))). This defines a morphism O_{D}^{×} ⟶ Aut(Def).

In the preceding definition one has to verify the isomorphism class of g(G(g^{−1}(X), g^{−1}(Y))) is independent of the choice of the lift of g, but this is easy to verify.

3.6 Disgression about the universal group law on Lubin-Tate spaces

Once more, using Cartier theory one can prove the following

Theorem 4. Let f(T) = ∑_{i≥0} a_{i}T^{p^{i}} ∈ (ℚ[x_{1}, …, x_{n−1}])[[T]] be the unique solution of the following functional equation

f(T) = T + ∑_{k=1}^{n−1} \frac{f(x_{i}T^{p^{i}})}{p} + \frac{f(T^{p^{n}})}{p}

then F (X, Y) = f^{−1}(f(X) + f(Y)) lies in ℤ_{(p)}[[x_{1}, …, x_{n−1}]][[T]] and is a universal group law over Lubin-Tate space.

4 From one-dimensional formal group laws to p-divisible groups

Until now we’ve been working with formal group laws that are rather concrete objects. To go further in the study of Lubin-Tate spaces, we will switch to a more abstract but more powerful object that are p-divisbile groups. Those objects are better suited for the study of higher dimensional formal groups too. We begin with explaining the bridge between one dimensional group laws and p-divisible groups. We refer to the appendix ? ? ? for facts about finite flat group schemes.

Recall the following form of Weierstrass division/factorization :

Lemma 9. Let A be a complete local ring with maximal ideal 𝔪 and f = ∑_{k≥0} a_{k}T^{k} ∈ A[[T]] s.t f mod 𝔪 ≠ 0. Let n ∈ ℕ be such that a_{n} ∈ A^{×} and ∀k < n a_{k} ∈ 𝔪. Then for each g ∈ A[[T]] ∃!Q, R with Q ∈ A[[T]], R ∈ A[T], deg R < n and

g = Qf + R

Moerover ∃g ∈ A[T] a degre n unitary polynomial and h ∈ A[[T]]^{×} s.t.

f = gh

Proof. The division assertion is an easy approximation argument, the case when A is a field being evident.

The factorization assertion is, as usual, deduced from the division one by applying it to T^{n}, writting T^{n} = Qf + R and constating Q ∈ A[[T]]^{×}.

Thus for such an f there is an isomorphism of A-algebras

A[[T]]/(f(T)) ≃ A[T]/(g(T))

that is a finite free A-module.

Lemma 10. Let A be a complete local ring with a car. p residue field k. Let F be a one-dimensional formal group law over A s.t. F ⊗_{A} k has finite height h. Then for each k ∈ ℕ^{∗}

F [p^{k}] = Spec(A[[T]]/([p^{k}]_{F} (T)))

is a (commutative) finite localy free Spec(A)-group scheme of order p^{kh}.

Proof. Let B = A[[T]]/([p^{k}]_{F} (T)) equiped with the T -adic topology. The formal scheme Spf(B) with (T) being a definition ideal, has a structure of formal group scheme given by the formal group law F (X, Y). In fact, [p^{k}]_{F} (F (X, Y)) = F ([p^{k}]_{F} (X), [p^{k}]_{F}) and thus the composed morphism of A-algebras

A[[T]] ⟶ A[[X, Y]] ↠ A[[X, Y]]/([p^{k}]_{F} (X), [p^{k}]_{F} (Y)) = B ˆ⊗_{A}B
T ⟼ F (X, Y)

verfies [p^{k}]_{F} (F (X, Y)) ∈ ([p^{k}]_{F} (X), [p^{k}]_{F} (Y)) in A[[X, Y]] and thus factorises as a morphism

B ⟶ B ˆ⊗_{A}B

Now, from the preceding lemma and the definition of height one deduces B is a finite A-module. From this one deduces easily that B ˆ⊗_{A}B = B ⊗_{A} B. In fact if [p^{k}]_{F} = gh as in the preceding lemma with g a unitary polynomial and h a unit then B ≃ A[T]/(g) and B ˆ⊗_{A}B = B ⊗_{A} B = A[X, Y]/(g(X), g(Y)).

Remark 13. One has to be careful with the following : in the last lemma, for R an A-algebra in general

F [p^{k}] ⊗_{A} R ≠ Spec(R[[T]]/([p^{k}]_{F ⊗_{A}R}(T)))

For example if A is an unequal caracteristic discrete valuation ring then F [p^{k}] ⊗_{A} A[\frac{1}{p}] is étale allthough A[\frac{1}{p}][[T]]/([p^{k}]_{F} (T)) is T -adic. Spf(A[\frac{1}{p}][[T]]/([p^{k}]_{F} (T))) is the formal completion of F [p^{k}]⊗_{A} A[\frac{1}{p}] along its zero section and is thus equal to Spec(A[\frac{1}{p}]). This phenomenon will become clearer in a few moment when we will study the Newton stratfication.

By definition, when we will say a sequence of group schemes is exact this will mean exact as a sheaf sequence on the fppf site.

Lemma 11. With the hypothesis of the last lemma for each 1 ≤ l ≤ k the sequence of finite flat group schemes

0 ⟶ F [p^{l}] ⟶ F [p^{k}] ⟶^{p^{l}} F [p^{k−l}] ⟶ 0

is exact.

Proof. The sequence

0 ⟶ F [p^{l}] ⟶ F [p^{k}] ⟶^{p^{l}} F [p^{k−l}]

is easily seen to be exact (even on the big Zariski site). By couting ranks one concludes ( ? ? ?).

Lemma 12. Suppose moreover A is artinian. Consider the formal scheme Spf(A[[T]]) with (T) being a definition ideal. It is a formal group scheme, the group law being given by F (X, Y), and thus an ind-group scheme. Then the natural morphism of ind-group schemes

lim_{⟶}_{k} F [p^{k}] ⟶ Spf(A[[T]])

is an isomorphism where F [p^{k}] is a radicial group scheme over Spec(A).

Proof. Let’s first prove F [p^{k}] is radicial. Since A is artinian and the A-module A[[T]]/([p^{k}]_{F}) is of finite type it is of finite length. Thus if I = T.A[[T]]/([p^{k}]_{F}) is the augmentation ideal of A[[T]]/([p^{k}]_{F}), the sequence (I^{k})_{k≥0} is stationary for k >> 0, but A[[T]]/([p^{k}]_{F}) being I-adicaly complete one deduces the augmentation ideal I is nilpotent and thus the morphism F [p^{k}] ⟶ Spec(A) is radicial.

This proves moreover that for each k ∈ ℕ there exists l ∈ ℕ such that T^{l} ∈ [p^{k}]_{F} (T).A[[T]]. Now to finish to verify that our morphism of ind-group schemes is an isomorphism we have to

prove that for each k ∈ ℕ there exists l ∈ ℕ such that [p^{l}]_{F} (T) ∈ T^{k}.A[[T]]. But

[p]_{F} ∈ T^{p^{h}}A[[T]] + T 𝔪[[T]]

but

f(T) ∈ T^{p^{h}}A[[T]] + T 𝔪[[T]] ⟹ f^{∘k}(T) ∈ T^{p^{h}}A[[T]] + T 𝔪^{k}[[T]]

(compute in A[T]/(T^{p^{h}})).

Thus, when A is artinian the formal group scheme Spf(A[[T]]) and the inductive system of finite flat group schemes (F [p^{k}])_{k} determine each other.

5 p-divisible groups : basics

5.1 p-divisible groups : definition

Let’s fix a base scheme S. For an abelian scheaf G we note G[p^{k}] the kernel of the multiplication by p^{k}.

A p-divisible group is a particular type of ind-group scheme.

Definition 17 (Grothendieck). A p-divisible group over S is an abelian fppf sheaf G over S such that :

– p : G ⟶ G is an epimorphism

– The natural monomorphism lim_{⟶}_{k} G[p^{k}] ⟶ G is an isomorphism, that is to say G is p^{∞}-

torsion

– For each k ∈ ℕ^{∗} G[p^{k}] is representable by a finite localy free group scheme The height of G[p] (as a localy constant function on S) is called the height of G.

Of course, as usual for sheaves, when we say G is p^{∞}-torsion this means that any section is localy killed by a power of p (here it is sufficient to restrict one-self to Zariski localy, and thus globaly on a quasicompact scheme).

Lemma 13 (Equivalent definition). A p-divisible group over S “is the same” as an inductive system of finite localy free group schemes (G_{k})_{k≥1} s.t. for each k there is an exact sequence

0 ⟶ G_{1} ⟶ G_{k} ⟶^{×p} G_{k−1} ⟶ 0

where the map G_{k} ⟶^{×p} G_{k−1} ↪ G_{k} is multiplication by p on G_{k}.

Proof. To (G_{k})_{k≥1} one associates G = lim_{⟶}_{k} G_{k} and to G one associates the system (G[p^{k}])_{k≥1}.

Lemma 14 (Equivalent definition not using fppf topology). A p-divisible group is an inductive system of finite flat group schemes (G_{k})_{k≥1} such that for all k G_{k} is killed by p^{k}, G_{k} = G_{k+1}[p^{k}] and if h is the heigth of G_{1} then the heigth of G_{k} is kh.

Proof. This is a consequence of the fact that in the exact sequence

0 ⟶ G_{1} ⟶ G_{k} ⟶^{×p} G_{k−1}

the rank of the middle term is the product of the rank of other terms and thus ? ? ?

0 ⟶ G_{1} ⟶ G_{k} ⟶^{×p} G_{k−1} ⟶ 0

is exact.

The preceding definition is the original one given by Tate.

Example 4. – ℚ_{p}/ℤ_{p} or more generaly (ℚ_{p}/ℤ_{p})^{h} – μ_{p^{∞}} which is the inductive system formed by the μ_{p^{k}} for k ∈ ℕ^{∗}.

– If A is an abelian scheme over S then A[p^{∞}] is a p-divisible group over S

– It follows from last section that if F is a one dimensional formal group law over a complete local ring R with residue field of car. p then the (F [p^{k}])_{k≥1} form a p-divisible group over Spec(R)

Remark 14. The category of p-divisible groups is ℤ_{p}-linear and pseudo-abelian (any idempotent has a kernel and cokernel). This allows one to construct more p-divisible groups from abelian varities (the category of ab. var. being only ℤ-linear) in the following way : take A an abelian variety with an action of an order O_{D} in a division algebra D over a number field and suppose D ⊗ ℚ_{p} is split and O_{D} ⊗ ℤ_{p} is a maximal order. Then A[p^{∞}] has an action of O_{D} ⊗ ℤ_{p} that is a product of matrix algebras over the ring of integers of finite extensions of ℚ_{p}. Then one can apply idempotents in those matrix algebras as in Morita equivalence and obtain new p-divisible groups.

5.2 Example : the étale case

Definition 18. A p-divisible group G is étale if G[p] is étale, or equivalently if the G[p^{k}] for all k are étale.

The equivalence results from the short exact sequences

O ⟶ G[p] ⟶ G[p^{k}] ⟶^{p} G[p^{k−1}] ⟶ 0

the fact that an extension of an étale group by an étale one is still étale and induction.

Example 5. If p is invertible over S then any p-divisible group on S is étale.

Example 6. For any h ∈ ℕ^{∗} (ℚ_{p}/ℤ_{p})^{h} is an étale p-divisible group.

Recall a smooth p-adic sheaf on S_{ét} is a projective system (F_{k})_{k≥1} where F_{k} is localy free ℤ/p^{k}ℤ-sheaf on S_{ét} and F_{k+1} ⊗ ℤ/p^{k}ℤ ⟶^{∼} F_{k} is an isomorphism.

Lemma 15. The category of étale p-divisible groups over S is equivalent to the category of smooth ℤ_{p}-adic sheaves over S_{ét}.

Proof. In one direction the equivalence is given by G ⟼the projective system given by the G[p^{k}] and with transition mapping G[p^{k+1}] ⟶^{p} G[p^{k}]. This projective system can be non-rigorously written ” lim_{⟵}_{k} G[p^{k}]”.

In the other direction the equivalence is “F ⟼ F ⊗ℚ_{p}/ℤ_{p}” which means if F = (F_{k})_{k≥1} then

one associates the inductive system (F_{k})_{k≥1} with transition mappings F_{k} = F_{k+1} ⊗ ℤ/p^{k}ℤ ⟶^{Id⊗p} F_{k+1} where p : ℤ/p^{k}ℤ ⟶ ℤ/p^{k+1}ℤ.

Let K be a separably closed field. Suppose S is connected and let ξ ∈ S(K) be a geometric point. Set Γ = π_{1}(S, ξ). Recall there is an equivalence between smooth p-adic sheaves on S_{ét} and continouous representations of Γ in finite type free ℤ_{p}-modules. This equivalence is given by

(F_{k})_{k≥1} ⟼ lim_{⟵}_{k} F_{k,ξ}

Corollary 6. There is an category equivalence between étale p-divisible groups over S and continouous representations of Γ in finite type free ℤ_{p}-modules. The rank of the ℤ_{p}-module equals the height of the p-divisible group.

Example 7. The ℤ_{p}-modules that are free of finite type and have a trivial action of Γ correspond to p-divisible groups isomorphic to (ℚ_{p}/ℤ_{p})^{h}.

Example 8. Suppose p is invertible on the base S. Then the p-divisible group μ_{p^{∞}} is étale. The corresponding one dimensional representation of Γ is the cyclotomic character ℤ_{p}(1).

Definition 19. The smooth ℤ_{p}[Γ]-module attached to an étale p-divisible group is called its Tate module and is denoted T_{p}(G).

The notation T_{p}(G) is ambigous because there no ξ in it. Concretely

T_{p}(G) = lim_{⟵}_{k} G[p^{k}](K)
= Hom(ℚ_{p}/ℤ_{p}, G ×_{S} Spec(K))

5.3 Cartier duality

If G is a p-divisible group then one defines its Cartier dual G^{D} by setting G^{D}[p^{k}] = G[p^{k}]^{D} and the inclusion G^{D}[p^{k}] ↪ G^{D}[p^{k+1}] is the Cartier dual of multiplication by p. This is a p-divisible group since cartier duality transforms the exact sequence

0 ⟶ G[p] ⟶ G[p^{k+1}] ⟶^{p} G[p^{k}] ⟶ 0

in

0 ⟶ G[p^{k}]^{D} ⟶ G[p^{k+1}]^{D} ⟶ G[p]^{D} ⟶ 0

Example 9. In the same way we defined an étale p-divisible group one can define a p-divisible group of multiplicative type by saying G[p] is of multiplicative type that is to say localy isomorphic to some μ_{p}^{h}. Then p-divisible groups of multiplicative type are Cartier duals to the étale one.

Take the notations of the last section on fundamental groups. Tori T over S split after a finite étale base change are equivalent to discrete representations of Γ = π_{1}(S, ξ) on finite type free ℤmodules. This equivalence is given by T ↦ X^{∗}(T_{ξ}). For such a T T [p^{∞}] is of multiplicative type and

T_{p}(T [p^{∞}]^{D}) = X^{∗}(T) ⊗ ℤ_{p}

Example 10. If A is an S-abelian scheme and A^{∨} = \underline{Ext}^{1}(A, 𝔾_{m}) is the dual abelian scheme then

A^{∨}[p^{∞}] = A[p^{∞}]^{D}

In fact from the exact sequence

0 ⟶ A[p^{k}] ⟶ A ⟶^{p^{k}} A ⟶ 0

one deduces by applying \underline{Hom}(−, 𝔾_{m}) the exact sequence

Hom(A, 𝔾_{m}) = 0 ⟶ A[p^{k}]^{D} ⟶ A^{∨} ⟶^{p^{k}} A^{∨} ⟶ 0

If λ : A ⟶ A^{∨} is a polarization it is symetric, λ^{∨} = λ, and induces a map λ : A[p^{∞}] ⟶ A[p^{∞}]^{D} that verifies λ^{∨} = −λ (when you turn triangles in derived categories you have to change the signs !) which expresses the antisymetry of Weil pairing.

5.4 Isogenies

Definition 20. An isogeny between two p-divisible groups G_{1} and G_{2} is an epimorphism f : G_{1} ⟶ G_{2} such that ker f is represented by a finite localy free group scheme. The height of ker f is called the height of the isogeny.

For example, p : G ⟶ G is an isogeny.

In fact one has the following :

Lemma 16. Let G be a p-divisible group over S and H a subgroup that is finite localy free over S (thus H ⊂ G[p^{n}] for n >> 0). Then G/H is a p-divisible group and the morphism G ↠ G/H is an isogeny.

Proof. The only non-trivial thing to prove is that foral all n the (G/H)[p^{n}] are finite localy free group schemes. But if N is such that H ⊂ G[p^{N}] then if H′ makes the following square cartesian

\xymatrix{H′ \ar[r] \ar@{^{(}->}[d] & H \ar@{^{(}->}[d] \\ G[p^{N+n}] \ar[r]^{p^{n}} & G[p^{N}]}

then H′ is finite localy free over S because the bottom horizontal morphism is finite localy free since it is an epimorphism of finite localy free group schemes and thus the top horizontal morphism is finite localy free which implies H′ is finite localy free over H. Then (G/H)[p^{n}] = H′/H.

Thus, up to isomorphisms, isogenies are given by finite localy free sub-group schemes of the G[p^{n}], n ≥ 1.

If f : A_{1} ⟶ A_{2} is an isogeny of abelian schemes then the induced morphisms A_{1}[p^{∞}] ⟶ A_{2}[p^{∞}] is an isogeny. And this induces a bijection between isogenies from

Remark 15. For any two p-divisible groups G_{1}, G_{2} the group Hom(G_{1}, G_{2}) is a ℤ_{p}-module without p-torsion.

Lemma 17. Suppose S is connected or quasicompact then a morphism bewteen p-divisible groups f : G_{1} ⟶ G_{2} is an isogeny iff there exists a morphism g : G_{2} ⟶ G_{1} and an integer N such that g ∘ f = p^{N}Id_{G_{2}} and f ∘ g = p^{N}Id_{G_{1}}.

Proof. Suppose f is an isogeny. Then let N ∈ ℕ be such that ker f ⊂ G_{1}[p^{N}]. Then, f being an epimorphism f induces an isomorphism

G_{1}/ ker f ⟶^{∼} G_{2}

But the morphism p^{N} : G_{1}/ ker f ⟶ G_{1}/ ker f lifts to a morphism

\xymatrix{G_{1}/ ker f \ar[r]^{p^{N}} \ar@{-->}[dr]^{p^{N}} & G_{1}/ ker f \\  & G_{1} \ar[u]}

And thus there is a composed morphism g : G_{2} ≃ G_{1}/ ker f ⟶^{p^{N}} G_{1} that verifies g ∘ f = p^{N}. This implies (f ∘ g) ∘ f = (p^{N}Id) ∘ f, but f being an epimorphism this implies f ∘ g = p^{N}.

Reciprocaly, let f and g verifie f ∘ g = p^{N} and g ∘ f = p^{N}. Then f is an epimorphism since p^{N} is. Moreover ker f ⊂ G_{1}[p^{N}]. Then the morphisms

\xymatrix{G_{1}[p^{N}] \ar[r]^{f_{|G_{1}[p^{N}]}} & G_{2}[p^{N}] \ar[l]^{g_{|G_{1}[p^{N}]}}}

verifie

ker f_{|G_{1}[p^{N}]} = Im g_{|G_{2}[p^{N}]}

In fact the inclusion Im g_{|G_{2}[p^{N}]} ⊂ ker f_{|G_{1}[p^{N}]} is immediate and if x ∈ G_{1}[p^{N}] ∩ ker f fppf localy x = p^{N}y = g(f(y)) for somme y. But p^{N}f(y) = f(g(f(y))) = f(x) = 0. Thus f(y) ∈ G_{1}[p^{N}]. Now ker f = ker f_{|G_{1}[p^{N}]} is representable by a finite S-group scheme of finite presentation. Moreover there is an epimorphism g_{|G_{2}[p^{N}]} : G_{2}[p^{N}] ⟶ ker f. This implies ( ? ? ?) that ker f is finite flat over S.

Corollary 7. There is an isogeny G_{1} ⟶ G_{2} iff there is one G_{2} ⟶ G_{1}. The relation to be isogenous is reflexive and defines an equivalence relation on the set of isomorphism classes of p-divisible groups over S.

Proof. Apply the preceding lemma to verfiy reflexivity and that the composition of two isogenies is an isogeny.

Example 11. If G is an étale p-divisible group then all isogenous p-divisible groups are étale. A morphism f : G_{1} ⟶ G_{2} between étale p-divisible groups is an isogeny iff the induced morphism f_{∗} : T_{p}(G_{1}) ⟶ T_{p}(G_{2}) is injective and makes T_{p}(G_{1}) a lattice inside T_{p}(G_{2}) that is to say the cokernel of f_{∗} is torsion.

If G is étale the isomorphism classes of couples (G′, f) where f : G ⟶ G′ is an isogeny are in bijection with Γ-stable lattices in T_{p}(G)[\frac{1}{p}] containing the lattice T_{p}(G) (where Γ is the fundamental group). The kernel of the isogeny f : G ⟶ G′ is the finite étale group-scheme associated to the torsion Γ-module T_{p}(G′)/f_{∗}T_{p}(G).

6 Formal Lie groups

6.1 Definition

Definition 21. A (commutative) formal Lie group over S is a formal group scheme G over S s.t. localy on S there is an isomorphism of pointed S-formal schemes G ≃ Spf(O_{S}[[T_{1}, …, T_{d}]]) for some d, where G is pointed by its unit section and Spf(O_{S}[[T_{1}, …, T_{d}]]) by the section given by ∀i T_{i} ↦ 0.

In particular G is an affine formal scheme over S.

Example 12. Let G be a smooth group scheme over S. Then Ĝ, its formal completion along its zero section, is a formal Lie group. This is for example the case if G is an abelian scheme, a torus or a vector bundle.

If G is a formal Lie group on S we denote ω_{G} the localy free O_{S}-module that is the conormal sheaf of the unit section or if you want e^{∗}Ω_{G/S}^{1} with e the unit section. Then if p : G ⟶ S

Ω_{G/S}^{1} ≃ p^{∗}ω_{G}

which expresses the fact that the cotangent bundle is trivial, as it-is on any Lie group. Then

p^{−1}ω_{G} ⊂ p^{∗}ω_{G} = Ω_{G/S}^{1}

is the sub-sheaf of translation invariant differential forms that is to say the sub-scheaf of Ω_{G/S}^{1} of section α such that

m^{∗}α = pr_{1}^{∗}α + pr_{2}^{∗}α

whrere m : G×_{S} G ⟶ G is the multiplication and pr_{1}, pr_{2} : G×_{S} G ⟶ G are the two projections. We note Lie G = ω_{G}^{∗}. If G = Spf(A) with augmentation ideal I there is a natural isomorphism of augmented O_{S}-

algebras

Ŝym_{O_{S}}ω_{G} ⟶^{∼} Gr_{I}A

6.2 Connection with formal group laws

The connection with one-dimensional formal Lie groups is given by the following :

Lemma 18. There is a category equivalence between one dimensional formal group laws over R and the category whose objects are couples (G, r) where G is a one dimensional formal Lie group,

r : R[[T]] ⟶^{∼} Γ(Spec(R), O_{G})

is an isomorphism of augmented R-algebras and morphisms are Hom((G_{1}, r_{1}), (G_{2}, r_{2})) = Hom(G_{1}, G_{2}).

Thus, via the functor (G, r) ↦ G, the category of formal group laws is a fibered groupoïd over the category of formal Lie groups. The fiber category over a formal Lie group G is a groupoïd attached to the group of pointed automorphisms of the R-formal scheme Spf(R[[T]]).

6.3 Formal Lie groups as fppf-sheaves

We view S as a formal scheme, (0) being a definition ideal. Definition 22. Let 𝔛 be an S-formal scheme. For U an S-scheme we set 𝔛(U) = Hom_{S}(U, 𝔛) where U is viewed as a formal scheme with definition ideal (0).

This defines an fppf sheaf on S still denoted 𝔛. If 𝔛 = lim_{⟶}_{k} X_{k} is a presentation of 𝔛 as an-ind-scheme then the sheaf 𝔛 is the inductive limit of the sheaves associated to the X_{k}.

More clearly : if I is a definition ideal of 𝔛 and U is quasicompact 𝔛(U) = lim_{⟶}_{k} Hom_{S}(U, Spec(O_{𝔛}/I^{k}))

If G is a formal group-scheme, for example a formal Lie group, we wtill note G for the associated fppf sheaf.

6.4 Formal completion of an abelian sheaf along its zero section

Definition 23. A closed immersion T ↪ U is called a nil-immersion of order ≤ k if the ideal I defining it verfies I^{k+1} = (0). Definition 24. Let F be an abelian fppf sheaf on S. For all k ∈ ℕ^{∗} we note Inf^{k}F for the set-sheaf associated to the sub-presheaf of F

U ⟶ {s ∈ F(U) | ∃nil-immersion T ↪ U of order ≤ k s.t. s_{|T} = 0}

Remark 16. The given presheaf being a sub-presheaf of F it is separated. Thus the associated sheaf if easy to explicit. Example 13. If G is an S-group scheme with augmentation ideal I then Inf^{k}G = Spec(O_{G}/I^{k+1}).

For all k Inf^{k}F is a subsheaf of F. One verifies Inf^{k}F + Inf^{k′}F ⊂ inf^{k+k′+1}F. Definition 25. We note F̂ = lim_{⟶}_{k} Inf^{k}F as an abelian sub-sheab of F. It is called the formal completion of F along its zero section. Example 14. If G is a formal group scheme then G = Ĝ. Remark 17. The quotient F/F̂ is formaly net and F̂ is the smallest abelian sub-sheaf of F such that F/F̂ is formaly net. Thus we have written canonicaly F as an extension of a formally net sheaf by an “infinitesimal” one.

6.5 A caracterisation of formal Lie groups in caracteristic p

Let S be a caracteristic p scheme that is to say s.t. p.O_{S} = 0. Definition 26. For a sheaf F on S_{fppf} we note

F^{(p)} = F rob^{∗}F

where F rob : S ⟶ S is the Frobenius morphism. We note

F : F ⟶ F^{(p)}

for the morphism that associates to any U over S the morphism

F(U) ⟶ F(U′)
s ⟼ F rob^{∗}s

where if U ⟶^{p} S then U′ = U is the S-scheme defined by q = F rob ∘ p and

\xymatrix{U′ \ar[rr]^{F rob} \ar[dr]_{q} &  & U \ar[dl]^{p} \\  & S}

Definition 27. An fppf abelian sheaf G on S is said to be F -divisible if it satisfies the analogous conditions to the one satisfied by a p-divisible group but with p replaced by F :

– F : G ⟶ G^{(p)} is an epimorphism

– If ∀n G(n) = ker F^{n} then G is F^{∞}-torsion : the natural morphism lim_{⟶}_{n} G(n) ⟶ G is an

isomorphism

– For all n G(n) is represented by a finite localy free S-group scheme

We now note for F a sheaf F(n) for the kernel of F^{n} : F ⟶ F^{(p^{n})}.

Theorem 5 (Messing). An fppf abelian sheaf G is F -divisible iff it is represented by a formal Lie group over S.

Proof. Of course one direction is easy : if G is a formal Lie group it easily satisfies the given conditions. In fact it sufficies to verify the following for the formal affine space 𝔸̂^{d} = Spf(R[[R_{1}, …, T_{d}]]) :

– We have (𝔸̂^{d})^{(p)} = 𝔸̂^{p} and the morphism F : 𝔸̂^{d} ⟶ 𝔸̂^{d} is given by F^{∗}T_{i} = T_{i}^{p}

– F : 𝔸̂^{d} ⟶ 𝔸̂^{d} is faithfully flat of finite presentation thus an fppf epimorphism

– Forall n the reciprocial image by F of the zero section in 𝔸̂^{d} is Spec(R[T_{1}, …, T_{d}](T_{1}^{p}, …, T_{d}^{p})) and is thus a finite localy free scheme

– Any section of 𝔸̂^{d} over a quasicompact scheme is clearly killed by a power of F The other direction is based one the following proposition :

Proposition 12. Let G be a finite localy free S-group scheme killed by F^{n}. Then the following are equivalent :

– Localy on S G ≃ Spec(O_{S}[T_{1}, …, T_{d}]/(T_{1}^{p^{n}}, …, T_{d}^{p^{n}}))

– For all 0 ≤ i ≤ n the morphism F^{i} : G ⟶ G(n − i)^{(p^{i})} is an epimorphism

Proof. In the second condition we can replace epimorphism by flat since G is flat over S and F^{i} is always surjective (use ? ? ?).

Now it is easy to verify that the first condition implies the second one.

In the other direction, if the second condition is verified let’s first check the first condition when S is the spectrum of a perfect field k. In this case one knows

G ≃ Spec(k[T_{1}, …, T_{d}]/(T_{1}^{p^{a_{1}}} , …, T_{d}^{p^{a_{d}}}))

for some a_{i} ∈ ℕ. But then it is an easy commutative algebra computation to check that if for all 0 ≤ i ≤ n F^{i} : G ⟶ G(n − i)^{(p^{i})} is flat then a_{1} = ⋯ = a_{d} = n.

Now for S general, to the local choice of a lifting to I of a generating system of ω_{G} = I/I^{2} is associated a morphism localy on S

O_{S}[T_{1}, …, T_{d}]/(T_{1}^{p^{n}}, …, T_{d}^{p^{n}}) ⟶ O_{G}

(use the fact that G is killed by F^{n}) that is an isomorphism on geometric fibers of S by the field case. This is thus an isomorphism since both sides are finite localy free over O_{S}.

The proof that if G is F -divisible then it is a formal Lie group is now easy since one can check easily that for all n the G(n) satisfy the hypothesis of the last proposition. Thus, localy on S, if I is the augmentation ideal of G(n) (which does not depend on n) then any lift of a base of I/I^{2} to I gives compatibles isomorphisms for varying n

G(n) ⟶^{∼} Spec(O_{S}[T_{1}, …, T_{d}]/(T_{1}^{p^{n}}, …, T_{d}^{p^{n}}))

7 p-divisible groups in caracteristic p

For any finite flat group scheme G over a caracteristic p scheme there is a Verschiebung morphism V : G ⟶ G^{(p)} verifying F V = p and V F = p. This defines for any p-divisible group G over a caracteristic p base a morphism V : G ⟶ G^{(p)} verifying F V = p and V F = p.

Now as a corolary of lemma 17 we have

Lemma 19. The morphisms F and V are isogenies.

Theorem 6 (Messing). Let G be a p-divisible group on a scheme S s.t. p.O_{S} = 0. Then if ∀n G(n) = ker F^{n}

lim_{⟶}_{n} G(n) = Ĝ

is a formal Lie group on S.

Proof. The equality lim_{⟶}_{n} G(n) = Ĝ is easily checked once one knows lim_{⟶}_{n} G(n) is a formal Lie

group. In fact the inclusion Ĝ ⊂ lim_{⟶}_{n} G(n) is always true and if lim_{⟶}_{n} G(n) is a formal Lie group

then it is infinitesimal thus the other inclusion is verified.

Let’s prove lim_{⟶}_{n} G(n) is a formal Lie group by applying theorem 5. It is clearly of F^{∞}-

torsion. Now since F is an isogeny it is clear ∀n G(n) is finite localy free over S and F is an epimorphism.

8 Differential calculus and deformation theory on syntomic schemes

8.1 Cotangent complex

Let S ba a scheme and p : X ⟶ S an S-scheme. If p is smooth then the differential calculus of X/S is governed by Ω_{X/S}^{1} a localy free O_{X}-module of rank the relative dimension of X over S. If X is not smooth over S anymore the O_{X}-module Ω_{X/S}^{1} is not the good object anymore.

Definition 28 (Grothendieck-Illusie). The cotangent complex 𝕃_{X/S}^{•} ∈ 𝔻^{[−∞,0]}(X, O_{X}) is the “derived functor” of Ω^{1} :

𝕃_{X/S}^{•} = Ω_{A_{•}/p^{−1}O_{S}}^{1} ⊗_{A_{•}} O_{X} ∈ 𝔻^{[−∞,0]}(X, O_{X})

where :

– A_{•} is a particular simplicial resolution of O_{X} as a p^{−1}O_{S}-algebra : A_{•} is a simplicial p^{−1}O_{S}- algebra s.t. ∀i ∈ ℕ π_{i}(A_{•}) = 0 if i > 0 and π_{0}(A_{•}) ⟶^{∼} O_{S}. The π_{i} here being the “local” homotopy groups (those are sheaves on X).

– The simplicial complex Ω_{A_{•}/p^{−1}O_{S}}^{1} ⊗_{A_{•}} O_{X} is seen as a complexe of O_{X}-modules in negative degrees via the Dold-Kan equivalence between simplicial objects and complexes in negative degres in abelian categories.

– The simplical p^{−1}O_{S}-algebra A_{•} is constructed in the following way : consider the adjoint functors couple

\xymatrix{p^{−1}O_{S} − algebras \ar[r]^{F} & p^{−1}O_{S} − modules \ar[l]^{G}}

where F is the forget functor and G(M) = O_{S}[M]. Then to this couple is associated a simplicial resolution

(A_{n})_{n≥0} = ((G ∘ F)^{∘n}(O_{X}))_{n≥0}

the edge and faces map being given by the adjonction morphisms

\{{G ∘ F ⟶ Id \atop Id ⟶ F ∘ G}

Then for all n A_{n} is a “formaly smooth” O_{S}-algebra (O_{S}[M] is “formaly smooth”) and 𝕃_{X/S}^{•} = (Ω_{A_{•}/O_{S}}^{1}) ⊗_{A_{•}} O_{X}.

This cotangent complex is augmented

𝕃_{X/S}^{•} ⟶ Ω_{X/S}^{1}

and this augmentation gives an isomorphism

H^{0}(𝕃_{X/S}^{•}) ⟶^{∼} Ω_{X/S}^{1}

Of course 𝕃_{X/S}^{•} is functorial in X ⟶ S as is Ω_{X/S}^{1}.

One of the first results of the theory is the following :

Proposition 13. Let

\xymatrix{X \ar[rr]^{f} \ar[dr] &  & Y \ar[dl] \\  & S}

be a diagram of schemes. There is a distinguished triangle

\xymatrix{f^{∗}𝕃_{Y/S}^{•} \ar[rr] &  & 𝕃_{X/S}^{•} \ar[dl] \\  & 𝕃_{X/Y}^{•} \ar[ul]^{+1}}

This triangle generalises the usual short exact sequence

f^{∗}Ω_{Y/S}^{1} ⟶ Ω_{X/S}^{1} ⟶ Ω_{X/Y}^{1} ⟶ 0

that is obtained by applying H^{0} to the preceding triangle.

There are particular cases :

Proposition 14. – Let X be of finite type over S. The scheme X is smooth over S iff the augmentation morphism 𝕃_{X/S}^{•} ⟶ Ω_{X/S}^{1} is an isomorphism in 𝔻(X, O_{X}) thath is to say ∀i < 0 H^{i}(𝕃_{X/S}^{•}) = 0

– Let X ↪ Y be a closed immersion defined by the ideal I. Then

τ_{≥−1}𝕃_{X/Y}^{•} ≃ I/I^{2}[1]

– In the preceding if moreover X ↪ Y is a regular immersion then

𝕃_{X/Y}^{•} ≃ I/I^{2}[1]

Example 15. In the triangle of the last proposition if the morphism f is a closed immersion then one finds the usual exact sequence

I/I^{2} ⟶ f^{∗}Ω_{Y/S}^{1} ⟶ Ω_{X/S}^{1} ⟶ 0

Example 16. If f is smooth then one finds back the fact that the sequence

0 ⟶ f^{∗}Ω_{Y/S}^{1} ⟶ Ω_{X/S}^{1} ⟶ Ω_{X/Y}^{1} ⟶ 0

is exact

The following corollary will be fundamental for the study of group schemes.

Corollary 8. If X is a localy complete intersection scheme over S then 𝕃_{X/S}^{•} is a perfect complex in 𝔻^{[−10]}(X, O_{X}).

8.2 Obstruction to smoothness

We have seen that H^{0}(𝕃_{X/S}^{•}) ⟶^{∼} Ω_{X/S}^{1}. Thus, for any O_{X}-module M

Ext_{O_{X}}^{0} (𝕃_{X/S}^{•}, M) ≃ Der_{O_{S}}(O_{X}, M)

Recall (EGA ? ? ?) one can define the O_{X}-module Extalg_{O_{S}}(O_{X}, M) classifying isomorphism classes of extensions of O_{S}-algebras of O_{X} by the squared zero ideal M, the identity class being the one of the dual numbers algebra O_{X} ⊕ M.

We can go a step further now :

Theorem 7. For all O_{X}-modules M there is natural isomorphism

Ext_{O_{X}}^{1} (𝕃_{X/S}^{•}, M) ⟶^{∼} Extalg_{O_{S}}(O_{X}, M)

And we find back the short exact sequence of EGA ? ? ?.

Let T_{0} ↪ T be a closed immersion of S-schemes defined by an ideal I s.t. I^{2}. It thus corresponds to a class

c ∈ Ext_{O_{T_{0}}}^{1} (𝕃_{T_{0}/S}^{•}, I)

We want to study the application reduction modulo I : X(T) ⟶ X(T_{0}).

Suppose thus we are given f : T_{0} ⟶ T an S-morphism. We want to know if it lifts to an element of X(T) :

\xymatrix{T_{0} \ar[r]^{f} \ar@{^{(}->}[d] & X \ar[d] \\ T \ar[r] & S}

This is equivalent to fill the following diagram of O_{S}-algebras

\xymatrix{ &  &  & f^{−1}O_{X} \ar[d]^{f^{∗}} \ar@{-->}[dl] \\ 0 \ar[r] & I \ar[r] & O_{T} \ar[r] & O_{T_{0}} \ar[r] & 0}

or equivalently to know wether or not the clas of the extension of O_{T_{0}} by I becomes trivial by pullback from O_{T_{0}} to f^{−1}O_{X} that is to say if through the morphism

Ext_{O_{T_{0}}}^{1} (𝕃_{T_{0}/S}^{•}, I) ⟶ Ext_{f^{−1}O_{X}}^{1} (f^{−1}𝕃_{X/S}^{•}, I) ≃ Ext_{O_{X}}^{1} (𝕃_{X/S}^{•}, rf_{∗}I)

the class c goes to zero (we suppose Rf_{∗} has finite cohomological dimension on quasi-coherent sheaves, for example quasicompact and quasi-separated).

Moerover if such an extension exists the set of such liftings of our element of X(T_{0}) to an element of X(T)is a torsor under Ext^{0}(𝕃_{X/S}^{•}, Rf_{∗}I).

In particular if T_{0} is affine (and X is separated over S which implies f is affine) this obstruction lies in

Ext_{O_{X}}^{1} (𝕃_{X/S}^{•}, f_{∗}I)

If X is localy a complete intersection over S, 𝕃_{X/S}^{•} is a perfect complex and

Ext_{O_{X}}^{1} (𝕃_{X/S}^{•}, f_{∗}I) ≃ H^{1}(X, (𝕃_{X/S}^{•})^{∨} ⊗^{𝕃}_{O_{X}} f_{∗}I)

9 p-divisible groups and formal Lie groups

Theorem 8 (Grothendieck-Messing). Let G be a p-divisible group over a base S such that p is localy nilpotent on S. Then G is fomaly smooth and Ĝ is a formal Lie group.

In particular, one can define for such a G ω_{G} and Lie G that are localy free O_{S}-module of finite rank. Localy on S, we have ω_{G} = ω_{G[p^{n}]} for n >> 0, the conormal sheaf associated to the unit section of H[p^{n}], and the inclusion H[p^{n}] ↪ H[p^{n+1}] induces isomorphisms ω_{H[p^{n+1}]} ⟶^{∼} ω_{H[p^{n}]} for n >> 0. In fact one has

∀n ω_{H[p^{n}]} ≃ ω_{H}/p^{n}ω_{H}

Theorem 9. Let S and G be as in the preceding theorem. The following are equivalent : – G = Ĝ

– G[p] is radicial

– for all n G[p^{n}] is radicial In this case G is said to be a formal p-divisible group. Moerover the following are equivalent :

– Ĝ is p-divisible that is to say is a formal p-divisible group

– G[p] is an extension of an étale finite finite group scheme by a finite localy free radicial one

0 ⟶ G[p]^{0} ⟶ G[p] ⟶ G[p]^{ét} ⟶ 0

– idem for all G[p^{n}]

– The function s ⟼ the separable rank of the fiber G[p]_{s} is localy constant If this is the case then Ĝ = lim_{⟶}_{n} G[p^{n}]^{0} is a formal p-divisible group, G^{ét} = lim_{⟶}_{n} G[p^{n}]^{ét} is an

étale one and G is an extension

0 ⟶ Ĝ ⟶ G ⟶ G^{ét} ⟶ 0

10 Classification over caracteristic p perfect fields

10.1 The Dieudonné functor

Let k be a caracteristic p perfect field. Let W (k) be the Witt vectors and σ its Frobenius.

Definition 29. A cristal over k is a couple (M, V) where M is a finite free W (k)-module and V : M ⟶ M is a σ^{−1}-linear mapping verifying

pM ⊂ V (M) ⊂ M

Remark 18. An equivalent definition is to take a triple (M, F, V) where F is σ-linear, V is σ^{−1}-linear and F V = p.

Theorem 10 (Dieudonné). There is a category equivalence between p-divisible groups over Spec(k) and cristals over k.

In fact, more precisely, there is a covariant functor 𝔻 from p-divisible groups over a perfect field k to finite free W (k)-modules. This functor is compatible with base change : if k_{1} ⟶ k_{2} then 𝔻(− ⊗_{k_{1}} k_{2}) ⟶^{∼} W (k_{2}) ⊗_{W (k_{1})} 𝔻(−). The Frobenius morphism F : G ⟶ G^{(p)} induces 𝔻(F) : 𝔻(G) ⟶ W (k) ⊗_{W (k),σ} 𝔻(G) and thus a σ^{−1}-linear mapping V : 𝔻(G) ⟶ 𝔻(G). In the same way there is a σ-linear mapping F : 𝔻(G) ⟶ 𝔻(G) induced by the Verschiebung. Then since V F = p on the p-divisible group we have F V = p on 𝔻(G). Thus to a G there is associated a cristal (𝔻(G), F, V).

Moreover the rank of 𝔻(G) is equal to the height of G and [𝔻(G) : V 𝔻(G)] = dim(G).

Remark 19. We take the covariant definition of the Dieudonné modules. There is a contravariant one. Both are linked through Cartier duality.

Remark 20. The Lie algebra functor can be deduced from 𝔻 thanks to the relation

Lie G = 𝔻(G)/V 𝔻(G)

Moreover

Lie G^{D} = V 𝔻(G)/p𝔻(G)

and the extension

0 ⟶ V 𝔻(G)/p𝔻(G) ⟶ 𝔻(G)/p𝔻(G) ⟶ 𝔻(G)/V 𝔻(G) ⟶ 0

is the Lie algebra of the universal vector extension of G. We will see in the sequel that in fact 𝔻(G) can be defined as the Lie algebra of the universal extension of a lifting of G to W (k).

One of the corollaries of the construction of the Dieudonné functor is :

Fact 1. A morphism f : G_{1} ⟶ G_{2} is an isogeny iff 𝔻(f) is injective and Im 𝔻(f) ia a lattice in 𝔻(G_{2}), or equivalently 𝔻(f) induces an isomorphism bewtween 𝔻(G_{1}) ⊗ ℚ and 𝔻(G_{2}) ⊗ ℚ. Moreover the height of the isogeny f is equal to the length of coker(𝔻(f)).

10.2 Three definitions of the Dieudonné functor

Here we give a brief summary of the different approaches to define the Dieudonné module of a p-divisible group.

10.2.1 Via co-Witt vectors

Let for all n ∈ ℕ^{∗} W_{n} be the group scheme of truncated Witt vectors. One has W_{n} = cokerV^{n} where W is the functor of Witt vectors and V : W ⟶ W is the Verschiebung. As a scheme one has W_{n} ≃ 𝔸^{n}. One defines the co-Witt vectors as the fppf sheaf

CW (−) = lim_{⟶}_{n} W_{n}(−)

where the transition morphisms are given by the Verschiebung V : W_{n} ⟶ W_{n+1}. The idea is to use an analog of the Pontryagin duality M ↦ Hom(M, ℚ_{p}/ℤ_{p}) where CW play the role of the dualizing object. Then one puts for a p-divisible group over Spec(k)

𝔻(G) = Hom(G, CW)

In fact this is not the good definition. One has to modify it a little bit using unipotent co-Witt vectors CW^{u} to catch the unipotent part of the p-divisible group.

This Dieudonné module is not the one we spoke before : it is the contravariant one (both are linked through Cartier duality).

10.3 Via Cartier theory

10.4 Via the evaluation of a crystal

10.5 Classification of isocrystals

Definition 30. An isocristal over k is a couple (N, V) where N is a finite dimensional W (k)_{ℚ}- vector space and V : N ⟶^{∼} N is a σ^{−1}-linear isomorphism. A morphism of isocrystals is a W (k)_{ℚ}-linear morphism commuting to the action of V.

Thus to any crystal over k is associated an isocrystal. And thanks to the given caracterisation of isogenies one has Corollary 9. The functor G ↦ 𝔻(G)_{ℚ} is a fully faithfull functor from the category of p-divisible groups up to isogeny (the cateory of p-divisible groups where we have inverted the class of maps given by isogenies, in the sens of localization of categories) to the category of isocrystals over k.

We will see in a few moment what is the esential image of this functor.

Definition 31. Let λ ∈ ℚ. An isocrystal (N, V) is isoclinic with slope λ if there exists a lattice M in N such that V^{s}M = p^{r}M where λ = r/s.

Theorem 11 (Dieudonné-Manin). The category of isocrystals over k is a direct orthogonal sum indexed by ℚ of the categories of isoclinic isocrystals with some given slope in ℚ. In down to earth terms : for any (N, V) there is a V -invariant decomposition

N = ⨁_{λ∈ℚ} N_{λ}

were (N_{λ}, V) is isoclinic (zero for only a finite number of λ ∈ ℚ) and if λ_{1} ≠ λ_{2} then

Hom((N_{λ_{1}}, V), (N_{λ_{2}}, V)) = 0

If k is algebraically closed then this category is semi-simple, the simple objects being indexed by ℚ : for exach slople λ there exists a unique simple isocrystal isoclinic with slope λ.

The sets of λ s.t. N_{λ} ≠ 0 are called the slopes of (N, V).

Corollary 10. The functor G ⟼ 𝔻(G)_{ℚ} induces an equivalence between p-divisible groups over k up to isogeny and isocrystals whose slope lie bewteen 0 and 1.

One know G is étale iff F : G ⟶ G is an isomorphism. Thus :

Fact 2. G is étale iff 𝔻(G)_{ℚ} is isoclinic with slope 0. G is a formal p-divisible group iff 𝔻(G)_{ℚ} doesn’t have the zero slope.

Thus, in the extension

0 ⟶ Ĝ ⟶ G ⟶ G^{ét} ⟶ 0

𝔻(Ĝ)_{ℚ} ⊂ 𝔻(G)_{ℚ} is the direct sum of the isoclinic parts with slope > 0.

Remark 21. This is very particular to the zero slope case that the zero slope part of 𝔻(G)_{ℚ} splits without inverting p at the level of 𝔻(G).

If λ_{1}, …, λ_{r} are the slopes of 𝔻(G) with multiplicity (a_{1}, …, a_{r}) then if ∀i λ_{i} = \frac{d_{i}}{h_{i}} with d_{i} ∧ h_{i} = 1 then

ht(G) = ∑_{i} a_{i}h_{i} dim(G) = ∑_{i} a_{i}d_{i}

Sometimes the number a_{i}h_{i} is called the multiplicity of λ_{i}.

Definition 32. The Newton polygon attached to the data (λ_{i}, a_{i}) is the unique convex polygon whose breakpoints are in ℤ^{2} that begins at (0, 0) and whose slpopes are the λ_{i} each with multiplicity a_{i}h_{i}.

Thus this Newton polygon, if attached to (N, V), ends at (ht(N, V), dim(N, V)).

Corollary 11. Let k be algebraically closed. The isogeny classes of height h dimension d p-divisible groups over k are in bijection with convex polygons whose breakpoints are in ℕ^{2}, whose slopes are in [01], that begin at (0, 0) and end at (h, d).

Fig. 1 – The possible Newton polygons of a one dimensional p-divisible group

10.6 Example in the case of Lubin-Tate groups

Using the preceding results one can find back the classification of one dimensional formal p-divisible groups over \overline{𝔽}_{p} by their height.

In fact the possible Newton polygons of a one dimensional p-divisible group over \overline{𝔽}_{p} are shown in the following figure

One sees they are parametrized by an integer h ∈ {0, …, n − 1} where a p-divisible group such that its Newton polygon is attached to the integer h is an extension

0 ⟶ H^{0} ⟶ H ⟶ H^{ét} ⟶ 0

where H^{ét} is étale of height h and H^{0} is a formal one dimensional p-divisibile group with height n − h.

Moerover one verifies there is a unique isomorphism class of formal p-divisible groups H of height n. Their crystal is given in the following way. The W (k)-module 𝔻(H) has a base (e_{1}, …, e_{n}) such that

∀1 ≤ i ≤ n − 1 V.e_{i} = e_{i+1} and V.e_{n} = pe_{1}

that is to say the matrix of V is

(\begin{matrix}0 & … & 0 & p \\ 1 & & & 0 \\ & ... & & ⋮ \\ & & 1 & 0 \end{matrix})

Then End(H) = End_{(W (k),V)}(𝔻(H)) = O_{D} where D is a division algebra with invariant 1/n. If ℚ_{p^{n}} = W (𝔽_{p^{n}})[\frac{1}{p}] is the degre n unramified extension of ℚ_{p} thent

D = ℚ_{p^{n}}[Π] O_{D} = ℤ_{p^{n}}[π]

where Π^{n} = p and ∀x ∈ ℚ_{p^{n}} Πx = x^{σ}Π. Concretely,

∀1 ≤ i ≤ n − 1 Π.e_{i} = e_{i+1} and Π.e_{n} = pe_{1}

and ∀x ∈ ℤ_{p^{n}} its action on 𝔻(H) is x.e_{i} = σ^{−i}(x)e_{i}.

11 Algebraization theorem

Let A be an I-adic ring.

Recall (EGA III) the following corollary of Grothendieck’s algebraization theorem :

Theorem 12 (Grothendieck). Define an abelian scheme over Spf(A) as being a compatible system of abelian schemes over the Spec(A/I^{n}) for n ∈ ℕ^{∗} (a cartesian section of the fibered category of abelian schemes over the Spec(A/I^{n}), n ≥ 1). Define a polarization of an abelian scheme over Spf(A) as being a compatible system of polarizations.

Then the functor “formal completion along V (I)” induces a category equivalence between polarizable abelian schemes over Spec(A) and polarizable abelian schemes over Spf(A).

We’re going to see the same property is satisfied by p-divisible groups.

Definition 33. Let 𝔛 be a formal scheme with ideal of definition I. A p-divisible group over 𝔛 is a compatible system of p-divisible groups over the Spec(O_{X}/I^{n}), n ≥ 1.

This means we have forall n ≥ 1 a p-divisible group G_{n} over Spec(O_{X}/I^{n}) and isomorphisms G_{n+1} mod I^{n} ⟶^{∼} G_{n} satisfying the usual cocyle relation.

Proposition 15 (Messing). The functor G ⟼ (G mod I^{n})_{n≥1} induces a category equivalence between p-divisible groups over Spec(A) and over Spf(A).

12 Rigidity of quasi-isogenies

Let’s remark for two p-disvisible groups G_{1}, G_{2} Hom(G_{1}, G_{2}) is a ℤ_{p}-module without torsion.

Proposition 16. Let S_{0} ↪ S be an immersion defined by a localy nilpotent ideal. Suppose p is localy nilpotent on S. Let G, H be two p-divisible groups over S and G_{0}, S_{0} their reduction to S_{0}. Then the reduction to S_{0} induces an injection

Hom_{S}(G, H) ↪ Hom_{S}(G_{0}, H_{0})

and if moreover S is quasi-compact there exists N ∈ ℕ s.t.

p^{N}Hom_{S_{0}}(G_{0}, H_{0}) ⊂ Hom_{S}(G, H)

Proof. By devissage one is reduced to the case when the immersion i : S_{0} ↪ S is defined by a squared zero ideal I.

We have now

Hom_{S_{0}}(G_{0}, H_{0}) = Hom_{S_{0}}(i^{∗}G, i^{∗}H) = Hom_{S}(G, i_{∗}i^{∗}H)

There is an adjonction morphism H ⟶ i_{∗}i^{∗}H and the morphism Hom_{S}(G, H) ⟶ Hom_{S}(G_{0}, H_{0}) is nothing else than the morphism induced by applying Hom_{S}(G, −) to this adjunction morphism. The formal smoohtness of H implies H ⟶ i_{∗}i^{∗}H is an epimorphism. Thus we have an exact sequence

0 ⟶ ker(H ⟶ i_{∗}i^{∗}H) ⟶ H ⟶ i_{∗}i^{∗}H ⟶ 0

But from the following lemma we have

ker(H ⟶ i_{∗}i^{∗}H) ≃ i_{∗}\underline{Hom}_{O_{S_{0}}} (ω_{H_{0}}, I)

This gives the exact sequence

Hom_{S}(G, i_{∗}\underline{Hom}_{O_{S_{0}}} (ω_{H_{0}}, I)) ⟶ Hom_{S}(G, H) ⟶ Hom_{S}(G_{0}, H_{0}) ⟶ Ext^{1}(G, i_{∗}\underline{Hom}_{O_{S_{0}}} (ω_{H_{0}}, I))

But p is an epimorphism on G and localy nilpotent on i_{∗}\underline{Hom}_{O_{S_{0}}} (ω_{H_{0}}, I) thus the left term in this sequence is zero. Now if p^{N} is zero on S then the right term is killed by p^{N} since it kills i_{∗}\underline{Hom}_{O_{S_{0}}} (ω_{H_{0}}, I).

Lemma 20. Let i : S_{0} ↪ S be a closed immersion defined by an ideal I such that I^{2} = (0). Let H be a group-scheme over S and H_{0} = i^{∗}H its reduction modulo I. Let H ⟶ i_{∗}i^{∗}H be the adjonction morphism that associates to s ∈ H(U) s_{|U×_{S}S_{0}} ∈ H(U ×_{S} S_{0}) = H_{0}(U ×_{S} S_{0}). Then as fppf abelian sheaves over S

ker(H ⟶ i_{∗}i^{∗}H) ≃ i_{∗}\underline{Hom}_{O_{S_{0}}} (ω_{H_{0}}, I)

where ω_{H_{0}} is the conormal quasi-coherent sheaf associated to the unit section of H_{0}.

Proof. Easy.

13 Grothendieck Messing deformation theory

13.1 Universal vector extension

13.2 Motivation

We know the Hodge filtration of the H^{1} of an abelian variety over ℂ determines the deformation theory of this abelian variety. For example is S is a smooth analytic space over ℂ and A ⟶ S is a princiaply polarized abelian variety for each s ∈ S if one trivializes the Betti relative cohomology of A ⟶ S in a a neighborhood of s the Hodge filtration the local trivialization of Betti defines an holomorphic map U ⟶ H ⊂ Gr where U is a neighborhood of s and H ⊂ Gr is Siegel space in its associated Grassmanian. Then the germs (A ×_{S} V)_{V} where V goes through neigborhoods of s ins S is a versal deformation of A_{s} iff the tangent map to U ⟶ H at s is an isomorphism that is to say it is a local isomorphism at s.

Let A be an abelian variety of ℂ. Consider its Hodge filtration

0 ⟶ Γ(A, Ω_{A}^{1}) ⟶ H_{dR}^{1}(A) ⟶ H^{1}(A, O_{A}) ⟶ 0

and Γ(A, Ω^{1}) = ω_{A} is the vectorspace of translation invariant differential forms, H^{1}(A, O_{A}) = ω_{A^{∨}}^{∗} where A^{∨} is the dual abelian variety.

There is a imbedding H_{B}^{1}(A, ℤ) ⊂ H_{dR}^{1}(A) given by the comparison theorem between De Rham and Betti cohomolohy. Moreover this imbeding composed with the projection H_{dR}^{1}(A) ⟶ H^{1}(A, O_{A}) is still an embedding and

A^{∨}(ℂ) = H^{1}(A, O_{A})/H_{B}^{1}(A, ℤ)

as can be easily verified by writting A = V/Λ, A^{∨} = V^{∗}/Λ^{∨}, ω_{A^{∨}} ≃ V, H_{dR}^{1}(A) ≃ V^{∗} ⊗_{ℝ} ℂ.

Now we’re looking for a geometric way to find back the Hodge filtration. Let’s look at the following extension of holomorphic Lie groups

0 ⟶ ω_{A} ⟶ H_{dR}^{1}(A)/H_{B}^{1}(A, ℤ) ⟶ A^{∨}(ℂ) ⟶ 0

obtained by taking the quotient of the Hodge filtation by the discrete H_{B}^{1}(A, ℤ). It is an extension of the abelian variety A^{∨} by the vector bundle ω_{A}.

Moreover, since we took the quotient by a discrete subgroup, one can find back the Hodge filtration from this extension by applying the Lie algebra functor to it.

In fact one can prove :

Fact 3. The preceding extension is algebraic and it is universal among extension of A^{∨} by a vector bundle.

which gives an intrinsec definition of it.

We are going to do the same for p-divisible groups, apart from the fact that the theory we’re going to developp is covariant that is to say is expressed in terms of the “De Rham homology”.

13.3 Universal vector extension of a p-divisible groups

Let S be a scheme on which p is localy nilpotent.

Definition 34. Let M be a quasi-coherent O_{S}-module. We note \underline{M} for the associated fppf sheaf.

Remark 22. If M is localy free of finite rank over S then \underline{M} is represented by a vector bundle but in general it is not representable by a scheme.

Definition 35. Let G be a p-divisible group over S. A vector extension of G is an extension of fppf sheaves

0 ⟶ \underline{M} ⟶ E ⟶ G ⟶ 0

where M is a quasi-coherent O_{S}-module. A morphism of vector extensions is a morphism of digagrams

[figure: 0; \underline{M_{1}}; E_{1}; G; 0; 0; \underline{M_{2}}; E_{2}; G; 0]

such that the left square is co-cartesian that is to say the morphisms induce an isomorphism between the push-out of the upper extension by \underline{M_{1}} ⟶ \underline{M_{2}} and the bottom extension.

The category of vector extensions of G is rigid in the sens that there is at most one morphism from a vector extension to another one. In fact, two morphisms differ from an element of Hom(G, \underline{M}). But since p is an epimorphism on G and localy niltpotent on \underline{M} we have Hom(G, \underline{M}) = 0.

Thus this has a meaning to speak about a universal vector extension that is to say an initial object in the category of vector extensions.

From now we note M for \underline{M} since there’s no ambiguity.

Proposition 17. There exists a universal vector extension

0 ⟶ V (G) ⟶ E(G) ⟶ G ⟶ 0

Moreover V (G) = ω_{G^{D}} and is thus a vector bundle.

Proof. We have to prove the functor M ⟼ Ext^{1}(G, \underline{M}) is representable. This is local on S and we thus can suppose p^{N}O_{S} = 0 for an N ∈ ℕ. The exact sequence

0 ⟶ G[p^{N}] ⟶ G ⟶^{p^{N}} G ⟶ 0

induces

0 ⟶ Hom(G, \underline{M}) ⟶^{p^{N}} Hom(G, \underline{M}) ⟶ Hom(G[p^{N}], \underline{M}) ⟶ Ext^{1}(G, \underline{M}) ⟶^{p^{N}} Ext^{1}(G, \underline{M})

But in the preceding sequence both maps p^{N} are zero since p^{N} is zero on \underline{M}. Thus there is an isomorphism

Ext^{1}(G, \underline{M}) ≃ Hom(G[p^{N}], \underline{M})

but ( ? ? ?)

Hom(G[p^{N}], \underline{M}) ≃ Hom_{O_{S}}(ω_{G[p^{N}]^{D}}, \underline{M})

Thus the extension obtained py push-out in the following diagram

\xymatrix{0 \ar[r] & G[p^{N}] \ar[r] \ar[d] & G \ar[r]^{p^{N}} & G \ar[r] & 0 \\  & ω_{G[p^{N}]^{D}}}

where G[p^{N}] ⟶ ω_{G[p^{N}]^{D}} is the morphisms that associates to an x ∈ G[p^{N}] = (G[p^{N}]^{D})^{D}, x : G[p^{N}]^{D} ⟶ 𝔾_{m}, the element x^{∗} \frac{dT}{T} ∈ ω_{G[p^{N}]^{D}}, is universal.

Example 17. The uniersal vector extension of ℚ_{p}/ℤ_{p} is

0 ⟶ 𝔾_{a} ⟶ (𝔾_{a} ⊕ ℚ_{p}) /ℤ_{p} ⟶ ℚ_{p}/ℤ_{p} ⟶ 0

where ℤ_{p} ↪ 𝔾_{a} ⊕ ℚ_{p} is the diagonal embeding.

Of course the universal vector extension is functorial in G : for all f : G_{1} ⟶ G_{2} there is a morphism

\xymatrix{0 \ar[r] & V (G_{1}) \ar[r] \ar[d]^{h} & E(G_{1}) \ar[d] \ar[r] & G_{1} \ar[r] \ar[d]^{f} \ar[r] & 0 \\ 0 \ar[r] & V (G_{2}) \ar[r] & E(G_{2}) \ar[r] & G_{2} \ar[r] \ar[r] & 0}

where h = (f^{D})^{∗} : ω_{G_{1}^{D}} ⟶ ω_{G_{2}^{D}}. This is easily checked using the pullback by f of the bottom extension and the universality property of the upper extension.

Proposition 18. The fppf-sheaf E(G) is formaly smooth and Ê(G) is a formal Lie group and is an extension of formal Lie groups

0 ⟶ V̂(G) ⟶ Ê(G) ⟶ Ĝ ⟶ 0

where V̂(G), the formal completion of a vector bundle along its zero section, is Zariski localy on S isomorphic to a sum of copies of 𝔾̂_{a}.

Moreover the following sequence of O_{S}-modules is exact

0 ⟶ V (G) ⟶ Lie E(G) ⟶ Lie G ⟶ 0

Proof. This is not difficult, essentialy by writting E(G) = lim_{⟶}_{n} E_{n} where E_{n} is the reciprocal

image of G[p^{n}] and

0 ⟶ V (G) ⟶ E_{n} ⟶ G[p^{n}] ⟶ 0

thus, E_{n} is a fppf-torsor over G[p^{n}] under a smooth affine scheme and is thus representable by a smooth G[p^{n}]-scheme.

Remark 23. By functoriality of the universal extension, for a p-divisible group over a formal scheme such that p is in its definition ideal there is associated a universal vector extension.

13.4 Messing’s crystal

Let S be a scheme on which p is localy nilpotent. Let NCRIS(S) be the absolute big Zariski nilpotent cristalline site of S whose objects are (U ↪ T, γ) where U is an S-scheme, U ↪ T is a nil-immersion defined by an ideal I equiped with nilpotent divided powers γ. Let O_{S}^{CRIS} be the structural sheaf of this site.

Theorem 13 (Messing). There exists a crystal E in localy free O_{S}^{CRIS}-modules on NCRIS(S) such that ∀(U ↪ T, γ) ∈ NCRIS(S) for any lifting G̃_{U} of G ×_{S} U to a p-divisible group over T there exists a canonical isomorphism

E_{(U↪T)} ⟶^{∼} Lie E(G̃_{U})

Here to explain the word canonical it would need to go inside the construction of this crystal. This is done in the following section.

Definition 36. We will note 𝔻(G) for the crystal of G.

Remark 24. In fact Messing proves E(G̃_{U}) defines a crystal in fppf sheaves. The preceding crystal is deduced by applying the Lie algebra functor.

13.5 Construction of Messing’s crystal

13.5.1 The exponential map

Theorem 14 (Messing). Let S_{0} ↪ S be an immersion defined by an ideal I equiped with nilpotent divided powers γ. Let H be a formal Lie group over S and M a localy free of finite rank O_{S}-module. Let G_{0} and M_{0} be the reductions moduli I of G and M. Then there exists a morphism

exp : Hom_{O_{S}}(M, I.Lie H) ⟶ ker(Hom_{S}(M, H) ⟶ Hom_{S_{0}}(M_{0}, H_{0}))

natural in H, M and (S, I, γ) such that if I^{2} = pI = 0 then via the natural identification between ker(Hom_{S}(M, H) ⟶ Hom_{S_{0}}(M_{0}, H_{0})) and Hom_{O_{S}}(M, I.Lie H) then

exp f = ∑_{i≥0}(−1)^{i}Π^{i} ∘ f

Wehre Π : I ⊗ Lie H ⟶ I ⊗ Lie H is defined by Π = γ_{p} ⊗ α where γ_{p} : I ⟶ I is Frob-linear and α is the F rob-linear morphism induced at the level of the Lie algebra Lie H by V : H^{(p)} ⟶ H if H is p-divisible and more generaly by the p-exponentiation of invariant derivations on H for general H (the operation defining the restricted Lie algebra structure on Lie (H)).

We won’t give the proof of this theorem. The proof given in ? ? ? is not realy natural. A more natural one is given in ? ? ? using Cartier theory. The points consists in proving that with the hypothesis of the theorem there is an isomorphism

log : H(I) ⟶^{∼} I.Lie H

(depending on the divided powers on I allthough it is not in the notations). This can be done for the infinite dimensional formal group given by the formal completion of the Witt vectors Ŵ(−). Then Cartier theory gives a resolution of any formal group by Witt vectors

Ŵ^{m} ⟶ Ŵ^{n} ⟶ H ⟶ 0

and this enables one to construct such a logarithm isomorphism using the one for Witt vectors.

In the case of a one dimension formal group law the concrete statement about the existence of this logarithm is the following.

Theorem 15. Let R ba a ring and I an ideal in R equiped with nilpotent divided powers (γ_{n})_{n}. Let F ∈ R[[X, Y]] be a formal group law over R. Let ω = f(T)dT be a generator of the invariant differential forms on F. Let f(T) = ∑_{n≥0} a_{n}T^{n}. For x ∈ I put

log(x) = ∑_{n≥0} a_{n}(n − 1)!γ_{n}(x)

Then log induces an isomorphism of groups

log : (I, +_{F}) ⟶^{∼} (I, +)

Example 18. If F is 𝔾̂_{m} then log(x) = ∑_{n≥0}(n − 1)!γ_{n}(x). Where (n − 1)!γ_{n}(x) is the analog of “x^{n}/n”.

Now let’s note exp : I.Lie H ⟶ H(I) for the inverse of log. To construct

exp : Hom_{O_{S}}(M, I.Lie H) ⟶ Hom_{S}(M, H)

one uses the fact for any flat S scheme T the divided powers extend to I.O_{T}. Thus since M is represented by a flat S-scheme one concludes thanks to Yoneda lemma.

13.5.2 The crystal

Here we explain how using the exponential morphism Messing proves the preceding theorem.

Theorem 16 (Messing). Let S_{0} ↪ S be a closed immersion defined by an ideal equiped with nilpotent divided powers. Suppose p is localy nilpotent on S. Let G, resp. H, be two p-divisible groups over S and G_{0}, resp. H_{0} their reduction to S_{0}. Suppose given a morphism f : G_{0} ⟶ H_{0}. Then there exists a unique morphism g : E(G) ⟶ E(H) reducing to E(f) on S_{0} such that for any linear morphism u : V (G) ⟶ V (H) lifting V (f) : V (G_{0}) ⟶ V (H_{0}) in the following diagram

\xymatrix{0 \ar[r] & V (G) \ar[r]^{i} \ar[d]^{u} & E(G) \ar[d]^{g} \ar[r] & G \ar[r] & 0 \\ 0 \ar[r] & V (H) \ar[r]^{j} & E(H) \ar[r] & H \ar[r] & 0}

j ∘ u − g ∘ i : V (G) ⟶ E(H) (that reduces to zero on S_{0}) is an exponential (with values in the formal Lie group Ê(H)).

This theorem proves for any p-divisible group G_{0} over S_{0}, for two lifts G, G′ there exists a unique isomorphism E(G) ⟶^{∼} E(G′) deforming the identity and satisfying the conditions of the preceding theorem. Moerover, still thanks to the preceding theorem, this construction is functorial in G, proving the fact that this defines a crystal.

The proof of the preceding theorem is by devissage to the case when the ideal I defininig he immersion S_{0} ↪ S verifies I^{2} = pI = 0. Then one uses the explicit formula given for the exponential map in this case and the universal property of the universal vector extension.

13.6 Deformation theory

Let S_{0} ↪ S be a divided powers immersion as in the preceding theorem.

For each p-divisible group G_{0} over S_{0} we can consider its crystal 𝔻(G_{0}). Its evaluation 𝔻(G_{0})_{(S_{0}⟶^{Id}S_{0})}

is identified with Lie E(G_{0}) and is thus filtered by the sub O_{S_{0}}-module V (G_{0}) = Fil 𝔻(G)_{(S_{0}⟶^{Id}S_{0})}

that is a localy free, localy direct summand in 𝔻(G_{0})_{(S_{0}⟶^{Id}S_{0})}.

To each lifting G of G_{0} over S is associated localy free localy direct factor filtration V (G) of the evaluation of the crystal 𝔻(G_{0})_{(S_{0}↪S)}, filtration thtat reduces modulo de divided powers ideal

I to Fil 𝔻(G)_{(S_{0}⟶^{Id}S_{0})}.

Definition 37. Let C be the category whose objects are couples (G_{0}, Fil) where G_{0} is a p-divisible group over S_{0} and Fil is a localy free localy direct summand O_{S}-module in 𝔻(G_{0})_{(S_{0}↪S)} such that its reduction to S_{0} is V (G_{0}) ⊂ 𝔻(G_{0})_{(S_{0}⟶^{Id}S_{0})} ; and Hom_{C}((G_{0}, Fil), (G′_{0}, Fil′)) consists in

morphisms f from G_{0} to G′_{0} such that the induced crystal morphism 𝔻(f) verifies

𝔻(f)_{(S_{0}↪S)}(Fil) ⊂ Fil′

Theorem 17. The functor from the category of p-divisible group over S to C that associates to G the couple (G_{0}, Fil) where G_{0} is the reduction to S_{0} of G and Fil is the vector part V (G) is a category equivalence.

Remark 25. The case when S_{0} ↪ S is defined by a squared zero ideal has been obtained by Grothendieck using the theory of the cotangent complex ( ? ? ?). The more general case with divided powers is due to Messing.

13.7 The squared zero case

This case had allready been obtained by Grothendieck using the cotangent complex theory.

Corollary 12. Let S_{0} ↪ S be an immersion defined by an ideal I s.t. I^{2} = (0). Let G_{0} be a p-divisible group over S_{0}. Then the set of liftings of G_{0} to a p-divisible group over S is a principal homogenous space under

ω_{G^{D}}^{∗} ⊗ ω_{G}^{∗} ⊗ I

Proof. The tangent space to the deformation functor of p-divisible groups is identified to the one of a Grassmanian thanks to Grothendieck-Messing’s deformation theory.

Here is a more conceptual restatement of the preceding results

Theorem 18. The stack 𝔛 of p-divisible groups is formaly smooth. Let G be the universal pdivisible group over 𝔛. Then restricted to schemes on which p is localy nilpotent the tangent bundle to this stack is Lie G^{D} ⊗ Lie G.

Example 19. Let O be an inequal caracteristic discrete valuation ring with residue field of caracteristic p. Then if p ≠ 2, pO has nilpotent divided powers. Thus the category of p-divisible groups over Spf(O) being equivalent to the one over Spec(O) there is a fully faithfull functor from the category of p-divisible groups over Spec(O) to the category of couples (G_{0}, Fil) where G_{0} is a p-divisible group over O/pO and Fil is a direct factor filtration of the evaluation of its crystal on O ↠ O/pO.

If O is unramified over ℤ_{p} one obtains thus a fully faithfull functor from the category of p-divisibile groups over O to filtered Dieudonné modules. After inverting p one obtains a fully faithfull functor from p-divisible groups over O to filtered isocrystals. This works the same when the absolute ramification index e verifies e ≤ p − 1 since then the ideal pO is equiped with divided powers. The determination of the essential image of this functor is Fontaine’s theory of filtered admissible modules in a partciular case.

13.8 Application to the deformation spaces of p-divisible groups

Here we are going to refind and generalize theorem ? ? ? on Lubin-Tate deformation spaces.

Theorem 19. Let k be a caracteristic p perfect field. Let ℍ be a dimension d and height h pdivisible group over Spec(k) Let Def_{ℍ} be the functor from artinian rings with residue field k to sets that associates to A the isomorphism classes of couples (H, ρ) where H is a p-divisible group over A and ρ : ℍ ⟶^{∼} H ⊗_{A} k is an isomorphism.

Then Def_{ℍ} is pro-representable :

Def_{ℍ} ≃ Spf (W (k)[[T_{1}, …, T_{d(h−d)}]])

Proof. We apply theorem ? ?. We know the functor F is formaly smooth. The computation of the tangent space has been done through Grothendieck-Messing deformation theory.

14 The period morphism to projective space : first definition

14.1 Motivation

Consider the functor F on smooth analytic spaces that associates to S the isomorphism classes of couples (E, ρ) where E ⟶^{p} S is an elliptic curve over S and ρ : \underline{ℤ}^{2} ⟶^{∼} R^{1}p_{∗}\underline{ℤ} is a rigidification of its Betti cohomology. Then F is representable by Poincaré upper/lower half plane ℂ \ ℝ = H^{±}. The universal elliptic curve is described as O_{ℙ^{1}}(1)_{|H^{±}}/\underline{ℤ}^{2} where \underline{ℤ}^{2} ⟶ O_{ℙ^{1}}(1) is induced by \underline{ℤ}^{2} ⟶ O_{ℙ^{1}}^{2} ↠ O_{ℙ^{1}}(1).

Now consider the period morphism H^{±} ⟶ ℙ^{1}(ℂ) defined by (E, p) ⟼ Fil O_{S}^{2} where Fil O_{S}^{2} is defined by the Hodge filtration of E coupled with the rigidification ρ :

ω_{E/S} = FilH_{dR}^{1}(E/S) ⊂ H_{dR}^{1}(E/S) = R^{1}p_{∗}\underline{ℤ} ⊗ O_{S} ≃ O_{S}^{2}

This morphism is a local ismorphism and even an embeding.

We are going to do the same for Lubin-Tate spaces : here the couple (E, ρ) is replaced by (H, ρ) where ρ is a rigidification of the p-adic Betti cohomology : the isogeny class of the reduction. The associated period morphism will be étale thanks to Grothendieck-Messing deformation theory, but not a local isomorphism.

14.2 Notations

Let ℍ be a height n one dimensional formal p-divisible group over \overline{𝔽}_{p}. Let

𝔛 ≃ Spf(W (k)[[x_{1}, …, x_{n−1}]])

be the associated Lubin-Tate space clasiffying deformations of ℍ. We note (H, ρ) for the universal deformation, H being a p-divisible group over 𝔛 and

ρ : ℍ ⟶^{∼} H mod (p, x_{1}, …, x_{n−1})

We have allready described the crystal (𝔻(ℍ), V) with its action of the maximal order O_{D} of the division algebra D in section ? ? ?.

Set L = W (k)_{ℚ}. We note 𝔛^{rig} for the generic fiber of 𝔛 as a rigid space

𝔛^{rig} ≃ 𝔹^{∘}^{n−1} = {(x_{1}, …, x_{n−1}) ∈ 𝔸^{n} | ∀i v(x_{i}) > 0 }

As a rigid space over L 𝔛^{rig} is a growing-up union of closed balls

𝔛^{rig} = ⋃_{a∈ℕ^{∗}} 𝔹^{n−1}(0, p^{−1/a})

Where

𝔹^{n−1}(0, p^{−1/a}) = \{(x_{1}, …, x_{n−1}) | ∀i v(x_{i}) ≥ \frac{1}{a} \}
= Sp(A_{a})

where A_{a} is the affinoïde alegbra

A_{a} = L < X_{1}, …, X_{n−1}, T_{1}, …, T_{n−1} > /(X_{i}^{a} − pT_{i})_{1≤i≤n−1}

and the morphism Sp(A_{a}) ↪ Sp(A_{b}) for a ≤ b is given by

A_{b} ⟶ A_{a}
X_{i} ⟼ X_{i}
T_{i} ⟼ X_{i}^{b−a}T_{i}

Recall there is a left action of O_{D}^{×} on 𝔛 over Spf(W (k)) via

∀g ∈ O_{D}^{×} = Aut(ℍ) ∀(H_{0}, ρ_{0}) ∈ 𝔛 g.(H_{0}, ρ_{0}) = (H_{0}, ρ_{0} ∘ g^{−1})

One can prove (we won’t use this lemma)

Lemma 21. The action of O_{D}^{×} is continous in the sens that if 𝔪 = (p, x_{1}, …, x_{n−1}) for all k ∈ ℕ^{∗} there exits an open compact subgroup K ⊂ O_{D}^{×} such that K acts trivialy on Spec(W (k)[[x_{1}, …, x_{n−1}]]/𝔪^{k}).

Proof. By rigidity of quasi-isogenies ∀g ∈ O_{D} ∃N ∈ ℕ^{∗} p^{N}g lifts to an endomorphism of H mod 𝔪^{k}. But O_{D} being a ℤ_{p}-module of finite type

∃N ∈ ℕ^{∗} ∀g ∈ O_{D} p^{N}g ∈ End(H mod 𝔪^{k}) ⊂ End(ℍ)

Thus End(H mod 𝔪^{k}) being p-adicaly complete ring

∀g ∈ O_{D} Id + p^{N}g ∈ Aut(H mod 𝔪^{k})

and Id + p^{N}O_{D} is a compact open subroup of O_{D}^{×}.

14.3 Statement of the main theorem

Let Lie E(H) be the universal vector extension of H on 𝔛. It is a free O_{𝔛}-module of rank n. It has a flitration

0 ⟶ ω_{H^{D}} ⟶ Lie E(H) ⟶ ω_{H}^{∗} ⟶ 0

where ω_{H^{D}} is free of rank n − 1 and ω_{H}^{∗} free of rank 1. All those free modules are equivariant O_{D}^{×}-vector bundles on 𝔛. As O_{D}^{×}-vector bundles they are non-trivial at all as they are if we forget the action of O_{D}^{×}. Let

Lie E(H)^{rig}

be the O_{D}^{×}-equivariant rigid analytic vector bundle over 𝔛^{rig}.

Theorem 20. Lie E(H)^{rig} is a flat O_{D}^{×}-vector bundle : there is an isomorphism

Lie E(H)^{rig} ≃ 𝔻(ℍ)_{ℚ} ⊗_{L} O_{𝔛^{rig}}

14.4 First proof via the rigidity of quasi-isogenies

For each a ∈ ℕ^{∗} consider the following p-adic integral model of A_{a}

A_{a} = O_{L} < X_{1}, …, X_{n−1}, T_{1}, …, T_{n−1} > /(X_{i}^{a} − pT_{i})_{1≤i≤n−1}

thus A_{a} = A_{a}[\frac{1}{p}]. Equip A_{a} with the p-adic topology. Then there is a morphism

Spf(A_{a}) ⟶ 𝔛

associated to the continous O_{L}-algebras morphism

W (k)[[x_{1}, …, x_{n−1}]] ⟶ A_{a}
x_{i} ⟼ X_{i}

that induces the inclusion Sp(A_{a}) ↪ 𝔛^{rig} on the generic fiber. Let (H_{a}, ρ_{a}) be the pullback of (H, ρ) to Spf(A_{a}) where

ρ_{a} : ℍ ×_{Spec(\overline{𝔽}_{p})} Spec(\overline{𝔽}_{p}[T_{1}, …, T_{n−1}]) ⟶^{∼} H_{a} mod (p, X_{1}, …, X_{n−1})

Now observe the ideal (X_{1}, …, X_{n−1}) is nilpotent in A_{a}/pA_{a}. Thus by rigidity of quasi-isogenies there exists two morphisms

\xymatrix{ℍ ×_{Spec(\overline{𝔽}_{p})} Spec(A_{a}/pA_{a}) \ar[r]^{f_{1}} & H_{a} ×_{Spec(A_{a})} Spec(A_{a}/pA_{a}) \ar[l]^{f_{2}}}

and an integer N ∈ ℕ^{∗} such that

f_{1} ∘ f_{2} = p^{N}Id f_{2} ∘ f_{1} = p^{N}Id

Now suppose p ≠ 2. Then the ideal pA_{a} of A_{a} is equiped with nilpotent divided powers (if p ≠ 2 replace 2A_{a} by 4A_{a}). Consider the crystal of

ℍ ×_{Spec(\overline{𝔽}_{p})} Spec(A_{a}/pA_{a})

evaluated on the P.D. thickening

Spec(A_{a}/pA_{a}) ↪ Spec(A_{a})

and noted

𝔻(ℍ)_{(Spec(A_{a}/pA_{a})↪Spec(A_{a}))}

There is a morphism of P.D. thickenings

\xymatrix{Spec(A_{a}/pA_{a}) \ar@{^{(}->}[r] \ar[d] & Spec(A_{a}) \ar[d] \\ Spec(\overline{𝔽}_{p}) \ar@{^{(}->}[r] & Spec(W (\overline{𝔽}_{p}))}

Thus by the crystal property there is a canonical isomorphism

𝔻(ℍ)_{(Spec(A_{a}/pA_{a})↪Spec(A_{a}))} ≃ 𝔻(ℍ) ⊗_{O_{L}} A_{a}

where 𝔻(ℍ) without any subscript means the Dieudonné module of ℍ: 𝔻(ℍ) = 𝔻(ℍ)_{(Spec(\overline{𝔽}_{p})↪Spec(W (\overline{𝔽}_{p})))}. Now the crystal of H_{a} ×_{Spec(A_{a})} Spec(A_{a}/pA_{a}) evaluated on the P.D. thickening Spec(A_{a}/pA_{a}) ↪ Spec(A_{a}) is

Lie E(H_{a})

By functoriality of the crystal functor f_{1} and f_{2} induce

\xymatrix{𝔻(ℍ) ⊗_{O_{L}} A_{a} \ar[r]^{𝔻(f_{1})} & Lie E(H_{a}) \ar[l]^{𝔻(f_{2})}}

such that 𝔻(f_{1}) ∘ 𝔻(f_{2}) = 𝔻(f_{1} ∘ f_{2}) = 𝔻(p^{N}Id) = p^{N}Id and 𝔻(f_{2}) ∘ 𝔻(f_{1}) = p^{N}Id. Thus there is a canonical isomorphism

Lie E(H_{a})[\frac{1}{p}] ≃ 𝔻(ℍ) ⊗_{O_{L}} A_{a}

induced by p^{k}𝔻(f_{1}) where k ∈ ℤ and p^{k}f_{1} ≡ ρ_{a} modulo (X_{1}, …, X_{n−1}).

Now those morphisms are compatible for varying a : for a ≤ b there is a restriction morhpism A_{a} ⟶ A_{b} and the following diagramm commutes

\xymatrix{Lie E(H_{b})[\frac{1}{p}] \ar[r]^{∼} \ar[d] & 𝔻(ℍ) ⊗ A_{b} \ar[d] \\ Lie E(H_{a})[\frac{1}{p}] \ar[r]^{∼} & 𝔻(ℍ) ⊗ A_{a}}

This follows from the commutativity of the following diagramm of P.D. thickenings

\xymatrix{ & Spec(A_{b}/pA_{b}) \ar@{^{(}->}[r] \ar@{-->}[ddl] & Spec(A_{b}) \ar[ddl] \\ Spec(A_{a}/pA_{a}) \ar@{^{(}->}[r] \ar[ur] \ar[d] & Spec(A_{a}) \ar[d] \ar[ur] \\ Spec(\overline{𝔽}_{p}) \ar@{^{(}->}[r] & Spec(W (\overline{𝔽}_{p}))}

Now let’s verify this isomorphism is O_{D}^{×}-equivariant. For each g ∈ O_{D}^{×} for each a ∈ ℕ^{∗} ∃b ∈ ℕ^{∗} such that

g(𝔹(0, p^{−1/a})) ⊂ 𝔹(0, p^{−1/b})

Thus the associated co-morphism induces

g^{∗} : A_{b} ⟶ A_{a}

But one can verify ∀c A_{c} = {f ∈ A_{c} | ∥f∥_{A_{c},∞} ≤ 1} where ∥f∥_{A_{c},∞} = sup{|f(x)| | x ∈ Sp(A_{c})}. Thus g^{∗} induces a morphism at the level of the integral models

g^{∗} : A_{b} ⟶ A_{a}

since ∀f ∈ A_{b} ∥g^{∗}f∥_{A_{a},∞} ≤ ∥f∥_{A_{b},∞}. But now for the quasi-isogeny

ℍ ×_{Spec(\overline{𝔽}_{p})} Spec(A_{b}/pA_{b}) ⟶^{ρ_{b}} H_{b} ×_{Spec(A_{b})} Spec(A_{b}/pA_{b})

if one takes the pullback g^{∗}ρ_{b} of ρ_{b} to Spec(A_{a}/pA_{a}) via g : Spec(A_{a}/pA_{a}) ⟶ Spec(A_{b}/pA_{b}) one has a commutative diagramm

[figure: ℍ ×_{Spec(\overline{𝔽}_{p})} Spec(A_{a}/pA_{a}); g^{∗}ρ_{b}; g^{−1}; H_{a} ×_{Spec(A_{a})} Spec(A_{a}/pA_{a}); ℍ ×_{Spec(\overline{𝔽}_{p})} Spec(A_{a}/pA_{a}); ρ_{a}; H_{a} ×_{Spec(A_{a})} Spec(A_{a}/pA_{a})]

where g^{−1} ∈ Aut(ℍ). From this it follows the isomorphism

h_{a} : 𝔻(ℍ) ⊗_{O_{L}} A_{a} ⟶^{∼} Lie E(H_{a})[\frac{1}{p}]

we onstructed before verifies for g^{∗} : A_{b} ⟶ A_{a} the following diagram commutes

[figure: 𝔻(ℍ) ⊗ A_{a}; g^{∗}h_{b}; 𝔻(g)^{−1}⊗Id; Lie E(H_{a})[\frac{1}{p}]; 𝔻(ℍ) ⊗ A_{a}; h_{a}; Lie E(H_{a})[\frac{1}{p}]]

14.5 Second proof via Dwork’s trick

Let

∇ : Lie E(H) ⟶ Lie E(H) ⊗_{O_{𝔛}} Ω_{𝔛/W (\overline{𝔽}_{p})}^{1}

be the Gauss Manin connection. Let’s look at the system of differential equations

(Lie E(H)^{rig})^{∇=0}

as a sheaf on the grothendieck topology of admissible open subsets where ∇v = 0 means ∀1 ≤ i ≤ n_{1} ∇_{\frac{∂}{∂x_{i}}} v = 0. One wants to prove Lie E(H)^{rig} is generated by the solutions of this system as in the Riemann Hilbert correspondence bewteen local systems of ℂ-vector spaces and vector bundles equiped with an integrable connection on a complexe manifold S

\xymatrix{F \ar@{|->}[r] & (F ⊗_{ℂ} O_{S}, Id ⊗ d) \\ E^{∇=0} & (E, ∇) \ar@{|->}[l]}

and if S is simply connected E^{∇=0} is trivial and thus E is generated by it global solutions.

Let’s first see on

𝔹(0, p^{−1/(p−1)})

Lie E(H)^{rig} is generated by its flat section. In fact if

B = O_{L} < X_{1}, …, X_{n−1}, T_{1}, …, T_{n−1} > /(X_{i}^{p−1} − pT_{i})_{1≤i≤n−1}

then Spf(B) is an integral model of 𝔹(0, p^{−1/(p−1)}) such that the ideal

(p, X_{1}, …, X_{n−1})

in B is equiped with topologicaly nilpotent divided powers (p ≠ 2). Thus there is an isomorphism

Lie(E(H)) ⊗_{O_{𝔛}} O_{Spf(B)} ≃ 𝔻(ℍ) ⊗_{O_{L}} O_{Spf(B)}

Now let E be the crystal of H mod p and E^{(p)} its pullback by the Frobenius. There are two morphisms

\xymatrix{E \ar[r]^{V} & E^{(p)} \ar[l]^{F}}

such that F V = p and V F = p induced by

\xymatrix{G \ar[r]^{F} & G^{(p)} \ar[l]^{V}}

Let φ : 𝔛 ⟶ 𝔛 be the lifting of the Frobenius morphism defined by

∀i φ^{∗}x_{i} = x_{i}^{p}

and φ = σ on W (\overline{𝔽}_{p}). Then the main constatation is

∀0 < ϵ < 1 φ^{rig}(𝔹(0, ϵ)) ⊂ 𝔹(0, ϵ^{p})

Thus

∀0 < ϵ < 1 ∃i (φ^{rig})^{i} (𝔹(0, ϵ)) ⊂ 𝔹(0, p^{−1/(p−1)})

or

𝔛^{rig} = ⋃_{i≥1}(φ^{rig})^{−i} (𝔹(0, p^{−1/(p−1)}))

Let’s set

∀k ≥ 1 B_{k} = O_{L} < X_{1}, …, X_{n−1}, T_{1}, …, T_{n−1} > /(X_{i}^{k(p−1)} − pT_{i})_{1≤i≤n−1}

thus Spf(B_{k})^{rig} = 𝔹(0, p^{− \frac{1}{k(p−1)}} ) and the morphism Spf(B_{k}) ⟶ Spf(B_{k+1}) given by X_{i} ↦ X_{i} and T_{i} ↦ X_{i}^{p−1}T_{i} induces the inclusion 𝔹(0, p^{− \frac{1}{k(p−1)}} ) ↪ 𝔹(0, p^{− \frac{1}{(k+1)(p−1)}} ) on the generic fiber

Moreover φ induces

∀k ≥ 1 φ : Spf(B_{pk}) ⟶ Spf(B_{k})

Now for all i ∈ ℕ^{∗}

(E^{(p^{i})})_{(Spec(B_{p^{i}} /pB_{p^{i}})↪Spec(B_{p^{i}}))} ≃ (φ^{i})^{∗}E_{(Spec(B_{1}/pB_{1})↪Spec(B_{1}))}

since there is a morphism of P.D. thickenings

\xymatrix{Spec(B_{p^{i}}/pB_{p^{i}}) \ar[d]^{F rob} \ar@{^{(}->}[r] & Spec(B_{p^{i}}) \ar[d]^{φ} \\ Spec(B_{1}/pB_{1}) \ar@{^{(}->}[r] & Spec(B_{1})}

Now consider the following sequence of morphisms

[figure: Lie E(H) ⊗_{O_{𝔛}} O_{Spf(B_{p^{i}})}; E_{(Spec(B_{p^{i}} /pB_{p^{i}})↪Spec(B_{p^{i}}))}; V^{i}; (E^{(p^{i})})_{(Spec(B_{p^{i}} /pB_{p^{i}})↪Spec(B_{p^{i}}))}; 𝔻(ℍ) ⊗_{O_{L},σ^{i}} O_{Spf(B_{p^{i}})}; F^{i}⊗Id; (𝔻(ℍ) ⊗_{O_{L}} O_{Spf(B_{1})}) ⊗_{O_{Spf(B_{1})},(φ^{i})^{∗}} O_{Spf(B_{p^{i}})}; (φ^{i})^{∗}E_{(Spec(B_{1}/pB_{1})↪Spec(B_{1}))}; ≃; 𝔻(ℍ) ⊗_{O_{L}} O_{Spf(B_{p^{i}})}]

Note the composite

Δ_{i} : Lie E(H) ⊗_{O_{𝔛}} O_{Spf(B_{p^{i}})} ⟶ 𝔻(ℍ) ⊗_{O_{L}} O_{Spf(B_{p^{i}})}

Then consider

[figure: \frac{1}{p^{i}} Δ_{i} : Lie E(H)^{rig} ⊗_{O_{𝔛^{rig}}} O_{Sp(B_{p^{i}} [ \frac{1}{p}])}; 𝔻(ℍ) ⊗_{O_{L}} O_{Sp(B_{p^{i}} [ \frac{1}{p}])}; _{(Lie E(H)^{rig})_{|𝔹(0,p^{− \frac{1}{p^{i}(p−1)}} )}}; 𝔻(ℍ) ⊗ O_{𝔹(0,p^{− \frac{1}{p^{i}(p−1)}} )}]

This is an isomorphism since we have defines Δ_{i} as Δ_{i} = F^{i} ⊗ Id ∘ α_{i} ∘ V^{i} for some isomorphism α_{i} and then \frac{1}{p^{i}} F^{i} ∘ α_{i}^{−1} ∘ V^{i} ⊗ Id is a reciproc isomorphism to \frac{1}{p^{i}} Δ_{i}.

Furthermore one can verify easily the isomorphisms \frac{1}{p^{i}} Δ_{i} are compatible when i varies (use once more F V = p).

14.6 The link between both proofs

Let S_{0} ↪ S be a nilpotent immersion of quasi-compact 𝔽_{p}-schemes. Let G be a p-divisible group over S and set G_{0} = G ×_{S} S_{0}. Suppose there exists a section

\xymatrix{S_{0} \ar@{^{(}->}[r] & S \ar[l]_{ϵ}}

Lemma 22. For i >> 0 G^{(p^{i})} ≃ G_{0}^{(p^{i})}×_{S_{0},ϵ}, S Proof. easy Thus for i >> 0 there is an isogeny

δ_{i} : G ⟶^{F^{i}} G^{(p^{i})} ≃ G_{0}^{(p^{i})} ×_{S_{0},ϵ} S ⟶^{V^{i}×Id} G_{0} ×_{S_{0},ϵ} S

and thus a degre 0 quasi-isogeny : \frac{1}{p^{i}} δ_{i}.

Now apply this to S = Spec(A_{k}/pA_{k}) and S_{0} = V (X_{1}, …, X_{n−1}).

14.7 The period morphism

We have a filtration by localy free sheaves

0 ⟶ V (H)^{rig} ⟶ Lie E(H)^{rig} ⟶ \underbrace{Lie (H)^{rig}}_{rank 1} ⟶ 0

that is O_{D}^{×}-equivariant. Moreover we have an O_{D}^{×}-equivariant isomorphism

Lie E(H)^{rig} ≃ 𝔻(ℍ) ⊗ O_{𝔛^{rig}}

Definition 38. We note π̆ : 𝔛^{rig} ⟶ ℙ (𝔻(ℍ)) for the O_{D}^{×}-equivariant rigid analytic morphism induced by

𝔻(ℍ) ⊗ O_{𝔛^{rig}} ≃ Lie E(H)^{rig} ↠ Lie (H)^{rig}

15 First properties of the period morphism

15.1 Étalness

Proposition 19. The period morphism π̆ is étale.

Proof. This is a consequence of Grothendieck-Messing deformation theory. One has to check for any affinoïd algebra A equiped with a squared zero ideal I the diagram

\xymatrix{𝔛^{rig}(A) \ar[r] \ar[d] & ℙ(𝔻(ℍ))^{rig}(A) \ar[d] \\ 𝔛^{rig}(A/I) \ar[r] & ℙ(𝔻(ℍ))^{rig}(A/I)}

is cartesian. This is a bit technical because one has to chose models for Spf(A)...this is done in ? ? ?. But at the end this is a consequence of Grothendieck-Messing deformation theory.

Remark 26. There are different notions of étalness for morphisms between rigid spaces. For example the inclusion of the unit ball 𝔹(0, 1) ↪ 𝔸^{1} is not étale in Berkovich’s theory but is in the classical theory of Tate (since it is an open immersion). In terms of classical rigid spaces the difference is between étale morphisms and overconvergent étale one (the overconvergent one being the étale morphisms in Berkovich’s theory). But for morphisms between 𝔛^{rig} and ℙ^{n} both notions coïncide (in terms of Berkovich’s langage : ∂𝔛^{an} = ∅ and ∂ℙ^{n} = ∅ thus for any f : 𝔛^{an} ⟶ ℙ^{n} ∂(𝔛^{an}/ℙ^{n}) = ∅).

15.2 Fibers

Let K|L be a valued extension for a valuation v : K ⟶ ℝ∪{+∞} and such that K is complete for this valuation.

Proposition 20. Let x = (H, ρ) ∈ 𝔛^{rig}(K) = 𝔛(O_{K}). Then the fiber

π̆^{−1}(π̆(x)) ⊂ 𝔛^{rig}(K)

is the set of (H′, ρ′) such that there exists an isogeny f : H ⟶ H′ over Spec(O_{K}) and an integer N ∈ ℕ s.t. the following diagram commutes

\xymatrix{ℍ ⊗_{\overline{𝔽}_{p}} O_{K}/pO_{K} \ar[r]^{ρ} \ar[d]^{p^{N}} & H mod p \ar[d]^{f mod p} \\ ℍ ⊗_{\overline{𝔽}_{p}} O_{K}/pO_{K} \ar[r]^{ρ′} & H′ mod p}

or equivalently such that the quasi-isogeny ρ′ ∘ ρ^{−1} between H mod p and H′ mod p lifts to a quasi-isogeny between H and H′ over Spec(O_{K}).

Proof. If the quasi-isogeny ρ′ ∘ ρ^{−1} over Spf(O_{K}) lifts to a quasi-isogeny f over Spec(O_{K}) then it induces an isomorphism

\xymatrix{Fil Lie E(H)[\frac{1}{p}] \ar@{^{(}->}[r] \ar[dd]^{≃}_{f_{∗}} & Lie E(H)[\frac{1}{p}] \ar[dd]^{≃}_{f_{∗}} \ar@{-->}[dr]^{≃} \\  &  & 𝔻(ℍ)[\frac{1}{p}] ⊗ K \\ Fil Lie E(H′)[\frac{1}{p}] \ar@{^{(}->}[r] & Lie E(H′)[\frac{1}{p}] \ar@{-->}[ur]^{≃}}

by functoriality of the universal vector extension. Thus they have the same image by the period morphism. Reciprocaly, if (H′, ρ′) has the same image by the period morphism as (H, ρ) then the induced morphism at the level of the evaluation of the crystals of H mod p and H′ mod p on O_{K} ↠ O_{K}/pO_{K} by the quasi-isogeny ρ′ ∘ ρ^{−1} induces an isomorphism like in the preceding diagram. Thus for N >> 0 p^{N}ρ′ ∘ ρ^{−1} induces a commutative diagram

\xymatrix{Fil Lie E(H) \ar@{^{(}->}[r] \ar[d] & Lie E(H) \ar[d]^{𝔻(p^{N} ρ′∘ρ^{−1})=p^{N} 𝔻(ρ′∘ρ^{−1})} \\ Fil Lie E(H′) \ar@{^{(}->}[r] & Lie E(H′)}

and thus by Messing’s theorem p^{N}ρ′ ∘ ρ^{−1} lifts to O_{K}. In the same way for N >> 0 p^{N}ρ ∘ ρ′^{−1} lifts to O_{K}.

Lemma 23. Let H be a p-divisible group over O_{K}. Then the isomorphism classes of couples (H′, f) where H′ is a p-divsible group over O_{K} and f : H ⟶ H′ is a quasi-isogeny over Spec(O_{K}) are in bijection with Gal(\overline{K}|K)-stable lattices inside V_{p}(H).

Proof. Such isomorphism classes are in bijection with equivalence classes of couples (N, H) where N ∈ ℤ and H is a finite localy free sub-group scheme of H over O_{K} and where the equivalence relation is the one generated by (N, H) is equivalent to (N − 1, p^{−1}(H)). To (N, H) is associated p^{N}× the isogeny H ⟶ H/N.

But since O_{K} is a valuation ring, for any k ∈ ℕ^{∗} the map H ⟼ H ⊗_{O_{K}} K from finite flat sub-group schemes of H[p^{k}] to the one of H[p^{k}]⊗_{O_{K}} K is a bijection. The reverse bijection is given by schematical closure.

Thus couples (N, H) are in bijection with couples (N, W) where N ∈ ℤ and W ⊂ T_{p}(H) ⊗ ℚ_{p}/ℤ_{p} is a Gal(\overline{K}|K)-stable finite sub-group. Now equivalence classes of such couples are in bijection with Galois-stable lattices in V_{p}(H). To Λ is associated (N, p^{−N}Λ/T_{p}(H)) for N >> 0.

Corollary 13. Suppose K = \overline{K}. Let x = (H, ρ) ∈ 𝔛^{rig}(K) = 𝔛(O_{K}). Then the fiber

π̆^{−1}(π̆(x)) ⊂ 𝔛^{rig}(K)

is a homogenous space under GL(V_{p}(H))^{1} the stabilizer of x being GL(T_{p}(H)). This fiber is thus in bijection with

GL_{n}(ℚ_{p})^{1}/GL_{n}(ℤ_{p})

16 An explicit formula for the period morphism

For any ring R we note W (R) fo its witt vectors. There is an augmentation W (R) ↠ R that sends [x_{0}, …] to x_{0}. The associated augmentation ideal is equiped with divided powers (functorialy in R, as usual is suficies to verify it in a universal situation when R is torsion free, R = ℤ[X_{i}]_{i≥0}). This divided powers are compatible with the one on the ideal (p) of ℤ_{(p)} . If R is a p-adic ring then W (R) is p-adic and if p ≠ 2 the divided powers on the augmentation ideal of W (R) are topologicaly nilpotent. Moerover W (R) is canonicaly equiped with a litfting of Frobenius F : W (R) ⟶ W (R) relative to W (R) ⟶ R/pR :

\xymatrix{W (R) \ar[r] \ar[d]^{F} & R/pR \ar[d]^{F rob} \\ W (R) \ar[r] & R/pR}

and if I_{R} = ker(W (R) ⟶ R) then (I_{R}, p) is equiped with topologicaly nilpotent divided powers (p ≠ 2) since the divided powers on I_{R} are compatible with the one on (p).

Theorem 21 (Zink). There exists a system of formal coordinates (x_{1}, …, x_{n−1}) on 𝔛 s.t. if R = W (\overline{𝔽}_{p})[[x_{1}, …, x_{n−1}]] and H is the universal deformation over 𝔛 then the evaluation of the crystal of H over W (R) relative to the P.D. thickening W (R) ↠ R is

𝔻(H)_{W (R)↠R} = W (R)e_{1} ⊕ ⋯ ⊕ W (R)e_{n}

the associated morphism 𝔻(H)_{W (R)↠R} ⟶ 𝔻(H)_{R⟶^{Id}R} = Lie E(H) ⟶ Lie H giving the filtration of the filtered module is identified with the projection

W (R)e_{1} ⊕ … W (R) ⊕ e_{n} ↠ Re_{1}

Moerover the associated Frobenius morphism associated to the Verschiebung V : G^{(p)} ⟶ G, the P.D. thickening W (R) ↠ R/pR and the lifting of Frobenius F on W (R)

[figure: 𝔻(H)_{W (R)↠R} ⊗_{W (R),F} W (R); F robenius; 𝔻 ((H ⊗_{R} R/pR)^{(p)})_{W (R)↠R/pR}; 𝔻(V); 𝔻(H ⊗_{R} R/pR)_{W (R)↠R/pR}; 𝔻(H)_{W (R)↠R}]

has as a matrix in the bases (e_{i} ⊗ 1)_{i} and (e_{i})_{i}

(\begin{matrix}[x_{1}] & p[x_{2}] & … & p[x_{n−1}] & p \\ 1 & 0 & … & 0 & 0 \\ 0 & p & & 0 & 0 \\ & 0 & ... & ⋮ & ⋮ \\ & & & p & 0 \end{matrix})

where [x_{i}] ∈ W (R) is the Teichmüller lift of x_{i}.

In this coordinates if

(\begin{matrix}x_{1} & px_{2} & … & px_{n−1} & p \\ 1 & 0 & … & 0 & 0 \\ 0 & p & & 0 & 0 \\ & 0 & ... & ⋮ & ⋮ \\ & & & p & 0 \end{matrix})

is the reduction of the preceding matrix through W (R) ↠ R and

B = (\begin{matrix}0 & 0 & … & 0 & p \\ 1 & 0 & … & 0 & 0 \\ 0 & p & & 0 & 0 \\ & 0 & ... & ⋮ & ⋮ \\ & & & p & 0 \end{matrix})

is the matrix of the Frobenius of the crystal of ℍ then one has the following formula for the period morphism

π̆ = lim_{k→+∞}[1 : 0 : …: 0].AA^{(σ)} … A^{(σ^{k−1})}.B^{−k}

where A^{(σ^{k})} is for the matrix A where one replaces all x_{i} by x_{i}^{p^{k}}. This limit has to be taken in the sens of the Frechet topology of Γ(𝔛^{rig}, O_{𝔛^{rig}}) defined by uniform convergence on each closed ball with radius < 1.

16.1 The GL_{2}-case

Theorem 22. Let 𝔛 = Spf(W (\overline{𝔽}_{p})[[x]]) be the Lubin-Tate space for n = 2. The period morphism π̆ = [f_{0} : f_{1}] induces an isomorphism from the open ball

\{x ∈ 𝔛^{rig} | v(x) > \frac{1}{p + 1}\}

to the open ball

\{[y_{0} : y_{1}] ∈ ℙ^{1} | v(y_{1}/y_{0}) > \frac{1}{p + 1}\}

moreover on this ball we have

v(f_{1}(x)/f_{0}(x)) = v(x)

17 The Lubin-Tate tower

17.1 Motivation

As allready remarked, the upper/lower Poincaré half plane H^{±} = ℂ \ ℝ represents the functor from smooth analytic spaces S to sets that associates to S the isomorphism classes of couples (E, ρ) where E ⟶^{p} S is an elliptic curve and ρ : \underline{ℤ}^{2} ⟶^{∼} R^{1}p_{∗}\underline{ℤ}. Instead of trivialising the Betti cohomology one can trivialse it partialy : for N ≥ 3 one can consider the functor X(N) that associates to S the isomorphism classes of couples (E, ρ) where E is as before and

ρ : (\underline{Z}/N\underline{ℤ})^{2} ⟶^{∼} R^{1}p_{∗}\underline{ℤ}/N\underline{ℤ}

This is the level N modular curve.

On Lubin-Tate spaces the situation is more complicated : there are two fiber functors on pdivisible groups in unequal caracteristic, the Dieudonné functor (crystalline cohomology of the special fiber) and the Tate module functor (the p-adic étale cohomology of the generic fiber). We have allready trivialized the Dieudonné functor on Lubin-Tate spaces by specifying the isomorphism class of our p-divisible group on the special fiber. Now, our space is not simply connected as is H^{±}. We still can trivialize the Tate-module, at least partialy like in the case of X(N).

17.2 Level structures on the generic fiber

Let 𝔛 = Spf(R) be the Lubin-Tate space of deformations of the p-divisible group ℍ of height n. Let (H, ρ) be the universal deformation. We still note H for the associated p-divisible group over Spec(R).

Let’s consider the Tate module T_{p}(H) of H ×_{Spec(R)} Spec(R[\frac{1}{p}]). We can see it as an étale p-adic local system on 𝔛^{rig} : the system (H[p^{k}]^{rig})_{k≥1}.

Definition 39. Let X = 𝔛^{rig}. We define forall k ≥ 1 X_{k} to be the functor on rigid analytic spaces over 𝔛^{rig} that associates to S ⟶ X the sets of isomorphism of étale sheaves

η : (p^{−k}ℤ/ℤ)^{n} ⟶^{∼} H[p^{k}]^{rig} ×_{X} S

called a level k structure.

Lemma 24. X is represented by a finite étale Galois covering of X with Galois group GL_{n}(ℤ/p^{k}ℤ).

Proof. There is a right action of GL_{n}(ℤ/p^{k}ℤ) on X by η ↦ η∘g. After a finite étale covering of X trivializing H[p^{k}] that is to say X′ ⟶ X such that H[p^{k}]×_{S} X′ ≃ (p^{−k}ℤ/ℤ)^{n}, X_{k} ×_{X} X′ becomes a GL_{n}(ℤ/p^{k}ℤ)-Galois cover over X′. Thus by finite étale descent this proves the assertion.

We thus have a tower of étale coverings

… ⟶ X_{k+1} ⟶ X_{k} ⟶ … ⟶ X_{1} ⟶ X

with an action of GL_{n}(ℤ_{p}) wich make it “pro-Galois covering of X with Galois group GL_{n}(ℤ_{p})”.

17.3 Drinfeld level structures

Definition 40. Let (A, 𝔪) be a complete local ring with caratetristic p residue field. Let F ∈ A[[X, Y]] be a height n formal group law on R. A level k Drinfeld structure on F is a group morphism

η : (p^{−k}ℤ/ℤ)^{n} ⟶ F [p^{k}](R)

that is to say a group morphism η : (p^{−k}ℤ/ℤ)^{n} ⟶ (𝔪, +_{F}) such that

∏_{x∈(p^{−k}ℤ/ℤ)^{n}}(T − η(x)) | [p^{k}]_{F} (T)

Remark 27. The preceding definition depends only on the formal group and not on the group law since for any f ∈ A[[T]], f(0) = 0 and f′(0) ∈ A^{×}

∏_{x∈(p^{−k}ℤ/ℤ)^{n}}(T − η(x)) | [p^{k}]_{F} (T) ⟺ ∏_{x∈(p^{−k}ℤ/ℤ)^{n}}(T − f(η(x))) | f ∘ [p^{k}]_{F} ∘ f^{−1}

Hence the preceding definition makes sens for a formal p-divisible group H without any particular choice of an isomorphism of pointed formal schemes Spf(A[[T]]) ⟶^{∼} H.

We will use repeteadly the following form of Weierstrass division

Lemma 25. Let A be an I-adicaly compete ring. Let g = ∑_{i≥0} a_{i}T^{i} ∈ A[[T]]. Suppose ∃n a_{n} ∈ A^{×} and ∀i < n a_{i} ∈ I. Then for all f ∈ A[[T]] ∃Q, R such that Q ∈ A[[T]], R ∈ A[T], deg R < n and

f = Qg + R

Lemma 26. Let F (X, Y) be a formal group law over a ring B. Then ∃w ∈ B[[X, Y]]^{×} such that

X −_{F} Y = (X − Y) × w

Proof. We apply Weierstrass division theorem in the ring C = B[[Y]], complete for the Y -adic topology. The series X − Y ∈ C[[X]] verifies that the coefficient of X is 1, a unit, and −Y is topologicaly nilpotent. Thus we can apply Weierstrass division

∃f ∈ C[[X]] ∃g ∈ C X −_{F} Y = (X − Y)f + g

But puting X = Y one sees g = 0. Thus X −_{F}Y = (X −Y)f. Now since X −_{F}Y ≡ X −Y mod deg 2

one sees f(0) = 1 and thus f is a unit.

Remark 28. Last lemma is false for X +_{F} Y and X + Y as one sees in the case of 𝔾̂_{m}.

Corollary 14. In the definition of a Drinfeld level structure one can replace

∏_{x∈(p^{−k}ℤ/ℤ)^{n}}(T − η(x))

by

∏_{x∈(p^{−k}ℤ/ℤ)^{n}}(T −_{F} η(x))

Proposition 21. With the hypothesis of the preceding definition for a morphism

η : (p^{−k}ℤ/ℤ)^{n} ⟶ (𝔪, +_{F})

the following are equivalent – η is a Drinfeld level structure

– [p^{k}]_{F} (T) = u × ∏_{x∈(p^{−k}ℤ/ℤ)^{n}}(T − η(x)) where u ∈ A[[T]]^{×}.

– ∏_{x∈(p^{−1}ℤ/ℤ)^{n}}(T − η(x)) | [p]_{F} (T)

– [p]_{F} (T) = v × ∏_{x∈(p^{−1}ℤ/ℤ)^{n}}(T − η(x)) with v ∈ A[[T]]^{×}.

Proof. The second assertion clearly implies the first. Let’s see the first assertion implies the second one. If

[p^{k}]_{F} (T) = f × ∏_{x∈(p^{−k}ℤ/ℤ)^{n}}(T − η(x))

for somme f ∈ A[[T]] then by reducing modulo the maximal ideal of A, using the fact that [p^{k}]_{F} is congruent to unit×T^{p^{n}} modulo 𝔪 and degre ≥ p^{n} +1, one sees f(0) is a unit and thus f ∈ A[[T]]^{×}. Thus the two first assertiion are equivalent.

In the same way one proves assertions three and four are equivalent.

Let’s prove assertion three implies the first one. First by the preceding lemma we have to prove

∏_{x∈(p^{−1}ℤ/ℤ)^{n}}(T −_{F} η(x)) | [p]_{F} (T) ⟹ ∏_{x∈(p^{−k}ℤ/ℤ)^{n}}(T −_{F} η(x)) | [p^{k}]_{F} (T)

Thus let’s suppose

[p]_{F} (T) = f(T) × ∏_{x∈(p^{−1}ℤ/ℤ)^{n}}(T −_{F} η(x))

for some f(T). Let’s prove by induction on i, starting at i = 1 and finishing at i = k, the following assertion

∏_{x∈(p^{−i}ℤ/ℤ)^{n}}(T −_{F} η(x)) | [p^{i}]_{F} (T)

Suppose it is verified for i − 1. Then

[p]_{F} (T) = f(T)× ∏_{x∈(p^{−1}ℤ/ℤ)^{n}}(T −_{F}η(x)) ⟹ [p^{i}]_{F} (T) = f([p^{i−1}]_{F} (T))× ∏_{x∈(p^{−1}ℤ/ℤ)^{n}}([p^{i−1}]_{F} (T)−_{F}η(x))

If we choose for each x ∈ (p^{−1}ℤ/ℤ)^{n} an element x̃ ∈ (p^{−i}ℤ/ℤ)^{n} such that p^{i−1}x̃ = x then

[p^{i}]_{F} (T) = f([p^{i−1}]_{F} (T)) × ∏_{x∈(p^{−1}ℤ/ℤ)^{n}}([p^{i−1}]_{F} (T) −_{F} [p^{i−1}]_{F} (η(x̃)))
= f([p^{i−1}]_{F} (T)) × ∏_{x∈(p^{−1}ℤ/ℤ)^{n}}[p^{i−1}]_{F} (T −_{F} η(x̃))

By the induction hypothesis

∃g(T) [p^{i−1}]_{F} = g(T) × ∏_{y∈(p^{−i+1}ℤ/ℤ)^{n}}(T −_{F} η(y))

And thus

[p^{i}]_{F} (T) = f([p^{i−1}]_{F} (T)) × ∏_{{x∈(p^{−1}ℤ/ℤ)^{n} \atop y∈(p^{−i+1}ℤ/ℤ)^{n}}} g(T −_{F} η(x̃)) \underbrace{(T −_{F} η(x̃) −_{F} η(y))}_{T −_{F}η(x̃+y)}

and thus

∏_{{x∈(p^{−1}ℤ/ℤ)^{n} \atop y∈(p^{−i+1}ℤ/ℤ)^{n}}} (T −_{F} η(x̃ + y)) | [p^{i}]_{F} (T)

But the family of x̃ + y, with x, y as in the preceding formula, is exactly (p^{−i}ℤ/ℤ)^{n}.

Now let’s prove assertion two implies assertion three. Thus suppose

[p^{k}]_{F} (T) = w × ∏_{x∈(p^{−k}ℤ/ℤ)^{n}}(T −_{F} η(x))

with w ∈ A[[T]]^{×}. Writte

[p]_{F} (T) = ∏_{x∈(p^{−1}ℤ/ℤ)^{n}}(T −_{F} η(x)) × Q(T) + R(T)

for the Weierstrass division of [p]_{F} by the polynomial ∏_{x∈(p^{−1}ℤ/ℤ)^{n}}(T −_{F} η(x)) that is to say

Q ∈ A[[T]] and R ∈ A[T] deg R < p^{n}. As before one verifies Q ∈ A[[T]]^{×}. Moreover by reducing modulo 𝔪, since [p]_{T} ≡ unit × T^{p^{n}} mod deg > p^{n} we have R ∈ 𝔪[T]. Putting T = 0 one finds

R ∈ T 𝔪[T]

Now suppose R ∈ 𝔪^{k}[T] for some k ≥ 1. We will now prove R ∈ 𝔪^{k+1}[T]. This will allow us to conclude since then R ∈ (∩_{k≥1}𝔪^{k})[T] = (0).

Thus, let’s compute in A/𝔪^{k+1}[[T]]. Then we prove by induction on i for 1 ≤ i ≤ k

[p^{i}]_{F} ≡ ∏_{x∈(p^{−i}ℤ/ℤ)^{n}}(T −_{F} η(x)) × Q_{i}(T) + R(a_{i}T^{p^{(i−1)n}}) mod (𝔪^{k+1}[[T]], T^{p^{in}}𝔪^{k}[[T]])

where Q_{i} ∈ A[[T]]^{×} and a_{i} ∈ A^{×}. In fact if it is verified for i − 1 that is to say

[p^{i−1}]_{F} ≡ ∏_{x∈(p^{−i+1}ℤ/ℤ)^{n}}(T −_{F} η(x)) × Q_{i−1}(T) + R(a_{i−1}T^{p^{(i−2)n}}) mod (𝔪^{k+1}[[T]], T^{p^{(i−1)n}}𝔪^{k}[[T]])

then

[p^{i}]_{F} ≡ ∏_{x∈(p^{−i+1}ℤ/ℤ)^{n}}[p]_{F} (T −_{F}η(x̃))×Q_{i−1}∘[p]_{F} +R(a_{i−1}[p]_{F}^{p^{(i−2)n}}) mod (𝔪^{k+1}[[T]], [p]_{F}^{p^{(i−1)n}}𝔪^{k}[[T]])

where x̃ ∈ (p^{−i}ℤ/ℤ)^{n} is such that px̃ = x. But the ideal (𝔪^{k+1}[[T]], [p]_{F}^{p^{(i−1)n}}𝔪^{k}[[T]]) is equal to (𝔪^{k+1}[[T]], T^{p^{in}}) since [p]_{F} ≡ unit × T^{p^{n}} mod (𝔪[[T]], T^{p^{n}+1}𝔄[[T]]). Now in the same way one has

R(a_{i−1}[p]_{F}^{p^{(i−2)n}}) ≡ R(unit × T^{p^{(i−1)n}})

thus, we have modulo (𝔪^{k+1}[[T]], T^{p^{in}}𝔪^{k}[[T]])

[p^{i}]_{F} ≡ ∏_{x∈(p^{−i+1}ℤ/ℤ)^{n}} ( ∏_{y∈(p^{−1}ℤ/ℤ)^{n}}(T −_{F} η(x̃ + y)).Q(T −_{F} η(x̃)) + R(T −_{F} η(x̃))) .Q([p]_{F})+R(a_{i}T^{p^{(i−1)n}})

Now R(T −_{F} η(x̃)) ≡ R(T), R(T)^{2} ≡ 0 and Q(T −_{F} η(x̃))R(T) ≡ Q(T)R(T). Thus

[p^{i}]_{F} ≡ Q([p]_{F}) ∏_{{x∈(p^{−i+1}ℤ/ℤ)^{n} \atop y∈(p^{−1}ℤ/ℤ)^{n}}} (T −_{F} η(x̃ + y))Q(T −_{F} η(x̃)) + p^{i−1}Q([p]_{F})R(T) + R(a_{i}T^{p^{(i−1)n}})

But p^{i−1} ∈ 𝔪 and thus p^{i−1}R(T) ≡ 0. We thus obtain the induction hypothesis for i easily. Thus we now have

[p^{k}]_{F} ≡ ∏_{x∈(p^{−k}ℤ/ℤ)^{n}}(T −_{F} η(x)) × Q_{k}(T) + R(a_{k}(T^{p^{(k−1)n}})) mod (𝔪^{k+1}[[T]], T^{p^{kn}}𝔪^{k}[[T]])

where Q_{k} ∈ A[[T]]^{×} and a_{x}k ∈ A^{×}. And by hypothesis

[p^{k}]_{F} (T) = w × ∏_{x∈(p^{−k}ℤ/ℤ)^{n}}(T −_{F} η(x))

with w ∈ A[[T]]^{×}. Using Q_{k} and w are units plus the fact that deg R < p^{n} one concludes R ≡ 0 mod 𝔪^{k+1}[T].

17.4 Drinfeld integral models

Proposition 22. Consider the functor on artin local ring with residue field \overline{𝔽}_{p} that associates to A the set of isomorphism classes of triples (H, ρ, η) where H is a p-divisible group over A,

ρ : ℍ ⟶^{∼} H ⊗_{A} \overline{𝔽}_{p}

and

η : (p^{−k}ℤ/ℤ)^{n} ⟶ H[p^{k}](A)

is a Drinfeld level structure. This functor is pro-representable by a formal scheme 𝔛_{k} = Spf(R_{k}) where R_{k} is a complete local ring.

Proof. By associating (H, ρ) to (H, ρ, η) one sees this functor is over Lubin-Tate space 𝔛 = Spf(R). Moreover if F (X, Y) ∈ R[[X, Y]] is a universal formal group law over R this functor is a subfunctor of

𝔜 = F [p^{k}] ×_{𝔛} ⋯ ×_{𝔛} F [p^{k}]

where F [p^{k}] = Spf(R[[T]]/([p^{k}]_{F})). We have 𝔜 = Spf(B) with

B = R[[T_{1}]]/([p^{k}]_{F} (T_{1})) ˆ⊗_{R} … ˆ⊗_{R}R[[T_{n}]]/([p^{k}]_{F} (T_{n}))

and over 𝔜 there ia a universal η :

η : (p^{−k}ℤ/ℤ)^{n} ⟶ H ×_{𝔛} 𝔜

given by e_{i} ↦ T_{i} where (e_{i})_{1≤i≤n} is the canonical base of (p^{−k}ℤ/ℤ)^{n}. Now our functor is the subfunctor of 𝔜 where this η satisfies Drinfeld divisibility condition. First let’s remark B is a complete local ring finite over R. In fact By Weierstrass R[[T]]/([p^{k}]_{F} (T)) is a finite R-algebra thus B is finite over R (finiteness implies one can replace ˆ⊗ by ⊗) and

B ≃ R[[T_{1}, …, T_{n}]]/([p]_{F} (T_{1}), …, [p]_{F} (T_{n}))

with each [p^{k}]_{F} (T_{i}) in the maximal ideal (𝔪, T_{1}, …, T_{n}) of the complete local ring R[[T_{1}, …, T_{n}]] with 𝔪 being the maximal ideal of R, which implies B is local and complete.

Now we can apply Weierstrass division theorem to the ring B, [p^{k}]_{F} (U) ∈ B[[U]] and ∏_{x∈(p^{−k}ℤ/ℤ)^{n}}(U− T_{i}) :

[p^{k}]_{F} (U) = ∏_{x∈(p^{−k}ℤ/ℤ)^{n}}(U − T_{i}).Q(U) + R(U)

where R(U) ∈ B[U] and deg R < p^{nk}. Writte

R = ∑_{i=0}^{p^{nk}−1} a_{i}U^{i}

Reducing modulo the maximal ideal of B the preceding division on sees forall i a_{i} lies in this maximal ideal. Thus

R_{k} = B/(a_{1}, …, a_{p^{nk}−1})

is a complete local ring such that Spf(R_{k}) represents our functor.

Theorem 23. For all k the complete local rings R_{k} are regular with a system of parameter given by (η(p^{−k}e_{1}), …, η(p^{−k}e_{n})), η being the universal level structure and (e_{1}, …, e_{n}) the canonical basis of ℤ^{n}.

Proof. We have seen in the preceding proof that R_{k} is a finite R-algebra. Since dim R = n we deduce dim R_{k} = n. Thus it suffices to prove

R_{k}/(η(p^{−k}e_{1}), …, η(p^{−k}e_{n})) = \overline{𝔽}_{p}

But by proposition ? ? ? and the preceding proof giving an explicit construction of R_{k}

R_{k} = R_{1}[[T_{1}, …, T_{n}]]/ (([p^{k−1}]_{F} (T_{1}) − η(p^{−1}e_{1})), …, ([p^{k−1}]_{F} (T_{n}) − η(p^{−1}e_{n})))

where F is the universal formal group law and on R_{k}

∀i η(p^{−k}e_{i}) = T_{i}

Thus

R_{k}/(η(p^{−k}e_{1}), …, η(p^{−k}e_{n})) = R_{1}/(η(p^{−1}e_{1}), …, η(p^{−1}e_{n}))

We thus have to prove the assertion for R_{1}. But now thanks to ? ? ? we can choose formal coordinates (x_{1}, …, x_{n−1}) on R such that

[p]_{F} = pu_{0}T + x_{1}u_{1}T^{p} + ⋯ + x_{n−1}u_{n−1}T^{p^{n−1}} + u_{n}T^{p^{n}}

with ∀i u_{i} ∈ W (\overline{𝔽}_{p})[[x_{1}, …, x_{n−1}]][[T]]^{×}. Writting Drinfeld divisibility condition we find

unit × T^{p^{n}} = pu_{0}T + x_{1}u_{1}T^{p} + ⋯ + x_{n−1}u_{n−1}T^{p^{n−1}} + u_{n}T^{p^{n}} modulo (η(p^{−1}e_{1}), …, η(p^{−1}e_{n}))

and thus

∀i x_{i} ∈ (η(p^{−1}e_{1}), …, η(p^{−1}e_{n})) and p ∈ (η(p^{−1}e_{1}), …, η(p^{−1}e_{n}))

But, as seen in the explicit given construction of R_{1} from R, R_{1} as an R-algebra is generated by (η(p^{−1}e_{1}), …, η(p^{−1}e_{n})). Thus

R_{1}/(η(p^{−1}e_{1}), …, η(p^{−1}e_{n})) = R/(p, x_{1}, …, x_{n−1}) = \overline{𝔽}_{p}

Remark 29. This proof consists just in verifying any Lubin-Tate group with a trivial Drinfeld level structure is trivial that is to say isomorphic to ℍ since Spf(R_{k}/((η(p^{−k}e_{1}), …, η(p^{−k}e_{n}))) is the moduli space of (H, ρ, η) with trivial η.

Corollary 15. Putting R = R_{0}, forall k′ ≥ k ≥ 0 R_{k′} is a finite flat R_{k}-algebra.

Proof. We have allready seen it is finite. But both ring being regular it is flat since any finite type morphism between noetherian regular rings with fibers of the same dimension is flat.

Now there is a right action of GL_{n}(ℤ/p^{k}ℤ) on 𝔛_{k} over 𝔛 by η ⟼ η ∘ g.

Proposition 23. Spec(R_{k}[\frac{1}{p}]) ⟶ Spec(R[\frac{1}{p}]) with its action of GL_{n}(ℤ/p^{k}ℤ) is the moduli space of trivializations of the finite étale group scheme H[p^{k}] ⊗_{R} R[\frac{1}{p}].

Proof. Writte [p^{k}]_{F} = u(T) × f(T) with u an invertible series and f(T) a polynomial. After inverting p Drinfeld condition becomes : all η(x) form distinct roots of the separable polnyomial f(T).

A Finite flat group schemes

All schemes will be separated. If S is a scheme we will consider the associated fppf topology. It is the topology associated to all schemes over S and whose covering morphisms are generated by faithfully flat of finite presentation morphisms of S-schemes (by topology associated to we mean the topology associated to the preceding pretopology). We refer to [SGA 3, tome I, exposé IV] for generalities on Grothendieck topologies and quotients of sheaves.

In the following notes we will work with general hypothesis like “of finite presentation” or “finite localy free”. For a first approach the reader can suppose he is working over a noetherian base and all its schemes are of finite type. This is what the author of this note used to do until the day he really had to use the theory over non-noetherian bases.

A.1 General results of algebraic geometry

A.1.1 Finite flat/finite localy free

Recall the following very basic lemma from commutative algebra, since we use it without quoting.

Lemma 27. Let f : X ⟶ S. Then X is finite localy free over S iff f is finite flat of finite presentation.

In particular if S is localy noetherian finite flat over S is equivalent to finite localy free over S.

Recall moreover that a finite morphism f : X ⟶ S is of finite presentation iff f_{∗}O_{X} is localy of finite presentation as an O_{S}-module. That is to say a finite A-algebra B is of finite presentation as an A-algebra iff it is of finite presentation as an A-module.

A.1.2 Flatness fiber by fiber

Recall :

Theorem 24 (Grothendieck, EGA IV Coro. 11.3.11). Let

\xymatrix{X \ar[rr]^{f} \ar[dr]_{g} &  & Y \ar[dl]^{h} \\  & S}

be a diagram of localy of finite presentation morphisms of schemes. Then the following are equivalent

– g is flat and ∀s ∈ S f_{s} : X_{s} ⟶ Y_{s} is flat

– f is flat and h is flat at all points of f(X)

A.1.3 Monomorphisms

Proposition 24 (EGA IV, proposition 8.1.5). A proper morphism of schemes is a monomorphism iff it is a closed immersion.

Corollary 16. Let S be a scheme. A morphism from a finite S-group-scheme to a finite type S-groups-scheme is a monomorphism iff it is a closed immersion iff the kernel is trivial.

Remark 30. We refer to [SGA 3 exposé XVI, chapter 1] for examples of monomorphisms of group schemes that are not immersions.

A.1.4 Connected components of a finite flat morphism

Recall for a quasi-finite morphism f : X ⟶ S and s ∈ S the separable rank of the fiber X_{s} is |X_{s}(k(s)^{sep})| where k(s)^{sep} is a separable closure of k(s). One has X_{s} ≃ Spec(A) where A is a finite k(s)-algebra. It is thus artinian isomorphic to a finite product

A ≃ ∏_{i∈I} B_{i}

where each B_{i} is artinian local with residue field k_{i} that is a finite extension of k(s). Then the separable rank of X_{s} is

∑_{i∈I}[k_{i} : k(s)]^{sep}

where [. : .]^{sep} means separable degre.

Lemma 28 ([?], lemma 4.8 p.63). Let f : X ⟶ S be finite and localy free. Then there exists a factorization

f : X ⟶^{g} Y ⟶^{h} S

with h and g finite localy free, h étale and g radicial surjective iff the separable degre of the fibers of f is localy constant. If this is the case then this factorization is unique up to a unique isomorphism and functorial in X/S.

We refer to [?] for the details of the proof, but let’s point out that when S = Spec(k) is the spectrum of a field the proof is the following. Let A ≃ ∏_{i∈I} B_{i} be a finite k-algebra as before with B_{i} local artinian and residue field k_{i}. Then if for each i k′_{i} is the separable closure of k in k_{i} thanks to the fact that k′_{i}|k is separable (and thus étale) there exists a unique lift

\xymatrix{B_{i} \ar@{->>}[r] & k_{i} \\  & k′_{i} \ar@{^{(}->}[u] \ar@{^{(}->}[ul]}

of k-algebra morphisms from k′_{i} into B_{i}. Then Y = Spec(∏_{i∈I} k′_{i}) and at the level of algebras the factorization is

k ⟶ ∏_{i∈I} k′_{i} ⟶ ∏_{i∈I} B_{i} = A

Here is a particular case of [EGA IV, proposition 15.5.1] :

Proposition 25. Suppose S is localy noetherian. Let X ⟶ S be finite flat. Then the function S ∋ s ⟼ (separable rank of X_{s}) is lower semi-continuous.

This means for each n ∈ ℕ the sets of s ∈ S such that |X_{s}(k(s)^{sep})| ≥ n is open in S. The proof of [EGA IV, proposition 15.5.1] is, using the fact one konw this function is allready constructible, by reduction to the case when S is the spectrum of a discrete valuation ring S = Spec(O). Then one has to prove the separable rank of the special fiber of X/O is greater than the one of its generic fiber.

Corollary 17. Let S be noetherian. Let X ⟶ S be finite flat. Then there exists a “découpage” of X into dijsoints localy closed subsets

|X| = ∐_{1≤i≤N} Z_{i}

such that ∀k ⋃_{i≤k} Z_{i} is closed and if Z_{i} stands for the associated reduced localy closed subscheme

of X then over Z_{i} there is a factorization

X ×_{S} Z_{i} ⟶ Y_{i} ⟶ Z_{i}

as in lemma 28.

A.2 Quotients

This is one of the main theorem of [SGA 3, tome I, exposé V].

Theorem 25 (Grothendieck). Let S be a scheme. Let G be a finite localy free S-group scheme acting on an S-scheme X via q : G ×_{S} X ⟶ X. Suppose each orbit of G is contained in an affine open subset of X that is to say if p : G ×_{S} X ⟶ X is the projection ∀x ∈ X the set q(p^{−1}(x)) is contained in an affne open subset. Then the fppf quotient sheaf G\X is representable by an S-scheme. Moreover :

– The quotient scheme G\X coincide with the quotient in the category of ringed spaces

– The morphism X ⟶ G\X is integral

– If X is affine so is G\X

– If G acts freely on X then X ⟶ G\X is a finite localy free morphism and the morphism

G ×_{S} X ⟶ X ×_{G\X} X
(g, x) ⟼ (x, g.x)

is an isomorphism. Thus X ⟶ G\X is an fppf G-torsor.

In the preceding theorem by quotient in the category of ringed spaces we mean : consider the diagram of ringed spaces

\xymatrix{G ×_{S} X \ar[r]^{q} \ar[r]_{p} & X}

then the quotient ringed spaces has as underlying topological space the quotient Y of the diagram

\xymatrix{|G ×_{S} X| \ar[r]^{q} \ar[r]_{p} & |X|}

in the category of topological spaces and has as a structural sheaf the subsheaf of δ_{∗}O_{X}, where δ : |X| ⟶ Y is the projection, consisting of elements x such that q^{∗}x = p^{∗}x.

If S = Spec(R), G = Spec(A) and X = Spec(B) then in the diagram

\xymatrix{B \ar[r]^{p^{∗}} \ar[r]_{q^{∗}} & A ⊗_{R} B}

(where p^{∗}b = 1 ⊗ b) if

C = {b ∈ B | q^{∗}b = p^{∗}b}

we have

G\X = Spec(C)

In the preceding theorem when we say G acts freely on X we mean : for all S-scheme U G(U) acts freely on X(U), and it suffices to verify it for U = X and Id : X ⟶ X that is to say the square

\xymatrix{X \ar[r]^{Id} \ar[d]^{(e,Id)} & X \ar[d]^{Δ_{X/S}} \\ G ×_{S} X \ar[r]^{(p,q)} & X ×_{S} X}

is cartesian (where e is the unit section and Δ_{X/S} the diagonal).

Remark 31. In the preceding theorem if G acts freely on X and X is of finite presentation over S then G\X is of finite presentation over S. This results from the fact that X ⟶ G\X being finite localy free it is of finite presentation and EGA IV proposition 1.4.3 (v).

Corollary 18. Let G be an S-group scheme of finite presentation and H ⊂ G a sub-group-scheme finite localy free over S such that H is distinguished in G. Then G/H is representable by a finite presentation S-group scheme and G ⟶ G/H is finite localy free.

A.3 epimorphism/faithfully flat

Any faithfully flat of finite presentation morphism of schemes is an fppf epimorphism but in general the converse is false : take for example X a scheme, Y an X-scheme that is not flat over X and the morphism X ∐ Y ⟶ X that is the identity on X and the structural morphism of Y on Y. Since it has a section it is an epimorphism but is not flat.

Proposition 26. Let G be a finite localy free S-group scheme and H an S-group scheme of finite presentation. Let f : G ⟶ H be a morphism of S-group schemes. Then the following are equivalent

– f is an fppf epimorphism

– f is faithfully flat

– ∀s ∈ S f_{s} : G_{s} ⟶ H_{s} is an épimorphism

– ∀s ∈ S f_{s} : G_{s} ⟶ H_{s} is faithfully flat And if one of those conditions is verified then H is finite localy free over S.

Proof. We use the fact that if s ∈ S and X is an S-scheme then if f : Spec(k(s)) ⟶ S and F is the fppf sheaf associated to X f^{∗}F is represented by the Spec(k(s))-scheme X_{s}. This is due to the fact we took all schemes in the definition of the fppf topology (for example this fact is false in general for the small étale site). Thus in particular if X ⟶ Y is an epimorphism then ∀s ∈ S X_{s} ⟶ Y_{s} is an epimorphism.

By Grothendieck’s theorem on flatness fiber by fiber everything is now reduced to prove that if S is the spectrum of a field then f is an epimorphism iff it is faithfully flat. Of course if it is faithully flat is an epimorphism. Now suppose it is an epimorphism. The kernel of f is a finite flat group scheme since S is the spectrum of a field. The morphism of sheaves G/ ker f ⟶ H is an isomorphism. But by grothendieck’s theorem on quotients by finite flat group schemes the morphism G ⟶ G/ ker f is faithully flat, which proves the result.

A.4 Exact sequences

Proposition 27. Consider an exact sequence

0 ⟶ G_{1} ⟶ G_{2} ⟶ G_{3} ⟶ 0

of finite S-group schemes that are of finite presentation over S. Then – If G_{2} is localy free over S so are G_{1} and G_{3}

– If G_{1} and G_{3} are localy free over S so is G_{2}

Proof. If G_{2} is localy free over S then by the preceding proposition so is G_{3} and moreover the morphism G_{2} ⟶ G_{3} is faithfully flat which implies G_{1} is finite localy free.

If G_{1} and G_{3} are localy free then G_{2} being a G_{1}-torsor over G_{3} thus if flat over G_{3} thus over S.

Lemma 29. Consider an exact sequence

0 ⟶ G_{1} ⟶^{u} G_{2} ⟶^{v} G_{3}

of finite localy free S-group schemes. Then the sequence

0 ⟶ G_{1} ⟶^{u} G_{2} ⟶^{v} G_{3} ⟶ 0

is exact iff

rank(G_{2}) = rank(G_{1}) × rank(G_{3})

Proof. If it is exact then G_{2} being a G_{1}-torsor over G_{3} we have G_{1} ×_{S} G_{2} ⟶^{∼} G_{2} ×_{G_{3}} G_{2}. Thus rank_{S}(G_{1}) × rank_{S}(G_{2}) = rank_{G_{3}}(G_{2})^{2}. But rank_{S}(G_{2}) = rank_{S}(G_{3}) × rank_{G_{3}}(G_{2}). Thus combining both equality rank(G_{2}) = rang(G_{1}) × rank(G_{3}).

Reciprocaly, there is a monomorphism coker(u) ↪ G_{3} that is a closed immersion since coker(u) is finite localy free. Moreover by the preceding case and the hypothesis rank(coker(u)) is equal to rank(G_{3}). Thus, a closed immersion of finite localy free S-schemes of the same rank being an isomorphism we conclude.

A.5 Étale/connected finite flat group-schemes

Let G be a finite localy free S-group scheme.

We say G is étale, resp. radicial, if the morphism G ⟶ S is étale, resp. radicial.

We remark if ω_{G} is the conormal sheaf of the unit section of G then Ω_{G/S}^{1} ≃ p^{∗}ω_{G} where p : G ⟶ S.

Lemma 30. Write G = Spec(O_{S} ⊕ I) where I is the augmentation ideal of G.

The group G is étale iff ω_{G} = I/I^{2} = 0.

The group G is radicial iff I is (localy on S) a nil-ideal.

A.6 Finite flat group-schemes over fields

Let k be a field.

Lemma 31. Let G be finite group-scheme over k. Then

Lemma 32. Let G be a finite group scheme over k. Then there is an exact sequence

0 ⟶ G^{0} ⟶ G ⟶ G^{ét} ⟶ 0

where G^{0} is connected and G^{ét} étale. If k is perfect then G^{0} = G_{red}. If k is separably closed this sequence is split.

Proof.

A.7 The order of a finite flat group-scheme

In this section all group-schemes are commutative.

Definition 41. For G a finite localy free group-scheme over S we note |G| the order of G that is to say the rank of O_{G} as a localy free O_{S}-module.

Proposition 28 (Deligne). Any finite localy free G over S is killed by its order.

The proof is given in ? ? ?. When S is the spectrum of a field there is a more elementary proof. In fact if S = Spec(k), if k has caracteristic 0 then G is étale and this is the usual statement for abstract finite groups (after going to the algebraic closure of k). If k has caracteristic p, one can suppose k is algebraically closed. Then G ≃ G^{0} × G^{ét} and one is reduced to the statement for G connected : G = G^{0}. Then if G = Spec(k ⊕ I) were I is the augmentation ideal of the unit section then multiplication by p on G induces multiplication by p ∈ k on I/I^{2} and is thus zero. Thus if [p] ∈ End(G) is multiplication by p then [p]^{∗}I ⊂ I^{2}. With this one easily concludes.

Remark 32. The preceding proof over a field implies at least that for any base S any finite localy free G over S is torsion, which is weaker than Deligne’s statement but sufficient for most applications.

Lemma 33. Let G finite localy free over S and N ∈ ℕ its annihilator that is to say Nℤ = ker(ℤ ⟶ End(G)). Then the prime divisors of N and |G| are the same.

Proof. By the preceding proposition N| |G|. Let now p be a prime factor of |G|. Suppose p ∤ N. Then p : G ⟶ G is an isomorphism. We want to see this is impossible. By specialization we can suppose S is the spectrum of an algebraicaly closed field. Then G ≃ G^{0} × G^{ét}. If ℓ is the caracteristic of this field G^{0} is killed by a power of ℓ. Thus G^{0} = (0) or ℓ ≠ p. If G^{0} = (0) we are reduced to the case of G = G^{ét} an abstract finite group and it is clear p : G^{ét} ⟶ G^{ét} canot be an isomorphism. Suppose now ℓ ≠ p. We know the order of G^{0} is a power of ℓ ( ? ? ?). Thus p||G^{ét}|. Thus as before this is impossible by the case of usual abstract finite groups.