5.2 Some properties
Lemma 5.2.1. Let R be an integral domain over a field k which is infinite di-mensional as k-vector space and let M be a finite free R-module. Assume that we have a ring automorphism σ : R → R which restricts to a ring automorph-ism of k, i.e., σ(k) = k and we have a σ-linear morphism ϕ : M → M, i.e., ϕ(rm) = σ(r)ϕ(m) for all r ∈ R and m ∈ M. If the dimension (as k-vector space) of the cokernel of ϕ is finite, then ϕ is injective.
Proof. Take an R-basis of M, say m1,· · ·, mn. For every i = 1,· · · , n write It follows thatϕ is the following composition of morphisms:
M →∼= Rn →σn Rn→ρ Rn→∼= M
where the first and last morphisms are the R-linear isomorphisms given by the choice of the basis {m1,· · · , mn} and ρ is the R-linear morphism given by the matrix (rji). The morphism σn : Rn →Rn is a σ-linear isomorphism. It follows from these observations that ϕ is injective if and only if ρ is injective and that the cokernel of ϕ is isomorphic (as R-module) to the cokernel of ρ. So, we have reduced the problem to the case, where ϕ is an R-linear morphism and not only σ-linear. So, we assume that ϕ:M →M is an R-linear morphism.
We claim that cokernel of ϕ is a torsion R-module. Indeed, take a non-zero el-ement m ∈ Coker(ϕ) and an infinite set of k-linearly independent elements of R, say {r1, r2,· · · }. Since dimk(Coker(ϕ))< ∞ we have that �n
i=1xi(rim) = 0 for some n ∈ N and a subset {x1, x2,· · · , xn} �= {0} of k. Since ri are linearly independent, the sum r := �n
i=1xiri is not zero but rm = 0, which shows that m is a torsion element. This proves the claim.
Let us denote the fraction field of R by Q. Cokernel of ϕ being a torsion R-module implies that the tensor product Coker(ϕ)⊗RQ is zero. It follows that ϕ⊗Id : M ⊗RQ → M ⊗R Q is surjective. The Q-vector space M ⊗RQ has finite dimension, which implies that the morphism ϕ⊗Id is injective too. By flatness of Q over R, we have Ker(ϕ)⊗R Q = Ker(ϕ ⊗Id) = 0. Since M is a torsion-free R-module, its submodule Ker(ϕ) is also torsion-free and therefore
embeds in Ker(ϕ)⊗RQ which is a trivial module. Hence Ker(ϕ) = 0 and ϕ is injective.
Lemma 5.2.2. Let O =Fq�π� and M be a π-divisible O-module scheme over a perfect field k containing Fq. Then the contravariant Dieudonn´e module of M, D(M), is a finite free module over O⊗Fq k ∼=k�π�.
Proof. Let us write M = �
Mn where Mn are kernels of πn and write D (respectively Dn) for D(M) (respectively D(Mn)). Then D = lim
←− Dn. The Dieudonn´e module Dn is finite over W(k), the ring of Witt vectors on k, but p· M= 0, which implies that p·D= 0 and thus Dn is finite over W(k)/p=k.
Let d1,· · · , dr be elements in D whose images in D1 is a basis over k and define a morphism k�π�r →D by sending basis elements todi. This morphism induces morphisms (k�π�/(πn))r → D/πnD ∼= Dn which are surjective (since modulo π they are surjective) an so, being an inverse limit of surjective morphisms, k�π�r →D is surjective. This implies that D is a finite module over k�π�. The action of π on M is surjective and therefore its action on D is injective. It follows that D is a torsion-freek�π�-module and hence is free over it, since k�π� is a principal ideal domain.
Theorem 5.2.3. Finite dimensional π-divisible O-module schemes are formally smooth.
Proof. Note that we may assume that the base scheme is an algebraically closed field k.
Let M be a π-divisible O-module scheme over k. If k is has characteristic dif-ferent from p, then Mis ´etale and so it is smooth. So, we can assume that the characteristic of k is p. Sincek is perfect, Msplits into the ´etale and connected factors and so, Mis smooth if and only if both the ´etale and connected parts are smooth. The ´etale factor is smooth and so we may assume that Mis connected.
As for connected formal schemes, being smooth is equivalent to the Frobenius morphism being an epimorphism, we will show that the Frobenius morphism is an epimorphism.
In the mixed characteristic case, by Remark 5.1.6, every π-divisible module is p-divisible. Since the multiplication by p factors through Frobenius, i.e., p = FM ◦VM, and multiplication by p is an epimorphism, we see that FM is an epimorphism too.
Now, assume that O has characteristic p and so O = Fq�π�. By Lemma 5.2.2, the contravariant Dieudonn´e module of M, D := D(M) is a finite free k�π� -module. Denote by σ:k →kthe Frobenius morphism ofk, i.e.,σ(x) =xp for all x∈k. It has a natural extension to k�π� by sending π to itself, and also denote
5.2. SOME PROPERTIES 69 this extension by σ. Since the action of π on Mis a morphism of formal group schemes, it commutes with the Frobenius ofMand thus the Frobenius morphism of D is σ-linear morphism. The dimension of the tangent space of M is equal to the dimension ofM and the tangent space of Mis isomorphic to the dual of the cokernel of the Frobenius ofD. This shows that the cokernel of D has finite dimension over k. It follows from Lemma 5.2.1 that Frobenius of D is injective and therefore the Frobenius of Mis an epimorphism. Hence the smoothness of M.
Remark 5.2.4. What we have shown in the last Theorem is that the Frobe-nius morphism of the contravariant Dieudonn´e module of aπ-divisibleO-module scheme is injective. We will see in the next lemma that similarly, the Verschiebung of the covariant Dieudonn´e module of aπ-divisibleO-module scheme is injective as well.
Lemma 5.2.5. Let Mbe a finite dimensional π-divisible O-module scheme over a perfect field of characteristicp, then the Verschiebung morphism of the covariant Dieudonn´e module of M is injective.
Proof. LetDand respectivelyDi (for any natural numberi) denote the covari-ant Dieudonn´e module ofMand respectively ofMi, the kernel ofπi :M→M. Let us also denote by Ki the kernel of the Verschiebung of Di. We know that Ki ∼= Lie(Mi), and so dimkKi ≤ dimkLie(M) < ∞. Since the inclusion η:Di �→Di+j (j a natural number) is compatible with the Verschiebungen, it in-duces a morphism between kernels ofV, which we denote also by η:Ki �→Ki+j. Similarly, the epimorphism ζ :Di+j �Di induces the morphism ζ :Ki+j →Ki, and as we have seen before, the composition Ki+j
→ζ Ki
�→η Ki+j is the multi-plication by πj (we have seen it for the composition of the morphisms between Dieudonn´e modules, but as Ki is a submodule of Di, this is also true for the morphisms between these kernels).
Now, since the dimension ofKi is bounded above, and we have inclusions Ki �→ Ki+1, there exists a natural number n0, such that for all i ≥ n0, we have dimkKi = dimkKn0 and so η : Ki �→ Ki+j is an isomorphism for any natu-ral number j and any i ≥ n0. We claim that the morphisms ζ : Ki+n0 → Ki
are zero for all i ≥ 1. Indeed, the composition Ki+n0 → Ki �→ Ki+n0 is the multiplication by πn0, and so, composed with the inclusion Kn0 �→Ki+n0 is zero (note that Kn0 ⊂Dn0 and Dn0 is killed by πn0), which implies that the compo-sition Kn0 �→Ki+n0 → Ki is zero (Ki �→Ki+n0 is injective). But, the morphism Kn0 �→ Ki+n0 is actually an isomorphism by the choice of n0, and therefore the morphism Ki+n0 →Ki is zero and the claim is proved.
For every natural number i, we have an exact sequence 0 → Ki → Di
→V Di. Taking the inverse limit overiwith the transition morphismsζ, and recalling that
D is the inverse limit ofDi, we obtain the exact sequence 0→lim
←− Ki →D→V D (note that the inverse limit is a left exact functor). But, since for every i≥1, the transition morphism Ki+n0 → Ki is zero, it follows that the inverse limit lim
←− Ki
is trivial, and hence V :D→D is injective.
Theorem 5.2.6. Let S be a scheme and Ma π-divisible O-module scheme over S. Then the height h(M) :S →Q≥0 takes integer values.
Proof. Since the height is invariant under base change, we may assume thatSis the spectrum of an algebraically closed field k. The order of finite group schemes is multiplicative with respect to exact sequences and it follows from Remark 5.1.4 that the height of M is the sum of the heights of its connected and ´etale parts.
So, we prove the statement for ´etale and connected π-divisible O-modules.
Assume that M is ´etale. Then by Remark 5.1.4, M1 is also ´etale, and since k is separably closed, M1 is a constant group scheme, and so, the order of M1 is equal to the order of the group M1(k), which is a module over O/π = Fq. The height of Mis by definition equal to logq|M1|= logq|M1(k)|. But being a vector space over the field Fq, the order of M1(k) is a power of q and therefore the height of Mis a natural number.
Now, assume that M is connected and so k has characteristic p. By Theorem 5.2.3, M is smooth and therefore it is a commutative formal Lie group. Again, we consider the problem in mixed and equal characteristic cases separately. If O has characteristic zero, then by Remark 5.1.6, Mis a p-divisible group of height ef h(M). Note that forp-divisible groups, the height is always a natural number, which follows from its definition (the residue degree is 1), i.e.,ef h(M)∈N. From Theorem 1 of [Wat74], we know that the height of M, regarded as a p-divisible group, is divisible by the degree of the extension K/Qp, which is ef. Therefore, h(M) is a natural number.
Now, assume that O has characteristic p and therefore is isomorphic to Fq�π�. The Dieudonn´e module of M, D :=D(M), is a finitely generated module over the ring k⊗FpO (see Lemma 5.2.2). This ring is isomorphic to the product
�
Gal(Fq/Fp)
k⊗Fq O = �
jmodf
k⊗Fq O,
where we identify the Galois group Gal(Fq/Fp) with the group Z/fZ by sending the Frobenius to 1 and the isomorphism is given explicitly by
θ :k⊗FpFq�π�→ �
jmodf
k⊗Fq Fq�π�= �
jmodf
k�π�
5.3. EXTERIOR POWERS 71