Chapter VII: Level structures on Dieudonné modules
7.4 Integral domain base
Let nowRbe an perfect integral domain of characteristicp. DenoteK :=Frac(R) andηthe generic point ofS=SpecR. The ringRinjects intoK. MoreoverW(R)is an integral domain which injects intoW(K)([Shi14], Lemma 3.7).
Lemma 7.4.1. K is flat overRandW(K)is flat overW(R).
Proof of lemma: The first part is basic commutative algebra. The second part is more subtle. From [Shi14], Proposition 5.2 it follows thatW(K)is flat overW(R). Indeed, Proposition 5.2 in loc. cit. says that
W(K)w W(B)(p)∧
The proof of this proposition shows that W(B)(p) is Noetherian. Thus, W(K) is obtained via a localization and a completion of a Noetherian ring, and both these
operations are flat.
Lemma 7.4.2. If R → S is a flat ring homomorphism of perfect Fp-rings, then Wm(R) →Wm(S)is flat for anym ≥ 1.
Proof of lemma: We apply Theorem 1 in Bourbaki, Alg. com., III, §5 [Bou85].
We reproduce below the part we need. We have to verify that we are justified in applying the result.
Using the notation of the Bourbaki lemma, we haveA=Wm(R),M =Wm(S)andJ = (p). Since the rings are perfectJis indeed nilpotent. M/JM =Wm(S)/pWm(S)w S and A/J=Wm(R)/pWm(R)w R. By assumptionR→ Sis flat, thusM/JMis flat over A/J.
Finally gri(A)= piWm(R)/pi+1Wm(R)w R, sinceRis perfect. Similarly, gri(M)= piWm(S)/pi+1Wm(S) w S. Thus, the requirement gr(A) ⊗gr
0(A) gr0(M) → gr(M) being bijective is indeed met, as it simply says thatR⊗RS→ Sis bijective.
Lemma 7.4.3(Bourbaki ACIII§5, Th.1). Let Abe a commutative ring,Jan ideal of A, and Man A-module. Consider the following properties:
(i) M is a flatA-module.
(iv) M/JM is a flat A/J-module, and the canonical homomorphismgr(A) ⊗gr
0(A)
gr0(M) → gr(M)is bijective.
Then (i) implies (iv). If moreoverJis nilpotent, then (i) and (iv) are equivalent.
Taking truncations we haveWm(R),→Wm(K)(since at the level of sets, this is just Rm ,→ Km). Consider the diagram (the subscriptcstands for constant):
Mc,R =Wm(R)⊕h
MG,R =Wm(R)⊕h
oo A
Mc,R(σ) =(Wm(R)⊕h)(σ)
0 w id0
OO
MG,R(σ) = (Wm(R)⊕h)(σ)
Ns(B)
OO
Froboo sA oo ker(A⊗1) 3a
kerNs(B)
OO
Mc,K =Wm(K)⊕h oo A MG,K =Wm(K)⊕h
Mc,K(σ) = (Wm(K)⊕h)(σ)
w
OO
MG,K(σ) = (Wm(K)⊕h)(σ)
Ns(BK)
OO
Froboo sAK oo ker(Frobs AK)
kerNs(BK)
OO
The bottom-right part of the diagram is obtained by base-changing to the generic point, i.e. using the morphism SpecK →SpecR. The vertical maps arelinear. All parallelograms in the diagram are commutative (by functoriality of the Frobenius, Dieudonné module and base change).
Assume first that αis a level structure. We want to show that ker
Frobs(A) = kerNs(B).
Let a ∈ ker FrobsA. Then the image ea of a in Wm(K)⊕h(σ) is in the kernel of Frobs AK. Since αK is a level structure for the truncated p-divisible group GK over SpecK, from the main theorem in the case where the base is a field, proved above, we have ker(FrobsAK) = kerNs(BK). So ˜a ∈ kerNs(BK). But the map Wm(R)⊕h → Wm(K)⊕h is injective because it is simply induced by the inclusion R,→ K (so on the level of sets is just Rm ,→ Km). Thusa ∈kerNs(B), and thus ker Frobs A= kerNs(B).
Assume now that ker FrobsA = kerNs(B). We want to prove that α is a level structure.
We shall need the following simple result from commutative algebra:
Lemma 7.4.4. Given a module homomorphism f : M → N ofR-modules and a flat R-algebraS, then(ker f) ⊗RS =ker(f ⊗1S).
Proof of lemma: LetM0be the kernel of f,N0the image andPthe cokernel. We then have a diagram
0 //M0 //M //
N //P //0
0 //N0
>>//0
The top row is exact, and the two arising short exact sequences are exact. Tensoring withS, the short exact sequences stay exact. If a tensorwinM⊗RSgoes to 0 inN⊗RS under f ⊗1S, then from the injectivity ofN0⊗RS→ N⊗RSit follows that it goes to zero inN0⊗RS. Thus, ker(f⊗1)= ker(M⊗RS→ N0⊗RS)= M0⊗S =(ker f) ⊗RS,
as wanted.
Since W(K) is flat over W(R) (and Wm(K) flat over Wm(R)), it follows the maps induced by base change: Ns(BK) and Frobs AK have kernels equal to ker(Ns(B)) ⊗W(R) W(K) and ker FrobsA ⊗W(R) W(K) respectively. By assump- tion, ker Frobs A = kerNs(B), hence kerNs(BK) = ker Frobs AK. But our main theorem over a perfect field precisely says that this equality is equivalent to the group homomorphismαK :(p−mZ/Z)h
K →GK induced byαbegin a level structure.
Thus the condition ker FrobsA= kerNs(B)implies thatαK is a level structure on GK. It will suffice then to prove the following statement:
Proposition 7.4.1. Letα:(Z/pmZ)h
S →Gbe a homomorphism of group schemes overS = SpecR, with Ra perfect integral domain. Let K be the fraction field of
RandαK the homomorphism obtained by base-change toSpecK. IfαK is a level structure onGK, thenαis a level structure onG.
In other words, the above proposition says that being a level structure is a closed condition. In fact, we can replace(Z/pmZ)hby anarbitrary(abstract) finite abelian groupH and viewα:HS →Gas a set of|H |sections. Moreover,Gneed not be a truncatedp-divisible group. We keep, however, the essential assumption thatRis a (perfect)integral domain.
Proof. LetG= SpecBfor a locally freeR-algebraB. We can assume that Bis free over R, of rankN. Choose a basisb1, . . . ,bN. Thenb1⊗1, . . . ,bN ⊗1 form a basis forB⊗RK overK.
Letϕ1, . . . , ϕN :S →Gbe a set of sections, corresponding to ring homomorphisms f1, . . . ,fN : B→ R. By assumption,(ϕi)K form a full set of sections forGK overK. This is equivalent to the fact (see [KM85], page 38), that the ‘‘universal’’ element
Z0= Õ
Ti(bi⊗1) ∈ (B⊗RK) ⊗K K[T1, . . . ,TN]satisfies Norm(Z0)=Ö
(fi)K(Z0) (7.3)
as an equality in(B⊗RK)[T1, . . . ,TN].
Nowϕ1, . . . , ϕN form a full set of sections forGoverRif and only if the ‘‘universal’’
elementZ =Õ
Tibi ∈ B⊗RR[T1, . . . ,TN]satisfies Norm(Z)=Ö
fi(Z). (7.4)
So the proposition comes down to checking the equivalence of (7.3) and (7.4).
The proposition then follows because Z0 = Z ⊗ 1, so the linear map that is multiplication by Z0 has the ‘‘same’’ matrix (with respect to basis {bi ⊗ 1}) as multiplication by Z does with respect to basis {bi}, so has the same norm /
characteristic polynomial.