In this section, we combine the results of the last two sections to prove that the exterior powers of π-divisible O-module schemes over any base field exist.
Theorem 5.6.1. LetMbe aπ-divisible O-module scheme of heighthover a base field k. Assume that the dimension of M is at most 1. Fix a positive natural number j. Then the jth-exterior power of M in the category of π-divisible O -module schemes over k exists, and has height �h
j
�. If the dimension of M is 1, then �
O
jM has dimension�h−1
j−1
�, otherwise, it has dimension zero. Moreover, for every positive natural number n, we have (�
O
jM)n∼=�
O
j(Mn).
Proof. If the characteristic ofk is different from por the dimension ofMis zero, then Mis ´etale and the statements of the theorem follow from Proposition 5.4.1 and Remark 5.4.2. So, we can assume that the characteristic of k is p and the dimension of M is 1. The statements of the theorem now follow from Theorem 5.5.34.
Chapter 6
Multilinear Theory of Displays
In this chapter, after recalling basic definitions, constructions and results of the theory of display developed by Zink, we develop a multilinear theory of displays, which, in the next chapter, will be related to the multilinear theory of group schemes developed by Pink. For more details on displays, we refer to [Zin02].
Unless otherwise specified,R is a ring andFR:W(R)→W(R) is the Frobenius morphism of the ring of Witt vectors with coefficient in R, and VR : W(R) → W(R) its Verschiebung. We denote by IR the image of the Verschiebung. We sometimes denote the Frobenius and Verschiebung without the superscript R, when it is clear what is meant.
6.1 Recollections
Definition 6.1.1. A 3n-display over Ris a quadruple P = (P, Q, F, V−1), where P is a finitely generated W(R)-module, Q ⊆ P is a submodule and F, V−1 are FR-linear morphisms F : P → P and V−1 : Q → P, subject to the following axioms:
(i) IRP ⊆ Q ⊆ P and there is a decomposition of P into the direct sum of W(R)-modules P = L⊕T, called a normal decomposition, such that Q=L⊕IRT.
(ii) V−1 : Q → P is an FR-linear epimorphism (i.e., the W(R)-linearization (V−1)� :W(R)⊗FR,W(R)Q→P is surjective).
(iii) For any x∈P and w∈W(R) we have
V−1(VR(w)x) =wF(x).
Remark 6.1.2. Note that from the last axiom, it follows that F is uniquely determined by V−1. Indeed, we have for every x∈P:
F(x) = V−1(VR(1)x).
101
It follows also from this relation and FR-linearity of V−1, that for every y ∈ Q, we have
F(y) = V−1(VR(1)y) =FRVR(1)V−1(y) =pV−1(y).
Construction 6.1.3. According to lemma 10, p.14 of [Zin02], there exists a unique W(R)-linear map V� : P → W(R)⊗F,W(R) P, satisfying the following equations:
V�(wF(x)) =pw⊗x, w∈W(R), x∈P and
V�(wV−1(y)) =w⊗y, w∈W(R), y ∈Q.
If we denote byF� :W(R)⊗F,W(R)P →P theW(R)-linearization of F :P →P, we have the properties:
F�◦V�=p.IdP and V�◦F� =p.IdW(R)⊗F,W(R)P. (1.4) Denote by Vn� the composition
P −−−→V� W(R)⊗F P −−−−−−→Id⊗FV� W(R)⊗F2 P →· · ·−−−−−−−→Id⊗F nV� W(R)⊗FnP.
Construction 6.1.5. Let P = (P, Q, F, V−1) be a 3n-display over a ringR and let ϕ :R→S be a ring homomorphism. We are going to construct a 3n-display, which will be the 3n-display obtained from P by base change with respect to ϕ :R→S. Set PS := (PS, QS, FS, VS−1), where:
• PS is W(S)⊗W(R)P,
• QS is the kernel of the morphism W(S)⊗W(R)P →S⊗RP/Q,
• FS :PS →PS is the morphism FS⊗F and
• VS−1 : QS → PS is the unique W(S)-linear homomorphism, which satisfies the following properties:
VS−1(w⊗y) = FS(w)⊗V−1(y), w∈W(S), y ∈Q and
VS−1(VS(w)⊗x) = x⊗F(x), w∈W(S), x∈P.
If P = L⊕T is a normal decomposition of P, then one can show that PS = LS ⊕ TS is a normal decomposition of PS, where LS := W(S)⊗W(R) L and TS =W(S)⊗W(R)T and that we have QS =LS⊕IS⊗W(R)T (note that ISTS = IS ⊗W(R)T). For the details of this construction, in particular to see why this construction produces a 3n-display, refer to Definition 20 and the discussions following it, p.20 of [Zin02].
6.1. RECOLLECTIONS 103 Definition 6.1.6. LetP = (P, Q, F, V−1) be a 3n-display over R. Assume that p is nilpotent in R. Then P is called display if it satisfies the nilpotence or V-nilpotence condition, i.e., if there exists a natural number N such that the morphism
VN� :P →W(R)⊗FN,W(R)P is zero moduloIR+pW(R).
Remark 6.1.7.
1) Ifpis not nilpotent inR, but is topologically nilpotent, one defines a display as follows, however, in the sequel, we will only work with displays over rings where p is nilpotent. Let R be a linearly topologized ring, with topology given by a filtration by ideals:
R =a0 ⊇a1 ⊇· · ·⊇an ⊇. . . ,
such that aiaj ⊆ ai+j. By assumption, p is nilpotent in R/a1, and hence in every ring R/ai. We also assume that R is complete and separated with respect to this filtration. In particular it follows that R is a p-adic ring.
We call a 3n-displayP adisplay, if the 3n-display obtained fromP by base change over R/a1 is a display.
2) Displays are sometimes called nilpotent displays, whereas 3n-displays are
“not necessarily nilpotent”. In order to emphasize that displays satisfy V-nilpotence condition, we will also sometimes stress the adjective “nilpo-tent”.
For more details on the following construction, refer to Example 14, p.16 of [Zin02].
Construction 6.1.8. Letkbe a perfect field of characteristicp > 0. A Dieudonn´e module over k is an Ek-module M, which is finitely generated and free asW (k)-module. It is therefore equipped with two operators F :M →M and V :M → M, which are respectively F : W(k) → W(k) and F−1 : W(k) → W(k) linear, such thatF V =V F =p. We obtain fromM a 3n-displayP = (P, Q, F, V−1), by settingP :=M, Q:=V M and F beingF :M →M and finallyV−1 :V M →M being the inverse of V : M → V M (note that since F V = p and M is a free W(k)-module,V is injective). Moreover,P is a display, if and only if the morph-ism V : M/pM → M/pM is nilpotent. Thus, if G is a p-divisible group over k, the Dieudonn´e module, D∗(G), of G gives rise to a 3n-display. Since we are working with the covariant Dieudonn´e theory, we observe thatGis connected, if and only if the corresponding 3n-display is a (nilpotent) display.
Definition 6.1.9. LetR be a ring and P = (P, Q, F, V−1) a 3n-display over R.
The tangent module of P, denoted byT(P), is theR-module P/Q.
Definition 6.1.10. LetRbe a ring and P = (P, Q, F, V−1) a 3n-display overR.
The rank of P, is the rank of its tangent module over R and the height of P, is the rank of P overW(R).
Remark 6.1.11.
1) Using the normal decompositionP =L⊕T and Q=L⊕IRT, we observe that the tangent module ofP is isomorphic toT /IRT, which is a projective R-module and therefore the rank of the tangent module overR is equal to the rank of T over W(R). It follows that the height of P is equal to the sum of ranks of Land T over W(R).
2) IfR is a perfect field of characteristicp > 0 andP is the display associated to a connected p-divisible group G, then the tangent module of P, which is an R-vector space, is canonically isomorphic to the tangent space of the p-divisible groupG. The rank and height ofP are respectively equal to the dimension of G and its height.
The following construction is extracted without proofs from example 23, p. 22 of [Zin02].
Construction 6.1.12. Let P = (P, Q, F, V−1) be a 3n-display over a ring R with p.R = 0. Denote by Frob: R → R the absolute Frobenius morphism of R, i.e., Frob(r) = rp for any r ∈ R. Denote by P(p) = (P(p), Q(p), F, V−1) the base change of P with respect to Frob. More precisely, we have
P(p) =W(R)⊗F,W(R)P and
Q(p)=IR⊗F,W(R)P + Im(W(R)⊗F,W(R)Q→W(R)⊗F,W(R)P).
The operators F, V−1 are uniquely determined by the relations:
F(w⊗x) = F(w)⊗F(x), w∈W(R), x∈P, V−1(V w⊗x) =w⊗F(x), w∈W(R), x∈P and
V−1(w⊗y) = F(w)⊗V−1(y), w∈W(R), y ∈Q.
The mapV� in Construction 6.1.3, satisfiesV�(P)⊆Q(p) and using the fact that P is generated as a W(R)-module by elements V−1(y) with y ∈ Q, one shows that V� commutes with F and V−1 and therefore induces a homomorphism of 3n-displays
F rP :P →P(p).
6.1. RECOLLECTIONS 105 Similarly, F� satisfies F�(Q(p))⊆ IRP and commutes with F and V−1 and thus induces a homomorphism of 3n-displays
V erP :P(p)→P. From the equations (1.4), we obtain the equations
F rP ◦V erP =p.IdP(p) and V erP ◦F rP =p.IdP.
For the next construction, we fix a 3n-display P = (P, Q, F, V−1) over a ring R, wherepis topologically nilpotent, with a fixed normal decompositionP =L⊕T, and a nilpotent R-algebra N. Set S :=R⊕N. This has a natural structure of anR-algebra. This construction is a recapitulation of some of the constructions and results in section 3 of [Zin02].
Construction 6.1.13. Set P�(N) := W�(N)⊗W(R) P and Q(� N) := P�N ∩QS, whereQS is the base change of Q (as in 3n-display), i.e.,
QS = Ker(W(S)⊗W(R)P →S⊗RP/Q).
Then, one sees that
Q(� N) =W�(N)⊗W(R)L⊕I�N ⊗W(R)T.
We extend the maps F : P →P and V−1 :Q→ P to mapsF :P�(N)→P�(N) andV−1 :Q(� N)→P�(N) as follows. We setF :=FN⊗F, whereFN :W�(N)→
�W(N) is the Frobenius morphism. We letV−1 act on�W(N)⊗W(R)LasF⊗V−1 and on I�N ⊗W(R) T as V−1 ⊗F (since the action of V on the Witt vectors is injective, the map V−1 :I�N →W�(N) is well-defined). If we want to look at P�N and Q�N as functors on NilR, we denote P�N byG0P(N) and Q�N by G−1P (N).
Denote by BTP(N) the cokernel of the map V−1 −Id :G−P1(N) →G0P(N). By theorem 81, p. 77 of [Zin02], the following sequence is exact:
0−→G−P1(N)−−−−−−→V−1−Id G0P(N)−→BTP(N)−→0
and the functorBTP : NilR →Abis a finite dimensional formal group. Moreover, if P is a display (nilpotent), then BTP is a p-divisible group (corollary 89, p.83 of [Zin02]). By corollary 86, p. 81 of [Zin02], the construction ofBTP commutes with base change of 3n-displays. Now, assume that pR = 0. By functoriality, the Frobenius map F rP : P → P(p) gives rise to a map BTP(F rP) : BTP → BTP(p) ∼= BTP(p), where by BTP(p) we mean the base change of the formal group BTP with respect to the extension Frob:R→R. Proposition 87, p.81 of [Zin02]
states that this homomorphism is the Frobenius morphism of the formal group BTP. Similarly, the induced morphism BTP(V erP) : BTP(p) ∼=BTP(p) → BTP is the Verschiebung of BTP.
For future reference, we summarize these results in the following proposition.
Proposition 6.1.14. Let P be a 3n-display over a ring R, then:
• BTP is a finite dimensional formal group and the construction P � BTP commutes with base change, i.e., if R →S is a ring homomorphism, then there exists an canonical isomorphism (BTP)S ∼=BTPS.
• If p is nilpotent in R and P is nilpotent, then BTP is an infinitesimal p-divisible group.
• If pR = 0, and P is nilpotent, then the Frobenius and respectively Ver-schiebung morphisms of the p-divisible group BTP are BTP(F rP) and re-spectively BTP(V erP).
Notations 6.1.15. For a nilpotent R-algebra N, we denote by [b] the class of an element b ∈ G0P(N) modulo (V−1 −Id)G−P1(N). If [b] is annihilated by pn, we write [b]n to emphasize the fact that this element belongs to the kernel of pn. In this case, pnb belongs to the subgroup (V−1 −Id)G−1P (N) of G0P(N), and therefore, since V−1−Id : G−P1(N)→ G0P(N) is injective, there exists a unique element ngP(b)∈G−P1(N) with (V−1−Id)(ngP(b)) =pnb.
Remark 6.1.16. It follows from the construction ofBTP that for any nilpotent R-algebra N, any w ∈ �W(N) and any x ∈ P, we have [F w ⊗x] = [w⊗V x]
and [V w⊗x] = [w⊗F x]. Indeed, by Construction 6.1.13, we know that (V−1− Id)(w⊗V x) = F w⊗x−w⊗V xand that (V−1−Id)(V w⊗x) =w⊗F x−V w⊗x.
We will need theorem 103, p.94 of [Zin02], which states:
Theorem 6.1.17. Let R be an excellent local ring or a ring such that R/pR is of finite type over a field k. The functor P �→ BTP gives an equivalence of categories between the category of (nilpotent) displays over R and infinitesimal p-divisible groups over R.