= g1 =exp
∑
r i=1µi⊗Xi
! .
Hence, for all i we have Ad ρ(σ)Xi = Xi. So Xi ∈ CentG(ρ) = Lie(Zρ) = Z where Z = Lie(ZG); therefore, from γ1∈ Z ⊗mwe conclude g1 =expG(γ1) ∈ZbG(A/m2).
Assume that if the element gn−1 =g (mod mn) ∈Gb(A/mn)commutes with ρ, then gn−1 ∈ ZbG(A/mn). Let gn ∈Gb(A/mn+1)commuting with ρ. Consider the exponential exact sequence
0→mn/mn+1⊗ G −−→expG Gb(A/mn+1)−→rn Gb(A/mn) →0
by the inductive assumption, where rn(gn) = gn−1 ∈ ZbG(A/mn). Moreover, by smoothness of bZG, we can find zn ∈ ZbG(A/mn+1)such that rn(zn) = gn−1. Then z−n1gn ∈ ker(rn) and z−n1gn=exp(γn). As before, consider a basis{µi}of mn/mn+1over k. The components Xi ∈ G of γnare inZ; hence, exp(γn) ∈ZbG(A/mn+1). So gn=znexp(γn) ∈ZbG(A/mn+1)and we can conclude g=lim←−gn ∈ZbG(A).
This concludes the checking of verification of condition (H4).
3.7. Verification of condition (H3)
We recall a fundamental fact of deformation theory.
PROPOSITION3.3. There is a canonical isomorphism of k-vector space tD −→∼ H1(Γ, Ad(ρ)).
Remark 3.4. The action ofΓ onG is given by the composed map Γ−→ρ G(k)−→Ad GL(G).
The k-vector spaceG with this action of Γ is usually called the adjoint representation of ρ, and denoted Ad(ρ).
Proof. Let π(ρ) ∈D(k[ε]). The maximal ideal(ε)of k[ε]is principal of square 0. So G −→∼ Gb(k[ε])
X 7→ exp(εX).
A special feature of the infinitesimal extension k[ε] → k is that it admits a section. So we can lift ρ to k[ε]and we can compare ρ and ρ. This define X(σ) ∈ G by
ρ(σ) =exp(εX(σ))ρ(σ). Moreover, σ7→X(σ)is a 1-cocycle for the adjoint action
ρ(στ) = exp(εX(στ))ρ(στ) ρ(σ) = exp(εX(σ))ρ(σ) ρ(τ) = exp(εX(τ))ρ(τ)
ρ(σ)ρ(τ) = exp(εX(σ))ρ(σ) ·exp(εX(τ))ρ(σ)−1ρ(στ)
= exp[ε(X(σ) +Ad ρ(σ)X(τ))]ρ(στ). Conversely, given an 1-cocycle X :Γ→ G
ρ(σ) =exp(εX(σ))ρ(σ) defines a deformation of ρ over k[ε], hence a class π(ρ) ∈D(k[ε]).
Furthermore, if X is a coboundary, we have X(σ) = (Ad ρ(σ) −1)Y for some Y ∈ G. We have the following computation
ρ(σ) =exp(εX(σ))ρ(σ) = exp[ε(Ad ρ(σ)Y−Y)]ρ(σ)
= exp(−εY) ·exp(εAd ρ(σ)Y)ρ(σ)
= exp(−εY)ρ(σ)exp(εY). By using of the isomorphism again
G −→∼ Gb(k[ε]) Y 7→ exp(εY)
we conclude that X is a coboundary if and only if ρ is conjugate to ρ in bG(k[ε]).
To finish the proof, note that the zero element of tD is π(ρ)and it is sent to 0∈H1(Γ,G). By this proposition, in order to check Schlessingers condition (H3), we must check the finiteness of H1(Γ, Ad(ρ)). By the inflation-restriction exact sequence, we may assume thatΓ acts trivially onG
0→H1S(L/F, Ad(ρ))−→Inf H1S(F, Ad(ρ))−→Res H1S(L, Ad(ρ)) Then note that
H1S(L, Ad(ρ)) =Hom(ΓL,S,G) = Hom(ΓL,S, k) ⊗ G
= Hom(Fr(ΓL,S), k) ⊗ G where
Fr(ΓL,S):=ΓL,S/(ΓL,S,ΓL,S)ΓL,Sp
is the p-Frattini quotient ofΓL,S. It is canonically isomorphic to the Galois group of the maximal S-ramified abelian extension of L of exponent p. It remains to notice that if L is big enough (i.e., it contains all p-th roots of unity), this Galois group is Kummer dual to the group of S-units of F modulo its p-th power
O×L,S/O×L,S,p
which is finite by Dirichlet’s theorem on S-units. Note that this proof can be axiomatized to obtain the finiteness of H1(Γ, Ad(ρ))for arbitrary profinite group Γ satisfying the condition that its p-Frattini quotient and those of its open subgroups are finite.
We can state as a theorem the results proven in this section.
THEOREM3.5. Let G be a smooth group scheme overOwhose center is also smooth overO. LetΓ be a profinite group whose p-Frattini quotient is finite as well as those of its open subgroups. Let ρ :Γ→ G(k)be a Galois representation such that the connected component of its centralizer is contained in the center of G:
(Centr)0k: Zρ0= Z0
G.
Then the deformation functorD of ρ is pro-representable. We will denote by (Ru, ρu)the unique uni-versal pair.
Remark 3.6. Let Sp be the set of places of F above a rational number p and S∞ be the set of non-archimedean places of F. Let(p,∞)denote the condition
(p,∞): Spand S∞are contained in S.
Then the assumption concerning profinite is satisfied forΓ = Gal(FS/F)for a finite set S of places satisfying(p,∞)of a number field F, orΓ=Gal(F/F)for a local field F.
Remark 3.7. The assumption on the centralizer can be reformulated as H0(Γ, Ad(ρ)) = Z.
Remark 3.8. From now on, we always assume that the reductive group G and its center ZG
are smooth overOand Zρ0= Z0
G, and assume that the profinite group satisfies the p-finiteness condition.
§ 4. The Universal Deformation: Properties
In previous section, we have proven that if G and ZGare smooth overOand Zρ0 = Z0
G, and if Γ satisfies the p-finiteness condition, then the deformation problem D of ρ
D : CNL0O ; Sets
(A, ϕ) ; {ρ:Γ→G(A)|ρ⊗Ak=ρ}/ bG(A).
is pro-representable. So there exists a unique universal pair (Ru, π(ρ)u) such that for any π(ρ) ∈D(A), there exists a unique α : Ru → A such that
ρu⊗αAGb∼(A)ρ.
4.1. Functorial properties
In this §, we only deal with absolutely irreducible residual representation.
4.1.1. Change of range— FixΓ and k. Let W=W(k)be the ring of Witt vectors of k. Let
δ/W : G/W →G0/W be a homomorphism of group schemes. Let
ρ0 :Γ→ G(k)
be the residue representation composed with δ/k. The composition with δ brings deformation of ρ to deformations with ρ0. If ρ and ρ0 are absolutely irreducible and Ru and R0u are the corresponding universal deformation rings, then composition with δ induces a morphism
rδ : R0u →Ru.
The system of morphisms δ7→rδ has the homomorphic property:
(i) r1= 1
(ii) rδ1 ·rδ2 =rδ1δ2.
(Conjugation) In particular, if
δg : G/W →G/W0
is given by the conjugation with a fixed element g∈ G(W), we obtain an isomorphism rδg : Ru → R0u
Clearly, the isomorphism rδg depends only upon the image of g, the reduction of g. But since ρ is absolute;y irreducible, an application of Schur’s lemma guarantees that the image of g is completely determined by the pair(ρ, ρ0), where ρ0 is obtained from ρ composing with δg. Consequently, rδg depends only on the pairs(ρ, ρ0).
(Duality) Let
τ: G/W →G/W
be the outer automorphism “transpose-inverse”. Then if ρ∗is the composition with τ, i.e., the contragredient representation, and if R, R∗ are the universal deformation rings of ρ and ρ∗ respectively, we have morphisms
R −−−→rτ R∗ −−−→rτ R
which can easily be seen to be two-sided inverse of one another. This establishes canonical identifications of R and R∗.
(Determinant) Let
δ=det : G/W →GL1/W
be the determinant homomorphism. We obtain a natural homomorphism
R0u −−−→rδ R to which we shall return later.
4.1.2. Tensor product— Let ρ1:Γ→ G1(k)and ρ2 :Γ→G2(k)be two residual representations, and let
ρ1⊗ρ2 :Γ→G1(k) ×G2(k) denote their tensor product.
To any pair of deformations of ρ1 to A1 and of ρ2 to A2, we can associate a
defor-mation of ρ1⊗ ρ2 to the completed tensor product A1⊗bWA2 defined by the following: Let m := ker(A1⊗WA2 →k). One sees that m = m1⊗W A2+A1⊗W m2 where mi ⊂ Ai are the maximal ideals for i=1, 2. Then
A1⊗bWA2 = lim←−
n
(A1⊗W A2)/mn
= lim←−
n
(A1/mn1) ⊗W(A2/mn2),
and if mb ⊂ A1⊗bWA2 denotes the closure of m, one sees that A1⊗bWA2 is again a complete noetherian local ring with maximal idealmb and with residue field k.
Let ρ1, ρ2, and ρ1⊗ρ2be absolutely irreducible. We get a natural homomorphism
R(ρ1⊗ρ2) −−−−→h(ρ1,ρ2) R1⊗bWR2
where R(ρ)refers to the universal deformation ring. The system f homomorphisms(ρ1, ρ2) 7→
h(ρ1, ρ2)satisfies evident commutativity and associativity properties.
(Contraction with a lifting of ρ1) Now let ρ1 : Γ → G1(W)be a deformation of ρ1 to W.
Thus ρ1is induced from the deformation of ρ1via a unique homomorphism hρ1 : R(ρ1) →W.
Define
h(ρ1, ρ2): R(ρ1⊗ρ2) →R(ρ2) to be the composition of h(ρ1, ρ2)with hρ1 ⊗1.
From the associative property referred to above, one sees that the following diagram R(ρ0⊗ρ1⊗ρ2)
h(ρ0,ρ1⊗ρ2)
vvnnnnnnnnnnnn h(ρ0⊗ρ1,ρ2) N ''N NN NN NN NN N R(ρ1⊗ρ2)
h(ρ1,ρ2) //R(ρ2)
commutes, where the relevant residual representations are absolutely irreducible, and ρ0, ρ1
are deformations of ρ0, ρ1respectively, to W.
(Twisting by a character) In the special case where ρ1is 1-dimensional, we refer to h(ρ1, ρ2) as the twisting morphism by ρ1, and sometimes denote it simply h(ρ1). From the commutative triangle displayed above, one sees that the twisting morphisms are isomorphisms in the cate-gory CNLWand enjoy the homomorphic property in the variable ρ1.
Now let ¯ρ be any absolutely irreducible residual representation, and let ρ0 be the tensor product of ρ with a 1-dimensional residual representation ρ1. Let ρ1 denote the Teichm ¨uller lifting to W×of the character ρ1, and let
r(ρ, ρ0) →R(ρ)
be the twisting isomorphism h(ρ1, ρ). This enables us to define canonical isomorphisms r(ρ0, ρ) for any pair of twist-equivalent residual representations such that(ρ0, ρ) 7→r(ρ0, ρ)possesses the homomorphic property.
4.1.3. Change of domain— Let
Γ→ϕ Γ0
be a continuous homomorphism between profinite groups satisfying p-finiteness condition.
Let
ρ0 :Γ0 →G(k)
be a residual representation. Let ρ be the residual representation
ρ:Γ→G(k)
obtained by composing ρ0 with ϕ. Suppose that both ρ and ρ0 are absolutely irreducible. Set R, and R0 be the corresponding universal deformation ring. Then, composition with ϕ brings deformation of ρ0to deformations of ρ and therefore induces a homomorphism
r(ϕ): R→R0.
The system ϕ7→ r(ϕ)is homomorphic in ϕ. If ϕ is surjective, then for all A∈ Ob(CNLO), r(ϕ)induces an injection
Homlocalg(R0, A) ,→Homlocalg(R, A).
4.1.4. Change of field— Let ι : k ,→ k0 be a morphism of finite fields of characteristic p. Let rho :Γ → G(k)be an absolutely irreducible residual representation and let ρ0 : Γ → G(k0)be the representation obtained from ρ by extension of scalars via ι.
Then tensoring with W(k0)over W(k)brings deformations of ρ to deformations of ρ0 and therefore induces a natural morphism
r(ι): R0 →R⊗W(k)W(k0) where R= R(ρ)and R0 = R(ρ0).
The morphism r(ι)induces an isomorphism on Zariski tangent spaces.
4.2. 1-dimensional split torus
For G = GL1 and ρ : Γ → k×, the assumptions (SmC) and (Centr) are trivially fulfilled. We will compute R=Ruand ρuin this §.
Consider deformation ρ of ρ to A, i.e., a character
ρ:Γ→ A×.
Since A ∈ Ob(CNLO), the reduction morphism A → k has a multiplicative lifting ωA called the Teichm ¨uller lifting which is functorial: if A→α B→k, then α◦ωA=ωB.
Write ρ(σ) = ω◦ρ(σ) · hρi(σ)with hρi(σ) ≡ 1 (mod m). Since 1+mis pro-p-abelian, the character hρi factors through the maximal p-abelian quotient Γp,ab of Γ. We define φρ : O[[Γp,ab]] → A as the unique localO-algebra homomorphism such that for all γ∈ Γp,ab
φρ([γ]) = hρi(γ)
where[γ]denotes the corresponding element of γ in the group ring.
PROPOSITION4.1. For G=GL1, the universal pair(Ru, ρu)is given by Ru = O[[Γp,ab]]
and
ρu :Γ → O[[Γp,ab]]
σ 7→ ω(ρ(σ)) · [σp,ab] where σ7→σp,abis the projectionΓ→Γp,ab.
Proof. Take any deformation ρ of ρ to A, i.e., ρ :Γ→ A×, we get a localO-algebra homomor-phism φρ :O[[Γp,ab]] → A uniquely determined by the condition
φρ([γ]) = hρi(γ). But we have
φρ(ρu(σ)) = φρ(ω(ρ(σ)) · [σp,ab])
= ω(ρ(σ))φρ([σp,ab])
= ω(ρ(σ))hρi(σ)
= ρ(σ),
i.e., φρ◦ρu = ρ. ThusO[[Γp,ab]]is the universal deformation ring and ρu defined above is the universal deformation of ρ.