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arXiv:1012.1290v1 [math.NT] 6 Dec 2010
INVERSE PROBLEMS FOR DEFORMATION RINGS
FRAUKE M. BLEHER, TED CHINBURG, AND BART DE SMIT
Abstract. Let W be a complete local commutative Noetherian ring with residue field k of
positive characteristic p. We study the inverse problem for the versal deformation rings RW(Γ,V )
relative to W of finite dimensional representations V of a profinite group Γ over k. We show that
for all p and n ≥ 1, the ring W[[t]]/(pnt,t2) arises as a versal deformation ring. This ring is not
a complete intersection if pnW ?= {0}, so we obtain an answer to a question of M. Flach in all
characteristics. We also study the ‘inverse inverse problem’ for the ring W[[t]]/(pnt,t2); this is to
determine all pairs (Γ,V ) such that RW(Γ,V ) is isomorphic to this ring.
1. Introduction
Suppose Γ is a profinite group and that V is a continuous finite dimensional representation of Γ
over a field k of characteristic p > 0. Let W be a complete local commutative Noetherian ring with
residue field k. In §2 we recall the definition of a deformation of V over a complete local commutative
Noetherian W-algebra with residue field k. It follows from work of Mazur and Schlessinger [14, 17]
that V has a Noetherian versal deformation ring RW(Γ,V ) if the p-Frattini quotient of every open
subgroup of Γ is finite. Without assuming this condition, de Smit and Lenstra proved in [11] that V
has a universal deformation ring RW(Γ,V ) if EndkΓ(V ) = k. The ring RW(Γ,V ) is a pro-Artinian
W-algebra, but it need not be Noetherian. In this paper we consider the following inverse problem:
Question 1.1. Which complete local commutative Noetherian W-algebras R with residue field k
are isomorphic to RW(Γ,V ) for some Γ and V as above?
It is important to emphasize that in this question, Γ and V are not fixed. Thus for a given R, one
would like to construct both a profinite group Γ and a continuous finite dimensional representation
V of Γ over k for which RW(Γ,V ) is isomorphic to R. We will be most interested in the case of
finite groups Γ in this paper, for which RW(Γ,V ) is always Noetherian.
One can also consider the following inverse inverse problem:
Question 1.2. Suppose R is a complete local commutative Noetherian W-algebra with residue field
k. What are all profinite groups Γ and all continuous finite dimensional representations V of Γ
over k such that R∼= RW(Γ,V )?
The goal of this paper is to answer Questions 1.1 and 1.2 for the rings R = W[[t]]/(pnt,t2). More
precisely, we prove the following main results Theorem 1.3 and Theorem 1.4.
Theorem 1.3. For all fields k and rings W as above, and for all n ≥ 1, there is a representation
V of a finite group Γ over k having a universal deformation ring RW(Γ,V ) which is isomorphic to
W[[t]]/(pnt,t2). In particular, this ring is not a complete intersection if pnW ?= {0}.
Theorem 1.4. Let k be perfect and let W = W(k) be the ring of infinite Witt vectors over k.
Then there exists a complete classification, given in Theorem 3.2, of all profinite groups Γ and all
continuous finite dimensional representations V of Γ over k with EndkΓ(V ) = k such that
Date: December 7, 2010.
2000 Mathematics Subject Classification. Primary 11F80; Secondary 11R32, 20C20,11R29.
The first author was supported in part by NSF Grant DMS0651332. The second author was supported in part
by NSF Grant DMS0801030. The third author was funded in part by the European Commission under contract
MRTN-CT-2006-035495.
1
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2FRAUKE M. BLEHER, TED CHINBURG, AND BART DE SMIT
• if K is the kernel of the Γ-action on V then V is projective as a module for Γ/K, and
• the universal deformation ring RW(Γ,V ) is isomorphic to W[[t]]/(pnt,t2) and the universal
deformation of V is faithful as a representation of Γ.
In [7], B¨ ockle gives a survey of recent results on presentations of deformation rings and of ap-
plications of such presentations to arithmetic geometry. In particular, [7] discusses how one can
show that deformation rings are complete intersections as well as the relevance of presentations to
arithmetic, e.g. to Serre’s conjectures in the theory of modular forms and Galois representations.
The problem of constructing representations having universal deformation rings which are not
complete intersections was first posed by M. Flach [9]. The first example of a representation of this
kind was found by Bleher and Chinburg when char(k) = 2; see [4, 5]. A more elementary argument
proving the same result was given in [8]. Theorem 1.3 gives an answer to Flach’s question for all
possible residue fields of positive characteristic.
As of this writing we do not know of a complete local commutative Noetherian ring R with
perfect residue field k of positive characteristic which cannot be realized as a versal deformation
ring of the form RW(k)(Γ,V ) for some profinite Γ and some representation V of Γ over k.
There is an extensive literature concerning explicit computations of universal deformation rings
(often with additional deformation conditions). See [7], [3], [1, 2] and their references for an intro-
duction to this literature. Theorem 1.3 and the formulation of the inverse problem in Question 1.1
first appeared in [6]. In subsequent work on the inverse problem, Rainone found in [16] some other
rings which are universal deformation rings and not complete intersections; see Remark 4.3.
The sections of this paper are as follows.
In §2 we recall the definitions of deformations and of versal and universal deformation rings and
describe how versal deformation rings change when extending the residue field k (see Theorem 2.2).
In §3 we consider arbitrary perfect fields k of characteristic p and we take W = W(k). In
Theorem 3.2, which implies Theorem 1.4, we give a sufficient and necessary set of conditions on a
representation˜V of a finite group Γ over k for the universal deformation ring RW(k)(Γ,˜V ) to be
isomorphic to R = W(k)[[t]]/(pnt,t2). The proof that these conditions are sufficient involves first
showing that RW(k)(Γ,˜V ) is a quotient of W(k)[[t]] by proving that the dimension of the tangent
space of the deformation functor associated to˜V is one. We then construct an explicit lift of˜V over
R and show that this cannot be lifted further to any small extension ring of R which is a quotient
of W(k)[[t]].
In §4 we prove Theorem 1.3. We use Theorem 2.2 to reduce the proof of Theorem 1.3 to the case
in which k = Fp= Z/p and W = W(k) = Zp. In the latter case we provide explicit examples using
twisted group algebras of the form E[G0] where E = Fp2 and G0= Gal(E/Fp).
Acknowledgments: The authors would like to thank M. Flach for correspondence about his
question. The second author would also like to thank the University of Leiden for its hospitality
during the spring of 2009 and the summer of 2010.
2. Deformation rings
Let Γ be a profinite group, and let k be a field of characteristic p > 0. Let W be a complete local
commutative Noetherian ring with residue field k. We denote byˆC the category of all complete
local commutative Noetherian W-algebras with residue field k. Homomorphisms inˆC are continuous
W-algebra homomorphisms which induce the identity map on k. Define C to be the full subcategory
of Artinian objects inˆC. For each ring A inˆC, let mAbe its maximal ideal and denote the surjective
morphism A → A/mA= k inˆC by πA. If α : A → A′is a morphism inˆC, we denote the induced
morphism GLd(A) → GLd(A′) also by α.
Let d be a positive integer, and let ρ : Γ → GLd(k) be a continuous homomorphism, where GLd(k)
has the discrete topology. By a lift of ρ over a ring A inˆC we mean a continuous homomorphism
τ : Γ → GLd(A) such that πA◦ τ = ρ. We say two lifts τ,τ′: Γ → GLd(A) of ρ over A are
strictly equivalent if one can be brought into the other by conjugation by a matrix in the kernel of
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INVERSE PROBLEMS FOR DEFORMATION RINGS3
πA: GLd(A) → GLd(k). We call a strict equivalence class of lifts of ρ over A a deformation of ρ
over A and define Defρ(A) to be the set of deformations [τ] of lifts τ of ρ over A. We then have a
functor
ˆHρ:ˆC → Sets
which sends a ring A inˆC to the set Defρ(A). Moreover, if α : A → A′is a morphism inˆC, then
ˆHρ(α) : Defρ(A) → Defρ(A′) sends a deformation [τ] of ρ over A to the deformation [α ◦ τ] of ρ
over A′.
Instead of looking at continuous matrix representations of Γ, we can also look at topological
Γ-modules as follows. Let V = kdbe endowed with the continuous Γ-action given by composition
of ρ with the natural action of GLd(k) on V , i.e. V is the d-dimensional topological kΓ-module
corresponding to ρ. A lift of V over a ring A ∈ˆC is then a pair (M,φ) consisting of a finitely
generated free A-module M on which Γ acts continuously together with a Γ-isomorphism φ : k ⊗A
M → V of (discrete) k-vector spaces. We define DefV(A) to be the set of isomorphism classes
[M,φ] of lifts (M,φ) of V over A. We then have a functor
ˆFV :ˆC → Sets
which sends a ring A inˆC to the set DefV(A). Moreover, if α : A → A′is a morphism inˆC,
thenˆFV(α) : DefV(A) → DefV(A′) sends a deformation [M,φ] of V over A to the deformation
[A′⊗A,αM,φα] of V over A′, where φαis the composition k⊗A′ (A′⊗A,αM)∼= k⊗AM
functorsˆFV andˆHρare naturally isomorphic.
One says that a ring R = RW(Γ,ρ) (resp. R = RW(Γ,V )) inˆC is a versal deformation ring for
ρ (resp. for V ) if there is a lift ν : Γ → GLd(R) of ρ over R (resp. a lift (U,φU) of V over R) such
that the following conditions hold. For all rings A inˆC, the map
φ
− → V . The
fA: HomˆC(R,A) → Defρ(A) (resp. fA: HomˆC(R,A) → DefV(A))
which sends a morphism α : R → A inˆC to the deformationˆHρ(α)([ν]) (resp. ˆFV(α)([U,φU])) is
surjective. Moreover, if k[ǫ] is the ring of dual numbers with ǫ2= 0, then fk[ǫ]is bijective. (Here
the W-algebra structure of k[ǫ] is such that the maximal ideal of W annihilates k[ǫ].) We call the
deformation [ν] (resp. [U,φU]) a versal deformation of ρ (resp. of V ) over R. By Mazur [15, Prop.
20.1],ˆHρ(resp.ˆFV) is continuous, which means that we only need to check the surjectivity of fA
for Artinian rings A in C. The versal deformation ring R = RW(Γ,ρ) (resp. R = RW(Γ,V )) is
unique up to isomorphism if it exists.
If the map fAis bijective for all rings A inˆC, then we say R = RW(Γ,ρ) (resp. R = RW(Γ,V ))
is a universal deformation ring of ρ (resp. of V ) and [ν] (resp. [U,φU]) is a universal deformation
of ρ (resp. of V ) over R. This is equivalent to saying that R represents the deformation functorˆHρ
(resp.ˆFV) in the sense thatˆHρ(resp.ˆFV) is naturally isomorphic to the Hom functor HomˆC(R,−).
We will suppose from now on that Γ satisfies the following p-finiteness condition used by Mazur
in [14, §1.1]:
Hypothesis 2.1. For every open subgroup J of finite index in Γ, there are only a finite number of
continuous homomorphisms from J to Z/p.
It follows by [14, §1.2] that for Γ satisfying Hypothesis 2.1, all finite dimensional continuous
representations V of Γ over k have a versal deformation ring. It is shown in [11, Prop. 7.1] that if
EndkΓ(V ) = k, then V has a universal deformation ring.
A proof of the following base change result is given in an appendix (see §5). For finite extensions
of k, this was proved by Faltings (see [19, Ch. 1]).
Theorem 2.2. Let Γ, k, W and ρ be as above. Let k′be a field extension of k. Suppose W′
is a complete local commutative Noetherian ring with residue field k′which has the structure of
a W-algebra, in the sense that we fix a local homomorphism W → W′. Let ρ′: Γ → GLd(k′)
be the composition of ρ with the injection GLd(k) ֒→ GLd(k′). Then the versal deformation ring
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4FRAUKE M. BLEHER, TED CHINBURG, AND BART DE SMIT
RW′(Γ,ρ′) is the completion R′of Ω = W′⊗WRW(Γ,ρ) with respect to the unique maximal ideal
mΩof Ω.
3. The inverse inverse problem for R = W[[t]]/(pnt,t2)
Throughout this section we make the following assumptions.
Hypothesis 3.1. Let k be an arbitrary perfect field of characteristic p > 0 and let W be the ring
W(k) of infinite Witt vectors over k. Let Γ be a profinite group satisfying Hypothesis 2.1. Let
d be a positive integer and let ˜ ρ : Γ → GLd(k) be a continuous representation of Γ. Denote the
corresponding kΓ-module by˜V . Let K = Ker(˜ ρ) and define G = Γ/K, so that G is a finite group.
Let π : Γ → G be the natural surjection. Let ρ : G → GLd(k) be the representation whose inflation
to Γ is ˜ ρ, and denote the kG-module corresponding to ρ by V . Suppose V is a projective kG-module
and that EndkG(V ) = k. Let n ≥ 1 be a fixed integer and define A = W/(Wpn). Let VA be a
projective AG-module such that k ⊗AVA is isomorphic to V as a kG-module. Let MAbe the free
A-module HomA(VA,VA), so that MAis a projective AG-module. Define
M = k ⊗AMA= Homk(V,V ).
If L is an AG-module, we will also view L as an (Z/pn)G-module via restriction of operators from
AG to (Z/pn)G.
Theorem 3.2. Assume Hypothesis 3.1. The following statements (i) and (ii) are equivalent:
(i) The universal deformation ring RW(Γ,˜V ) is isomorphic to W[[t]]/(pnt,t2) and the universal
deformation of˜V as a representation of Γ is faithful.
(ii) The following conditions hold:
(a) The group K is a finitely generated (Z/pn)G-module.
(b) Writing K additively, the group Hom(Z/p)G(K/pK,M) is a one-dimensional k-vector
space with respect to the k-vector space structure induced by M.
(c) There is an injective homomorphism ψ : K → MAin Hom(Z/pn)G(K,MA) whose image
is not contained in pMA.
(d) Either
• there exist g,h ∈ K with ψ(g) ◦ ψ(h) ?≡ ψ(h) ◦ ψ(g) mod pMA, or
• p = 2 and there exists x ∈ K of order 2 with ψ(x) ◦ ψ(x) ?≡ 0 mod 2MA.
Note that Theorem 3.2 implies Theorem 1.4. To show Theorem 1.3, we construct in Section 4
examples for which the conditions in Theorem 3.2(ii) are satisfied.
The following Remark 3.3 and Lemma 3.4 play an important role when proving the equivalence
of (i) and (ii) in Theorem 3.2. For any G-module L, we denote by˜L the Γ-module which results by
inflating L via the natural surjection π : Γ → G.
Remark 3.3. Since VAis a projective AG-module which is a lift of V over A, there exists a matrix
representation ρW: G → GLd(W) whose reduction mod pnW is a matrix representation ρA: G →
GLd(A) for VA, and whose reduction mod pW is the matrix representation ρ : G → GLd(k) for V .
Let R = W[[t]]/(pnt,t2). We have an exact sequence of multiplicative groups
(3.1)1 → (1 + tMatd(R))∗→ GLd(R) → GLd(W) → 1
resulting from the natural isomorphism R/tR = W. The isomorphism tR → A = W/pnW defined
by tw → w mod pnW for w ∈ W ⊂ R gives rise to isomorphisms of groups
(1 + tMatd(R))∗∼= Matd(A)+∼= MA= HomA(VA,VA)
where Matd(A)+is the additive group of Matd(A). Hence we obtain a short exact sequence of
profinite groups
(3.2)
(3.3)1 → MA→ GLd(R) → GLd(W) → 1
where the homomorphism MA→ GLd(R) results from (3.1) and (3.2).
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INVERSE PROBLEMS FOR DEFORMATION RINGS5
The conjugation action of ρW(G) ⊂ GLd(W) on (1+tMatd(R))∗which results from (3.1) factors
through the homomorphism ρW(G) → ρA(G) ⊂ GLd(A) = AutA(VA).
with the action of G on MA = HomA(VA,VA) in (3.2) coming from the action of G on VA via
ρA: G → GLd(A).
This action coincides
Lemma 3.4. Let ρW, ρA and R be as in Remark 3.3.
homomorphisms ψ : K → MAand ρR: Γ → GLd(R) such that there is a commutative diagram
Suppose there exist continuous group
(3.4)
1
??K
ψ
??
??Γ
ρR
??
π
??G
ρW
??
??1
1
??MA
??GLd(R)
??GLd(W)
??1
where the bottom row is given by (3.3).
Suppose R′is a W-algebra inˆC which is a small extension of R, in the sense that there is an
exact sequence
0 → J → R′ ν− → R → 0(3.5)
in which ν is a continuous W-algebra homomorphism and dimk(J) = 1. Define M′
of the homomorphism GLd(R′) → GLd(W) resulting from the composition of R′ν− → R with R → W.
Let E = (1 + Matd(J))∗. There is a natural exact sequence of groups
1 → E → M′
A→ MA→ 1.
There is a continuous representation ρR′ : Γ → GLd(R′) which lifts ρRif and only if there is a
homomorphism ψ′: K → M′
Awhich lifts ψ.
Ato be the kernel
(3.6)
Proof. The natural short exact sequence (3.6) results from the observation that M′
elements in GLd(R′) whose image in GLd(R) under ν lies in MA, viewed as a subgroup of GLd(R)
via (3.3).
The group E = (1 + Matd(J))∗is naturally isomorphic to˜ M = Homk(˜V ,˜V ) as a kΓ-module,
since J has k-dimension 1. In particular, K acts trivially on E.
Since M is a projective kG-module, we have Hi(G,H0(K,˜
Because Hom(K,M) is isomorphic to a direct summand of a kG-module that is induced from
the trivial subgroup of G, Hom(K,M) is cohomologically trivial.
Hi(G,Hom(K,M)) = 0 for all i > 0. This implies that the Hochschild-Serre spectral sequence
for H2(Γ,˜
M) degenerates to give
H2(Γ,˜
M) = H0(G,H2(K,˜
Aconsists of all
M)) = Hi(G,M) = 0 if i > 0.
Hence Hi(G,H1(K,˜
M)) =
(3.7)M)) = H2(K,˜
M)G.
But this means that the restriction homomorphism
H2(Γ,E) → H2(K,E)
is injective. Since the obstruction to the existence of a lift ρR′ of ρRis an element ω ∈ H2(Γ,E)
whose restriction to K gives the obstruction to the existence of a lift ψ′of ψ, this completes the
proof of Lemma 3.4.
?
Remark 3.5. For later use, we now analyze small extensions R′of R = W[[t]]/(pnt,t2) which are
themselves quotients of W[[t]]. Suppose I is an ideal of W[[t]] that is contained in the ideal (pnt,t2)
such that the natural surjection ν : R′→ R is a small extension as in (3.5). Since J = (pnt,t2)/I
is isomorphic to k, it follows that I contains the product ideal
(pnt,t2) · (p,t) = (pn+1t,pt2,t3)
in W[[t]]. Now (pnt,t2)/(pn+1t,pt2,t3) is a two-dimensional vector space over k with a basis given
by the classes of pnt and t2. Since dimk((pnt,t2)/I) = 1 and (pn+1t,pt2,t3) ⊂ I, there exist a,b ∈ W
such that
I = (pn+1t,pt2,t3,apnt + bt2)(3.8)
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6FRAUKE M. BLEHER, TED CHINBURG, AND BART DE SMIT
and at least one of a or b is a unit.
Suppose first that b is a unit. Then t2= −b−1apnt in R′= W[[t]]/I. Hence
I = (pn+1t,t2+ b−1apnt) (3.9)
since pt2= −b−1apn+1t ∈ I and t3= −b−1apnt2∈ Wpt2⊂ I. Moreover,
R′= W[[t]]/I = W[[t]]/(pn+1t,t2+ b−1apnt) = W ⊕ (Wt/Wpn+1t). (3.10)
Now suppose b ∈ pW, so that a must be a unit. Then
(3.11)I = (pnt,pt2,t3)
since bt2∈ Wpt2lies in I, so (apnt + bt2) − bt2= apnt ∈ I and a is a unit in W. Moreover,
R′= W[[t]]/I = W[[t]]/(pnt,pt2,t3) = W ⊕ (Wt/Wpnt) ⊕ (Wt2/Wpt2). (3.12)
3.1. Proof that (ii) implies (i) in Theorem 3.2. Throughout this subsection, we assume that
condition (ii) of Theorem 3.2 holds. As before, if L is a G-module we denote by˜L the Γ-module
which results by inflating L via the natural surjection π : Γ → G.
Lemma 3.6. One has dimkH1(Γ,˜
RW(Γ,˜V ) of˜V has dimension 1. The ring RW(Γ,˜V ) is a quotient of W[[t]].
M) = 1. The tangent space of the universal deformation ring
Proof. Since M is a projective kG-module, we have Hi(G,H0(K,˜
Therefore the Hochschild-Serre spectral sequence for H1(Γ,˜
H1(Γ,˜
M) = H0(G,H1(K,˜
M)) = Hi(G,M) = 0 if i > 0.
M) degenerates to give
(3.13)M)) = H0(G,Hom(K,M)) = Hom(K,M)G.
Writing K additively and using that M has exponent p, we have from condition (ii)(b) of Theorem
3.2 that
Hom(K,M)G= Hom(K/pK,M)G= Hom(Z/p)G(K/pK,M)∼= k.(3.14)
On putting together (3.13) and (3.14), we conclude from [15, Prop. 21.1] that there is a natural
isomorphism
?
(3.15)t˜V=defHomk
m
m2+ pRW(Γ,˜V ),k
?
→ H1(Γ,˜
M) = k
where t˜Vis the tangent space of the deformation functor of˜V and m is the maximal ideal of the
universal deformation ring RW(Γ,˜V ). This implies
?
m2+ pRW(Γ,˜V )
so there is a continuous surjection of W-algebras W[[t]] → RW(Γ,˜V ).
dimk
m
?
= 1
?
Lemma 3.7. Let ρW, ρAand R be as in Remark 3.3. There exists a lift ρR: Γ → GLd(R) of the
representation ˜ ρ : Γ → GLd(k) for˜V such that ρRlies in a commutative diagram of the form (3.4)
where ψ : K → MAis as in condition (ii)(c) of Theorem 3.2. Let γ : RW(Γ,˜V ) → R be the unique
continuous W-algebra homomorphism corresponding to the isomorphism class of the lift ρR. Then
γ is surjective. There is a W-algebra surjection µ : W[[t]] → RW(Γ,˜V ) whose composition with γ is
the natural surjection W[[t]] → R = W[[t]]/(pnt,t2). The kernel of µ is an ideal of W[[t]] contained
in (pnt,t2).
Proof. The obstruction to the existence of ρR is an element of H2(G,MA). This group is trivial
since MA is projective, so ρR exists. Since ρW is a lift of the matrix representation ρ : G →
GLd(k) = Autk(V ) over W, we find that ρRis a lift of ρ ◦ π = ˜ ρ over R.
The ring k[ǫ] of dual numbers over k is isomorphic to R/pR = k[[t]]/(t2), and γ is surjective if
and only if it induces a surjection
RW(Γ,V )
m2+ pRW(Γ,V )−→
(3.16)
γ :
R
m2
R+ pR=
R
pR
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INVERSE PROBLEMS FOR DEFORMATION RINGS7
where m is the maximal ideal of RW(Γ,V ). If γ is not surjective, its image is k. Thus to prove that
γ is surjective, it will suffice to show that the composition ρR/pRof ρRwith the natural surjection
GLd(R) → GLd(R/pR) = GLd(k[ǫ]) is not a matrix representation of the trivial lift of˜V over k[ǫ].
However, the kernel of the action of Γ on this trivial lift is K ⊂ Γ, while ρR/pRis not trivial on K
because of condition (ii)(c) of Theorem 3.2. Hence γ must be surjective.
The tangent space of the deformation functor of V is one dimensional by Lemma 3.6, so (3.16)
is in fact an isomorphism. Let r be any element of RW(Γ,V ) such that γ(r) is the class of t in
R = W[[t]]/(pnt,t2). We then have a unique continuous W-algebra homomorphism µ : W[[t]] →
RW(Γ,V ) which maps t to r. Since (γ ◦µ)(t) is the class of t in R, we se that γ ◦µ is surjective. So
because γ is an isomorphism, Nakayama’s lemma implies that µ : W[[t]] → RW(Γ,V ) is surjective.
?
We now complete the proof that (ii) implies (i) in Theorem 3.2. Let R = W[[t]]/(pnt,t2) and let
ρR: Γ → GLd(R) be the lift of ˜ ρ from Lemma 3.7. Let ψ : K → MAbe the injective (Z/pn)G-
module homomorphism from condition (ii)(c) of Theorem 3.2. Since ρR lies in a commutative
diagram of the form (3.4) and ψ and ρWare both injective, it follows that ρRis faithful.
Let R′= W[[t]]/I be a small extension of R as in Remark 3.5, so that ν : R′→ R is the natural
surjection. Let M′
composition R′
homomorphism ψ′: K → M′
Awhich lifts ψ.
Suppose to the contrary that such a homomorphism ψ′exists. Write K additively and M′
multiplicatively. Define S to be the union of {0} with the set of Teichm¨ uller lifts in W = W(k) of
the elements of k∗. Let g ∈ K be arbitrary. Then there exist unique
Abe the kernel of the homomorphism GLd(R′) → GLd(W) resulting from the
ν− → R → W. By Lemmas 3.4 and 3.7, it is enough to show that there is no group
A
α0(g),α1(g),...,αn−1(g) ∈ Matd(S)
such that
(3.17)ψ(g) = α0(g) + pα1(g) + ··· + pn−1αn−1(g).
Moreover, since ψ′lifts ψ, we have
ψ′(g) ≡ 1 + tψ(t)mod (pnt,t2)Matd(R′).
By Remark 3.5, there exist a,b ∈ W such that I is as in (3.8) and such that one of the alternatives
(3.9) or (3.11) holds. Suppose first that b is a unit in (3.8) and we have alternative (3.9). By (3.10),
it follows that there exists a unique β(g) ∈ Matd(S) such that
(3.18)ψ′(g) = 1 + tα0(g) + ptα1(g) + ··· + pn−1tαn−1(g) + pntβ(g).
If a ∈ pW in (3.9), it follows that t2= 0 = pn+1t in R′. Therefore, since (pn)g = 0Kbecause of
condition (ii)(a) of Theorem 3.2, we have
(3.19) 1 = ψ′(g)pn=?1 + tα0(g) + ptα1(g) + ··· + pn−1tαn−1(g) + pntβ(g)?pn
when b is a unit and a ∈ pW. Thus pntα0(g) = 0. Since alternative (3.9) holds, this means that
α0(g) = 0, which implies by (3.17) that ψ(g) ∈ pMA. Since g was an arbitrary element of K, this
is a contradiction to condition (ii)(c) of Theorem 3.2. Hence the case when b is a unit and a ∈ pW
in (3.8) cannot occur.
If both b and a are units in (3.8), then by (3.9) we have pn+1t = 0 and t2= −b−1apnt in R′.
Suppose h is another element of K. Because pt2= 0 in R′, it follows from (3.18) that
= 1 + pntα0(g)
(3.20)ψ′(g) · ψ′(h) − ψ′(h) · ψ′(g) = −(b−1a)pnt[α0(g) · α0(h) − α0(h) · α0(g)].
If b is not a unit in (3.8), then a has to be a unit, and alternative (3.11) holds. By (3.12), it
follows that there exists a unique β(g) ∈ Matd(S) such that
(3.21)ψ′(g) = 1 + tα0(g) + ptα1(g) + ··· + pn−1tαn−1(g) + t2β(g).
Page 8
8FRAUKE M. BLEHER, TED CHINBURG, AND BART DE SMIT
Because pt2= 0 = t3in R′in this case, it follows from (3.21) that
(3.22)ψ′(g) · ψ′(h) − ψ′(h) · ψ′(g) = t2[α0(g) · α0(h) − α0(h) · α0(g)].
Because K is abelian, we must have ψ′(h+g) = ψ′(g+h), and thus ψ′(g)·ψ′(h)−ψ′(h)·ψ′(g) = 0.
Therefore, it follows from (3.20) (resp. (3.22)) that
(3.23)α0(g) · α0(h) ≡ α0(h) · α0(g) mod pMatd(W) for all g,h ∈ K.
This implies by (3.17) that for all g,h ∈ K we have
(3.24)ψ(g) ◦ ψ(h) ≡ ψ(h) ◦ ψ(g)mod pMA
where ◦ stands for the composition of elements in MA= HomA(VA,VA). In other words, the first
case in condition (ii)(d) cannot occur. Therefore, we must have that p = 2 and that there exists
an element x ∈ K of order 2 such that ψ(x) ◦ ψ(x) ?≡ 0 mod 2MA. Replacing g = x in (3.18)
(resp. (3.21)) and using that ψ′(x) · ψ′(x) = ψ′(x + x) = ψ′(0K) = 1 shows that in both cases
α0(x) · α0(x) ≡ 0 mod pMatd(W). By (3.17), this means that ψ(x) ◦ ψ(x) ∈ pMA= 2MA. Since
this is a contradiction to condition (ii)(d) of Theorem 3.2, this completes the proof of (ii) implies
(i) in Theorem 3.2.
3.2. Proof that (i) implies (ii) in Theorem 3.2. Throughout this subsection, we assume that
condition (i) of Theorem 3.2 holds. Let ρW, ρA and R = W[[t]]/(pnt,t2) be as in Remark 3.3.
By assumption, RW(Γ,˜V ) is isomorphic to R. Since the natural surjection R → W which sends t
to 0 is the unique morphism inˆC from R to W, there exists a universal lift ρR: Γ → GLd(R) of
˜ ρ : Γ → GLd(k) over R such that ρRfollowed by GLd(R) → GLd(W) is equal to ρW◦π. This implies
that the image of K under ρRlies inside (1 + tMatd(R))∗. Let ψ : K → MAbe the restriction of
ρRto K followed by the isomorphism (1+tMatd(R))∗ ∼= MAfrom (3.2). We obtain that ρRlies in
a commutative diagram of the form (3.4).
Since ρRis faithful by assumption, ψ is an injective group homomorphism. In particular, K is an
abelian group which is annihilated by pn, and hence a (Z/pn)G-module. As seen in Remark 3.3, the
conjugation action of ρW(G) ⊂ GLd(W) on (1 + tMatd(R))∗factors through the homomorphism
ρW(G) → ρA(G) ⊂ GLd(A) = AutA(VA). Since this action coincides with the action of G on
MA= HomA(VA,VA) in (3.2) coming from the action of G on VAvia ρA: G → GLd(A), it follows
that ψ is an injective homomorphism in Hom(Z/pn)G(K,MA). Let ρR/pRbe the composition of ρR
with the natural surjection GLd(R) → GLd(R/pR) = GLd(k[ǫ]). If the image of ψ is contained in
pMA, it follows that ρR/pRfactors through G. Since V is a projective kG-module, this implies that
ρR/pRis a matrix representation of the trivial lift of˜V over k[ǫ]. Since R/pR∼= k[ǫ] is the universal
deformation ring associated to mod p lifts of ˜ ρ, this is a contradiction. Hence the image of ψ is not
contained in pMA, giving condition (ii)(c) of Theorem 3.2.
Writing K additively, it follows from Hypothesis 2.1 that K/pK is a finitely generated elementary
abelian p-group. Since K/pK is the Frattini quotient of K, this implies that K is finitely generated
as a Z/pn-module, which is condition (ii)(a) of Theorem 3.2.
By assumption, R/pR∼= k[ǫ], which implies H1(Γ,˜
projective kG-module by Hypothesis 3.1, we see as in (3.13) that H1(Γ,˜
Hom(K,M)G= Hom(Z/p)G(K/pK,M), this gives condition (ii)(b) of Theorem 3.2.
Suppose condition (ii)(d) of Theorem 3.2 fails. We will show that then ρRcan be lifted from R
to the small extension R′= W[[t]]/I, where
M)∼= k since R = RW(Γ,˜V ). Because M is a
M) = Hom(K,M)G. Since
I = (pnt,pt2,t3),
so we are in case (3.11) of Remark 3.5. Let J = (pnt,t2)/I. By Lemma 3.4, it is enough to show
that ψ can be lifted to a homomorphism ψ′: K → M′
Awhere M′
Alies in a short exact sequence
1 → (1 + Matd(J))∗→ M′
A→ MA→ 1.
In what follows, we write K additively and M′
MAwith (1 + tMatd(R))∗.
Amultiplicatively. Moreover using (3.2), we identify
Page 9
INVERSE PROBLEMS FOR DEFORMATION RINGS9
If p ?= 2, define ψ′: K → M′
Ato be the exponential function of (tψ(g)) mod I. In other words,
ψ′(g) = 1 + tψ(g) +t2
2[ψ(g) ◦ ψ(g)].
Since we assume that condition (ii)(d) fails, i.e. the image of ψ is commutative mod pMA with
respect to map composition, it follows that ψ′is a group homomorphism which lifts ψ.
If p = 2, we use that K is a finitely generated (Z/2n)-module. Let x1,...,xr be a minimal set
of generators of K. We will show that ψ′may be defined by letting
(3.25)ψ′(xj) = 1 + tψ(xj)
for 1 ≤ j ≤ r and by extending ψ′additively to all of K. Since ψ is a group homomorphism and
2t2= 0 in R′, we have
ψ′(xj)2
ψ′(xj)2i
=1 + tψ(2xj) + t2[ψ(xj) ◦ ψ(xj)] and
1 + tψ((2i)xj)for 2 ≤ i ≤ n.=
For p = 2, the failing of condition (ii)(d) means that not only the image of ψ is commutative mod
2MAwith respect to map composition, but also that ψ(x) ◦ ψ(x) ≡ 0 mod 2MAfor all x ∈ K of
order 2. Hence it follows that if xj has order 2 then ψ′(xj)2= 1. Therefore, we can extend (3.25)
additively to obtain a group homomorphism ψ′: K → M′
This completes the proof of (i) implies (ii) in Theorem 3.2.
Awhich lifts ψ.
4. The inverse problem for R = W[[t]]/(pnt,t2)
In this section, we use Theorem 3.2 to prove Theorem 1.3. We first establish a special case.
Theorem 4.1. Let k = Fp, W = W(k) = Zp, n ≥ 1 and A = W/pnW = Z/pn. Let E = Fp2 and
let G0= Gal(E/k). Define G = E∗×G0, where G0acts on E∗by restricting the natural action of
G0on E to E∗. The natural action of G0and E∗on V = E makes V into a projective and simple
kG-module. The endomorphism ring M = Endk(V ) is isomorphic to the twisted group ring V [G0]
as k-algebras. There exists a simple projective kG-module V′such that
(4.26)M∼= V′⊕ kG0
as kG-modules. Let K = V′
Let Γ be the semidirect product K×δG where δ : G → Aut(K) is the group homomorphism given
by the G-action on the (Z/pn)G-module K = V′
the universal deformation ring RW(Γ,˜V ) is isomorphic to W[[t]]/(pnt,t2).
Abe a projective AG-module such that k ⊗AV′
A∼= V′as kG-modules.
A. If˜V is the inflation of V to a kΓ-module, then
Proof. Let VA be a projective AG-module such that k ⊗AVA∼= V as kG-modules. Let MA =
EndA(VA). We prove that G, K, M and MAsatisfy the conditions in Theorem 3.2(ii).
If p = 2, then G is isomorphic to the symmetric group S3on 3 letters and V is the unique simple
projective kG-module, up to isomorphism. If p ≥ 3, then the order of G is relatively prime to p and
V is also a simple projective kG-module.
Since V = E is a Galois algebra over k with Galois group G0, it follows that M = Endk(V ) is
isomorphic to the twisted group ring E[G0] as k-algebras. This isomorphism defines a kG-module
structure on E[G0] by conjugation as follows. Let G0= ?σ?, let E∗= ?ζ? and let x = b0+ b1σ ∈
E[G0], so b0,b1∈ E. Then σ.x = σxσ−1= (b0)p+ (b1)pσ and ζ.x = ζxζ−1= b0+ b1ζ1−pσ.
We have E[G0] = E + Eσ as k-vector spaces. The above G-action on E[G0] implies that both
E and Eσ are kG-submodules of E[G0]. It follows for example from the normal basis theorem that
E∼= kG0as kG-modules, where E∗⊂ G acts trivially by conjugation on E. Thus to prove (4.26) it
suffices to show that V′= Eσ is a simple projective kG-module. Since V is a projective kG-module,
so are M, E[G0] and V′. Considering the action of E∗= ?ζ? on Eσ, we see that the action of ζ
has eigenvalue ζ1−p. Since ζ1−plies in Fp2 − Fp, it follows that V′= Eσ is a simple projective
kG-module.
Page 10
10FRAUKE M. BLEHER, TED CHINBURG, AND BART DE SMIT
For all p, let K = V′
Theorem 3.2(ii). Define Γ = K×δG where δ : G → Aut(K) is the group homomorphism given by
the G-action on the (Z/pn)G-module K = V′
as (Z/p)G-modules, it follows that
Hom(Z/p)G(K/pK,M)∼= Hom(Z/p)G(V′,V′⊕ kG0)∼= k
giving condition (b) of Theorem 3.2(ii). Since K = V′
that HomAG(K,MA) is a projective A-module H such that H/pH = HomkG(K/pK,M)∼= k.
Therefore, HomAG(K,MA)∼= A and there exists an injective AG-module homomorphism ψ ∈
HomAG(K,MA) whose image is not contained in pMA. Since A = Z/pn, this gives condition (c)
of Theorem 3.2(ii). By the above calculations in the twisted group algebra E[G0], we see that the
image of ψ mod pMA is isomorphic to Eσ. Since for example (σ)(ζσ) = ζp?= ζ = (ζσ)(σ), we
obtain that the image of ψ mod pMAis not commutative with respect to the multiplication in the
ring MA. This gives condition (d) of Theorem 3.2(ii). Therefore, it follows from Theorem 3.2 that
RW(Γ,˜V ) is isomorphic to W[[t]]/(pnt,t2).
A, so K is a finitely generated (Z/pn)G-module, giving condition (a) of
A. Since by our above calculations, M∼= (K/pK)⊕kG0
Aand MAare projective AG-modules, it follows
?
Remark 4.2. If p > 3, we can replace the group G in Theorem 4.1 by the symmetric group S3and
V by the 2-dimensional simple projective kS3-module. It follows then that M = Homk(V,V )∼=
k[Z/2] ⊕ V as kG-modules, which means that we can take V′= V and K = VAin this case.
Remark 4.3. As mentioned in the introduction, in subsequent work on Question 1.1, Rainone proved
in [16] that if p > 3 and 1 ≤ m ≤ n, the ring Zp[[t]]/(pn,pmt) is a universal deformation ring relative
to W = Zp. These rings and the rings of Theorems 1.3 and 4.1 form disjoint sets of isomorphism
classes. Rainone’s work gave the first negative answers to two questions of Bleher and Chinburg
(Question 1.2 of [5] and Question 1.1 of [3]). Later we observed that Theorem 4.1 also gives a
negative answer to Question 1.2 of [5] when p > 2.
Completion of the Proof of Theorem 1.3. Let k, p, W and n be as in Theorem 1.3. By Theorem
4.1, there is a finite group Γ and a representation V0 of Γ over Fp such that EndFpG(V0) = Fp
and the universal deformation ring RZp(Γ,V0) is isomorphic to Zp[[t]]/(pnt,t2). Let V = k ⊗Fp
V0. Then EndkG(V )∼= k ⊗FpEndFpG(V0)∼= k. By Theorem 2.2, the universal deformation ring
RW(Γ,V ) is isomorphic to the completion of W ⊗ZpZp[[t]]/(pnt,t2) with respect to its maximal
ideal. This completion is isomorphic to W[[t]]/(pnt,t2). It remains to show that this ring is not a
complete intersection if pnW ?= {0}. This is clear if W is regular. In general, if one assumes that
W[[t]]/(pnt,t2) is a complete intersection, then W is a quotient S/I for some regular complete local
commutative Noetherian ring S and a proper ideal I of S. If S′= S[[t]], then W[[t]]/(pnt,t2) = S′/I′
when I′is the ideal of S′generated by I, pnt and t2. Since dimW[[t]]/(pnt,t2) = dimW, we obtain
by [13, Thm. 21.1] that
dimk(I′/mS′I′) = dimS′− dim(S′/I′) = dimS + 1 − dim(S/I) ≤ dimk(I/mSI) + 1.
Using power series expansions, we see that dimk(I′/mS′I′) = dimk(I/mSI)+2 if pnW ?= {0}. Since
this contradicts (4.27), W[[t]]/(pnt,t2) is not a complete intersection if pnW ?= {0}. This completes
the proof of Theorem 1.3.
(4.27)
Remark 4.4. To construct more examples to which Theorem 3.2 applies, there are two fundamental
issues. One must construct a group G and a projective kG-module V for which both the left kG-
module structure and the ring structure of M = Homk(V,V ) can be analyzed sufficiently well to
be able to produce a G-module K having the properties in the Theorem. When one can identify
the ring Homk(V,V ) with a twisted group algebra, as in the proof of Theorem 4.1, this can be very
useful in checking condition (ii)(d) of Theorem 3.2. A natural approach to analyzing the kG-module
structure of M is to note that the Brauer character ξM of M is the tensor product ξV ⊗ ξV∗ of
the Brauer characters of V and its k-dual V∗. For example, if V is induced from a representation
X of a subgroup H of G, then ξV is given by the usual formula for the character of an induced
representation. If dimk(X) = 1, the analysis of the ring structure of M becomes a combinatorial
problem using X and coset representatives of H in G.
Page 11
INVERSE PROBLEMS FOR DEFORMATION RINGS11
5. Appendix: Proof of Theorem 2.2
We assume the notation in the statement of Theorem 2.2. Let R = RW(Γ,ρ). Recall that
Ω = W′⊗WR and R′is the completion of Ω with respect to its unique maximal ideal mΩ. Define
ˆC′to be the category of all complete local commutative Noetherian W′-algebras with residue field
k′. Let ν : Γ → GLd(R) be a versal lift of ρ over R, and let ν′: Γ → GLd(R′) be the lift of ρ′over
R′defined by ν′(g) = (1 ⊗ ν(g)i,j)1≤i,j≤dfor all g ∈ Γ.
The first step is to show that if A′∈ Ob(C′) is an Artinian W′-algebra with residue field k′and
τ′: Γ → GLd(A′) is a lift of ρ′over A′, then there is a morphism α : R′→ A′inˆC′such that
[τ′] = [α◦ν′]. Since A′is Artinian, HomˆC′(R′,A′) is equal to the space Homcont(Ω,A′) of continuous
W′-algebra homomorphisms which induce the identity map on the residue field k′. Because of
Hypothesis 2.1, one can find a finite set S ⊆ Γ such that τ′(S) is a set of topological generators for
the image of τ′. Since ρ′and ρ have the same image in GLd(k) ⊂ GLd(k′), there exists for each g ∈ S
a matrix t(g) ∈ Matd(W) such that all entries of the matrix τ′(g)−t(g) lie in the maximal ideal mA′
of A′. Let T ⊆ mA′ be the finite set of all matrix entries of τ′(g) − t(g) as g ranges over S. Then
there is a continuous homomorphism f : W[[x1,...,xm]] → A′with m = #T and {f(xi)}m
Since A′has the discrete topology, the image B of f must be a local Artinian W-algebra with
residue field k. Since τ′(S) is a set of topological generators for the image of τ′, it follows that τ′
defines a lift of ρ over B. Because ν : Γ → GLd(R) is a versal lift of ρ over the versal deformation
ring R = RW(Γ,ρ) of ρ, there is a morphism β : R → B inˆC such that τ′: Γ → GLd(B) is conjugate
to β ◦ ν by a matrix in the kernel of πB : GLd(B) → GLd(B/mB) = GLd(k). Let β′: R → A′
be the composition of β with the inclusion B ⊂ A′. Define α : R′→ A′to be the morphism in
ˆC′corresponding to the continuous W′-algebra homomorphism Ω = W′⊗WR → A′which sends
w′⊗ r to w′· β′(r) for all w′∈ W′and r ∈ R. It follows that α satisfies [τ′] = [α ◦ ν′].
The second step is to show that when k′[ǫ] is the ring of dual numbers overk′, then HomˆC′(R′,k′[ǫ])
is canonically identified with the set Defρ′(k′[ǫ]) of deformations of ρ′over k′[ǫ]. Since k′[ǫ] is Ar-
tinian, it suffices to show that Homcont(Ω,k′[ǫ]) is identified with Defρ′(k′[ǫ]). Let
i=1= T.
(5.28)T(W′,Ω) =
mΩ
m2
Ω+ Ω · mW′
andT(W,R) =
mR
m2
R+ R · mW
so that we have natural isomorphisms Homcont(Ω,k′[ǫ])∼= Homk′(T(W′,Ω),k′) and HomˆC(R,k[ǫ])∼=
Homk(T(W,R),k). Since Ad(ρ′) = k′⊗kAd(ρ), we have from [15, Prop. 21.1] that there are natural
isomorphisms
Defρ′(k′[ǫ]) = H1(Γ,Ad(ρ′)) = k′⊗kH1(Γ,Ad(ρ)) = k′⊗kDefρ(k[ǫ]).
Hence it suffices to show that the natural homomorphism µ : k′⊗kT(W,R) → T(W′,Ω) is an
isomorphism of k′-vector spaces. Since mW is finitely generated, one can reduce to the case when
W = k, by considering generators α of mW and successively replacing W by W/(Wα) and R by
R/(Rα). One then divides W′and Ω further by ideals generated by generators for mW′ to be able
to assume that W′= k′. However, the case when W = k and W′= k′is obvious, since then
T(k′,Ω) = mΩ/m2
?mR/m2
References
Ω∼= k′⊗k
R
?= k′⊗ T(k,R). This completes the proof of Theorem 2.2.
[1] F. M. Bleher, Universal deformation rings and dihedral defect groups. Trans. Amer. Math. Soc. 361 (2009),
3661–3705.
[2] F. M. Bleher, Universal deformation rings and generalized quaternion defect groups. Adv. Math. 225 (2010),
1499–1522.
[3] F. M. Bleher and T. Chinburg, Universal deformation rings and cyclic blocks. Math. Ann. 318 (2000), 805–836.
[4] F. M. Bleher and T. Chinburg, Universal deformation rings need not be complete intersections. C. R. Math.
Acad. Sci. Paris 342 (2006), 229–232.
[5] F. M. Bleher and T. Chinburg, Universal deformation rings need not be complete intersections. Math. Ann. 337
(2007), 739–767.
[6] F. M. Bleher, T. Chinburg and B. de Smit, Deformation rings which are not local complete intersections, March
2010. arXiv:1003.3143
Page 12
12FRAUKE M. BLEHER, TED CHINBURG, AND BART DE SMIT
[7] G. B¨ ockle, Presentations of universal deformation rings. In: L-functions and Galois representations, 24–58,
London Math. Soc. Lecture Note Ser., 320, Cambridge Univ. Press, Cambridge, 2007.
[8] J. Byszewski, A universal deformation ring which is not a complete intersection ring. C. R. Math. Acad. Sci.
Paris 343 (2006), 565–568.
[9] T. Chinburg, Can deformation rings of group representations not be local complete intersections? In: Problems
from the Workshop on Automorphisms of Curves. Edited by Gunther Cornelissen and Frans Oort, with contri-
butions by I. Bouw, T. Chinburg, Cornelissen, C. Gasbarri, D. Glass, C. Lehr, M. Matignon, Oort, R. Pries and
S. Wewers. Rend. Sem. Mat. Univ. Padova 113 (2005), 129–177.
[10] H. Darmon, F. Diamond and R. Taylor, Fermat’s Last Theorem. In : R. Bott, A. Jaffe and S. T. Yau (eds),
Current developments in mathematics, 1995, International Press, Cambridge, MA., 1995, pp. 1–107.
[11] B. de Smit and H. W. Lenstra, Explicit construction of universal deformation rings. In: G. Cornell, J. H.
Silverman and G. Stevens (eds), Modular Forms and Fermat’s Last Theorem (Boston, MA, 1995), Springer-
Verlag, Berlin-Heidelberg-New York, 1997, pp. 313–326.
[12] A. Grothendieck,´El´ ements de g´ eom´ etrie alg´ ebrique, Chapitre IV, Quatri´ eme Partie. Publ. Math. IHES 32 (1967),
5–361.
[13] H. Matsumura, Commutative Ring Theory. Cambridge Studies in Advanced Mathematics, Vol. 8, Cambridge
University Press, Cambridge, 1989.
[14] B. Mazur, Deforming Galois representations. In: Galois groups over Q (Berkeley, CA, 1987), Springer-Verlag,
Berlin-Heidelberg-New York, 1989, pp. 385–437.
[15] B. Mazur, An introduction to the deformation theory of Galois representations. In: G. Cornell, J. H. Silverman
and G. Stevens (eds), Modular Forms and Fermat’s Last Theorem (Boston, MA, 1995), Springer-Verlag, Berlin-
Heidelberg-New York, 1997, pp. 243–311.
[16] R. Rainone, On the inverse problem for deformation rings of representations. Master’s thesis, Universiteit Leiden,
B. de Smit thesis advisor, June 2010. http://www.math.leidenuniv.nl/en/theses/205/
[17] M. Schlessinger, Functors of Artin Rings. Trans. of the AMS 130 (1968), 208–222.
[18] J. P. Serre, Corps Locaux. Hermann, Paris, 1968.
[19] A. Wiles, Modular elliptic curves and Fermat’s last theorem. Ann. of Math. 141 (1995), 443–551.
F.B.: Department of Mathematics, University of Iowa, Iowa City, IA 52242-1419
E-mail address: frauke-bleher@uiowa.edu
T.C.: Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395
E-mail address: ted@math.upenn.edu
B.deS: Mathematisch Instituut, University of Leiden, P.O. Box 9512, 2300 RA Leiden, The Netherlands
E-mail address: desmit@math.leidenuniv.nl
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