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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.

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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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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)