# Inverse Problems for deformation rings

**Abstract**

Let $\mathcal{W}$ 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 $R_{\mathcal{W}}(\Gamma,V)$ relative to
$\mathcal{W}$ of finite dimensional representations $V$ of a profinite group
$\Gamma$ over $k$. We show that for all $p$ and $n \ge 1$, the ring
$\mathcal{W}[[t]]/(p^n t,t^2)$ arises as a universal deformation ring. This
ring is not a complete intersection if $p^n\mathcal{W}\neq\{0\}$, so we obtain
an answer to a question of M. Flach in all characteristics. We also study the
`inverse inverse problem' for the ring $\mathcal{W}[[t]]/(p^n t,t^2)$; this is
to determine all pairs $(\Gamma, V)$ such that $R_{\mathcal{W}}(\Gamma,V)$ is
isomorphic to this ring.

arXiv:1012.1290v1 [math.NT] 6 Dec 2010

INVERSE PROBLEMS FOR DEFORMATION RINGS

FRAUKE M. BLEHER, TED CHINBURG, AND BART DE SMIT

Abstract. Let Wbe a complete local commutative Noetherian ring with residue ﬁeld kof

positive characteristic p. We study the inverse problem for the versal deformation rings RW(Γ, V )

relative to Wof ﬁnite dimensional representations Vof a proﬁnite group Γ over k. We show that

for all pand n≥1, the ring W[[t]]/(pnt, t2) arises as a versal deformation ring. This ring is not

a complete intersection if pnW 6={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 Vis a continuous ﬁnite dimensional representation of Γ

over a ﬁeld kof characteristic p > 0. Let Wbe a complete local commutative Noetherian ring with

residue ﬁeld k. In §2 we recall the deﬁnition of a deformation of Vover a complete local commutative

Noetherian W-algebra with residue ﬁeld k. It follows from work of Mazur and Schlessinger [14, 17]

that Vhas 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 Rwith residue ﬁeld k

are isomorphic to RW(Γ, V )for some Γand Vas above?

It is important to emphasize that in this question, Γ and Vare not ﬁxed. Thus for a given R, one

would like to construct both a proﬁnite group Γ and a continuous ﬁnite dimensional representation

Vof Γ over kfor 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 Ris a complete local commutative Noetherian W-algebra with residue ﬁeld

k. What are all proﬁnite groups Γand all continuous ﬁnite dimensional representations Vof Γ

over ksuch 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 kand rings Was above, and for all n≥1, there is a representation

Vof a ﬁnite group Γover khaving 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 6={0}.

Theorem 1.4. Let kbe 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 Γand all

continuous ﬁnite dimensional representations Vof Γover kwith EndkΓ(V) = ksuch 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

2 FRAUKE M. BLEHER, TED CHINBURG, AND BART DE SMIT

•if Kis the kernel of the Γ-action on Vthen Vis 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 Vis 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 Rwith

perfect residue ﬁeld kof 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 Vof 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 kof characteristic pand 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 ˜

Vof a ﬁnite group Γ over kfor 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 ˜

Vis one. We then construct an explicit lift of ˜

Vover

Rand show that this cannot be lifted further to any small extension ring of Rwhich 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=Fp2and 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 kbe a ﬁeld of characteristic p > 0. Let Wbe a complete local

commutative Noetherian ring with residue ﬁeld k. We denote by ˆ

Cthe category of all complete

local commutative Noetherian W-algebras with residue ﬁeld k. Homomorphisms in ˆ

Care continuous

W-algebra homomorphisms which induce the identity map on k. Deﬁne Cto be the full subcategory

of Artinian objects in ˆ

C. For each ring Ain ˆ

C, let mAbe its maximal ideal and denote the surjective

morphism A→A/mA=kin ˆ

Cby πA. If α:A→A′is a morphism in ˆ

C, we denote the induced

morphism GLd(A)→GLd(A′) also by α.

Let dbe a positive integer, and let ρ: Γ →GLd(k) be a continuous homomorphism, where GLd(k)

has the discrete topology. By a lift of ρover a ring Ain ˆ

Cwe mean a continuous homomorphism

τ: Γ →GLd(A) such that πA◦τ=ρ. We say two lifts τ , τ′: Γ →GLd(A) of ρover Aare

strictly equivalent if one can be brought into the other by conjugation by a matrix in the kernel of

INVERSE PROBLEMS FOR DEFORMATION RINGS 3

πA: GLd(A)→GLd(k). We call a strict equivalence class of lifts of ρover Aa deformation of ρ

over Aand deﬁne Defρ(A) to be the set of deformations [τ] of lifts τof ρover A. We then have a

functor

ˆ

Hρ:ˆ

C → Sets

which sends a ring Ain ˆ

Cto the set Defρ(A). Moreover, if α:A→A′is a morphism in ˆ

C, then

ˆ

Hρ(α) : Defρ(A)→Defρ(A′) sends a deformation [τ] of ρover Ato 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 ρwith the natural action of GLd(k) on V, i.e. Vis the d-dimensional topological kΓ-module

corresponding to ρ. A lift of Vover a ring A∈ˆ

Cis then a pair (M, φ) consisting of a ﬁnitely

generated free A-module Mon which Γ acts continuously together with a Γ-isomorphism φ:k⊗A

M→Vof (discrete) k-vector spaces. We deﬁne Def V(A) to be the set of isomorphism classes

[M, φ] of lifts (M , φ) of Vover A. We then have a functor

ˆ

FV:ˆ

C → Sets

which sends a ring Ain ˆ

Cto the set DefV(A). Moreover, if α:A→A′is a morphism in ˆ

C,

then ˆ

FV(α) : DefV(A)→DefV(A′) sends a deformation [M , φ] of Vover Ato the deformation

[A′⊗A,α M, φα] of Vover A′, where φαis the composition k⊗A′(A′⊗A,α M)∼

=k⊗AMφ

−→ V. The

functors ˆ

FVand ˆ

Hρare naturally isomorphic.

One says that a ring R=RW(Γ, ρ) (resp. R=RW(Γ, V )) in ˆ

Cis a versal deformation ring for

ρ(resp. for V) if there is a lift ν: Γ →GLd(R) of ρover R(resp. a lift (U, φU) of Vover R) such

that the following conditions hold. For all rings Ain ˆ

C, the map

fA: Hom ˆ

C(R, A)→Defρ(A) (resp. fA: Hom ˆ

C(R, A)→DefV(A))

which sends a morphism α:R→Ain ˆ

Cto the deformation ˆ

Hρ(α)([ν]) (resp. ˆ

FV(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 Wannihilates 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], ˆ

Hρ(resp. ˆ

FV) is continuous, which means that we only need to check the surjectivity of fA

for Artinian rings Ain 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 Ain ˆ

C, 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 Rrepresents the deformation functor ˆ

Hρ

(resp. ˆ

FV) in the sense that ˆ

Hρ(resp. ˆ

FV) is na turally isomor phic to the Hom functor Hom ˆ

C(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 Jof ﬁnite index in Γ, there are only a ﬁnite number of

continuous homomorphisms from Jto Z/p.

It follows by [14, §1.2] that for Γ satisfying Hypothesis 2.1, all ﬁnite dimensional continuous

representations Vof Γ over khave a versal deformation ring. It is shown in [11, Prop. 7.1] that if

EndkΓ(V) = k, then Vhas 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,Wand ρbe 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

aW-algebra, in the sense that we ﬁx a local homomorphism W → W ′. Let ρ′: Γ →GLd(k′)

be the composition of ρwith the injection GLd(k)֒→GLd(k′). Then the versal deformation ring

4 FRAUKE 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 kbe an arbitrary perfect ﬁeld of characteristic p > 0and let Wbe the ring

W(k)of inﬁnite Witt vectors over k. Let Γbe a proﬁnite group satisfying Hypothesis 2.1. Let

dbe 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 Gis a ﬁnite group.

Let π: Γ →Gbe the natural surjection. Let ρ:G→GLd(k)be the representation whose inﬂation

to Γis ˜ρ, and denote the kG-module corresponding to ρby V. Suppose Vis a projective kG-module

and that EndkG(V) = k. Let n≥1be a ﬁxed integer and deﬁne A= W/(Wpn). Let VAbe a

projective AG-module such that k⊗AVAis isomorphic to Vas 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 Lis an AG-module, we will also view Las 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 ˜

Vas a representation of Γis faithful.

(ii) The following conditions hold:

(a) The group Kis a ﬁnitely generated (Z/pn)G-module.

(b) Writing Kadditively, 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 ∈Kwith ψ(g)◦ψ(h)6≡ ψ(h)◦ψ(g) mod pMA, or

•p= 2 and there exists x∈Kof order 2with ψ(x)◦ψ(x)6≡ 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 ˜

Lthe Γ-module which results by

inﬂating Lvia the natural surjection π: Γ →G.

Remark 3.3.Since VAis a projective AG-module which is a lift of Vover 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 →wmod pnW for w∈W⊂Rgives rise to isomorphisms of groups

(3.2) (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

(3.3) 1 →MA→GLd(R)→GLd(W) →1

where the homomorphism MA→GLd(R) results from (3.1) and (3.2).

INVERSE PROBLEMS FOR DEFORMATION RINGS 5

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). This action coincides

with the action of Gon MA= HomA(VA, VA) in (3.2) coming from the action of Gon VAvia

ρA:G→GLd(A).

Lemma 3.4. Let ρW,ρAand Rbe as in Remark 3.3. Suppose there exist continuous group

homomorphisms ψ:K→MAand ρR: Γ →GLd(R)such that there is a commutative diagram

(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 in ˆ

Cwhich is a small extension of R, in the sense that there is an

exact sequence

(3.5) 0 →J→R′ν

−→ R→0

in which νis a continuous W-algebra homomorphism and dimk(J) = 1. Deﬁne M′

Ato be the kernel

of the homomorphism GLd(R′)→GLd(W) resulting from the composition of R′ν

−→ Rwith R→W.

Let E= (1 + Matd(J))∗. There is a natural exact sequence of groups

(3.6) 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.

Proof. The natural short exact sequence (3.6) results from the observation that M′

Aconsists of all

elements in GLd(R′) whose image in GLd(R) under νlies 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 Jhas k-dimension 1. In particular, Kacts trivially on E.

Since Mis a projective kG-module, we have Hi(G, H0(K, ˜

M)) = Hi(G, M ) = 0 if i > 0.

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. Hence Hi(G, H 1(K, ˜

M)) =

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

(3.7) H2(Γ,˜

M) = H0(G, H2(K, ˜

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 Kgives 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 Iis an ideal of W[[t]] that is contained in the ideal (pnt, t2)

such that the natural surjection ν:R′→Ris a small extension as in (3.5). Since J= (pnt, t2)/I

is isomorphic to k, it follows that Icontains 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 kwith a basis given

by the classes of pntand t2. Since dimk((pnt, t2)/I) = 1 and (pn+1t, pt2, t3)⊂I, there exist a, b ∈W

such that

(3.8) I= (pn+1t, pt2, t3, apnt+bt2)

6 FRAUKE M. BLEHER, TED CHINBURG, AND BART DE SMIT

and at least one of aor bis a unit.

Suppose ﬁrst that bis a unit. Then t2=−b−1apntin R′= W[[t]]/I. Hence

(3.9) I= (pn+1t, t2+b−1apnt)

since pt2=−b−1apn+1t∈Iand t3=−b−1apnt2∈Wpt2⊂I. Moreover,

(3.10) R′= W[[t]]/I = W[[t]]/(pn+1t, t2+b−1apnt) = W ⊕(Wt/Wpn+1t).

Now suppose b∈pW, so that amust be a unit. Then

(3.11) I= (pnt, pt2, t3)

since bt2∈Wpt2lies in I, so (apnt+bt2)−bt2=apnt∈Iand ais a unit in W. Moreover,

(3.12) R′= W[[t]]/I = W[[t]]/(pnt, pt2, t3) = W ⊕(Wt/Wpnt)⊕(Wt2/Wpt2).

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 Lis a G-module we denote by ˜

Lthe Γ-module

which results by inﬂating Lvia the natural surjection π: Γ →G.

Lemma 3.6. One has dimkH1(Γ,˜

M) = 1. The tangent space of the universal deformation ring

RW(Γ,˜

V)of ˜

Vhas dimension 1. The ring RW(Γ,˜

V)is a quotient of W[[t]].

Proof. Since Mis a pro jective kG-module, we have Hi(G, H0(K, ˜

M)) = Hi(G, M ) = 0 if i > 0.

Therefore the Hochschild-Serre spectral sequence for H1(Γ,˜

M) degenerates to give

(3.13) H1(Γ,˜

M) = H0(G, H1(K, ˜

M)) = H0(G, Hom(K, M )) = Hom(K, M )G.

Writing Kadditively and using that Mhas exponent p, we have from condition (ii)(b) of Theorem

3.2 that

(3.14) Hom(K, M )G= Hom(K/pK, M )G= Hom(Z/p)G(K/pK, M )∼

=k.

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=def Homkm

m2+pRW(Γ,˜

V), k→H1(Γ,˜

M) = k

where t˜

Vis the tangent space of the deformation functor of ˜

Vand mis the maximal ideal of the

universal deformation ring RW(Γ,˜

V). This implies

dimkm

m2+pRW(Γ,˜

V)= 1

so there is a continuous surjection of W-algebras W[[t]] →RW(Γ,˜

V).

Lemma 3.7. Let ρW,ρAand Rbe as in Remark 3.3. There exists a lift ρR: Γ →GLd(R)of the

representation ˜ρ: Γ →GLd(k)for ˜

Vsuch 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)→Rbe the unique

continuous W-algebra homomorphism corresponding to the isomorphism class of the lift ρR. Then

γis surjective. There is a W-algebra surjection µ: W[[t]] →RW(Γ,˜

V)whose composition with γis

the natural surjection W[[t]] →R= W[[t]]/(pnt, t2). The kernel of µis an ideal of W[[t]] contained

in (pnt, t2).

Proof. The obstruction to the existence of ρRis an element of H2(G, MA). This group is trivial

since MAis projective, so ρRexists. Since ρWis a lift of the matrix representation ρ:G→

GLd(k) = Autk(V) over W, we ﬁnd that ρRis a lift of ρ◦π= ˜ρover R.

The ring k[ǫ] of dual numbers over kis isomorphic to R/pR =k[[t]]/(t2), and γis surjective if

and only if it induces a surjection

(3.16) γ:RW(Γ, V )

m2+pRW(Γ, V )−→ R

m2

R+pR =R

pR

INVERSE PROBLEMS FOR DEFORMATION RINGS 7

where mis the maximal ideal of RW(Γ, V ). If γis not surjective, its image is k. Thus to prove that

γis surjective, it will suﬃce to show that the composition ρR/pR of ρRwith the natural surjection

GLd(R)→GLd(R/pR) = GLd(k[ǫ]) is not a matrix representation of the trivial lift of ˜

Vover k[ǫ].

However, the kernel of the action of Γ on this trivial lift is K⊂Γ, while ρR/pR is not trivial on K

because of condition (ii)(c) of Theorem 3.2. Hence γmust be surjective.

The tangent space of the deformation functor of Vis one dimensional by Lemma 3.6, so (3.16)

is in fact an isomorphism. Let rbe any element of RW(Γ, V ) such that γ(r) is the class of tin

R= W[[t]]/(pnt, t2). We then have a unique continuous W-algebra homomorphism µ: W[[t]] →

RW(Γ, V ) which maps tto r. Since (γ◦µ)(t) is the class of tin R, we se that γ◦µis surjective. So

because γis 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 ˜ρfrom Lemma 3.7. Let ψ:K→MAbe the injective (Z/pn)G-

module homomorphism from condition (ii)(c) of Theorem 3.2. Since ρRlies in a commutative

diagram of the form (3.4) and ψand ρWare both injective, it follows that ρRis faithful.

Let R′= W[[t]]/I be a small extension of Ras in Remark 3.5, so that ν:R′→Ris the natural

surjection. Let M′

Abe the kernel of the homomorphism GLd(R′)→GLd(W) resulting from the

composition R′ν

−→ R→W. By Lemmas 3.4 and 3.7, it is enough to show that there is no group

homomorphism ψ′:K→M′

Awhich lifts ψ.

Suppose to the contrary that such a homomorphism ψ′exists. Write Kadditively and M′

A

multiplicatively. Deﬁne Sto be the union of {0}with the set of Teichm¨uller lifts in W = W(k) of

the elements of k∗. Let g∈Kbe arbitrary. Then there exist unique

α0(g), α1(g),...,αn−1(g)∈Matd(S)

such that

(3.17) ψ(g) = α0(g) + p α1(g) + ···+pn−1αn−1(g).

Moreover, since ψ′lifts ψ, we have

ψ′(g)≡1 + t ψ(t) mod (pnt, t2) Matd(R′).

By Remark 3.5, there exist a, b ∈W such that Iis as in (3.8) and such that one of the alternatives

(3.9) or (3.11) holds. Suppose ﬁrst that bis 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 + t α0(g) + pt α1(g) + ···+pn−1t αn−1(g) + pnt β(g).

If a∈pW in (3.9), it follows that t2= 0 = pn+1 tin R′. Therefore, since (pn)g= 0Kbecause of

condition (ii)(a) of Theorem 3.2, we have

(3.19) 1 = ψ′(g)pn=1 + t α0(g) + pt α1(g) + ···+pn−1t αn−1(g) + pnt β(g)pn

= 1 + pnt α0(g)

when bis a unit and a∈pW. Thus pnt α0(g) = 0. Since alternative (3.9) holds, this means that

α0(g) = 0, which implies by (3.17) that ψ(g)∈pMA. Since gwas an arbitrary element of K, this

is a contradiction to condition (ii)(c) of Theorem 3.2. Hence the case when bis a unit and a∈pW

in (3.8) cannot occur.

If both band aare units in (3.8), then by (3.9) we have pn+1t= 0 and t2=−b−1apntin R′.

Suppose his another element of K. Because pt2= 0 in R′, it follows from (3.18) that

(3.20) ψ′(g)·ψ′(h)−ψ′(h)·ψ′(g) = −(b−1a)pnt[α0(g)·α0(h)−α0(h)·α0(g)] .

If bis not a unit in (3.8), then ahas 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 + t α0(g) + pt α1(g) + ···+pn−1t αn−1(g) + t2β(g).

8 FRAUKE 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 Kis 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 ∈Kwe 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∈Kof order 2 such that ψ(x)◦ψ(x)6≡ 0 mod 2MA. Replacing g=xin (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,ρAand 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 in ˆ

Cfrom Rto W, there exists a universal lift ρR: Γ →GLd(R) of

˜ρ: Γ →GLd(k) over Rsuch that ρRfollowed by GLd(R)→GLd(W) is equal to ρW◦π. This implies

that the image of Kunder ρRlies inside (1 + tMatd(R))∗. Let ψ:K→MAbe the restriction of

ρRto Kfollowed 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, ψis an injective group homomorphism. In particular, Kis 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 Gon

MA= HomA(VA, VA) in (3.2) coming from the action of Gon VAvia ρA:G→GLd(A), it follows

that ψis an injective homomorphism in Hom(Z/pn)G(K, MA). Let ρR/pR be the composition of ρR

with the natural surjection GLd(R)→GLd(R/pR) = GLd(k[ǫ]). If the image of ψis contained in

pMA, it follows that ρR/pR factors through G. Since Vis a projective kG-module, this implies that

ρR/pR is a matrix representation of the trivial lift of ˜

Vover k[ǫ]. Since R/pR ∼

=k[ǫ] is the universal

deformation ring associated to mod plifts of ˜ρ, this is a contradiction. Hence the image of ψis not

contained in pMA, giving condition (ii)(c) of Theorem 3.2.

Writing Kadditively, 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 Kis 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,˜

M)∼

=ksince R=RW(Γ,˜

V). Because Mis a

projective kG-module by Hypothesis 3.1, we see as in (3.13) that H1(Γ,˜

M) = Hom(K, M )G. Since

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

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 ψcan 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 Kadditively and M′

Amultiplicatively. Moreover using (3.2), we identify

MAwith (1 + tMatd(R))∗.

INVERSE PROBLEMS FOR DEFORMATION RINGS 9

If p6= 2, deﬁne ψ′:K→M′

Ato be the exponential function of (t ψ(g)) mod I. In other words,

ψ′(g) = 1 + t ψ(g) + t2

2[ψ(g)◦ψ(g)].

Since we assume that condition (ii)(d) fails, i.e. the image of ψis commutative mod pMAwith

respect to map composition, it follows that ψ′is a group homomorphism which lifts ψ.

If p= 2, we use that Kis a ﬁnitely generated (Z/2n)-module. Let x1,...,xrbe a minimal set

of generators of K. We will show that ψ′may be deﬁned by letting

(3.25) ψ′(xj) = 1 + t ψ(xj)

for 1 ≤j≤rand by extending ψ′additively to all of K. Since ψis a group homomorphism and

2t2= 0 in R′, we have

ψ′(xj)2= 1 + t ψ(2 xj) + t2[ψ(xj)◦ψ(xj)] and

ψ′(xj)2i= 1 + t ψ((2i)xj) for 2 ≤i≤n.

For p= 2, the failing of condition (ii)(d) means that not only the image of ψis commutative mod

2MAwith respect to map composition, but also that ψ(x)◦ψ(x)≡0 mod 2MAfor all x∈Kof

order 2. Hence it follows that if xjhas order 2 then ψ′(xj)2= 1. Therefore, we can extend (3.25)

additively to obtain a group homomorphism ψ′:K→M′

Awhich lifts ψ.

This completes the proof of (i) implies (ii) in Theorem 3.2.

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≥1and A= W/pnW = Z/pn. Let E=Fp2and

let G0= Gal(E/k). Deﬁne G=E∗×G0, where G0acts on E∗by restricting the natural action of

G0on Eto E∗. The natural action of G0and E∗on V=Emakes Vinto 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′

Abe a projective AG-module such that k⊗AV′

A∼

=V′as kG-modules.

Let Γbe the semidirect product K×δGwhere δ:G→Aut(K)is the group homomorphism given

by the G-action on the (Z/pn)G-module K=V′

A. If ˜

Vis the inﬂation of Vto a kΓ-module, then

the universal deformation ring RW(Γ,˜

V)is isomorphic to W[[t]]/(pnt, t2).

Proof. Let VAbe a projective AG-module such that k⊗AVA∼

=Vas kG-modules. Let MA=

EndA(VA). We prove that G,K,Mand MAsatisfy the conditions in Theorem 3.2(ii).

If p= 2, then Gis isomorphic to the symmetric group S3on 3 letters and Vis the unique simple

projective kG-module, up to isomorphism. If p≥3, then the order of Gis relatively prime to pand

Vis also a simple pro jective kG-module.

Since V=Eis a Galois algebra over kwith 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=hσi, let E∗=hζiand let x=b0+b1σ∈

E[G0], so b0, b1∈E. Then σ.x =σxσ−1= (b0)p+ (b1)pσand 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

Eand Eσ are kG-submodules of E[G0]. It follows for example from the normal basis theorem that

E∼

=kG0as kG-modules, where E∗⊂Gacts trivially by conjugation on E. Thus to prove (4.26) it

suﬃces to show that V′=Eσ is a simple projective kG-module. Since Vis a projective kG-module,

so are M,E[G0] and V′. Considering the action of E∗=hζion 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.

10 FRAUKE M. BLEHER, TED CHINBURG, AND BART DE SMIT

For all p, let K=V′

A, so Kis a ﬁnitely generated (Z/pn)G-module, giving condition (a) of

Theorem 3.2(ii). Deﬁne Γ = K×δGwhere δ:G→Aut(K) is the group homomorphism given by

the G-action on the (Z/pn)G-module K=V′

A. Since by our above calculations, M∼

=(K/pK)⊕kG0

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′

Aand MAare projective AG-modules, it follows

that HomAG(K, MA) is a projective A-module Hsuch that H/pH = HomkG (K/pK, M )∼

=k.

Therefore, HomAG(K, MA)∼

=Aand 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 ψmod pMAis isomorphic to Eσ. Since for example (σ)(ζσ) = ζp6=ζ= (ζ σ)(σ), we

obtain that the image of ψmod 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).

Remark 4.2.If p > 3, we can replace the group Gin Theorem 4.1 by the symmetric group S3and

Vby the 2-dimensional simple projective kS3-module. It follows then that M= Homk(V, V )∼

=

k[Z/2] ⊕Vas kG-modules, which means that we can take V′=Vand 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,Wand nbe as in Theorem 1.3. By Theorem

4.1, there is a ﬁnite group Γ and a representation V0of Γ over Fpsuch 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 6={0}. This is clear if Wis regular. In general, if one assumes that

W[[t]]/(pnt, t2) is a complete intersection, then Wis a quotient S/I for some regular complete local

commutative Noetherian ring Sand a proper ideal Iof S. If S′=S[[t]], then W[[t]]/(pnt, t2) = S′/I′

when I′is the ideal of S′generated by I,pntand t2. Since dim W[[t]]/(pnt, t2) = dim W, we obtain

by [13, Thm. 21.1] that

(4.27) dimk(I′/mS′I′) = dim S′−dim (S′/I′) = dim S+ 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 6={0}. Since

this contradicts (4.27), W[[t]]/(pnt, t2) is not a complete intersection if pnW 6={0}. This completes

the proof of Theorem 1.3.

Remark 4.4.To construct more examples to which Theorem 3.2 applies, there are two fundamental

issues. One must construct a group Gand a projective kG-module Vfor 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 Khaving 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 Mis to note that the Brauer character ξMof Mis the tensor product ξV⊗ξV∗of

the Brauer characters of Vand its k-dual V∗. For example, if Vis induced from a representation

Xof a subgroup Hof G, then ξVis given by the usual formula for the character of an induced

representation. If dimk(X) = 1, the analysis of the ring structure of Mbecomes a combinatorial

problem using Xand coset representatives of Hin G.

INVERSE PROBLEMS FOR DEFORMATION RINGS 11

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′⊗WRand R′is the completion of Ω with respect to its unique maximal ideal mΩ. Deﬁne

ˆ

C′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 ρover 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′in ˆ

C′such that

[τ′] = [α◦ν′]. Since A′is Artinian, Hom ˆ

C′(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 ρhave 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 granges over S. Then

there is a continuous homomorphism f:W[[x1,...,xm]] →A′with m= #Tand {f(xi)}m

i=1 =T.

Since A′has the discrete topology, the image Bof fmust 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 ρover B. Because ν: Γ →GLd(R) is a versal lift of ρover the versal deformation

ring R=RW(Γ, ρ) of ρ, there is a morphism β:R→Bin ˆ

Csuch that τ′: Γ →GLd(B) is conjugate

to β◦νby a matrix in the kernel of πB: GLd(B)→GLd(B/mB) = GLd(k). Let β′:R→A′

be the composition of βwith the inclusion B⊂A′. Deﬁne α:R′→A′to be the morphism in

ˆ

C′corresponding to the continuous W′-algebra homomorphism Ω = W′⊗WR→A′which sends

w′⊗rto w′·β′(r) for all w′∈ W′and r∈R. It follows that αsatisﬁes [τ′] = [α◦ν′].

The second step is to show that when k′[ǫ] is the ring of dual numbers over k′, then Hom ˆ

C′(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 Def ρ′(k′[ǫ]). Let

(5.28) T(W′,Ω) = mΩ

m2

Ω+ Ω ·mW′

and T(W, R) = mR

m2

R+R·mW

so that we have natural isomorphisms Homcont(Ω, k ′[ǫ]) ∼

=Homk′(T(W′,Ω), k′) and Hom ˆ

C(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 mWis ﬁnitely generated, one can reduce to the case when

W=k, by considering generators αof mWand successively replacing Wby W/(Wα) and Rby

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=kand W′=k′is obvious, since then

T(k′,Ω) = mΩ/m2

Ω∼

=k′⊗kmR/m2

R=k′⊗T(k, R). This completes the proof of Theorem 2.2.

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B. de Smit thesis advisor, June 2010. http://www.math.leidenuniv.nl/en/theses/205/

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[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

- CitationsCitations11
- ReferencesReferences16

- "Traditionally, universal deformation rings are studied when Λ is equal to a group algebra kG, where G is a finite group and k has positive characteristic p (see e.g., [10, 12, 13, 14, 15, 16, 17, 18, 19] and their references). This approach has led to the solution of various open problems, e.g., the construction of representations whose universal deformation rings are not local complete intersections (see [10, 14, 15]). On the other hand, in [20, 22, 43] , universal deformation rings for certain selfinjective algebras, which are not Morita equivalent to a block of a group algebra, were discussed. "

[Show abstract] [Hide abstract]**ABSTRACT:**Let $\Lambda$ be a finite-dimensional algebra over a fixed algebraically closed field $\mathbf{k}$ of arbitrary characteristic, and let $V$ be a finitely generated $\Lambda$-module. It follows from results obtained by F.M. Bleher and the second author that $V$ has a well-defined versal deformation ring $R(\Lambda, V)$, which is a complete local commutative Noetherian $\mathbf{k}$-algebra with residue field $\mathbf{k}$. The second author also proved that if $\Lambda$ is a Gorenstein $\mathbf{k}$-algebra and $V$ is a Cohen-Macaulay $\Lambda$-module whose stable endomorphism ring is isomorphic to $\mathbf{k}$, then $R(\Lambda, V)$ is universal. In this article we prove that the isomorphism class of a versal deformation ring is preserved under singular equivalence of Morita type between Gorenstein $\mathbf{k}$-algebras. These singular equivalences of Morita type were introduced by X. W. Chen and L. G. Sun in an unpublished manuscript and then discussed by G. Zhou and A. Zimmermann in an article entitled "On singular equivalences of Morita type", which was published in J. Algebra during 2013.- "3.1.2]). This approach has recently led to the solution of various open problems, e.g., the construction of representations whose universal deformation rings are not local complete intersections (see [3, 6, 7]). On the other hand, in [10, 11, 21], universal deformation rings for certain self-injective algebras, which are not Morita equivalent to a block of a group algebra, were discussed . "

[Show abstract] [Hide abstract]**ABSTRACT:**Let $\mathbf{k}$ be an algebraically closed field, let $\Lambda$ be a finite dimensional $\mathbf{k}$-algebra and let $V$ be a $\Lambda$-module with stable endomorphism ring isomorphic to $\mathbf{k}$. If $\Lambda$ is self-injective then $V$ has a universal deformation ring $R(\Lambda,V)$, which is a complete local commutative Noetherian $\mathbf{k}$-algebra with residue field $\mathbf{k}$. Moreover, if $\Lambda$ is also a Frobenius $\mathbf{k}$-algebra then $R(\Lambda,V)$ is stable under syzygies. We use these facts to determine the universal deformation rings of string $\Lambda_N$-modules with stable endomorphism ring isomorphic to $\mathbf{k}$, where $N\geq 1$ and $\Lambda_N$ is a self-injective special biserial $\mathbf{k}$-algebra whose Hochschild cohomology ring is a finitely generated $\mathbf{k}$-algebra as proved by N. Snashall and R. Taillefer.- "Traditionally, universal deformation rings are studied when Λ is equal to a group algebra kG, where G is a finite group and k has positive characteristic p (see e.g., [9, 11, 12, 13, 14, 15, 16, 17, 18] and their references). This approach has led to the solution of various open problems, e.g., the construction of representations whose universal deformation rings are not local complete intersections (see [9, 13, 14]). On the other hand, in [19, 20, 33], universal deformation rings for certain self-injective algebras, which are not Morita equivalent to a block of a group algebra, were discussed. "

[Show abstract] [Hide abstract]**ABSTRACT:**Let $\mathbf{k}$ be an algebraically closed field, and let $\Lambda$ be a finite dimensional $\mathbf{k}$-algebra. We prove that if $\Lambda$ is a Gorenstein algebra, then every finitely generated Cohen-Macaulay $\Lambda$-module $V$ whose stable endomorphism ring is isomorphic to $\mathbf{k}$ has a universal deformation ring $R(\Lambda,V)$, which is a complete local commutative Noetherian $\mathbf{k}$-algebra with residue field $\mathbf{k}$, and which is also stable under taking syzygies. We investigate a particular non-self-injective Gorenstein algebra $\Lambda_0$, which is of infinite global dimension and which has exactly three isomorphism classes of finitely generated indecomposable Cohen-Macaulay $\Lambda_0$-modules $V$ whose stable endomorphism ring is isomorphic to $\mathbf{k}$. We prove that in this situation, $R(\Lambda_0,V)$ is isomorphic either to $\mathbf{k}$ or to $\mathbf{k}[[t]]/(t^2)$.

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