Inverse Problems for deformation rings

Article (PDF Available)inTransactions of the American Mathematical Society 365(11) · December 2010with16 Reads
DOI: 10.1090/S0002-9947-2013-05848-5 · Source: arXiv
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, V )
relative to Wof finite dimensional representations Vof a profinite group Γ over k. We show that
for all pand n1, 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, 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, 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, V ) if EndkΓ(V) = k. The ring RW, 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, 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, V ) is isomorphic to R. We will be most interested in the case of
finite groups Γ in this paper, for which RW, 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, 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 n1, there is a representation
Vof a finite group Γover khaving a universal deformation ring RW, 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, 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), 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),˜
V) to be
isomorphic to R=W(k)[[t]]/(pnt, t2). The proof that these conditions are sufficient involves first
showing that RW(k),˜
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 AA/mA=kin ˆ
Cby πA. If α:AAis 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 πAτ=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:AAis a morphism in ˆ
C, then
ˆ
Hρ(α) : Defρ(A)Defρ(A) sends a deformation [τ] of ρover Ato the deformation [ατ] 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 Aˆ
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:kA
MVof (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:AAis a morphism in ˆ
C,
then ˆ
FV(α) : DefV(A)DefV(A) sends a deformation [M , φ] of Vover Ato the deformation
[AA,α M, φα] of Vover A, where φαis the composition kA(AA,α M)
=kAMφ
V. The
functors ˆ
FVand ˆ
Hρare naturally isomorphic.
One says that a ring R=RW, ρ) (resp. R=RW, 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:RAin 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, ρ) (resp. R=RW, 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, ρ) (resp. R=RW, 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 kbe a field extension of k. Suppose W
is a complete local commutative Noetherian ring with residue field kwhich 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, ρ)is the completion Rof Ω = WWRW, ρ)with respect to the unique maximal ideal
mof .
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:GGLd(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 n1be a fixed integer and define A= W/(Wpn). Let VAbe a
projective AG-module such that kAVAis isomorphic to Vas a kG-module. Let MAbe the free
A-module HomA(VA, VA), so that MAis a projective AG-module. Define
M=kAMA= 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,˜
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:KMAin 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 xKof 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:GGLd(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:GGLd(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 wWRgives 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 MAGLd(R)GLd(W) 1
where the homomorphism MAGLd(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:GGLd(A).
Lemma 3.4. Let ρW,ρAand Rbe as in Remark 3.3. Suppose there exist continuous group
homomorphisms ψ:KMAand 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 Ris a W-algebra in ˆ
Cwhich is a small extension of R, in the sense that there is an
exact sequence
(3.5) 0 JRν
R0
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 Rν
Rwith RW.
Let E= (1 + Matd(J)). There is a natural exact sequence of groups
(3.6) 1 EM
AMA1.
There is a continuous representation ρR: Γ GLd(R)which lifts ρRif and only if there is a
homomorphism ψ:KM
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,˜
M) degenerates to give
(3.7) H2,˜
M) = H0(G, H2(K, ˜
M)) = H2(K, ˜
M)G.
But this means that the restriction homomorphism
H2, E)H2(K, E )
is injective. Since the obstruction to the existence of a lift ρRof ρRis an element ωH2, E)
whose restriction to Kgives the obstruction to the existence of a lift ψof ψ, this completes the
proof of Lemma 3.4.
Remark 3.5.For later use, we now analyze small extensions Rof 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:RRis 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=b1apntin R= W[[t]]/I. Hence
(3.9) I= (pn+1t, t2+b1apnt)
since pt2=b1apn+1tIand t3=b1apnt2Wpt2I. Moreover,
(3.10) R= W[[t]]/I = W[[t]]/(pn+1t, t2+b1apnt) = W (Wt/Wpn+1t).
Now suppose bpW, so that amust be a unit. Then
(3.11) I= (pnt, pt2, t3)
since bt2Wpt2lies in I, so (apnt+bt2)bt2=apntIand 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,˜
M) = 1. The tangent space of the universal deformation ring
RW,˜
V)of ˜
Vhas dimension 1. The ring RW,˜
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,˜
M) degenerates to give
(3.13) H1,˜
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,˜
V), kH1,˜
M) = k
where t˜
Vis the tangent space of the deformation functor of ˜
Vand mis the maximal ideal of the
universal deformation ring RW,˜
V). This implies
dimkm
m2+pRW,˜
V)= 1
so there is a continuous surjection of W-algebras W[[t]] RW,˜
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:KMAis as in condition (ii)(c)of Theorem 3.2. Let γ:RW,˜
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,˜
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 ρπ= ˜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, V )
m2+pRW, V )R
m2
R+pR =R
pR
INVERSE PROBLEMS FOR DEFORMATION RINGS 7
where mis the maximal ideal of RW, 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 KΓ, 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, 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, V ) which maps tto r. Since (γµ)(t) is the class of tin R, we se that γµis surjective. So
because γis an isomorphism, Nakayama’s lemma implies that µ: W[[t]] RW, 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:KMAbe 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:RRis the natural
surjection. Let M
Abe the kernel of the homomorphism GLd(R)GLd(W) resulting from the
composition Rν
RW. By Lemmas 3.4 and 3.7, it is enough to show that there is no group
homomorphism ψ:KM
Awhich lifts ψ.
Suppose to the contrary that such a homomorphism ψexists. 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 gKbe arbitrary. Then there exist unique
α0(g), α1(g),...,αn1(g)Matd(S)
such that
(3.17) ψ(g) = α0(g) + p α1(g) + ···+pn1αn1(g).
Moreover, since ψlifts 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) + ···+pn1t αn1(g) + pnt β(g).
If apW 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) + ···+pn1t αn1(g) + pnt β(g)pn
= 1 + pnt α0(g)
when bis a unit and apW. 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 apW
in (3.8) cannot occur.
If both band aare units in (3.8), then by (3.9) we have pn+1t= 0 and t2=b1apntin 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) = (b1a)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) + ···+pn1t αn1(g) + t2β(g).
8 FRAUKE M. BLEHER, TED CHINBURG, AND BART DE SMIT
Because pt2= 0 = t3in Rin 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 xKof 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,˜
V) is isomorphic to R. Since the natural surjection RW 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 ρWπ. This implies
that the image of Kunder ρRlies inside (1 + tMatd(R)). Let ψ:KMAbe 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:GGLd(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,˜
M)
=ksince R=RW,˜
V). Because Mis a
projective kG-module by Hypothesis 3.1, we see as in (3.13) that H1,˜
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:KM
Awhere M
Alies in a short exact sequence
1(1 + Matd(J))M
AMA1.
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:KM
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 ψis 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 ψmay be defined by letting
(3.25) ψ(xj) = 1 + t ψ(xj)
for 1 jrand by extending ψadditively 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 in.
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 xKof
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:KM
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,n1and A= W/pnW = Z/pn. Let E=Fp2and
let G0= Gal(E/k). Define G=E×G0, where G0acts on Eby restricting the natural action of
G0on Eto E. The natural action of G0and Eon 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 Vsuch that
(4.26) M
=VkG0
as kG-modules. Let K=V
Abe a projective AG-module such that kAV
A
=Vas kG-modules.
Let Γbe the semidirect product K×δGwhere δ:GAut(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,˜
V)is isomorphic to W[[t]]/(pnt, t2).
Proof. Let VAbe a projective AG-module such that kAVA
=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 p3, 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, b1E. Then σ.x =σxσ1= (b0)p+ (b1)pσand ζ .x =ζ1=b0+b1ζ1pσ.
We have E[G0] = E+as k-vector spaces. The above G-action on E[G0] implies that both
Eand are kG-submodules of E[G0]. It follows for example from the normal basis theorem that
E
=kG0as kG-modules, where EGacts trivially by conjugation on E. Thus to prove (4.26) it
suffices to show that V=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 ζ1p. Since ζ1plies in Fp2Fp, 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:GAut(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,˜
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 mn, 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, V0) is isomorphic to Zp[[t]]/(pnt, t2). Let V=kFp
V0. Then EndkG (V)
=kFpEndFpG(V0)
=k. By Theorem 2.2, the universal deformation ring
RW, 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 Iis the ideal of Sgenerated by I,pntand t2. Since dim W[[t]]/(pnt, t2) = dim W, we obtain
by [13, Thm. 21.1] that
(4.27) dimk(I/mSI) = dim Sdim (S/I) = dim S+ 1 dim (S/I )dimk(I /mSI) + 1.
Using power series expansions, we see that dimk(I/mSI) = 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 ξVξVof
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, ρ). Recall that
Ω = WWRand Ris the completion of Ω with respect to its unique maximal ideal m. Define
ˆ
Cto 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 ρover
Rdefined by ν(g) = (1 ν(g)i,j )1i,jdfor all gΓ.
The first step is to show that if AOb(C) is an Artinian W-algebra with residue field kand
τ: Γ GLd(A) is a lift of ρover A, then there is a morphism α:RAin ˆ
Csuch that
[τ] = [αν]. Since Ais 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 SΓ such that τ(S) is a set of topological generators for
the image of τ. Since ρand ρhave the same image in GLd(k)GLd(k), there exists for each gS
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 TmAbe 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]] Awith m= #Tand {f(xi)}m
i=1 =T.
Since Ahas 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, ρ) of ρ, there is a morphism β:RBin ˆ
Csuch that τ: Γ GLd(B) is conjugate
to βνby a matrix in the kernel of πB: GLd(B)GLd(B/mB) = GLd(k). Let β:RA
be the composition of βwith the inclusion BA. Define α:RAto be the morphism in
ˆ
Ccorresponding to the continuous W-algebra homomorphism Ω = WWRAwhich sends
wrto w·β(r) for all w∈ Wand rR. It follows that αsatisfies [τ] = [αν].
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 ρover 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) = m
m2
+ 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) = kkAd(ρ), we have from [15, Prop. 21.1] that there are natural
isomorphisms
Defρ(k[ǫ]) = H1,Ad(ρ)) = kkH1,Ad(ρ)) = kkDefρ(k[ǫ]).
Hence it suffices to show that the natural homomorphism µ:kkT(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/(). One then divides Wand further by ideals generated by generators for mWto be able
to assume that W=k. However, the case when W=kand W=kis obvious, since then
T(k,Ω) = m/m2
=kkmR/m2
R=kT(k, R). This completes the proof of Theorem 2.2.
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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
    • "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.
    Full-text · Article · Aug 2016
    • "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.
    Full-text · Article · May 2016
    • "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 $\mathbf{k}[[t]]/(t^2)$.
    Article · Apr 2016
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