GHOSTS IN MODULAR REPRESENTATION THEORY
SUNIL K. CHEBOLU, J. DANIEL CHRISTENSEN, AND J'AN MIN'A~C
Abstract.A ghost over a finite group G is a map between modular represen-
tations of G which is invisible in Tate cohomology. Motivated by the fai*
*lure of
the generating hypothesis_the statement that ghosts between finite-dimen*
*sional
G-representations factor through a projective_we define the compact ghos*
*t num-
ber of kG to be the smallest integer l such that the composition of any *
*l ghosts
between finite-dimensional G-representations factors through a projectiv*
*e. In this
paper we study ghosts and the compact ghost numbers of p-groups. We begi*
*n by
showing that a weaker version of the generating hypothesis, where the ta*
*rget of
the ghost is fixed to be the trivial representation k, holds for all p-g*
*roups. We do
this by proving that a map between finite-dimensional G-representations *
*is a ghost
if and only if it is a dual ghost. We then compute the compact ghost num*
*bers of
all cyclic p-groups and all abelian 2-groups with C2 as a summand. We ob*
*tain
bounds on the compact ghost numbers for abelian p-groups and for all 2-g*
*roups
which have a cyclic subgroup of index 2. Using these bounds we determine*
* the
finite abelian groups which have compact ghost number at most 2. Our met*
*hods
involve techniques from group theory, representation theory, triangulate*
*d category
theory, and constructions motivated from homotopy theory.
Contents
1. Introduction 2
2. The generating hypothesis 4
2.1. The generating hypothesis with target k 4
2.2. Ghosts and duality 6
3. Universal ghosts 8
3.1. Periodic modules 11
4. The ghost projective class 11
4.1. Projective classes 11
4.2. The ghost projective class 12
5. Computing compact ghost numbers 14
5.1. Cyclic p-groups 15
5.2. The Klein four group 16
____________
Date: March 30, 2007.
2000 Mathematics Subject Classification. Primary 20C20, 20J06; Secondary 55P4*
*2.
Key words and phrases. Stable module category, generating hypothesis, ghost m*
*ap, projective
class, Jennings' theorem, nilpotency index.
2 SUNIL K. CHEBOLU, J. DANIEL CHRISTENSEN, AND J'AN MIN'A~C
5.3. The quaternion group 17
5.4. Compact ghost numbers of abelian groups 18
6. Nilpotency index 19
References 22
1.Introduction
Let G be a finite p-group and let k be a field of characteristic p. Recall th*
*at the
stable module category StMod (kG) of G is the tensor triangulated category obta*
*ined
from the abelian category of (left) kG-modules by killing the projectives. The *
*ob-
jects of StMod (kG) are the left kG-modules, and the space of morphisms between
kG-modules M and N, denoted Hom__kG(M, N), is the k-vector space of kG-module
homomorphisms modulo those maps that factor through a projective module. The
category stmod (kG) is obtained similarly from the category of finite-dimension*
*al
left kG-modules. A ghost in the stable module category of G is a map between
kG-modules that is trivial in Tate cohomology. In [3], we formulated the genera*
*ting
hypothesis (GH) for kG as the statement that all ghosts between finite-dimensio*
*nal
kG-modules are trivial in the stable module category, i.e., they factor through*
* a
projective. (This formulation was motivated by the famous classical generating
hypothesis of Peter Freyd [11] in the stable homotopy category which states that
there no non-trivial maps between finite spectra that are trivial in stable hom*
*otopy
groups.) We have shown in [3] that the GH holds for kG, where G is a non-trivial
finite p-group and k is a field of characteristic p, if and only if G is either*
* C2 or C3.
Motivated by the failure of the GH, we proceed in two natural directions. The*
* first
one addresses the GH with the trivial representation k as the target. More prec*
*isely,
we show that in the stable module category of any p-group, a map M ! k from a
finite-dimensional module to the trivial module is stably trivial if it induces*
* the zero
map in Tate cohomology. We give two proofs of this result in Section 2. One of *
*them
uses well-known duality theorems (Spanier-Whitehead duality and Tate duality) to
show that a map between finite-dimensional modules over a p-group is a ghost if*
* and
only if it is a dual ghost. (A dual ghost is a map that induces the zero map on*
* the
functor Hom__kG(-, *k).) It follows that the GH with target k holds.
The second direction we take measures the degree to which the GH fails in p-g*
*roups
other than C2 and C3. In order to measure the degree of failure of the GH, we d*
*efine
the compact ghost number of kG to be the smallest non-negative integer l such t*
*hat
the composition of any l ghosts between finite-dimensional kG-modules is trivia*
*l.
(Note that, in this terminology, C2 and C3 are the only p-groups with compact g*
*host
number 1.) A concept that will be key to our analysis of compact ghost numbers *
*and
related invariants is that of a projective class. Loosely speaking, a projectiv*
*e class
GHOSTS IN MODULAR REPRESENTATION THEORY 3
in a triangulated category is a collection of objects and a collection of maps *
*each of
which is the "orthogonal complement" of the other, and such that there are enou*
*gh
projectives; see Section 4 for the precise definition. We define the ghost proj*
*ective
class, which is a projective class where the collection of maps is the ghosts. *
*When
the trivial representation is periodic we show that a ghost projective class ca*
*n be
constructed in stmod(kG). Moreover, there is an operation on the ghost projecti*
*ve
class which corresponds to composition of ghosts, and thus it is tailor-made fo*
*r the
problem of computing compact ghost numbers.
Working in the framework of projective classes, in Section 5 we compute the
compact ghost numbers of some p-groups. We show that the compact ghost number
of kCpris d(pr-1)=2e, where dxe is the smallest integer that is greater than or*
* equal
to x. If G is a finite abelian 2-group with C2 as a summand, then the compact g*
*host
number of kG is shown to be one less than the nilpotency index of the Jacobson
radical J(kG) of kG. Computing the compact ghost number of an arbitrary group
algebra seems to be a hard problem. Nevertheless, we are able to obtain bounds
on compact ghost numbers. For an arbitrary finite p-group G, we show that the
nilpotency index of J(kG) is an upper bound for the compact ghost number of
kG. Thus we obtain bounds on the compact ghost numbers of some group algebras
by computing the nilpotency indices of their Jacobson radicals using a theorem *
*of
Jennings [14]. (However, the bounds thus obtained are not necessarily sharp; s*
*ee
Section 6 for details.) In this way, we show that if G is a group of order 2n w*
*hich
has a cyclic subgroup of index 2, then any composition of 2n-1 ghosts in stmod(*
*kG)
is trivial. In particular, this applies to the dihedral, semidihedral, modular*
* and
quaternion groups of order 2n. Moreover, this bound turns out to be the sharpest
uniform bound for the compact ghost numbers of the groups in question. Experien*
*ce
tells us that finding lower bounds for compact ghost numbers is much harder than
finding upper bounds. We obtain reasonable lower bounds for the compact ghost
numbers of the group algebras of abelian p-groups. We use these bounds to show
that the only abelian p-groups with compact ghost number 2 are C4, C2 C2, and
C5.
Some other results of this paper that are worth mentioning are our applicatio*
*ns
of universal ghosts, in Section 3, where we give several criteria for the exist*
*ence of a
non-trivial ghost out of a given representation.
The proofs of the aforementioned results involve a pleasant mix of methods fr*
*om
group theory, representation theory and triangulated category theory.
Similar results on phantom maps (maps between kG-modules which factor through
a projective when restricted to finite-dimensional modules) in the stable module
category appear in the work of Benson and Gnacadja; see [2]. For example, in the
stable module category of the Klein four group they show that the composition o*
*f any
two phantoms is trivial if k is countable, and the composition of any three pha*
*ntoms
4 SUNIL K. CHEBOLU, J. DANIEL CHRISTENSEN, AND J'AN MIN'A~C
is trivial if k is uncountable. Similarly, they show that for the quaternion gr*
*oup the
composition of any three phantoms is trivial. There is also more recent work by
Hovey, Lockridge, and Puninski [13] on the GH in the derived category of a ring.
The paper is organised as follows. In Section 2, we prove the GH when the tar*
*get
is k, and we use Tate duality and Spanier-Whitehead duality to study dual ghost*
*s. In
Section 3, we construct universal ghosts and use them to produce various condit*
*ions
which guarantee the existence of non-trivial ghosts. In Section 4, we introduce*
* the
notion of compact ghost number and obtain various inequalities involving compact
ghost lengths, compact generating lengths, nilpotency indices, etc. We compute *
*the
compact ghost numbers for some p-groups in Section 5. In Section 6, we compute
the nilpotency indices of some important families of p-groups to obtain bounds *
*on
compact ghost numbers.
We assume throughout that the group G is finite, and quite often that it is a
p-group. The characteristic of the field k is always assumed to divide the orde*
*r of
the group. For example, when we write kC3, the reader will understand that the
characteristic of k is 3. The words "kG-module" and "G-representation" are used
interchangeably. We use the categorical word "retract" to mean "direct summand".
When we speak of suspensions of a kG-module M, we mean iM for any integer i,
as an object of StMod (kG). We write e iM for the projective-free part of iM, a
well-defined kG-module. When we speak of Heller shifts of M, we mean e iM for
any integer i.
We would like to thank Dave Benson [3] for giving us Propositions 5.1 and 5.10
which helped a great deal in strengthening some of our results. We also thank M*
*ark
Hovey and Keir Lockridge for helpful conversations about this work.
2.The generating hypothesis
A map OE: M ! N between kG-modules is said to be a ghost if the induced map
Hom__kG( ik, M) -! Hom__kG( ik, N)
between the Tate cohomology groups is zero for each integer i. (Recall that the*
* Tate
group bHi(G, M) of G with coefficients in M is isomorphic to Hom__( ik, M).) If*
* G is
a finite p-group and k is a field of characteristic p, then "the generating hyp*
*othesis
for kG" is the statement that all ghosts between finite-dimensional kG-modules *
*are
trivial in the stable module category. In [3] we have shown that the only non-t*
*rivial
p-groups for which the GH holds are the cyclic groups C2 and C3.
2.1. The generating hypothesis with target k. While the generating hypoth-
esis generally fails in the stable module category, we show that a weak version*
* of
it holds for all p-groups. We begin with motivation coming from homotopy theory
for studying this weak version. Devinatz [10] proved the following partial affi*
*rma-
tive result on the generating hypothesis for the p-local stable homotopy catego*
*ry of
GHOSTS IN MODULAR REPRESENTATION THEORY 5
spectra where p is an odd prime: If OE: X ! S0 is a map from a finite spectrum *
*to
the sphere spectrum such that ss*(OE) = 0, then the K(p)-localisation of OE is *
*trivial,
where K(p)periodic complex K-theory localised at p.
Motivated by this result, we consider "the GH with target k", which is the st*
*ate-
ment that every map M ! k from a finite-dimensional kG-module M to the trivial
representation k that induces the zero map in Tate cohomology is trivial in the*
* stable
module category. We show that the GH with target k holds for all p-groups. In f*
*act,
we give two proofs of this fact below; see Corollaries 2.3 and 2.9.
Lemma 2.1. Let G be a p-group and let A be a projective-free kG-module. Then
the norm map
N :A -! A
is trivial.
P Recall that the norm map is given by multiplication by the norm element n =
g2G g in kG. The ideal generated by the norm element is the unique non-zero
minimal ideal of kG [1, p. 92].
Proof.Pick a non-zero element t in A. If ann(t) = 0, that would be mean that
t spans a free kG-submodule of the projective-free module A, a contradiction, so
ann(t) 6= 0. But if ann(t) 6= 0, it contains the unique non-zero minimal ideal,*
* so it
contains the norm element n, showing that nt = 0.
Proposition 2.2. Let G be a p-group and let f :U ! V be a ghost between project*
*ive-
free kG-modules. Then we have the following.
(1) UG is contained in Ker(f). (UG is the G-invariant submodule of U.)
(2) Im(f) is contained in JV . (J denotes the Jacobson radical of kG.)
Proof.Since f is a ghost, the induced map in Tate cohomology is zero. In partic*
*ular,
the maps
G V G
bH0(G, f): _U____ -! ______
Im (N) Im (N)
and
Hb-1(G, f): Ker(N)_- ! Ker(N)_
JU JV
are zero. (Here Im(N) and Ker(N) are respectively the image and kernel of norm
maps.) Since the norm map is trivial on any projective-free module, the above m*
*aps
can be written as
(2.1) Hb0(G, f): UG -! V G
and
U V
(2.2) bH-1(G, f): ___ -! ___.
JU JV
6 SUNIL K. CHEBOLU, J. DANIEL CHRISTENSEN, AND J'AN MIN'A~C
Both of the above maps are zero. The first part of the proposition follows from*
* (2.1)
and the second part from (2.2).
Corollary 2.3. The GH with target k holds for p-groups.
Proof.Let f :M ! k be a ghost in stmod(kG). M is isomorphic in stmod(kG) to
a projective-free kG-module. Therefore we may assume that M is projective-free.
Then, by Proposition 2.2, the image of f :M ! k is contained in J(k), which is
zero. So we are done.
Let us define a map f :M ! N to be an l-ghost if it is a composition of l
ghosts. Recall that associated to a kG-module M, one has the socle (ascending)
series SociM and the radical (descending) series Rad iM. For i = 1, Soc1M := MG*
* ,
the G-invariant submodule of M, and for i > 1, SociM is defined inductively by
SociM=Soci-1M ~=(M=Soci-1M)G . Rad iM := JiM for all i 0, where Ji denotes
the ith power of the Jacobson radical of kG. See [1] for some properties of th*
*ese
series. Proposition 2.2 can now be generalised as follows.
Corollary 2.4. Let G be a p-group and let f :M ! N be an l-ghost between
projective-free kG-modules. Then we have the following.
(1) Socl(M) is contained in Ker(f).
(2) Im(f) is contained in Rad l(M).
Proof.This follows by a straightforward induction using Proposition 2.2.
2.2. Ghosts and duality. A map d: M ! N between kG-modules is called a dual
ghost if the induced map
Hom__kG(M, ik) - Hom__kG(N, ik)
is zero for all i. Recall that for every kG-module L, there is a corresponding *
*dual
kG-module L* := Hom k(L, k) with the G-action defined as follows: for g in G, a*
*nd
OE in L*, (gOE)(x) := OE(g-1x).
Proposition 2.5. Let G be a finite group. A map f :M ! N between kG-modules
is a dual ghost if and only if f* :N* ! M* is a ghost.
Proof.Consider the natural isomorphism
Hom kG(N, Hom k(T, k)) ~=Hom kG(T, Hom k(N, k)),
where T is a Tate resolution of k and N is any kG-module; see [1, Proposition 3*
*.1.8],
for instance. Since Hom k(T, k) is a complete injective resolution of k, taking*
* homol-
ogy of the chain complexes in the last isomorphism gives natural isomorphisms
Hom__kG(N, -n k) ~=Hom__kG( nk, N*).
This implies that a map d: M ! N is a dual ghost if and only if d*: N* ! M* is a
ghost.
GHOSTS IN MODULAR REPRESENTATION THEORY 7
The second isomorphism used in the proof of the above proposition is Spanier-
Whitehead duality for the stable module category.
We now use Tate duality to show that one can use group cohomology (Hbi(G, -),
i 0) to detect ghosts.
Proposition 2.6. Let G be a p-group. A map f :M ! N between finite-dimensional
kG-modules is a ghost if and only if the following two conditions hold:
(1) bHi(G, f): bHi(G, M) ! bHi(G, N) is zero for all i 0.
(2) bHi(G, f*): bHi(G, N*) ! bHi(G, M*) is zero for all i 0.
Proof.Clearly it suffices to show that statement (ii) is equivalent to the stat*
*ement:
Hb-i-1(G, f): bH-i-1(G, M) ! bH-i-1(G, N) is zero for all i 0. Recall that Ta*
*te
duality [5] gives a natural isomorphism
bH-i-1(G, L) ~=(Hbi(G, L*))*
for any finite-dimensional module L. Thus, for each i 0, we have the followi*
*ng
commutative diagram where the vertical maps are induced by f:
~=
Hb-i-1(G, M) ____//_(Hbi(G, M*))*
| |
| |
fflffl| ~ fflffl|
Hb-i-1(G, N) _=__//_(Hbi(G, N*))*.
Since the two horizontal maps are isomorphisms, the left vertical map is zero i*
*f and
only if the right vertical map is zero. Finally, by the faithfulness of the vec*
*tor space
duality functor, the right vertical map is zero if and only if statement (ii) h*
*olds. So
we are done.
Combining Spanier-Whitehead duality and Tate duality gives us the following
interesting result.
Theorem 2.7. A map f :M ! N between finite-dimensional kG-modules is a ghost
if and only if it is a dual ghost.
Proof.By Proposition 2.5, we know that f :M ! N is a dual ghost if and only if
f* :N* ! M* is a ghost. And by Proposition 2.6, f* is a ghost if and only if
(1) bHi(G, f*): bHi(G, N*) ! bHi(G, M*) is zero for all i 0, and
(2) bHi(G, f): bHi(G, M) ! bHi(G, N) is zero for all i 0.
(We get the second statement from the fact that double dual f** is naturally is*
*o-
morphic to f.) The last two statements are in turn equivalent, again by Proposi*
*tion
2.6, to the statement that f is a ghost.
Remark 2.8. The analogue of Theorem 2.7 fails in the derived category of a com-
mutative ring. For example, in D(Z), let X be the cone of the map Z p!Z, and let
Y = Z. Then the map f :X ! Y which projects onto the top class, i.e., the map
8 SUNIL K. CHEBOLU, J. DANIEL CHRISTENSEN, AND J'AN MIN'A~C
that is the identity in degree 1 and zero elsewhere, is easily seen to be a non*
*-trivial
ghost. However, f is not a dual ghost. In fact, the composition X !f Y != Y (= *
* Z)
is just f, which is non-trivial.
The first part of the following corollary gives an alternative proof of Corol*
*lary 2.3.
Corollary 2.9. Let M be a finite dimensional kG-module. Then we have the fol-
lowing.
(1) If f :M ! ik is a ghost, then f is stably trivial. In other words, the *
*GH
with target k holds.
(2) If f : ik ! M is a dual ghost, then f is stably trivial.
Proof.If f :M ! ik is a ghost, then by Theorem 2.7, f is also a dual ghost, so*
* the
composition M -f! ik -=! ik, which is just f, is stably trivial.
The proof of the second statement is similar.
Remark 2.10. Using the results in this section, we can show that "the GH with
domain L" (the statement that every ghost in stmod(kG) with domain L is trivial)
holds if and only if "the GH with target L*" holds. This generalises Corollary *
*2.9.
We leave the easy details to the reader.
3.Universal ghosts
A ghost : M ! N between kG-modules is said to be a universal ghost if every
ghost out of M factors through . (Such a map should technically be called weak*
*ly
universal, since we do not assume the factorisation is unique.) We will show t*
*hat
there exists a universal ghost out of any given kG-module. Let M be a kG-module.
We assemble all the homogeneous elements in bH*(G, M) into a map
M
|j|k -! M,
j2Hb*(G,M)
where |j| is the degree of j. Completing this map to an exact triangle in StMod*
* (kG),
we get
M M
(3.1) |j|k -! M -! UM .
j2Hb*(G,M)
We now recall a couple of easily established facts (see [7] for proofs) which*
* we will
need in the sequel.
Proposition 3.1 ([7]). The map M :M ! UM is a universal ghost out of M.
Our next proposition characterises modules out of which all ghosts vanish.
Proposition 3.2 ([7]). Let M be a kG-module. Then the following are equivalent
statements in the stable module category:
GHOSTS IN MODULAR REPRESENTATION THEORY 9
(1) All ghosts out of M are trivial.
(2) The universal ghost M :M ! UM is trivial.
(3) M is a retract of a direct sum of suspensions of the trivial representat*
*ion.
Moreover, if M is finite-dimensional, (3) can be replaced with the condition: M*
* is a
finite direct sum of suspensions of the trivial representation.
Now suppose that M is a finite-dimensional kG-module such that Hb*(G, M) is
finitely generated as a graded module over Hb*(G, k). (This happens, for exampl*
*e,
when k is periodic, that is, ik is stably isomorphic to k for some i 6= 0.) We*
* will
show that a universal ghost out of M can be constructed in the category stmod(k*
*G);
that is, the target module of the universal ghost out of M can be chosen to be *
*finite-
dimensional as well. This is done as follows. Let {vj} be a finite set of homog*
*eneous
generators for bH*(G, M) as an bH*(G, k)-module. These generators can be assemb*
*led
into a map M
|vj|k -! M
j
in StMod (kG). This map can then be completed to a triangle
M M
|vj|k -! M -! FM .
j
By construction, it is clear that the first map in the above triangle is surjec*
*tive on
the functors Hom__( lk, -) for each l. Therefore, the second map M must be a
ghost. Thus we have the following proposition.
Proposition 3.3. Suppose M is a finite-dimensional kG-module such that
Hb*(G, M) is finitely generated as a graded module over Hb*(G, k). Then the map
M : M ! FM in the above triangle is a universal ghost out of M. Furthermore,
FM is compact and thus can be chosen to be finite-dimensional.
Proof.Universality of M is easy to see. For the last statement, since the sum
is finite, j |vj|k is compact. And since the category of compact objects form*
*s a
triangulated subcategory of StMod (kG), FM is compact as well.
Corollary 3.4. Suppose G is a finite group such that the trivial representation*
* k is
periodic, and let M be a finite-dimensional kG-module. Then a universal ghost o*
*ut
of M can be constructed in stmod(kG).
Proof.Since M is finite-dimensional, each Tate group bHi(G, M) is finite-dimens*
*ional.
And since k is periodic, so is the Tate cohomology of M. Therefore, by combining
these two facts, we conclude that the Tate cohomology of M is finitely generated
over bH*(G, k). Now Proposition 3.3 applies.
Corollary 3.5. Let M be a finite-dimensional module such that bH*(G, M) is fini*
*tely
generated as a graded module over bH*(G, k). Then the following are equivalent *
*state-
ments in the stable module category:
10 SUNIL K. CHEBOLU, J. DANIEL CHRISTENSEN, AND J'AN MIN'A~C
(1) All ghosts out of M are trivial.
(2) All ghosts out of M into finite-dimensional modules are trivial.
(3) The universal ghost M :M ! FM is trivial.
(4) M is a finite direct sum of suspensions of the trivial representation.
Proof.The implications (1) ) (2) ) (3) ) (4) ) (1) are all clear. For (2) ) (3)
we use that FM is isomorphic to a finite-dimensional module (by Proposition 3.*
*3)
and for (3) ) (4) we use Krull-Schmidt.
We now give some applications of universal ghosts. For the remainder of this
subsection, G will be a finite p-group. We begin with a characterisation of fi*
*nite-
dimensional indecomposable projective-free representations that are isomorphic *
*to a
Heller shift of the trivial representation.
Proposition 3.6. Let G be a finite p-group and let M be a finite-dimensional in-
decomposable projective-free kG-module. Then all ghosts out of M are trivial if*
* and
only if M ~=e ik for some integer i. Moreover, if bH*(G, M) is finitely generat*
*ed as
a graded module over Hb*(G, k), then M ~= e ik if and only if all ghosts out of*
* M
into finite-dimensional modules are trivial.
Proof.By Proposition 3.2, we know that all ghosts out of M are trivial if and o*
*nly if
M is isomorphic (in the stable module category) to a finite direct sum of suspe*
*nsions
of the trivial representation. Since M is projective-free, this implies that M*
* is
isomorphic as a kG-module to a finite direct sum e ik. But M is assumed to be
indecomposable, so the Krull-Schmidt theorem tells us that M is isomorphic to e*
* ik
for some i as desired. The last statement now follows from Corollary 3.5.
Corollary 3.7. Let G be a finite p-group and let M be a finite-dimensional inde*
*com-
posable projective-free kG-module. If dim(M) is not congruent to +1 or -1 modulo
|G|, then there exists a non-trivial ghost out of M. Moreover, if bH*(G, M) is *
*finitely
generated as a graded module over bH*(G, k), then there exists a non-trivial gh*
*ost out
of M whose target is finite-dimensional.
Proof.By the previous proposition it suffices to show that M AE e ik for any i.*
* This
will be shown by proving that under the given hypothesis the dimensions of the
Heller shifts e ik are congruent to +1 or -1 modulo |G|. Recall that e 1k is de*
*fined
to be the kernel of the augmentation map, so we have a short exact sequence
0 -! e 1k -! kG -! k -! 0,
which tells us that dim(e 1k) -1 modulo |G|. Inductively, it is clear from t*
*he
short exact sequences
0 -! e i+1k -! (kG)t- ! e ik -! 0
that dim(e ik) (-1)i modulo |G| for i 0. (Here (kG)t, for some t, is a mini*
*mal
projective cover of e ik.) Also, since e ik ~= (e -ik)* in Mod (kG), it follow*
*s that
GHOSTS IN MODULAR REPRESENTATION THEORY 11
dim(e ik) (-1)i modulo |G| for each integer i. In particular, M AE e ik for a*
*ny
integer i.
3.1. Periodic modules. Recall that a representation M of a group G is said to be
periodic if there exists a positive integer l such that lM is stably isomorphi*
*c to M.
For example, the trivial representation k of a cyclic group satisfies e 2k ~=k.
Proposition 3.8. Let G be a finite p-group such that the trivial representation*
* k is
not periodic. If M is a finite-dimensional non-projective periodic kG-module, t*
*hen
there exists a non-trivial ghost out of M.
For example, if G is the Klein four group V4 = C2 C2, then the trivial repr*
*e-
sentation k is not periodic, but every even-dimensional indecomposable module M
is periodic and there are infinitely many such modules up to isomorphism. See, *
*for
example, [8, Section 2].
Proof.It follows from the Krull-Schmidt theorem that if M is finite-dimensional*
* and
periodic, so are the indecomposable summands of M. So we assume without loss of
generality that M is indecomposable and projective-free. By Proposition 3.6, it*
* is
enough to show that M AE e ik for any i. But if M ~=e ik and M is periodic, then
k is periodic as well, which is a contradiction.
Remark 3.9. By a result of Artin and Tate [6, p. 262], we know that the only fi*
*nite p-
groups whose Tate cohomology is periodic are the cyclic p-groups and the quater*
*nion
groups. Thus we have proved:
Corollary 3.10. Let G be a finite p-group which is not cyclic and not a quatern*
*ion
group. If M is a finite-dimensional non-projective periodic kG-module, then th*
*ere
exists a non-trivial ghost out of M.
4.The ghost projective class
In order to measure the degree to which the GH fails, it is natural to consid*
*er
the smallest integer l such that the composition of any l ghosts between finite-
dimensional kG-modules is trivial. This integer will be called the compact gho*
*st
number of kG and can be best understood using the concept of a projective class.
So we begin with a recollection of the notion of a projective class in a triang*
*ulated
category. A good reference for this is [9], where projective classes were stud*
*ied in
the stable homotopy category and the derived category of a ring.
4.1. Projective classes. Let T denote a triangulated category. A projective cla*
*ss
in T is a pair (P, G) where P is a class of objects and G is a class of maps in*
* T which
satisfy the following properties:
(1) The class of all maps X ! Y such that the composite P ! X ! Y is zero
for all P in P and all maps P ! X is precisely G.
12 SUNIL K. CHEBOLU, J. DANIEL CHRISTENSEN, AND J'AN MIN'A~C
(2) The class of all objects P such that the composite P ! X ! Y is zero for
all maps X ! Y in G and all maps P ! X is precisely P.
(3) For each object X there is a cofibre sequence P ! X ! Y with P in P and
X ! Y in G.
The maps in G form an ideal in T . That is, if f and g are parallel maps in *
*G,
then f + g is in G, and if f, g, and h are composable with g in G, then both fg*
* and
gh are also in G.
Once we have a projective class as defined above, we can form "derived" proje*
*ctive
classes in a natural way as follows. The powers Gn of the ideal G form a decrea*
*sing
filtration of the maps in T , and each Gn is part of a projective class. The co*
*rrespond-
ing classes of objects are obtained as follows. Let P1 = P and inductively defi*
*ne Pn
to be the collection of retracts of objects M that appear in a triangle
A -! M -! B,
where A belongs to P1 and B belong to Pn-1. The classes Pn form an increasing
filtration of the objects in T . It is a fact [9, Theorem 1.1] that (Pn, Gn) is*
* a projective
class for each n. We set P0 to be the collection of zero objects in T and G0 t*
*o be
the collection of all maps in T . (P0, G0) is the trivial projective class.
4.2. The ghost projective class. Now we specialise to the stable module category
to define the ghost projective class in StMod (kG). For simplicity, we assume *
*that
G is a p-group in this section. The ideal G consists of the class of ghosts. *
*The
associated class P of objects consists of retracts of direct sums of suspension*
*s of the
trivial representation.
Proposition 4.1. The pair (P, G) is a projective class in StMod (kG).
Proof.It is clear that P and G are orthogonal, i.e., the composite P ! M !h N is
zero for all P in P, for all h in G, and all maps P ! M. So by [9, Lemma 3.2] it
remains to show that for all kG-modules M, there exists a triangle P ! M ! N
such that P is in P and M ! N is in G. The universal ghost (3.1)out of M has
this property, so we are done.
In some special cases one can also build a ghost projective class in stmod (k*
*G).
Let Pc denote the collection of finite direct sums of suspensions of k and Gc t*
*he
class of ghosts in stmod(kG). (Note that the collection Pc is already closed un*
*der
retractions by the Krull-Schmidt theorem.) Then we have the following propositi*
*on.
Proposition 4.2. Let G be a finite p-group such that the Tate cohomology of each
finite-dimensional kG-module is finitely generated as a graded module over the *
*ring
Hb*(G, k). Then (Pc, Gc) is a projective class in stmod (kG). In particular, *
*this
applies to groups whose trivial representation is periodic.
GHOSTS IN MODULAR REPRESENTATION THEORY 13
Proof.Orthogonality of Pc and Gc is clear. We have already seen in Proposition *
*3.3
that under the given hypothesis a universal ghost out of a finite-dimensional m*
*odule
can be constructed within stmod(kG). So the proposition follows from [9, Lemma
3.2].
If the pair (Pc, Gc) forms a projective class in stmod(kG), it will be referr*
*ed to as
the compact ghost projective class. Whether or not it forms a projective class,*
* we can
define Pmc and Gmcas in the previous subsection. A finite-dimensional kG-module*
* is
said to have compact generating length m if it belongs to Pmc but not to Pm-1c,*
* and
compact ghost length m if it is the domain of a non-zero map in Gm-1c but not in
Gmc.
We now prove a sequence of inequalities.
Proposition 4.3. Let G be a finite p-group and M a finite-dimensional kG-module.
Then
compact ghost length ofM compact generating length ofM.
Moreover, equality holds if (Pc, Gc) is a projective class.
Proof.The inequality follows by induction, using the technique found in the pro*
*of
of [9, Prop. 3.3]. The equality follows directly from [9, Prop. 3.3].
Proposition 4.4. Let G be a finite p-group and M a finite-dimensional kG-module.
Then
compact generating length ofM radical length ofM.
Proof.The Jacobson radical J = J(kG) of kG is nilpotent. The radical length of M
is the smallest integer h such that JhM = 0. This gives the radical or Lowey se*
*ries
for M:
M ! JM ! . .!.Jh-1M ! JhM = 0.
Note that J annihilates each successive quotient and hence each of them is a di*
*rect
sum of trivial representations. This shows that the compact generating length i*
*s at
most the radical length.
Recall that the nilpotency index of the Jacobson radical J(kG) of kG is the s*
*mall-
est integer m such that J(kG)m = 0.
Proposition 4.5. Let G be a finite p-group and M a projective-free kG-module.
Then
radical length ofM < nilpotency index ofJ(kG) |G|.
Proof.Let m be the nilpotency index of J(kG). We begin by noting that since
G is a p-group, the last non-zero power J(kG)m-1 of kG is the unique non-zero
minimal ideal in kG; see [1, p. 92]. For the first inequality, it is enough to *
*show that
J(kG)m-1 M = 0. Let x be an element of M. Since M is projective-free, Ann (x) 6*
*= 0.
14 SUNIL K. CHEBOLU, J. DANIEL CHRISTENSEN, AND J'AN MIN'A~C
Thus Ann (x) contains the unique non-zero minimal ideal J(kG)m-1 of kG. That is,
J(kG)m-1 x = 0.
Since the powers of J(kG) form a strictly decreasing series, then the nilpote*
*ncy
index of J(kG) is at most dimk kG = |G|.
These inequalities show that for each p-group G, the compact ghost lengths and
compact generating lengths of finite-dimensional kG-modules are uniformly bound*
*ed
above. So we define the compact generating number of kG to be the least upper
bound of the compact generating lengths of all finite-dimensional kG-modules, a*
*nd
the compact ghost number of kG to be the least upper bound of the compact ghost
lengths of all finite-dimensional kG-modules. We don't know if the compact gene*
*r-
ating number and compact ghost number of kG depend on the field k.
Combining the above results gives:
Theorem 4.6. Let G be a finite p-group. Then
compact ghost number ofkG compact generating number ofkG
< nilpotency index ofJ(kG) |G|.
In particular, the compact generating number and compact ghost number of the gr*
*oup
algebra of any finite p-group are finite, and any composition of |G| - 1 ghosts*
* in
stmod (kG) is trivial.
Remark 4.7. A similar argument involving the projective class (P, G) shows that*
* any
composite of m ghosts in StMod (kG) is trivial, where m is the nilpotency index*
* of
J(kG).
Proposition 4.8. Let H be a subgroup of a finite p-group G. Then
compact ghost number ofkH compact ghost number ofkG.
Proof.In [3], we have shown that the induction functor
Ind_:stmod(kH) -! stmod(kG),
which sends a kH-module M to M"G := kG kH M, preserves ghosts and non-
trivial maps. It follows that the compact ghost number of kH is no more than th*
*at
of kG.
5.Computing compact ghost numbers
We now investigate the problem of computing compact ghost numbers and com-
pact generating numbers of some specific groups. The following lemma that we
learned from Dave Benson [3] will be very helpful in computations.
Proposition 5.1. Let G be a finite p-group and let M be a kG-module. If an elem*
*ent
` belonging to J(kG) is central in kG, then the map
` :M -! M
GHOSTS IN MODULAR REPRESENTATION THEORY 15
is a ghost. In particular, if g 2 G is central, then the map
g - 1: M -! M
is a ghost.
Proof.The proof of [3, Lemma 2.2] applies without change.
5.1. Cyclic p-groups. Recall that for cyclic groups, the trivial representation*
* is
periodic, and therefore (Pc, Gc) forms a projective class. So the compact ghost*
* length
and compact generating length are same for modules over cyclic p-groups.
Proposition 5.2. All finite-dimensional k Cpr-modules have compact ghost length
at most d(pr - 1)=2e.
Here dye denotes the smallest integer that is greater than or equal to y.
Proof.Since the characteristic of k is p, we have kCpr ~=k[x]=(xpr), with x cor*
*re-
sponding to oe-1, where oe is a generator of Cpr. A finite-dimensional indecomp*
*osable
projective-free module over k[x]=(xpr) is of the form k[x]=xi for 1 i pr - *
*1. It
is also clear that e(k[x]=(xi)) ~=k[x]=(xpr-i). This tells us that the compact *
*ghost
length of k[x]=(xi) is the same as that of k[x]=(xpr-i). For 1 i d(pr - 1)=*
*2e, we
show that the compact ghost length of k[x]=xi is at most i. For this it is enou*
*gh to
observe that we have short exact sequences
0 -! k -! k[x]=xi- ! k[x]=xi-1- ! 0
of modules over k[x]=xpr, for 2 i d(pr - 1)=2e.
Proposition 5.3. There exists a composable sequence of d(pr - 1)=2e - 1 ghosts *
*in
stmod (kCpr) whose composite is non-trivial.
Proof.Recall that kCpr ~=k[x]=(xpr). Let h: k[x]=xd ! k[x]=xd be multiplication
by x = oe - 1, where d = d(pr - 1)=2e. By Proposition 5.1, h is a ghost. To s*
*ee
that hd-1 is non-trivial, we have to show that it cannot factor through the pro*
*jective
cover k[x]=xpri k[x]=xd, i.e., that we cannot have a commutative diagram
xd-1 d
k[x]=xd__________________//k[x]=x
K K ss9999s
K K ssss
K%% rss
k[x]=xp .
By considering the images of the generator of the left-hand cyclic module in the
above diagram, one can easily see that the existence of such a factorisation wo*
*uld
mean that
(d - 1) + (d - 1) pr - 1,
or, equivalently, that
d(pr - 1)=2e - 1 + d(pr - 1)=2e - 1 pr - 1.
16 SUNIL K. CHEBOLU, J. DANIEL CHRISTENSEN, AND J'AN MIN'A~C
It is straightforward to verify that this inequality fails for all primes p and*
* all positive
integers r. So we are done.
Combining these two propositions, we get the following theorem.
Theorem 5.4. The compact ghost number of kCpr is d(pr - 1)=2e.
Corollary 5.5. The GH holds for kCpr if and only if pr is equal to 1, 2 or 3.
Proof.Recall that the GH holds for kG precisely when the compact ghost number
of kG is 0 or 1. So, by Theorem 5.4, we conclude that the GH holds for kCpr if *
*and
only if d(pr - 1)=2e = 0 or 1. The last equation holds if and only if pr = 1, *
*2 or
3.
See [3] for a different proof of the above corollary.
5.2. The Klein four group.
Proposition 5.6. Let M be a finite-dimensional indecomposable projective-free k*
*V4-
module. Then we have the following:
(1) If M is odd-dimensional, then it has compact generating length one.
(2) If M is even-dimensional, then it has compact generating length two.
Proof.It is well-known that the odd-dimensional indecomposable modules are pre-
cisely the Heller shifts of the trivial representation; see [1], for instance. *
*So they all
have compact generating length one by definition. Since M is projective-free, *
*one
can show using the classification of the indecomposable kV4 modules (e.g., [1])*
*, or
directly, that there is a short exact sequence
0 -! MV4 -! M -! MV4 -! 0,
where the invariant submodule MV4 and the coinvariant module MV4 are both direct
sums of the trivial representation k. Thus M has compact generating length at
most two. Moreover, if M is even-dimensional and indecomposable, then M is not
isomorphic to e ik for any i. In particular, M cannot have compact generating l*
*ength
one. So we are done.
Theorem 5.7. The compact ghost number and the compact generating number of
kV4 are both two.
Proof.Since every finite-dimensional module is a sum of indecomposables, the st*
*ate-
ment about compact generating number follows from the above proposition. Since
the compact ghost number is at most the compact generating number, we only have
to show that the compact ghost number is bigger than one. This follows from [3]
because there we showed that the GH fails for stmod(kV4).
GHOSTS IN MODULAR REPRESENTATION THEORY 17
5.3. The quaternion group. By our main result on the GH in [3], we know that
the GH fails for the quaternion group Q8 of order 8. Now we give bounds on the
compact ghost number.
Proposition 5.8. The compact ghost number of kQ8 is at least two and at most
four.
Proof.Since the trivial representation of Q8 is periodic, we know that the comp*
*act
ghost projective class exists in stmod(kQ8). Therefore the compact ghost number
and the compact generating number of kQ8 are the same. The nilpotency index of
J(kQ8) can be shown to be 5 (see Proposition 6.1), so by Theorem 4.6, we know
that the compact ghost number of kQ8 is at most 4. We have already seen that the
GH fails for Q8, so the compact ghost number is at least 2.
In the following example we will give another disproof of the GH for the grou*
*p Q8
by exhibiting an explicit finite-dimensional module with compact ghost length t*
*wo.
This example should also illustrate some of the ideas surrounding projective cl*
*asses.
Example 5.9. A minimal presentation for Q8 is given by
Q8 = .
The structure of the left kQ8-module J(kQ8)3 can be obtained using Jennings' th*
*e-
orem [14] or otherwise. It is shown in the diagram below:
(x-1)(ffl-1)(y-1)(ffl-1)
o?? o"
??? """"
???"""
o
(y-1)(x-1)(ffl-1)
Here a bullet is a one-dimensional k-vector space, the southwest line segment c*
*orre-
sponds to the action of x - 1, the southeast line segment corresponds to the ac*
*tion
of y - 1, and if no line segment emanates from a bullet in a given direction, t*
*hen the
corresponding action is trivial. ffl is the central element x2(= y2).
It is clear from the diagram that the invariant submodule of J(kQ8)3 is one-
dimensional and therefore we conclude that J(kQ8)3 is indecomposable (see, for
example, [7, Lemma 3.2]). Also note that J(kQ8)3 is projective-free. Moreover, *
*the
dimension of J(kQ8)3 is 3, which is neither +1 nor -1 modulo 8 (= |Q8|). Thus
by Corollary 3.7 we know that there exists a non-trivial ghost in stmod(kQ8) wh*
*ose
domain is J(kQ8)3. In particular, the compact ghost length of J(kQ8)3 is at lea*
*st 2.
On the other hand the compact generating length of J(kQ8)3 is at most 2 because
we have a short exact sequence of kQ8-modules
0 -! k -! J(kQ8)3 -! k k -! 0.
18 SUNIL K. CHEBOLU, J. DANIEL CHRISTENSEN, AND J'AN MIN'A~C
Since the compact ghost length is always less than or equal to the compact gene*
*rating
length, we conclude that the compact ghost length and the compact generating le*
*ngth
of J(kQ8)3 are both 2.
5.4. Compact ghost numbers of abelian groups. We begin with an extremely
useful proposition. This is a slight generalisation of a result we learnt from*
* Dave
Benson that appeared in [3].
Proposition 5.10. Let GPbe a finite p-group and let H be a non-trivialPproper
subgroup of G. Let ` = gffgg be a central element of kG such that h2H ffh 6*
*= 0.
Then multiplication by ` on kH "G is stably non-trivial, where kH is the trivia*
*l kH-
module. In particular, if g is a central element in G-H, then multiplication by*
* g -1
on kH "G is a non-trivial ghost.
We include a proof, since this is slightly more general than [3, Lemma 2.3]. *
*We
do not require H to be normal and we include more general `.
Proof.Recall that kH "G denotes the induced module kG kH kH , and that induction
is both left and right adjoint to restriction. These adjunctions give rise to n*
*atural
kH-linear maps kH ! kH "G#H , sending x to 1 x, and kH "G#H ! kH , sending
g x to x if g 2 H and to 0 otherwise.
To show that ` :kH "G ! kH "G is stably non-trivial, it is enough to show that
`#H :kH "G#H ! kH "G#H is stably non-trivial. For this, it is enough to show th*
*at
the composite
`#H G
kH ! kH "G#H -! kH " #H ! kH
P
is stably non-trivial. But this composition is multiplication by h2H ffh, whi*
*ch is
non-zero by assumption. And since H is non-trivial, all non-zero maps kH ! kH a*
*re
stably non-trivial.
The last statement follows from the first part of this proposition, combined *
*with
Proposition 5.1.
Theorem 5.11. Let G be an abelian p-group and let m denote the nilpotency index
of J(kG). Then we have
m - pr-1(p - 1) compact ghost number ofkG m - 1,
where pr is the order of the smallest cyclic summand of G.
Proof.We have already seen the upper bound, so we only have to establish the lo*
*wer
bound. Let Cprbe the smallest cyclic summand of G, so that for some integers ri*
* r,
G = Cpr Cpr1 . . .Cprt.
Let H be the subgroup of order p in Cpr and let kH be the trivial kH-module. Set
M = kH "G. We will produce m - pr-1(p - 1) - 1 ghosts M ! M whose composite
is stably non-trivial. This will give the desired lower bound for the compact g*
*host
GHOSTS IN MODULAR REPRESENTATION THEORY 19
number. Let g be a generator for Cpr and let gi be a generator for Cprifor each*
* i.
By Proposition 5.1, the map
r-1-1 pr1-1 prt-1
` = (g - 1)p (g1 - 1) . .(.gt- 1) :M ! M
is a composite of
(pr-1 - 1) + (pr1- 1) + . .+.(prt- 1)
ghosts. One can see directly (or using Jennings' formula, Section 6) that the n*
*ilpo-
tency index of J(kG) is
m = 1 + (pr - 1) + (pr1- 1) + . .+.(prt- 1).
Thus ` is a composite of m - pr-1(p - 1) - 1 ghosts. Now to see thatP` is stab*
*ly
non-trivial on M, it is enough to note that if ` 2 kG is written g2G ffgg, th*
*en
ffh = 0 for h 2 H unless h = e. So Proposition 5.10 applies.
We derive some easy corollaries.
Corollary 5.12. Let G be an abelian 2-group which has C2 as a summand. Then
the compact ghost number of kG is one less than the nilpotency index of J(kG).
Proof.In this case, both the lower bound and the upper bound for the compact
ghost number of kG are one less than the nilpotency index of J(kG).
Corollary 5.13. Let G be an elementary abelian 2-group of rank l, i.e., G ~=(C2*
*)l.
Then the compact ghost number of kG is l.
Proof.The nilpotency index of J(k(C2)l) is easily shown to be l + 1.
Theorem 5.14. Let G be an abelian p-group. The compact ghost number of kG is
2 if and only if G is C4, C2 C2, or C5.
Proof.By Theorem 5.4 and Corollary 5.13 we know that the three given groups have
compact ghost number 2. Now we prove the converse. An easy exercise using the
structure theorem tells us that if |G| > 5, then G contains one of the followin*
*g groups
as a subgroup: Cpl(pl> 5), C2 C2 C2, Cp Cp (p > 2), or C2 C4. It is eas*
*ily
seen using the lower bound in Theorem 5.11 that the compact ghost number of each
of the above groups is at least 3. Therefore, by Proposition 4.8, the compact g*
*host
number of G is also at least 3. So if the compact ghost number of G is at most *
*2,
then |G| should be at most 5. We know that C2 and C3 have compact ghost number
1, and the only remaining groups of order at most 5 are C4, C5 and C2 C2.
6.Nilpotency index
Let G be a finite p-group. Recall that we have proved that the compact ghost
number of kG is less than the nilpotency index of J(kG) (the smallest integer m
such that J(kG)m = 0). This bound is crude because the successive quotients in
the radical series for a kG-module M are built using only copies of k and not u*
*sing
20 SUNIL K. CHEBOLU, J. DANIEL CHRISTENSEN, AND J'AN MIN'A~C
suspensions of k (see proof of Proposition 4.4). Nevertheless, it is good to ha*
*ve some
bound on the compact ghost number, so we compute the nilpotency indices for some
important families of p-groups.
We begin by recalling a beautiful formula of Jennings [14] that gives the nil*
*potency
index of J(kG). The dimension subgroups Fi of G are defined by
Fi:= {g 2 G : g - 1 2 J(kG)i} for i 1.
These form a descending chain of normal subgroups in G
F1 F2 F3 . . .Fd Fd+1,
with F1 = G and Fd+1 trivial. Define integers ei by pei= [Fi : Fi+1] for 1 i *
* d.
Then Jennings' formula states that the nilpotency index m of J(kG) is given by
Xd
(6.1) m = 1 + (p - 1) i ei.
i=1
When doing specific computations with examples it is helpful to have the recurs*
*ive
formula
F1 = G and
(6.2)
Fn = [G, Fn-1] (Fdn_pe)p forn > 1,
where [G, Fn-1] is the subgroup generated by the commutators aba-1b-1 for a in G
and b in Fn-1, dxe denotes the smallest integer that is greater than or equal t*
*o x,
and (Fdn_pe)p denotes the subgroup generated by {gp : g 2 Fdn_pe}. See [1, Sect*
*ion 3.14]
for a proof of this recursive formula.
Note that the recursive formula shows that the dimension subgroups Fi are ind*
*e-
pendent of the field k, and so by (6.1)it follows that the nilpotency index of *
*J(kG)
is independent of k.
Using the above results, we can compute the nilpotency indices for several fa*
*milies
of p-groups.
Proposition 6.1. Let G be a non-cyclic group of order 2n which has a cyclic sub-
group of index 2. Then the nilpotency index of J(kG) is 2n-1 + 1.
Proof.Let C2n-1= be the cyclic subgroup of index 2. Then G is an extension
of cyclic groups:
1 -! C2n-1,-! G -ss!C2 -! 1.
Let y be an element in G which does not belong to C2n-1. Then clearly G is gene*
*rated
by x and y. Moreover, y2 belongs to the subgroup generated by x2: since ss(y2) *
*= 1,
y2 certainly belongs to C2n-1 (the subgroup generated by x), so y2 = xl for some
integer l. If l is odd, then xlwill generate C2n-1, which then implies that y h*
*as order
2n, a contradiction to the hypothesis that G is not cyclic. So l has to be even*
*, that
is, y2 belongs to the subgroup generated by x2.
GHOSTS IN MODULAR REPRESENTATION THEORY 21
We now compute the dimension subgroups of G using the formula (6.2). F1 = G,
so F2 = [G, G] G2. Note that the quotient group G=G2 has the property that every
element squares to the identity. Such a group is always abelian. In particula*
*r,
[G, G] G2, and therefore F2 = G2. Since y2 belongs to the subgroup generated *
*by
x2, it follows that F2 = . We now claim that Fi = , where l is the uni*
*que
integer such that 2l-1 < i 2l. Assuming this claim we finish the proof. Note
that the claim implies that the dimension subgroups Fiare constant in the inter*
*vals
(2l-1, 2l] and that they are of the form:
n-2
G ) ) = ) = = = ) . .=. ) 1.
The sequence (ei) is therefore (2, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1,*
* . .,.0, 1), that is,
e1 = 2, ei= 1 whenever i is a power of 2 and 1 < i 2n-2, and ei is zero other*
*wise.
The nilpotency index formula gives:
m = 1 + [(1)2 + (2)1 + (22)1 + (23)1 + . .+.(2n-2)1]
= 1 + [2 + 21 + 22 + 23 + . .+.2n-2]
= 1 + [2 + 2n-1 - 2]
= 1 + 2n-1.
Now we prove our claim: Fi = , where l is the unique integer such that
2l-1 < i 2l. Fi is a subgroup of F2 = , so Fi = for some di with
2 | di| 2n-1. Since the subgroups Fi are decreasing, di| di+1 for all i. We h*
*ave the
formula Fi= [G, Fi-1] (Fd_i2e)2. We will show that
[G, Fi-1] (Fi-1)2 (Fd_i2e)2.
The second inclusion is easy: note that the subgroups Fiare decreasing, so for *
*i 2,
, ss
i 2 2
i - 1 __ =) Fi-1 Fd_ie=) (Fi-1) (Fd_ie) .
2 2 2
Now we prove that first inclusion. Since the subgroup generated by x has index *
*2,
it is normal in G, so yxy-1 = xc for some c. Note that c has to be coprime to 2*
*n-1,
hence c has to be odd. Now consider the commutator yxdi-1y-1x-di-1. We have
yxdi-1y-1x-di-1= (yxy-1)di-1x-di-1= xcdi-1x-di-1= x(c-1)di-1.
Since c - 1 is even, x(c-1)di-1belongs to (Fi-1)2.
Thus the formula for Fi simplifies to Fi = (Fd_i2e)2. This translates to di =*
* 2dd_i2e
with d2 = 2. It is easy to see that these equations imply that di = l, where l*
* is
the unique integer such that 2l-1 < i 2l. This proves our claim and so we are
done.
Theorem 6.2. If G is a group of order 2n which has a cyclic subgroup of index 2,
then any composition of 2n-1 ghosts in stmod(kG) is trivial.
22 SUNIL K. CHEBOLU, J. DANIEL CHRISTENSEN, AND J'AN MIN'A~C
Proof.If G is cyclic of order 2n, then by Theorem 5.4 we know that its compact *
*ghost
n-1 *
* n-1
number is d2___2e. A quick calculation tells us that the last expression is jus*
*t 2 .
So this takes care of the cyclic case. If G is not cyclic, then the previous pr*
*oposition
in conjunction with Theorem 4.6 tells us that the compact ghost number is at mo*
*st
2n-1. So we are done.
Remark 6.3. It is clear from the proof of the above proposition that in fact 2n*
*-1 is
the sharpest uniform bound for the compact ghost numbers of the group algebras *
*of
2-groups which have a cyclic subgroup of index 2.
Corollary 6.4. Let G be a group of order 2n that is isomorphic to one of the
following: the quaternion group (Q2n), the dihedral group (D2n), the semidihedr*
*al
group (SD2n), or the modular group (M2n). Then any composition of 2n-1 ghosts
in stmod(kG) is trivial.
Proof.These groups have presentations
n-1 2 2n-2 -1 -1
Q2n =
n-1 2 -1 -1
D2n =
n-1 2 -1 -1+2n-2
SD2n =
n-1 2 -1 1+2n-2
M2n =
In each case, the subgroup generated by x has index 2 and so the previous theor*
*em
applies.
Remark 6.5. Note that the group D22 is the Klein four group V4. Therefore it is
interesting to note that the above corollary gives a different proof of the fac*
*t that
the composition of any two ghosts in stmod(kV4) is trivial; see Theorem 5.7 whe*
*re
we have used projective classes and the structure of kV4-modules to establish t*
*his
fact.
Remark 6.6. It is a well-known theorem [4, Ch. 8, pp. 134-135] that every non-
abelian 2-group that has a cyclic subgroup of index 2 is a dihedral, semidihedr*
*al,
modular or quaternion group.
References
[1]D. J. Benson. Representations and cohomology. I, volume 30 of Cambridge Stu*
*dies in Advanced
Mathematics. Cambridge University Press, Cambridge, 1998.
[2]D. J. Benson and G. Ph. Gnacadja. Phantom maps and purity in modular repres*
*entation theory.
I. Fund. Math., 161(1-2):37-91, 1999. Algebraic topology (Kazimierz Dolny, *
*1997).
[3]David Benson, Sunil K. Chebolu, J. Daniel Christensen, and J'an Min'a~c. Th*
*e generating
hypothesis for the stable module category of a p-group. J. Algebra, 310(1):*
*428-433, 2007.
arXiv:math.RT/0611403.
[4]W. Burnside. Theory of groups of finite order. Dover Publications Inc., New*
* York, 1955. 2d ed.
GHOSTS IN MODULAR REPRESENTATION THEORY 23
[5]Jon F. Carlson. Projective resolutions and degree shifting for cohomology a*
*nd group rings. In
Representations of algebras and related topics (Kyoto, 1990), volume 168 of*
* London Math. Soc.
Lecture Note Ser., pages 80-126. Cambridge Univ. Press, Cambridge, 1992.
[6]Henri Cartan and Samuel Eilenberg. Homological algebra. Princeton Landmarks*
* in Mathemat-
ics. Princeton University Press, Princeton, NJ, 1999. With an appendix by D*
*avid A. Buchs-
baum, reprint of the 1956 original.
[7]Sunil K. Chebolu, J. Daniel Christensen, and J'an Min'a~c. Groups which do *
*not admit ghosts.
To appear in Proc. Amer. Math. Soc., 2007. arXiv:math.RT/0610423.
[8]Sunil K. Chebolu, J'an Min'a~c, and John Swallow. Representations of the Kl*
*ein's four group and
Galois theory. 2006. In preparation.
[9]J. Daniel Christensen. Ideals in triangulated categories: phantoms, ghosts *
*and skeleta. Adv.
Math., 136(2):284-339, 1998.
[10]Ethan S. Devinatz. K-theory and the generating hypothesis. Amer. J. Math., *
*112(5):787-804,
1990.
[11]Peter Freyd. Stable homotopy. In Proc. Conf. Categorical Algebra (La Jolla,*
* Calif., 1965), pages
121-172. Springer, New York, 1966.
[12]Gilles Ph. Gnacadja. Phantom maps in the stable module category. J. Algebra*
*, 201(2):686-702,
1998.
[13]Mark Hovey, Keir Lockridge, and Gena Puninski. The generating hypothesis in*
* the derived
category of a ring. To appear in Mathematische Zeitschrift, 2007. arXiv:mat*
*h.AT/061020.
[14]S. A. Jennings. The structure of the group ring of a p-group over a modular*
* field. Trans. Amer.
Math. Soc., 50:175-185, 1941.
Department of Mathematics, University of Western Ontario, London, ON N6A 5B7,
Canada
E-mail address: schebolu@uwo.ca
Department of Mathematics, University of Western Ontario, London, ON N6A 5B7,
Canada
E-mail address: jdc@uwo.ca
Department of Mathematics, University of Western Ontario, London, ON N6A 5B7,
Canada
E-mail address: minac@uwo.ca