GROUPS WHICH DO NOT ADMIT GHOSTS
SUNIL K. CHEBOLU, J. DANIEL CHRISTENSEN, AND J'AN MIN'A~C
Abstract.A ghost in the stable module category of a group G is a map bet*
*ween
representations of G that is invisible to Tate cohomology. We show that *
*the only
non-trivial finite p-groups whose stable module categories have no non-t*
*rivial
ghosts are the cyclic groups C2 and C3. We compare this to the situation*
* in
the derived category of a commutative ring. We also determine for which *
*groups
G the second power of the Jacobson radical of kG is stably isomorphic to*
* a
suspension of k.
1.Introduction
Let G be a p-group and let k be a field of characteristic p. A natural home
for the cohomology of modular representations of G is the stable module category
StMod (kG) of G. It is the category obtained from the category of left kG-modul*
*es by
killing the projectives. It has the structure of a tensor triangulated category*
* with the
trivial representation k as the unit object and as the loop (desuspension) fu*
*nctor.
The space of morphisms from M to N in StMod (kG) is denoted Hom_kG(M, N)
and consists of the kG-module homomorphisms modulo those that factor through a
projective module. Note that a map of kG-modules that factors through a project*
*ive
clearly induces the zero map in Tate cohomology. A natural question is whether *
*the
converse is true. Namely, if f :M ! N is a map of kG-modules such that the indu*
*ced
map in Tate cohomology
Hom_kG( *k, M) -! Hom_kG( *k, N)
is trivial, then does f factor through a projective module? Equivalently, is ev*
*ery such
f trivial in the stable module category? In this paper we investigate the group*
*s G
for which the above question always has an affirmative answer. Our main theorem
states:
Theorem 1.1. Let G be a non-trivial finite p-group and let k be a field of char*
*ac-
teristic p. All maps which are trivial in Tate cohomology factor through a proj*
*ective
kG-module if and only if G is either C2 or C3.
A map between kG-modules is said to be a ghost if the induced map in Tate
cohomology Hom_kG( *k, -) is trivial. Using this terminology, the main theorem
says that the only finite p-groups for which all ghosts are trivial in the stab*
*le module
category are C2, C3 and the trivial group. It is interesting and surprising to*
* note
that the answer is so simple.
____________
Date: January 5, 2007.
2000 Mathematics Subject Classification. Primary 20C20, 20J06; Secondary 55P*
*42.
Key words and phrases. Ghost map, stable module category, derived category, *
*Jennings' theorem,
generating hypothesis.
1
2 SUNIL K. CHEBOLU, J. DANIEL CHRISTENSEN, AND J'AN MIN'A~C
We now explain the strategy of the proof. We construct a (weakly) universal g*
*host
out of a given kG-module M, and from this, we deduce that all ghosts out of M v*
*anish
if and only if M is a retract of a direct sum of suspensions of k. In particula*
*r, when
M is a finite-dimensional indecomposable module in StMod(kG), all ghosts out of*
* M
vanish if and only if M ~= ik in StMod (kG) for some i. Therefore, we determine
the finite p-groups G which admit an indecomposable kG-module that is not stably
isomorphic to any ik. A formula of Jennings [7], which computes the nilpotency
index of the Jacobson radical of kG, plays an important role in this investigat*
*ion.
The proof (sketched above) of our main result yields some interesting additio*
*nal
results. For instance, Proposition 3.7 characterises finite p-groups G for whic*
*h the sec-
ond power of the Jacobson radical of kG is isomorphic in the stable module cate*
*gory
to a suspension of k. In addition, the formal material in Section 2 has implica*
*tions
in other settings, as we illustrate in Section 4.
We now explain how we were led to these results. An old conjecture of Peter
Freyd [6] in homotopy theory called the generating hypothesis (GH) claims that a
map OE: X ! Y between finite spectra that induces the zero map on stable homoto*
*py
groups is null-homotopic. It is one of the most important unsolved problems in
stable homotopy theory. In order to gain some insight into this deep problem, i*
*t is
natural to examine its analogues in algebraic settings such as the derived cate*
*gory of
a commutative ring and the stable module category of a finite group. The GH in *
*the
derived category D(R) of a ring R is the statement that a map OE: X ! Y between
perfect complexes that induces the zero map in homology is chain-homotopic to t*
*he
zero map. Keir Lockridge [8] showed that the GH holds in the derived category o*
*f a
commutative ring R if and only if R is a von Neumann regular ring.
In the stable module category, the GH is the statement that a map OE: M ! N
between finite-dimensional kG-modules is trivial in StMod(kG) if the induced ma*
*p in
Tate cohomology is trivial. This paper takes a first step towards the GH by stu*
*dying
the variant in which the modules are not assumed to be finite-dimensional. The
theorem above implies that the GH is true for C2 and C3, but does not answer the
question for other groups, since it does not guarantee the existence of ghosts *
*between
finite-dimensional modules. Somewhat surprisingly, this variant of the GH turns*
* out
to be equivalent to the GH. Indeed, motivated by the main result of this paper,
we have shown in joint work with Dave Benson [3] that the GH holds for a non-
trivial finite p-group G if and only if G is either C2 or C3. This result can b*
*e used to
deduce the main theorem stated above. However, the methods used in the two pape*
*rs
are completely different. In this paper, we use techniques inspired from homoto*
*py
theory and classical group theory, while the techniques in [3] are more represe*
*ntation
theoretic, relying heavily on the induction and restriction. Moreover, we shou*
*ld
emphasise that the methods in [3] do not give the additional results mentioned *
*above.
We end the introduction by posing a riddle to the reader: Which finite p-grou*
*ps
are like a finite product of fields? We solve this riddle in the last section u*
*sing a result
of Lockridge, which is an analogue of our main theorem for the derived category*
* of
a commutative ring. Thus the riddle sets a context for our main theorem, both in
representation theory and commutative algebra.
GROUPS WHICH DO NOT ADMIT GHOSTS 3
2.Ghosts in triangulated categories
Let T be a triangulated category which admits arbitrary coproducts and let S*
* be
a distinguished object. (If T is tensor triangulated, we always take S to be th*
*e unit
object of T .) If X and Y are objects in T , then [X, Y ]* will denote the grad*
*ed abelian
group of maps from X to Y , and ss*(X) will stand for [S, X]*. A map OE: M ! N *
*in
T is a ghost if the induced map
ss*(M) OE*-!ss*(N)
is trivial. We say that OE: M ! N is a universal ghost if OE is a ghost and if *
*every ghost
out of M factors through OE. (Such a map should technically be called weakly un*
*iversal
because we don't assume the factorisation to be unique.) We begin by showing the
existence of a universal ghost out of any given object in T . This technique is*
* well-
known in homotopy theory, but we include the details in this section to keep the
paper self-contained.
Let M be an object in T . We assemble all the homogeneous elements of ss*(M)
into a map a
|j|S -! M
j 2 ss*(X)
from a coproduct of suspensions of the unit object, where |j| is the degree of *
*j.
Completing this map to an exact triangle in T , we get
a M
(2.1) |j|S -! M -! UM .
j 2 ss*(X)
Proposition 2.1. The map M :M ! UM is a universal ghost out of M.
Proof.It is clear from its construction that M is a ghost. Now suppose f :M !*
* N
is any ghost. Then for each j in ss*(M), the composite
|j|S -j!M -f! N
is null. Therefore, so is the composite
a f
|j|S -! M -! N.
j 2 ss*(M)
Now (2.1)is an exact triangle and therefore f factors through M .
Proposition 2.1 essentially says that ghosts form part of a projective class*
*. See [5]
for more details.
Recall that an object C in T is said to be compact if the natural map
M a
[C, Xff] -! [C, Xff]
ff ff
is an isomorphism for all set-indexed collections of objects Xffin T . An objec*
*t X
in T is said to be indecomposable if X 6= 0 and a decomposition X ~=A q B in T
implies that either A or B is the zero object.
We say that T has the Krull-Schmidt property if the following two conditions*
* hold:
o Each compact object in T can be decomposed uniquely into indecomposable
objects.
o The distinguished object S in T is compact and indecomposable.
4 SUNIL K. CHEBOLU, J. DANIEL CHRISTENSEN, AND J'AN MIN'A~C
Our next proposition characterises the objects in T out of which all ghosts van*
*ish.
Proposition 2.2. Let T be a triangulated category which admits arbitrary coprod*
*ucts
and let S be a distinguished object. Then the following are equivalent for an o*
*bject
M in T :
(1) All ghosts out of M are trivial.
(2) The universal ghost M :M ! UM is trivial.
(3) M is a retract of a coproduct of suspensions of S.
Moreover, if M is compact, then (3) is equivalent to:
(30)M is a retract of a finite coproduct of suspensions of S.
If M is compact and T has the Krull-Schmidt property, then (3) is equivalent to:
(300)M is a finite coproduct of suspensions of S.
Proof.(1) ) (2) is trivial, for M :M ! UM is itself a ghost. Now to see that *
*(2)
implies (3), suppose the universal ghost M is trivial. That means that (2.1)is*
* a split
triangle. In particular, M is a retract of q iS. So there exists a map M -j! q*
* iS
such that the composite a
M -j! iS -! M
ss*(M)
is the identity in T . Now if M is compact, then j factors through a finite cop*
*roduct.
Therefore M is a retract of a finite coproduct of suspensions of S. If T has *
*the
Krull-Schmidt property, then it follows that M is a finite coproduct of suspens*
*ions
of S. Finally (3) ) (1) is clear.
For reference we record the following corollaries which are immediate from P*
*ropo-
sition 2.2.
Corollary 2.3. Let T be a triangulated category which admits arbitrary coproduc*
*ts
and which has the Krull-Schmidt property, and let S be a distinguished object. *
*If M
is a compact indecomposable object in T such that M AE iS for any i, then the*
*re
exists a non-trivial ghost out of M.
Corollary 2.4. Let T be a triangulated category which admits arbitrary coproduc*
*ts
and let S be a distinguished object. Every ghost in T is trivial if and only if*
* T is the
collection of retracts of coproducts of suspensions of S.
In the next section, we use these corollaries to determine when the stable m*
*odule
category of a finite p-group has no non-trivial ghosts. In the following sectio*
*n, we do
the same for the derived category of a commutative ring.
3. Stable module categories
We begin with some preliminaries. Let G be a finite group and let k be a fie*
*ld. The
stable module category StMod(kG) of G is the category obtained from the categor*
*y of
left kG-modules by killing off the projectives. The space of morphisms from M t*
*o N in
StMod (kG) is denoted Hom_kG(M, N) and consists of the kG-module homomorphisms
modulo those that factor through a projective module. Thus a map in the stable
module category is trivial if and only if it factors through a projective modul*
*e. A key
fact [2] is that the Tate cohomology groups can be described as groups of morph*
*isms
GROUPS WHICH DO NOT ADMIT GHOSTS 5
in StMod(kG): Hbi(G, M) ~=Hom_( ik, M). StMod (kG) has the structure of a tensor
triangulated category, where the trivial representation k is the unit object an*
*d is
the loop (desuspension) functor. ( M is defined to be the kernel of a projectiv*
*e cover
of M. This is well-defined in the stable module category.) We denote by e i(M) *
*the
projective-free part of iM, which is a well-defined kG-module. For more facts *
*about
kG-modules and StMod(kG), we refer the reader to Carlson's excellent lecture no*
*tes
[4].
From now on we work exclusively with finite p-groups and assume that the char-
acteristic of k is p. We begin by proving the easy direction of our main theore*
*m.
Proposition 3.1. If G is either C2 or C3, then StMod(kG) has no non-trivial gho*
*sts.
Proof.The group rings kC2 ~=k[x]=(x2) and kC3 ~=k[x]=(x3) are Artinian principal
ideal rings. It is a fact [9, p. 170] that every module over an Artinian princi*
*pal ideal
ring is a direct sum of cyclic modules. We will use this fact to show that eve*
*ry
projective-free kG-module is a direct sum of suspensions of k. The result will *
*then
follow from Corollary 2.4.
First consider the group C2. By the above fact, we know that every module ov*
*er
the ring k[x]=(x2) is a direct sum of copies of k and k[x]=(x2). In particular,*
* every
projective-free k[x]=(x2)-module is a direct sum of copies of k. Now consider t*
*he group
C3. In this case, the above fact implies that every projective-free k[x]=(x3)-m*
*odule
is a direct sum of copies of k and k[x]=(x2). But note that
e(k) := ker k[x]=(x3) i k ~=k[x]=(x2).
In both cases we have shown that every projective-free kG-module is a direct su*
*m of
suspensions of k. So we are done.
We now collect some facts about finite p-groups that we need in the sequel.
Lemma 3.2. Let G be a finite p-group and let M be a finite-dimensional non-zero
kG-module. Then the invariant submodule MG of M is non-zero. Thus there is
only one simple kG-module, namely the trivial module k. Moreover, if MG is one-
dimensional, then M is indecomposable.
Proof.The proof of the first statement is an easy exercise; see [1, 3.14.1]. Fo*
*r the last
statement, suppose to the contrary that M ~=A B, with A and B non-zero. Then
we have that MP ~=AP BP . This shows that MP is at least two-dimensional, for
by the first part of the lemma, both AP and BP are at least one-dimensional. Th*
*is
contradiction completes the proof.
Lemma 3.3. Let G be a finite p-group and let I be a non-trivial, proper ideal o*
*f kG.
Then I is an indecomposable projective-free kG-module. In particular, the powe*
*rs
Ji(kG) which are non-zero are indecomposable projective-free kG-modules.
Proof.We first show that I is projective-free. If I has a projective submodule,*
* then
since projective modules over finite p-groups are free, that would mean that I *
*should
have k-dimension at least |G|, which is not possible since I is proper. To prov*
*e that
I is indecomposable,Tit suffices to show (by Lemma 3.2) that IG is one-dimensio*
*nal.
Note that IG = I (kG)G . It is easyPto see that (kG)G is the one-dimensional
subspace generated by the norm element g2Gg. We also know from Lemma 3.2
that IG is non-zero. Therefore IG is a one-dimensional submodule.
6 SUNIL K. CHEBOLU, J. DANIEL CHRISTENSEN, AND J'AN MIN'A~C
Lemma 3.4. Let G be a finite p-group. Then, for all integers i, we have
dim(e ik) (-1)i mod |G|.
Proof.Recall that e 1k is 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 th*
*e 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 follows*
* that
dim(e ik) (-1)i modulo |G| for each integer i.
We now introduce a formula of Jennings. Let G be a finite p-group and let J(k*
*G)
be the Jacobson radical of kG. Since kG is a local Artinian ring, it follows th*
*at J(kG)
is nilpotent. So there exists a smallest integer m such that J(kG)m = 0. This i*
*nteger
will be called the nilpotency index of J(kG), and it can be shown to be indepen*
*dent
of the field k. Very closely related to the powers of the Jacobson radical are*
* the
dimension subgroups of G, which we now define. The dimension subgroups Fi of G
are defined by
Fi:= {g 2 G : g - 1 2 Ji(kG)} 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 a formula due to Jennings [7] states that the nilpotency index m of J(kG) *
*is
given by
Xd
(3.1) m = 1 + (p - 1) i ei.
i=1
Moreover, e1 is the minimal number ofPgenerators for G. From the definition of *
*the
numbers ei, it is clear that |G| = p iei.
Proposition 3.5. Let G be a non-trivial finite p-group that is not isomorphic t*
*o C2 or
C3. Then there exists a finite-dimensional indecomposable projective-free kG-mo*
*dule
that is not isomorphic to e ik for any i. In particular, there exists a non-tri*
*vial ghost
in StMod (kG).
Proof.It is well known that there are indecomposable projective-free modules ov*
*er
the Klein four group (C2 C2) which are not of the form e i(k). In fact, every *
*even-
dimensional projective-free indecomposable k(C2 C2)-module has this property.
Such modules are known to exist; see [1, Thm. 4.3.3], for instance. Therefore, *
*we can
assume that G is not any one of the groups C2, C3 and C2 C2.
Consider the module J2(kG), which we denote by J2 for brevity. By Lemma 3.3,
we know that J2 is an indecomposable projective-free kG-module. Let |G| = pn for
some positive integer n. Since the dimension of e ik is +1 or -1 modulo |G| = pn
(see Lemma 3.4), we will be done if we can show that the congruence class mod pn
GROUPS WHICH DO NOT ADMIT GHOSTS 7
of dim(J2) is different from +1 and -1. In fact, we will show that when G is no*
*t one
of the above 3 groups, then
1 < dim(J2) < pn - 1.
Note that J2 ( J. For, otherwise, Nakayama's lemma would imply that J = 0, a
contradiction. Therefore the second inequality is clear. Now we establish the*
* first
inequality. We have two cases to consider here:
Case 1: Suppose dimJ2 = 0. Then the nilpotency index m of J is 2. So by Jenning*
*s'
formula we have
2 = 1 + (p - 1)[e1 + 2e2 + . .+.ded].
This means 1 = (p - 1)[e1 + 2e2 + . .+.ded]. Recall that e1 is the minimal numb*
*er
of generators of G, so e1 > 0. Therefore,Pthe last equation holds if and only i*
*f p = 2,
e1 = 1 and ei= 0 for i 2. Since |G| = p iei, it follows that J2 = 0 if and o*
*nly if
G = C2. But G 6= C2, by assumption. So this case cannot arise.
Case 2: Suppose dimJ2 = 1. That means J2 = k, therefore J(J2) = J(k) = 0. So
the nilpotency index m of J is 3. By Jennings' formula we have
3 = 1 + (p - 1)[e1 + 2e2 + . .+.ded].
This means 2 = (p - 1)[e1+ 2e2+ . .+.ded]. Here there are two possibilities. Ei*
*ther
p = 3, e1 = 1, and ei= 0 for all i 2, or p = 2, e1 = 2, and ei= 0 for all i *
* 2. In
the former, we have |G| = 3, so G ~=C3, and in the latter, |G| = 4 and G is gen*
*erated
by 2 elements, so G ~=C2 C2. But G was assumed to not be one of these groups,
so this case also cannot arise.
Since both cases are ruled out, we have proved the first inequality.
Finally, the last statement of the proposition follows from Corollary 2.3.
The main step of the proof of Proposition 3.5 is essentially the classificati*
*on of
p-groups with nilpotency index at most 3. This is known to be an easy consequen*
*ce
of Jennings' formula, but we have included a proof to keep the paper self-conta*
*ined.
Combining Propositions 3.1 and 3.5, we have proved our main theorem:
Theorem 3.6. Let G be a non-trivial finite p-group and let k be a field of char*
*ac-
teristic p. Then StMod (kG) has no non-trivial ghosts if and only if G is eithe*
*r C2
or C3.
We extract the following interesting result from the proof of Proposition 3.*
*5.
Proposition 3.7. Let G be a non-trivial finite p-group. Then J2(kG) ~= e ik in
Mod (kG) for some i if and only if G is isomorphic to C3 or C2 C2.
Proof.The only if part follows from the proof of Proposition 3.5. For the conve*
*rse,
if G is C3, then every finite-dimensional indecomposable projective-free module*
* (and
hence J2(kG)) is of the form e ik, for some i; see the proof of Proposition 3.1*
*. If G
is C2 C2, then J2(kG) ~=k. This completes the proof of the proposition.
8 SUNIL K. CHEBOLU, J. DANIEL CHRISTENSEN, AND J'AN MIN'A~C
4.Derived categories
Let R be a commutative ring and let D(R) be its (unbounded) derived category.
This has the structure of a tensor triangulated category with R (viewed as a ch*
*ain
complex concentrated in degree zero) as the unit object. Observe that a map OE:*
* X !
Y in D(R) is a ghost if and only if the induced map H*(OE): H*(X) ! H*(Y ) in
homology is zero. The natural question is to characterise commutative rings R f*
*or
which D(R) has no non-trivial ghosts. This has been done by Lockridge in [8]. We
include a proof below for the reader's convenience and also to illustrate the r*
*esults
in Section 2.
Theorem 4.1. [8] Let R be a commutative ring. Then D(R) has no non-trivial
ghosts if and only if R is a finite product of fields.
Proof.Let R be a finite product of fields, F1xF2x. .x.Fl, say. Then every R-mod*
*ule
splits naturally as a direct sum of modules over the subrings Fi. It follows th*
*at D(R)
is equivalent to D(F1)xD(F2)x. .x.D(Fl). Now note that the derived category of a
field F is equivalent to the category of Z-graded F -vector spaces. From this i*
*t follows
that every object in D(R) is a retract of direct sum of suspensions of R. There*
*fore,
by Corollary 2.4, D(R) does not have any non-trivial ghosts. Conversely, suppo*
*se
there are no non-trivial ghosts in D(R). Then it is not hard to see that for e*
*very
pair of R-modules M and N, we have ExtiR(M, N) = 0 for each i > 0. This implies
that every R-module is projective. Therefore R is semi-simple; see [10, Thm. 4.*
*2.2].
Since commutative semi-simple rings are precisely finite direct products of fie*
*lds (by
the Artin-Wedderburn theorem), we are done.
The answer to the riddle posed in the introduction should be now clear to the
reader. The only non-trivial finite p-groups that are like a finite product of *
*fields are
C2 and C3.
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 Jon F. Carlson. Products in negative cohomology. J. Pure A*
*ppl. Algebra,
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*e generating
hypothesis for the stable module category of a p-group. Journal of Algebra,*
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* Calif., 1965),
pages 121-172. Springer, New York, 1966.
[7]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.
[8]Keir Lockridge. The generating hypothesis in the derived category of R-modu*
*les. Journal of
Pure and Applied Algebra, 208(2):485-495, 2007.
[9]D. W. Sharpe and P. V'amos. Injective modules. Cambridge University Press, *
*London, 1972.
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GROUPS WHICH DO NOT ADMIT GHOSTS 9
Department of Mathematics, University of Western Ontario, London, ON, Canada
E-mail address: schebolu@uwo.ca
Department of Mathematics, University of Western Ontario, London, ON, Canada
E-mail address: jdc@uwo.ca
Department of Mathematics, University of Western Ontario, London, ON, Canada
E-mail address: minac@uwo.ca