THE GENERATING HYPOTHESIS FOR THE STABLE MODULE
CATEGORY OF A p-GROUP
DAVID J. BENSON, SUNIL K. CHEBOLU, J. DANIEL CHRISTENSEN, AND J'AN MIN'A~C
Abstract.Freyd's generating hypothesis, interpreted in the stable module*
* cate-
gory of a finite p-group G, is the statement that a map between finite-d*
*imensional
kG-modules factors through a projective if the induced map on Tate cohom*
*ology
is trivial. We show that Freyd's generating hypothesis holds for a non-t*
*rivial finite
p-group G if and only if G is either C2 or C3. We also give various cond*
*itions which
are equivalent to the generating hypothesis.
1. Introduction
The generating hypothesis (GH) is a famous conjecture in homotopy theory due*
* to
Peter Freyd [6]. It states that a map between finite spectra that induces the z*
*ero map
on stable homotopy groups is null-homotopic. If true, the GH would reduce the s*
*tudy of
finite spectra X to the study of their homotopy groups ss*(X) as modules over s*
*s*(S0).
Therefore it stands as one of the most important conjectures in stable homotopy*
* theory.
This problem is notoriously hard; despite serious efforts of homotopy theorists*
* over the
last 40 years, the conjecture remains open, see [4, 5]. Keir Lockridge [9] show*
*ed that
the analogue of the GH holds in the derived category of a commutative ring R if*
* and
only if R is a von Neumann regular ring (a ring over which every R-module is fl*
*at).
More recently, Hovey, Lockridge and Puninski have generalised this result to ar*
*bitrary
rings [7]. Lockridge's result [9] applies to any tensor triangulated category w*
*here the
graded ring of self maps of the unit object is graded commutative and is concen*
*trated in
even degrees. Note that this condition is not satisfied by the stable homotopy *
*category
of spectra. So in order to better understand the GH for spectra, we formulate a*
*nd solve
the analogue of Freyd's GH in the stable module category of a finite p-group. H*
*ere the
ring of self maps of the unit object (the trivial representation k) is non-zero*
* in both
even and odd degrees.
To set the stage, let G be a non-trivial finite p-group and let k be a field*
* of characteris-
tic p. Consider the stable module category StMod(kG) of G. It is the category o*
*btained
from the category of kG-modules by killing the projectives. The objects of StMo*
*d(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. StMod (kG) has the structur*
*e of a
tensor triangulated category with the trivial representation k as the unit obje*
*ct and
as the loop (desuspension) functor. The category stmod(kG) is defined similarly*
* using
the finite-dimensional kG-modules. A key fact [1] is that the Tate cohomology g*
*roups
____________
Date: 28 November 2006.
2000 Mathematics Subject Classification. Primary 20C20, 20J06; Secondary 55P*
*42.
Key words and phrases. Generating hypothesis, stable module category, ghost *
*map.
2 DAVID J. BENSON, SUNIL K. CHEBOLU, J. DANIEL CHRISTENSEN, AND J'AN MIN'A~C
can be described as groups of morphisms in StMod (kG): Hbi(G, M) ~=Hom_( ik, M).
In this framework, the GH for kG is the statement that a map OE: M ! N between
finite-dimensional kG-modules is trivial in stmod(kG) if the induced map in Tat*
*e coho-
mology Hom_( ik, M) ! Hom_( ik, N) is trivial for each i. Maps between kG-modul*
*es
that are trivial in Tate cohomology will be called ghosts. It is shown in [2] t*
*hat there
are no non-trivial ghosts in StMod(kG) if and only if G is cyclic of order 2 or*
* 3. The
methods in [2] do not yield ghosts in stmod(kG). In this paper, we use inductio*
*n to
build ghosts in stmod(kG). Our main theorem says:
Theorem 1.1. Let G be a non-trivial finite p-group and let k be a field of char*
*acteristic
p. There are no non-trivial maps in stmod(kG) that are trivial in Tate cohomolo*
*gy if
and only if G is either C2 or C3. In other words, the generating hypothesis hol*
*ds for
kG if and only if G is either C2 or C3.
Note that the theorem implies that the GH for p-groups does not depend on the
ground field k, as long as its characteristic divides the order of G.
We now explain the strategy of the proof of our main theorem. We begin by sh*
*owing
that whenever the GH fails for kH, for H a subgroup of G, then it also fails fo*
*r kG.
We then construct non-trivial ghosts over cyclic groups of order bigger than 3 *
*and over
Cp Cp. It can be shown easily that the only finite p-groups that do not have *
*one of
these groups as a subgroup are the cyclic groups C2 and C3. And for C2 and C3 we
show that the GH holds.
For a general finite group G, the GH is the statement that there are no non-*
*trivial
ghosts in the thick subcategory generated by k. When G is not a finite p-group,*
* our
argument does not necessarily produce ghosts in thick(k) and the GH is an open *
*problem.
In the last section we give conditions on a finite p-group equivalent to the*
* GH. One of
them says that the GH holds for kG if and only if the category stmod(kG) is equ*
*ivalent
to the full subcategory of finite coproducts of suspensions of k. We also show *
*that if
the GH holds for a finite p-group, then the Tate cohomology functor bH*(G, -) f*
*rom
stmod(kG) to the category of graded modules over the ring bH*(G, k) is full.
Throughout we assume that the characteristic of k divides the order of the f*
*inite group
G. For example, when we write kC3, it is implicitly assumed that the characteri*
*stic of
k is 3. We denote the desuspension of M in StMod(kG) by (M), or by G (M) when
we need to specify the group in question. All modules are assumed to be left mo*
*dules.
2. Proof of the main theorem
Suppose H is a subgroup of G. A natural question is to ask how the truth or *
*falsity
of the GH for H is related to that for G. We begin by addressing this question.
Proposition 2.1. Let H be a subgroup of a finite group G. If OE is a ghost in s*
*tmod(kH),
then OE"G is ghost in stmod(kG). Moreover, if OE is non-trivial in stmod(kH), t*
*hen so
is OE"G in stmod(kG).
Proof.It is well known that the restriction Res_GHand induction Ind_GHfunctors *
*form
an adjoint pair of exact functors; see [8, Cor. 5.4] for instance. Therefore, *
*for any
kH-module L, we have a natural isomorphism
Hom_kH(( iGk)#H , L) ~=Hom_kG( iGk, L"G).
GENERATING HYPOTHESIS FOR THE STABLE MODULE CATEGORY 3
But since ( iGk)#H ~= iHk in stmod(kH), the above isomorphism can be written as
Hom_kH ( iHk, L) ~=Hom_kG( iGk, L"G).
The proposition now follows from the naturality of this isomorphism. The second*
* state-
ment follows from the observation that OE is a retract of OE"G#H .
Proposition 2.1 implies that if G is a finite p-group, then the GH fails for *
*kG whenever
it fails for a subgroup of G.
We now state two lemmas which will be needed in proving our main theorem.
Lemma 2.2. Let G be a finite p-group and let x be a central element in G. Then *
*for
any kG-module M, the map x - 1: M ! M is a ghost.
Proof.Since x is a central element, multiplication by x - 1 defines a kG-linear*
* map.
We have to show that for all n and all maps f : nk ! M, the composition nk -f!
M x-1-!M factors through a projective. To this end, consider the commutative di*
*agram
f
nk _____//M
x-1 || |x-1|
fflffl| fflffl|
nk __f__//M.
Note that x - 1 acts trivially on k, so functoriality of shows that the left *
*vertical
map is stably trivial. By commutativity of the square, the desired composition *
*factors
through a projective.
Lemma 2.3. Let G be a finite p-group and let H be a non-trivial proper normal s*
*ubgroup
of G. If x is a central element in G - H, then multiplication by x - 1 on kH "G*
* is a
non-trivial ghost, where kH is the trivial kH-module. In particular, the GH f*
*ails for
k(Cp Cp).
Proof.Since kH "G#H is a trivial kH-module, non-triviality of x - 1 is easily s*
*een by
restricting to H. The fact that x - 1 is a ghost follows from Lemma 2.2. The *
*last
statement follows because kH "G is finite-dimensional.
Proof of Theorem 1.1.If G ~= C2 and chark = 2, then kC2 ~=k[x]=(x2), so by the
structure theorem for modules over a PID it is clear that every kG-module is st*
*ably
isomorphic to a sum of copies of k. Similarly, if G ~=C3 and chark = 3, then on*
*e sees
that every kG-module is stably isomorphic to a sum of copies of k and (k). It *
*follows
that there are no non-trivial ghosts between finite-dimensional kG-modules if G*
* is either
C2 or C3.
Now suppose that G is not isomorphic to C2 or C3. It suffices to show that i*
*n these
cases the GH fails for some subgroup of G. It is an easy exercise to show that *
*if G is
not isomorphic to C2 or C3, then G either has a cyclic subgroup of order at lea*
*st four,
or a subgroup isomorphic to Cp Cp for some prime p. In Lemma 2.3 we have seen
that the GH fails for k(Cp Cp). We will be done if we can show that the GH fa*
*ils for
cyclic groups of order at least 4.
So let G be a cyclic group of order at least 4. Let oe be a generator for G *
*and let M
be a cyclic module of length two generated by U, so we have (oe - 1)2U = 0. Con*
*sider
4 DAVID J. BENSON, SUNIL K. CHEBOLU, J. DANIEL CHRISTENSEN, AND J'AN MIN'A~C
the map h: M ! M which multiplies by oe - 1:
UoOOOOO U oO
oe-1| OOhOOO |oe-1
O| OO''OO|
o o .
It is not hard to see that h is non-trivial, i.e., that it does not factor thro*
*ugh the
projective cover of M; this is where we use the hypothesis |G| 4. The fact th*
*at h is a
ghost follows from Lemma 2.2.
3. Conditions equivalent to the generating hypothesis
Theorem 3.1. The following are equivalent statements for a non-trivial finite p*
*-group
G.
(1) G is isomorphic to C2 or C3.
(2) There are no non-trivial ghosts in stmod(kG). That is, the GH holds for *
*kG.
(3) There are no non-trivial ghosts in StMod(kG).
(4) stmod(kG) is equivalent to the full subcategory of the collection of fin*
*ite coprod-
ucts of suspensions of k.
(5) StMod(kG) is equivalent to the full subcategory of arbitrary coproducts *
*of sus-
pensions of k.
Proof.We have already seen that the statements (2) and (4) are equivalent to (1*
*).
The implications (5) ) (3) ) (2) are obvious. So we will be done if we can show
that (1) ) (5). This follows immediately from the following more general fact,*
* due
to Crawley and J'onsson [3], which was also proved independently by Warfield [1*
*0]. It
states that if G has finite representation type (i.e., the Sylow p-subgroups ar*
*e cyclic),
then every kG-module is a direct sum of finite-dimensional kG-modules.
We now state a dual version of the previous theorem. 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.
Theorem 3.2. The following are equivalent statements for a non-trivial finite p*
*-group
G.
(1) G is isomorphic to C2 or C3.
(20)There are no non-trivial dual ghosts in stmod(kG).
(30)There are no non-trivial dual ghosts in StMod(kG).
(40)stmod(kG) is equivalent to the full subcategory of the collection of fin*
*ite products
of suspensions of k.
(50)StMod(kG) is equivalent to the full subcategory of retracts of arbitrary*
* products
of suspensions of k.
Proof.Every finite-dimensional kG-module M is naturally isomorphic to its doubl*
*e dual
M**. Therefore, the exact functor M 7! M* gives a tensor triangulated equivale*
*nce
between stmod(kG) and its opposite category. This shows that (20) , (2). In a*
*ny
additive category finite coproducts and finite products are the same, therefore*
* (40) ,
GENERATING HYPOTHESIS FOR THE STABLE MODULE CATEGORY 5
(4). Thus, statements (1), (20), and (40) are equivalent. We will be done if we*
* can show
that (50) ) (30) ) (1) ) (50).
(50) ) (30): Fix an arbitrary kG-module M. We have to show that there are no
non-trivial dual ghosts out of M. Consider the full subcategory of all modules *
*X such
that there is no non-trivial dual ghost from M to X. This subcategory clearly c*
*ontains
arbitrary products of suspensions of k and is closed under taking retractions. *
* So by
assumption the subcategory has to be StMod(kG).
(30) ) (1): (30) clearly implies (20). But we have already observed that (20)*
* ) (2) )
(1).
(1) ) (50): We know that (1) ) (5). It remains to show that (5) ) (50). Let M*
* be
any kG-module. By (5), M is a coproduct sk of suspensions of k. We will comp*
*lete
the proof by showing that the canonical map
M Y
sk -! sk
is a split monomorphism in StMod(kG). By (5), the fibre F of this map is a copr*
*oduct
tk of suspensions of k. Since the objects tk are compact, one can show that*
* the
map F ! sk is zero and therefore the desired splitting exists.
We end with a final observation. In the stable homotopy category of spectra,*
* the
GH says that the stable homotopy functor ss*(-) from the category of finite spe*
*ctra to
the category of graded modules over the homotopy ring ss*(S0) of the sphere spe*
*ctrum
is faithful. Freyd showed [6] that if the GH is true, then ss*(-) is also full*
*. So it is
natural to ask if the same is true in other algebraic settings in which the GH *
*is being
studied. Very recently, Hovey, Lockridge and Puninski [7] have given an exampl*
*e of
ring R for which the homology functor H*(-) from the category of perfect comple*
*xes
of R-modules to the category of graded R-modules is faithful, but not full. It*
* turns
out that from this point of view, the stable module category of a group behaves*
* more
like the stable homotopy category of spectra than the derived category of a rin*
*g. More
precisely, we have the following result.
Theorem 3.3. Let G be a finite p-group and let k be a field of characteristic p*
*. If the
GH holds for G, then the functor bH*(G, -) from stmod(kG) to the category of gr*
*aded
modules over the graded ring bH*(G, k) is full.
Proof.We know by Theorem 1.1 that G has to be either C2 or C3. Therefore every
finite-dimensional kG-module M is stably isomorphic to a finite sum of suspensi*
*ons of
k. In particular, bH*(G, M) is a free bH*(G, k)-module of finite rank. It follo*
*ws that the
induced map
Hom_kG(M, X) -! Hom bH*(G,k)(Hb*(G, M), bH*(G, X))
is an isomorphism for all kG-modules X. Since M was an arbitrary finite-dimensi*
*onal
kG-module, we have shown that the functor bH*(G, -) is full, as desired.
6 DAVID J. BENSON, SUNIL K. CHEBOLU, J. DANIEL CHRISTENSEN, AND J'AN MIN'A~C
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Department of Mathematical Sciences, University of Aberdeen, Meston Building,
King's college, Aberdeen AB24 3UE, Scotland, UK
E-mail address: \/b\e/n\s/o\n/d\j/\ (without the slashes) at maths dot abdn *
*dot ac dot uk
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