FREYD'S GENERATING HYPOTHESIS WITH ALMOST SPLIT
SEQUENCES
JON F. CARLSON, SUNIL K. CHEBOLU, AND J'AN MIN'A~C
Abstract.Freyd's generating hypothesis for the stable module category of*
* a non-
trivial finite group G is the statement that a map between finitely gene*
*rated kG-
modules that belongs to the thick subcategory generated by k factors thr*
*ough a
projective if the induced map on Tate cohomology is trivial. In this pa*
*per we
show that Freyd's generating hypothesis fails for kG when the Sylow p-su*
*bgroup of
G has order at least 4 using almost split sequences. By combining this w*
*ith our
earlier work, we obtain a complete answer to Freyd's generating hypothes*
*is for the
stable module category of a finite group. We also derive some consequenc*
*es of the
generating hypothesis.
1. Introduction
The second and third authors have studied the generating hypothesis (GH) and*
* re-
lated questions for the stable module category of a finite group in a series of*
* papers
[5, 2, 6, 7] with Benson and Christensen, and have obtained many partial result*
*s. In
this paper we give a complete solution to the generating hypothesis for the sta*
*ble mod-
ule category of a finite group. We begin by recalling the statement of the GH. *
*Let G be
a finite group and let k be a field of characteristic p. We will work in the st*
*able module
category stmod(kG) of kG. Recall that this is the tensor triangulated category *
*obtained
from the category of finitely generated left kG-modules by factoring out the pr*
*ojective
modules. In the stable module category, Hom_kG(A, B) stands for the space of ma*
*ps be-
tween modules A and B, and will denote the translation functor. Loosely speak*
*ing,
the generating hypothesis claims that if a module L generates a subcategory the*
*n the
functor Hom_kG( *L, -) detects trivial maps (maps that factor through a project*
*ive) in
the subcategory. We make this precise in the case that is of interest to us, na*
*mely the
subcategory generated by the trivial representation k. Let thickG(k) denote the*
* thick
subcategory generated by k. That is, the smallest full subcategory of stmod(kG)*
* that
contains the trivial module k and is closed under exact triangles and direct su*
*mmands.
The Generating Hypothesis*(GH) for a group ring kG is the statement that the Ta*
*te
cohomology functor ^H(G, -) ~=Hom_kG( *k, -)
*
thickG(k) -! ^H(G, k)-modules
*
M 7! ^H(G, M)
____________
Date: June 12, 2008.
2000 Mathematics Subject Classification. Primary 20C20, 20J06; Secondary 55P*
*42.
Key words and phrases. Tate cohomology, generating hypothesis, stable module*
* category, ghost
map, almost split sequence.
The first author is partially supported by a grant from the NSF and the thir*
*d author is supported
from the NSERC.
2 JON F. CARLSON, SUNIL K. CHEBOLU, AND J'AN MIN'A~C
*
detects trivial maps in thickG(k), i.e., the Tate cohomology functor ^H(G, -) o*
*n thickG(k)
is faithful. Note that when G is a p-group, then thick(k) is the entire stable*
* module
category stmod(kG).
It is shown in [2] that the GH holds for kP when P is a p-group if and only i*
*f P is C2
or C3. It is natural then to conjecture that for an arbitrary finite group G th*
*e GH holds
for kG if and only if the Sylow p-subgroup of G is either C2 or C3. In [6] this*
* conjecture
has been proved using block theory for groups which have periodic cohomology. I*
*n this
paper, we show that the GH fails for groups with non-periodic cohomology. In fa*
*ct, we
show that the GH fails whenever the Sylow p-subgroup of G has order at least 4.*
* Thus
we have a complete solution to the Freyd's generating hypothesis in the stable *
*module
category:
Theorem 1.1. Let G be a finite group and let k be a field of characteristic p t*
*hat divides
the order of G. Then the GH holds for kG if and only if a Sylow p-subgroup of G*
* is
isomorphic to either C2 or C3.
Our main tool in showing the failure of the GH is the Auslander-Reiten theor*
*y of
almost split sequences. In contrast with our earlier counting techniques [6] wh*
*ich only
show the existence of a counter-example to the GH, our current treatment with a*
*l-
most split sequences has the advantage that it produces an explicit and simple *
*counter-
example whenever the GH fails.
For the interested reader we mention that there has been great interest in g*
*enerating
hypothesis in other triangulated categories including the stable homotopy categ*
*ory of
spectra [8] where it originated, but also in the derived categories of rings [1*
*0, 9].
Throughout the paper G will denote a finite non-trivial group, and k will be*
* a field
of characteristic p which divides the order of G. We use the standard facts abo*
*ut the
stable module category of kG which can be found in [3].
The first author thanks the Alexander von Humboldt Foundation for support and
the RWTH in Aachen for their hospitality while part of this paper was written.
2.The generating hypothesis
A ghost in thick(k) is a map between kG-modules in thick(k) that induces the*
* trivial
map in Tate cohomology. Our goal is to show that there are non-trivial ghosts i*
*n thick(k)
whenever the Sylow p-subgroup of G has order at least 4. Our main tool in showi*
*ng the
existence of these non-trivial ghosts is an almost split sequence (a.k.a. Ausla*
*nder-Reiten
sequences) which we now define.
A short exact sequence
ffl: 0 -! A -! B -! C -! 0
of finitely generated kG-modules is an almost split sequence if A and C are ind*
*ecom-
posable kG-modules, and ffl is a non-split sequence with the property that ever*
*y map
M ! C which is not a split epimorphism factors through the middle term B [1]. I*
*t is
a theorem of Auslander and Reiten that given any finitely generated indecomposa*
*ble
non-projective kG-module C, there exists a unique (up to isomorphism of short e*
*xact
sequences) almost sequence terminating in C. Moreover, for symmetric algebras,*
* the
first term A of the almost split sequence ending in C is shown to be isomorphic*
* to 2C.
The theorem that we are now interested in is the following.
FREYD'S GENERATING HYPOTHESIS WITH ALMOST SPLIT SEQUENCES 3
Theorem 2.1. Let M and N be two non-projective indecomposable modules in stmod(*
*kG)
such that N 6~= i(M) for any i. Then there exist a non-trivial map OE: N -! N *
*in
stmod(kG) such that the induced map
* _____// *
OE* : dExtkG(M, N) EdxtkG(M, N)
is the zero map.
Proof.Consider the almost split sequence
0 -! 2N -! B -! N -! 0
ending in N. This short exact sequence represents a distinguished triangle
2N -! B -! N -OE! N
in the stable category. We will show that the map OE: N - ! N has the desired
properties. Almost split sequences are, by definition, non-split short exact se*
*quences,
and therefore the boundary map OE in the above triangle must be a non-trivial m*
*ap in
the stable category.
The next thing to be shown is that the map OE: N -! N induces the zero map on
i
the functors Hom_kG( iM, -) ~=dExt(M, -) for all i. To this end, consider any *
*map
f : iM -! N. We have to show that the composite
iM -f! N -OE! N
is trivial in the stable category. Consider the following diagram
_iM_
____
_____|f|_____
""_____|fflfflOE
2N _____//B_____//N______// N
where the bottom row is our distinguished triangle. The map f : iM -! N cannot
be a split epimorphism by the given hypothesis, therefore by the defining prope*
*rty of
an almost split sequence, the map f factors through the middle term B as shown *
*in
the above diagram. Since the composite of any two successive maps in a distingu*
*ished
triangle is zero, the composite OE O f is also zero by commutativity. So we are*
* done.
Corollary 2.2. If there is an indecomposable non-projective module N in thickG *
*(k)
which is not isomorphic to ik for any i, then the GH fails for kG.
* *
Proof.We apply the previous theorem with M = k. Since dExtkG(k, N) ~=^H(G, N), *
*the
existence of an indecomposable nonprojective module N in thickG(k) with the pro*
*perty
that N 6~= nk for any n implies (by the above theorem) the existence*of a non-t*
*rivial
map OE : N - ! N in thickG(k) such that the induced map of ^H(G, k)-modules,
^H*(G, N) -! ^H*(G, N), is the zero map. In other words OE: N -! N is a non-tr*
*ivial
ghost in thick(k). Therefore the GH fails for kG.
It is not hard to produce modules that satisfy the conditions laid out in th*
*e above
corollary. Specifically we have the following.
Theorem 2.3. Suppose that the Sylow p-subgroup P of G has order is at least 4. *
*Then
the GH fails for kG.
4 JON F. CARLSON, SUNIL K. CHEBOLU, AND J'AN MIN'A~C
Proof.We divide the proof into two cases. First suppose that H *(G, k) is peri*
*odic,
implying that the P isneither cyclic or quaternion. By Tate duality, there must*
* exist a
nonzero element i 2 ^H(G, k) for some n which is an odd integer (not divisible *
*by 2).
Then we have an exact sequence
Ei: 0 ____//_Li___//_ nki__//_k___//_0
where i in the sequence is a cocycle representing the cohomology element i, and*
* Li
is the kernel of i. In the case that P is cyclic, nk when restricted to P is *
*a direct
sum of a projective module and a uniserial module of dimension |P | - 1. Becaus*
*e the
cohomology element i is not zero, its restriction to P is not zero and the rest*
*riction of
Li is the direct sum of a projective module and a uniserial module of dimension*
* |P | - 2.
Consequently, Li must be indecomposable and moreover, because its dimension is *
*not
1 or -1 modulo |P |, Li is not isomorphic to nk for any n. So by corollary 2.2*
*, the GH
fails.
If P is a quaternion group, then again by Tate duality, we can assume that n *
*is positive
and congruent to -1 modulo 4. When we restrict Li to P we get the direct sum of*
* a
projective module and a copy of Rad( -1k) which is easily seen to be indecompos*
*able.
So by the same argument as before, the GH fails.
Now suppose that the cohomology ring H*(G, k) is not periodic. This means tha*
*t the
maximal ideal spectrum VG (k) of the cohomology ring H*(G, k) has Krull dimensi*
*on at
least two. This time we choose i 2 Hn(G, k) with the property that n > 0 and i *
*not
nilpotent. Then we construct the sequence Ei and the module L = Li exactly as *
*in
the periodic case. The support variety of the module L is equal to VG (L) = VG *
*(i), the
collection of maximal ideals that contain the element i [4]. (Note here that we*
* do not
need to assume that the field k is algebraically closed, though the proof of th*
*e statement
about the variety of Li requires extending the scalars to the algebraically clo*
*sed case.)
Because i is not nilpotent, VG (L) is a proper subvariety of VG (k) and the sam*
*e statement
will hold for any direct summand of L. Hence, if U is any indecomposable direct
summand of L, we must have that U 6~= n(k) for all n, simply because the support
varieties are different. Consequently by Corollary 2.2, the GH cannot hold for *
*kG.
3.Consequences of the GH
We now derive some consequences of the GH. First of all, we would like to po*
*int out
that our main result implies that the GH for kG depends only on the characteris*
*tic of
the field k. In other words, if k1 and k2 are two fields of characteristic p wh*
*ich divides
the order of G, then the GH holds for k1G if and only if it holds for k2G. It i*
*s not clear
how one would prove this fact directly.
The dual generating hypothesis is also a natural problem to ask. That is, in*
*stead of
using the (covariant) Tate cohomology functor Hom_kG( *k, -), we can use the (c*
*on-
travariant) dual Tate cohomology functor Hom_kG(-, *k), and ask if this contra*
*variant
functor detects trivial maps in thickG(k). The duality functor M 7! M* sets a t*
*ensor
triangulated equivalence between thickG(k) and its opposite category thickG(k)o*
*pp. In
particular, the GH holds for kG if and only if the dual GH holds for kG. Combin*
*ing
this with our previous results we have:
Theorem 3.1. Then the following assertions are equivalent.
FREYD'S GENERATING HYPOTHESIS WITH ALMOST SPLIT SEQUENCES 5
(1) The Sylow p-subgroup of G is either C2 or C3.
(2) The functor Hom_kG( *k, -) is faithful on thickG(k).
(3) The functor Hom_kG(-, *k) is faithful on thickG(k).
(4) thickG(k) consists of finite direct sums of suspensions of k.
Proof.(1) =) (2) is shown in [6], (2) =) (4) is shown in Corollary 2.2, (4) =) *
*(2) is
trivial, and (2) =) (1) is shown in Theorem 2.3. Finally the equivalence of sta*
*tements
(2) and (3) is shown in the paragraph preceding this theorem.
Our final result is motivated by a result of Peter Freyd [8] which states th*
*at if the
stable homotopy functor on the category of finite spectra is faithful then it i*
*s also full.
We now prove the analogue of this statement for the stable module category. Th*
*is
generalizes Theorem 3.3 of [2] where we established the same result for p-group*
*s.
Theorem 3.2. If the GH holds for kG,*then the Tate cohomology functor from thic*
*kG(k)
to the category of modules over ^H(G, k) is also full.
Proof.If the GH holds for kG, then from the equivalence (2 () 4) of Theorem 3.1*
* we
know that thickG(k) is made*up of finite direct*sums of suspensions of k. In pa*
*rticular,
for each M in thickG(k), ^H(G, M) is a free ^H(G, k)-module of finite rank. It *
*follows
that the induced map
* *
Hom_kG(M, X) -! Hom ^H*(G,k)(^H (G, M), ^H(G, X))
is an isomorphism for all kG-modules X. Since*M was an arbitrary kG-module in
thickG(k), we have shown that the functor ^H(G, -) is full, as desired.
The class of groups for which the GH has an affirmative answer (namely, grou*
*ps
whose Sylow p-subgroup is C2 or C3) although small, has some interesting candid*
*ates.
This includes interesting simple groups. For example, it has the smallest simpl*
*e group
A5 of order 60 in characteristic 3. Another interesting example in characterist*
*ic 3 is the
smallest Janko group J1 which has order 175560. In fact, J1 is the only sporadi*
*c simple
group for which the GH holds.
4. Symmetric algebras
We end the paper by pointing out that much of the work here applies for any *
*finite
dimensional symmetric algebra. Note that if A is such an algebra, we have almos*
*t split
sequences ending in any finitely generated indecomposable non-projective A-modu*
*le N:
0 -! 2N -! E -! N -! 0.
Therefore our proofs generalize immediately to give the following result.
Theorem 4.1. Let A be a finite dimensional symmetric k-algebra. Fix a finitely *
*gen-
erated non-projective indecomposable A-module L. Then the following are equival*
*ent.
(1) The functor Hom_A( *L, -) is faithful on thick(L), the thick subcategory*
* gener-
ated by L in stmod(A).
(2) thick(L) consists of modules isomorphic to finite direct sums of suspens*
*ions of
L.
We sketch a proof and leave the details to the reader.
6 JON F. CARLSON, SUNIL K. CHEBOLU, AND J'AN MIN'A~C
Proof.Clearly (2) implies (1). For the other direction, suppose that (2) fails.*
* Then we
have a an indecomposable non-projective object N in thick(L) such that N AE iL*
* for
any i. Then the map N -! N corresponding to the almost split sequence ending in
N is shown (exactly as before) to be a non-trivial map that is invisible to the*
* functor
Hom_A( *L, -). This shows that (1) fails. So we are done.
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Department of Mathematics, University of Georgia, Athens, GA 30602.
E-mail address: jfc@math.uga.edu
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: minac@uwo.ca