AUSLANDERREITEN SEQUENCES AS APPETIZERS FOR
HOMOTOPISTS AND ARITHMETICIANS
SUNIL K. CHEBOLU AND J'AN MIN'A~C *
Dedicated to Professor John Labute with admiration, respect and friend*
*ship.
Abstract.We introduce AuslanderReiten sequences for group algebras and *
*give
several recent applications. The first part of the paper is devoted to s*
*ome funda
mental problems in Tate cohomology which are motivated by homotopy theor*
*y. In
the second part of the paper we interpret AuslanderReiten sequences in *
*the context
of Galois theory and connect them to some important arithmetic objects.
Contents
1. Introduction 1
2. AuslanderReiten sequences 2
3. Stable module category 4
4. Krause's proof of the existence of AR sequences 5
5. AuslanderReiten duality 6
6. Representations of V4modules 7
7. Tate cohomology 8
7.1. Modules with finitely generated Tate cohomology 9
7.2. Counterexamples to Freyd's generating hypothesis 9
8. Galois theoretic connections 10
9. Towards the AuslanderReiten translate 2F2 12
10. Generators and relations for 2F2 13
11. The AuslanderReiten sequence ending in F2 15
12. Conclusion 16
References 16
1. Introduction
In the past the second author had an opportunity to discuss with John Labute*
* some
beautiful aspects of modular representation theory and they both came to the co*
*nclusion
that the techniques from modular representation theory could be applied effecti*
*vely to
____________
Date: February 28, 2009.
2000 Mathematics Subject Classification. Primary 20C20, 20J06; secondary 55P*
*42.
Key words and phrases. AuslanderReiten sequence, Tate cohomology, generatin*
*g hypothesis, Klein
four modules, stable module category, Galois groups, Galois cohomology.
J'an Min'a~c was supported in part by Natural Sciences and Engineering Resea*
*rch Council of Canada
grant R3276A01.
2 SUNIL K. CHEBOLU AND J'AN MIN'A~C *
study problems in number theory, particularly in field theory and Galois theory*
*. But
unfortunately some of these techniques from representation theory are not well*
*known
to people working in number theory.
Several times recently, we ran into AuslanderReiten sequences in our work an*
*d we
began to appreciate the power and elegance of these sequences. These sequences*
* are
some special nonsplit short exact sequences which are very close to being spli*
*t sequences.
They were introduced by Auslander and Reiten in the early 70s [AR75 ], and they*
* have
been among the most important tools from modular representation theory.
The goal of this article is to introduce AuslanderReiten sequences and to gi*
*ve some
applications. This article is by no means a historical survey of AuslanderReit*
*en theory;
it focuses only on some selective applications from our recent joint work. The *
*treatment
here will be particularly suitable for homotopy theorists and number theorists.*
* We
hope that the readers who are not specialists in representation theory might be*
* inspired
further to learn more about AuslanderReiten sequences and investigate possible*
* Galois
field theoretic applications.
We would like to thank the referee for his/her valuable suggestions which hav*
*e helped
us to improve the exposition of our paper
2.AuslanderReiten sequences
Throughout we let G denote a finite group and k a field of characteristic p *
*that
divides the order of G. We denote by the group algebra kG. Although Auslander
Reiten theory is often studied in the broader context of Artin algebras, for si*
*mplicity
we restrict ourselves to the most interesting case of group algebras.
Definition 2.1. A short exact sequence of modules
ffl: 0 ! M ff!E fi!N ! 0
is an AuslanderReiten (AR) sequence if the following three conditions are sati*
*sfied:
o ffl is a nonsplit sequence.
o ff is left almost split, i.e., any map OE: M ! M0that is not a split mon*
*omorphism
factors through E via ff.
o fi is right almost split, i.e., any map _ :N0 ! N that is not a split ep*
*imorphism
factors through E via fi.
_N0
____
__________
ff _""____fflfflfi
0 _____//M_____//E____//___N__//0
____
OE_________
fflffl""____
M0
There are several other equivalent definitions of an AuslanderReiten sequen*
*ce. The
one we have given here is perhaps the most symmetric of all. Note that a split *
*short
exact sequence satisfies the last two conditions. This justifies why Auslander*
*Reiten
sequences are also called almost split sequences. Further, we note that these c*
*onditions
also imply that the terms M and N in an AR sequence have to be indecomposable a*
*nd
nonprojective.
AUSLANDERREITEN SEQUENCES 3
Example 2.2. (Oneup and Onedown) Let us start with a very simple example. Let
= k[x]=xp where p is the characteristic of k. Observe that is a group algeb*
*ra kG
with a G cyclic group of order p. Then for each integer i strictly between 0 an*
*d p, the
short exact sequence
0 ! k[x]=xi(j,ss)!k[x]=xi+1 k[x]=xi1! k[x]=xi ! 0
is easily shown (exercise left to the reader) to be an AuslanderReiten sequenc*
*e. Here j
is the obvious injection into the first summand of the middle term, and ss is t*
*he obvious
surjection on the second summand.
The remarkable result is the existence of these sequences:
Theorem 2.3. [ARS95 ] Given any indecomposable nonprojective module N, there
exists a unique (up to isomorphism of short exact sequences) AuslanderReiten s*
*equence
ffl ending in N. Moreover the first term M of the ARsequence ffl is isomorphic*
* to 2N.
In Section 4 we present a nonstandard proof due to Krause of the existence *
*of AR
sequences. This uses the Brown representability theorem from algebraic topology*
*. For
now, we assume existence and show uniqueness.
Suppose we have two ARsequences ending in an indecomposable nonprojective *
*kG
module L:
0 ! M ff!E fi!L ! 0
0 fi0
0 ! M0 ff!E0 ! L ! 0
The defining property of AR sequences then gives the following commutative diag*
*ram.
fi
0_____//M__ff_//__E___//_L___//0
___ _____ 
f _______ _____ =
fflffl___fflffl___fflfflff0fi0
0_____//M0____//__E0__//_L___//0
___ ____ 
g ______ _____ =
fflffl____fflffl____fflfflfffi
0_____//M_____//_E____//L____//0
The composite OE := gf cannot be nilpotent, otherwise fi would be a split epimo*
*rphism.
(This is an easy diagramchase exercise.) Now since M is a module of finite len*
*gth, for
sufficiently large values of n, we must have
M ~=Im(OEn) Ker(OEn).
Since M is indecomposable, and OE is not nilpotent, it follows that M ~=Im (OEn*
*) and
0 = Ker(OEn). This shows that OEn, and hence OE, is an isomorphism. Therefore*
* f is
the left inverse of g. Interchanging the roles of the two ARsequences above, i*
*t follows
similarly that f is the right inverse of g. Thus we have M ~= M0. The five le*
*mma
now tells us that E ~=E0. Thus the two ARsequences are isomorphic as short exa*
*ct
sequences.
4 SUNIL K. CHEBOLU AND J'AN MIN'A~C *
3.Stable module category
We now introduce the stable module category. We will work mostly in this cat*
*egory
for the first half of the paper. The stable module category StMod (kG) of G is *
*the cate
gory obtained from the category Mod (kG) of (left) kGmodules by killing the pr*
*ojective
modules. More precisely, it is the category whose objects are kGmodules and wh*
*ose
morphisms are equivalence classes of kGmodule homomorphisms, where two homomor
phisms are equivalent if their difference factors through a projective module (*
*such a map
is called projective). If M and N are two kGmodules, then we write Hom_kG(M, N*
*) for
the kvector space of maps between M and N in the stable module category. It is*
* easy to
see that a kGmodule is projective if and only if it is the zero object in StMo*
*d(kG). The
compact stable category stmod(kG) is the full subcategory of StMod(kG) consisti*
*ng of
the finitely generated (equivalently, finitedimensional) kGmodules.
The category StMod (kG) admits a natural tensor triangulated structure which*
* we
now explain. If M is a kGmodule then define its desuspension M to be the ker*
*nel
of a surjective map from a projective module to M. By Schanuel's Lemma, the new
module thus obtained is welldefined up to projective summands and therefore de*
*fines
an endofunctor
: StMod(kG) ! StMod(kG).
Using the fact that kG is a Frobenius algebra (injectives and projective coinci*
*de) one
can show that is an equivalence. The inverse functor 1 sends M to the coker*
*nel of
an injective map from M to an injective module. This allows one to talk about n*
*egative
powers of .
The importance of the stable module category can be seen in the following us*
*eful and
wellknown fact.
Proposition 3.1. Let M and N be kGmodules. If n is any positive integer, then
Hom_kG( n M, N) ~=ExtnkG(M, N) ~=Hom_kG(M, n N).
In particular, group cohomology is elegantly encoded in the stable module ca*
*tegory.
In fact, allowing n to be any integer, one recovers the Tate cohomology ring of*
* G in
this way as the (graded) ring of self maps of the trivial representation. Unde*
*r this
isomorphism, the multiplication in the Tate ring of G corresponds to compositio*
*n in
Hom_( *k, k).
Now we describe the triangles in the stable module category. These arise fro*
*m short
exact sequences in the module category in the following way. Suppose
0 ! L ! M ! N ! 0
is a short exact sequence of kGmodules. This short exact sequence determines a*
* unique
class in Ext1kG(N, L) which under the isomorphism given in the above propositio*
*n cor
responds to a map N ! 1L. Thus we get a diagram
L ! M ! N ! 1L. (*)
A triangle in StMod(kG) is any such diagram which is isomorphic to a diagram wh*
*ich
arises as above (*). These triangles and the suspension functor satisfy the st*
*andard
axioms for a triangulated category; see [Car96] for example.
The tensor product of two kGmodules (with diagonal Gaction on the tensor p*
*roduct
of kvector spaces) descends to a wellbehaved tensor product in the stable mod*
*ule
AUSLANDERREITEN SEQUENCES 5
category. (This follows from the fact that if M and N are kGmodules with eithe*
*r M
or N projective, then M k N is projective.) The trivial representation k of G *
*serves
as a unit for the tensor product.
4.Krause's proof of the existence of AR sequences
The proof of the existence of an AuslanderReiten sequence can be found in [*
*ARS95 ].
We present a different avatar of AuslanderReiten sequences in the stable modul*
*e cate
gory, and prove their existence following Krause [Kra00].
Definition 4.1. A triangle A ff!B fi!C fl! 1A in StMod(kG) is an Auslander
Reiten triangle if the following three conditions hold.
o fl 6= 0.
o Every map f :A ! A0which is not a section factors through ff.
o Every map g :C0 ! C which is not a retraction factors through fi.
The following theorem is crucial in what follows. This is the Brown represen*
*tability
theorem for StMod(kG).
Theorem 4.2. (Brown) A contravariant functor F :StMod(kG) ! Ab that is exact
and sends coproducts to products is representable, i.e., there exists a module *
*MF in
StMod (kG) such that
F () ~=Hom_kG(, MF ).
Now fix a finitely generated nonprojective kGmodule Z that has a local end*
*omor
phism ring, and set = Hom_kG(Z, Z). Let I be the injective envelope of =rad*
* .
Consider the functor
Hom (Hom_kG(Z, ), I).
It is not hard to see that this is a contravariant functor that is exact and se*
*nds coproducts
to products. So by the Brown representability theorem this functor is represent*
*able:
(4.1) : Hom (Hom_kG(Z, ), I) ~=Hom_kG(, TZI).
Finally consider the composite ! : ss! =rad ,! I, and denote by fl the m*
*ap
(!). Note that fl is a map from Z ! TZI. Extending this map to a triangle in
StMod (kG), we get
TZI ! Y ! Z fl!TZI.
Theorem 4.3 ([Kra00]). The triangle TZI ff!Y fi!Z fl!TZI is an Auslander
Reiten triangle.
Proof.fl 6= 0: Recall that fl = (!), where is an isomorphism of functors, an*
*d ! is a
nonzero map. Therefore fl 6= 0.
fi has the right almost split property: That is, given a map ae: Z0 ! Z tha*
*t is not a
retraction, we have to show that there exists a map ae0:Z0 ! Y making the foll*
*owing
diagram commutative.
Z0_
0______
ae___ae____
""____fflffl_fl
TZI ____//_Y____//Z____//_TZI
6 SUNIL K. CHEBOLU AND J'AN MIN'A~C *
For this, it is enough to know that flae = 0, since the bottom row in the above*
* diagram is
an exact triangle. Since ae is not a retraction, ImHom_kG (Z, Z0) ae*!Hom_kG(Z*
*, Z) =
does not contain the identity map. Consequently Im(ae*) is contained in rad .
The naturality of the isomorphism yields the following commutative diagram.
~=
Hom (Hom_kG(Z, Z), I)___//_Hom_kG(Z, TZI)
 
 
fflffl ~= fflffl
Hom (Hom_kG(Z, Z0),_I)__//Hom_kG(Z0, TZI)
Note that ! maps to zero under the left vertical map and by the commutativity o*
*f the
diagram, we see that fl also maps to zero under the right vertical map. Thus fl*
*ae = 0, as
desired.
ff has the left almost split property: For this, it turns out that it suffice*
*s to show
that Hom_kG(TzI, TZI) is a local ring; see [Kra00] for details.
Hom_kG(TzI, TZI)~= Hom (Hom_kG(Z, TZI), I)
~= Hom (Hom (Hom_kG(Z, Z), I), I)
~= Hom (Hom ( , I), I)
~= Hom (I, I)
The injective module I is indecomposable since =rad is simple, and therefo*
*re
Hom (I, I) is a local ring. So we are done.
5. AuslanderReiten duality
The isomorphism given in eq. 4.1 is just another avatar of AuslanderReiten *
*duality
which we now explain. Let M and L be finitely generated kGmodules. Then the
AuslanderReiten duality says that there is a nondegenerate bilinear form (fun*
*ctorial
in both variables)
(, ): Hom_kG(L, M) x Hom_kG(M, L) ! k.
To explain this pairing, first recall that M is defined by the short exact seq*
*uence
0 ! M j! P q! M ! 0,
where P is a minimal projective cover of M. Given f :L ! M and g :M ! L, we
then get the following diagram.
P _____________//POO
q j
fflfflg f 
M ____//_L___//_ M
The top horizontalPself map of P is in the image of the norm map j :EndkP ! End*
*kGP
which sends OE to g2Gg . OE. Now we define (f, g) = tr(`), where j(`) = jfgq.
To see the connection between the AuslanderReiten sequence and the natural *
*iso
morphism 4.1 which was responsible for the existence of AuslanderReiten triang*
*les,
consider the special case of AuslanderReiten duality when M = k. This gives a *
*pairing
(natural in L):
Hom_kG(L, k) x Hom_kG(k, L) ! k.
AUSLANDERREITEN SEQUENCES 7
Or equivalently, a natural isomorphism of functors
Hom k(Hom_kG(k, ), k) ~=Hom_kG(, k)
This is precisely the isomorphism 4.1 when Z = k. Since the representing objec*
*t is
unique, we get TkI = k. Thus the AuslanderReiten triangle ending in k has the*
* form
2k ! M ! k ! k.
In the next three sections we give three different applications of Auslander*
*Reiten
sequences from our recent work.
6.Representations of V4modules
Consider the Klein four group V4 = C2 C2. We use AR sequences to prove the
following intriguing result which also plays a crucial role in the proof of the*
* classification
of the indecomposable V4 representations.
Theorem 6.1. Let M be a projectivefree V4module. Then we have the following.
(1) If l is the smallest positive integer such that l(k) is isomorphic to a*
* submodule
of M, then l(k) is a summand of M.
(2) Dually, if l is the smallest positive integer such that l(k) is isomor*
*phic to a
quotient module of M, then l(k) is a summand of M.
Proof.First note that the second part of this lemma follows by dualising the fi*
*rst part;
here we also use the fact that ( lk)* ~= lk. So it is enough to prove the firs*
*t part.
The AR sequences in the category of kV4modules are of the form (see [Ben84,*
* Ap
pendix, p 180]):
0 ! l+2k ! l+1k l+1k ! lk ! 0 l 6= 1
0 ! 1k ! kV4 k k ! 1 k ! 0
To prove the first part of the above lemma, let l be the smallest positive i*
*nteger such
that lk embeds in a projectivefree V4module M. If this embedding does not sp*
*lit,
then by the property of an almost split sequence, it should factor through l1*
*k l1k
as shown in the diagram below.
0 ____//_"lk___//_`_l1k__ l1k__////_ l2k_//0
 ___________
 ___f_g____
fflfflww_______
M
Now if either f or g is injective, that would contradict the minimality of l, s*
*o they
cannot be injective. So both f and g should factor through l2k l2k as sho*
*wn in
the diagrams below.
0 ____//_ l1k___//_ l2k_ l2k___////_ l3k_//0
______
f ______________
fflffl(f1wf2)w__________
M
8 SUNIL K. CHEBOLU AND J'AN MIN'A~C *
0_____// l1k____// l2k l2k___//_//_ l3k//_0
______
g  _______________
fflffl(g1wg2)w__________
M
Proceeding in this way we can assemble all the lifts obtained using the almos*
*t split
sequences into one diagram as shown below.
"
lkO______________________________________________//"M`11______*
*_________________________44______________99_____==___
 _____________________________________*
*________________________________
 ______________________________________________*
*_______
fflffl_______________________________________________________*
*__
l1k " l1k` ___________________________________________
 __________________ _________________
 _____________ _____________
fflffl________ _________________
( l2k l2k) " (`l2k l2k) ___________________
____ _____
 _______ ______
 _______ ______
fflffl _________ _______
.. ________ ______
." ` __________ ________
 _________ ________
 ________ _____
fflffl____ _______
( 1k 1k) .".`.( 1k 1k)__________
___
 ______
 ______
 _____
 ______
 ______
 _____
fflffl____
(kV4 k k) . . .(kV4 k k)
So it suffices to show that for a projectivefree M there cannot exist a factor*
*isation of
the form "
lkO_______//_"M`::_____
 _________
 _______
fflfflOE____
(kV4)s kt
where l is a positive integer. It is not hard to see that the invariant submodu*
*le ( lk)G of
lk maps into ((kV4)s)G . We will arrive at a contradiction by showing ((kV4)s)*
*G maps
to zero under the map OE. Since ((kV4)s)G ~=((kV4)G )s it is enough to show tha*
*t OE maps
each (kV4)G to zero. (kV4)G is a onedimensional subspace, generated by an elem*
*ent
which we denote as v. If v maps to a nonzero element, then it is easy to see t*
*hat the
restriction of OE on the corresponding copy of kV4 is injective, but M is proje*
*ctivefree, so
this is impossible. In other words OE(v) = 0 and that completes the proof of th*
*e lemma.
7. Tate cohomology
Recall that the Tate cohomology of G with coefficients in a module M is give*
*n by
^H*(G, M) = Hom_kG( *k, M). In our recent joint work with Jon Carlson we focuse*
*d on
two fundamental questions ([CCM07b ] and [CCM07a ]) about Tate cohomology whi*
*ch
AUSLANDERREITEN SEQUENCES 9
we discuss in the next two subsections. As we shall see, AuslanderReiten sequ*
*ences
play an important role in answering both of these questions.
7.1. Modules with finitely generated Tate cohomology. Let M be a finitely gen
erated kGmodule. A wellknown result of EvensVenkov states that the ordinary *
*co
homology H*(G, M) is finitely generated as a module over H*(G, k). So it is a *
*very
natural question*to investigate whether the same is true for Tate*cohomology. *
*That
is, whether ^H(G, M) is finitely generated as a module over ^H(G, k). As shown*
* in
[CCM07b ] Tate cohomology is seldom finitely generated. However, there is an *
*inter
esting family of modules arising from AR sequences whose Tate cohomology is fin*
*itely
generated.
To start, let N be a finitely generated indecomposable nonprojective kGmod*
*ule
that is not isomorphic to ik for any i. Consider the AuslanderReiten sequence
0 ! 2N ! M ! N ! 0
ending in N. Assume that N has finitely generated Tate cohomology, then we shal*
*l see
that the middle term M also has finitely generated Tate cohomology.
Consider the connecting map OE: N ! N in the exact triangle
2N ! M ! N OE! N
corresponding to the above AuslanderReiten sequence. We first argue that OE in*
*duces
the zero map in Tate cohomology. To this end, let f : ik ! N be an arbitrary ma*
*p. We
want to show that the composite ik f!N !OE N is zero in the stable category. N*
*ow
observe that the assumption on N implies that f is not a split retraction, ther*
*efore the
map f factors through the middle term M. Since the composition of any two succe*
*ssive
maps in an exact triangle is zero, it follows that OEf = 0.
Since the boundary map OE induces the zero map in Tate cohomology, the resul*
*ting
long exact sequence in Tate cohomology breaks into short exact sequences:
* 2 * *
0 ! ^H(G, N) ! ^H(G, M) ! ^H(G, N) ! 0.
It is now clear that if N has finitely generated Tate cohomology, then so does *
*M.
In [CCM07b ] we have also shown that the middle term of the AR sequence end*
*ing
in k has finitely generated Tate cohomology.
These results imply that if we have a sequence of finitely generated indecom*
*posable
nonprojective kGmodules
N1, N2, . .,.Ns, s 2 N
* *
such that ^H(G, N1) is finitely generated as a module over ^H(G, k) and Ni+1 is*
* a
summand of a middle*term of the AuslanderReiten sequence*which ends in Ni, 1 *
* i < s,
then all modules ^H(G, Ni) are finitely generated over ^H(G, k).
7.2. Counterexamples to Freyd's generating hypothesis. Motivated by the cel
ebrated generating hypothesis (GH) of Peter Freyd in homotopy theory [Fre66] an*
*d its
analogue in the derived category of a commutative ring [HLP07 , Loc07], we have*
* formu
lated in [CCM07a ] the analogue of Freyd's GH in the stable module category st*
*mod(kG)
of a finite group G, where k is a field of characteristic p. This is the statem*
*ent that the
Tate cohomology functor detects trivial maps (maps that factor through a projec*
*tive
10 SUNIL K. CHEBOLU AND J'AN MIN'A~C *
module) in the thick subcategory generated by k. We studied the GH and related *
*ques
tions in special cases in a series of papers: [CCM08a , BCCM07 , CCM08b , CCM*
*07c ].
We have finally settled the GH for the stable module category in joint work wit*
*h Jon
Carlson. The main result of [CCM07a ] is:
Theorem 7.1. [CCM07a ] The generating hypothesis holds for kG if and only if t*
*he
Sylow psubgroup of G is either C2 or C3.
Maps in the thick subcategory generated by k that induce the zero map in Tate
cohomology are called ghosts. Thus nontrivial ghosts give counterexamples to t*
*he GH.
After much research on this, we were led to a big revelation when we discovered*
* that a
good source of nontrivial ghosts come from AuslanderReiten sequences.
To explain this in more detail, consider an AR sequence
0 ! 2N ! B ! N ! 0
ending in N. This short exact sequence represents a exact triangle
2N ! B ! N OE! N
in the stable category. We will show that the map OE: N  ! N is a nontrivial
ghost. First of all, AR sequences are, by definition, nonsplit short exact seq*
*uences, and
therefore the boundary map OE in the above triangle must be a nontrivial map i*
*n the
stable category. The next thing to be shown is that the map OE: N ! N induces
i
the zero map on the functors Hom_kG( ik, ) ~= dExt(k, ) for all i. Arguing a*
*s in
section 7.1, consider any map f : ik ! N. We have to show that the composite
ik f!N OE! N
is trivial in the stable category. Consider the following diagram
__ik
____
_____f____
""_____fflfflOE
2N _____//B_____//N_____//_ N
where the bottom row is our exact triangle. The map f : ik ! N cannot be a spl*
*it
epimorphism if we choose N to be an indecomposable nonprojective module in the
thick subcategory generated by k that is not isomorphic to ik for any i. Then *
*by the
defining property of an AR 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*
* exact
triangle is zero, the composite OEOf is also zero by commutativity. Thus the mo*
*ral of the
story is that in order to disprove the GH for kG, we just have to find an indec*
*omposable
nonprojective module in the thick subcategory generated by k that is not isomo*
*rphic
to ik for any i. This was the strategy we used to disprove the GH for kG when *
*the
Sylow psubgroup has order at least 4.
8. Galois theoretic connections
In this section we provide the motivation for some of the arithmetic objects*
* which will
appear in the later sections when we study AuslanderReiten sequences in the co*
*ntext
of Galois theory. To set the stage, we begin with our notation.
AUSLANDERREITEN SEQUENCES 11
Let F be a field of characteristic not equal to 2 and let Fsepdenote the sepa*
*rable
closure of F . We shall introduce several subextensions of Fsep.
o F (2)= compositum of all quadratic extensions of F .
o F (3)= compositum of all quadratic extensions of F (2), which are Galois *
*over
F .
o F {3}= compositum of all quadratic extensions of F (2).
o Fq = compositum of all Galois extensions K=F such that [K : F ] = 2n, for*
* some
positive integer n.
All of these subextensions are Galois and they fit in a tower
F F (2) F (3) F {3} Fq Fsep.
We denote their Galois groups (over F ) as
GF ! Gq ! G{3}F! G[3]F(= GF ) ! G[2]F(= E) ! 1.
Q 2
Observe that G[2]Fis just C2, where I is the dimension of F *=F * over F2. *
* F *
i2I
denotes the multiplicative subgroup of F . GF is the absolute Galois group of F*
* . The
absolute Galois group of fields are in general rather mysterious objects of gre*
*at interest.
(See [BLMS07 ] for restrictions on the possible structure of absolute Galois gr*
*oups.) Al
though the quotients Gq are much simpler we are far from understanding their st*
*ructure
in general. F {3}and its Galois group over F are considerably much simpler and*
* yet
they already contain substantial arithmetic information of the absolute Galois *
*group.
To illustrate this point, consider W F the Witt ring of quadratic forms; see *
*[Lam05 ]
for the definition. Then we have the following theorem.
Theorem 8.1. [MS96 ] Let F and L be two fields of characteristic not 2. Then W *
*F ~=
W L (as rings) implies that GF ~=GL as pro2groups. Further if wepassume_addit*
*ionally
in thepcase_when each element of F is a sum of two squares that 12 F if and *
*only
if 1 2 L, then GF ~=GL implies W F ~=W L.
Thus we see that GF essentially controls the Witt ring W F and in fact, GF c*
*an be
viewed as a Galois theoretic analogue of W F . In particular, GF detects order*
*ings of
fields. (Recall that P is an ordering of F if P is an additively closed subgrou*
*p of index
2 in F *.) More precisely, we have:
Theorem 8.2. [MS90 ] There is a 11 correspondence between the orderings of a f*
*ield F
and cosets {oe (GF )  oe 2 GF \ (GF )and oe2 = 1}. Here (GF ) is the Frattini*
* subgroup
of GF , which is just the closed subgroup of GF generated by all squares in GF *
*. The
correspondence is as follows:
p __ p __
oe (GF ) ! Poe= {f 2 F * oe( f) = f}.
This theorem was generalised considerably for detecting additive properties *
*of mul
tiplicative subgroups of F *in [MMS04 ]. In this paper (see [MMS04 , Section *
*8]) it was
shown that GF can be used also for detecting valuations on F .
Also in [AKM99 , Corollary 3.9] it is shown that GF ~=GL if and only if k*(*
*F ) ~=k*(L).
Here k*(A) denotes the Milnor Ktheory (mod 2) of a field A. In particular, in *
*[AKM99 ,
Theorem 3.14] it is shown that if R is the subring of H*(GF , F2) generated by *
*one
dimensional classes, then R is isomorphic to the Galois cohomology H*(GF , F2) *
*of F .
Thus we see that GF also controls Galois cohomology and in fact H*(GF , F2) con*
*tains
12 SUNIL K. CHEBOLU AND J'AN MIN'A~C *
some further substantial information about F which H*(GF , F2) does not contain*
*. In
summary, GF is a very interesting object. On the one hand GF is much simpler th*
*an GF
or Gq, yet it contains substantial information about the arithmetic of F . In [*
*BLMS07 ,
Section 2] the case of p > 2 was also considered, and the definition of GF was *
*extended to
fields which contain a primitive pthroot of unity. The key reason for restrict*
*ing to p = 2
in earlier papers stems from the interest in quadratic forms. Nevertheless a nu*
*mber of
interesting properties also hold for GF in the case p > 2.
9.Towards the AuslanderReiten translate 2F2
Because GF contains considerable information about the arithmetic of F , it *
*is a
natural question to ask for properties of Gq, or more precisely Gq(F ), determi*
*ned by
GF . First of all, it is possible that for two fields F1 and F2, we have
GF1 ~=GF2 but Gq(F1) AE Gq(F2).
Indeed, set F1 = C((t1))((t2)) to be a field of formal power series in t2 over *
*the field
of formal power series in t1 over C and F2 the field Q5 of 5adic numbers. Then
GF1 = GF2 = C4 x C4, but Gq(F1) = Z2 x Z2 (Z2 is the additive group of 2adic
numbers), while Gq(F2) = see [MS96 , Koc02]. Howeve*
*r, in this
case G{3}F1~=G{3}F2, as they are both isomorphic to C4 x C4; see [AGKM01 ].
The very interesting question about possible groups G{3}Fwhen G[3]Fis given,*
* is cur
rently investigated by Min'a~c, Swallow and Topaz [MST ]. In order to tackle th*
*is question
it is important to understand the universal case when Gq(F ) is a free pro2gr*
*oup. From
now on we further assume that F *=F *2is finitedimensional. (This is the most *
*impor
tant case to understand, as G{3}Fin general is the projective limit of its fini*
*te quotients.)
Let n be the dimension. Then we have F *=F *2= 2n. In this case we have an exte*
*nsion
1 ! A ! G{3}F! E ! 1
Q n {3} Q m
where E = 1C2 and A = Gal(GF =F (2)) ~= 1 C2 with m = (n  1)2n + 1. Set
R = F (2)for simplicity. Indeed from Kummer theory we know that
A ~=Hom F2(R*=R* 2, F2).
Then also by Kummer theory the minimal number of topological generators of G(2)*
*q:=
Gal(Fq=F (2)) is equal to the dimension of R*=R* 2. From the topological versi*
*on of
Schreier's theorem we know that this number d(G(2)q) is given by d(G(2)q) = (n*
*1)2n+1,
because G(2)qis the open subgroup of index 2n of Gq and d(Gq) = n. (See [Koc02,
Example 6.3]) Thus
dim F2A = dim F2R*=R* 2
= dim F2H1(G(2)q, F2)
= d(G(2)q)
= (n  1)2n + 1.
Moreover, A is a natural F2Emodule where the Eaction is induced by conjuga*
*tion
in G{3}F. The next result gives a completely arithmetical interpretation of the*
* Auslander
Reiten translate in our case. We write 2EF2 in order to stress that we are wor*
*king in
the category of F2Emodules.
AUSLANDERREITEN SEQUENCES 13
Theorem 9.1. [Gas54]
2EF2 ~=A.
We sketch in the next section a direct proof of this theorem as it also provi*
*des some
information about the structure of our module A. After that we shall consider t*
*he AR
sequence
0 ! 2F2 ! M ! F2 ! 0
and interpret M and also its associated group AR(E) which will be defined later*
* below
as a certain Galois group.
10. Generators and relations for 2F2
Q n Q n
Recall that E = 1C2 = 1. We shall first use the definition of 2F2 *
*over F2E
to describe 2F2 via generators and relations and then we shall see that indeed*
* A has
this presentation. This, in turn, will imply that A ~= 2F2.
Let Paibe a free F2E module of rank 1 generated by aifor 1 i n, so Pai= *
*F2Eai.
Then recall that 2F2 is defined by the short exact sequence:
0 ! 2F2 ! ni=1Pai(= P ) _!F2E !fflF2 ! 0,
where ffl: F2E ! F2 is the augmentation map ffl(oei) = 1 for all i, and _(ai) =*
* oei 1. We
shall denote oei 1 by aeifor simplicity. Exactness at F2 is trivial and exactn*
*ess at F2E
is a standard exercise which we leave to the reader. Observe that 2F2 is deter*
*mined
only up to a projective summand. But since we insist that 2F2 is projectivefr*
*ee, we
actually get it up to an isomorphism of F2Emodules.
Note that 2F2 = Ker_, and it certainly contains the F2Esubmodule of P gene*
*rated
by
S = {aeiai, aejai+ aeiaj, 1 i < j n.}
We claim that the F2E submodule W generated by S is the entireQmodule 2F2. Fir*
*st
observe that I = 1F2 = Kerffl has dimension 2n  1 and aeT = i2Taei, where T*
* is
a nonempty subset of {1, 2, . .,.n} form a basis of I. This allows us to pick*
* a nice
F2vector space section of _: V the vector space span of the set
U = {aeTai 1 i n, andT {i + 1, i + 2, . .,.n}}.
It is not hard to see that P splits as Ker_ V as F2vector spaces. Therefore*
* it is
enough to show that W ker_ together with V generate the entire module P over
F2. Set Q := W V , the vector span of V and W over F2. Since {a1, a2, . .,.an*
*} Q
and {a1, a2, . .,.an} generates P as an F2Emodule, it is sufficient to show th*
*at Q is an
F2E submodule of P . Because W is an F2Esubmodule of Q it is enough to show th*
*at
for each v 2 U and aelwe also have ff := aelv in Q. Write v = aeTai for some T *
*and i. If
l = i, then ff belongs to W . If l > i, then ff belongs to U [ {0}. In particul*
*ar, in both
cases, ff 2 Q. If l < i, then
ff = aeTaelai
= aeT(aelai+ aeial) + aeTaeial
= w + v
14 SUNIL K. CHEBOLU AND J'AN MIN'A~C *
where the first summand belongs to W and the second summand belongs to U [{0} *
* V .
Hence ff belongs to Q, as desired. This shows that 2F2 is generated over F2E b*
*y the
set S.
Let us denote:
[ai, aj, at1, . .,.atr] := aetraetr1. .a.et1(aeiaj+ aejai), and
[ai2, at1, . .,.atr] := aetraetr1. .a.et1aeiai,
where all the subscripts range over the set {1, 2, . .,.n}. Then we have the fo*
*llowing
identities. Below oe denotes an arbitrary permutation of the set {1, 2, 3, . .*
*,.s}, and
bi= atifor any i.
(1) [ai, aj] = [aj, ai]
(2) [ai, aj, b1, . .,.bs] = [ai, aj, boe(1), . .,.boe(s)]
(3) [a2i, b1, . .,.bs] = [a2i, boe(1), . .,.boe(s)]
(4) [ai, aj, ar] + [aj, ar, ai] + [ar, ai, aj] = 0
(5) [a2i, ar] = [ai, ar, ai]
(6) [ai, aj, b1, . .,.bff, . .,.bff, . .,.bs] = 0
Then 2F2 P is a vector span over F2 of [ai, aj, at1, . .a.tr] and [a2i, a*
*t1, . .a.tr] as
above. In fact from the short exact sequence
0 ! 2F2 ! P _! F2E !fflF2 ! 0,
we see that dimF2 2F2 = n2n  2n + 1 = 2n(n  1) + 1. With what follows using t*
*he
above identities we see that the set
X := {[cij, at1, . .,.atr]  1 i j n, i < t1 < . .<.tr n}
generates 2F2 over F2. Here cij= [ai, aj] if i < j or a2iif i = j. Observe that
Xn
X = (n  i + 1)2ni.
i=1
Indeed for a fixed i there are n  i + 1 possibilities for j and there are 2ni*
* possibilities
for the number of subsets of {i + 1, . .,.n} for the choices of t1, . .,.tr. Al*
*so we have
Xn Xn (n + 1)Y n(Y  1)  (Y n+1 1)
(n  i + 1)Y ni= (Y ni+1)0= ___________________________2.
i=1 i=1 (Y  1)
P n
Plugging Y = 2, we see that X = i=1(n  i + 1)2ni = 2n(n  1) + 1. Hence X*
* is
the basis of 2F2 over F2. On the other hand, a detailed investigation of ident*
*ities in
G[3]F(see [GT67 , MST ]) shows that a basis of A over F2 is
Z = {[dij, oet1, . .o.etr]  1 i j n, i < t1 < . .<.tr n},
dij= [oei, oej] if i < j, dii= oe2i, and [oe1, oe2, . .,.oel] = [. .[.[[oe1, oe*
*2], oe3], . .,.oel]. In fact,
writing oei in place of ai above, the same set of 6 identities hold in G[3]F. *
*This shows
that the map OE: 2F2 ! A defined by OE(a2i) = oe2i, and OE([ai, aj]) = [oei, *
*oej] is a well
defined surjective F2E homomorphism and because of dimensional reasons it has t*
*o be
an isomorphism.
AUSLANDERREITEN SEQUENCES 15
11.The AuslanderReiten sequence ending in F2
Finally we want to obtain the AuslanderReiten sequence ending in F2:
0 ! 2F2 ! M j! F2 ! 0.
We will get a sequence of this form which is not split. Then simply because
Ext1F2E(F2, 2F2) = F2
the nonsplit sequence we obtain has to be an AuslanderReiten sequence; see [C*
*R87 ,
Proposition (78.28)].
Observe from our proof of our description of 2F2 P , that a := aenaen1 .*
* .a.e2a1
belongs to P  2F2. Let M := 2F2 + {a, 0} P . (Because aeia belongs to 2F2*
* for
each i = 1, 2, . .,.n, we see that M is indeed an F2Emodule.) Now, let j :M ! *
*F2
be defined by j( 2F2) = 0 and j(a) = 1. Observe that since ae1a belongs to 2F2*
* and
aeia = 0 for i 2, we obtain that j is indeed an F2Ehomomorphism.
We claim that the short exact sequence
0 ! 2F2 ! M j! F2 ! 0
is an AuslanderReiten sequence. Suppose to the contrary, this sequence splits.*
* Then
the coset a + 2F2 would contain a trivial element t = a + m, m 2 2F2 (i.e., t*
*he action
of each aeion t is 0). But t 2 Soc(P ) 2F2. Hence tm = a belongs to 2F2, w*
*hich is
a contradiction. (Here Soc(P ) is the socle of P which is the submodule of E co*
*nsisting
of the fixed elements of P under the action of E.) SoQwe are done.Q
We now introduce AuslanderReiten groups. Let E = ni=1C2 = ni=1 as *
*before.
Assume that n 2. Consider the AuslanderReiten sequence
0 ! 2F2 h!M ! F2 ! 0.
Consider the long exact sequence in cohomology associated to the above AR seque*
*nce.
As observed before, all connecting homomorphisms are zero except the first one.*
* In
particular, we have a short exact sequence:
0 ! H2(E, 2F2) ! H2(E, M) ! H2(E, F2) ! 0.
Furthermore, H2(E, 2F2) ~=F2. The unique nonzero element of H2(E, 2F2) corre
sponds to a short exact sequence
1 ! 2F2 ! G{3}F! E ! 1
where F *=F *2 = 2n and Gal(Fq=F ) is a free pro2group. Note that in our ca*
*se G{3}F
does not depend on F , but only on the number n of the rank of the free absolute
pro2group GF . Therefore we shall denote it by G{3}(n). Now consider the foll*
*owing
commutative diagram
1_____// 2F2____//G{3}(n)___//E____//1

h  =
fflffl fflffl fflffl
1______//M_______//T (n)____//E____//1
Here T (n) is the pushout of G{3}(n) and M along 2F2. The second row correspon*
*ds
to the image of the nontrivial element of H2(E, 2F2) in H2(E, M).
16 SUNIL K. CHEBOLU AND J'AN MIN'A~C *
Definition 11.1. We call T (n) the AuslanderReiten group associated to E.
Observe that we have obtained a Galois theoretic model for our AuslanderReit*
*en
sequence:
0 ! 2F2 ! M ! F2 ! 0.
Indeed, let K=L be a Galois extension as above such that Gal(K=F ) = T (n). The*
*n we
have a short exact sequence
0 ! M ! T (n) ! G ! 0
where AR Galois group T (n) is generated by G{3}(n) and o subject to the relati*
*ons
o2 = 1, [o, oei] = 1 for 2 i n, and [o, oe1] = [oe21, oe2, . .,.oen]. Let L*
* be the fixed field
of M. Then the Galois group Gal(K=L) ~=M is the middle term of our AuslanderRe*
*iten
sequence. Further, let H be a subgroup generated by oe1, . .,.oen. Then H ~=G{3*
*}(n)
and H \ M ~= 2F2. Hence our AuslanderReiten sequence has the form
0 ! H \ Gal(K=L) ! Gal(K=L) ! F2 ! 0.
12.Conclusion
We hope that our hot and mild appetizers will inspire our readers in their c*
*hoices
of further delightful, tasty, and mouthwatering main dishes. John Labute's sk*
*illful
interplay between Lie theoretic and group theoretic methods in his past work (s*
*ee for
example [Lab67]), as well as in his recent work ([Lab06]) provides us with insp*
*iration for
exploiting the connections between modular representation theory and Galois the*
*ory.
We are now lifting our glasses of champagne in honour of John Labute and we are
wishing him happy further research, and pleasant further AuslanderReiten encou*
*nters.
References
[AGKM01] Alejandro Adem, Wenfeng Gao, Dikran B. Karagueuzian, and J'an Min'a~c.*
* Field theory and
the cohomology of some Galois groups. J. Algebra, 235(2):608635, 2001.
[AKM99] Alejandro Adem, Dikran B. Karagueuzian, and J'an Min'a~c. On the cohom*
*ology of Galois
groups determined by Witt rings. Adv. Math., 148(1):105160, 1999.
[AR75] Maurice Auslander and Idun Reiten. Representation theory of Artin alge*
*bras. III. Almost
split sequences. Comm. Algebra, 3:239294, 1975.
[ARS95] Maurice Auslander, Idun Reiten, and Sverre O. Smalo. Representation th*
*eory of Artin
algebras, volume 36 of Cambridge Studies in Advanced Mathematics. Camb*
*ridge University
Press, Cambridge, 1995.
[BCCM07] David J. Benson, Sunil K. Chebolu, J. Daniel Christensen, and J'an Min*
*'a~c. The generating
hypothesis for the stable module category of a pgroup. J. Algebra, 31*
*0(1):428433, 2007.
[Ben84] David Benson. Modular representation theory: new trends and methods, v*
*olume 1081 of
Lecture Notes in Mathematics. SpringerVerlag, Berlin, 1984.
[BLMS07] Dave Benson, Nicole Lemire, J'an Min'a~c, and John Swallow. Detecting *
*propgroups that
are not absolute Galois groups. J. Reine Angew. Math., 613:175191, 20*
*07.
[Car96] Jon F. Carlson. Modules and group algebras. Lectures in Mathematics ET*
*H Z"urich.
Birkh"auser Verlag, Basel, 1996. Notes by Ruedi Suter.
[CCM07a] Jon F. Carlson, Sunil K. Chebolu, and J'an Min'a~c. Freyd's generating*
* hypothesis using
almost split sequences. 2007. preprint.
[CCM07b] Jon F. Carlson, Sunil K. Chebolu, and J'an Min'a~c. Finite generation *
*of Tate cohomology.
2007. preprint.
[CCM07c] Sunil K. Chebolu, J. Daniel Christensen, and J'an Min'a~c. Freyd's gen*
*erating hypothesis for
groups with periodic cohomology. 2007. preprint, arXiv:0710.3356.
AUSLANDERREITEN SEQUENCES 17
[CCM08a]Sunil K. Chebolu, J. Daniel Christensen, and J'an Min'a~c. Groups which*
* do not admit
ghosts. Proc. Amer. Math. Soc., 136(4):11711179, 2008.
[CCM08b]Sunil K. Chebolu, J. Daniel Christensen, and J'an Min'a~c. Ghosts in mo*
*dular representation
theory. Advances in Mathematics, 217:27822799, 2008.
[CR87] Charles W. Curtis and Irving Reiner. Methods of representation theory. *
*Vol. II. Pure and
Applied Mathematics (New York). John Wiley & Sons Inc., New York, 1987.*
* With appli
cations to finite groups and orders, A WileyInterscience Publication.
[Fre66] Peter Freyd. Stable homotopy. In Proc. Conf. Categorical Algebra (La Jo*
*lla, Calif., 1965),
pages 121172. Springer, New York, 1966.
[Gas54] Wolfgang Gasch"utz. "Uber modulare Darstellungen endlicher Gruppen, die*
* von freien Grup
pen induziert werden. Math. Z., 60:274286, 1954.
[GT67] Narain D. Gupta and Se'an J. Tobin. On certain groups with exponent fou*
*r. Math. Z.,
102:216226, 1967.
[HLP07] Mark Hovey, Keir Lockridge, and Gena Puninski. The generating hypothesi*
*s in the derived
category of a ring. Mathematische Zeitschrift, 256(4):789800, 2007.
[Koc02] Helmut Koch. Galois theory of pextensions. Springer Monographs in Math*
*ematics.
SpringerVerlag, Berlin, 2002. With a foreword by I. R. Shafarevich, Tr*
*anslated from the
1970 German original by Franz Lemmermeyer, With a postscript by the aut*
*hor and Lem
mermeyer.
[Kra00] Henning Krause. AuslanderReiten theory via Brown representabilit*
*y. KTheory,
20(4):331344, 2000. Special issues dedicated to Daniel Quillen on the *
*occasion of his six
tieth birthday, Part IV.
[Lab67] John P. Labute. Classification of Demu~shkin groups. Canad. J. Math., 1*
*9:106132, 1967.
[Lab06] John Labute. Mild propgroups and Galois groups of pextensions of Q. *
*J. Reine Angew.
Math., 596:155182, 2006.
[Lam05] T. Y. Lam. Introduction to quadratic forms over fields, volume 67 of Gr*
*aduate Studies in
Mathematics. American Mathematical Society, Providence, RI, 2005.
[Loc07] Keir Lockridge. The generating hypothesis in the derived category of R*
*modules. Journal
of Pure and Applied Algebra, 208(2):485495, 2007.
[MMS04] Louis Mah'e, J'an Min'a~c, and Tara L. Smith. Additive structure of mul*
*tiplicative subgroups
of fields and Galois theory. Doc. Math., 9:301355 (electronic), 2004.
[MS90] J'an Min'a~c and Michel Spira. Formally real fields, Pythagorean fields*
*, Cfields and W
groups. Math. Z., 205(4):519530, 1990.
[MS96] J'an Min'a~c and Michel Spira. Witt rings and Galois groups. Ann. of Ma*
*th. (2), 144(1):3560,
1996.
[MST] J'an Min'a~c, John Swallow, and Adam Topaz. The structure of Galois met*
*aabelian groups
of exponent 4. (In preparation.)
Department of Mathematics, Illinois State University, Normal, IL 61790, USA
Email address: schebol@ilstu.edu
Department of Mathematics, University of Western Ontario, London, ON N6A 5B7,
Canada
Email address: minac@uwo.ca