AUSLANDER-REITEN 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 Auslander-Reiten 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 Auslander-Reiten sequences in * *the context of Galois theory and connect them to some important arithmetic objects. Contents 1. Introduction 1 2. Auslander-Reiten sequences 2 3. Stable module category 4 4. Krause's proof of the existence of AR sequences 5 5. Auslander-Reiten duality 6 6. Representations of V4-modules 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 Auslander-Reiten translate 2F2 12 10. Generators and relations for 2F2 13 11. The Auslander-Reiten 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. Auslander-Reiten 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 Auslander-Reiten sequences in our work an* *d we began to appreciate the power and elegance of these sequences. These sequences* * are some special non-split 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 Auslander-Reiten sequences and to gi* *ve some applications. This article is by no means a historical survey of Auslander-Reit* *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 Auslander-Reiten 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.Auslander-Reiten 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 Auslander-Reiten (AR) sequence if the following three conditions are sati* *sfied: o ffl is a non-split 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 _""____fflffl|fi 0 _____//M_____//E____//___N__//0 ____ OE||_________ fflffl|""____ M0 There are several other equivalent definitions of an Auslander-Reiten 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 non-projective. AUSLANDER-REITEN SEQUENCES 3 Example 2.2. (One-up and One-down) 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]=xi-1-! k[x]=xi- ! 0 is easily shown (exercise left to the reader) to be an Auslander-Reiten 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 non-projective -module N, there exists a unique (up to isomorphism of short exact sequences) Auslander-Reiten s* *equence ffl ending in N. Moreover the first term M of the AR-sequence ffl is isomorphic* * to 2N. In Section 4 we present a non-standard 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 AR-sequences ending in an indecomposable non-projective * *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___fflffl|ff0fi0 0_____//M0____//__E0__//_L___//0 ___ ____ | g ______ _____ |= fflffl____fflffl____fflffl|fffi 0_____//M_____//_E____//L____//0 The composite OE := gf cannot be nilpotent, otherwise fi would be a split epimo* *rphism. (This is an easy diagram-chase 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 AR-sequences 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 AR-sequences 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) kG-modules by killing the pr* *ojective modules. More precisely, it is the category whose objects are kG-modules and wh* *ose morphisms are equivalence classes of kG-module 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 kG-modules, then we write Hom_kG(M, N* *) for the k-vector space of maps between M and N in the stable module category. It is* * easy to see that a kG-module 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, finite-dimensional) kG-modules. The category StMod (kG) admits a natural tensor triangulated structure which* * we now explain. If M is a kG-module 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 well-defined 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 well-known fact. Proposition 3.1. Let M and N be kG-modules. 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 kG-modules. 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 kG-modules (with diagonal G-action on the tensor p* *roduct of k-vector spaces) descends to a well-behaved tensor product in the stable mod* *ule AUSLANDER-REITEN SEQUENCES 5 category. (This follows from the fact that if M and N are kG-modules 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 Auslander-Reiten sequence can be found in [* *ARS95 ]. We present a different avatar of Auslander-Reiten 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 non-projective kG-module 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 non-zero 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. Auslander-Reiten duality The isomorphism given in eq. 4.1 is just another avatar of Auslander-Reiten * *duality which we now explain. Let M and L be finitely generated kG-modules. Then the Auslander-Reiten duality says that there is a non-degenerate 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|| fflffl|g 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 Auslander-Reiten sequence and the natural * *iso- morphism 4.1 which was responsible for the existence of Auslander-Reiten triang* *les, consider the special case of Auslander-Reiten duality when M = k. This gives a * *pairing (natural in L): Hom_kG(L, k) x Hom_kG(k, L) -! k. AUSLANDER-REITEN 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 Auslander-Reiten 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 V4-modules 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 projective-free V4-module. 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 kV4-modules 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 projective-free V4-module M. If this embedding does not sp* *lit, then by the property of an almost split sequence, it should factor through l-1* *k l-1k as shown in the diagram below. 0 ____//_"lk___//_`_l-1k__ l-1k__////_ l-2k_//0 | ___________ | ___f_g____ fflffl|ww_______ 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 l-2k l-2k as sho* *wn in the diagrams below. 0 ____//_ l-1k___//_ l-2k_ l-2k___////_ l-3k_//0 ______ f|| ______________ fflffl|(f1wf2)w__________ M 8 SUNIL K. CHEBOLU AND J'AN MIN'A~C * 0_____// l-1k____// l-2k l-2k___//_//_ l-3k//_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|_______________________________________________________* *__ l-1k " l-1k` ___________________________________________ | __________________ _________________ | _____________ _____________ fflffl|________ _________________ ( l-2k l-2k) " (`l-2k l-2k) ___________________ ____ _____ | _______ ______ | _______ ______ fflffl| _________ _______ .. ________ ______ ." ` __________ ________ | _________ ________ | ________ _____ fflffl|____ _______ ( 1k 1k) .".`.( 1k 1k)__________ ___ | ______ | ______ | _____ | ______ | ______ | _____ fflffl|____ (kV4 k k) . . .(kV4 k k) So it suffices to show that for a projective-free M there cannot exist a factor* *isation of the form " lkO_______//_"M`::_____ | _________ | _______ fflffl|OE____ (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 one-dimensional subspace, generated by an elem* *ent which we denote as v. If v maps to a non-zero 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* *ctive-free, 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 AUSLANDER-REITEN SEQUENCES 9 we discuss in the next two subsections. As we shall see, Auslander-Reiten 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 kG-module. A well-known result of Evens-Venkov 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 non-projective kG-mod* *ule that is not isomorphic to ik for any i. Consider the Auslander-Reiten 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 Auslander-Reiten 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 non-projective kG-modules 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 Auslander-Reiten 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 p-subgroup 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 non-trivial 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 non-trivial ghosts come from Auslander-Reiten 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 non-trivial ghost. First of all, AR sequences are, by definition, non-split short exact seq* *uences, and therefore the boundary map OE in the above triangle must be a non-trivial 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|____ ""_____fflffl|OE 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 non-projective 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 non-projective 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 p-subgroup 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 Auslander-Reiten sequences in the co* *ntext of Galois theory. To set the stage, we begin with our notation. AUSLANDER-REITEN 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 pro-2-groups. 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 1-1 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 K-theory (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 pth-root 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 Auslander-Reiten 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 5-adic numbers. Then GF1 = GF2 = C4 x C4, but Gq(F1) = Z2 x Z2 (Z2 is the additive group of 2-adic 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 pro-2-gr* *oup. From now on we further assume that F *=F *2is finite-dimensional. (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 F2E-module where the E-action 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 F2E-modules. AUSLANDER-REITEN 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 projective-fr* *ee, we actually get it up to an isomorphism of F2E-modules. Note that 2F2 = Ker_, and it certainly contains the F2E-submodule 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 non-empty subset of {1, 2, . .,.n} form a basis of I. This allows us to pick* * a nice F2-vector 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 F2-vector 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 F2E-module, it is sufficient to show th* *at Q is an F2E submodule of P . Because W is an F2E-submodule 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] := aetraetr-1. .a.et1(aeiaj+ aejai), and [ai2, at1, . .,.atr] := aetraetr-1. .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)2n-i. i=1 Indeed for a fixed i there are n - i + 1 possibilities for j and there are 2n-i* * 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 n-i= (Y n-i+1)0= ___________________________2. i=1 i=1 (Y - 1) P n Plugging Y = 2, we see that |X| = i=1(n - i + 1)2n-i = 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. AUSLANDER-REITEN SEQUENCES 15 11.The Auslander-Reiten sequence ending in F2 Finally we want to obtain the Auslander-Reiten 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 non-split sequence we obtain has to be an Auslander-Reiten sequence; see [C* *R87 , Proposition (78.28)]. Observe from our proof of our description of 2F2 P , that a := aenaen-1 .* * .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 F2E-module.) 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 F2E-homomorphism. We claim that the short exact sequence 0 -! 2F2 -! M -j! F2 -! 0 is an Auslander-Reiten 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 t-m = 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 Auslander-Reiten groups. Let E = ni=1C2 = ni=1 as * *before. Assume that n 2. Consider the Auslander-Reiten 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 non-zero 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 pro-2-group. 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 pro-2-group 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 non-trivial 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 Auslander-Reiten group associated to E. Observe that we have obtained a Galois theoretic model for our Auslander-Reit* *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 Auslander-Re* *iten sequence. Further, let H be a subgroup generated by oe1, . .,.oen. Then H ~=G{3* *}(n) and H \ M ~= 2F2. Hence our Auslander-Reiten 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 mouth-watering 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 Auslander-Reiten 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):608-635, 2001. 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Additive structure of mul* *tiplicative subgroups of fields and Galois theory. Doc. Math., 9:301-355 (electronic), 2004. [MS90] J'an Min'a~c and Michel Spira. Formally real fields, Pythagorean fields* *, C-fields and W- groups. Math. Z., 205(4):519-530, 1990. [MS96] J'an Min'a~c and Michel Spira. Witt rings and Galois groups. Ann. of Ma* *th. (2), 144(1):35-60, 1996. [MST] J'an Min'a~c, John Swallow, and Adam Topaz. The structure of Galois met* *a-abelian groups of exponent 4. (In preparation.) Department of Mathematics, Illinois State University, Normal, IL 61790, USA E-mail address: schebol@ilstu.edu Department of Mathematics, University of Western Ontario, London, ON N6A 5B7, Canada E-mail address: minac@uwo.ca