Cochain sequences and the Quillen category of a coclass family
aa r X i v : . [ m a t h . G R ] M a y COCHAIN SEQUENCES AND THE QUILLEN CATEGORY OFA COCLASS FAMILY
BETTINA EICK AND DAVID J. GREEN
Abstract.
We introduce the concept of an infinite cochain sequence and ini-tiate a theory of homological algebra for them. We show how these sequencessimplify and improve the construction of infinite coclass families (as introducedby Eick and Leedham-Green) and how they apply in proving that almost allgroups in such a family have equivalent Quillen categories. We also includesome examples of infinite families of p -groups from different coclass familiesthat have equivalent Quillen categories. Introduction
Coclass theory was initiated by Leedham-Green and Newman [14]. The funda-mental aim of this theory is to classify and investigate finite p -groups using thecoclass as primary invariant. The infinite coclass families of finite p -groups offixed coclass as defined by Eick and Leedham-Green [10] are considered a steptowards these aims. Their definition is based on a splitting theorem for a certaintype of second cohomology group.Various interesting properties of the infinite coclass families have been deter-mined. For example, the automorphism groups and the Schur multiplicators ofthe groups in one family can be described simultaneously for all groups in thefamily, see [6, 5] and [7]. It is conjectured that almost all groups in an infi-nite coclass family have isomorphic mod- p cohomology rings. This conjecture isstill open, but it is underlined by computational evidence obtained by Eick andKing [9] and by our earlier result [8] saying that the Quillen categories of almostall groups in an infinite coclass family are equivalent. The proof of the lattertheorem uses a splitting theorem for cohomology groups.In this paper we derive a generalization of the splitting theorems obtained andused in [10] and [8], and describe the splitting at the cocycle level. Based on this,we introduce the concept of an infinite cochain sequence and take the first stepstowards the development of a theory of homological algebra for them.We then show that the infinite coclass families of [10] can be defined using theinfinite cochain sequence. This way of defining the families is more explicit thanthe definition in [10], since it is based on cocycles rather than just cohomology Date : 7 May 2015.Green received travel assistance from DFG grant GR 1585/6-1. classes. This difference is significant; for example, it is useful for the investigationof the Quillen categories of the groups in a coclass family. Further, we use theinfinite cochain sequences to give a new, more conceptual proof of our maintheorem in [8] on the Quillen categories of the groups in an infinite coclass family.In final part of this paper we give some examples of groups from differentcoclass families with equivalent Quillen categories. Let q = p ℓ for a prime p ,let Z p denote the p -adic integers and consider the irreducible action of C q on T = Z ( p − p ℓ − p (this is unique up to equivalence). Then G q = T ⋊ C q is an infinitepro- p -group of coclass ℓ . For ℓ = 1 it is the unique infinite pro- p -group of coclass1 (or of maximal class) and for ℓ > p -group of coclass ℓ .The main line groups associated with an infinite pro- p -group G of coclass r are the infinitely many lower central series quotients G/γ i ( G ) that have coclass r ; this infinite sequence is not necessarily a coclass family itself, but it consistsof finitely many different coclass families. The skeleton groups associated withan infinite pro- p -group G of coclass r form a significantly larger family of groupscontaining the main line groups and they play an important role in coclass theory;we refer to [13, Sec. 8.4] or [11, Sec. 3] for details. Theorem 1.1. (1)
For a arbitrary, fixed prime p , the Quillen categories of almost all mainline groups associated with G p are pairwise equivalent. (2) The Quillen categories of almost all skeleton groups associated with G are pairwise equivalent.Proof. (1): See Section 8.1 for odd p ; for p = 2 the main line groups are thedihedral 2-groups, and the result is well known, see e.g. [8, Sec. 9].(2): See Section 8.2. (cid:3) Remark.
Theorem 1 (1) can be made more explicit: the Quillen categories of thequotients G p /γ i ( G p ) are equivalent for all i ≥ p + 1. Note that G /γ i ( G ) haveisomorphic mod-3 cohomology rings for i ∈ { , , } , but the cohomology ring for i = 4 is different, see [9].2. Infinite cochain sequences
Preliminaries.
We work throughout with the normalized standard resolu-tion, see [12, p. 8]. That is, a cochain f ∈ C n ( G, M ) is a map f : G n → M withthe additional property that f ( g , . . . , g n ) is zero if any g i is the identity element.We denote the coboundary operator by ∆ : C n ( G, M ) → C n +1 ( G, M ). Recallthat the coboundary of a 2-cochain is given by∆ f ( g , g , g ) = g f ( g , g ) − f ( g g , g ) + f ( g , g g ) − f ( g , g ) OCHAIN SEQUENCES AND THE QUILLEN CATEGORY 3 and more generally the coboundary of an n -cochain is∆ f ( g , . . . , g n +1 ) = g f ( g , . . . , g n +1 )+ n X i =1 ( − i f ( g , . . . , g i − , g i g i +1 , g i +2 , . . . , g n +1 )+ ( − n +1 f ( g , . . . , g n ) . The n -cocycles are the elements of Z n ( G, M ) = ker (cid:0) C n ( G, M ) ∆ −→ C n +1 ( G, M ) (cid:1) , and the n -coboundaries are the elements of B n ( G, M ) = Im (cid:0) C n − ( G, M ) ∆ −→ C n ( G, M ) (cid:1) . Since ∆ = 0 it follows that B n ( G, M ) ⊆ Z n ( G, M ), and we set H n ( G, M ) = Z n ( G, M ) /B n ( G, M ). Elements of H n ( G, M ) are called cohomology classes; if f is an n -cocycle, then its cohomology class is f + B n ( G, M ) ∈ H n ( G, M ). Remark . By transfer theory, | G | · H n ( G, M ) = 0 for all n ≥
1: see e.g. [3,Proposition 3.6.17].2.2.
Splitting theorems.
From now on for the remainder of this section we fixthe following notation.
Notation . Let G be a finite p -group with m = log p ( | G | ), let R be a com-mutative ring, let M an RG -module and let N be a submodule of M withAnn N ( p ) = { } .We need the following generalization of [8, Theorem 7], which is itself a gener-alization of [10, Theorem 18]. Theorem 2.3.
We use Notation 2.2 and let n ≥ and r ≥ m . Then there is asplitting H n ( G, M/p r N ) ∼ = H n ( G, M ) ⊕ H n +1 ( G, N ) which is natural with respect to restriction to subgroups of G .Notation. Projection M ։ M/p r N induces maps C n ( G, M ) ։ C n ( G, M/p r N )of cochain modules and H n ( G, M ) → H n ( G, M/p r N ) of cohomology modules.We shall denote all three maps by pro r . Proof.
Recall that | G | = p m and define i r : N → M : x p r x . Consider thelong exact sequence in group cohomology induced by the following short exactsequence of coefficient modules0 −−−→ N i r −−−→ M pro r −−−→ M/p r N −−−→ . The proof of [8, Theorem 7] readily generalizes, showing that if n ≥ r ≥ m then0 −−−→ H n ( G, M ) pro r −−−→ H n ( G, M/p r N ) con r −−−→ H n +1 ( G, N ) −−−→ B. EICK AND D. J. GREEN is a split short exact sequence, where con r is the connecting homomorphism. (cid:3) We now describe how Theorem 2.3 works at the cocycle level.
Proposition 2.4.
We use Notation 2.2 and let n ≥ , pick ρ ∈ Z n ( G, M ) and η ∈ Z n +1 ( G, N ) . (1) There is a (not necessarily unique) n -cochain σ ∈ C n ( G, N ) such that ∆( σ ) = p m η . (2) For every r ≥ m and for every choice of σ in (1) , the induced cochain pro r ( ρ + p r − m σ ) lies in Z n ( G, M/p r N ) . (3) For every r ≥ m and for every choice of σ in (1) , the cohomology class pro r ( ρ + p r − m σ ) + B n ( G, M/p r N ) ∈ H n ( G, M/p r N ) is the unique class corresponding via the isomorphism of Theorem 2.3 to (cid:0) ρ + B n ( G, M ) , η + B n +1 ( G, N ) (cid:1) ∈ H n ( G, M ) ⊕ H n +1 ( G, N ) . Proof. (1): p m η is a coboundary, since p m H n +1 ( G, N ) = 0 by Remark 2.1.(2): pro r and ∆ commute, and ∆( ρ + p r − m σ ) = p r η lies in the kernel of pro r .(3): The proof of [8, Theorem 7] says that the component maps of the isomor-phism H n ( G, M/p r N ) → H n ( G, M ) ⊕ H n +1 ( G, N ) are the connecting homomor-phism con r : H n ( G, M/p r N ) → H n +1 ( G, N ) and the map H n ( G, M/p r N ) π ∗ −→ H n ( G, M/p r − m N ) (pro r − m ) − −−−−−−→ H n ( G, M ) , where π : M/p r N → M/p r − m N is the projection map x + p r N x + p r − m N . As π ∗ pro r ( ρ + p r − m σ ) = pro r − m ( ρ + p r − m σ ) = pro r − m ( ρ ) , the image in H n ( G, M ) is ρ + B n ( G, M ).Recall from e.g. the proof of [13, Theorem 9.1.5] that con r is constructed asthe composition Z n ( G, M/p r N ) (pro r ) − −−−−−→ C n ( G, M ) ∆ −→ Z n +1 ( G, M ) ( i r ) − ∗ −−−→ Z n +1 ( G, N ) , with i r as in the proof of Theorem 2.3. So pro r ( ρ + p r − m σ ) ρ + p r − m σ p r η = i r ( η ) η . Uniqueness follows. (cid:3) The definition of cochain sequences.
Using the ideas of Proposition 2.4we now define infinite cochain sequences.
Definition.
We use Notation 2.2 and let n ≥ r ≥
0. We call a sequence( α r ) r ≥ r of cochains α r ∈ C n ( G, M/p r N ) a cochain sequence if there are cochains ρ ∈ C n ( G, M ) and σ ∈ C n ( G, N ) and an ω ∈ { , , . . . , r } such that α r = pro r ( ρ + p r − ω σ ) ∈ C n ( G, M/p r N ) for all r ≥ r . OCHAIN SEQUENCES AND THE QUILLEN CATEGORY 5
Note that the cochain sequences defined by ( ρ, σ ; r , ω ) and ( ρ ′ , σ ′ ; r ′ , ω ′ ) areequal if and only if r = r ′ and the cochains pro r ( ρ + p r − ω σ ) and pro r ( ρ ′ + p r − ω ′ σ ′ )are equal as elements of C n ( G, M/p r N ) for all r ≥ r .Often r will be clear from the context. We then also write α • for ( α r | r ≥ r )and M/p • N for ( M/p r N | r ≥ r ). If α • is induced from ( ρ, σ ; r , ω ), then wealso write α • = pro • ( ρ + p •− ω σ ) Notation.
Often r and N will be fixed from the context. We then denote M r = M/p r N , we write M • for ( M r | r ≥ r ), and we denote with C nr ( G, M • ) the setof all cochain sequences which start at r .2.4. Homological algebra for cochain sequences.
Our next aim is to developsome elementary homological algebra for cochain sequences.
Notation . We continue to use the Notation 2.2, imposing minor additionalrestrictions. We assume from now on that R is a noetherian integral domain and p a prime number, which in R is neither zero nor a unit. Further, M is a finitelygenerated RG -module which is free as an R -module. Then T r p r M = { } byKrull’s Theorem [2, 10.17], and Ann N ( p ) = { } . Lemma 2.6.
The set C nr ( G, M • ) of cochain sequences is an R -module.Proof. Let α • be defined by ( ρ, σ ; r , ω ), and β • by ( ρ ′ , σ ′ ; r , ω ′ ). Then α r + β r = pro r (cid:16) ρ + ρ ′ + p r − ℓ ( p ℓ − ω σ + p ℓ − ω ′ σ ′ ) (cid:17) for ℓ = max( ω, ω ′ ),and so α • + β • is the cochain sequence defined by ( ρ + ρ ′ , p ℓ − ω σ + p ℓ − ω ′ σ ′ ; r , ℓ ).And for x ∈ R , xα • is the cochain sequence defined by ( xρ ; xσ ; r , ω ). (cid:3) Lemma 2.7.
Let α • ∈ C nr ( G, M • ) be the cochain sequence defined by ( ρ ; σ ; r , ω ) . (1) Either α • = 0 , that is α r = 0 for all r ≥ r ; or α r = 0 for all sufficientlylarge r . (2) α • = 0 if and only if ρ = 0 in C n ( G, M ) and σ lies in p ω C n ( G, N ) .Proof. If ρ = 0 and σ is not divisible by p ω , then p r − ω σ is not divisible by p r ,and so α r is non-zero for all r . If ρ = 0 then there is k such that pro k ( ρ ) = 0 in C n ( G, M/p k N ), and hence α r = 0 for all r ≥ k + ω . (cid:3) Notation.
Let α • ∈ C nr ( G, M • ). We define the level of α • to be the smallest valueof ω such that α • is defined by ( ρ, σ ; r , ω ) for some ρ, σ . Remark.
By definition, the cochain sequence defined by ( ρ, σ ; r , ω ) has level atmost ω . Note that ( ρ, σ ; r , ω ) and ( ρ, pσ ; r , ω + 1) define the same cochainsequence. Thus the level of the cochain sequence defined by ( ρ, σ ; r , ω ) can bestrictly smaller than ω . B. EICK AND D. J. GREEN
Definition.
Define C nr ( G, M • ) ∆ → C n +1 r ( G, M • ) by (∆ α ) r := ∆( α r ). That is, if α • is defined by ( ρ, σ ; r , ω ), then ∆( α • ) is defined by (∆( ρ ) , ∆( σ ); r , ω ). Further,write Z nr ( G, M • ) = ker(∆) and B n +1 r ( G, M • ) = Im(∆).The map ∆ is R -linear and satisfies ∆ = 0. Thus B nr ( G, M • ) ⊆ Z nr ( G, M • ). Remark.
By Lemma 2.7 we have ∆( α • ) = 0 if and only if ∆( ρ ) = 0 and ∆( σ ) isdivisible by p ω . So we may rephrase Proposition 2.4 as follows: Corollary 2.8.
Let n ≥ and r ≥ m . For every ¯ ρ ∈ H n ( G, M ) and every ¯ η ∈ H n +1 ( G, N ) there is a cocycle sequence α • ∈ Z nr ( G, M • ) of level at most m such that for every r ≥ r the cohomology class α r + B n ( G, M r ) ∈ H n ( G, M r ) corresponds under the isomorphism of Theorem 2.3 to ( ¯ ρ, ¯ η ) ∈ H n ( G, M ) ⊕ H n +1 ( G, N ) . (cid:3) Lemma 2.9.
Let n ≥ . Suppose that α • ∈ Z nr ( G, M • ) has level ω ≤ r − m .The following statements are equivalent: (1) α r ∈ B n ( G, M r ) for some value r ≥ r of r . (2) α r ∈ B n ( G, M r ) for every r ≥ r . (3) α • ∈ B nr ( G, M • ) . (4) α • = ∆( β • ) for some β • ∈ C n − r ( G, M • ) of level at most ω + m .Proof. The implications (4) ⇒ (3) ⇒ (2) ⇒ (1) are clear. (1) ⇒ (4): Let α • = pro • ( ρ + p •− ω σ ). Since α r ∈ B n ( G, M r ) there are φ ∈ C n − ( G, M ) and ψ ∈ C n ( G, N ) such that ρ + p r − ω σ = ∆( φ ) + p r ψ , and hence p r − ω ( σ − p ω ψ ) = ∆( φ ) − ρ . By Lemma 2.7 we may replace σ by σ − p ω ψ without altering α • . Hence p r − ω σ = ∆( φ ) − ρ . Now, the right hand side is a cocycle, since α • ∈ Z n ( G, M • ) means that ρ isa cocycle. Hence the left hand side lies in Z n ( G, N ). So σ ∈ Z n ( G, N ) byregularity of p , and therefore since p m H n ( G, N ) = 0 there is χ ∈ C n − ( G, N )with ∆( χ ) = p m σ . Hence ρ = ∆( λ ) for λ = φ − p r − ω − m χ ∈ C n − ( G, M ). So α • = ∆( β • ) for β • = pro • ( λ + p •− ω − m χ ). (cid:3) The next result will be needed in the proof of Lemma 4.4.
Lemma 2.10.
Let n ≥ and r ≥ r ≥ m . For each z ∈ Z n ( G, M r ) there isan α • ∈ Z nr ( G, M • ) of level at most m with α r = z .Proof. Let ξ be the element of H n ( G, M ) ⊕ H n +1 ( G, N ) corresponding to z + B n ( G, M r ) ∈ H n ( G, M r ) under the isomorphism of Theorem 2.3. By Corol-lary 2.8 there is some β • = pro • ( ρ + p •− m σ ) ∈ Z nr ( G, M • ) such that β r + B n ( G, M r ) corresponds to ξ for every r ≥ r . Hence z − β r ∈ B n ( G, M r ).Pick λ ∈ C n − ( G, M r ) with z = ∆( λ ) + β r , and choose ¯ λ ∈ C n − ( G, M )with pro r (¯ λ ) = λ . For ρ ′ = ρ + ∆(¯ λ ) ∈ Z n ( G, M ) we then have z = α r for α • = pro • ( ρ ′ + p •− m σ ). (cid:3) OCHAIN SEQUENCES AND THE QUILLEN CATEGORY 7 Coclass families of p -groups Coclass families are certain infinite families of finite p -groups of fixed coclass.Their construction has been introduced in [10] based on a version of Theorem2.3. Here we exhibit a construction based on Proposition 2.4. The constructiondiffers from [10] in that it uses cocycles rather than their corresponding cocycleclasses and thus is slightly more explicit. This difference will be essential in ourlater applications.Every coclass family of p -groups of coclass r is associated with an infinitepro- p -group S of coclass r . The structure of the infinite pro- p -groups of finitecoclass is well investigated. For example, it is known that for each such group S there exists natural numbers l and d so that the l -th term of the lower centralseries γ l ( S ) satisfies that γ l ( S ) ∼ = Z dp , where Z p denotes the p -adic numbers, and S/γ l ( S ) is a finite p -group of coclass r . The integer d is an invariant of S calledthe dimension of S . The integer l is not an invariant; in fact, each integer largerthan l can be used instead of l . The subgroup γ l ( S ) is often denoted by T andcalled a translation subgroup of S . Its subgroup series defined by T = T and T i +1 = [ T i , S ] satisfies [ T i : T i +1 ] = p for i ∈ N . Thus the series T = T > T > . . . is the unique series of S -normal subgroups in T , and T is called a uniserial S -module. We refer to [13] for many more details on the structure of the infinitepro- p -groups of coclass r .Given S and l , we write S i = S/γ l + i ( S ) for i ∈ N . Then S , S , . . . is aninfinite sequence of finite p -groups of coclass r . This sequence is called the mainline associated with S . The main line is not necessarily a coclass family it-self, but it always consists of d coclass families and finitely many other groups.More precisely, there exists an integer h ≥ l so that the d infinite sequences( S h + i , S h + i + d , S h + i +2 d , . . . ) for 0 ≤ i < d are coclass families. Note that the group S can be viewed as an extension of S h + i + jd by its natural module T h + i + jd for each h, i and j and the group S h + i + jd + k can be viewed as an extension of S h + i + jd byits natural module T h + i + jd /T h + i + jd + k for each h, i, j and k .For each coclass family ( G , G , . . . ) associated with the infinite pro- p -group S there exists an integer k so that each group G j is a certain extension of S h + i + jd with its natural module T h + i + jd /T h + i + jd + k . The extensions can be choosen so thatthe main line group S h + i + jd +1 is not a quotient of G j . In this case the integer k is an invariant of the coclass family called its distance to the main line.To describe the groups in a coclass family explicitly, it is more convenient to usea different type of extension construction. Instead of describing a group G j in acoclass family as extension of an associated main line group S h + i + jd by its naturalmodule of fixed size p k , we describe each G j as an extension of a fixed main linegroup S ℓ for some suitable ℓ by a module of variable size M j := T ℓ /T ℓ + jd . It isnot difficult to observe that T ℓ + jd = p j T ℓ and thus the group M j is isomorphic toa direct product of d copies of cyclic groups of order p j . B. EICK AND D. J. GREEN
We now use Proposition 2.4 to exhibit a complete construction for a coclassfamily ( G , G , . . . ) associated with the infinite pro- p -group S of coclass r . Forthis purpose let m = log p ( S ℓ ) = r + ℓ − j ≥ m + 1. Let ρ ∈ Z ( S ℓ , T ℓ )so that S is an extension of S ℓ with T ℓ via ρ . Definition.
There exists η ∈ Z ( S ℓ , T ℓ ) so that G j is an extension of S ℓ by M j via τ j where τ • = pro • ( ρ + p •− m σ ) and ∆( σ ) = p m η .The definition of coclass families asserts that for each coclass family thereexists an η yielding this family. It may happen that different cocycles η , η yieldcoclass families with pairwise isomorphic groups; for example, this is the caseif η ≡ η mod B ( S ℓ , T ℓ ). We note that every η ∈ Z ( S ℓ , M j ) yields a coclassfamily via the above construction.The significance of coclass families is underlined by the fact that for ( p, r ) =(2 , r ) or ( p, r ) = (3 ,
1) all but finitely many p -groups of coclass r are containedin a coclass family.4. Cochain sequences and elementary abelians
We now apply the results of Section 2.4 to the coclass family G • of Section 3.In the language of Notation 2.5 this means that M = T . It would be natural totake N = T as well, but for technical reasons we shall actually take N = pT .Hence M • = M/p • N = T /p • +1 T , and G • +1 = M • .P with extension cocycle τ • ∈ Z r ( P, M • ).Recall that if N E G and U ≤ G/N , then a lift of U is a subgroup ¯ U ≤ G suchthat the projection map G → G/N maps ¯ U isomorphically to U .Suppose that we are given H ≤ G r +1 . Setting K := H ∩ M r and Q := HM r /M r ,we see that H is an extension H = K.Q , with Q ≤ P , and K a Q -submoduleof M r . If K has a complement C in H – which is certainly the case if H iselementary abelian –, then C ≤ G r +1 is a lift of Q .4.1. Extension theory.
We recall some details of extension theory, see e.g. [3, § G be a finite group and M a left Z G -module. Recall that every groupextension Γ = M.G can be constructed using a 2-cocycle τ ∈ Z ( G, M ): theunderlying set is M × G , with multiplication( t , g )( t , g ) = ( t + g t + τ ( g , g ) , g g ) . Associativity is equivalent to the cocycle condition. The extension splits as asemidirect product Γ = M ⋊ G if and only τ ∈ B ( G, M ). If τ = ∆( f ) then G ( f ) = { ( − f ( g ) , g ) | g ∈ G } ≤ Γ is a lift of G , and every lift arises thus. It simplifies Remark 4.3 and especially Lemma 5.1 if
M/p r N and M/N have the sameelementary abelian subgroups.
OCHAIN SEQUENCES AND THE QUILLEN CATEGORY 9
Lemma 4.1. If f, f ′ ∈ C ( G, M ) satisfy ∆( f ) = τ = ∆( f ′ ) then f ′ − f ∈ Z ( G, M ) and moreover f ′ − f ∈ B ( G, M ) ⇐⇒ G ( f ) and G ( f ′ ) are conjugate by an element of M .Proof. This is Proposition 3.7.2 of [3]. Observe from the proof of that result thatconjugation by elements of M induces every coboundary. (cid:3) Lifting elementary abelians.Lemma 4.2.
Let Q ≤ P and suppose r ≥ m . Then the three following state-ments are equivalent: (1) Q has has a lift ¯ Q r ≤ G r +1 for all r ≥ r . (2) Q has has a lift ¯ Q r ≤ G r +1 for at least one r ≥ r . (3) τ • | Q = ∆( f • ) for some cochain sequence f • ∈ C r ( Q, M • ) of level atmost m .Proof. Let H r ≤ G r +1 be the subgroup with M r ≤ H r and H r /M r = Q . Then H r = M r .Q with extension class τ r | Q , and Q has a lift ¯ Q r ≤ G r +1 if and only if τ r | Q lies in B ( Q, M r ). Now apply Lemma 2.9. (cid:3) Now suppose that E ≤ G r +1 is elementary abelian. Setting K = E ∩ M r and U = EM r /M r ≤ P as above, we have E = K × ¯ U for a lift ¯ U ≤ G r +1 of U .Recall that ¯ U = U ( φ ) for some φ ∈ C ( U, M r ) with ∆( φ ) = τ r | U . Notation.
We shall need to refer to several different maps between cohomologymodules. Let L ⊆ M be a submodule. • Inclusion
L ֒ → M induces H n ( G, L ) inc −→ H n ( G, M ). • mul r : M/L → M/p r L , x + L p r x + p r L induces H n ( G, M/L ) mul r −−→ H n ( G, M/p r L ).Note that mul r ◦ mul s = mul r + s , and pro r ( ρ + p r − m σ ) = pro r ( ρ )+mul r − m pro m ( σ ). Remark . Since K ≤ M r = T /p r +1 T is elementary abelian, it follows that K ≤ Ω ( M r ) = p r T /p r +1 T (mul r ) − −−−−−→ ∼ = T /pT . So K = mul r ( W ) for some W ≤ T /pT . Since E is abelian, we have [ ¯ U , K ] = 1,which is equivalent to W ≤ ( T /pT ) U . Notation. E is the set of all triples ( U, f • , W ) with U ≤ P an elementary abelian; f • ∈ C r ( U, M • ) a cochain sequence of level at most 2 m such that ∆( f • ) = τ • | U ;and W ≤ ( T /pT ) U . Lemma 4.4.
Suppose that r ≥ r ≥ m . Every elementary abelian E ≤ G r +1 has the form E = E r ( U, f • , W ) := mul r ( W ) × U ( f r ) for some ( U, f • , W ) ∈ E . Proof.
We saw above that E = mul r ( W ) × U ( φ ) with U ≤ P elementary abelian; W ≤ ( T /pT ) U ; and φ ∈ C ( U, M r ) with ∆( φ ) = τ r | U . As U ( φ ) is a lift of U in G r +1 , there is f • ∈ C r ( U, M • ) of level at most 2 m with ∆( f • ) = τ • | U by Lemma 4.2. Hence φ − f r ∈ Z ( U, M r ), so by Lemma 2.10 there is z • ∈ Z r ( U, M • ) of level at most m with z r = φ − f r . So ( U, f • + z • , W ) ∈ E , and E = mul r ( W ) × U ( f r + z r ). (cid:3) Change of module
The following technical lemma is required in the proofs of Proposition 6.1 andLemma 7.1. We revert to Notation 2.5, and consider the case of two submodules
L, N ⊆ M satisfying the condition pL ⊆ N ⊆ L . Example.
If (
U, f • , W ) ∈ E , then W ≤ T /pT , and so W = L/pT for some pT ⊆ L ⊆ T . Hence pL ⊆ N ⊆ L , since N = pT .We shall investigate the cohomology maps induced by the short exact sequence0 → L/N mul r −−→ M/p r N → M/p r L → . As we now have to distinguish between two different projection maps, we shalldenote them by M pro Nr −−−→ M/p r N and M pro Lr −−→ M/p r L . Lemma 5.1.
Suppose the RG -submodule L ⊆ M satisfies pL ⊆ N ⊆ L . (1) Assume r ≥ . Then j ∗ ◦ i ∗ = 0 for the chain maps C ∗ ( G, L/N ) i ∗ −→ C ∗ r ( G, M/p • N ) j ∗ −→ C ∗ r ( G, M/p • L ) given by i n ( c ) r = mul r ( c ) and j n ( α • ) r = α r + p r L . (2) Suppose that α • ∈ Z nr ( G, M/p • N ) satisfies j n ( α • ) ∈ B nr ( G, M/p • L ) . If α • has level ω ≤ r − m , then α • = i n ( c ) • + ∆(pro N • ( κ + p •− ( ω + m ) λ )) for some c ∈ Z n ( G, L/N ) , κ ∈ C n − ( G, M ) and λ ∈ C n − ( G, L ) .Proof. (1): Pick ¯ c ∈ C n ( G, L ) with pro L (¯ c ) = c , then p ¯ c ∈ C n ( G, N ) and i n ( c ) • = pro N • (0 + p •− · p ¯ c ) ∈ C nr ( G, M/p • N ) , with level 1 ≤ r .If α • = pro N • ( ρ + p •− ω σ ) ∈ C nr ( G, M/p • N ), then j n ( α • ) = pro L • ( ρ + p •− ω σ ).Clearly i ∗ , j ∗ are chain maps. And j n i n = 0, since c takes values in L .(2): Let α • = pro N • ( ρ + p •− ω σ ), so j n ( α • ) = pro L • ( ρ + p •− ω σ ). By Lemma 2.9we have j n ( α • ) = ∆( γ • ) for some γ • ∈ C n − r ( G, M/p • L ) of the form γ • = pro L • ( κ + p •− ω − m λ ) with κ ∈ C n − ( G, M ), λ ∈ C n − ( G, L ).Applying Lemma 2.7 we have ρ = ∆( κ ), and p m σ = ∆( λ ) + p ω + m ¯ c for some ¯ c ∈ C n ( G, L ). OCHAIN SEQUENCES AND THE QUILLEN CATEGORY 11
From α • ∈ Z nr ( G, M/p • N ) it follows that ∆( σ ) takes values in p ω N , and so ∆(¯ c )takes values in N . So c := pro N (¯ c ) lies in Z n ( G, L/N ), and i n ( c ) r = pro Nr ( p r ¯ c ).Hence α • = pro N • ( ρ + p •− ω σ )= pro N • (∆( κ + p •− ( ω + m ) λ ) + p • ¯ c )= i n ( c ) + ∆(pro • ( κ + p •− ( ω + m ) λ )) . (cid:3) Morphisms in the Quillen category
Notation.
Consider the triple (
U, f • , W ) ∈ E . Recall from Section 2 that G r +1 has underlying set M r × P , and that U ( f r ) = { ( − f r ( u ) , u ) | u ∈ U } . So thesubgroup E r ( U, f • , W ) ≤ G r +1 of Lemma 4.4 is E r ( U, f • , W ) = { ( p r w − f r ( u ) , u ) | u ∈ U, w ∈ W } . Let j fr : W × U → E r ( U, f • , W ) be the isomorphism ( w, u ) ( p r w − f r ( u ) , u ). Proposition 6.1.
Suppose r ≥ m and m ≥ . For ( U, f • , W ) , ( U ′ , f ′• , W ′ ) ∈ E the set of isomorphisms W × U → W ′ × U ′ of the form W × U j fr −−−→ E r ( U, f • , W ) conjugation in G r +1 −−−−−−−−−−−→ E r ( U ′ , f ′• , W ′ ) ( j f ′ r ) − −−−−→ W ′ × U ′ is independent of r . For the proof we need two lemmas. Observe that mul r − r embeds M r = T /p r +1 T in G r +1 as p r − r T /p r +1 T ≤ M r . Lemma 6.2.
Suppose that m ≥ and r ≥ m . Let ( U, f • , W ) ∈ E . (1) Aut M r ( E r ( U, f • , W )) = Aut mul r − r ( M r ) ( E r ( U, f • , W )) . (2) The subgroup N = { x ∈ M r | mul r − r ( x ) ∈ N G r +1 ( E r ( U, f • , W )) } depends on neither r nor f • . Nor does the action of N on W × U obtainedby using j fr to identify E r ( U, f • , W ) with W × U . Write ¯ W for the module pT ⊆ ¯ W ⊆ T with W = ¯ W /pT . Proof. (1): Conjugation by t ∈ T fixes M r U ( f r ) /M r and W pointwise, and isdescribed by ∆( t ) ∈ B ( U, M r ): ( t, ( p r w − f r ( u ) , u ) = ( p r w − ∆( t )( u ) − f r ( u ) , u ) . If t normalizes mul r ( W ) × U ( f r ) ≤ G r +1 then ∆( t ) must take values in p r ¯ W .Hence ∆( t ) ∈ p r Z ( U, ¯ W ) ⊆ p r − m B ( U, ¯ W ). So there is ¯ v ∈ ¯ W such that∆( t ) = p r − m ∆(¯ v ), and pro r ( p r − m ¯ v ) = mul r − r pro r ( p r − m ¯ v ) ∈ mul r − r ( M r ) hasthe same conjugation action as t .(2): Conversely if v = ¯ v + p r +1 T ∈ M r then mul r − r ( v ) normalizes mul r ( W ) × U ( f r ) if and only if ∆( p r − r ¯ v ) ∈ Z ( U, T ) takes values in p r ¯ W , that is if ∆( v ) = mul r ( z ) for some z ∈ Z ( U, W ). And the action on W × U is then ( w, u ) ( w − z ( u ) , u ). (cid:3) Lemma 6.3.
Suppose r ≥ m and g ∈ P . Let ( U, f • , W ) , ( g U, f ′• , g W ) ∈ E .Define χ r ∈ C ( g U, M r ) by χ r ( v ) = ( g f r )( v ) − τ r ( g, v g ) + τ r ( v, g ) − f ′ r ( v ) . Then χ • ∈ Z r ( g U, M • ) is a cocycle sequence of level at most m .Proof. For c ∈ C n ( H, M ) we of course define g c ∈ C n ( g H, M ) by( g c )( k , . . . , k n ) = g c ( k g , . . . , k gn ) . Everything is of level at most 2 m . Since ∆( f ′• ) = τ • | g U and ∆( g f • ) = g ( τ • | U ) wehave∆( χ r )( v , v ) = g τ r ( v g , v g ) − τ r ( g, v g ) + τ r ( g, v g v g ) − τ r ( g, v g )+ τ r ( v , g ) − τ r ( v v , g ) + τ r ( v , g ) − τ r ( v , v )= ∆( τ r )( g, v g , v g ) − ∆( τ r )( v , g, v g ) + ∆( τ r )( v , v , g ) = 0 . (cid:3) Proof of Proposition 6.1. If ( t,g ) E r ( U, f • , W ) = E r ( U ′ , f ′• , W ′ ) then g U = U ′ and g W = W ′ . So we assume that g ∈ P is fixed, and consider which t ∈ M r satisfy ( t,g ) E r ( U, f • , W ) = E r ( g U, f ′• , g W ), and which isomorphisms arise in this way.The map F : W × U → g W × g U given by j fr , then conjugation by ( t, g ), and then( j f ′ r ) − must have the form F ( w, u ) = ( g w − π ( g u ) , g u ) for some π ∈ Z ( g U, g W ).So we consider g, π to be fixed and ask for which values of r there is t + p r +1 T ∈ M r realising this F . The condition on t, i can be phrased thus:( t, g )( p r w − f r ( u ) , u ) = ( p r · g w − p r π ( g u ) − f ′ r ( g u ) , g u )( t, g )Equality in g U is immediate. We are left with the following condition in M r : t + p r · g w − g ( f r ( u )) + τ r ( g, u ) = p r ( g w − π ( g u )) − f ′ r ( g u ) + ( g u ) t + τ r ( g u, g ) . So with χ r as in Lemma 6.3 we have p r π ( g u ) = ( g f )( g u ) − τ r ( g, u ) + τ r ( g u, g ) − f ′ r ( g u ) + ∆( t )( g u )= ( χ r + ∆( t ))( g u ) . That is, F is realisable for this r if and only if p r π − χ r ∈ B ( g U, M r ) . (1)Since π takes values in g W = g ¯ W /pT , a necessary condition for any such F tobe realisable is that χ r + p r · g ¯ W ∈ B ( g U, T /p r · g ¯ W ) . (2)Since χ • ∈ Z r ( g U, M • ) has level at most 2 m and r ≥ m + m , we deduce fromLemma 2.9 that (2) is either satisfied for all r , or for none.If (2) is satisfied then we apply Lemma 5.1 with G = g U , α • = χ • and L = g ¯ W ,hence L/N = g W . Note that χ • has level at most 2 m ≤ r − m . We conclude OCHAIN SEQUENCES AND THE QUILLEN CATEGORY 13 that there are c ∈ Z ( g U, g W ), κ ∈ C ( g U, T ) and λ ∈ C ( g U, g ¯ W ) with χ • =mul • ( c ) + ∆(pro • ( κ + p •− m λ )). We conclude that if we take π = c then Eqn. (1)is solvable for all r . That is, this one map F : W × U → g W × g U is independentof r . But all other maps for this value of g correspond to a M r -automorphismof U × W followed by F , and we saw in Lemma 6.2 that these isomorphisms areindependent of r too. (cid:3) Corollary 6.4.
Suppose e ≥ m and m ≥ . For ( U, f • , W ) , ( U ′ , f ′• , W ′ ) ∈ E thefollowing statements are equivalent: (1) E r ( U, f • , W ) and E r ( U ′ , f ′• , W ′ ) are G r +1 -conjugate for some r . (2) E r ( U, f • , W ) and E r ( U ′ , f ′• , W ′ ) are G r +1 -conjugate for every r .Proof. They are G r +1 -conjugate if and only if the set of isomorphisms in Propo-sition 6.1 is non-empty. But this set does not depend on r . (cid:3) Wrapping up the main theorem
Lemma 7.1.
Suppose r ≥ m . Let ( U, f • , W ) ∈ E . For each V ≤ W × U thereis ( U ′ , f ′• , W ′ ) ∈ E such that ∀ r : j fr ( V ) = E r ( U ′ , f ′• , W ′ ) . Moreover the map κ V : W ′ × U ′ j f ′ r −→ E r ( U ′ , f ′• , W ′ ) ֒ → E r ( U, f • , W ) ( j fr ) − −−−−→ W × U has image V and is independent of r .Proof. Take W ′ = V ∩ W and U ′ = { u ∈ U | uW ∈ V W/W } ≤ U . Then V /W ′ ∼ = U ′ and W ′ is a direct factor of V , so there is c ∈ Z ( U ′ , W ) with V = { ( w − c ( u ) , u ) | w ∈ W ′ , u ∈ U ′ } . Then j fr ( V ) = { ( p i + e − w − p i + e − c ( u ) − f r ( u ) , u ) | w ∈ W ′ , u ∈ U ′ } . Done with f ′• = f • | U ′ + i ( c ) • in the terminology of Lemma 5.1. In particular, κ V ( w, u ) = ( w − c ( u ) , u ). (cid:3) Corollary 7.2.
Suppose r ≥ m and m ≥ . For ( U, f • , W ) , ( U ′ , f ′• , W ′ ) ∈ E the set of homomorphisms W × U → W ′ × U ′ of the form W × U j fr −−−→ E r ( U, f • , W ) morphism in A p ( G r +1 ) −−−−−−−−−−−−−→ E r ( U ′ , f ′• , W ′ ) ( j f ′ r ) − −−−−→ W ′ × U ′ is independent of r .Proof. Every such map is an isomorphism to some V ≤ W ′ × U ′ . Lemma 7.1:For some ( U ′′ , f ′′• , W ′′ ) have j f ′ r ( V ) = E r ( U ′′ , f ′′• , W ′′ ) for all r . Proposition 6.1:The set I V of isomorphisms of the form W × U j fr −−−→ E r ( U, f • , W ) conjugation in G r +1 −−−−−−−−−−−→ E r ( U ′′ , f ′′• , W ′′ ) ( j f ′′ r ) − −−−−→ W ′′ × U ′′ is independent of r . But φ κ V ◦ φ is a bijection from I V to the set of homo-morphisms of the form W × U j fr −−−→ E r ( U, f • , W ) morphism in A p ( G r +1 ) −−−−−−−−−−−−−→ E r ( U ′ , f ′• , W ′ ) ( j f ′ r ) − −−−−→ W ′ × U ′ whose image is V . (cid:3) Proposition 7.3.
Suppose e ≥ m and m ≥ . Choose a subset E ⊆ E suchthat for every conjugacy class C of elementary abelian subgroups in G r +1 thereis exactly one ( U, f • , W ) ∈ E such that E r ( U, f • , W ) lies in C . Define C r to bethe full subcategory of the Quillen category A p ( G r +1 ) on the E r ( U, f • , W ) with ( U, f • , W ) in E . Then: (1) C r is a skeleton of A p ( G r +1 ) for every r ≥ r . (2) The categories C r are all isomorphic to each other.Hence the Quillen categories A p ( G r +1 ) are all equivalent to each other.Proof. E exists by Lemma 4.4. (1): Need to show that each conjugacy class C in G r +1 contains E r ( U, f • , W ) for precisely one ( U, f • , W ) ∈ E . Corollary 6.4:at most one such triple. Lemma 4.4: there is some ( U ′ , f ′• , W ′ ) ∈ E such that E r ( U ′ , f ′• , W ′ ) lies in C . By construction of E there is ( U, f • , W ) ∈ E suchthat E ( U, f • , W ), E ( U ′ , f ′• , W ′ ) are G -conjugate. So E r ( U, f • , W ) lies in C byCorollary 6.4.(2): For r, r ′ ≥ r and ( U, f • , W ) ∈ E have isomorphism λ frr ′ : E r ( U, f • , W ) ( j fr ) − −−−−→ W × U j fr ′ −→ E r ′ ( U, f • , W ) , mit λ fr ′ r = (cid:16) λ frr ′ (cid:17) − . For a morphism E r ( U, f • , W ) φ −→ E r ( U ′ , f ′• , W ′ ) in C r , define F ( φ ) in C r ′ thus: F ( φ ) : E r ′ ( U, f • , W ) λ fr ′ r −−→ E r ( U, f • , W ) φ −→ E r ( U ′ , f ′• , W ′ ) λ f ′ rr ′ −−→ E r ′ ( U ′ , f ′• , W ′ ) . This is a bijection C r ( E r ( U, f • , W ) , E r ( U ′ , f ′• , W ′ )) → C r ′ ( E r ′ ( U, f • , W ) , E r ′ ( U ′ , f ′• , W ′ ))by Corollary 7.2, and it is functorial since F (Id W × U r ( f ) ) = λ frr ′ Id λ fr ′ r = Id W × U r ′ ( f ) ,and for E r ( U ′ , f ′• , W ′ ) ψ −→ E r ( U ′′ , f ′′• , W ′′ ) in C r have F ( ψ ) F ( φ ) = λ f ′′ rr ′ ψλ f ′ r ′ r ◦ λ f ′ ir ′ φλ fr ′ r = λ f ′′ rr ′ ψφλ fr ′ r = F ( ψφ ) . This establishes (2). The last part follows from (1) and (2). (cid:3) Examples
Main line maximal class groups. If p is an odd prime then G ( p,
1) con-sists of one infinite tree, together with the isolated point C p : so there is onlyone uniserial p -adic space group of coclass one. We recall the construction of themain line groups from Example 3.1.5(ii) of [13].The p th local cyclotomic number field is K = Q p ( θ ), where θ has minimalpolynomial Φ p ( X ) = X p − X − . The ring of integers in K is O = Z p [ θ ], a free Z p -module of rank p − , θ, . . . , θ p − . The coclass one uniserial p -adic OCHAIN SEQUENCES AND THE QUILLEN CATEGORY 15 space group is then G := O ⋊ C p , where the generator τ of C p acts as multiplicationby θ ; that is, τ v = θv for v ∈ O .The valuation ring O has unique maximal ideal α O , where α = θ −
1. So γ i ( G ) = α i − O for i ≥
2; and by considering Φ p ( X + 1) one observes that p O = α p − O . Since 1 , α, . . . , α p − is a Z p -basis of O , this means that O /α O ∼ = F p ,and hence γ i ( G ) /γ i +1 ( G ) ∼ = C p for i ≥ G i = G/γ i ( G ). These main line groupsfall into p − ≤ r ≤ p − i th group in the r thfamily is G r +( p − i . From [8] (see also Proposition 7.3) we know that all groupsin one coclass family have equivalent Quillen categories. But here a strongerresult holds: all p − Lemma 8.1.
For this group G = O ⋊ C p , the Quillen category of G/γ i ( G ) isindependent (up to equivalence of categories) of i for i ≥ p + 1 .Remark. For p = 3, the first author and S. King [9] have shown that G/γ ( G ), G/γ ( G ) and G/γ ( G ) have isomorphic cohomology rings; and that these differfrom the cohomology ring of G/γ ( G ) ∼ = 3 . Proof. If v ∈ O then ( vτ ) p = (Φ p − ( θ ) · v ) τ p = 1, and so vτ has order p . So since p O = α p − O , there are two kinds of order p elements of G/γ i ( G ): • Elements of the form vτ r γ i ( G ), with v ∈ O and 1 ≤ r ≤ p −
1; and • Elements of γ i − p +1 ( G ) /γ i ( G ) ∼ = ( C p ) p − .Moreover the conjugacy class of vτ in G is { wτ | w ∈ v + α O} ; and the centralizerof vτ γ i ( G ) in G/γ i ( G ) is elementary abelian of rank 2, generated by vτ γ i ( G ) and γ i − ( G ) /γ i ( G ). So as τ acts on γ i − p +1 ( G ) /γ i ( G ) = α i − p O /α i − O as multiplicationby 1+ α , the objects of the Quillen category form the following equivalence classes: • The class of h v j τ γ i ( G ) , γ i − ( G ) /γ i ( G ) i ∼ = C p for some fixed transversal v , . . . , v p of O /α O ; • The class of h v j τ γ i ( G ) i ∼ = C p for the same transversal v , . . . , v p ; and • The conjugacy classes of subgroups of γ i − p +1 ( G ) /γ i ( G ) ∼ = O /α p − O ∼ = C p − p under the action of C p given by multiplication by 1 + α .So the equivalence classes of objects admit a description which is independentof i . From this description and the fact that h v j τ γ i ( G ) , γ i − ( G ) /γ i ( G ) i has nor-malizer h v j τ γ i ( G ) , γ i − ( G ) /γ i ( G ) i , it follows that the morphisms between theserepresentatives also admit a description which is independent of i . (cid:3) Since p ∈ α O we may always take the transversal v , . . . , v p to be 0 , , . . . , p − p = 3 and i = 4 the Quillen category has skeleton1 h α i h τ i h τ i h τ ih α , α i h τ, α i h τ, α i h τ, α ih α i where the three automorphisms of each rank two elementary abelian are omittedfor clarity. Specifically, the three maps h τ i → h τ, α i are 2 τ (2 + λα ) τ for λ = 0 , ,
2; and three three autormorphisms of h τ, α i fix α and act on 2 τ asone of these three maps.8.2. A more substantial example.
Together with Leedham–Green, Newmanand O’Brien, the first author studied the 3-groups of coclass two in [11]. Inparticular they construct the skeleton groups in the four coclass trees (out ofsixteen) whose branches have unbounded depth. Here we consider the skeletongroups in one of these unbounded depth trees: the tree associated to the pro-3-group which they denote as R (see their Theorem 4.2(a)).We briefly recall the construction of the skeleton groups R j − ,γ,m from [11,Sect. 5]. Let j ≥
7. Let K = Q ( θ ) be the ninth local cyclotomic number field,so θ is a root of Φ ( X ) = X + X + 1. Let O be the ring of integers in K ;then O = Z [ θ ] is free as a Z -module, with basis 1 , θ, . . . , θ . Moreover, O is alocal ring, with maximal ideal p = ( θ − O . Observing that ( θ − and 3 areassociates, one sees that 3 O = p .We now recall the twisting map p ∧ p → O , which we shall denote by γ . Notehowever that in [11] it is called ϑ . It is the map γ ( x ∧ y ) = σ ( x ) σ − ( y ) − σ − ( x ) σ ( y ) , where the automorphism σ r ∈ Gal( K/ Q ) is given by σ r ( θ ) = θ r . Lemma 5.1of [11] shows that γ ( p i ∧ p j ) = p i + j + ε for ε = ( i ≡ j (mod 3)2 otherwise . Pick j ≥ T = p j − , T ℓ = p j − ℓ . Then γ ( T ∧ T ) = T j , and γ ( T j ∧ T ) = T k for k = ( j | j j − ∤ j . OCHAIN SEQUENCES AND THE QUILLEN CATEGORY 17
Now pick a unit c ∈ O × and set γ = cγ . For any m ∈ { j, j + 1 , . . . , k } onedefines T j − ,γ,m to be the group with underlying set T /T m and product( x + T m ) ∗ ( y + T m ) = (cid:18) x + y + 12 γ ( x ∧ y ) (cid:19) + T m . Finally one sets R j − ,γ,m = T j − ,γ,m ⋊ C , where C = h τ i has order 9 and actson T via τ v = θv for v ∈ T . Note that | R j − ,γ,m | = 3 m +2 . Lemma 8.2.
Let v, w ∈ T . (1) ( v + T m ) r = rv + T m in T j − ,γ,m for all r ∈ Z . (2) The order elements of R j − ,γ,m are: • Elements of the form ( v + T m ) τ r , with v ∈ T and r ∈ { , } ; • Elements of the form v + T m with v ∈ T m − . (3) If v + T m has order then γ ( v ∧ w ) ∈ T m for all w ∈ T . Hence Ω ( T j − ,γ,m ) ≤ Z ( T j − ,γ,m ) .Proof. (1): Follows by induction, since γ ( v ∧ rv ) = rγ ( v ∧ v ) = 0.(2): Firstly, [( v + T m ) τ ] = (1+ θ + θ ) v + T m = 0. Secondly: ( v + T m ) = 3 v + T m .This is zero for v ∈ − T m = T m − .(3): Suppose that v ∈ T m − and w ∈ T . Then γ ( v ∧ w ) lies in T j + m − ε , with ε ∈ { , } . Since ε ≥ j ≥
7, this means that γ ( v ∧ w ) lies in T m . (cid:3) Lemma 8.3. (1)
The orbit of ( v + T m ) τ under conjugation by T j − ,γ,m is (cid:8) ( v ′ + T m ) τ (cid:12)(cid:12) v ∈ v + T (cid:9) . (2) ( v + T m ) τ and ( w + T m ) τ are conjugate in R j − ,γ,m if and only if v + T and w + T lie in the same orbit under the action of C on T /T . (3) The action of R j − ,γ,m on T m − /T m factors through C , and coincides wththe action of C on T /T via the isomorphism v + T ( θ − m − v + T m . (4) C T j − ,γ,m (( v + T m ) τ ) = T m − /T m .Proof. (1): Since T m and the image of γ lie in T j ⊆ T we have ( w + T m ) [( v + T m ) τ ] = [( w + T m ) ∗ ( v + T m ) ∗ ( − θ w + T m )] τ ∈ (cid:0) v + (1 − θ ) w + T (cid:1) τ . Since p = ( θ − O and 3 O = p it follows that (1 − θ ) T = (1 − θ ) T = p T = T .So for each v ′ ∈ v + T we find w ∈ T with ( w + T m ) [( v + T m ) τ ] = ( v ′′ + T m ) τ and v ′′ ∈ v ′ + T . If we now adjust w by adding u ∈ T r , then since γ ( T ∧ T r ) = T j + r − ε ⊆ T r +6 we alter v ′ by an element of (1 − θ ) u + T r +6 . So if the error v ′′ − v ′ lies in T s +3 , then with one correction we can reduce to an error in T s +6 .Iterating reduces the error to an element of T m .(2): Follows from (1). (3): The action factors by Lemma 8.2 (3). Thesecond statement follows, since C acts as multiplication by θ .(4): Follows from (3), since T /T is the subspace of T /T consisting of elementsfixed by τ . (cid:3) Let d be the number of orbits for the action of C on T /T . Pick v , . . . , v d ∈ T such that the v i + T fom a set of orbit representatives for this action. Lemma 8.4.
Every maximal elementary abelian subgroup of R j − ,γ,m is conjugateto precisely one of the following: (1) d rank four groups of the form V i = h ( v i + T m ) τ i × T m − /T m ; (2) V = T m − /T m of rank six.If U ≤ V i is not contained in V , then it is not conjugate to a subgroup of anyother V j .Proof. Any elementary abelian outside T m − /T m must contain some element ofthe form ( v + T m ) τ and is therefore contained in h ( v + T m ) τ i × C T j − ,γ,m (( v + T m ) τ ), that is h ( v + T m ) τ i× T m − /T m . Since m ≥ j ≥ m − ≥ (cid:3) Theorem 8.5.
Up to equivalence of categories, the Quillen category of the skele-ton group R j − ,γ,m is independent of j, γ, m .Proof. V is a normal subgroup, and Lemma 8.3 (3) describes the conjugationaction. So by the last part of Lemma 8.4 it suffices to show that if U ≤ V i isnot contained in V , then the set of homomorphisms U → V i lying in the Quillencategory is independent of j, γ, m .So U = h ( v + T m ) τ i × A for some A ≤ T m − /T m and some v ∈ v i + T m − .Consider conjugation by ( u + T m ) τ r : by Lemma 8.3 this can only send ( v + T m ) τ to an element of V if θ r v i lies in v i + T ; and if θ r v i does lie there, then byadjusting u we may send ( v + T m ) τ to any element of the form ( v ′ + T m ) τ with v ′ ∈ v i + T m − . Moreover, the restriction to A of conjugation by ( u + T m ) τ r onlydepends on r . (cid:3) The generalized quaternion groups.
Let G be a finite group, and k afield of characteristic p . Write¯ H ∗ ( G, k ) = lim E ∈ A p ( G ) H ∗ ( E, k ) . Quillen [16, Th. 6.2] proved that the induced homomorphism φ G : H ∗ ( G, k ) → ¯ H ∗ ( G, k ) induces a homeomorphism between prime ideal spectra.Our result shows that if G r is a coclass family, then ¯ H ∗ ( G r , k ) is independentof r . However this does not mean that the map φ G r is an isomorphism for large r .The (generalised) quaternion groups Q n ( n ≥
3) provide a good example. One easily verifies that d = 11. OCHAIN SEQUENCES AND THE QUILLEN CATEGORY 19
The quaternion groups form a coclass sequence. The mod-2 cohomology ring H ∗ ( Q n , F ) is well-known : H ∗ ( Q , F ) ∼ = F [ x, y, z ] / ( x + xy + y , x y + xy ) H ∗ ( Q n , F ) ∼ = F [ x, y, z ] / ( x + xy, y ) ( n ≥ , with x, y ∈ H and z ∈ H . Since H ( G, F ) = Hom( G, F ) and all ordertwo elements lie in the Frattini subgroup, it follows that x, y ∈ ker( φ Q n ). In fact¯ H ∗ ( Q n , F ) ∼ = F [ z ], since z restricts to the central C as t ∈ H ∗ ( C , F ) ∼ = F [ t ]:see Rusin’s construction of z as a top Stiefel–Whitney class [17, p. 316]. So both H ∗ ( Q n , F ) and ¯ H ∗ ( Q n , F ) are constant for n ≥
4, but φ Q n is never injective.In fact one can demonstrate that φ Q n is never injective without even knowingthe cohomology of Q n . Recall that a class x ∈ H n ( G, k ) is called essential if itsrestriction to every proper subgroup
H < G vanishes: so if G is not elementaryabelian, then every essential class lies in the kernel of φ G . Now, Adem andKaragueuzian showed [1] that H ∗ ( G, F p ) is Cohen–Macaulay and has non-zeroessential elements if and only if G is a p -group and all order p elements are central.As Q n satisfies this group-theoretic condition, it follows that ker( φ Q n ) = 0. References [1] A. Adem and D. Karagueuzian. Essential cohomology of finite groups.
Comment. Math.Helv. , 72(1):101–109, 1997.[2] M. F. Atiyah and I. G. Macdonald.
Introduction to commutative algebra . Addison-WesleyPublishing Co., Reading, Mass.-London-Don Mills, Ont., 1969.[3] D. J. Benson.
Representations and Cohomology. I . Cambridge Studies in Advanced Math.,vol. 30. Cambridge University Press, Cambridge, second edition, 1998.[4] H. Cartan and S. Eilenberg.
Homological algebra . Princeton University Press, Princeton,N. J., 1956.[5] M. Couson.
On the character degrees and automorphism groups of finite p -groups by coclass .PhD thesis, TU Braunschweig, 2013.[6] B. Eick. Automorphism groups of 2-groups. J. Algebra , 300(1):91–101, 2006.[7] B. Eick and D. Feichtenschlager. Computation of low-dimensional (co)homology groups forinfinite sequences of p -groups with fixed coclass. Internat. J. Algebra Comput. , 21(4):635–649, 2011.[8] B. Eick and D. J. Green. The Quillen categories of p -groups and coclass theory. Israel J.Math. , 206(1):183–212, 2015.[9] B. Eick and S. King. The isomorphism problem for graded algebras and its applicationto mod- p cohomology rings of small p -groups. Submitted, Mar. 2015. arXiv:1503.04666[math.RA].[10] B. Eick and C. Leedham-Green. On the classification of prime-power groups by coclass. Bull. London Math. Soc. , 40:274–288, 2008. To our knowledge the earliest references are [4, p. 253-4] for the additive structure and[15, p. 244] for the ring structure. By 1987, Rusin [17, p. 316] could quote the result withoutneeding to give a reference. [11] B. Eick, C. R. Leedham-Green, M. F. Newman, and E. A. O’Brien. On the classificationof groups of prime-power order by coclass: the 3-groups of coclass 2.
Internat. J. AlgebraComput. , 23(5):1243–1288, 2013.[12] L. Evens.
The cohomology of groups . Oxford Univ. Press, Oxford, 1991.[13] C. R. Leedham-Green and S. McKay.
The structure of groups of prime power order , vol-ume 27 of
London Mathematical Society Monographs. New Series . Oxford University Press,Oxford, 2002. Oxford Science Publications.[14] C. R. Leedham-Green and M. F. Newman. Space groups and groups of prime-power order.I.
Arch. Math. (Basel) , 35(3):193–202, 1980.[15] H. J. Munkholm. Mod 2 cohomology of D n and its extensions by Z . In Conf. on AlgebraicTopology (Univ. of Illinois at Chicago Circle, Chicago, Ill., 1968) , pages 234–252. Univ. ofIllinois at Chicago Circle, Chicago, Ill., 1969.[16] D. Quillen. The spectrum of an equivariant cohomology ring. I, II.
Ann. of Math. (2) , 94:573–602, 1971.[17] D. J. Rusin. The mod-2 cohomology of metacyclic 2-groups.
J. Pure Appl. Algebra , 44(1-3):315–327, 1987.
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