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K(2)-LOCAL SPHERE AT THE PRIME 2
Hans-Werner Henn
To cite this version:
Hans-Werner Henn. THE CENTRALIZER RESOLUTION OF THE K(2)-LOCAL SPHERE AT THE PRIME 2. 2018. �hal-01697478�
THE PRIME 2.
HANS-WERNER HENN
Abstract. LetG2be the Morava stabilizer group at the prime 2. We construct a resolution of theK(2)-local sphere at the prime 2 in terms of certain homotopy fixed point spectra which are closely related to the spectrum of topological modular forms. This resolution is in certain ways analogous to the centralizer resolution of theK(n)-local sphere constructed in [18] ifpis an odd prime andn=p−1.
Contents
1. Introduction 2
1.1. Preliminaries on Morava stabilizer groups atn=p= 2 2
1.2. Main results 5
2. Important finite subgroups for Morava stabilizer groups atn=p= 2 7
3. The mod-2 cohomology algebra ofP S21 10
3.1. Quillen’sF-isomorphism for the mod-pcohomology of a profinite group 10
3.2. The Quillen category ofP S21 11
3.3. Quillen’sF-isomorphism forP S21 and the mod-2 cohomology algebra ofP S12 13
4. Algebraic centralizer resolutions 15
4.1. Galois-twisted modules 15
4.2. The algebraic centralizer resolution forPG12 andPG2 17
5. Realizing the centralizer resolutions 22
5.1. Preliminaries on Morava modules 22
5.2. Realizing the centralizer resolution forG12 and forG2 27
References 31
Date: January 31, 2018.
The origin of this paper goes back to the early 2000’s and was stimulated by the joint work with P. Goerss, M. Mahowald and C. Rezk [14]. Some of the results were announced in [18] but proofs of the existence of the centralizer resolutions were never published. Recent work by Beaudry [2], [3], [4] by Bobkova and Goerss [5] and a joint project with Beaudry and Goerss [6] have underlined the importance of these resolutions. The author apologizes for the delay in making these results available and he is happy to acknowledge helpful discussions with Goerss, Mahowald, Rezk, Beaudry and Bobkova which have lead to this research and to improvements of the original results and simplifications of the proofs.
1
1. Introduction
Letpbe a prime, letn >0 be an integer and let K(n) be then-th MoravaK-theory atp.
The category of K(n)-local spectra is a basic building block of the stable homotopy category ofp-local spectra and theK(n)-localization of the sphere,LK(n)S0, plays a central role in this category. The homotopy ofLK(n)S0 can be studied by the Adams-Novikov spectral sequence, and by [10] this spectral sequence can be identified with the homotopy fixed point spectral sequence for the action of the extended Morava stabilizer group Gn on En. Here En denotes the 2-periodic Landweber exact spectrumEnwhose coefficients in degree 0 classify deformations (in the sense of Lubin and Tate) of a suitable formal group law Γn of heightnoverFpnandGn
is the automorphism group of Γn in the category of formal group laws (cf [28]). TheE2-page of this spectral sequence is given by the continuous cohomology Hcts∗ (Gn,(En)∗) of Gn with coefficients in (En)∗. It becomes therefore interesting to find resolutions of the trivial module for the groupGn from which one can calculate this continuous cohomology.
Ifpis large with respect tonthen theE2-page satisfiesEs,∗2 = 0 for∗> n2and the spectral sequence collapses at itsE2-page. In the sequel we concentrate on the casen= 2 because the casen= 1 is well understood and very little is understood in explicit terms ifn >2.
Forn= 2 the spectral sequence collapses if and only if p >3. In these cases the homotopy of LK(2)S0 has been calculated in [29] without using the point of view of group cohomology.
The results have been reinterpreted in [7] and an independant calculation for the Moore space has been carried out in [24] by using an explicit projective resolution of length 4 of the trivial G2-moduleZp.
Ifn= 2 andp≤3 the mod-pcohomological dimension of the groupG2 is infinite and there cannot be any projective resolution of the trivialG2-moduleZp of finite length. However, for p= 3 very useful resolutions of the trivial G2-moduleZp of length 4 in terms of more general modules and corresponding topological resolutions ofLK(2)S0 exist; a “duality resolution” has been constructed in [14] and a “centralizer resolution” in [18]. These resolutions complement each other and they have been crucial in recent progress of our understanding of K(2)-local homotopy theory at the prime 3. In particular they have been used for proving the chromatic splitting conjecture forn= 2 [13], for determining Hopkins’ Picard group ofK(2)-local spectra [22], [15] and for identifying the Brown-Comenetz dual of theK(2)-local sphere [16].
If n = 2 and p = 2 our understanding is less complete although the chromatic splitting conjecture has already been successfully analyzed in [4] and [6] by heavily using the algebraic and topological duality resolution for an important subgroup S12 of G2. The existence of an algebraic duality and an algebraic centralizer resolution of length 3 forS12was already announced in [18], as well as a topological centralizer resolution for the homotopy fixed point spectrum E2hS12, in all cases without proofs. For the algebraic duality resolution the construction was finally established in [2] and the construction of its topological counterpart was given in [5].
The latter paper relied heavily on the existence of both the algebraic and topological centralizer resolution forS12for which no proof has been published yet. The main purpose of this paper is to fill this gap in the literature and extend the announced results from the groupS12to S2and even toG2. Such extensions appear to be impossible for the algebraic and topological duality resolutions.
1.1. Preliminaries on Morava stabilizer groups at n=p= 2.
1.1.1. Let Γ be a formal group law of height n defined over Fp, let q = pn and assume that the automorphism groupSn(Γ) := AutFq(Γ) is isomorphic toSn :=Sn(ΓH), the automorphism
group of the Honda formal group law.1 Because the formal group law is defined over Fp the Galois group Gal of the extension Fp ⊂Fq acts on Sn(Γ) and we get extended automorphism groups
Gn(Γ) =Sn(Γ)oGal.
For n =p= 2 there are two important candidates for Γ. In fact, there are two particularly interesting formal group laws Γ of height 2 over the prime field F2: the Honda formal group law ΓH, i.e. the [2]-typical formal group law with [2]-series [2]ΓH(x) =x4, and the formal group law ΓE of the supersingular elliptic curve over F2 with affine equation y2+y = x3. In the remainder of this introduction Γ always refers to either ΓH or to ΓE.
IfF2denotes the algebraic closure of F2 then the endomorphism rings of both formal group laws satisfy
EndF4(Γ)∼= EndF2(Γ),
and because both formal group laws become isomorphic overF2 their endomorphism rings are already isomorphic over F4. Consequently the automorphisms groups S2(Γ) = AutF4(Γ) of these two formal group laws over the fieldF4 are abstractly isomorphic. If Γ = ΓH this group is the classical second Morava stabilizer group atp= 2 and usually denoted S2, andG2(Γ) is usually called the extended Morava stabilizer group and denoted G2. While the groups S2(Γ) are abstractly isomorphic this ceases to be true for the groupsG2(Γ) (cf. Lemma 2.2).
The endomorphism rings EndF4(Γ) contain W, the ring of Witt vectors of F4. They are generated as a non-commutative W-algebra by the endomorphism ξΓ ∈ EndF4(Γ) given by ξΓ(x) =x2. In order to describe the endomorphism rings more explicitly we denote the image ofw∈Wwith respect to the lift of the Frobenius automorphism ofF4byσwand we abbreviate ξΓsimply byξ. Then the canonical algebra map from the free non-commutativeW-algebraWhξi generated byξto EndF4(Γ) induces an isomorphism
(1.1) Whξi/(ξw−σwξ, ξ2−2u)∼= EndF4(Γ) where
(1.2) u=
(1 Γ = ΓH
−1 Γ = ΓE .
An explicit isomorphism between the two rings is given by theW-algebra map which sendsξ to ξy where we can take for y any element inW with the property yyσ =−1 (cf. [2] for an explicit choice ofy).
The ideal generated byξis a two-sided maximal idealmwith quotientF4and the endomor- phism rings are complete with respect to them-adic topology. This also defines a filtration on the groupS2(Γ) indexed by half integers i2 ≥0 given by
Fi 2 :=Fi
2S2(Γ) :={g∈S2(Γ)| g≡1 mod (ξi)}
and successive quotients
Fi 2/Fi+1
2
∼= (
F×4 i= 0 F4 i= 0. The group
S2(Γ) :=Fi 2S2(Γ) is a profinite 2-group, the normal 2-Sylow subgroup ofS2(Γ).
1This is equivalent to the endomorphismξΓgiven byξΓ(x) =xpsatisfyingξΓn=pu(cf. Remark 5.2).
Inverting 2 in the endomorphism rings gives two isomorphic division algebras which we denote byD2(Γ). They containQ2as their center and are of dimension 4 overQ2. The division algebras are equipped with a valuation
v:D2(Γ)× → 1 2Z
which extends the valuation onQp which is normalized byv(p) = 1.
The group of unitsD2(Γ)× ofD2(Γ) containsS2(Γ) as the group of elements of valuation 0 and from (1.1) it is clear that the action of the Galois group onS2(Γ) is realized by conjugation byξΓ in D2(Γ)×. Therefore we get canonical isomorphisms
(1.3) G2(Γ)∼=D2(Γ)×/hξΓ2i ∼= (
D2(Γ)×/h2i Γ = ΓH
D2(Γ)×/h−2i Γ = ΓE .
The groups S2(Γ) and G2(Γ) contain −1 as unique central element of order 2 and dividing out by the subgroupC2generated by it gives us quotient groups which we will denote PS2(Γ) andPG2(Γ). From (1.3) it is clear we have isomorphisms
PG2(ΓH)∼=D2(ΓH)×/h2,−1i ∼=D2(ΓE)×/h−2,−1i ∼=PG2(ΓE).
1.1.2. From (1.1) we see that EndF4(Γ) is a freeW-module with basis 1 andξ. Right multipli- cation inducesW-linear maps and the determinant gives a multiplicative homomorphism
det : EndF4(Γ)→W
which, in fact takes its values inZ2. It is explicitly given as follows: ifa, b∈Wthen det(a+bξΓ) =aaσ−2ubbσ
withu= 1 as in (1.2). This determinant induces an epimomorphism det :S2(Γ)→Z×2
which is often also called the reduced norm. Finally we get an epimorphism given as composition G2(Γ) =S2(Γ)oGaldet×id−→ Z×2 ×Gal→Z×2 →Z×2/{±1}
in which the second and third part are given as the obvious projections. LetG12(Γ) be the kernel of this composition and S12(Γ) resp. S21 its intersection with S2(Γ) resp. S2(Γ). We observe that the action of Gal on S2(Γ) leaves S12(Γ) invariant and G12(Γ) is equal to the semidirect productS12(Γ)oGal. By the definition ofG12(Γ) it is clear that every finite subgroup ofG2(Γ) is contained inG12(Γ).
The central element−1 = 1−uξ2Γ(whereuis as in (1.2)) is contained inS21(Γ) and generates a central subgroupC2 of order 2. IfH is any closed subgroup ofG2(Γ) containingC2then we will denote the quotientH/C2 byP H.
1.1.3. The groupsS12(Γ),PS12(Γ),G12(Γ),PG12(Γ) andP S21(Γ) contain certain finite subgroups which figure in the statements of our main results. In all cases except that of G12(Γ) the isomorphism type of the ambient group is independent of Γ and only when we discuss finite subgroups ofG12(Γ) the choice of Γ matters. In the other cases we will therefore from now on omit Γ from our notation.
IfF is a finite subgroup ofG12(Γ) which contains the centralC2 and for whichF0:=F∩S12
is of index 2 inF then we have a commutative diagram of groups with exact rows
(1.4)
1 −→ F0 −→ F −→ Gal −→ 1
↓ ↓ ↓=
1 −→ P F0 −→ P F −→ Gal −→ 1 .
In the following table we give a list of closed subgroupsF ⊂G12(Γ) and the corresponding groups P F ⊂ PG12, F0 ⊂ S12, P F0 ⊂PS12 and P F0∩P S21 ⊂P S21 which will be relevant for stating our main results. Subgroups ofP S21 will play an important role in section 4.2.
(1.5)
F G12(Γ) G48(Γ) G048(Γ) G12(Γ) C8 C2×Gal P F PG12 S4 S04 S3 C4 Gal F0 S12 G24 G024 C6 C4 C2 P F0 PS12 A4 A04 C3 C2 {1}
P F0∩P S21 P S21 E2 E20 {1} C2 {1}
We refer to Section 2, in particular Lemma 2.2, Lemma 2.3, Lemma 2.4 and Lemma 2.5 for more details on this table. Here we are content to explain that in this tableCn denotes a cyclic group of ordern, Gal is the Galois group of the extensionF2⊂F4,SnandS0ndenote symmetric groups onnletters,A4andA04alternating groups on 4 letters andE2andE20 groups isomorphic toC2×C2. The groupsG24 andG024 are groups of order 24 both isomorphic toSL2(F3). The isomorphism type of the groupsF =G48(Γ), F=G048(Γ) andF =G12(Γ) depends on Γ. The first two are maximal subgroups of G12(Γ) of order 48 which are non-conjugate in G12(Γ) but become conjugate inG2(Γ). In fact, we have (cf. Lemma 2.2)
G48(Γ)∼=G048(Γ)∼=
(GL2(F3) Γ = ΓE
O48 Γ = ΓH
whereO48denotes the binary octahedral group. For the groupsG12(Γ) we get (cf. Lemma 2.3) G12(Γ)∼=
(C2×S3 Γ = ΓE C3oC4 Γ = ΓH
whereC3oC4denotes the semidirect of C3 withC4acting non-trivially onC3. 1.2. Main results.
Let G be a profinite group, let X be a profinite G-set such that X = limiXi with finite G-setsXi and letWbe the ring of Witt vectors for a finite fieldkof order q=pn for a prime pand an integer n >0. We define
(1.6) W[[X]] = limi,kW/pk[[Xi]] .
Suppose thatGis equipped with a continuous homomorphismφ:G→Gal to the Galois group Gal of the extensionFp⊂Fq.
TheGalois-twisted completed group ring Wφ[[G]] ofGis the W-moduleW[[G]] with multi- plication induced by (w1g1)(w2g2) =w1g1w2g1g2 ifg1, g2 ∈G,w1, w2 ∈W and if g1w2 is the result of the Galois action ofφ(g1) onw2. A p-profinite Wφ[[G]]-module will also be called a Galois-twisted p-profinite G-module, or simply a Galois-twisted profinite G-module ifp is un- derstood from the context. In order to keep notation simple we will write W[[G]] instead of Wφ[[G]].
Analogous to [18] we introduce relative homological algebra in the context of Galois-twisted p-profiniteG-modules. LetF(G) be the set of conjugacy classes of finite subgroups of Gand
assume that F(G) is a finite set. A Galois-twisted p-profiniteG-module P will be called F- projective if it is a direct summand in a module of the formL
(F)W[[G]]⊗W[F]M where each MF is ap-profinite2 W[F]-module and the direct sum is indexed by conjugacy classes of finite subgroups ofG. In the sequel we will also writeM ↑GF instead ofW[[G]]⊗W[F]M.
The class ofF-projective Galois-twistedp-profiniteG-modules determines in the usual way a class ofF-exact sequences: a sequence of Galois-twistedp-profiniteG-modulesM0→M →M00 is calledF-exact if the compositionM0→M00is trivial and
HomW[[G]](P, M0)→HomW[[G]](P, M)→HomW[[G]](P, M00)
is an exact sequence of abelian groups for eachF-projective Galois-twistedp-profiniteG-module P.
AnF-resolution of a Galois-twistedp-profiniteG-module M is a sequence of Galois-twisted p-profiniteG-modules
. . .→P1→P0→M →0
where each Pi is F-projective and each 3-term subsequence is F-exact. We note that F- exactness is equivalent to the complex being split when restricted to any finite subgroup of G.
Here is the main algebraic result of this paper in whichWis now the ring of Witt vector of F4 and the subgroups ofPG2are those of table (1.5).
Theorem 1.1. There exists anF-resolution of the trivial Galois-twisted profinitePG2-module W
0−→W↑PSG32 −→∂4 W↑PSG32⊕W↑PGalG2−→∂3 W↑PGalG2⊕W↑PSG32⊕W↑PC4G2
∂2
−→W↑PSG2
3 ⊕W↑PCG2
4 ⊕W↑PSG2
4 ⊕W↑PSG02 4
∂1
−→W↑PSG2
4 ⊕W↑PSG02 4
−→ε W.
The main work towards establishing this theorem is the following result.
Theorem 1.2. There exists anF-resolution of the trivial Galois-twisted profinitePG12-module W
0−→W↑PSG12
3
∂3
−→W↑PGalG12 −→∂2 W↑PSG12
3 ⊕W↑PCG12
4
∂1
−→W↑PSG12
4 ⊕W↑PSG012 4
−→ε W .
Remark 1.3. a) The resolutions for the group PG12 resp. for PG2 can be considered as resolutions for G12(Γ) resp. G2(Γ) via the obvious projections G12 → PG12 resp. G2 →PG2. In terms of the table (1.5) this has the effect of replacing a summand in the resolution of the formW↑PP FG12 resp. W↑PP FG12 byW↑GF12 resp. W↑GF12. Unlike forPG12 resp. forPG2the resulting resolutions forG12(Γ) resp. G2(Γ) will depend on the choice of Γ.
b) Restricted toS12(Γ) the resolution forG12(Γ) is an untwistedF- resolution of Wwhich is a W-linear extension of the algebraic centralizer resolution announced in [18] and used in [5].
We refer to Remark 3.3 for a justification of the terminology centralizer resolution.
Next we will describe the topological analogues of these algebraic resolutions. As in [18] we call a sequence of spectra
(1.7) X•:∗ →X−1−→α0 X0→X1−→α1 . . .
acomplex of spectraif the composite of two consecutive maps is null-homotopic. Such a complex is called a a resolutionofX−1 if in addition each of the maps αi :Xi−1 →Xi, i≥0, can be factored as Xi−1 −→βi Wi −→γi Xi such that Wi−1 γ−→i−1 Xi−1 −→βi Wi is a cofibration for every
2The assumption thatMisp-profinite was regrettably missing in [18].
i≥0 (with W−1 := ∗). We say that the resolution is of length nif Wn 'Xn and Xi ' ∗ if i > n.
Here are the main topological results of this paper. In their statementsE2should really read E2(Γ) whereE2(Γ) is the 2-periodic Landweber exact spectrum whose coefficients in degree 0 classify deformations (in the sense of Lubin and Tate) of Γ. In order to keep notation readable we will nevertheless simply writeE2instead ofE2(Γ). By the Goerss-Hopkins-Miller theorem Γ acts onE2, in particular there exist homotopy fixed point spectraE2hF for all finite subgroups ofG2(Γ) and by [10] also for all closed subgroups.
Theorem 1.4. Let Γ be eitherΓH orΓE. Then there exists a resolution ofEhG
1 2(Γ) 2
∗ →Eh2G12(Γ)→E2hG48(Γ)∨EhG
0 48(Γ)
2 →E2hG12(Γ)∨E2hC8 →E2C2×Gal→E2hG12(Γ)→ ∗. Theorem 1.5. LetΓ be eitherΓH orΓE. Then there exist a resolution ofLK(2)S0'E2hG2(Γ)
∗ →LK(2)S0→E2hG48(Γ)∨EhG
0 48(Γ)
2 →E2hG12(Γ)∨E2hC8∨E2hG48(Γ)∨EhG
0 48(Γ) 2
→E2C2×Gal∨E2hG12(Γ)∨E2hC8 →E2hG12(Γ)∨E2C2×Gal→E2hG12(Γ)→ ∗.
Remark 1.6. a) BecauseG48(Γ) andG048(Γ) are conjugate subgroups ofG2(Γ), the homotopy fixed point spectraE2hG48(Γ) andEhG
0 48(Γ)
2 have the same homotopy type.
b) There are corresponding resolutions forEhS
1 2
2 and E2hS2 which are obtained by replacing E2hF byEhF2 0 whereF and F0 are the finite subgroups of table (1.5).
The paper is organized as follows. In section 2 we discuss the finite subgroups of the Morava stabilizer groups atn=p= 2 which figure in our main results and in section 3 we study the mod-2 cohomology algebra ofP S21 via its restriction to the cohomology of elementary abelian 2-subgroups. Section 4 contains the construction of the algebraic centralizer resolutions and in section 5 we show how to realize the algebraic resolutions topologically.
2. Important finite subgroups for Morava stabilizer groups atn=p= 2 In this section we will elaborate on table (1.5) and describe more explicitly the relevant finite subgroups. We remark that in the general case of any primepand any heightnfinite subgroups of Sn have been studied by Hewett in [20] and [21] and finite subgroups of Gn(Γ) have been studied by Bujard [9].
We will start by recalling from [2] the description of explicit maximal subgroups G24 and G024ofS2and we prefer to work withS2(ΓH) and writeS instead ofξH.
Letω be a third root of unity inW× and let
(2.1) π:= 1 + 2ω .
By Hensel’s Lemma the element −7 ∈Z2 has two square roots inZ2. We pick the one which satisfies√
−7≡1 + 4 mod (8) and let
(2.2) α:= 1−2ω
√−7 .
We note that πand αboth belong to S2 and the reduced norm ofαis−1 while the reduced norm ofπis 3.
The following lemma is proved by direct calculation (cf. Lemma 2.4.3 of [2]).
Lemma 2.1. Let i:= 1
3(1 + 2ω2)(1−αS), j :=1
3(1 + 2ω2)(1−αω2S), k:= 1
3(1 + 2ω2)(1−αωS). Then the elements{±1,±i,±j,±k}form a subgroup ofS12which is isomorphic to the quaternion groupQ8. This subgroup is invariant by conjugation byω, more precisely
j=ωiω−1, k=ωjω−1 i=ωkω−1 . Furthermore
ω=−1
2(1 +i+j+k).
We let G24 be the subgroup generated by Q8 and ω. It is isomorphic to the semidirect product ofQ8 withC3,
(2.3) G24∼=Q8oC3 .
It is easy to verify that the 16 elements ofG24which are not inQ8are the elements of the form
1
2(±1±i±j±k) so that
(2.4) G24={±1,±i,±j,±k,1
2(±1±i±j±k)}
We also note that the center ofG24 is the subgroup{±1}andQ8is a characteristic subgroup.
Lemma 2.2. Let Γ be eitherΓE or ΓH.
a) The subgroup ofG2(Γ)generated byG24and the image of1+iis a maximal finite subgroup G48(Γ) ofG2(Γ)of order48.
b)G48(Γ)is a subgroup ofG12(Γ).
c) The quotientP G48(Γ)is isomorphic toS4 independent ofΓ.
d) There are isomorphisms G48(ΓE)∼=GL2(F3) andG48(ΓH)∼=O48. The groupsGL2(F3) andO48 are not isomorphic.
e) The intersection G48(Γ)∩S12 is G24, P G24 is isomorphic to A4 and P G24∩P S21 is the 2-Sylow subgroup of A4, isomorphic toC2×C2.
Proof. a) It is easy to see, for example from (2.4), that the element 1 +inormalizes the group G24. The order of 1 +ias element ofD×2 is clearly infinite. However, because of (1 +i)2= 2i and because of (1.3), its square inG2(Γ) is an element ofS2(Γ), equal toiif Γ = ΓH and equal to −i if Γ = ΓE. Because G24 is a maximal finite subgroup of S2 of order 24 it follows that G48(Γ) is a maximal finite subgroup ofG2(Γ) and is of order 48.
b) Any finite subgroup ofG2(Γ) is contained inG12(Γ).
c) For F a subgroup of Glet NG(F) resp. CG(F) denote the normalizer resp. centralizer ofF inG. Conjugation inD×2 induces a monomorphism fromN
D×2(Q8)/C
D×2(Q8) to Aut(Q8), the group of automorphisms of Q8. The latter group is well known to be isomorphic toS4
and the subgroupA4 ofS4 is realized by conjugation inG24/C2 =P G24. The element 1 +i belongs toN
D×2(Q8) and it is easy to check that conjugation by it does not belong toA4. Hence conjugation induces an epimorphismP G48(Γ)→Aut(Q8)∼=S4 which for cardinality reasons has to be an isomorphism.
d) The automorphism group of the elliptic curve with equation y2+y = x3 over F4 is isomorphic to G24 (cf. [30]). This group injects into the automorphism group of the formal group law over F4. Because the elliptic curve is already defined over F2 we get an injection
G24oGal→G2(ΓE) and the image is G48(ΓE). It is elementary to verify that the group of F4-points of the elliptic curve is of order 9, isomorphic toZ/3×Z/3 and thatG24oGal realizes all automorphisms ofE[3]. HenceG48(ΓE) is isomorphic toGL2(F3).
Next it is easy to construct an isomorphism between O48 and G48(ΓH) which restricts to the identity on G24; in fact, O48 can be realized within the classical unit quaternions such that G24 corresponds to the subgroup which contains the elements of (2.4) and the element (1 +i)∈G48(ΓH) corresponds to the element √1
2(1 +i)∈O48.
In order to see that GL2(F3) and O48 are not isomorphic it is enough to see that their 2- Sylow subgroups are not isomorphic. In the case ofGL2(F3) this is the semidihedral group of order 16 while in the case ofO48this is the generalized quaternion group of order 16 and these two groups of order 16 are not isomorphic.
e) This is now obvious.
Then we define
(2.5) G024:=πG24π−1, G048(Γ) :=πG48(Γ)π−1 .
The groupsG24 andG024 are known to be non-conjugate inS12 and, up to conjugacy, they are the two maximal finite subgroups of S12 (cf. [2]). Consequently G48(Γ) and G048(Γ) are non-conjugate in G12(Γ) and, up to conjugacy, they are the two maximal finite subgroups of G12(Γ). Likewise,S4 andS04 are non-conjugate inPG12(Γ) and, up to conjugacy, they are the two maximal finite subgroups ofPG12(Γ).
Lemma 2.3. Let Γ be eitherΓE or ΓH.
a) The subgroup of G2(Γ) generated by C6 = h−ωi and the image of j−k is a subgroup G12(Γ) ofG2(Γ)of order12.
b)G12(Γ)is a subgroup ofG12(Γ).
c) The quotientP G12(Γ)is isomorphic toS3 independent ofΓ.
d) There are isomorphisms G12(ΓE)∼=C2×S3 andG12(ΓH)∼=C3oC4.
e) The intersectionG12(Γ)∩S12 isC6,P C6 is isomorphic toC3andP C6∩P S12 is the trivial group.
Proof. a) The element j−k normalizes the subgroupC6 generated by −ω. In fact, a direct calculation in the division algebra using thatω=−12(1 +i+j+k) shows
(j−k)ω(j−k)−1=ω2 .
The order ofj−kas element ofD×2 is clearly infinite. However, because of (j−k)2=−2, its square inG2(Γ) is an element ofS2(Γ), equal to 1 if Γ = ΓE and equal to−1 if Γ = ΓH. Then it is clear thatG12(Γ) is of order 12.
b) Any finite subgroup ofG2(Γ) is contained inG12(Γ).
c) This is immediate from the calculation in part a). The image ofω in P G12(Γ) generates a normal subgroup of order 3 and the image ofj−kis of order 2 and acts non-trivially on the image ofω.
d) This follows because the image ofj−kinG12(Γ) is of order 2 in the case of Γ = ΓEand
of order 4 in the case of Γ = ΓH.
The following two lemmas are elementary and their proof is left to the reader.
Lemma 2.4. Let Γ be eitherΓE or ΓH.
a) The subgroup of G2(Γ) generated by 1 +i is a subgroup G8 of G2(Γ) of order 8 which containsC4=hii and is, up to isomorphism, independent ofΓ.
b)G8 is a subgroup ofG12(Γ).
c) The quotientP G8 is isomorphic toC4.
d) The intersectionG8∩S12 is the subgroup C4 generated byi,P C4is isomorphic toC2 and
P C4∩P S21=P C4.
Lemma 2.5. Let Γ be eitherΓE or ΓH.
a) The subgroup ofG2(Γ) generated by−1 and the Galois group is a subgroupG4 ofG2(Γ) of order4 which is isomorphic toC2×Gal independent ofΓ.
b)G4 is a subgroup ofG12(Γ).
c) The quotientP G4 is isomorphic toGal.
d) The intersection G4∩S12 is the subgroupC2 generated by −1 and P C2 =P C2∩P S21 is
the trivial group.
3. The mod-2 cohomology algebra ofP S21
3.1. Quillen’sF-isomorphism for the mod-pcohomology of a profinite group.
LetGbe a profinite group and letpbe a fixed prime. The continuous cohomologyHc∗(G;Fp) ofGwith coefficients in the trivial moduleFp will be abbreviated byH∗(G;Fp), or simply by H∗G if p is understood from the context. We recall that if G is the (inverse) limit of finite groupsGi thenH∗G= colimiH∗Gi.
We will assume that H∗G is finitely generated asFp-algebra. By work of Lazard [23] it is known that this holds for many interesting profinite groups, for example for profinitep-analytic groups likeGL(n,Zp), the general linear groups over thep-adic integers, or the automorphism groups of formal group laws over finite fields.
In case H∗Gis finitely generated as Fp-algebra Quillen has shown [26] that there are only finitely many conjugacy classes of elementary abelianp-subgroups ofG(i.e. groups isomorphic to (Z/p)nfor some natural numbern). In other words, the following categoryA(G) is equivalent to a finite category: objects of A(G) are all elementary abelian p-subgroups of G, and if E1
andE2 are elementary abelianp-subgroups of G, then the set of morphisms from E1 toE2 in A(G) consists precisely of those homomorphisms α: E1 −→ E2 of abelian groups for which there exists an elementg ∈ Gwith α(e) = geg−1 for all e∈ E1. The assignment E 7→H∗E determines a functor from the opposite categoryA∗(G)op to gradedFp-algebras.
Theorem 3.1. (Quillen)[26]Let Gbe a profinite group and assumeH∗Gis a finitely gener- ated Fp-algebra. Then the canonical map
qG:H∗G→limA(G)opH∗E is anF-isomorphism, in other words q has the following properties.
• Ifx∈KerqG, thenxis nilpotent.
• Ify∈limA(G)opH∗E then there exists an integer n withypn∈Imq.
In the sequel we will callA(G) the Quillen category ofG.
Let A∗(G) be the full subcategory of A(G) whose objects are all elementary abelian p- subgroups except the trivial subgroup. The centralizer CG(E) of an elementary abelian p- subgroupE is a closed subgroup and hence inherits a natural profinite structure fromG. The assignment E 7→ H∗CG(E) extends to a functor from A∗(G) to graded Fp - algebras and the restriction homomorphisms H∗G −→ H∗CG(E) (for E running through the non-trivial elementary abelianp-subgroups ofG) induce a canonical mapρ:H∗G−→limA∗(G)H∗CG(E).
The main result of [17] reads as follows.
Theorem 3.2. LetGbe a profinite group and assumeH∗Gis a finitely generatedFp - algebra.
Then the canonical mapρ:H∗G−→limA∗(G)H∗CG(E)has finite kernel and cokernel.
Remark 3.3. a) In our current approach this theorem is is no longer needed. However, it played a crucial role in our initial approach to construct resolutions forPS12 and is utimately the reason for naming our resolutions centralizer resolutions. Furthermore, in [18] the theorem played a crucial role for constructing algebraic centralizer resolutions at odd primes, which as the algebraic resolutions of this paper areF-resolutions in the sense of Section 1.2.
b) Theorem 3.2 is not useful ifG contains central elements of orderp, because then H∗G appears in the limit. In these cases one can use the theorem to study the quotient ofGby the maximal central elementary abelianp-subgroup ofGand this was the orign for considering the groupsP S21 andPS12.
3.2. The Quillen category of P S21.
We recall from section 2 thatS12contains two subgroups isomorphic toQ8and they give rise to two elementary abelian 2-subgroups E2 and E20 in PS12 which are contained in the normal 2-Sylow subgroupP S21.
The following result has a significant overlap with section 2.4 of [2].
Proposition 3.4. a) Up to conjugacy P S21 contains three elementary abelian 2-subgroups of rank 1 and two of rank2.
b) All automorphism groups of the category A(P S21) are trivial and there is exactly one morphism from each rank1 group to each of the rank 2 groups.
Proof. a) IfE is an elementary abelian 2-subgroup ofP S21then its inverse imageEe inS21is an extension ofE byZ/2. The structure of the possible finite 2 subgroups of the division algebra D2is explicitly known: in fact, any finite abelian subgroup must be cyclic and generates in the division algebra a cyclotomic extension the degree of which must divide 2. Hence any abelian 2-subgroup is cyclic of order 2 or 4 and this implies that any finite 2-subgroup is isomorphic to a subgroup ofQ8. In particular we see that the 2-rank ofEis either 1 or 2.
Now suppose that F1 and F2 are two elementary abelian 2-subgroups of rank 1 of P S21. Then Fe1 and Fe2 are two subgroups isomorphic to Z/4 and by the Skolem Noether theorem any isomorphism ϕ: Fe1 → Fe2 can be realized by conjugation by an element of u∈ D×2, i.e.
ϕ(x) =uxu−1 for any x∈Fe1. If we denote a generator of Fe1 byi then 1 +i∈D×n centralizes F1, so we can changeuby any power of (1 +i) and conjugation by u(1 +i)n will still giveϕ.
Because the valuation of 1 +iis 12 we can choose n such that (1 +i)nuis of valuation 0. In other words, we can suppose thatuis an element ofS2. Furthermore, the element 1 + 2i∈S2
has reduced norm 5 and is thus a topological generator of S2/S12. It also centralizes Fe1 and by multiplying u by a suitable p-adic power of 1 + 2i we can even assume that u is in S12. This implies that all rank 1 subgroups of PS12 are conjugate and therefore the quotient group S12/S12∼=F×4 which is generated by the image ofωacts transtively on theP S21-conjugacy classes of elementary abelian 2-subgroups of rank 1.
Thus there are either three or oneP S21-conjugacy classes of elementary abelian 2-subgroups of rank 1. If there was only one then conjugation byω would have to be the same as conjugation by an element in P S21 and this would mean that there is an element in S12 of the form ωu0 with u0 ∈ S21 whose image in PS12 centralizes F1, and hence ωu0 normalizes Fe1. However, NS1
2(Fe1)/CS1
2(Fe1) is isomorphic to a subgroup of ND×
2(Fe1)/CD×
2(Fe1)∼= Aut(Fe1)∼=C2
hence NS2(Fe1) contains the centralizer CS2(Fe1) ∼=Z2[i]× as an subgroup of index at most 2.
This implies that NS2(Fe1) is a profinite 2-group and cannot contain such an element which would have non-trivial image inF×4.
Next supposeF1 andF2are two elementary abelian 2-subgroups of rank 2 ofP S21. ThenFe1 and Fe2 are two subgroups ofS12 isomorphic toQ8 and again by the Skolem Noether theorem any isomorphism ϕ: Fe1 → Fe2 can be realized by conjugation by an element of u∈ D×2, i.e.
ϕ(x) =uxu−1 for anyx∈Fe1. In particular, we have an isomorphism ND×2(F1)/C
D×2(F1)∼= Aut(Q8)∼=S4 .
In order to determine the number of conjugacy classes of elementary abelian 2-subgroups of rank 2 ofP S21 we need to know something about the structure of the normalizerNS2(Q8). The centralizerC
D×2(Q8) is isomorphic to Q×2 and the quotientN
D×2(Q8)/C
D×2(Q8) is generated by the image of the group G24 and the element 1 +i (cf. the proof of part a) of Lemma 2.2).
Furthemore the centralizerCS2(Q8) is isomorphic toZ×2 and we get an isomorphism (3.1) NS2(Q8)∼=Z×2 ×C2G24
betweenNS2(Q8) and the central product Z×2 ×C2G24and an isomorphism (3.2) NS2(Q8)/CS2(Q8)∼=P G24=A4.
Because the normalizerN
D×2(F1) always contains an elementyof valuation 12, we can assume by changingu, if necessary, by a suitable power ofy that there is an isomorphismψ:Fe1→Fe2
which is realized by conjugation in S2. In particular, in S2 there is only one conjugacy class of subgroups isomorphic to Q8 and in PS2 there is only one conjugacy class of elementary abelian 2-subgroups of rank 2. This means that the group S2/S12 acts transitively on the set of conjugacy classes of subgroups of S12 which are isomorphic toQ8. Because the center acts trivially on the set of conjugacy classes and the image of the center inS2/S12is of index 2 there are at most two conjugacy classes ofQ8’s inS12. We claim that there are two of them given by Q8andπQ8π−1whereQ8is the 2- Sylow subgroup of the groupG24 of section 2 andπ∈S2is the element defined in (2.1). In fact, if they were conjugate thenπcould be written as product xnwithx∈S12andn∈NS2(Fe1). However, from (3.1) we see that the reduced norm of such an element is always a square inZ×2 and this contradicts the fact that the reduced norm ofπis 3.
b) It is clear that the automorphism groups of elementary abelian 2-subgroups of rank 1 are trivial. For the automorphisms of a rank 2 subgroup we note that (3.2) implies that the automorphism group AutA(PS2)(P Q8) isC3because conjugation by any element of the subgroup Q8 of G24 induces the trivial automorphism. This in turn implies that AutA(P S1
2)(P Q8) is trivial.
It remains to show that there is exactly one morphism from each rank 1 to each rank 2 object, or equivalently, that the three non-trivial elements in a rank 2 object are non-conjugate in P S21. If they were conjugate in P S21, then there would be an element in PS12 of the form ω±1xwithxinP S21which centralizes the rank 1 subgroup generated by one of these elements, respectively its preimage in S12 would normalize the preimage, and this contradicts what we
have seen in the proof of part a) above.
Remark 3.5. We can choose representatives E2 and E20 for the two conjugacy classes of elementary abelian 2-subgroups of rank 2 such thatE2∩E02is cyclic of order 2. In fact, if E2 is such thatEe2 is the subgroup ofG24 generatediandj then conjugation by 1 + 2ifixesiand carriesEe2 toEe20. InS12the groupEe2 is not conjugate toEe20, hence in the quotientPS12 we get thatE2 andE20 are non-conjugate and intersect in the subgroup generated by the image ofi.
3.3. Quillen’sF-isomorphism for P S21 and the mod-2 cohomology algebra of P S21. The inverse limit in Quillen’s Theorem 3.1 is always a subalgebra of the productQ
EH∗E where E runs through the maximal elementary abelian subgrous of G, up to conjugacy. By Theorem 3.4 there are, in the case of G = P S12, two of them, both or rank 2 with mod-2 cohomology both given byF2[x, y] withxandyof cohomological degree 1.
Proposition 3.6. There is an isomorphism of graded F2-algebras limA(P S1
2)H∗E∼={(p1, p2)∈F2[x, y]×F2[x, y]
p1−p2 is divisible by xy(x+y)}
Proof. IfE1 and E2 are two non-conjugate elementary abelian 2-subgroups of rank 2 ofP S21 then the non-trivial elements ofE1andE2belong to the three non-conjugate elementary abelian 2-subgroups F1, F2 and F3 of rank 1. This gives 6 morphisms in A(P S21), and if we choose the non-trivial elements of E1 and E2 as ej1, ej2 and ej3 for j = 1,2 then we have morphisms αi,j:Fi→Ej which send the nontrivial element ofFi to the elementeji ofEj.
Then the inverse limit is given by pairs of polynomials (p1, p2)∈H∗E1×H∗E2 such that α∗i,1p1 = α∗i,2p2 for i = 1,2,3, or if we identify H∗E1 with H∗E2 via the abstract group isomorphism which sends e1i to e2i for i = 1,2,3 thenp1−p2 must be divisible by the three
non-trivial elements inF2[x, y] and the claim follows.
The quotient homomorphism S21→S21/F1S21∼=F4 induces a surjection P S12 →F4 and the explicit form of the elementsi,j andkgiven in Lemma 2.1 shows that both subgroupsE2and E20 map isomorphically to this quotient.
Corollary 3.7. As a module over H∗(S21/F1S21) ∼=F2[x, y] the inverse limit is the free sub- module ofF2[x, y]×F2[x, y]generated by the classes (1,1)and(xy(x+y),0).
The following result describes the algebraic centralizer resolution of the trivialS12-odule Z2. It has been established in Theorem 1.2.1 and 1.2.6 of [2]. The subgroups ofS12 occuring in the statement are those of (1.5) andIS21is the augmentation ideal of the completed group algebra Z2[[S12]]. The notation used is analogous to that of Section 1.2. In other words, ifGis a profinite group andX is a profiniteG-set such thatX = limiXi with finiteS12-setsXi then we define (3.3) Z2[[X]] = limi,kZ/2k[[Xi]],
and ifF is a finite subgroup of Gand M is a Z2[F]-module then M ↑GF denotes the Z2[[G]]- moduleZ2[[S12]]⊗Z2[F]M.
