On Farrell-Tate cohomology of SL\_2 over S-integers
aa r X i v : . [ m a t h . K T ] D ec ON FARRELL–TATE COHOMOLOGY OF SL OVER S -INTEGERS ALEXANDER D. RAHM AND MATTHIAS WENDT
Abstract.
In this paper, we provide number-theoretic formulas for Farrell–Tate cohomologyfor SL over rings of S -integers in number fields satisfying a weak regularity assumption. Theseformulas describe group cohomology above the virtual cohomological dimension, and can beused to study some questions in homology of linear groups.We expose three applications, to (I) detection questions for the Quillen conjecture,(II) the existence of transfers for the Friedlander–Milnor conjecture,(III) cohomology of SL over number fields. Introduction
The cohomology of arithmetic groups like SL ( O K,S ) for O K,S a ring of S -integers in anumber field K has been long and intensively studied. In principle, the cohomology groups canbe computed from the action of SL on its associated symmetric space, but actually carryingout this program involves a lot of questions and difficulties from algebraic number theory. Inthe case of a group Γ of finite virtual cohomological dimension (vcd), Farrell–Tate cohomologyprovides a modification of group cohomology, which in a sense describes the obstruction for Γto be a Poincar´e-duality group. The Farrell–Tate cohomology of Γ can be described in terms offinite subgroups of Γ and their normalizers, and hence is more amenable to computation thangroup cohomology.The primary goal of the present paper is to provide explicit formulas for the Farrell–Tatecohomology of SL over many rings of S -integers in number fields, with coefficients in F ℓ , ℓ odd.These formulas generalize those which had previously been obtained in the case of imaginaryquadratic number rings by one of the authors, building on work of Kr¨amer, cf. [Rah13, Rah14],[Kr¨a80].We will state the main result after introducing some relevant notation. Let K be a globalfield, let S be a non-empty set of places containing the infinite ones and denote by O K,S thering of S -integers in K . Let ℓ be an odd prime different from the characteristic of K , and let ζ ℓ be some primitive ℓ -th root of unity.Assume ζ ℓ + ζ − ℓ ∈ K . We denote by Ψ ℓ ( T ) = T − ( ζ ℓ + ζ − ℓ ) T +1 the relevant quadratic factorof the ℓ -th cyclotomic polynomial, by K (Ψ ℓ ) = K [ T ] / Ψ ℓ ( T ) the corresponding K -algebra, by R K,S,ℓ = O K,S [ T ] / (Ψ ℓ ( T )) the corresponding order in K (Ψ ℓ ), and byNm : R × K,S,ℓ → O × K,S and Nm : Pic( R K,S,ℓ ) → Pic( O K,S ) . the norm maps on unit groups and class groups for the finite extension R K,S,ℓ / O K,S .With this notation, the following is our main result; the main part of the proof can be foundin Section 3.
Theorem 1.
Let ℓ be an odd prime. Let K be a global field of characteristic different from ℓ which contains ζ ℓ + ζ − ℓ , let S be a non-empty set of places containing the infinite ones and denoteby O K,S the ring of S -integers in K . Furthermore, assume one of the following conditions:(R1) ζ ℓ K and the prime ( ζ ℓ − ζ − ℓ ) is unramified in the extension O K,S / Z [ ζ ℓ + ζ − ℓ ] .(R2) ζ ℓ ∈ K and S ℓ ∈ S , i.e., S contains the places of K lying over ℓ .Then we have the following consequences: Date : May 22, 2018.2010
Mathematics Subject Classification.
Keywords:
Farrell–Tate cohomology, S -arithmetic groups, group homology. (1) b H • (SL ( O K,S ) , F ℓ ) = 0 .(2) The set C ℓ of conjugacy classes of elements in SL ( O K,S ) with characteristic polynomial Ψ ℓ ( T ) = T − ( ζ ℓ + ζ − ℓ ) T + 1 sits in an extension → coker Nm → C ℓ → ker Nm → . The set K ℓ of conjugacy classes of order ℓ subgroups of SL ( O K,S ) can be identifiedwith the orbit set K ℓ = C ℓ / Gal( K (Ψ ℓ ) /K ) , where the action of Gal( K (Ψ ℓ ) /K ) ∼ = Z / Z exchanges the two roots of Ψ ℓ ( T ) . There is a product decomposition b H • (SL ( O K,S ) , F ℓ ) ∼ = Y [Γ] ∈K ℓ b H • ( N SL ( O K,S ) (Γ) , F ℓ ) . This decomposition is compatible with the ring structure of b H • (SL ( O K,S ) , F ℓ ) .(3) If the class of Γ is not Gal( K (Ψ ℓ ) /K ) -invariant, then N SL ( O K,S ) (Γ) ∼ = ker Nm . Thereis an isomorphism of graded rings b H • ( N SL ( O K,S ) (Γ) , Z ) ( ℓ ) ∼ = F ℓ [ a , a − ] ⊗ F ℓ ^ (ker Nm ) , where a is a cohomology class of degree . In particular, this is a free module over thesubring F ℓ [ a , a − ] .(4) If the class of Γ is Gal( K (Ψ ℓ ) /K ) -invariant, then there is an extension → ker Nm → N SL ( O K,S ) (Γ) → Z / Z → . There is an isomorphism of graded rings b H • ( N SL ( O K,S ) (Γ) , Z ) ( ℓ ) ∼ = (cid:16) F ℓ [ a , a − ] ⊗ F ℓ ^ (ker Nm ) (cid:17) Z / , with the Z / -action given by multiplication with − on a and ker Nm . In particular,this is a free module over the subring F ℓ [ a , a − ] ∼ = b H • ( D ℓ , Z ) ( ℓ ) . Remark 1.1.
The conditions (R1) and (R2) exclude pathologies with the singularities over theplace ℓ and resulting non-invertibility of ideals. For global function fields, they are automaticallysatisfied whenever ℓ is different from the characteristic. Condition (R2) is a standard assumptionwhen dealing with the Quillen conjecture, which is one of the relevant applications of Theorem 1. Remark 1.2.
We recover as special cases the earlier results of Busch [Bus06]. For functionfields of curves over algebraically closed fields, there are similar formulas, cf. [Wen15a].
Remark 1.3.
The restriction to odd torsion coefficients is necessary. Only under this assump-tion, the formula has such a simple structure; for 2-torsion one has to take care of the possiblesubgroups A , S and A in PSL . Moreover, for F -coefficients, there is a huge differencebetween the cohomology of SL and PSL - only the cohomology of PSL has an easy descrip-tion and the computation of cohomology of SL depends on a lot more than just the finitesubgroups. Some results can be achieved, but the additional complications would only obscurethe presentation of the results.The main strategy of proof is the natural one: we use Brown’s formula which computes theFarrell–Tate cohomology with F ℓ -coefficients of SL ( O K,S ) in terms of the elementary abelian ℓ -subgroups and their normalizers. The main piece of information for the above theorem isthen the classification of conjugacy classes of finite subgroups of SL ( O K,S ), which – underthe conditions (R1) or (R2) – is directly related to number-theoretic questions about (relative)class groups and unit groups of S -integers in K and its cyclotomic extension. Another sec-ondary objective of the current paper is to reinterpret and generalize results on finite subgroupclassification in number-theoretic terms.The explicit formulas obtained allow to discuss a couple of questions concerning cohomologyof linear groups. These applications are based on the fact that Farrell–Tate cohomology andgroup cohomology coincide above the virtual cohomological dimension. In particular, Theorem 1provides a computation of group cohomology of SL ( O K,S ) above the virtual cohomologicaldimension.
N FARRELL–TATE COHOMOLOGY OF SL OVER S -INTEGERS 3 As a first application, the explicit formulas for the ring structure allow to discuss a conjectureof Quillen, cf. [Qui71], recalled as Conjecture 6.1 in Section 6 below. The following result isproved in Section 6; conclusions from it with consequences for Quillen’s conjecture have beendrawn in [RW15a].
Theorem 2.
With the notation of Theorem 1, the restriction map induced from the inclusion SL ( O K,S ) → SL ( C ) maps the second Chern class c ∈ H • cts (SL ( C ) , F ℓ ) to the sum of theelements a in all the components.As a consequence, Quillen’s conjecture for SL ( O K,S ) is true for Farrell–Tate cohomologywith F ℓ -coefficients for all K and S . This also implies that Quillen’s conjecture is true for groupcohomology with F ℓ -coefficients above the virtual cohomological dimension.However, if the number of conjugacy classes of order ℓ -subgroups is greater than two, therestriction map H • (SL ( O K,S ) , F ℓ ) → H • (T ( O K,S ) , F ℓ ) from SL ( O K,S ) to the group T ( O K,S ) of diagonal matrices is not injective. This result sheds light on the Quillen conjecture, showing that the Quillen conjecture holds ina number of rank one cases; it also sheds light on the relation between Quillen’s conjecture anddetection questions in group cohomology, where detection refers to injectivity of the restrictionmap to diagonal matrices. See [RW15a] for a more detailed discussion of these issues. It isalso worthwhile pointing out that the failure of detection as in the previous theorem, i.e., thenon-injectivity of the restriction mapH • (SL ( O K,S ) , F ℓ ) → H • (T ( O K,S ) , F ℓ )implies the failure of the unstable Quillen-Lichtenbaum conjecture as formulated in [AR13].This follows from the work of Dwyer and Friedlander [DF94]. Other examples of the failure ofdetection in function fields situations in a higher rank case are discussed in [Wen15b].Another interesting question in group cohomology is the existence of transfers, which wasdiscussed in [Knu01, Section 5.3] in the context of the Friedlander–Milnor conjecture. From theexplicit computations of Theorem 1, we can also describe the restriction maps on Farrell–Tatecohomology which allows to find examples for non-existence of transfers. The following resultis proved in Section 7: Theorem 3.
Let
L/K be a finite separable extension of global fields, let S be a non-empty finiteset of places of K containing the infinite places and let ˜ S be a set of places of L containing thoselying over S . Let ℓ be an odd prime different from the characteristic of K . Assume that O K,S and O L, ˜ S satisfy the conditions of Theorem 1.The restriction map b H • (SL ( O L, ˜ S ) , F ℓ ) → b H • (SL ( O K,S ) , F ℓ ) induced from the natural ring homomorphism O K,S → O L, ˜ S is compatible with the decompositionof Theorem 1 and is completely described by(1) the induced map on class groups Pic( R K,S,ℓ ) → Pic( R L, ˜ S,ℓ ) (2) the induced map on unit groups R × K,S,ℓ → R × L, ˜ S,ℓ .Let K = Q ( ζ ℓ ) be a cyclotomic field whose class group is non-trivial and has order prime to ℓ , e.g. ℓ = 23 . Denoting by H the Hilbert class field of K , the restriction map b H • (SL ( O H [ ℓ − ]) , F ℓ ) → b H • (SL ( O K [ ℓ − ]) , F ℓ ) is not surjective. Therefore, this is an example of a finite ´etale morphism for which it is notpossible to define a transfer in the usual K-theoretic sense. Finally, we want to note that the precise description of Farrell–Tate cohomology and therelevant restriction maps allows to consider the colimit of the groups b H • (SL ( O K,S ) , F ℓ ), where S runs through the finite sets of places of K . Using this, we investigate in Section 8 thebehaviour of Mislin’s extension of Farrell–Tate cohomology with respect to directed colimits: ALEXANDER D. RAHM AND MATTHIAS WENDT
Theorem 4. (1) There are cases where Mislin’s extension of Farrell–Tate cohomology does not com-mute with directed colimits. A simple example is given by the directed system of groups SL ( Z [1 /n ]) with n ∈ N .(2) The Friedlander–Milnor conjecture for SL ( Q ) is equivalent to the question if Mislin’sextension of Farrell–Tate cohomology commutes with the directed colimit of SL ( O K,S ) ,where K runs through all number fields and S runs through all finite sets of places. Results similar to the ones in the present paper can be obtained for higher rank groups, al-though there are more and more complications coming from the classification of finite subgroups.Away from the order of the Weyl group, the subgroup classification is easier. A discussion ofthe case SL will be done in the forthcoming paper [RW15b]. Structure of the paper:
We first recall group cohomology preliminaries in Section 2. Theproof of the main theorem, modulo the conjugacy classification, is given in Section 3. Sections 4and 5 establish the conjugacy classification of finite cyclic subgroups in SL ( O K,S ). Then wediscuss three applications of the results, to (I) non-detection in Section 6, (II) the existence oftransfers in Section 7 and (III) cohomology of SL over number fields in Section 8. Acknowledgements:
This work was started in August 2012 during a stay of the second namedauthor at the De Br´un Center for Computational Algebra at NUI Galway. We would like thankGuido Mislin for email correspondence concerning his version of Tate cohomology, and Jo¨elBella¨ıche for an enlightening MathOverflow answer concerning group actions on Bruhat–Titstrees. We thank Norbert Kr¨amer and an anonymous referee for very helpful comments on aprevious version of this paper.2.
Preliminaries on group cohomology and Farrell-Tate cohomology
In this section, we recall the necessary definitions of group cohomology and Farrell–Tatecohomology, introducing notations and results we will need in the sequel. One of the basicreferences is [Bro94].2.1.
Group cohomology.
Group cohomology is defined as the right derived functor of invari-ants Z [ G ] -mod → Z -mod : M M G . It can be defined algebraically by taking a resolution P • → Z of Z by projective Z [ G ]-modules and settingH • ( G, M ) := H • (Hom G ( P • , M )) . Alternatively, it can be defined topologically as the cohomology of the classifying space BG with coefficients in the local system associated to M .2.2. Farrell–Tate cohomology.
We shortly recall the definition and properties of Farrell–Tate cohomology, cf. [Bro94, chapter X]. Farrell–Tate cohomology is a completion of groupcohomology defined for groups of finite virtual cohomological dimension (vcd). Note that for K a number field, the groups SL ( O K,S ) have virtual cohomological dimension 2 r + 3 r + S f − r and 2 r are the numbers of real and complex embeddings of K , respectively, and S f is the number of finite places in S .For Γ a group with finite virtual cohomological dimension, a complete resolution is the datumof • an acyclic chain complex F • of projective Z [Γ]-modules, • a projective resolution ǫ : P • → Z of Z over Z [Γ], and • a chain map τ : F • → P • which is the identity in sufficiently high dimensions.The starting point of Farrell–Tate cohomology is the fact that groups of finite virtual coho-mological dimensions have complete resolutions. Definition 2.1.
Given a group Γ of finite virtual cohomological dimension, a complete reso-lution ( F • , P • , ǫ ) and a Z [Γ]-module M , Farrell–Tate cohomology of Γ with coefficients in M isdefined by b H • (Γ , M ) := H • (Hom Γ ( F • , M )) . N FARRELL–TATE COHOMOLOGY OF SL OVER S -INTEGERS 5 The functors b H • satisfy all the usual cohomological properties, cf. [Bro94, X.3.2], and in factFarrell–Tate cohomology can be seen as projective completion of group cohomology, cf. [Mis94].Important for our considerations is the fact [Bro94, X.3.4] that there is a canonical mapH • (Γ , M ) → b H • (Γ , M )which is an isomorphism above the virtual cohomological dimension.2.3. Finite subgroups in
PSL ( K ) . We first recall the classification of finite subgroups ofPSL ( K ), K any field. We implicitly assume that the order of the finite subgroup is prime tothe characteristic of K . The classification over algebraically closed fields is due to Klein. In the(slightly different) case of PGL ( K ), K a general field, the classification of finite subgroups canbe found in [Bea10].For K an algebraically closed field, in particular for K = C , Klein’s classification provides anexact list of isomorphism types (as well as conjugacy classes) of finite subgroups in PSL ( K ):any finite subgroup of PSL ( C ) is isomorphic to a cyclic group Z /n Z , a dihedral group D n ,the tetrahedral group A , the octahedral group S or the icosahedral group A .Over an arbitrary field K , we have the following classification, cf. [Ser72, 2.5], [Kr¨a80, Satz13.3] or [Bea10, proposition 1.1 and theorem 4.2]. Denote by ζ n some primitive n -th root ofunity. Proposition 2.2. (i)
PSL ( K ) contains a cyclic group Z /n Z if and only if ζ n + ζ − n ∈ K .(ii) PSL ( K ) contains a dihedral group D n if additionally the symbol (cid:0) ( ζ n − ζ − n ) , − (cid:1) issplit.(iii) PSL ( K ) contains A if and only if − is a sum of two squares, i.e., if the symbol ( − , − is split.(iv) PSL ( K ) contains S if and only if √ ∈ K and − is a sum of two squares.(v) PSL ( K ) contains A if and only if √ ∈ K and − is a sum of two squares. This result is best proved by considering the Wedderburn decomposition of K [ G ] and checkingfor (determinant one) two-dimensional representations among the factors.With the exception of Z / Z and the dihedral groups, all finite subgroups in PGL ( K ) areconjugate whenever they are isomorphic. For the dihedral groups D r , there is a bijectionbetween PGL ( K )-conjugacy classes and K × / (( K × ) · µ r ( K )) if ζ r ∈ K , cf. [Bea10, theorem4.2].2.4. Farrell–Tate cohomology of finite subgroups of SL ( C ) . We recall the well-knownformulas for group and Tate cohomology of cyclic and dihedral groups. We restrict to thecohomology with odd torsion coefficients, as our main results only use that case. The formulasbelow as well as corresponding formulas for the cohomology with F -coefficients can be found in[AM04]. Here, classes in square brackets are polynomial generators and classes in parenthesesare exterior classes; the index of a class specifies its degree in the graded F ℓ -algebra. • The cohomology ring for a cyclic group of order n with ℓ | n and ℓ odd is given by theformula H • ( Z /n Z , F ℓ ) ∼ = F ℓ [ a ]( b ) . The corresponding Tate cohomology ring is b H • ( Z /n Z , F ℓ ) ∼ = F ℓ [ a , a − ]( b ). Note thatcohomology with integral coefficients gets rid of the exterior algebra contribution whichcome from the universal coefficient formula. • The cohomology ring for a dihedral group of order 2 n with ℓ | n and ℓ odd is given bythe formula H • ( D n , F ℓ ) ∼ = F ℓ [ a ]( b ) . The corresponding Tate cohomology ring is b H • ( D n , F ℓ ) ∼ = F ℓ [ a , a − ]( b ). • The inclusions D ֒ → A , D ֒ → S and D ֒ → A induce isomorphisms in groupcohomology with F -coefficients. • The subgroup inclusion D → A induces an isomorphism in group cohomology with F -coefficients. ALEXANDER D. RAHM AND MATTHIAS WENDT Computation of Farrell–Tate cohomology
Brown’s formula.
We now outline the proof of the main result, Theorem 1. The essentialtool is Brown’s formula. For ℓ an odd prime, any elementary abelian ℓ -subgroup of SL ( O K,S )is in fact cyclic. This implies that Brown’s complex of elementary abelian ℓ -subgroups is adisjoint union of the conjugacy classes of cyclic ℓ -subgroups of SL ( O K,S ).By Brown’s formula for Farrell–Tate cohomology, cf. [Bro94, corollary X.7.4], we have b H • (SL ( O K,S ) , F ℓ ) ∼ = Y [Γ ≤ SL ] , Γ cyclic b H • ( C SL (Γ) , F ℓ ) N SL2 (Γ) /C SL2 (Γ) , where the sum on the right is indexed by conjugacy classes of finite cyclic subgroups Γ inSL ( O K,S ). Remark 3.1.
Alternatively, the Farrell–Tate cohomology of SL ( O K,S ) can be computed usingthe isotropy spectral sequence for the action of SL ( O K,S ) on the associated symmetric space X K,S . It is possible to show that the spectral sequence provides the same direct sum decompo-sition of Farrell–Tate cohomology as Brown’s formula above, because the ℓ -torsion subcomplexof X K,S is a disjoint union of classifying spaces for proper actions of the normalizers of cyclic ℓ -subgroup of SL ( O K,S ).3.2.
Cyclic subgrups, their centralizers and normalizers.
To prove Theorem 1, we needto describe the conjugacy classes of finite cyclic ℓ -subgroups Γ in SL ( O K,S ) and compute theircentralizers and normalizers. This is done in Sections 4 and 5.The classification proceeds by first setting up a bijection between conjugacy classes of elementsof order ℓ with characteristic polynomial T − ( ζ ℓ + ζ − ℓ ) T + 1 and classes of “oriented relativeideals”, cf. Definition 4.8. This bijection is essentially based on writing out representing matricesfor multiplication with ζ ℓ on suitable O K,S -lattices in K ∼ = K ( ζ ℓ ), cf. Proposition 4.10, andmostly follows classical arguments as in [LM33]. Under suitable regularity assumptions, namely(R1) and (R2) in Theorem 1, the set of oriented relative ideals can then be described in termsof relative class groups and the cokernel of norm maps on unit groups, cf. Section 4 for moredetails.Centralizers and normalizers of subgroups generated by elements of finite order can then alsobe determined very precisely. The main point is that the theory of algebraic groups puts strictconstraints on the possible shape of centralizers and normalizers for arithmetic subgroups ofalgebraic groups: the centralizers turn out to be groups of norm-one units, and the normalizersare either equal to the centralizers or Z / Z -extensions thereof. The precise results are proved inSection 5. The main result formulating the classification of finite cyclic subgroups in SL ( O K,S )is Theorem 5.8.Having the description of subgroups of order ℓ and their centralizers and normalizers, Parts(i) and (ii) of Theorem 1 follow directly from Theorem 5.8 and Brown’s formula mentionedabove.3.3. Farrell–Tate cohomology of centralizers and normalizers.
The description of therelevant normalizers for parts (iii) and (iv) of Theorem 1 can also be found in Theorem 5.8resp. Proposition 5.3. To prove the cohomology statements in parts (iii) and (iv) of Theorem 1,we provide the computation of the Farrell–Tate cohomology of the normalizers in the twopropositions below.First, we consider the case of cyclic subgroups whose conjugacy class is not Galois-invariant.In this case, the normalizer equals the centralizer and is isomorphic to the kernel of the normmap on units. As a group, the normalizer is then of the form Z /n Z × Z r for suitable n and r .The computation of the relevant Farrell–Tate cohomology is straightforward: Proposition 3.2.
Let A = Z /n Z × Z r , and let ℓ be an odd prime with ℓ | n . Then, with b , x , . . . , x r denoting classes in degree and a a class of degree , we have b H • ( A, F ℓ ) ∼ = b H • ( Z /n Z , F ℓ ) ⊗ F ℓ • ^ F rℓ ∼ = F ℓ [ a , a − ]( b , x , . . . , x r ) . N FARRELL–TATE COHOMOLOGY OF SL OVER S -INTEGERS 7 Proof.
We begin with a computation of group cohomology. In this case, the K¨unneth formulaimplies H • ( A, F ℓ ) ∼ = H • ( Z /n Z , F ℓ ) ⊗ F ℓ H • ( Z r , F ℓ ) . But H • ( Z r , F ℓ ) ∼ = V • F rℓ , again by iterated application of the K¨unneth formula. Therefore,the first isomorphism claimed above is true with group cohomology instead of Farrell–Tatecohomology.Now group cohomology and Farrell–Tate cohomology agree above the virtual cohomologicaldimension, which in this case is r . Moreover, the only finite subgroup of A is Z /n Z , and henceby [Bro94, theorem X.6.7] the group A has periodic Farrell–Tate cohomology. The latter inparticular means that there is an integer d such that b H i ( A, F ℓ ) ∼ = b H i + d ( A, F ℓ ) for all i . Thesetwo assertions imply that the formula we obtained for group cohomology above is also true forFarrell–Tate cohomology.The second isomorphism then just combines the first isomorphism with the formula for thecyclic groups discussed earlier in Section 2. (cid:3) Secondly, we discuss the case where the cyclic subgroup is Galois-invariant. Recall fromTheorem 5.8 that the group structure in this case is a semi-direct product ker Nm ⋊Z / Z ,where the action of Z / Z ∼ = Gal( K ( ζ ℓ ) /K ) on ker Nm is the natural Galois-action. Proposition 3.3.
The Hochschild–Serre spectral sequence associated to the semi-direct product ker Nm ⋊Z / Z degenerates and yields an isomorphism b H • (ker Nm ⋊Z / Z , F ℓ ) ∼ = b H • (ker Nm , F ℓ ) Z / Z . The
Gal( K ( ζ ℓ ) /K ) -action on ker Nm is by multiplication with − . The invariant classes arethen given by a ⊗ i tensor the even exterior powers plus a ⊗ (2 i +1)2 tensor the odd exterior powers. Remark 3.4.
The statements above provide an SL -analogue of the computations of Anton[Ant99]. 4. Conjugacy classification of elements of finite order
In this section, we will discuss the conjugacy classification of elements of finite order in SL over rings of S -integers in global fields, with some necessary augmentations. In the next section,we will use these results to provide a conjugacy classification of finite cyclic subgroups in SL .For a number field K , the general conjugacy classification of finite subgroups of SL ( O K ) is dueto Kr¨amer [Kr¨a80]. Special cases for totally real fields appeared before in the study of Hilbertmodular groups, cf. [Pre68] and [Sch75]. A more recent account of this can be found in [Mac06].We will discuss here a generalization of these results to rings of S -integers in number fields.Some of the necessary modifications for this have been considered in [Bus06]. Our expositioncan be seen as a geometric formulation of the classification result of Latimer-MacDuffee [LM33].4.1. Notation.
Throughout this section, we let K be a global field, S be a non-empty set ofplaces containing the infinite ones and consider the ring of S -integers O K,S in K . We fix anodd prime ℓ different from the characteristic of K , and assume that ζ ℓ + ζ − ℓ ∈ K .Any element of exact order ℓ in SL ( O K,S ) will have characteristic polynomial of the formΨ ℓ ( T ) = T − ( ζ ℓ + ζ − ℓ ) T + 1 for some choice ζ ℓ of primitive ℓ -th root of unity. We denoteby K (Ψ ℓ ) = K [ T ] / Ψ ℓ ( T ) the corresponding K -algebra, and by R K,S,ℓ = O K,S [ T ] / (Ψ ℓ ( T )) thecorresponding order in K (Ψ ℓ ). Note that K (Ψ ℓ ) is a field if ζ ℓ K , and is isomorphic to K × K if ζ ℓ ∈ K .There is an involution ι of K (Ψ ℓ ) given by sending ζ ℓ ζ − ℓ , i.e., exchanging the two roots ofthe polynomial Ψ ℓ ( T ). If K (Ψ ℓ ) is a field, then ι generates Gal( K (Ψ ℓ ) /K ); if K (Ψ ℓ ) ∼ = K × K ,it exchanges the two factors. In either case, K (Ψ ℓ ) ι is equal to K embedded as constantpolynomials. The involution ι restricts to R K,S,ℓ , because ζ ℓ is integral, and R ιK,S,ℓ = O K,S .Due to these statements, we will abuse notation and use Gal( K (Ψ ℓ ) /K ) ∼ = Z / Z ∼ = h ι i even if K (Ψ ℓ ) is not a field. ALEXANDER D. RAHM AND MATTHIAS WENDT
For definitions of class groups, structure of unit groups and other fundamental statementsfrom algebraic number theory, we refer to [Neu92]. We will denote the class group of an S -integer ring by Pic( O K,S ). Recall that for a finite extension A → B of commutative rings, thereare transfer maps on algebraic K-theory K • ( B ) → K • ( A ). Two special cases of such transfermaps will be interesting for us. On the one hand, specializing to rings of Krull dimension one,there is a norm map on class groups Nm ( B/A ) : Pic( B ) → Pic( A ). The kernel of Nm ( B/A )is called the relative class group Pic(
B/A ) of the extension
B/A . On the other hand, there is anorm map on unit groups Nm ( B/A ) : B × → A × .4.2. Structure of relevant rings.
We first describe the structure of the rings R K,S,ℓ . If K (Ψ ℓ )is a field, then R K,S,ℓ ∼ = O K,S [ ζ ℓ ] is an order in K (Ψ ℓ ). In general, it does not need to be amaximal order, hence it may fail to be a Dedekind ring. Proposition 4.1.
Assume K (Ψ ℓ ) is a field, and assume that the prime ( ζ ℓ − ζ − ℓ ) is unramifiedin the extension O K,S / Z [ ζ ℓ + ζ − ℓ ] . Then { , ζ ℓ } is a relative integral base for O K (Ψ ℓ ) , ˜ S over O K,S ,where ˜ S is the set of places of K (Ψ ℓ ) lying over the places in S . In particular, R K,S,ℓ ∼ = O K (Ψ ℓ ) , ˜ S is a Dedekind domain.Proof. We consider an element a + bζ ℓ ∈ K (Ψ ℓ ). We have the trace and norm of this elementgiven by Tr( a + bζ ℓ ) = 2 a + b ( ζ ℓ + ζ − ℓ ) , N( a + bζ ℓ ) = a + ( ζ ℓ + ζ − ℓ ) ab + b . The element a + bζ ℓ is integral over O K,S if and only if norm and trace are elements of O K,S .The discriminant of the basis (1 , ζ ℓ ) is concentrated at ℓ , so we only need to worry aboutdivisibility by elements in primes over ℓ . We need a case distinction.(1) If ℓ >
3, we have ζ ℓ + ζ − ℓ ζ ℓ − ζ − ℓ ).Then we consider the reduction O K,S / ( ζ ℓ − ζ − ℓ ). By assumption, this is a smooth algebra over Z [ ζ ℓ + ζ − ℓ ] / ( ζ ℓ − ζ − ℓ ) ∼ = F ℓ , in particular it has no nilpotent elements. We compute the universalexample of an F ℓ -algebra in which Tr = N = 0; the result is F ℓ [ A, B ] / ( A , B , AB ). In particular,the elements A and B have to be nilpotent modulo ( ζ ℓ − ζ − ℓ ), hence necessarily have to be zero.Hence we find ( ζ ℓ − ζ − ℓ ) divides A and B . Inductively, this deals with divisibility by powers of( ζ ℓ − ζ − ℓ ). Now the argument for elements which have non-zero reduction in O K,S / ( ζ ℓ − ζ − ℓ )is done in the same way: the latter, as an F ℓ -algebra is a product of field extensions, hence anyquotient of it will again have no nontrivial nilpotent elements. In particular, integrality of thenorm and trace implies that the element is already an O K,S -linear combination of 1 and ζ ℓ .(2) The argument for (1) does not work in case ℓ = 3. In this case 2 ≡ ζ + ζ − mod (3).In particular, the norm condition mod 3 is satisfied whenever the trace condition is satisfiedmod 3. However, in this case, we have K/ Q has discriminant coprime to 3 and Q ( ζ ) / Q hasdiscriminant a power of 3. Then the product of the integral bases for K and Q ( ζ ) is an integralbasis for the composite K ( ζ ). In particular, any integral element of K ( ζ ) is an O K,S -linearcombination of 1 and ζ , proving the claim. (cid:3) Example 4.2.
A simple example where R K,S,ℓ fails to be a maximal order, due to ramificationover ℓ is given as follows: take ℓ = 3, let K = Q ( √
3) and let S contain only the infinite places.In this case, O K = Z [ √
3] and R K,S,ℓ = Z [ √ , ζ ]. The element i = √ √ ζ Z [ √ ζ . This type of problems necessi-tates the discussion of non-invertible ideals. (cid:3) Corollary 4.3.
Assume K (Ψ ℓ ) is a field, and assume that the prime ( ζ ℓ − ζ − ℓ ) is unramified inthe extension O K,S / Z [ ζ ℓ + ζ − ℓ ] . Then R K,S,ℓ is a free O K,S -module of rank two. In particular,there exists an R K,S,ℓ -ideal with O K,S -basis.
N FARRELL–TATE COHOMOLOGY OF SL OVER S -INTEGERS 9 Next, we consider the case where K (Ψ ℓ ) ∼ = K × K is not a field. In this case, R K,S,ℓ is not aDedekind domain because it fails to be a domain. The structure of R K,S,ℓ is described by thefollowing:
Proposition 4.4.
Assume K (Ψ ℓ ) ∼ = K × K is not a field. Then the ring R K,S,ℓ ∼ = O K,S [ T ] / (( T − ζ ℓ )( T − ζ − ℓ )) sits in a fiber product square R K,S,ℓ ev ζℓ / / ev ζ − ℓ (cid:15) (cid:15) O K,Sπ (cid:15) (cid:15) O K,S π / / O K,S / ( ζ ℓ − ζ − ℓ ) . The maps denoted by ev x evaluate T x , and π : O K,S
7→ O
K,S / ( ζ ℓ − ζ − ℓ ) is the naturalreduction modulo ( ζ ℓ − ζ − ℓ ) .As a consequence, we have a ring isomorphism R K,S,ℓ ∼ = O K,S × O
K,S if S ℓ ⊆ S , i.e., S contains the places lying over ℓ . By convention, this includes the case where K is a globalfunction field of characteristic different from ℓ .Proof. The fiber product square is a special case of the following fiber product for two ideals
I, J in ring R : R/ ( I ∩ J ) / / (cid:15) (cid:15) R/I (cid:15) (cid:15)
R/J / / R/ ( I + J ) , specialized to R = O K,S [ T ], I = ( T − ζ ℓ ) and J = ( T − ζ − ℓ ). In this case, I ∩ J = (Ψ ℓ ( T )) and I + J = ( ζ ℓ − ζ − ℓ , T − ζ ℓ ).Note that in Z [ ζ ℓ ], we have ( ℓ ) = ( ζ iℓ − ℓ − for any 0 < i < ℓ . In particular, ζ ℓ − ζ − ℓ generates the maximal ideal of Z [ ζ ℓ ] lying over ( ℓ ). If S ℓ ⊆ S , then ζ ℓ − ζ − ℓ is invertible, andthe quotient ring O K,S / ( ζ ℓ − ζ − ℓ ) is trivial. In that case, the fiber product diagram shows that R K,S,ℓ is the direct product of two copies of O K,S . (cid:3) We can also note that as O K,S -module, R K,S,ℓ is always free of rank two, independent of theassumption S ℓ ⊆ S . Corollary 4.5.
The following map is an isomorphism of O K,S -modules: O K,S ⊕ O
K,S → O
K,S × O K,S / ( ζ ℓ − ζ − ℓ ) O K,S : ( a, b ) ( a, a + b ( ζ ℓ − ζ − ℓ )) . In particular, there always exists an ideal in R K,S,ℓ which has an O K,S -basis.
Remark 4.6.
Since R K,S,ℓ fails to be a domain, we cannot really speak about fractional idealor invertibility of ideals anyway. However, behaviour of ideals in the present situation is fairlysimilar to the case where K (Ψ ℓ ) is a field, and R K,S,ℓ ⊆ K (Ψ ℓ ) an order. We can define aconductor c = { x ∈ K × K | x · ( O K,S × O
K,S ) ⊆ R K,S,ℓ } = { ( a, b ) ⊆ R K,S,ℓ | a, b ∈ ( ζ ℓ − ζ − ℓ ) O K,S } . As an example of what can go wrong with ideals which are not coprime to the conductor, wewill discuss the case K = Z (deviating for once from the convention that ℓ is an odd prime). Inthis case, the ring R K,S,ℓ = Z × Z / Z Z , and the conductor is the ideal { ( a, b ) ∈ Z | a, b ∈ Z } .This ideal is in fact not a principal ideal, it is generated by (2 ,
0) and (0 , ,
0) and (0 ,
1) in Z fail to lie in R K,S,ℓ , the ideal generated by (2 ,
2) willcontain (4 ,
0) and (0 , ,
0) or (0 , c , we have c = (2 , c ,and the conductor is the only non-invertible (and non-principal) ideal class. Note also that theconductor, while not a projective R K,S,ℓ -module, is free as Z -module. This is very similar tothe situation of the order Z [ √−
3] in Z [ ζ ] from Example 4.2. Conjugacy classes of elements and oriented ideal classes.
As a next step, we canrelate conjugacy classes of elements with oriented relative ideal classes. Fix a global field K , a setplaces S and a prime ℓ as in Subsection 4.1. We fix a primitive ℓ -th root of unity ζ ℓ and denoteby C K,S,ℓ the set of conjugacy classes of order ℓ elements in SL ( O K,S ) whose characteristicpolynomial is Ψ ℓ ( T ) = T − ( ζ ℓ + ζ − ℓ ) T + 1. We will omit K and S whenever they are clearfrom the context.To relate these conjugacy classes of elements to data of a more number-theoretic nature, weconsider the following “oriented relative ideal classes”, i.e., classes of ideals whose determinant isprincipal, plus the additional datum of an element generating the determinant. Definitions likethe following have been used in the conjugacy classification of finite order elements in symplecticgroups over principal ideal domains, cf. [Bus06, Section 3]. Remark 4.7.
Note that requiring trivial norm (as in [Bus06]) is different from requiring trivialdeterminant (as in the definitions and results below). However, we will only consider thesituations in Corollaries 4.3 and 4.5 in which we know that the Steinitz class is trivial andtherefore norm and determinant are equivalent ideals.
Definition 4.8.
With the notation above, an oriented relative ideal of R K,S,ℓ is a pair ( a , a ),where a ⊆ K (Ψ ℓ ) is a fractional R K,S,ℓ -ideal together with a choice of generator a ∈ V O K,S a .This in particular implies that a ∼ = O K,S as O K,S -modules.Define the following equivalence relation on oriented relative ideals:( a , a ) ∼ ( b , b ) ⇐⇒ ∃ τ ∈ K (Ψ ℓ ) × , τ a = b , (m τ ∧ m τ )( a ) = b, where m τ denotes multiplication with τ on a . Note that (m τ ∧ m τ ) is multiplication with thenorm N K (Ψ ℓ ) /K ( τ ) on V K K (Ψ ℓ ) and maps V a to V b . The ∼ -equivalence class of an orientedrelative ideal ( a , a ) is denoted by [ a , a ].The set of oriented relative ideal classes f Pic( R K,S,ℓ / O K,S ) is defined as the set of ∼ -equivalenceclasses of oriented relative ideals of R K,S,ℓ . Remark 4.9.
We shortly comment on the choice of terminology.Including the qualifier “relative” is rather common terminology; the relative class group isthe kernel of the norm map Nm : Pic( B ) → Pic( A ) for a finite extension B/A of Dedekindrings. In the cases we consider, norm and determinant are equivalent ideals.Using the qualifier “oriented” is inspired from a more geometric way of stating the abovedefinition. In a function field situation, we would have a (possibly branched) degree 2 covering f : X → Y . A line bundle L on X gives rise to a rank 2 vector bundle f ∗ L on Y . Therequirement in the above definition means that f ∗ L has trivial determinant, and includes thedatum of a given trivialization. This means that f ∗ L is in fact an orientable vector bundle,with a given choice of orientation. This puts the class group considered above in the context ofalgebraic cohomology theories like oriented Chow groups.With the above terminology, we can now establish the following bijection between conjugacyclasses of elements and oriented relative ideal classes. The arguments follow those of [Bus06]. Proposition 4.10.
There is a bijection between f Pic( R K,S,ℓ / O K,S ) and C K,S,ℓ .Proof.
Part (1) and (2) of the proof set up the maps between these sets, and part (3) showsthat these maps are inverses of each other.(1) We first describe the map from elements of order ℓ with characteristic polynomial Ψ ℓ ( T )to oriented ideals. Let ρ ∈ SL ( O K,S ) be an element with characteristic polynomial Ψ ℓ ( T ). Viathe standard representation of SL , it acts on K , giving the latter the structure of a rank one K (Ψ ℓ )-module. Choosing an isomorphism K ∼ = K (Ψ ℓ ), the standard lattice O K,S ⊆ K givesrise to a finitely generated R K,S,ℓ -submodule O K,S ⊆ K ∼ = K (Ψ ℓ ), hence a fractional ideal a .Moreover, in the above, we have chosen a basis of R K,S,ℓ as O K,S -module, and this basis givesa generator a ∈ V O K,S a . N FARRELL–TATE COHOMOLOGY OF SL OVER S -INTEGERS 11 (1a) The assignment in (1) is independent of the choice of isomorphism K ∼ = K (Ψ ℓ ), up to ∼ -equivalence of oriented relative ideals. Any other isomorphism will be obtained by scalingwith λ ∈ K (Ψ ℓ ) × . This changes the fractional ideal a by multiplication with λ , and the volumeform a by multiplication with the norm N K (Ψ ℓ ) /K ( λ ). Hence it does not change the class of theassociated oriented relative ideal [ a , a ] in f Pic( R K,S,ℓ / O K,S ).(1b) More generally, SL ( O K,S )-conjugate elements are mapped to ∼ -equivalent oriented rel-ative ideals. Let ρ ∈ SL ( O K,S ) be an element with characteristic polynomial Ψ ℓ ( T ), and let A ∈ SL ( O K,S ), so that ρ and ρ ′ = A − ρA are conjugate via A . Choose an isomorphism φ : K → K (Ψ ℓ ) which maps ρ to the oriented relative ideal class [ φ ( O K,S ) , φ ( e ) ∧ φ ( e )].The isomorphism φ ′ = φ ◦ A : K → K (Ψ ℓ ) maps ρ ′ to the oriented relative ideal class[ φ ′ ( O K,S ) , φ ′ ( e ) ∧ φ ′ ( e )]. We claim that these ideal classes are equal. Since φ is O K,S -linearand det( A ) = 1, we find that these ideal classes are equal:[ φ ′ ( O K,S ) , φ ′ ( e ) ∧ φ ′ ( e )] = [ φ ( A · O K,S ) , φ ( A · e ) ∧ φ ( A · e )]= [ φ ( O K,S ) , det( A ) · ( φ ( e ) ∧ φ ( e ))]= [ φ ( O K,S ) , φ ( e ) ∧ φ ( e )] . Essentially, conjugation changes the K (Ψ ℓ )-structure of K , but it leaves invariant the standardlattice SL ( O K,S ) and the volume form.(2) Now we describe the map from oriented relative ideals to elements of order ℓ with charac-teristic polynomial Ψ ℓ ( T ). Let ( a , a ) be an oriented relative ideal. Then a ⊆ K (Ψ ℓ ) is a finitelygenerated R K,S,ℓ -submodule which is isomorphic to O K,S , because we have given an explicittrivialization a ∈ V O K,S a . Choosing an O K,S -basis of a with volume form a , we can writemultiplication with ζ ℓ ∈ R K,S,ℓ as a matrix in SL ( O K,S ).(2a) Any two bases of a with given volume a ∈ V O K,S a are SL ( O K,S )-conjugate. Therefore,the class in C K,S,ℓ of the matrix ρ ∈ SL ( O K,S ) associated to the oriented relative ideal ( a , a ) isindependent of the choice of basis.(2b) Assume that the oriented relative ideals ( a , a ) and ( b , b ) are ∼ -equivalent, i.e., thereexists τ ∈ K (Ψ ℓ ) × with τ a = b and (m τ ∧ m τ )( a ) = b . In particular, m τ induces an isomorphism a ∼ = b as R K,S,ℓ -modules. An O K,S -basis of a will be mapped by m τ to a basis of b , and thecorresponding representing matrices will be GL ( O K,S )-conjugate. Moreover, since the volumeforms correspond under the R K,S,ℓ -module isomorphism, the conjugating change-of-basis matrixwill already lie in SL ( O K,S ).(3) It is now easy to see that the maps constructed in (1) and (2) are inverses of each other.From a matrix, we get a fractional R K,S,ℓ -ideal a ⊆ K (Ψ ℓ ), with chosen basis in which Ψ ℓ actsvia the given matrix. Conversely, starting from an oriented relative ideal, we can choose an O K,S -basis and write out the representing matrix, which will then give back the R K,S,ℓ -modulestructure we started with. (cid:3)
Remark 4.11.
Our main interest in this paper is the conjugacy classification of finite ordersubgroups in SL ( O K,S ). However, there are similar results in the case GL , which actuallycan be formulated slightly easier. The basic correspondence between ideal classes in extensionsand conjugacy classes of elements was already described by Latimer–MacDuffee [LM33], andlater generalized by Taussky and Bender. A variation of [Ben67, theorem 1] shows that for aDedekind domain R , there is a bijective correspondence between conjugacy classes of elementsof order ℓ and ideal classes in R [ ζ ℓ ] which have an R -basis.The actual correspondence is fairly easy to setup, and is well explained in [LM33], or [Con12]for a more modern exposition. An ideal of R [ ζ ℓ ] with R -basis gives rise to a conjugacy classof elements in GL ( R ) by writing multiplication with ζ ℓ in some R -basis. Conversely, given anelement ρ ∈ GL ( R ), its action on R gives the latter the structure of fractional ideal for R [ ζ ℓ ].The arguments are similar to what we have done for SL above, only easier because there is nofixed orientation to carry around. Proposition 4.12.
Assume one of the following conditions:(R1) ℓ K and the prime ( ζ ℓ − ζ − ℓ ) is unramified in the extension O K,S / Z [ ζ ℓ + ζ − ℓ ] .(R2) ℓ ∈ K and S ℓ ⊆ S .Then there is a natural group structure on f Pic( R K,S,ℓ / O K,S ) , given by multiplication of idealsand volume forms.Proof. Multiplication is given by ideal multiplication and multiplication of volume forms. Thenatural element is the trivial ideal class oriented by 1. Here V O K,S R K,S,ℓ is viewed as thenatural sub- O K,S -module of V K K (Ψ ℓ ) ∼ = K . Since T ∈ R K,S,ℓ has norm 1, the basis (1 , T )maps to 1 ∧ T ∈ V O K,S R K,S,ℓ , which maps to 1 in V K K (Ψ ℓ ) ∼ = K . This implies that all theaxioms for a group operation are satisfied except the invertibility.In case (R1), Proposition 4.1 implies that R K,S,ℓ is a Dedekind ring. In particular, ideals areinvertible with respect to multiplication, the inverse of I is given by ι ( I ). Since the inverse ofan ideal has inverse norm, forming inverses preserves the property of having an O K,S -basis.In case (R2), R K,S,ℓ is not a Dedekind ring, but it is of the form O K,S × O
K,S . The ideals arethen given by pairs of ideals of O K,S , with entrywise multiplication. Since O K,S is a Dedekindring, ideals in R K,S,ℓ are invertible for the multiplication.In both cases, the volume form of I induces a volume form for ι ( I ). Since elements inthe image of the norm N R K,S,ℓ / O K,S provide volume forms ∼ -equivalent to 1, this provides theinverse. (cid:3) We will speak of the oriented relative class group of R K,S,ℓ / O K,S in those situations wherethe assumptions (R1) or (R2) are satisfied and the above proposition implies that the set f Pic( R K,S,ℓ / O K,S ) has a group structure.4.4.
Oriented class groups in the case ζ ℓ K . The next step is to identify the set oforiented relative ideal classes in terms of more conventional data from algebraic number theory.This will be done in the next subsections, handling the two cases ζ ℓ K and ζ ℓ ∈ K separately.As it turns out, the conjugacy classification of elements of finite order in SL ( O K,S ) is controlledmostly by kernels and cokernels of norm maps.
Proposition 4.13.
Under assumption (R1), the oriented relative class group f Pic( R K,S,ℓ ) sitsin an extension → coker (cid:16) Nm : R × K,S,ℓ → O × K,S (cid:17) → f Pic( R K,S,ℓ / O K,S ) → Pic( R K,S,ℓ / O K,S ) → . Proof.
The proof proceeds as in [Bus06, Proposition 3.10].We first note that there is a natural group homomorphism f Pic( R K,S,ℓ / O K,S ) → Pic( R K,S,ℓ )mapping an oriented relative ideal to its underlying R K,S,ℓ -ideal. The image consists exactlyof the ideal classes in R K,S,ℓ which have an O K,S -basis. In particular, under assumption (R1),trivial determinant and trivial norm are the same, cf. Corollary 4.3, and the map above inducesa surjection f Pic( R K,S,ℓ / O K,S ) → Pic( R K,S,ℓ / O K,S ) . We also get an injective group homomorphismcoker (cid:16) Nm : R × K,S,ℓ → O × K,S (cid:17) → f Pic( R K,S,ℓ / O K,S )by sending an element u ∈ O × K,S to the oriented relative ideal ( R K,S,ℓ , u ∧ T ) where u ∧ T isthe orientation corresponding to the O K,S -basis ( u, T ) of R K,S,ℓ . This map will factor throughthe quotient coker Nm and induce an injection as claimed by the definition of equivalence oforiented relative ideals.Exactness in the middle also follows directly from the definition of equivalence of orientedrelative ideals. (cid:3) N FARRELL–TATE COHOMOLOGY OF SL OVER S -INTEGERS 13 Remark 4.14.
The natural origin of the short exact sequence in Proposition 4.13 is the longexact sequence associated to the fiber sequence of K-theory spectrahofib Nm → K ( R K,S,ℓ ) Nm −→ K ( O K,S )(up to a discussion getting rid of the Z -summands in K ).We have now seen how to relate the number of classes of oriented relative ideals to number-theoretic data: the possible underlying ideals are counted via the relative class group, and thepossible orientations are counted via the cokernel of the norm map. It is possible to get evenmore information on these constituent pieces: for the relative class group, one could use classnumber formulas. On the other hand, the cokernel of the norm map on units can also beunderstood by generalizing the discussion of [Bus06, section 3.2] to get the following statement. Proposition 4.15.
The cokernel coker Nm ( R K,S,ℓ / O K,S ) of the norm-map on units is a Z / Z -module whose rank equals the number of inert places of the extension R K,S,ℓ / O K,S . Oriented class groups in the case ζ ℓ ∈ K . Understanding the oriented class group inthe case ζ ℓ ∈ K is easier. Proposition 4.16.
Under assumption (R2), we have an isomorphism f Pic( R K,S,ℓ / O K,S ) ∼ = Pic( O K,S ) . Proof.
The most important ingredient is Proposition 4.4, which implies that under assumption(R2), we have R K,S,ℓ ∼ = O K,S ×O K,S as rings. In particular, ideal classes of R K,S,ℓ are in bijectionwith pairs of ideal classes in O K,S , and an ideal has an O K,S -basis if and only if it is equivalentto one of the form ( I , I − ). We therefore get a surjection f Pic( R K,S,ℓ / O K,S ) → Pic( O K,S ). Toshow injectivity, we look at the possible orientations of the trivial ideal class. These are givenby generators of the determinant, so they differ by units in O K,S . Now scaling with a unit inthe image of the norm map R × K,S,ℓ → O × K,S does not change the orientation. However, the normmap is simply the multiplication map R × K,S,ℓ ∼ = ( O × K,S ) → O × K,S . Surjectivity of the norm mapthen implies that all orientations are equivalent, hence we get the isomorphism as claimed. (cid:3)
Remark 4.17.
It is still possible to describe the oriented class group in case (R2) is not satisfied.In this case, we have to restrict to those oriented ideals whose underlying ideals are invertible.The resulting class group is then given by an extension0 → coker (cid:16) O × K,S → (cid:0) O K,S / ( ζ ℓ − ζ − ℓ ) (cid:1) × (cid:17) → Pic( R K,S,ℓ ) → Pic( O K,S ) → . This is basically a form of Milnor patching for projective modules in the fiber square of Proposi-tion 4.4, with the cokernel-of-reduction on units classifying the possible gluing data. The exactsequence gives rise to a version of Dedekind’s formula for class groups of orders. In this case,there can also be non-trivial orientations coming from inert places over ( ζ ℓ − ζ − ℓ ). However, themost problematic part of understanding oriented relative ideals is the possible non-invertibilityof ideals as discussed in Remark 4.6. This is the reason for staying away from this sort of casesaltogether.5. Conjugacy classification of finite cyclic subgroups and descriptions ofnormalizers
In Section 4, we recalled the conjugacy classification of elements of order ℓ in SL ( O K,S ) withcharacteristic polynomial Ψ ℓ ( T ). What remains to be done is the description of the conjugacyclassification of subgroups of order ℓ and the description of their centralizers and normalizers.The difference between the classification of finite order elements and the classification of finitecyclic subgroups is completely controlled by the action of the “Galois group” Gal( K (Ψ ℓ ) /K ) ∼ = Z / Z . Again, the statements for SL ( Z [1 /n ]) can be found in [Bus06], and we provide somenecessary augmentations to deal with the general case SL ( O K,S ).In this section, we continue to use the notation set up at the beginning of Section 4. We willdenote by K K,S,ℓ the set of conjugacy classes of finite cyclic subgroups of order ℓ in SL ( O K,S ), and write K ℓ if the number ring is clear from the context. If we have any cyclic subgroup Γ oforder ℓ in SL ( O K,S ), then for any primitive ℓ -th root of unity ζ there will be an element of Γhaving characteristic polynomial Ψ ℓ ( T ) = T − ( ζ + ζ − ) T + 1. In particular, K ℓ is a quotientof C ℓ , and the difference appears whenever, for an element g , the normalizer N of the subgroup h g i acts non-trivially on this subgroup.5.1. Centralizers and norm one units.
We first consider the centralizers of elements of finiteorder. Again, we have to distinguish between the cases where K (Ψ ℓ ) is an extension field of L or where K (Ψ ℓ ) ∼ = K × K . Proposition 5.1.
Assume K (Ψ ℓ ) is a field and that condition (R1) is satisfied. If ρ is anelement of order ℓ and M ∈ SL ( O K,S ) centralizes h ρ i , then M is given by multiplication by anorm-one unit u ∈ ker Nm ⊆ R × K,S,ℓ . In particular, we have C SL ( O K,S ) ( h ρ i ) ∼ = ker (cid:16) R × K,S,ℓ Nm −→ O × K,S (cid:17) . Proof.
Under the correspondence set up in Section 4 (with ζ ℓ chosen such that Ψ ℓ ( T ) = T − ( ζ ℓ + ζ − ℓ ) T + 1 is the characteristic polynomial of ρ ), the element ρ corresponds toan oriented relative ideal ( a , a ), where a ⊆ K (Ψ ℓ ) is an invertible fractional R K,S,ℓ -ideal and a ∈ V O K,S a . The element ρ is represented as multiplication by ζ ℓ on a . Since the ring R K,S,ℓ is generated by ζ ℓ and elements from O K,S , any matrix M that commutes with multiplicationwith ζ ℓ necessarily commutes with multiplication with any element from R K,S,ℓ . We can thenconsider the associated algebraic group GL ( K ) acting on the two-dimensional K -vector space K (Ψ ℓ ) in which the matrices representing multiplication with elements of K (Ψ ℓ ) form a maximaltorus. Therefore, we see that any matrix M ∈ SL ( O K,S ) which commutes with ζ ℓ centralizes amaximal torus of GL ( K ) after embedding SL ( O K,S ) ⊆ GL ( K ). From the theory of algebraicgroups, we see that, as an element of GL ( K ), M must itself be an element of the maximaltorus, hence it necessarily is multiplication with a unit. The determinant of multiplication witha unit is given by the norm of the unit. Therefore, if an element M ∈ SL ( O K,S ) centralizes ζ ℓ ,then it is given by multiplication with a norm-one unit. (cid:3) Proposition 5.2.
Assume K (Ψ ℓ ) ∼ = K × K and that condition (R2) is satisfied. If ρ is anelement of order ℓ and M ∈ SL ( O K,S ) centralizes h ρ i , then M is given by multiplication by adiagonal matrix ( u, u − ) with u ∈ O × K,S . In particular, we have C SL ( O K,S ) ( h ρ i ) ∼ = O × K,S . Proof.
As before, the element T generates R K,S,ℓ as O K,S -algebra. Therefore, any matrix M ∈ SL ( O K,S ) commuting with multiplication by T will commute with multiplication byany element from R K,S,ℓ . Passing to the algebraic group SL ( K ), such a matrix M commutingwith multiplication by T again commutes with the whole maximal torus of SL ( K ). Therefore,it must already be a diagonal matrix, hence of the form diag( a, a − ) for a ∈ R × K,S,ℓ . (cid:3) Normalizers, dihedral overgroups and Galois action.
Next, we discuss normalizersof subgroups of finite order in SL ( O K,S ), as these are relevant data for the computation ofFarrell–Tate cohomology. Looking at the proofs of Propositions 5.1 and 5.2, we get the followingresult:
Proposition 5.3.
In the situation of Propositions 5.1 and 5.2, the Weyl group of the subgroup h ρ i generated by ρ is given by N SL ( O K,S ) ( h ρ i ) /C SL ( O K,S ) ( h ρ i ) ∼ = Stab Gal( K (Ψ ℓ ) /K ) ( I ρ ) , where I ρ is the ideal class corresponding to the element ρ .Proof. In each of the cases of Propositions 5.1 resp. 5.2, we see that an element centralizing ρ must centralize a maximal torus. Similarly, if an element normalizes the subgroup h ρ i , thenit already normalizes a maximal torus of SL ( K ). The theory of algebraic groups tells us thatthe normalizer of a maximal torus of SL ( K ) is of the form K × ⋊ Z / Z , with the finite group N FARRELL–TATE COHOMOLOGY OF SL OVER S -INTEGERS 15 quotient acting by inversion on K × . In particular, if an element normalizes h ρ i , but does notleave ρ invariant, then it must map ρ to ρ − . Then necessarily, we have I ρ ∼ = I σρ , i.e., theideal class I in f Pic( R K,S,ℓ / O K,S ) is invariant under the action of Gal( K (Ψ ℓ ) /K ) ∼ = Z / Z . Onthe other hand, if the Galois action does not leave the ideal class I ρ invariant, then ρ and ρ − cannot be conjugate, hence normalizer and centralizer agree in this case. (cid:3) Corollary 5.4.
In the situation of Proposition 5.3, a cyclic subgroup
Γ = h ρ i of order ℓ in SL ( O K,S ) is embeddable in a dihedral subgroup of SL ( O K,S ) precisely when the associatedideal I ρ is invariant under the action of Gal( K (Ψ ℓ ) /K ) .The conjugacy classes of dihedral overgroups of Γ are in bijection with the number of orien-tations of I − ⊗ I σ , which in turn is in bijection with the group ker (cid:16) Nm : R × K,S,ℓ → O × K,S (cid:17) ⊗ Z Z / Z of square residues of norm 1 units. Remark 5.5.
The above results on computations of normalizers also explain exactly how topass from the conjugacy classification for elements (with fixed characteristic polynomial) to theconjugacy classification for subgroups. What can happen is that the normalizer of the subgroup h ρ i identifies ρ and ρ − which have the same characteristic polynomial T − ( ζ ℓ + ζ − ℓ ) T + 1. Inparticular, the conjugacy classes of subgroups are given as orbit set of C K,S,ℓ ∼ = f Pic( R K,S,ℓ / O K,S )under the Gal( K (Ψ ℓ ) /K )-action.It remains to better understand the Galois action on the oriented relative class group. Thefollowing is just a consequence of writing out the definition of oriented ideals: Proposition 5.6. (1) Assume ζ ℓ K and assumption (R1) is satisfied. Then the ac-tion of Gal( K (Ψ ℓ ) /K ) on f Pic( R K,S,ℓ / O K,S ) is induced from sending an oriented ideal ( a , a ) ( ι ( a ) , a ′ ) , where a ′ is given as follows: represent the orientation a ∈ V O K,S a by a corresponding O K,S -basis s.th. a = x ∧ x . Then a ′ = ι ( x ) ∧ ι ( x ) , which is avolume form for ι ( a ) .(2) Assume ζ ℓ ∈ K and assumption (R2) is satisfied. Then the action of Gal( K (Ψ ℓ ) /K ) on f Pic( R K,S,ℓ / O K,S ) ∼ = Pic( O K,S ) is given by the inverse. In particular, the action of the Galois group on orientations is not the trivial one (as onecould think from the identification as quotient of O × K,S ). This implies that the Galois actiondoes not respect the group structure on orientations (viewed as cokernel of Nm ). Example 5.7.
We discuss the Galois action in the simplest case, namely elements and sub-groups of SL ( Z ) of order 3. In this case, we have O K,S = Z , R K,S,ℓ = Z [ ζ ] and condition(R1) is satisfied. The ring Z [ ζ ] is euclidean, so there is only a trivial ideal class. Moreover,the norm map on units is the trivial map Z [ ζ ] × ∼ = µ → µ ∼ = { , − } . Therefore, we find twoconjugacy classes of elements of exact order 3 in SL ( Z ), given by the two possible orientationsof Z [ ζ ] as free Z -module of rank two. We can choose a Z -basis (1 , ζ ) for Z [ ζ ]. The Galoisgroup Gal( Q ( ζ ) / Q ) will send this basis to (1 , ζ ), which is obviously orientation-reversing. Theresult is the obvious one - there is a unique conjugacy class of subgroups of order 3 in SL ( Z ). (cid:3) Explicit formulas.
Finally, we can combine the previous results into a result describingconjugacy classes of finite cyclic subgroups in SL ( O K,S ) with K a global field, under suitableregularity assumptions. Theorem 5.8.
Let K be a global field, and fix an odd prime ℓ different from the characteristicof K .(1) Assume ζ ℓ K and that assumption (R1) is satisfied. The set C ℓ of conjugacy classesof elements of order ℓ with characteristic polynomial Ψ ℓ ( T ) = T − ( ζ ℓ + ζ − ℓ ) T + 1 isnon-empty, has a group structure and sits in the extension → coker (cid:16) Nm : R × K,S,ℓ → O × K,S (cid:17) → C ℓ → ker (Nm : Pic( R K,S,ℓ ) → Pic( O K,S )) → . Additionally, the set C ℓ has an action of Gal( K (Ψ ℓ ) /K ) , obtained from its identificationwith oriented ideals in R K,S,ℓ . Denoting by K ℓ the conjugacy classes of subgroups oforder ℓ in SL ( O K,S ) , we have an isomorphism K ℓ ∼ = C ℓ / Gal( K (Ψ ℓ ) /K ) .A finite group Γ with [Γ] ∈ K ℓ is contained in a dihedral overgroup if and only if thecorresponding element I (Γ) in C ℓ is Gal( K (Ψ ℓ ) /K ) -invariant.If the corresponding element I (Γ) in C ℓ is Gal( K (Ψ ℓ ) /K ) -invariant, then the nor-malizer of Γ in SL ( O K,S ) is isomorphic to ker (cid:16) Nm : R × K,S,ℓ → O × K,S (cid:17) ⋊ Z / Z with the quotient Z / Z acting via inversion.If the corresponding element I (Γ) in C ℓ is not Gal( K (Ψ ℓ ) /K ) -invariant, then thenormalizer of Γ in SL ( O K,S ) is isomorphic to ker (cid:16) Nm : R × K,S,ℓ → O × K,S (cid:17) .(2) Assume ζ ℓ ∈ K and that assumption (R2) is satisfied. Then the conjugacy classes ofelements of order ℓ with characteristic polynomial Ψ ℓ ( T ) = T − ( ζ ℓ + ζ − ℓ ) T + 1 are inbijection with Pic( O K,S ) . Denoting the involution I 7→ I − by ι , the conjugacy classes K ℓ of subgroups of order ℓ are in bijection with the orbit set Pic( O K,S ) /ι .Any such finite group Γ is contained in a dihedral overgroup if and only if the corre-sponding element I (Γ) in Pic( O K,S ) is Z / Z -invariant.If the corresponding element I (Γ) in C ℓ is ι -invariant, then the normalizer of Γ in SL ( O K,S ) is isomorphic to O × K,S ⋊ Z / Z with Z / Z acting by inversion.If the corresponding element I (Γ) in C ℓ is not ι -invariant, then the normalizer of Γ in SL ( O K,S ) is isomorphic to O × K,S .Proof. (i) The conjugacy classification for elements is given by the bijection of Proposition 4.10.The exact sequence is Proposition 4.13. The description of centralizers and normalizers is givenin Proposition 5.1 and Proposition 5.3. The description of dihedral overgroups is Corollary 5.4.From the description of normalizers, we get the conjugacy classification of subgroups as claimed.Part (ii) is proved along the same lines, using Propositions 4.16 and 5.2 at the suitableplaces. (cid:3)
Remark 5.9.
See also [Wen15a] for similar results concerning SL ( K [ C ]) with C a smoothaffine curve over an algebraically closed field K . Remark 5.10.
Similar results can be obtained for GL ( O K,S ) with K a global field, with onlysmall modifications to the proofs. We only formulate the result.(1) Assume ζ ℓ ∈ K . Then the conjugacy classes [Γ] of ℓ -order subgroups in GL ( O K,S ) are inbijection with elements of Pic( O K,S ) /ι . Such a finite group Γ is contained in a dihedralovergroup if and only if the corresponding element I (Γ) in Pic( O K,S ) is Z / Z -invariant;in that case, all dihedral overgroups are conjugate. The normalizer of Γ in GL ( O K,S )is isomorphic to ( O × K,S ) ⋊ Z / Z if there is a dihedral overgroup, and isomorphic to( O × K,S ) otherwise.(2) Assume ζ ℓ K . Then the conjugacy classes [Γ] of ℓ -order subgroups in GL ( O K,S ) areclassified by elements of the orbit set Pic( R K,S,ℓ / O K,S ) / Gal( K (Ψ ℓ ) /K ). Such a finitegroup Γ is contained in a dihedral overgroup if and only if the corresponding element I (Γ) in Pic( R K,S,ℓ / O K,S ) is Z / Z -invariant; in that case, all dihedral overgroups areconjugate. The normalizer of Γ in GL ( O K,S ) is isomorphic to O K,S [ ζ ℓ ] × ⋊ Z / Z ifthere is a dihedral overgroup, and to O K,S [ ζ ℓ ] × otherwise. N FARRELL–TATE COHOMOLOGY OF SL OVER S -INTEGERS 17 Remark 5.11.
The above results are more or less immediate generalizations of the classificationin [Bus06]. The relation to the classification results of Prestel [Pre68], Schneider [Sch75], Kr¨amer[Kr¨a80] or Maclachlan [Mac06] is a bit more subtle to discuss. We restrict ourselves to mentionthe two major differences: one is due to the fact that the cited works consider the more generalsituation of possibly non-split quaternion algebras (instead of M ( A ) considered here). Thesecond difference is that the cited works provide much more elaborate formulas for the orders ofrelative class groups, where we are basically stopping at Proposition 4.13. For our applicationsto computations of Farrell–Tate cohomology we do not need the actual numbers, but the moreconceptual explanations provided by the results above.6. Application I: Quillen conjecture and non-detection
In this section, we want to discuss some consequences of our computations for a conjectureof Quillen as well as detection questions in group cohomology. In particular, we are going toprove Theorem 2.We first recall the conjecture stated in Quillen’s paper, cf. [Qui71, p. 591].
Conjecture 6.1 (Quillen) . Let ℓ be a prime number. Let K be a number field with ζ ℓ ∈ K ,and S a finite set of places containing the infinite places and the places over ℓ . Then thenatural inclusion O K,S ֒ → C makes H • (GL n ( O K,S ) , F ℓ ) a free module over the cohomology ring H • cts (GL n ( C ) , F ℓ ) ∼ = F ℓ [ c , . . . , c n ] . The range of validity of the conjecture has not yet been decided. Positive cases in whichthe conjecture has been established are n = ℓ = 2 by Mitchell [Mit92], n = 3, ℓ = 2 by Henn[Hen99], and n = 2, ℓ = 3 by Anton [Ant99].A related question is the following detection of cohomology classes on diagonal matrices: Definition 6.2 (Detection) . We say that detection of ℓ -cohomology classes is satisfied forGL n ( O K,S ) if the restriction morphism H • (GL n ( O K,S ) , F ℓ ) → H • (T n ( O K,S ) , F ℓ ) is injective,where T n is the group of diagonal matrices in GL n .Actually, all cases where the Quillen conjecture is known to be false can be traced to [HLS95,remark on p. 51], which shows that Quillen’s conjecture implies detection for GL n ( Z [1 / n ≥
32 and ℓ = 2. Dwyer’s bound was subsequently improved by Henn and Lannes to n ≥
14. At the prime ℓ = 3, Anton proved non-injectivity for n ≥
27, cf. [Ant99].Using the Farrell–Tate cohomology computations for SL ( O K,S ) in Theorem 1, we can nowdiscuss these questions - or weaker versions - in the case n = 2. Saying something aboutthe Quillen conjecture means studying the module structure of H • (SL ( O K,S ) , F ℓ ) over thecontinuous cohomology ring H • cts (SL ( C ) , F ℓ ) ∼ = F ℓ [ c ]. From our computations of Farrell–Tatecohomology, we can infer statements on the module structure of b H • (SL ( O K,S ) , F ℓ ), whichallows for group cohomology statements above the virtual cohomological dimension. Recallfrom Theorem 1, that the groups relevant for the computation of Farrell–Tate cohomology ofSL ( O K,S ) are abelian groups G = Z /ℓ × Z n or dihedral extensions of such. The following resultdescribes their Farrell–Tate cohomology as module over the relevant polynomial subrings. Proposition 6.3.
Let G = Z /ℓ × Z n .(1) Denoting by b , x , . . . , x n exterior classes of degree , and by a a polynomial class ofdegree , the cohomology ring H • ( G, F ℓ ) ∼ = F ℓ [ a ]( b , x , . . . , x n ) is a free module of rank n +2 over the subring F ℓ [ a ] .(2) Let Z / act via multiplication by − on all the generators. The invariant subring H • ( G, F ℓ ) Z / is a free module of rank n +1 over the subring F ℓ [ a ] .Proof. (1) is clear from the explicit formula given in Proposition 3.2; and 2 n +2 is the rankobtained from the basis consisting of all the wedge products of the set { a , b , x , . . . , x n } . (2) follows from this, the invariant ring is additively generated by a ⊗ (2 i +1)2 tensor the odd degreepart of V ( b , x , . . . , x n ) and a ⊗ (2 i )2 tensor the even degree part of V ( b , x , . . . , x n ). (cid:3) These statements now allow to formulate the following result, which we would like to see asa version of the Quillen conjecture for SL above the virtual cohomological dimension. Recallthat H • cts (SL ( C ) , F ℓ ) ∼ = F ℓ [ c ] is generated by the second Chern class c (which is a class indegree 4). This is the subring over which we have to express H • (SL ( O K,S ) , F ℓ ) as a free modulefor the Quillen conjecture. Theorem 6.4.
Let ℓ be an odd prime number. Let K be a number field, and let S be a finiteset of places containing the infinite places. Let G = SL ( O K,S ) . In the splitting of Theorem 1, b H • (SL ( O K,S ) , F ℓ ) ∼ = M [Γ] ∈K ℓ b H • ( N G (Γ) , F ℓ ) , denote by n the number of components where N G (Γ) is abelian, and by n the number ofcomponents where it is not. For each component group N G (Γ) , denote by a Γ the second Chernclass of the standard representation of Γ (which is a polynomial class in degree ).Then the restriction map induced by the natural inclusion O K,S ֒ → C is given as follows: F ℓ [ c ] → H • (SL ( O K,S ) , F ℓ ) → b H • (SL ( O K,S ) , F ℓ ) : c X [Γ] ∈K ℓ a Γ . Denoting by r the rank of the relative unit group of R K,S,ℓ / O K,S , the Farrell–Tate cohomology b H • (SL ( O K,S ) , F ℓ ) is a free module of rank r +1 (2 n + n ) over the Laurent polynomial subringgenerated by the image of c .Proof. All the subgroups Γ become conjugate in SL ( C ) and the same is true for the centralizers C G (Γ). The restriction map for all groups N G (Γ) then has to map c to the second Chern class ofthe standard representation of the cyclic or dihedral group, which is the element a Γ . The imageof the restriction map is then the diagonal subring generated by P a Γ . By Proposition 6.3, thecohomology ring is free as a module over this subring with the specified rank. (cid:3) As Γ runs through finite cyclic groups, this proves the decomposition in into a sum of squaresclaimed in Theorem 2.
Corollary 6.5.
Let K be a number field, let S be a finite set of places containing the infiniteones, and let ℓ be an odd prime.(1) (An analogue of ) The Quillen conjecture is true for the Farrell–Tate cohomology of SL ( O K,S ) . More precisely, the natural morphism F ℓ [ c ] ∼ = H • cts (SL ( C ) , F ℓ ) → H • (SL ( O K,S ) , F ℓ ) extends to a morphism φ : F ℓ [ c , c − ] → b H • (SL ( O K,S ) , F ℓ ) which makes b H • (SL ( O K,S ) , F ℓ ) a free F ℓ [ c , c − ] -module.(2) The Quillen conjecture holds for group cohomology H • (SL ( O K,S ) , F ℓ ) above the virtualcohomological dimension. Remark 6.6.
As a result, we can reformulate the Quillen conjecture for group cohomol-ogy as a relation between Farrell–Tate cohomology of SL ( O K,S ) and the Steinberg homol-ogy H • (SL ( O K,S ) , St SL ( O K,S ) ⊗ F ℓ ). This follows from the long exact sequence relating groupcohomology, Farrell–Tate cohomology and Steinberg homology, cf. [Bro94]: · · · → b H •− (Γ) → H n −• (Γ , St Γ ) → H • (Γ) → b H • (Γ) → · · · Vanishing of Steinberg homology for SL ( O K,S ) guarantees the Quillen conjecture by the aboveresult; however, it is possible that the Quillen conjecture is true even with non-vanishing Stein-berg homology. The Quillen conjecture fails whenever the mapH • (SL ( O K,S ) , F ℓ ) → b H • (SL ( O K,S ) , F ℓ ) N FARRELL–TATE COHOMOLOGY OF SL OVER S -INTEGERS 19 is not injective. Then there exist elements in group cohomology which - after multiplicationwith some power of the second Chern class - become trivial. Remark 6.7.
A similar result can be formulated for PGL ( O K,S ) if ℓ ∈ K , but we chose not tospell it out explicitly. A result for GL ( O K,S ) is not as easy to come by, for the following reason.The Farrell–Tate cohomology for SL ( O K,S ) and PGL ( O K,S ) is controlled by finite subgroupsand their normalizers. However, the central ℓ -subgroup of GL ( O K,S ) fixes the whole symmetricspace, so that computations of Farrell–Tate cohomology for GL are actually not significantlyeasier than the group cohomology computations.Next, we want to discuss the detection of cohomology classes, as well as its relation to theQuillen conjecture. Using our explicit computations, we can easily find examples where thecohomology of SL ( O K,S ) cannot be detected on the diagonal matrices. The following generalresult, deducing non-detection from non-triviality of suitable class groups, is very much in thespirit of Dwyer’s disproof of detection for GL ( Z [1 / Proposition 6.8.
Let ℓ be an odd prime number. Let K be a number field with ζ ℓ ∈ K , and let S be a finite set of places containing the infinite places. Assume that the class group Pic( O K,S ) has more than elements. Then the restriction map b H • (SL ( O K,S ) , F ℓ ) → b H • (T ( O K,S ) , F ℓ ) from SL ( O K,S ) to the diagonal matrices T ( O K,S ) is not injective.Proof. Under the assumptions of the proposition, Theorem 5.8(1) implies that the splitting ofTheorem 1 becomes b H • (SL ( O K,S ) , F ℓ ) ∼ = M [Γ] ∈ Pic( O K,S ) /ι b H • ( N G (Γ) , F ℓ ) . The group N G (Γ) is the normalizer in G = SL ( O K,S ) of the order ℓ subgroup representing [Γ].Recall that the Farrell–Tate cohomology b H • ( N G (Γ)) is obtained by making the cohomology ringsfrom Proposition 6.3 periodic for a ; in particular, all exterior products with an odd number offactors live in odd degree, products with even number of factors live in even degree. In bothcases, half of the elements is invariant under multiplication with −
1. For the trivial ideal class,we have a contribution of half the rank of b H • (T ( O K,S ) , F ℓ ). Under the assumption on theclass group, we either have two further ι -invariant ideal classes or a ι -orbit. In either case, theresulting direct summands in L [Γ] ∈ Pic( O K,S ) /ι b H • ( N G (Γ) , F ℓ ) yield a further contribution equalto the rank of b H • (T ( O K,S ) , F ℓ ). Therefore, the restriction map cannot be injective because therank of the source is bigger than the rank of the target. (cid:3) This completes the proof of Theorem 2.
Remark 6.9.
There are other cases in which non-detection results can be established. Theabove proposition is one of the easier ones to formulate, the cases where ζ ℓ K need some morecomplicated conditions. Example 6.10.
Let K = Q ( ζ ) and S = { (23) } ∪ S ∞ . The S -class group of K has order 3.The induced morphism b H • (SL ( O K,S ) , F ) → b H • (T ( O K,S ) , F )is not injective - the source has two copies of the cohomology of a dihedral extension of O × K,S ,but the target has only one copy of the cohomology of O × K,S . (cid:3) Example 6.11.
There are infinitely many counterexamples to detection at the prime 3.Let m be a positive square-free integer such that m ≡ Q ( √− m ) / Q and ramified in Q ( ζ ) / Q . In particular, there is only oneplace of Q ( √− m, ζ ) lying over the place v of Q , and this place is ramified in the extension Q ( √− m, ζ ) / Q ( √− m ). The extension Q ( √− m, ζ ) / Q ( √− m ) being ramified, the induced mapon class groups must be injective, by class field theory. Moreover, as noted above, there isa unique prime ideal p in O Q ( √− m,ζ ) lying above (3), and p = (3). The class group of O Q ( √− m,ζ ) [1 /
3] is obtained by killing the 2-torsion class of p , hence its size is either equal orhalf the size of the class group of O Q ( √− m,ζ ) .We conclude that half the class number of Q ( √− m ) is a lower bound for the size of the classgroup of O Q ( √− m,ζ ) [1 / m such that Q ( √− m ) has class number >
4, hence there are infinitely many S -integer rings of the form O Q ( √− m,ζ ) [1 /
3] whose class group has more than two elements. Each such ring R gives anexample where the restriction mapH • (GL ( R ) , F ) → H • (T ( R ) , F )fails to be injective, but for which the Quillen conjecture is true above the virtual cohomologicaldimension. (cid:3) The question for unstable analogues of the Quillen–Lichtenbaum conjecture was implicit in[DF94], and was raised explicitly in [AR13]. The results of [DF94] show that the unstableQuillen–Lichtenbaum conjecture in the situation of linear groups over S -integers implies detec-tion. The failure of detection as in Proposition 6.8 and Example 6.10 also implies the failure ofthe unstable Quillen–Lichtenbaum conjecture.The above results and their consequences for the Quillen conjecture are further discussed in[RW15a]. In short, the Quillen conjecture for Farrell–Tate cohomology is more related to thesubgroup structure of SL ( O K,S ) and actually happens to be almost tautologically true in suchsmall rank. On the other hand, the Quillen conjecture for group cohomology is more related to“something like cusp forms”, lying in the difference between Farrell–Tate cohomology and groupcohomology. Finally, detection questions, while a powerful method to provide counterexamplesto Quillen’s conjecture, are more related to the conjugacy classification of finite subgroups.Hence, for the rank one groups SL ( O K,S ), Quillen’s conjecture and detection questions areonly superficially related.7.
Application II: on the existence of transfers
Next, we are interested in the existence of transfer maps in Farrell–Tate cohomology as wellas group cohomology. Transfers in the cohomology of linear groups have been suggested as oneway of establishing the Friedlander–Milnor conjecture in [Knu01, section 5.3]. In this section, weshow examples that demonstrate the impossibility of defining transfers on group (co-)homologywith reasonable properties. In particular, we are going to prove Theorem 3. The general setupin this section will be the following:Let
L/K be a degree n extension of global fields, let S be a set of places of K and denote by ˜ S the set of places of L lying over S . Let ℓ be a prime differentfrom the characteristic of K .7.1. Definition of transfer maps.
We first recall various relevant notions of transfers. For afinite covering p : E → B of CW-complexes, there is a transfer map tr p : H • ( B ) → H • ( E ) suchthat the respective composition is multiplication with the degree: tr p ◦ p • = deg p . There aresimilar transfer maps for cohomology. More generally, the Becker–Gottlieb transfer provides awrong-way stable map Σ ∞ B → Σ ∞ E which induces the transfer on cohomology theories. Thiscan be applied to group (co-)homology to recover the classical definition of transfer: for H ⊂ G a finite-index subgroup of a group G and a G -module M , there are transfer mapstr GH : H • ( G, M ) → H • ( H, M ) and Cor GH : H • ( H, M ) → H • ( G, M ) , such that the following respective equalities hold where i : H → G denotes the inclusion of thesubgroup: i • ◦ tr GH = [ G : H ] and Cor GH ◦ i • = [ G : H ]These can be obtained from the (usual or Becker–Gottlieb) transfer applied to the finite covering BH → BG .Similar transfer maps can be considered in algebraic K-theory: for a finite flat map f : X → Y of schemes, there is a K-theory transfer tr f : K • ( X ) → K • ( Y ) such that the composition is the N FARRELL–TATE COHOMOLOGY OF SL OVER S -INTEGERS 21 multiplication with the degree: tr f ◦ f ∗ = deg f . There are a number of generalizations of thisconcept. We use one of the simpler ones, cf. [Knu01, section 5.3]: an abelian group-valuedfunctor F : Sch op → A b is said to admit transfers if for any finite flat morphism f : X → Y ,there is a homomorphism tr f : F ( X ) → F ( Y ) such that we have tr f ◦F ( f ) = deg f .We can apply this definition to group homology: for fixed i ∈ N , we have a functor on thecategory of smooth affine k -schemesH i (SL ( − ) , Z /ℓ ) : Sm opaff /k → A b : Spec k [ X ] H i (SL ( k [ X ]) , Z /ℓ ) . One can ask a similar question for cohomology, i.e., if for any finite flat map f : Spec k [ X ] → Spec k [ Y ] of affine schemes there is a transfer homomorphism tr f : H • (SL ( k [ Y ]) , Z /ℓ ) → H • (SL ( k [ X ]) , Z /ℓ ) such that the composition is multiplication with the degree: H • ( f ) ◦ tr f =deg f . It is an implicit question in the discussion of [Knu01, section 5.3] if group homologyfunctors as above can be equipped with transfers. To quote Knudson, cf. p.134 of loc.cit.:”Unfortunately, there appears to be no way to equip these functors with transfer maps.” Inthe following section, we want to make this precise: we will compute the restriction maps inFarrell-Tate cohomology, and from these computations it will be obvious that it is not possibleto define transfer maps satisfying the degree condition. Similar counterexamples for functionfields can be deduced from the computations of [Wen15a].7.2. Description of restriction maps.
To establish examples where transfers cannot exist,we need to determine the restriction maps for extensions of S -integer rings: b H • (SL ( O L, ˜ S ) , F ℓ ) → b H • (SL ( O K,S ) , F ℓ ) . From Theorem 1, we have a splitting of Farrell–Tate cohomology b H • (SL ( O K,S ) , F ℓ ) ∼ = M [Γ] ∈K b H • ( N G (Γ) , F ℓ ) , where the set K is given by the quotient of an oriented relative class group modulo the Galoisaction, and the group N G (Γ) is the normalizer of the order ℓ subgroup Γ < G = SL ( O K,S ).There are two types of components: if Γ is not contained in a dihedral group, then N G (Γ)is abelian, determined by a relative unit group. Otherwise, N G (Γ) is non-abelian and it is asemidirect product of a relative unit group with the finite group Z / Z acting via inversion.Now, given an extension of rings of S -integers O K,S → O L, ˜ S , the following changes in theclassification of finite subgroups can appear:(C 1) Several non-conjugate cyclic subgroups of order ℓ in SL ( O K,S ) can become conjugate inSL ( O L, ˜ S ).(C 2) A cyclic group which is not contained in a dihedral group in SL ( O K,S ) acquires a dihedralovergroup in SL ( O L, ˜ S ).(C 3) There appear several new cyclic subgroups of order ℓ in SL ( O L, ˜ S ) which are not conjugateto subgroups coming from SL ( O K,S ).The following proposition describes the restriction maps in Farrell–Tate cohomology in eachof the above three cases; it is a straightforward consequence of the above statements, andTheorem 1.
Proposition 7.1.
Fix an odd prime ℓ . Let K be a global field of characteristic different from ℓ , let S be a non-empty finite set of places containing the infinite ones. Let L/K be a finiteseparable extension of K , and let ˜ S be a finite set of places containing those places lying over S .(C 1) Assume that exactly the classes [Γ ] , . . . , [Γ m ] ∈ K ( K ) become identified to a single com-ponent [Γ] ∈ K ( L ) . Then the restriction map b H • ( N SL ( O L, ˜ S ) (Γ) , F ℓ ) → M [Γ i ] b H • ( N SL ( O K,S ) (Γ i ) , F ℓ ) is the sum of the natural maps induced from the inclusions N SL ( O K,S ) (Γ i ) → N SL ( O L, ˜ S ) (Γ) .(C 2) Assume that a cyclic group representing the class [Γ] ∈ K ( K ) is not contained in adihedral group over K , but is contained in a dihedral group over L . Then the restrictionmap b H • ( N SL ( O L, ˜ S ) (Γ) , F ℓ ) → b H • ( N SL ( O K,S ) (Γ) , F ℓ ) is given by the natural inclusion of fixed points.(C 3) The restriction map is trivial on the new components of K ( L ) . This yields claims (1) and (2) of Theorem 3.7.3.
Non-existence of transfers.Theorem 7.2.
The functor “Farrell–Tate cohomology of SL ” given by A b H • (SL ( A ) , F ℓ ) does not admit transfers. More precisely, there exists a finite flat morphism of commutativerings φ : A → B such that for no morphism tr : b H • (SL ( A ) , F ℓ ) → b H • (SL ( B ) , F ℓ ) we have φ • ◦ tr = deg φ .Proof. Let A → B be any extension O K,S → O L, ˜ S of degree prime to ℓ such that situation (C1) occurs with m >
1. For example, we can take a prime ℓ and a number field K with ζ ℓ ∈ K such that O K has non-trivial class group and the Hilbert class field L for O K has degree [ L : K ]prime to ℓ . An explicit example for this type of situation is given in Example 7.3 below.Let m > ℓ subgroups which become conjugate to Γ. ByProposition 7.1, the restriction map is b H • ( N SL ( O L, ˜ S ) (Γ) , F ℓ ) → m M i =1 b H • ( N SL ( O K,S ) (Γ i ) , F ℓ ) . The composition φ • ◦ tr cannot be surjective, because the restriction map has its image containedin the diagonal subring. By the previous choice of ℓ and [ L : K ], multiplication with the degree[ L : K ] will have full rank. Therefore, the two maps cannot be equal, no matter how we choosetr. (cid:3) Example 7.3.
To give a specific example, consider K = Q ( ζ ) with S = S ∞ ∪ S (23) . Its classnumber is 3, and its group of units is Z × Z / Z . Therefore, we have b H • (SL ( O Q ( ζ ) [1 / , F ) ∼ = b H • ( Z × Z / Z , F ) ⊕ b H • ( Z × Z / Z , F ) h− i i.e., the Farrell–Tate cohomology of SL ( O Q ( ζ ) [1 / − Z / Z modulo the inversion involution. The Hilbert class field of Q ( ζ ) is adegree 3 unramified extension H/ Q ( ζ ) such that the restriction map Pic( O Q ( ζ ) ) → Pic( H )is the trivial map. In particular, the restriction map on Farrell–Tate cohomology will factorthrough the diagonal map b H • (T ( O H [1 / ⋊ Z / , F ) → b H • ( Z × Z / Z , F ) ⊕ b H • ( Z × Z / Z , F ) h− i No matter how the transfer map b H • (SL ( O Q ( ζ ) [1 / , F ) → b H • (SL ( O H [1 / , F )is defined, the composition with the restriction map will not be multiplication with 3 on b H • (SL ( O Q ( ζ ) [1 / F , but there are many classeswhich are not in the image of the composition. (cid:3) N FARRELL–TATE COHOMOLOGY OF SL OVER S -INTEGERS 23 Remark 7.4.
Similarly, the situation (C 2) obstructs transfers because the inclusion of fixedpoints is not surjective. The situation (C 3) does not obstruct transfers.
Corollary 7.5.
The functor “group cohomology of SL ”, which is given by A H • (SL ( A ) , F ℓ ) , does not admit transfers.Proof. Farrell–Tate and group cohomology agree above the virtual cohomological dimension,which for SL over S -integers is finite. In the degrees above the vcd, the argument of Theorem 7.2also applies to group cohomology. (cid:3) This concludes the proof of Theorem 3.8.
Application III: cohomology of SL over number fields Using the formulas for Farrell–Tate cohomology from Theorem 1 and restriction maps fromSection 7, we can now compute the colimit of the Farrell–Tate cohomology groups over allpossible finite sets of places. This could be interpreted as “Farrell–Tate cohomology of SL overglobal fields”. The quotes are necessary as Farrell–Tate cohomology is only defined for groupsof finite virtual cohomological dimension. The goal of this section is to prove Theorem 4.8.1. Recollection on Mislin–Tate cohomology.
In [Mis94], Mislin has defined an extensionof Farrell–Tate cohomology to arbitrary groups. The basic idea behind the definitions in [Mis94]is to use satellites of group cohomology to kill projectives in the derived category of Z [ G ]-modules, and obtain a completed cohomology, cf. [Mis94, Section 2]. For groups of finitevirtual cohomological dimension, Mislin’s version of Tate cohomology agrees with Farrell–Tatecohomology, cf. [Mis94, Lemma 3.1]. Moreover, Mislin shows that for K a number field and G = GL n ( K ), his version of Tate cohomology can be identified with group homology, cf. [Mis94,Theorem 3.2]. The same argument also applies to SL .8.2. Colimit computations.
The following result is an immediate consequence of our earliercomputations, in particular Theorem 1 and Proposition 7.1.
Theorem 8.1.
Let K be a number field, and let ℓ be an odd prime. We have the following threecases:(1) If ζ ℓ + ζ − ℓ K , then lim S ⊇ S ∞ , finite b H • (SL ( O K,S ) , F ℓ ) = 0 . (2) If ζ ℓ + ζ − ℓ ∈ K and ζ ℓ K , then lim S ⊇ S ∞ , finite b H • (SL ( O K,S ) , F ℓ ) ∼ = Y Γ ∈K ℓ H • ( N SL ( K ) (Γ) , F ℓ )[( a ) − ] . In the above, K ℓ = coker (Nm : K ( ζ ℓ ) × → K × ) / Gal( K ( ζ ℓ ) /K ) denotes the set of con-jugacy classes of finite cyclic subgroups where the Galois action is the one described inProposition 5.6. The possible orientations of K ( ζ ℓ ) /K are given by the relative Brauergroup coker (Nm : K ( ζ ℓ ) × → K × ) ∼ = Br( K ( ζ ℓ ) /K ) ∼ = H ( Z / Z , K ( ζ ℓ ) × ) . For the nor-malizer, we have N SL ( K ) (Γ) ∼ = T( K ) (the maximal torus of SL ( K ) ) or N SL ( K ) (Γ) ∼ =N( K ) (the normalizer of the maximal torus in SL ( K ) ) depending on whether theclass of Γ is Z / -invariant or not. The [( a ) − ] indicates that the polynomial class a ∈ H ( N SL ( K ) (Γ) , F ℓ ) is ∪ -inverted.(3) If ζ ℓ ∈ K , then lim S ⊇ S ∞ , finite b H • (SL ( O K,S ) , F ℓ ) ∼ = H • (N( K ) , F ℓ )[( a ) − ] . Corollary 8.2.
There are cases where Mislin’s version of Farrell–Tate cohomology does notcommute with filtered colimits.
Proof.
For K = Q , we have ζ + ζ − ∈ Q and (( ζ − ζ − ) , −
1) = ( − , −
1) = −
1. In terms offinite subgroups of PSL ( Q ), there exist finite subgroups of order 3, but they are not containedin dihedral groups. By Theorem 8.1 above, we havelim S ⊇ S ∞ , finite b H • (SL ( O Q ,S ) , F ) = H • ( Q × , F )[ a − ] . In particular, the limit of Farrell–Tate cohomology groups in odd degrees containsH ( Q × , F ) ∼ = Hom F ( Q × / ( Q × ) , F ) , which is an infinite-dimensional F -vector space. However, by [Mis94, Theorem 3.2] there is anisomorphism b H (SL ( Q ) , F ) ∼ = H (SL ( Q ) , F ) , and the latter is trivial. Therefore, Mislin’s version of Farrell–Tate cohomology does not com-mute with directed colimits. (cid:3) For an extension of number fields
L/K , we can precisely describe the induced morphism onthe colimit of the Farrell–Tate homology - it is induced by the inclusion K × ֒ → L × . Againtaking a colimit, we arrive at the following “Farrell–Tate cohomology of SL ( Q )”: Corollary 8.3.
Let K be a number field and ℓ be an odd prime. We fix an algebraic closure Q of Q . Then we have lim Q ⊇ L ⊇ Q ,S ⊇ S ∞ b H • (SL ( O L,S ) , F ℓ ) ∼ = H • (N( Q ) , F ℓ )[( a ) − ] , where S runs through the finite sets of places of L containing the infinite places, and N( Q ) denotes the Q -points of the normalizer of a maximal torus of SL . Finally, we want to explain how these colimits over Farrell–Tate cohomology groups al-low to provide a reformulation of the Friedlander–Milnor conjecture. For a discussion of theFriedlander–Milnor conjecture, we refer to [Knu01, Chapter 5]. Recall that Milnor’s form ofwhat is now called the Friedlander–Milnor conjecture predicts that for a complex Lie group G , the natural change-of-topology map H • cts ( G, F ℓ ) → H • ( G δ , F ℓ ) is an isomorphism, wherethe source is continuous cohomology of the Lie group G with the analytic topology, and G δ is the group with the discrete topology. On the other hand, Friedlander’s generalized isomor-phism conjecture predicts that for each algebraically closed field K of characteristic differentfrom ℓ and each linear algebraic group G over K , another natural change-of-topology mapH • ´et ( BG K , F ℓ ) → H • ( BG ( K ) , F ℓ ) is an isomorphism. From the rigidity property of ´etale coho-mology, it follows that the Friedlander–Milnor conjecture for SL over Q is equivalent to thenatural restriction map H • cts (SL ( C ) , F ℓ ) → H • (SL ( Q ) , F ℓ )being an isomorphism. Note that the continuous cohomologyH • cts (SL ( C ) , F ℓ ) ∼ = F ℓ [ c ]is a polynomial ring generated by the second Chern class. Therefore, the Friedlander–Milnorconjecture for SL over Q can be reformulated as the claim thatH • (SL ( Q ) , F ℓ ) ∼ = F ℓ [ c ] . Note that the right-hand side can also be identified with the “Farrell–Tate cohomology ofSL ( Q )”, by Corollary 8.3 above. This allows to reformulate the Friedlander–Milnor conjecturefor SL over Q as the requirement that group cohomology and “Farrell–Tate cohomology” ofSL ( Q ) are isomorphic: Corollary 8.4.
The Friedlander–Milnor conjecture for SL over Q with F ℓ -coefficients, ℓ anodd prime, is equivalent to either one of the two following statements: N FARRELL–TATE COHOMOLOGY OF SL OVER S -INTEGERS 25 (1) The colimit of homology groups of Steinberg modules vanishes: lim Q ⊇ L ⊇ Q ,S ⊇ S ∞ H • (SL ( O L,S ) , St ( O L,S ); F ℓ ) = 0 . where as above S runs through the finite sets of places of L containing the infinite places.(2) Mislin’s version of Farrell–Tate cohomology commutes with the filtered colimit over theintermediate fields Q /L/ Q and their finite sets S of places.Proof. We denote by µ ℓ ∞ the group of ℓ -power roots of unity in Q . The group Q × /µ ℓ ∞ isuniquely ℓ -divisible. In particular, the inclusion µ ℓ ∞ ⊆ Q × induces an isomorphismH q (N( Q ) , F ℓ ) ∼ = H q ( µ ℓ ∞ ⋊ Z / , F ℓ ) ∼ = (cid:26) F ℓ , q ≡ , , otherwise , which is what is predicted by the Friedlander–Milnor conjecture.Let G = SL ( O L,S ). To prove Assertion (1), we take the filtered colimit of the long exactsequence · · · → b H •− ( G ) → H n −• ( G, St G ) → H • ( G ) → b H • ( G ) → · · · Group cohomology commutes with the filtered colimit, and by the above, the Friedlander–Milnorconjecture is equivalent to the fact that the colimit of the morphisms H • (SL ( O K,S ) → b H • ( G )is an isomorphism. Since filtered colimits are exact for F ℓ -vector spaces, the claim follows.Assertion (2) follows similarly. By [Mis94, Theorem 3.2], Mislin–Tate cohomology of SL ( K )agrees with group cohomology, so the Friedlander–Milnor conjecture is equivalent to the com-mutation of Farrell–Tate cohomology with the filtered colimit. (cid:3) Remark 8.5.
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E-mail address : [email protected]
Matthias Wendt, Fakult¨at Mathematik, Universit¨at Duisburg-Essen, Thea-Leymann-Strasse 9,45127, Essen, Germany
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