Homology of SL2 over function fields I: parabolic subcomplexes
aa r X i v : . [ m a t h . K T ] A p r HOMOLOGY OF SL OVER FUNCTION FIELDS I:PARABOLIC SUBCOMPLEXES
MATTHIAS WENDT
Abstract.
The present paper studies the homology of the groups SL ( k [ C ])and GL ( k [ C ]), where C = C \{ P , . . . , P s } is a smooth affine curve over an al-gebraically closed field k . It is well-known that these groups act on a product oftrees and the quotients can be described in terms of certain equivalence classesof rank two vector bundles on the curve C . There is a natural subcomplexconsisting of cells with non-trivial isotropy group. The paper provides explicitformulas for the equivariant homology of this “parabolic subcomplex”. Theseformulas also describe homology of SL ( k [ C ]) above degree s , generalizing aresult of Suslin for the case s = 1. Contents
1. Introduction 12. Preliminaries: trees, group actions, vector bundles 53. Decomposable bundles and parabolic subcomplex 94. Local structure of parabolic subcomplexes 125. Global structure of parabolic subcomplexes 176. Equivariant homology of the parabolic subcomplex 24Appendix A. Refined scissors congruence groups 32References 371.
Introduction
This paper is the first in a series of papers studying the homology of rank one lin-ear groups over function rings and function fields of curves, mostly over algebraicallyclosed fields. Of course, the groups SL and GL , their structure, their homologyand representation theory, have been subject to a lot of research in number-theoreticsituations. However, surprisingly little information is available on the structure ofSL in situations where the usual, more analytic, methods fail - for curves overinfinite base fields.That being said, the present paper still follows the standard path to computationof homology of linear groups over function rings of curves; the group SL ( k [ C ])acts on a product of trees, the building X C , and the isotropy spectral sequenceassociated to this action can be used to obtain information on group homology ofSL ( k [ C ]). In general, understanding the structure of the quotient SL ( k [ C ]) \ X C israther difficult, and this is one of the main reasons for the lack of group homologycomputations. There is, nevertheless, one part of group homology that is easierto understand: taking inspiration from the number-theoretic situation, we consider Date : April 2014.2010
Mathematics Subject Classification.
Key words and phrases. cohomology, linear groups, vector bundles.This work has been partially supported by the Alexander-von-Humboldt-Stiftung. a subcomplex of the building, called parabolic subcomplex P C , cf. Definition 3.6,consisting of cells with non-unipotent stabilizer. The equivariant homology of thissubcomplex is a function field analogue of Farrell-Tate homology, and sits in a longexact sequence, cf. Proposition 3.8 · · · → H SL ( k [ C ]) • ( P C ) → H • (SL ( k [ C ])) → H SL ( k [ C ]) • ( U C ) → · · · , where H SL ( k [ C ]) • ( U C ) is an analogue of cuspidal homology (or homology of theSteinberg module) in the number-theoretic situations. Moreover, the equivarianthomology of the parabolic subcomplex can be computed very explicitly in termsof the homology of (normalizers of) maximal tori and refined scissors congruencegroups RP • ( k ), cf. Section A. The following is the main result of the paper; itdescribes the equivariant homology of the parabolic subcomplex for SL ( k [ C ]) with k an algebraically closed field. For the proofs, cf. Lemma 5.2, Proposition 6.1 andProposition 6.3. Theorem 1.
Let k be an algebraically closed field, let C be a smooth projectivecurve over k , let P , . . . , P s ∈ C be closed points, and set C = C \ { P , . . . , P s } .Denoting by P C the parabolic subcomplex of the building X C , cf. Definition 3.6,we have the following formulas for the SL ( k [ C ]) -equivariant homology of P C :(1) The connected components of the parabolic subcomplex P C are indexed bythe quotient set K ( C ) = Pic( C ) /ι of the Picard group of C modulo theinvolution ι : L 7→ L − . The result is a direct sum decomposition: H SL ( k [ C ]) • ( P C , Z [1 / ∼ = M [ L ] ∈K ( C ) H SL ( k [ C ]) • ( P C ( L ) , Z [1 / . (2) If [ L ] ∈ K ( C ) is such that L| C = L| − C , then the homology of the component P C ( L ) is the homology of the group k [ C ] × : H SL ( k [ C ]) • ( P C ( L ) , Z [1 / ∼ = H • ( k [ C ] × , Z [1 / . (3) If [ L ] ∈ K ( C ) is such that L| C ∼ = L| − C , then there is a long exact sequence · · · → RP i +1 ( k ) ⊗ Z Z [1 / , k [ C ] × / ( k [ C ] × ) ] → H i ( ˜ SN , Z [1 / →→ H SL ( k [ C ]) i ( P C ( L ) , Z [1 / → RP i ( k ) ⊗ Z Z [1 / , k [ C ] × / ( k [ C ] × ) ] → · · · , where ˜ SN denotes the group of monomial matrices in SL ( k [ C ]) and RP i ( k ) are refined scissors congruence groups, cf. Section A. There is a similar result for the equivariant homology of the parabolic subcomplexfor the group PGL , cf. Proposition 6.1 and Proposition 6.4.The result is proved by explicitly computing the quotient SL ( k [ C ]) \ P C andthen working out the isotropy spectral sequence in detail. The analysis of thespectral sequence is significantly simplified by working with Z [1 / ( k ) to group homology of the normalizer of themaximal torus in SL ( k ) and refined scissors congruence groups RP i . The latterexact sequence would appear to be well-known, cf. [Dup01, Chapters 8, 15], butfor the sake of completeness we discuss the definition of the groups RP i as well asa proof of the abovementioned exact sequence in Section A.It has been known for some time, and surfaced particularly in the recent work ofKevin Hutchinson [Hut11a, Hut11b], that the action of square classes F × / ( F × ) isan extremely helpful tool in understanding group homology of SL ( F ). The resultabove exhibits a complete description as well as a geometric interpretation of thesquare-class action on the parabolic part of group homology. ARABOLIC SUBCOMPLEXES AND HOMOLOGY OF SL The above computations of the equivariant homology of the parabolic subcom-plex also imply formulas for group homology of SL ( k [ C ]) above the dimensionof the product of trees. The next result follows directly from a mod ℓ -version ofTheorem 1 and Proposition 3.8. Corollary 1.1.
Let k be an algebraically closed field, let C be a smooth projectivecurve over k , and set C = C \ { P , . . . , P s } . Let ℓ be an odd prime different fromthe characteristic of k . For i > s , there are isomorphisms H SL ( k [ C ]) i ( P C , Z /ℓ ) ∼ = H i (SL ( k [ C ]) , Z /ℓ ) . In particular, we have a direct sum decomposition for mod ℓ group homology of SL ( k [ C ]) above homological degree s H • >s (SL ( k [ C ]) , Z /ℓ ) ∼ = M [ L ] ∈K ( C ) , L| C = L| − C H • >s ( k [ C ] × , Z /ℓ ) ⊕ M [ L ] ∈K ( C ) , L| C ∼ = L| − C H SL ( k [ C ]) • >s ( P C ( L ) , Z /ℓ ) as well as an exact sequence describing the normalizer terms: · · · → RP i +1 ( k, Z /ℓ )[ k ( C ) × / ( k ( C ) × ) ] → H i ( ˜ SN , Z /ℓ ) → H SL ( k [ C ]) i ( P C ( L ) , Z /ℓ ) → RP i ( k, Z /ℓ )[ k ( C ) × / ( k ( C ) × ) ] → · · · Note that the above exact sequence is not simply the exact sequence of Theorem 1tensored with finite coefficients, but using a mod ℓ -version of the groups RP • ( k ),cf. Section A.The above theorem provides a generalization of a theorem of Suslin, cf. [Knu01,Theorem 4.5.7]; for s = 1 the above formula reduces to the one in loc.cit.:H • > (PGL ( k [ C ]) , Z /ℓ ) ∼ = M L∈K ( C ) , L =0 H • > (PGL ( k ) , Z /ℓ ) ⊕ M L∈K ( C ) , L6 =0 H • > ( k × , Z /ℓ ) . Note that the finite coefficients are necessary: on the curve C , there may existbundles with unipotent automorphism group (like Atiyah’s bundles F on an ellipticcurve). The automorphism groups influence the homology in degrees above s , butare not accounted for in the parabolic subcomplex P C . They do, however, notinfluence the mod ℓ homology above degree s because the additive group ( k, +) isuniquely ℓ -divisible.The definition of the parabolic subcomplex is functorial in the curve, hence mor-phisms of curves induce morphisms on equivariant homology of the correspondingparabolic subcomplexes. The induced morphisms can be explicitly described and,for a function field k ( C ), allow to define a “parabolic homology” of SL ( k ( C )) viathe obvious limit process. In analogy to the notation for Farrell-Tate homology, wedenote b H • (SL ( k ( C )) , Z /ℓ ) = colim S ⊆ C ( k ) H SL ( k [ C \ S ]) • ( P C \ S , Z /ℓ ) , where the index set of the colimit above is the set of finite sets of closed pointsof C , ordered by inclusion. The second main result of the paper now provides aformula for the parabolic homology of SL ( k ( C )) and establishes a rigidity resultfor it. It may not come as a surprise, but taking the limit to the algebraic closureof a function field, the parabolic homology has exactly the form predicted by theFriedlander-Milnor conjecture, cf. [FM84]. The result follows from Proposition 6.5and the above Theorem 1 resp. its corollary, cf. Proposition 6.7. MATTHIAS WENDT
Theorem 2.
Let k be an algebraically closed field, let C be a smooth curve over k and let ℓ be an odd prime different from the characteristic of k . Then we have thefollowing exact sequence · · · → RP i +1 ( k, Z /ℓ )[ k ( C ) × / ( k ( C ) × ) ] → H i (N( k ( C )) , Z /ℓ ) →→ b H i (SL ( k ( C )) , Z /ℓ ) → RP i ( k, Z /ℓ )[ k ( C ) × / ( k ( C ) × ) ] → · · · , where N( k ( C )) denotes the normalizer of a maximal torus in SL ( k ( C )) .(1) All classes of b H i (SL ( k ( C )) , Z /ℓ ) become constant over the quadratic clo-sure of k ( C ) .(2) Assume RP • ( k ) = 0 , i.e., Friedlander’s generalized isomorphism conjectureis true for SL over k . Then b H • (SL ( k ( C )) , Z /ℓ ) := colim K/k ( C ) finite b H • (SL ( K ) , Z /ℓ ) ∼ = H • (N( k ( C )) , Z /ℓ ) has exactly the form predicted by Friedlander’s isomorphism conjecture for SL over k ( C ) . In particular, Friedlander’s generalized isomorphism con-jecture for SL over k ( C ) is equivalent to vanishing of colim C/k H SL ( k [ C ]) • ( U C , Z /ℓ ) , where the colimit runs over all smooth affine curves over k . The above result provides two reformulations of Friedlander’s isomorphism con-jecture, one as a divisibility result for “the limit of cuspidal homology”, and oneas a detection result of homology on the normalizer of the maximal torus. Thesecond reformulation in turn is close to [Knu01, Corollary 5.2.10]. In general, wesee that the parabolic homology b H • (SL ( k ( C )) , Z /ℓ ) describes exactly the part ofthe homology of SL ( k ( C )) which can be detected on the normalizer N( k ( C )) andsome additional subgroups of SL ( k ( C )) isomorphic to SL ( k ). Moreover, we seethat the parabolic homology satisfies a much stronger rigidity than expected byFriedlander’s isomorphism conjecture: classes in the parabolic homology becometrivial already over quadratically closed fields. Hopefully, the above result helpsshed new light on the homology of SL over algebraically closed fields.Finally, we want to mention that related (and as it turns out structurally similar)computations in the number field case, i.e., computations of Farrell-Tate cohomol-ogy of SL ( O K,S ) with O K,S a ring of S -integers, are being developed in joint workwith Alexander D. Rahm.1.1. Structure of the paper:
We first recall preliminaries on trees and groupactions in Section 2. The definition and basic properties of the parabolic subcom-plex P C are given in Section 3. Then Section 4 works out the actions of stabilizerson links and the resulting local structure of the quotient of the parabolic subcom-plex, leading to a global description of the structure of the quotient SL ( k [ C ]) \ P C in Section 5. The structure of SL ( k [ C ])-equivariant homology of the parabolicsubcomplex P C is determined in Section 6. The appendix Section A provides arecollection of basic facts on the refined scissors congruence groups RP • ( k ).1.2. Acknowledgements:
The investigations reported in the paper started duringa stay at the De Br´un center for computational algebra at NUI Galway in August2012. I would like to thank Alexander D. Rahm for discussions about Farrell-Tate cohomology and his computations with Bianchi group [Rah13, Rah14], whichshaped my understanding of the structure and possible usefulness of the parabolicpart of group homology described in the paper. I would also like to thank Kevin
ARABOLIC SUBCOMPLEXES AND HOMOLOGY OF SL Hutchinson for explanations on his computations of homology of SL and residuemaps for refined Bloch groups in [Hut11a, Hut11b] as well as inspiring discussionson various topics related to group homology.2. Preliminaries: trees, group actions, vector bundles
In this section, we fix the notation for the paper, and recall preliminaries on rankone linear groups and (products of) trees associated to them. We also discuss thewell-known identification of the quotient of the building modulo the group actionin terms of vector bundles on the curve.We mostly follow the notation of [Ser80]. • k denotes a commutative field, • C an irreducible smooth projective curve over k , • C = C \ { P , . . . , P s } a smooth affine curve over k , with P , . . . , P s pairwisedistinct but not necessarily k -rational points of respective degrees d i =[ k ( P i ) : k ], • K = k ( C ) the function field of the curve C , • k [ C ] the ring of functions on the curve C , • for a closed point Q of C , v Q denotes the corresponding valuation, O Q thevaluation ring, k ( Q ) its residue field, deg Q = [ k ( Q ) : k ] its degree.2.1. Recollection on buildings.
We recall the definition and structure of theBruhat-Tits tree associated to SL over a field K with a valuation v . For detailson Bruhat-Tits trees, cf. [Ser80, II.1.1], for the more general theory of buildings,cf. [AB08]. All the statements below are standard and can be found in one of thesebooks. Definition 2.1.
Let K be a field equipped with a discrete valuation v . We denoteby O v the corresponding valuation ring with maximal ideal m v . We denote by π v achoice of uniformizer for v . Let V = K . A lattice L in V is a finitely generated O v -submodule of V which generates V . Two lattices L and L are called equivalent if there exists λ ∈ K × such that λL = L . We denote by Λ = [ L ] the equivalenceclass of L . To a lattice class Λ , we can assign a type v (det Λ) mod 2 ∈ Z / Z .The Bruhat-Tits tree associated to ( K, v ) is the following simplicial complex:the -simplices are equivalence classes of lattices. Lattice classes Λ and Λ areconnected by an edge if there exist representatives L i of Λ i such that π v L ⊂ L ⊂ L . The resulting simplicial complex is a tree, denoted by T v .There is an obvious action of GL ( K v ) on T v by setting GL ( K v ) × T v → T v : ( m, Λ) m Λ . Note that GL ( K v ) acts transitively on the vertices of the Bruhat-Tits tree for v , and SL ( K v ) acts transitively on the vertices of fixed type. The center actstrivially, i.e., the above actions on the building factor through actions of PGL ( K v )and PSL ( K v ), respectively.For each vertex x = [ L ] with representative lattice L , there is a bijection betweenthe link Lk( x ) and the lattices L ′ with π v L ⊂ L ′ ⊂ L , hence with one-dimensionalsubspaces of the two-dimensional k v -vector space L/π v L . Thus, the elements of thelink Lk( x ) are in bijection with the k v -points of the projective line P ( k v ), where k v is the residue field of the valuation v on K . In particular, the tree is homogeneous.We can be more precise about the correspondence between the link and P ( k v ):it is induced by mapping the points x ∈ A ( k v ) and x = ∞ to the lattice classes (cid:20) L · (cid:18) π v e x (cid:19)(cid:21) and (cid:20) L · (cid:18) π − v
00 1 (cid:19)(cid:21) , MATTHIAS WENDT where ˜ x is a lift of x ∈ k v to O v . This description of the correspondence of coursedepends on a choice of basis vector for the lines in k v , a choice of lift to L and achoice of completion to a basis of L .Let k be a field, let C be a smooth projective curve over k , and set C = C \{ P , . . . , P s } for a non-empty finite set of pairwise distinct closed points P , . . . , P s of C . Denote by v , . . . , v s the corresponding valuations on the function field K = k ( C ), and denote by T i the Bruhat-Tits tree associated to the valuation v i . Thenthe group SL ( k [ C ]) acts on X C := T × · · · × T s via the embeddingSL ( k [ C ]) ֒ → SL ( k ( C )) ֒ → SL ( k ( C ) v ) × · · · × SL ( k ( C ) v s ) . The product X C = T × · · · × T s is the Bruhat-Tits building associated to SL andthe smooth affine curve C . We view it as a cubical complex of dimension s whosenon-degenerate cubes are products of edges from the factors T i .To describe the local structure of the product, we consider the link of 0-simplices.Recall that for X a cubical complex and σ a 0-cube of X , the link Lk X ( σ ) of σ in X is the following simplicial complex: its 0-simplices are the 0-cubes of X connectedto σ via a 1-cube, and 0-simplices σ , . . . , σ m span an m -simplex if there exists an( m + 1)-cube containing σ, σ , . . . , σ m . For a vertex ( x , . . . , x s ) in T × · · ·× T s , thelink of ( x , . . . , x s ) is then the following simplicial complex: its set of 0-simplicesis the disjoint union F si =1 Lk T i ( x i ), and for each choice of index set I ⊆ { , . . . , s } of cardinality n and elements { y i ∈ Lk T i ( x i ) } i ∈ I , there is an n -simplex ( y , . . . , y n )corresponding to the ( n + 1)-cube spanned by ( x , . . . , x s ) and the y i . In particular,for n = s = 2, the link of ( x , x ) is the complete bipartite graph on the links of x and x .Again, we can be more precise about the lattices in the link: assume that thevertex ( x , . . . , x s ) is represented by the lattice classes ([ L ] , . . . , [ L s ]). Then foreach choice of index i ∈ { , . . . , s } and element α i ∈ P ( k ( P i )), the correspondingpoint in the link of ( x , . . . , x s ) is given by ([ L ] , . . . , [ L i · M i ] , . . . , [ L s ]), where M i is the corresponding matrix M i = (cid:18) π i e α i (cid:19) resp. (cid:18) π − i
00 1 (cid:19) . Buildings and vector bundles.
As discussed, for a smooth affine curve C = C \ { P , . . . , P s } , the groups GL ( k [ C ]) and SL ( k [ C ]) act on the Bruhat-Titsbuilding X C . It is well-known that the quotient can be described in terms of vectorbundles on the curve C , cf. [Ser80, Propositions II.2.4, II.2.5], [Stu76] and [Stu80].We recall the necessary steps. Definition 2.2.
Let ( L , . . . , L s ) be a tuple, in which L i is an O P i -lattice in V = K . We associate to this point in X C the unique coherent subsheaf E = E ( L , . . . , L s ) of the constant sheaf V given by taking the stalk of E at a point Q ∈ C to be L i if Q = P i and O Q otherwise.Two vector bundles E and E are called equivalent rel ∂C if there exist integers m , . . . , m s such that E ⊗ O C ( − P ) m ⊗ · · · ⊗ O C ( − P s ) m s ∼ = E , where O C ( − P ) is the ideal sheaf of functions vanishing at P . It is easy to see that E ( L , . . . , L s ) is the sheaf of sections of a rank two vectorbundle over C whose restriction to C is trivial. In the following, there will be nonotational distinction between the sheaf of sections and the vector bundle. It alsofollows easily that for scalars λ , . . . , λ s ∈ K × , we have E ( L , . . . , L s ) ⊗ O C ( − P ) v ( λ ) ⊗ · · · ⊗ O C ( − P s ) v s ( λ s ) ∼ = E ( λ L , . . . , λ s L s ) . ARABOLIC SUBCOMPLEXES AND HOMOLOGY OF SL From these remarks, we have the following identification of the quotient of thebuilding in terms of vector bundles.
Proposition 2.3.
The two assignments ( L , . . . , L s ) ([ L ] , . . . , [ L s ]) ∈ X C , and ( L , . . . , L s )
7→ E ( L , . . . , L s ) induce a bijection between(1) the -cells of the quotient SL ( k [ C ]) \ X C , and(2) equivalence (rel ∂C ) classes of pairs ( E , f ) where E is a rank two vectorbundle on C whose restriction to C is trivial, and f : det E → O C ( − P ) m ⊗ · · · ⊗ O C ( − P s ) m s is a fixed isomorphism for suitable m i .This bijection further induces (by forgetting the fixed determinant) a bijection be-tween(1) the -cells of the quotient GL ( k [ C ]) \ X C , and(2) equivalence (rel ∂C ) classes of rank two vector bundles E on C whose re-striction to C is trivial. Remark 2.4.
The vector bundle classification is usually stated in different form:the set of isomorphism classes of rank two vector bundles on C whose restrictionto C is trivial is parametrized by the double quotient GL ( k [ C ]) \ s Y i =1 GL ( k ( C ) v i ) ! / s Y i =1 GL ( O v i ) ! . In the building, we additionally divide out the centers of the groups GL ( k ( C ) v i ) which leads to the additional equivalence (rel ∂C ) on vector bundles. There is nodifference between using GL ( k ( C )) or GL ( k ( C ) v i ) in the above, the right cosetsof the corresponding maximal compact group are the same. Under the correspondence described above, the link of a point x = ([ L ] , . . . , [ L s ])corresponding to the vector bundle E = E ( L , . . . , L s ) is given by the vector bundles E ′ which arise from E by elementary transformations: there is an edge between thevector bundles E and E ′ if there is an embedding E ′ ֒ → E of E ′ as a subsheaf of E and the quotient E / E ′ is a torsion O C -module of length one, concentrated in oneof the points P , . . . , P s . Similarly, the n -simplices in the link of x are given by thechoice of n + 1 such elementary transformations at n + 1 pairwise distinct points in { P , . . . , P s } .The correspondence between vertices in the quotient of the building and equiva-lence classes of vector bundles also allows the identification of stabilizers of verticeswith automorphism groups of vector bundles. Proposition 2.5.
Let x = ([ L ] , . . . , [ L s ]) be a point in the building X C with asso-ciated vector bundle E = E ( L , . . . , L s ) . Then we have the following description ofstabilizers, where Stab( x ; G ) denotes the stabilizer of the vertex x in the group G :(1) Stab( x ; GL ( k [ C ])) ∼ = Aut( E ) × k × k [ C ] × , where k × → Aut( E ) denotes theembedding of the homotheties,(2) Stab( x ; PGL ( k [ C ])) ∼ = Aut( E ) /k × (3) Stab( x ; SL ( k [ C ])) ∼ = Aut( E , f ) , where Aut( E , f ) denotes those automor-phisms commuting with the map f : det E → O C ( − P ) m ⊗· · ·⊗O C ( − P s ) m s from Proposition 2.3.The stabilizers of a cube in any of the above groups are given by the intersection ofthe respective stabilizers of its vertices. The intersections are taken in the groups GL ( K ) , PGL ( K ) and SL ( K ) , respectively. MATTHIAS WENDT
Proof. (1) We first describe a homomorphismAut( E ) × k × k [ C ] × → Stab( x ; GL ( k [ C ])) . Note that the bundle E comes with an explicit embedding into K , in particu-lar the restriction of E to C comes with a given trivialization. An automorphism φ ∈ Aut( E ) can be restricted to a k [ C ]-linear automorphism of E| C , and the giventrivialization allows to write this as an element in GL ( k [ C ]). This produces ahomomorphism Aut( E ) → GL ( k [ C ]) which lands inside Stab( x ; GL ( k [ C ])) be-cause the automorphism preserves all the lattices. There is also a homomor-phism k [ C ] × → Stab( x ; GL ( k [ C ])) given by embedding k [ C ] × into the center ofGL ( k [ C ]) by u ∈ k [ C ] × (cid:18) u u (cid:19) . This also preserves the lattice classes [ L i ], hence stabilizes x . We can embed k × ֒ → Aut( E ) as the homotheties, and k × ֒ → k [ C ] × as constants. On k × , thehomomorphisms given above agree, hence we obtain the required homomorphism.This homomorphism is obviously injective.By definition, the elements of Stab( x ; GL ( k [ C ])) preserve the lattice classes[ L i ]. Moreover, any element of GL ( k [ C ]) preserves the standard lattice O Q for Q
6∈ { P , . . . , P s } . Therefore, any element of the stabilizer which preserves thelattices L i (as opposed to just the lattice classes) is in fact an automorphism of thevector bundle E . Now any element of the stabilizer can be factored as an elementwhich preserves the lattices L i and some central element, hence the homomorphismis surjective.(2) is obtained by dividing out the center on both sides. On the automorphismside, this is exactly dividing out the factor k [ C ] × . On the stabilizer side, this isprecisely the passage from GL to PGL .(3) is obtained by restricting to determinant 1.Finally, the building is a CAT(0)-space, and GL ( k [ C ]) acts via isometries.Therefore, if an element stabilizes vertices, it also stabilizes the cube they span.Stabilizers of cubes can then be computed by intersection of stabilizers of verticesinside the respective group. (cid:3) Recollection on Jacobians of curves.
We recall the Nagata exact sequencewhich describes units and Picard groups of smooth affine curves. This will be neededfor the classification of (geometrically) split rank two bundles later on.
Lemma 2.6.
Let C be a smooth projective curve over a field k , let P , . . . , P s beclosed points of degrees d i = [ k ( P i ) : k ] , and denote C = C \ { P , . . . , P s } . Thenthere is an exact sequence → O ( C ) × = k × → O ( C ) × → s M i =1 Z φ −→ Pic( C ) → Pic( C ) → . In particular:(1) There is an exact sequence → Pic ( C ) → Pic( C ) → Z / gcd( d , . . . , d s ) Z → , where Pic ( C ) = Pic ( C ) / (Im φ ∩ Pic ( C )) .(2) We have k [ C ] × ∼ = k × ⊕ Z dim ker φ . Proof.
The exact sequence is a special case of a localization sequence in K-theoryor Chow groups. A more elementary proof of the exact sequence may be found e.g.in [Ros73, Proposition 1].
ARABOLIC SUBCOMPLEXES AND HOMOLOGY OF SL The morphism φ : Z s → Pic( C ) sends the basis vector e i to the line bundle O C ( − P i ), cf. [Har77, Proposition II.6.5]. Moreover, the degree map provides thestandard exact sequence0 → Pic ( C ) → Pic( C ) → Z → . The composition deg ◦ φ : Z s → Pic( C ) deg −→ Z maps e i to deg( P i ), so its imageis generated by gcd( d , . . . , d s ). Thus ker(deg ◦ φ ) ∼ = Z s − consists exactly of thedegree zero divisors, and the map ker φ → Pic( C ) factors through Z s − → Pic ( C ).The rank of the latter map varies, according to possible rational equivalence re-lations between the points P , . . . , P s . This proves (1). The splitting and rankcomputation in (2) is obvious. (cid:3) Decomposable bundles and parabolic subcomplex
In this section, we use the identification of Proposition 2.5 to discuss the struc-ture of stabilizer subgroups of vertices in X C . Recalling some basics about automor-phism groups of rank two vector bundles, we see that the existence of a non-trivialnon-unipotent automorphism implies that the vector bundle is geometrically split.Based on this observation, we consider the subcomplex of the quotient which con-sists exactly of the cells which correspond to geometrically decomposable bundles.The equivariant cohomology of this complex is a version of Farrell-Tate cohomol-ogy, in fact agrees with it for finite base fields (and finite coefficients away from thecharacteristic).3.1. Automorphisms and decomposability of vector bundles.
We first recallthe structure of automorphism groups of rank two vector bundles. These resultsin particular imply that decomposability of a rank two vector bundle can be seenfrom the structure of the automorphism group.
Proposition 3.1.
Let k be a field, let C be a smooth projective curve over k , andlet E be a rank two vector bundle over C .(1) If E is geometrically indecomposable, then we have End( E k ) ∼ = k ⊕ N il .Every automorphism is a product of a central element and a unipotent ele-ment.(2) If E is geometrically decomposable, then there exists a separable extension L/k with [ L : k ] = 2 such that E L ∼ = L ⊕ L ′ with L and L ′ line bundles over C × k L . There are three possibilities:(a) If L ∼ = L ′ , then both L and L ′ as well as the splitting are defined over k . We have End( E ) ∼ = M ( k ) and Aut( E ) ∼ = GL ( k ) .(b) If L 6∼ = L ′ and the splitting is defined over k , then we have (assuming deg L ≥ deg L ′ ) Aut( E ) ∼ = (cid:26)(cid:18) a c b (cid:19) , | a, b ∈ k, ab = 0 , c ∈ H ( C, L ⊗ ( L ′ ) − ) (cid:27) . (c) If L 6∼ = L ′ and the splitting is not defined over k , then the automorphismgroup of E is the Weil restriction of a torus: Aut( E ) ∼ = R L/k ( G m ) . Proof. (1) is due to Atiyah. In a complex analytic setting, it can be found asProposition 15, its reformulation and Proposition 16 in [Ati57]. The algebraicresult is proved similarly using [Ati56].(2) can be found in [Ser80, p. 101]. (cid:3)
Corollary 3.2.
A rank two vector bundle is geometrically split if and only if thereexists a non-central non-unipotent automorphism.
Remark 3.3.
A similar pattern appears for the finite subgroups containing oddorder elements in arithmetic groups SL ( O K,S ) : case (a) is a dihedral group, case(b) is a diagonalizable finite cyclic group, and case (c) a non-diagonalizable finitecyclic group. Parabolic subcomplex and an exact sequence.
In this paper, we areinterested in the subcomplex of X C containing the decomposable bundles. Moreprecisely, we consider the subcomplex of X C containing exactly the cells which havea non-central non-unipotent element in their stabilizer. This is justified by the factthat the unipotent radicals of stabilizers will not be visible in homology with finitecoefficients away from the characteristic. Lemma 3.4.
Let k be a field, let C be a smooth projective curve over k , and denote C = C \ { P , . . . , P s } . The subset of the building X C consisting of the cells withnon-unipotent stabilizer is in fact a subcomplex. The action of GL ( k [ C ]) resp. SL ( k [ C ]) on the building restricts to an action on this subcomplex.Proof. As mentioned before, the stabilizers of n -cubes are the intersections of thestabilizers of their vertices. In particular, if an n -cube has non-unipotent stabilizer,then so do all its faces.If two cubes are conjugate by the GL ( k [ C ])- or the SL ( k [ C ])-action, then soare their stabilizers. (cid:3) Remark 3.5.
Note that this subcomplex is not necessarily a full subcomplex. Evenif all the vertices of a cube have non-unipotent stabilizers, the cube does not neces-sarily have a non-unipotent stabilizer. It frequently happens that the automorphismgroups of the vertices of a cube have trivial intersection inside
PGL ( K ) . Definition 3.6. • The subcomplex of the building consisting of cells withnon-unipotent stabilizer in
PGL ( k [ C ]) is denoted by P C and is called the parabolic subcomplex . • The quotient of the building X C modulo the subcomplex P C is denoted by U C and is called the unknown quotient . Remark 3.7.
The terminology “parabolic subcomplex” is supposed to underlinethat the stabilizers in the subcomplex are strongly related to parabolic (or betterparahoric) subgroups of GL ( K ) or SL ( K ) .The terminology “unknown quotient” is supposed to underline our complete lackof knowledge of its structure, homology etc. There are strong links to cuspidalphenomena in number theory: the quotient U C contains the information away frominfinity. In the case of finite base fields, the unknown quotients contains most of thecompactly supported cohomology. However, “cuspidal quotient” seems inappropriateterminology, overloaded as the term cuspidal is in number theory and representationtheory.Exploration of the structure of the great unknown will be the subject of furtherpapers in the series. It now follows from the definition that we have an exact sequence of chain com-plexes.
Proposition 3.8.
Let k be a field, let C be a smooth projective curve over k , anddenote C = C \ { P , . . . , P s } .(1) Denote by Γ one of the linear groups (P)GL ( k [ C ]) or (P)SL ( k [ C ]) . Thereis an exact sequence of Γ -chain complexes → C • ( P C ) → C • ( X C ) → C • ( U C ) → . ARABOLIC SUBCOMPLEXES AND HOMOLOGY OF SL In particular, there is a long exact sequence of Γ -equivariant homologygroups: · · · → H Γ • +1 (C • ( U C )) → H Γ • (C • ( P C )) → H Γ • (C • ( X C )) → H Γ • (C • ( U C )) → · · · Since X C is contractible, we also have H Γ • (C • ( X C )) ∼ = H • (Γ) .(2) Assume now that Γ is one of groups PGL ( k [ C ]) or (P)SL ( k [ C ]) , and let ℓ be an odd prime different from the characteristic of k . For i > s , we have H Γ i (C • ( U C ) , Z /ℓ ) = 0 . In particular, we have isomorphisms H Γ i (C • ( P C )) ∼ = H i (Γ , Z /ℓ ) . Proof. (1) is barely more than the definition of P C and U C .(2) By definition, the cells in U C have unipotent stabilizer in PGL ( k [ C ]) andPSL ( k [ C ]), hence they are uniquely ℓ -divisible groups by assumption. The sameholds for the stabilizers in SL ( k [ C ]) since the center has order 2. In particular, theisotropy spectral sequence computing H Γ • (C • ( U C ) , Z /ℓ ) is concentrated in the line q = 0 since only the groups H (Γ σ , Z /ℓ σ ) are non-trivial. The spectral sequencetherefore degenerates at the E -term and converges toH Γ • (C • ( U C ) , Z /ℓ ) ∼ = H • (C • (Γ \ U C ) , Z /ℓ ) . The quotient only has cells in dimension ≤ s , since the building only has dimension s . This proves the claim. (cid:3) Remark 3.9.
The exact sequence closely resembles the dual of the long exact se-quence relating group cohomology to Farrell-Tate cohomology and the homologyof the Steinberg module for groups of finite virtual cohomological dimension, cf. [Bro94] : · · · → b H •− (Γ) → H n −• (Γ , St Γ ) → H • (Γ) → b H • (Γ) → · · · For finite base fields k , the groups GL ( k [ C ]) and SL ( k [ C ]) have finite virtual ℓ -cohomological dimension if ℓ = char k and the long exact sequence of Proposition 3.8is the (dual of the) one for Farrell-Tate cohomology. For infinite base fields, thegroups GL ( k [ C ]) and SL ( k [ C ]) do no longer have finite virtual cohomologicaldimension. Nevertheless, we can see the homology of the parabolic subcomplex as areplacement of Farrell-Tate homology in this setting. Definition 3.10.
Let k be a field, let C be a smooth projective curve over k , anddenote C = C \ { P , . . . , P s } . Denote by Γ one of the linear groups (P)GL ( k [ C ]) or (P)SL ( k [ C ]) . The Γ -equivariant homology of the parabolic subcomplex P C iscalled the parabolic homology of Γ . Remark 3.11.
There is a version of Tate cohomology for arbitrary groups, dueto Mislin [Mis94] , which agrees with Farrell-Tate cohomology for groups of finitevirtual cohomological dimension. It is not clear if Mislin’s version of Tate coho-mology agrees with the equivariant cohomology of the parabolic subcomplex above.The difficulty is mainly in the different definitions: while Mislin’s version of Tatecohomology is defined via killing projectives in the derived category of Z [Γ] -modules,the parabolic subcomplex has only a very concrete geometric definition.A first step in comparing the two cohomologies would be to compute the para-bolic homology of a projective Z [Γ] -module which is possible using the methods ofthe present paper. It could probably be shown that a projective Z [SL ( k [ C ])] -modulewhose restriction to the constant group rings Z [SL ( k )] is injective, has trivial par-abolic homology. This would at least establish a morphism from a “relative” Mislin-Tate cohomology to parabolic cohomology. Functoriality.
The parabolic subcomplex is also functorial with respect tomorphisms of curves:
Proposition 3.12.
Let f : D → C be a finite morphism of smooth projective curvesover k , let P , . . . , P s be points on C , and let Q , . . . , Q t be points on D not in thepreimage of the P i . Set C = C \ { P , . . . , P s } and D = D \ ( { f − ( { P , . . . , P s } ) ∪{ Q , . . . , Q t } ) .(After possible subdivision of X C ) there is a natural morphism f ∗ : X C → X D which induces a morphism f ∗ : P C → P D of parabolic subcomplexes. The morphism f ∗ : P C → P D is equivariant with respect to the natural group homomorphism GL ( k [ C ]) → GL ( k [ D ]) . This assignment is functorial.There are induced morphisms f ∗ : GL ( k [ C ]) \ X C → GL ( k [ D ]) \ X D (and simi-larly for SL in place of GL resp. P in place of X ). In terms of the vector bundleinterpretation of the quotient, these morphisms are identified with pullback of vectorbundles.Proof. The morphism of curves induces a morphism of function fields f ∗ : k ( C ) → k ( D ). This is compatible with the valuations and hence induces a morphism s Y i =1 GL ( k ( C )) / ( s Y i =1 GL ( O P i ) · k ( C ) × ) → ˜ s + t Y i =1 GL ( k ( D )) / ( ˜ s + t Y i =1 GL ( O Q i ) · k ( D ) × ) , where the factor GL ( k ( C )) for the point P i is mapped diagonally to the factorsfor f − ( P i ). This morphism is obviously functorial. It is also obviously equivariantwith respect to GL ( k [ C ]) → GL ( k [ D ]), both groups embedded diagonally intoGL ( k ( C )) s and GL ( k ( D )) ˜ s + t , respectively.The cubical structure is slightly more complicated to take care of: a matrix foran elementary transformation (cid:18) π a i i α i (cid:19) is mapped to (cid:18) f ∗ ( π i ) a i f ∗ ( α i )0 1 (cid:19) g , and so maps a 1-cube to the diagonal of a g -cube, where g is cardinality of f − ( P i ).After suitably subdividing, the above map on vertices extends to a morphism ofcell complexes. This proves all the claims for the morphism f ∗ : X C → X D . Forthe parabolic subcomplexes, it then suffices to note that if a cell σ ∈ X C has anon-unipotent non-central subgroup in the stabilizer, then the same is true for itsimage: any non-unipotent non-central element in the stabilizer in GL ( k [ C ]) willhave non-unipotent non-central image in GL ( k [ D ]). Therefore f ∗ restricts to theparabolic subcomplexes, and all the auxiliary properties are inherited. (cid:3) Local structure of parabolic subcomplexes
In this section, we work out the local structure of the parabolic subcomplex P C resp. its quotient Γ \ P C , for Γ any of the linear groups (P)GL ( k [ C ]) or(P)SL ( k [ C ]). The general procedure is as follows: for a geometrically split bun-dle, we completely describe the action of its automorphism group on the link ofthe corresponding point in the building. The link Lk P ( x ) of the point x in P C isgiven by those simplices in Lk X ( x ) which are fixed by a non-unipotent non-centralelement of the stabilizer of x . The link of the image of x in the quotient Γ \ P C isthen given by the quotient Stab( x ; Γ) \ Lk P ( x ). These local computations will beused in the next section to describe the global structure of the quotient Γ \ P C .4.1. Action of the automorphism group on the links.
We first investigate theaction of automorphism groups of rank two bundles on the links of their correspond-ing points in X C . Let E be a vector bundle corresponding to the vertex x in X C . ByProposition 2.5, the stabilizer of x is essentially determined by the automorphism ARABOLIC SUBCOMPLEXES AND HOMOLOGY OF SL group of the vector bundle E . Combining this with Proposition 3.1, we see that x is in P C if and only if E is geometrically split. Moreover, Proposition 3.1 providesa detailed description of the isomorphism types of possible stabilizer groups.To obtain a description of the action of those stabilizer groups on the links,recall from Section 2.1 that for x a vertex of the building, the corresponding link isa disjoint union F si =1 P ( k ( P i )) with the “complete s -partite simplicial structure”.The point x is given by specifying O P i -lattices in V = K , or equivalently (usingthe standard basis of V ) by specifying matrices M i ∈ GL ( K v i ), for i ∈ { , . . . , s } .As in Section 2.1, the link of x can be described in terms of these data: the verticesin the i -th component P ( k ( P i )) are points corresponding to the tuples (cid:18) M , . . . , M i · (cid:18) π i ˜ α i (cid:19) , . . . , M s (cid:19) , ˜ α i ∈ O P i , and (cid:18) M , . . . , M i · (cid:18) π − i
00 1 (cid:19) , . . . , M s (cid:19) . Note that the point in the building only depends on the Q GL ( O P i ) · K × v i -rightcoset of the above, in particular, the first line above only depends on the residueclass α i of ˜ α i in A ( k ( P i )). The stabilizer of the point ( M , · · · , M s ) in the groupΓ = (P)GL ( k [ C ]) or Γ = (P)SL ( k [ C ]) is then exactly the subgroup of Γ leavinginvariant the Q GL ( O P i ) · K × v i -right coset of ( M , . . . , M s ). The next propositiondescribes the action of the stabilizer on the link by computing explicitly the actionof the stabilizer on the right cosets of matrices as given above. Lemma 4.1.
Let k be a field, let C be a smooth projective curve over k and set C = C \ { P , . . . , P s } . Let E ∼ = L ⊕ L ′ be a split vector bundle over C , and denoteby x the point of X C corresponding to E . Denote by Γ( C ) one of the linear groups (P)GL ( k [ C ]) or (P)SL ( k [ C ]) .(1) There exist closed points Q , . . . , Q t such that the restrictions of both L and L ′ to C ′ = C \ { P , . . . , P s , Q , . . . , Q t } are trivial.(2) There is an inclusion of buildings X C ֒ → X C ′ which is equivariant for thenatural homomorphism Γ( C ) → Γ( C ′ ) and induces an isomorphism onstabilizers Stab( x ; P Γ( C )) ∼ = Stab( x ; P Γ( C ′ )) .(3) As a consequence of (2), the inclusion Lk X C ( x ) ֒ → Lk X C ′ ( x ) is equivariantfor the action of Stab( x ; Γ( C )) .Proof. (1) First note that L − | C ∼ = L ′ | C because E| C is trivial. Because C isa smooth curve, the usual identification of divisor class group and Picard groupallows to write L ∼ = O C ( P si =1 n i P i + P tj =1 m j Q j ). The restriction of L to C ′ = C \ { P , . . . , P s , Q , . . . , Q t } is then obviously trivial, cf. also Lemma 2.6.For (2), we can consider the quotient description of the vertices of the building.The induced map is induced by inclusion of the first s factors: s Y i =1 GL ( K v i ) / (GL ( O P i ) · K × v i ) →→ s Y i =1 GL ( K v i ) / (GL ( O P i ) · K × v i ) × t Y j =1 GL ( K v j ) / (GL ( O Q j ) · K × v j )The map is evidently injective on vertices. Since the building is a cubical complex,i.e., every cube is uniquely determined by its vertices, the map is an inclusionof complexes. As the groups Γ( C ) and Γ( C ′ ) act via diagonal inclusion into theproduct Q GL ( K v i ), the equivariance of the inclusion of buildings is also clear.Similarly, it is clear that the map restricts to a homomorphism of stabilizers. Thestabilizers have been described in Proposition 2.5, and from that description it is clear that the map induced on stabilizers has to be an isomorphism - the relevantstabilizer groups are directly related to automorphism groups of vector bundles overthe curve C and independent of the choice of affine subcurve.(3) is a direct consequence of (2). (cid:3) Remark 4.2.
Note that the isomorphism on stabilizer groups in (2) above onlyholds for the projective (special) linear groups. In the case of GL , the stabilizergroup can become bigger, cf. Proposition 2.5. However, the k [ C ] × -part of the sta-bilizer acts trivially on the link anyway, so this does not affect application of theabove lemma to GL ( k [ C ]) . Proposition 4.3.
Let k be a field, let C be a smooth projective curve over k andset C = C \ { P , . . . , P s } . Let E ∼ = L ⊕ L ′ be a split vector bundle over C . Denoteby x the point of X C corresponding to E .There exist closed points Q , . . . , Q t of C such that the image of x in the doublequotient GL ( k [ C ′ ]) \ s Y i =1 GL ( K v i ) / (GL ( O P i ) · K × v i ) × t Y j =1 GL ( K v j ) / (GL ( O Q j ) · K × v j ) has a representative of the following form: (cid:18)(cid:18) π a
00 1 (cid:19) , . . . , (cid:18) π a t t
00 1 (cid:19)(cid:19) , X a i ≥ . (a) Assume L ∼ = L ′ , i.e., the stabilizer is conjugate to a diagonally embedded GL ( k ) . The action of GL ( k ) on the i -th component of the link is the stan-dard action of GL on P ( k ( P i )) , hence the action of GL ( k ) on the link is thediagonal standard action.(b) Assume L 6∼ = L ′ , i.e., the stabilizer is conjugate to the group B ( a , . . . , a t ) = (cid:26)(cid:18) a f b (cid:19) | a, b ∈ k, ab = 0 , v i ( f ) ≥ − a i (cid:27) . There are two cases for the action of the stabilizer on the i -th component ofthe link of x : if v i ( f ) = − a i , then the action is the “standard” action. If v i ( f ) > − a i , the action is trivial.Proof. We first prove the statement about the double coset representative. De-noting C ′ = C \ { P , . . . , P s , Q , . . . , Q t } as in Lemma 4.1, the double quotientwritten classifies exactly equivalence (rel ∂C ′ ) classes of rank two vector bundleson C whose restriction to C ′ is trivial. Therefore, the elements corresponding to E ∼ = L ⊕ L ′ and L ⊗ ( L ′ ) − ⊕ O are equal in the double quotient. It is easy to seethat the double coset representative given corresponds to the vector bundle O − s X i =1 a i P i − t X j =1 a s + j Q j ⊕ O Therefore, the statement follows if the a i are chosen such that O − s X i =1 a i P i − t X j =1 a s + j Q j ∼ = L ⊗ ( L ′ ) − . Note that it is necessary to enlarge the set { P , . . . , P s } (using Lemma 4.1) because L ⊗ ( L ′ ) − might not be trivial over C , whence it would not correspond to anelement in the double quotient k [ C ] × \ Q K × v i / Q O × P i .Next, we explain how to reduce the statements (a) and (b) to statements aboutthe given double coset representative. By Lemma 4.1 (3), the inclusion of links ARABOLIC SUBCOMPLEXES AND HOMOLOGY OF SL Lk X C ( x ) → Lk X C ′ ( x ) is equivariant for the action of the stabilizer of x . To de-termine this action, it thus suffices to determine the action of Stab( x ; GL ( k [ C ]))on the vertices of X C ′ . Moreover, if two points x and x ′ of X C ′ are GL ( k [ C ′ ])-conjugate, then so are their stabilizers as well as the actions of the stabilizers onthe respective links. We can therefore replace x by any other representative ofthe vertices of GL ( k [ C ′ ]) \ X C ′ . The vertices of the latter quotient are exactly thedouble cosets in the statement. Summing up, the link Lk X C ( x ) together with theaction of the stabilizer of x can be equivariantly embedded into the link of thechosen representative above in X C ′ .Now we come to the prove of part (a). From the above, we have that the stabilizerof x is GL ( k [ C ′ ])-conjugate to the stabilizer of the double coset (1 , . . . ,
1) where allentries are identity matrices. The stabilizer of this double coset inside GL ( k [ C ′ ])is obviously GL ( k ), embedded diagonally as constants. Up to right multiplicationwith elements from GL ( O P i ) · K × v i , we find (cid:18) a bc d (cid:19) · (cid:18) π − i
00 1 (cid:19) = (cid:18) aπ − i bcπ − i d (cid:19) ∼ (cid:18) π i ac (cid:19) , c = 0 (cid:18) a bc d (cid:19) · (cid:18) π i ˜ α (cid:19) = (cid:18) aπ i a ˜ α + bcπ i c ˜ α + d (cid:19) ∼ (cid:18) π i a ˜ α + bc ˜ α + d (cid:19) , c ˜ α + d = 0and the missing cases are dealt with similarly. But this is indeed the standardaction of GL ( k ) on P ( k ( P i )).(b) By what was said above, it suffices to compute the stabilizer and its actionon the link in the case of the diagonalized double coset representative.Up to right multiplication by elements of GL ( O P i ) · K × v i , we have the following (cid:18) a f b (cid:19) · (cid:18) π a i i
00 1 (cid:19) = (cid:18) aπ a i i f π a i i b (cid:19) ∼ (cid:18) π a i i
00 1 (cid:19) , whenever v i ( f ) ≥ − a i . From P a i ≥
0, it follows that the matrices above are theonly ones possibly stabilizing the representative. This shows that the stabilizer isindeed as described.We now study the action of the stabilizer B ( a , . . . , a t ) on the link of x . Forthe point ∞ , we have (cid:18) a f b (cid:19) (cid:18) π a i − i
00 1 (cid:19) = (cid:18) aπ a i − i f π a i − i b (cid:19) and there are two cases to consider: if v i ( f ) = − a i , then v i ( f π a i − i ) = −
1, and wecan rewrite the matrix using right multiplication with GL ( O P i ) · K × v i to the form (cid:18) π a i +1 i g (cid:19) , where v ( g ) = a i and the leading term is af with f the leading term of f (as inthe constant case). On the other hand, if v i ( f ) > − a i , then v i ( f π a i − i ) ≥
0, andright multiplication allows to rewrite this to diag( π a i − i , α ∈ P ( k ( P )) \ {∞} , we consider (cid:18) f (cid:19) (cid:18) π a i +1 i π a i i ˜ α (cid:19) = (cid:18) π a i +1 i π a i i ˜ αf π a i +1 i f π a i i ˜ α + 1 (cid:19) . The restriction a = b = 1 is only for simplification of exposition. In the case v i ( f ) = − a i , we can use right multiplication to transform the matrix to π a i +1 i π a i i · ˜ α fπ aii ˜ α ! . We can use further right multiplication to transform ˜ α fπ aii ˜ α to ˜ α c ˜ α with c theconstant term of f π a i i . If v i ( f ) > − a i , then this constant term is 0, and the actionis trivial.For v i ( f ) = − a i = 0, this is the standard action of the Borel B ( k ) on P ( k ( P i )).For v i ( f ) = − a i = 0 , the action is still the same, only that f acts via its constantcoefficient. If v i ( f ) > − a i , then the action is trivial. The claim is proved. (cid:3) Remark 4.4.
A similar result holds for bundles which are indecomposable butgeometrically split. Let E be such a bundle, with corresponding point x ∈ X C ,and let L/k be a splitting field for the bundle E . Then the morphism of curves φ : C × k L → C induces a morphism of buildings X C ֒ → X C × k L which embeds X C as the Gal(
L/k ) -invariants into X C × k L . The results of Proposition 4.3 abovethen imply that the action of the stabilizer of x on Lk X C ( x ) is also the standardone, coming from the embedding of Stab( x ; GL ( k [ C ])) as Gal(
L/k ) -invariants of Stab( x ; GL ( L [ C ])) . For this and other reasons, our main results are restricted tothe case of algebraically closed base fields. Local structure.
Having determined the action of vertex stabilizers on links,we are now ready to describe the local structure of the quotients Γ \ P C as subcom-plexes of the respective quotients of Γ \ X C , where as usual Γ is one of the lineargroups (P)GL ( k [ C ]) or (P)SL ( k [ C ]). For a point x ∈ X C , the link Lk Γ \ P C is thequotient Stab( x ; Γ) \ Lk P C ( x ). The following proposition now uses Proposition 4.3to determine Lk P C ( x ) as well as its quotient modulo Stab( x ; Γ). Proposition 4.5.
Let k be an algebraically closed field, let C be a smooth projectivecurve over k and denote C = C \ { P , . . . , P s } . Let E ∼ = L ⊕ L ′ be a split rank twovector bundle, and let x be the corresponding point in the building X C . Denote by Γ one of the linear groups (P)GL ( k [ C ]) or (P)SL ( k [ C ]) .Denote by A x the intersection of an apartment A with Lk X C ( x ) .(a) If L ′ ∼ = L − , then Lk Γ \ P ( x ) is isomorphic to the quotient of A x modulo theantipodal Z / -action. The star St Γ \ P ( x ) is isomorphic to a cone over (a cubicalcomplex of the homotopy type of ) RP s − . The stabilizer of x is Γ( k ) , everythingelse is stabilized (up to a unipotent group) by a maximal torus of Γ .(b) If L ′ = L − , then Lk Γ \ P ( x ) is isomorphic to A x . The stabilizer is (up to aunipotent group) a maximal torus in Γ( k ) .Proof. Recall that for a field F , the set P ( F ) is a building of type A . The groupGL ( F ) acts in the standard way on P ( F ). The standard apartment is the subset { , ∞} ⊆ P ( F ), its setwise stabilizer in GL ( F ) is the normalizer N ( F ) of thestandard maximal torus T ( F ), its pointwise stabilizer is the standard maximaltorus T ( F ).Now we consider the “complete n -partite simplicial complex” Lk X C ( x ) on thedisjoint union F si =1 P ( k ), with the diagonal action of GL ( k ). Note that A x is con-jugate to the full subcomplex of Lk X C ( x ) spanned by F si =1 { i , ∞ i } ⊆ F si =1 P ( k ).It is easy to see that the setwise stabilizer of A x is conjugate to the normalizerN ( k ) and the pointwise stabilizer is conjugate to the standard maximal torusT ( k ). Note that A x is a triangulation of the ( s − ( k ) / T ( k ) ∼ = Z / A x by the antipodal action.(a) By Proposition 4.3, we have Stab( x ; Γ) = Γ( k ) acting via the diagonal stan-dard action on the link F si =1 P ( k ). A simplex σ ∈ Lk X C ( x ) is contained in Lk P C ( x )if its stabilizer in GL ( k ) contains a non-central non-unipotent element. The latteris equivalent to σ being stabilized by some maximal torus of GL ( k ). But this is thecase precisely if the corresponding simplex is GL ( k )-conjugate to one in A x . Wesee that every simplex in Lk P C ( x ) is GL ( k )-conjugate to one in A x . As remarked ARABOLIC SUBCOMPLEXES AND HOMOLOGY OF SL above, the setwise stabilizer of the standard apartment is the normalizer of themaximal torus corresponding to A x , the pointwise stabilizer of the standard apart-ment is the maximal torus itself, and the Weyl group acts via the antipodal action.This proves the claim about the link. The claim about the star follows, because thestar is the cone over the link. The maximal torus stabilizes A x pointwise, so eachsimplex in Lk Γ \ P C ( x ) is stabilized at least by the maximal torus. As there are nosimplices stabilized by GL ( k ), the claim follows from the knowledge of stabilizersin Proposition 3.1.(b) By Proposition 4.3, we have Stab( x ; GL ( k [ C ])) ∼ = k n ⋊ T ( k ). The maximaltorus acts via the diagonal standard action on the link F si =1 P ( k ). Fixing a sectionof k n ⋊ T ( k ) → T ( k ), there are exactly two fixed points. Any two such choices ofsections yield actions which are conjugate (by the unipotent radical) to the actionwith fixed points exactly 0 and ∞ in each disjoint summand P ( k ). Then also thesimplices connecting these points are stabilized by the maximal torus. Furthermore,any simplex which is stabilized by some non-unipotent element is stabilized bysome such maximal torus. Summing up, we see that every simplex in Lk P C ( x ) isStab( x ; GL ( k [ C ]))-conjugate to a simplex in A x , and A x is a strict fundamentaldomain for the Stab( x ; GL ( k [ C ]))-action on Lk P C ( x ). This proves the claim.The same statements also are true for SL ( k [ C ]) instead of GL ( k [ C ]) used above. (cid:3) Remark 4.6.
The link Lk P C ( x ) is the preimage of these links under the projection P C → Γ \ P C . However, it is more difficult to describe explicitly due to unipotentsubgroups in the stabilizer of x . We ignored these in the local description of Γ \ P C because they will not be visible in the homology computations anyway. Remark 4.7.
The reason why we restricted to algebraically closed base fields inProposition 4.5 is the following: although we have identified in Proposition 4.3 thatthe action of GL ( k ) on P ( k ( P i )) is the standard one, this does not allow to givean easy description of the quotient Stab( x ; Γ) \ Lk P ( x ) .The phenomenon is already visible in the case where s = 1 and [ k ( P ) : k ] = 2 .The action of GL ( k ) on P ( k ( P )) is no longer transitive but has three orbits,one orbit isomorphic to P ( k ) , any of the two other orbits is a symmetric spacegeneralizing the upper half plane H . The stabilizer of points in P ( k ) is a Borelsubgroup, the stabilizer of points in the other orbits is a rank one non-split torus.In general, depending on the size of k ( P i ) and the number of degree subfields,there can be many relevant non-split tori in GL ( k ) fixing points in P ( k ( P i )) .The description of the quotient Stab( x ; Γ) \ Lk P ( x ) then strongly depends on theextension k ( P i ) /k . In any case, the quotient tends to be much more complicatedthan in the algebraically closed case handled in Proposition 4.5. Global structure of parabolic subcomplexes
After having determined the local structure of the parabolic subcomplex Γ \ P C ,we will now turn to the investigation of the global structure. The main results statethat Γ \ P C is a disjoint union of subcomplexes, indexed by classes of line bundlesover C , and the structure of the subcomplex associated to the line bundle L can beexplicitly described: if L 6∼ = L − then it is a torus related to the classifying space ofa maximal torus in GL ( K ), and if L ∼ = L − it is a slightly more complicated spacerelated to the classifying space of the normalizer of a maximal torus in GL ( K ).The topology of the quotient GL ( k [ C ]) \ X C - and in particular the homotopytype at infinity - was studied already by Stuhler in [Stu76] and [Stu80] in the casewhere k = F q . The results below can be seen as a version of his work: our resultswork over algebraically closed base fields k and describe a part of GL ( k [ C ]) \ X C slightly larger than a neighbourhood of infinity. The method of proof here is stillthe one already employed by Stuhler: we will write down model complexes, andthe local computations from Section 4 will imply that the model complexes areisomorphic to the components of Γ \ P C .5.1. Line bundles and connected components.Definition 5.1.
Let k be a field, let C be a smooth projective curve and denote C = C \{ P , . . . , P s } . We define K ( C ) to be the quotient of the Picard group Pic( C ) modulo the involution ι : L 7→ L − . As the notation suggests, the set K ( C ) is both related to the Kummer varietyassociated to the curve C as well as the set of komponents of Γ \ P C : Lemma 5.2.
Let k be an algebraically closed field, let C be a smooth projectivecurve and denote C = C \{ P , . . . , P s } . Denote by Γ one of the groups (P)GL ( k [ C ]) or (P)SL ( k [ C ]) . Then we have a bijection π (Γ \ P C ) ∼ = K ( C ) . Proof.
The set π is defined by taking the set of vertices of Γ \ P C and dividing bythe equivalence relation generated by 1-cells of Γ \ P C .We first write down a map (Γ \ P C ) → K ( C ): by Definition 3.6, Proposition 3.1and our assumption that k is algebraically closed, any vertex x of Γ \ P C correspondsto a split vector bundle E ∼ = L ⊕ L ′ . Recall from Proposition 2.3 thatdet E ∼ = L ⊗ L ′ ∼ = O ( a P ) ⊗ · · · ⊗ O ( a s P s )for suitable a i ∈ Z . Using Lemma 2.6, we see that the assignment(Γ \ P C ) → K ( C ) : x = [ L ⊕ L ′ ] [ L ]is well-defined. It does not depend on the choice of (rel ∂C )-equivalence class of arepresentative E of x and does not depend on the splitting.The map is clearly surjective: for any element l ∈ K ( C ), we can find a linebundle L on C such that l = {L| C , L − | C } . Then L ⊕ L − is a vector bundle on C with trivial determinant, hence corresponds to a point x ∈ Γ \ P C which is mappedto l by the map described above.It suffices to show injectivity. Let x and x be two points in Γ \ P C correspondingto vector bundles E and E which are both mapped to l ∈ K ( C ). This means thatwe have splittings E ∼ = L ⊕ L ′ , E ∼ = L ⊕ L ′ , with L | C ∼ = ( L | C ) ± . Up to equivalence (rel ∂C ) and switching summands, wecan assume L ∼ = L . But then we have L ′ ⊗ ( L ′ ) − ∼ = O ( a P ) ⊗ · · · ⊗ O ( a s P s ) . We need to find a chain of 1-cubes in Γ \ P C connecting E to E . It is easy to finda chain of elementary transformations connecting O ( a P ) ⊗ · · · ⊗ O ( a s P s ) to thetrivial line bundle, noting that O ( − P ) ֒ → O is an inclusion of line bundles whosequotient is a torsion sheaf of length one on the curve C . This provides a chain of1-cubes in Γ \ X C connecting E and E .We claim that this chain is contained in P C . Let L be any line bundle and con-sider the two bundles L ⊕ L − ⊗ O ( P ) and L ⊕ L − . We can choose a double cosetrepresentative of L ⊕ L − which only contains diagonal matrices. In particular,the stabilizer contains the diagonal maximal torus. The elementary transformationfrom L ⊕ L − to L ⊕ L − ⊗ O ( P ) is then given by the diagonal matrix diag(1 , π )where π is a uniformizer at P . In particular, this elementary transformation iscompatible with the action of the diagonal torus. In other words, identifying the ARABOLIC SUBCOMPLEXES AND HOMOLOGY OF SL link of L⊕L − with P with the standard action of the diagonal torus, the above el-ementary transformation corresponds to the point ∞ which is fixed by the diagonaltorus. Therefore, the 1-cube corresponding to the above elementary transformationactually lies in the subcomplex Γ \ P C . The general assertion is proved in just thesame way, only more notational effort is required to keep track of all the coefficients a i . Summing up, the path constructed between E and E lies inside Γ \ P C , and wehave thus shown that the map π (Γ \ P C ) → K ( C ) is injective. (cid:3) Remark 5.3.
For a number field K , one considers the action of SL ( O K ) on itscorresponding symmetric space X . In this situation, there is a well-known bijectionbetween the cusps of the locally symmetric space SL ( O K ) \ X and the class groupof K , going back (at least) to Siegel. The above computation of the connectedcomponents of Γ \ X C is a variation on this. The obvious change is that the classgroup has been replaced by the Picard group of the curve C . A non-obvious changeis the appearance of the involution ι . This comes from the fact that for each linebundle L with L 6∼ = L − , there are two components of the homotopy type at infinity,belonging to L and L − , which belong to the same connected component of P C . Model complexes of groups.
Now we come to the description of the con-nected components of Γ \ P C . There are two cases, depending on the line bundle L indexing the component: if L 6∼ = L − then the component is related to a “maximaltorus of Γ”, and if L ∼ = L − then the component is related to the “normalizer of amaximal torus of Γ”. The results are proved by first defining models for the cor-responding quotients and then provide isomorphisms to the respective connectedcomponents. Definition 5.4.
Let Z s ⊆ R s be the standard lattice. Denote by Z s the cubicalcomplex generated by it. We define the groups T = ( ( a , . . . , a s ) ∈ Z s | s X i =1 a i [ P i ] = 0 in Pic( C ) ) , and ST = T . The groups T and ST act on R s by translations and preserve the standard lattice Z s . Therefore, the translation actions of T and ST induces respective actions onthe cubical complex Z s .We define the groups N (resp. SN ) to be the subgroups of the euclidean group E( s ) generated by T (resp. ST ) and all point inversions in the points ( a , . . . , a s ) for which P si =1 a i [ P i ] = 0 in Pic( C ) . Remark 5.5.
Note that the complex Z s / T already appears in [Stu80] and is usedthere for the description of the homotopy type of PGL ( k [ C ]) \ X C at infinity. We note that the group T is a free abelian group whose rank equals the rankof the kernel of φ in Lemma 2.6, and in particular is isomorphic to the group ofnon-constant units in k [ C ]. There are many reasons for calling this group T - itis realized as a subgroup of translations, it is a subgroup of a maximal torus ofGL ( K ), and the quotient of Z s / T has the homotopy type of a torus.The next results will describe the structure of the connected components ofΓ \ P C as quotients of apartments modulo suitable torus- or normalizer-like groups. Lemma 5.6.
Let k be an algebraically closed field, let C be a smooth projectivecurve and denote C = C \ { P , . . . , P s } . Let L be a line bundle on C . There existsan apartment A L of X C and a morphism of cubical complexes φ L : Z s → A L suchthat(1) A L is contained in P C , (2) the -cells of the image of A L in GL ( k [ C ]) \ X C correspond to equivalence(rel ∂C ) classes of rank two bundles of the form ( L ⊗ O ( a P ) ⊗ · · · ⊗ O ( a s P s )) ⊕ L − , a i ∈ Z , (3) φ L is an isomorphism.Proof. Recall the double quotient description of the 0-cells of GL ( k [ C ]) \ X C :GL ( k [ C ]) \ s Y i =1 GL ( K ) / s Y i =1 (GL ( O P i ) · K × ) . Let ( M , . . . , M s ) ∈ GL ( K ) s be a representative of the double coset correspondingto the bundle L ⊕ L − . Denoting by T ( K ) the diagonal maximal torus of GL ( K ),we consider the maximal torusT( L ) = ( M T ( K ) M − , . . . , M s T ( K ) M − s ) ⊂ GL ( K ) s . We define A L to be the full cubical subcomplex of X C whose 0-cubes are the T( L )-orbit of ( M , . . . , M s ). By construction this is an apartment of X C , correspondingto a choice of frame in each of the s copies of K .(1) We see that the subgroup ( M T ( k ) M − , . . . , M s T ( k ) M − s ) ⊂ T( L ) pre-serves lattice classes in A L because T ( k ) ⊆ GL ( O P i ). In particular, this subgroupstabilizes A L pointwise . On the other hand, because ( M , . . . , M s ) represents a splitrank two bundle L ⊕ L − , the point ( M , . . . , M s ) is stabilized by a maximal torusof GL ( k ). From Proposition 4.3, we know that the action of the stabilizer of L ⊕ L − on the link is such that the stabilizer contains a maximal torus which fixespointwise the intersection of A L with the link Lk X C ( L ⊕ L − ). In particular, allthe vertices in the link Lk A L ( L ⊕ L − ) have the same stabilizer as L ⊕ L − . Aninductive argument shows that this is true for all vertices in A L . This implies thatthe intersection ( M T ( k ) M − , . . . , M s T ( k ) M − s ) ∩ GL ( k [ C ])contains a maximal torus of GL ( k ). Then all cubes of A L are stabilized by this,and we see that A L ⊂ P C .(2) is easy to see: up to Q si =1 (GL ( O P i ) · K × ), the elements of T s are of theform A = (cid:18)(cid:18) π a
00 1 (cid:19) , · · · , (cid:18) π a s s
00 1 (cid:19)(cid:19) . But then the elements of T( L ) · ( M , . . . , M s ) are those of the form (cid:18) M · (cid:18) π a
00 1 (cid:19) , · · · , M s · (cid:18) π a s s
00 1 (cid:19)(cid:19) , and these are exactly representatives for bundles of the form( L ⊗ O ( − a P ) ⊗ · · · ⊗ O ( − a s P s )) ⊕ L − , a i ∈ Z . Now the morphism φ L : Z s → A L is defined on 0-cubes as( a , . . . , a s ) (cid:18) M · (cid:18) π a
00 1 (cid:19) , · · · , M s · (cid:18) π a s s
00 1 (cid:19)(cid:19) , and we have seen above that this assignment indeed takes vertices to 0-cubes in A L .It is also easy to see that if vertices span a cube in Z s , then their φ L -images spana cube in A L . Therefore, the above definition of φ L indeed provides a morphism ofcubical complexes. (3) follows because A L is a Coxeter complex for the affine Weylgroup Z s ⋊ Z / s , and hence the morphism φ L has an obvious inverse mapping (cid:18) M · (cid:18) π a
00 1 (cid:19) , · · · , M s · (cid:18) π a s s
00 1 (cid:19)(cid:19) ( a , . . . , a s ) . ARABOLIC SUBCOMPLEXES AND HOMOLOGY OF SL (cid:3) Proposition 5.7.
Let k be an algebraically closed field, let C be a smooth projectivecurve and denote C = C \ { P , . . . , P s } . Let L be a line bundle on C such that L| C = L − | C .(1) The connected component of GL ( k [ C ]) \ P C containing the vector bundle L⊕L − is isomorphic to Z s / T . The stabilizers of cells are of the form ( k n ⋊ ( k × ) ) × k × k [ C ] × for some n depending on the cell, i.e., up to the unipotentradical all the cells have stabilizer T × k × k [ C ] × where T is a maximal torusin GL ( k ) .(2) The connected component of SL ( k [ C ]) \ P C containing the vector bundle L ⊕ L − is isomorphic to Z s / ST . The stabilizers of cells are of the form k n ⋊ k × for some n depending on the cell, i.e., up to the unipotent radicalall the cells have stabilizer a maximal torus in SL ( k ) .In both cases, the quotient has the homotopy type of a torus whose dimension equalsthe rank of k [ C ] × /k × .Proof. We give the argument for SL , the argument for GL is similar. The proof isstructured as follows: in step (1), we describe a homomorphism τ : ST → SL ( k [ C ])and show in step (2) that the map φ L from Lemma 5.6 is ρ -equivariant. Steps (3)to (5) deduce the claim from these facts.(1) Recall from Definition 5.4 that ST was defined as the subgroup of Z s contain-ing the tuples (2 a , . . . , a s ) with P si =1 a i [ P i ] = 0 in Pic( C ). This is a free abeliangroup isomorphic to k [ C ] × /k × : the condition on the sum in the Picard groupimplies that P a i [ P i ] is a principal divisor ( f ), and f ∈ k [ C ] × /k × . Choose anembedding u : ST → k [ C ] × which induces one such isomorphism ST ∼ = k [ C ] × /k × and define a homomorphism τ : ST → SL ( K ) s by sending (2 a , . . . , a s ) to( M , . . . , M s ) · (cid:18) u (2 a , . . . , a s ) 00 u (2 a , . . . , a s ) − (cid:19) · ( M − , . . . , M − s ) . By definition, τ (2 a , . . . , a s ) stabilizes the image of ( M , . . . , M s ) in SL ( k [ C ]) \ P C if P si =1 a i [ P i ] = 0 in Pic( C ): it sends the vector bundle L⊕L − to ( L⊗O ( − a P ) ⊗· · · ⊗ O ( − a s P s )) ⊕ L − , but these are both isomorphic under the above condition.In this case, τ (2 a , . . . , a s ) ∈ SL ( k [ C ]) because it stabilizes the double coset( M , . . . , M s ) in SL ( k [ C ]) \ Q GL ( K ) / ( Q GL ( O P i ) · K × ). We see that τ : ST → SL ( K ) s factors through a homomorphism ST → SL ( k [ C ]) which we will stilldenote τ .(2) We show that the embedding φ L from Lemma 5.6 is τ -equivariant. Theelement (2 a , . . . , a s ) acts on the model complex Z s by the evident translation.The element τ (2 a , . . . , a ) sends ( M , . . . , M s ) ∈ A L to (cid:18) M · (cid:18) π a π − a (cid:19) , · · · , M s · (cid:18) π a s s π − a s s (cid:19)(cid:19) which is also the translation by (2 a , . . . , a s ) on A L .(3) From (1) and (2) above we obtain a map of quotient cell complexes φ L : Z s / ST → SL ( k [ C ]) \ P C . We first show that this map is surjective onto the L -component of SL ( k [ C ]) \ P C .By Lemma 5.6 (2), we know that the 0-cells of the image are exactly the equivalence(rel ∂C ) classes of rank two bundles of the form ( L⊗O ( a P ) ⊗· · ·⊗O ( a s P s )) ⊕L − .For each ( a , . . . , a s ) ∈ Z s , the map φ L induces an isomorphism from the star of( a , . . . , a s ) in Z s to the star of φ L in SL ( k [ C ]) \ P C , by Proposition 4.5. Therefore,the map φ L is a surjection onto the L -component of SL ( k [ C ]) \ P C . (4) We show that the map φ L is injective. First, we consider 0-cells. Let( a , . . . , a s ) and ( b , . . . , b s ) be two vertices of Z s . The vector bundles associatedto these two vertices are( L ⊗ O ( a P ) ⊗ · · · ⊗ O ( a s P s )) ⊕ L − and ( L ⊗ O ( b P ) ⊗ · · · ⊗ O ( b s P s )) ⊕ L − . These vector bundles are isomorphic precisely when P ( a i − b i ) P i = 0 in Pic( C ). Butthen the two original vertices are ST -conjugate. This shows injectivity on 0-cells.Injectivity for n -cells now follows from injectivity of 0-cells and the fact - establishedin Proposition 4.5 and used in step (3) above - that φ L induces isomorphisms onstars.(5) Finally, the stabilizer statements follow from Proposition 2.5 and the factthat all vector bundles are split with non-isomorphic direct summands. It is alsoclear that the quotient modulo the group of translations has the homotopy type ofa torus of the rank claimed. (cid:3) Proposition 5.8.
Let k be an algebraically closed field, let C be a smooth projectivecurve and denote C = C \ { P , . . . , P s } . Let L be a line bundle on C such that L| C ∼ = L − | C .(1) The connected component of GL ( k [ C ]) \ P C containing the vector bundle L ⊕ L is isomorphic to Z s / N . The image of (0 , . . . , is stabilized by GL ( k ) × k × k [ C ] × . The stabilizers of all other cells are of the form ( k n ⋊ ( k × ) ) × k × k [ C ] × for some n depending on the cell, i.e., up to the unipotentradical all the other cells have stabilizer T × k × k [ C ] × where T is a maximaltorus in GL ( k ) .(2) The connected component of SL ( k [ C ]) \ P C containing the vector bundle L ⊕ L is isomorphic to Z s / SN . There are rk T points with stabilizer SL ( k ) , they are the ST -conjugacy classes of T -translates of (0 , . . . , . Thestabilizers of all other cells are of the form k n ⋊ k × for some n dependingon the cell, i.e., up to the unipotent radical all the other cells have stabilizera maximal torus in SL ( k ) .Proof. We give the argument for SL , the argument for GL is similar but easier.The proof structure is similar to the proof of Proposition 5.7.(1) We construct a homomorphism ν : SN → SL ( k [ C ]) as follows: on thesubgroup ST , the morphism ν agrees with τ . Now for P si =1 a i [ P i ] = 0 in Pic( C )and r ( a , . . . , a s ) the point-reflection with center ( a , . . . , a s ), the map ν is definedto be ( M , . . . , M s ) · (cid:18) u ( a , . . . , a s ) − u ( a , . . . , a s ) − (cid:19) · ( M − , . . . , M − s ) . Then ν ( r ( a , . . . , a s )) again stabilizes the image of (cid:18) M · (cid:18) π a (cid:19) , . . . , M s · (cid:18) π a s s (cid:19)(cid:19) in SL ( k [ C ]) \ P C under the condition P si =1 a i [ P i ] = 0. We see that ν : SN → SL ( K ) s factors through a homomorphism SN → SL ( k [ C ]) which we will stilldenote ν .(2) We show that the embedding φ L from Lemma 5.6 is ν -equivariant. Equiv-ariance on the subgroup ST follows from (2) in the proof of Proposition 5.7. Weneed to show that ν ( r ( a , . . . , a s )) is again a point-reflection. The definition of ν ( r ( a , . . . , a s )) shows that it sends the coset (cid:18) M · (cid:18) π b (cid:19) , . . . , M s · (cid:18) π b s s (cid:19)(cid:19) ARABOLIC SUBCOMPLEXES AND HOMOLOGY OF SL to the coset (cid:18) M · (cid:18) π b − a
00 1 (cid:19) , . . . , M s · (cid:18) π b s − a s s
00 1 (cid:19)(cid:19) . This proves the claim.(3) The equivariance from (1) and (2) shows that we obtain a map of quotientcell complexes φ L : Z s / SN → SL ( k [ C ]) \ P C . Surjectivity of φ L onto the L -component of SL ( k [ C ]) \ P C follows as in the proofof Proposition 5.7 from the results in Lemma 5.6 and Proposition 4.5.(4) For injectivity on 0-cells, let ( a , . . . , a s ) and ( b , . . . , b s ) be vertices of Z s which induce isomorphic vector bundles( L ⊗ O ( a P ) ⊗ · · · ⊗ O ( a s P s )) ⊕ L and ( L ⊗ O ( b P ) ⊗ · · · ⊗ O ( b s P s )) ⊕ L . The isomorphism of vector bundles implies that the line bundles O ( a P ) ⊗ · · · ⊗P ( a s P s ) and O ( b P ) ⊗ · · · ⊗ P ( b s P s ) are either isomorphic or inverses. The case ofisomorphisms is the one handled in (4) of the proof of Proposition 5.7. If they areinverses, then their tensor product is a trivial bundle, and the corresponding vectorbundles are related by a point reflection with center corresponding to the vectorbundle ( L ⊗ O (( a + b ) P ) ⊗ · · · ⊗ O (( a s + b s ) P s )) ⊕ L . The tuples ( a , . . . , a s ) and( b , . . . , b s ) are then SN -conjugate, and we get injectivity on 0-cells. Injectivity for n -cells then follows again from Proposition 4.5.(5) Finally, the stabilizer statements follow from Proposition 2.5 and the factthat all vector bundles are split with non-isomorphic direct summands. (cid:3) Functoriality.
From Proposition 3.12, we know that morphisms of curves f : C → D induce morphisms of parabolic subcomplexes and parabolic quotients.Now that we have precise knowledge of the structure of the parabolic quotients, wecan also describe precisely the effect of the induced morphisms: Proposition 5.9.
Let k be an algebraically closed field. Let f : D → C be a finitemorphism of smooth projective curves over k , let P , . . . , P s be points on C , and let Q , . . . , Q t be points on D not in the preimage of the P i . Set C = C \ { P , . . . , P s } and D = D \ ( { f − ( { P , . . . , P s } ) ∪ { Q , . . . , Q t } ) .Then we have the following assertions for the induced morphism f ∗ : P C → P D of Proposition 3.12:(1) The composition K ( C ) → π (GL ( k [ C ]) \ P C ) π ( f ∗ ) −→ π (GL ( k [ D ]) \ P D ) → K ( D ) is induced from pullback of line bundles f ∗ : Pic( C ) → Pic( D ) , where thefirst and last map are the bijections of Lemma 5.2.(2) The morphism from a given component to the image component can bedescribed on model complexes as follows: the map Z s → Z ˜ s + t maps the i -th coordinate diagonally to f − ( P i ) many coordinates of Z ˜ s + t .(3) The composition k [ C ] × /k × ∼ = T C → T D ∼ = k [ D ] × /k × is induced from thenatural map f ∗ : k [ C ] → k [ D ] on function rings.Proof. (1) is clear and follows because f ∗ on the vertices of the quotients is givenby pullback of vector bundles.(2) the map on model complexes is the same one as on the standard apartmentof the building, and hence the same as the one described in Proposition 3.12.(3) the map is induced from the inclusion of the units in the function fields andthe conjugation action on certain apartments. Therefore, it is the natural map. (cid:3) Equivariant homology of the parabolic subcomplex
In this section, we will use the description of the quotients Γ \ P C to computethe Γ-equivariant homology of P C , for Γ one of the linear groups (P)GL ( k [ C ])and (P)SL ( k [ C ]). The main results are Proposition 6.1 and Proposition 6.3 forthe two types of connected components: the components for non-2-torsion linebundles contribute the homology of k [ C ] × while the components for 2-torsion linebundles contribute groups related to a normalizer of k [ C ] × . Using these explicitdescriptions, we can also completely describe the morphisms on parabolic homologyinduced from any quasi-finite morphism of affine curves f : D → C .6.1. Recollection on equivariant homology.
We shortly recall equivariant ho-mology and the Borel isotropy spectral sequence for computing it, cf. [Bro94,Section VII] or [Knu01, Appendix A].A G -complex is a CW-complex X together with an action G × X → X whichis cellular. If X is a G -complex, then there is an action of G on the cellular chaincomplex C • ( X ), and the corresponding homology groupsH G • ( X, M ) := H • ( G, C • ( X ) ⊗ Z [ G ] M )are called equivariant homology groups of ( G, X ). If X is contractible (or even justacyclic), then the equivariant homology of X is group homology, i.e. H G • ( X, M ) ∼ =H • ( G, M ).For a G -complex X , the G -equivariant homology of X can be computed bymeans of the following spectral sequence, which arises from the stupid filtration ofC • ( X ) ⊗ Z [ G ] M : E p,q = M σ ∈ Σ p H q ( G σ , M σ ) ⇒ H Gp + q ( X, M ) . In the above, Σ p denotes a set of representatives for the G -action on C p ( X ). Themodule M σ is the coefficient G -module M twisted by the orientation character χ σ : G σ → Z / G σ . In the special case where each cellstabilizer fixes the cell pointwise, the orientation character is trivial. This canalways be achieved by a suitable subdivision of cells. The first differential is inducedfrom the boundary map of the complex X and inclusions of stabilizers, cf. [Bro94,VII.8]: d | H q ( G σ ,M σ ) : H q ( G σ , M σ ) M τ ⊆ σ H q ( G τ , M τ )In the following we consider the groups (P)GL ( k [ C ]) resp. (P)SL ( k [ C ]) actingon the cubical complex P C .6.2. Components of torus type.
We first evaluate the equivariant homology ofthe components corresponding to line bundles L with L 6∼ = L − . These componentsare called “of torus type” for two reasons. On the one hand, the quotient has thehomotopy type of a torus. On the other hand, the equivariant homology is mainlyinfluenced by the intersection of GL ( k [ C ]) with a maximal torus in GL ( K ) s . Proposition 6.1.
Let k be an algebraically closed field, let C be a smooth projectivecurve and denote C = C \ { P , . . . , P s } . Let L be a line bundle on C such that L| C = L| − C and denote by P C ( L ) the connected component of P C correspondingto L .(1) The T -equivariant inclusion Z s → P C from Proposition 5.7 induces iso-morphisms on homology H GL ( k [ C ]) • ( P C ( L ) , Z ) ∼ = H • (( k [ C ] × ) , Z )H PGL ( k [ C ]) • ( P C ( L ) , Z ) ∼ = H • ( k [ C ] × , Z ) ARABOLIC SUBCOMPLEXES AND HOMOLOGY OF SL (2) The ST -equivariant inclusion Z s → P C from Proposition 5.7 induces anisomorphism on homology H (P)SL ( k [ C ]) • ( P C ( L ) , Z ) ∼ = H • ( k [ C ] × , Z ) . Proof.
We give the argument for SL , the argument for GL is similar. Recall fromProposition 5.7 that we have an equivariant map Z s → P C ( L ). From the proofof Proposition 5.7, we see that the homomorphism τ : T → SL ( k [ C ]) extends toa homomorphism ˜ τ : k [ C ] × → SL ( k [ C ]). Using the stabilizer computations ofProposition 5.7, the map Z s → P C ( L ) is still equivariant with respect to ˜ τ , wherethe action of k [ C ] × on Z s is the one factoring through k [ C ] × → k [ C ] × /k × ∼ = T .The equivariant map above induces a morphism of isotropy spectral sequencescomputing the induced morphism on homologyH k [ C ] × • ( Z s , Z ) → H SL ( k [ C ]) • ( P C ( L ) , Z ) . By Proposition 5.7, the map Z s → P C ( L ) induces an isomorphism of quotient cellcomplexes. Any cell of Z s is stabilized by k × , and the cells of P C ( L ) are stabilizedby groups of the form k n ⋊ k × where the size of the unipotent group depends on thecell. Note that the action of k × on k n in the above semi-direct product is inducedfrom an embedding into a Borel subgroup of GL ( K ). For any cell σ in Z s , thecomposition k × ∼ = Stab( σ ; k [ C ] × ) → Stab( σ ; SL ( k [ C ])) ∼ = k n ⋊ k × p −→ k × is the identity. By [Knu01, Theorem 4.6.2], the inclusionStab( σ ; k [ C ] × ) → Stab( σ ; SL ( k [ C ]))induces an isomorphism in group homology. The morphism of equivariant spectralsequences is then an isomorphism on the E -page, and hence induces an isomor-phism on equivariant homology.The claim now follows easily, we simply compute H k [ C ] × • ( Z s , Z ): the model com-plex Z s is contractible, and the action of k [ C ] × factors as a free action of k [ C ] × /k × and a trivial action of k × . In particular, we have an isomorphismH k [ C ] × • ( Z s , Z ) ∼ = H k × • (( k [ C ] × /k × ) \Z s , Z ) . Note that the homology of the quotient X = ( k [ C ] × /k × ) \Z s is the homology of thefree abelian group k [ C ] × /k × , in particular it is torsion-free. The k × -equivarianthomology of X is defined as the homology of the Borel construction k × \ ( Ek × × X ).Since k × acts trivially on X , this quotient can be identified as Bk × × X . Inparticular, applying the K¨unneth formula, we see that the k × -equivariant homologyof X is the homology of the group k [ C ] × . (cid:3) Components of normalizer type, case SL . We next study the homologyof the components corresponding to 2-torsion line bundles. These componentsare called “of normalizer type” since the quotient is (away from 2) a classifyingspace for the intersection of SL ( k [ C ]) with the normalizer of a maximal torus inSL ( k ( C )). The argument in this case is more involved since there are points whichhave stabilizer SL ( k ), which is too large to just come from the normalizer of amaximal torus.We recall from Section 5 what we know about the global structure of thosenormalizer type components. We fix a line bundle L with L ∼ = L − and denoteby P C ( L ) the connected component of the parabolic subcomplex P C containingthe bundle L ⊕ L . Combining Lemma 5.6 and Proposition 5.8, there is a map φ L : Z s → A L ⊂ P C ( L ) and a homomorphism ν : SN → SL ( k [ C ]) such that φ L is ν -equivariant and induces an isomorphism of quotient complexes Z s / SN ∼ =SL ( k [ C ]) \ P C ( L ). From (1) in the proof of Proposition 5.8, we see that ν : SN → SL ( k [ C ]) can be extended to a group homomorphism ˜ ν : ˜ SN → SL ( k [ C ]), where˜ SN is the group of determinant 1 monomial (2 × k [ C ],i.e., ˜ SN = (cid:26)(cid:18) u − u − (cid:19) | u ∈ k [ C ] × (cid:27) ∪ (cid:26)(cid:18) u u − (cid:19) | u ∈ k [ C ] × (cid:27) . The group ˜ SN sits in an extension1 → k × → ˜ SN → SN → , the homomorphism ˜ ν is the natural embedding followed by conjugation as in (1) ofProposition 5.8, and ν is the composition of ˜ ν with a choice of section SN → ˜ SN .Of course, the subgroup k × above acts trivially on Z s and hence does not contributeto the quotient; its relevance for the isotropy spectral sequence comes from thecontribution to the stabilizer subgroups. Lemma 6.2.
Consider the above action of ˜ SN on Z s . With Z [1 / -coefficients,the isotropy spectral sequence for the ˜ SN -complex Z s degenerates at the E -pageand produces an isomorphism H ˜ SN• ( Z s , Z [1 / ∼ = H • ( ˜ SN , Z [1 / . Proof. (1) We first describe the stabilizer subgroups to infer some information onthe page E p,q = L σ ∈ Σ p H q ( ˜ SN σ , Z [1 / k × (em-bedded as constant diagonal matrices) acts trivially, hence every stabilizer containsat least a copy of k × . The group SN sits in an extension0 → ST → SN → k [ C ] × / ( k [ C ] × ) → . The subgroup ST acts via translations on Z s , hence acts without fixed points.The square classes in the quotient act as point inversions with suitable centers, asdescribed in Definition 5.4. Therefore, the vertices ( a , . . . , a s ) with P a i [ P i ] = 0have stabilizers k × ⋊ Z / Z / k × . All othervertices have k × as stabilizer. Because the action of the Z / Z s is viapoint inversions, the higher simplices also all have k × as stabilizer.(2) Note that the identity on Z s is equivariant for the inclusion ˜ ST ֒ → ˜ SN ,where ˜ ST is the group of determinant diagonal (2 × k [ C ]. The inclusion therefore induces a morphism of isotropy spectral sequencesconverging to the morphismH ˜ ST• ( Z s , Z [1 / → H ˜ SN• ( Z s , Z [1 / . The first group is the one discussed in the torus case above, and the isotropyspectral sequence for ˜ ST -equivariant homology of Z s degenerates at the E -page.In particular, the restrictions of the differentials to the homology of the stabilizersin ˜ ST is trivial.(3) Now we note that the quotient map Z s → Z s / ˜ ST is equivariant for thequotient homomorphism ˜ SN → ˜ SN / ˜ ST ∼ = k [ C ] × / ( k [ C ] × ) . We consider the k [ C ] × / ( k [ C ] × ) -equivariant homology of the quotient Z s / ˜ ST . Ob-viously, the stabilizers in this case are only 2-torsion. Therefore, the only non-trivialterms of the spectral sequence for k [ C ] × / ( k [ C ] × ) -equivariant homology of Z s / ˜ ST with Z [1 / -coefficients are concentrated in the q = 0-column. Hence the spectralsequence degenerates at the E -term, and all further differentials are trivial.(4) Combining points (2) and (3) above, we find that all differentials d r , r ≥ SN -equivariant homology of Z s haveto be trivial: the quotient map from (3) induces an isomorphism on the q = 0 ARABOLIC SUBCOMPLEXES AND HOMOLOGY OF SL part, and everything else is detected on the subgroup ˜ ST from (2). The claimfollows. (cid:3) Proposition 6.3.
Let k be an algebraically closed field, let C be a smooth projectivecurve and denote C = C \ { P , . . . , P s } . Let L be a line bundle on C which is -torsion in the Picard group of C , and denote by P C ( L ) the corresponding connectedcomponent of the parabolic subcomplex, cf. Proposition 5.8.Denote ˜ ν : ˜ SN → SL ( k [ C ]) the group homomorphism discussed above. The ˜ ν -equivariant map φ L : Z s → P C ( L ) induces a homomorphism H • ( ˜ SN , Z [1 / ∼ = H ˜ SN• ( Z s , Z [1 / → H SL ( k [ C ]) • ( P C ( L ) , Z [1 / such that there is a long exact sequence of Z [1 / -modules · · · → RP i +1 ( k ) ⊗ Z Z [1 / , k [ C ] × / ( k [ C ] × ) ] → H i ( ˜ SN , Z [1 / →→ H SL ( k [ C ]) i ( P C ( L ) , Z [1 / → RP i ( k ) ⊗ Z Z [1 / , k [ C ] × / ( k [ C ] × ) ] → · · · where the groups RP • ( k ) are the refined scissors configurations groups recalled inSection A.Proof. (1) As before, the ˜ ν -equivariance of the map φ L : Z s → P C ( L ) followsfrom Proposition 5.8 together with the remarks before Lemma 6.2. This impliesthe existence of an induced morphism on homologyH ˜ SN• ( Z s , Z [1 / → H SL ( k [ C ]) • ( P C ( L ) , Z [1 / . The identification of the source with group homology of ˜ SN follows from Lemma 6.2.The ˜ ν -equivariant map Z s ֒ → P C ( L ) also induces a morphism of the correspond-ing isotropy spectral sequences. To prove the remaining claims, we investigate thismorphism of isotropy spectral sequences, where we use the following indexing forthe isotropy spectral sequence: E p,q = M σ ∈ Σ p H q ( G σ , M σ ) ⇒ H Gp + q ( X, M )with differentials going d rp,q : E rp,q → E rp − r,q + r − . We denote E ip,q ( ˜ SN ) the ( p, q )-entry of the i -th page of the spectral sequence for ˜ SN -equivariant homology of Z s ,and similarly for E ip,q (SL ( k [ C ])).(2) From Proposition 5.8, the map Z s → P C ( L ) induces an isomorphism ofquotient cell complexes Z s / ˜ SN ∼ = SL ( k [ C ]) \ P C ( L ). In particular, the inducedmorphism E p, ( ˜ SN ) → E p, (SL ( k [ C ])) is an isomorphism of complexes and henceinduces an isomorphism on the ( q = 0)-column of the E -pages.(3) Next, we look at the stabilizers. For the ˜ SN -action on Z s , we discussed inthe proof of Lemma 6.2 that there are 2 rk ST vertices stabilized by a semi-directproduct k × ⋊ Z /
2, where the quotient is generated by the point inversion. For theaction of SL ( k [ C ]) on P C ( L ), there are 2 rk T vertices stabilized by SL ( k ). Themap Z s → P C ( L ) sends the former to the latter types of points, and the inducedmorphism on stabilizers is the inclusion of k × ⋊ Z / ( k ).All the other cells in Z s have stabilizer k × in ˜ SN . By Proposition 5.8, all othercells in P C ( L ) have stabilizer some semi-direct product k n ⋊ k × with the unipotentgroup k n depending on the cell. As in the torus case, the composition k × ∼ = ˜ SN σ ι −→ (SL ( k [ C ])) σ ∼ = k n ⋊ k × p −→ k × is the identity for any such cell σ in Z s . By [Knu01, Theorem 4.6.2], the inclusion˜ SN σ ֒ → (SL ( k [ C ])) σ induces an isomorphism in group homology. In particular,the map Z s → P C ( L ) induces an isomorphism on the ( p ≥ E -terms: the homology of the stabilizers changes only for the inclusion k × ⋊ Z / ֒ → SL ( k )of the normalizer of a maximal torus and this happens for special 0-cells only.(4) From (3) above, we see that the differentials d p ≥ , • agree and only the differ-ential d , • : L σ ∈ Σ H • ( G σ ) → L σ ∈ Σ H • ( G σ ) differs for the two spectral sequencesassociated to ˜ SN \Z s and SL ( k [ C ]) \ P C ( L ). Note that even this differential is thecomposition of the differential d , • ( ˜ SN ) with the map H • (N( k )) → H • (SL ( k )). Inparticular, the induced morphisms E p,q ( ˜ SN ) → E p,q (SL ( k [ C ])) are isomorphismsfor all p ≥ · · · → RP i +1 ( k ) ⊗ Z Z [1 / , k [ C ] × / ( k [ C ] × ) ] → E ,i ( ˜ SN ) →→ E ,i (SL ( k [ C ])) → RP i ( k ) ⊗ Z Z [1 / , k [ C ] × / ( k [ C ] × ) ] → · · · (5) In the next step, we want to establish an exact sequence describing themorphism of E -terms. First note that the map E ,i (SL ( k [ C ])) → RP i ( k ) ⊗ Z Z [1 / , k [ C ] × / ( k [ C ] × ) ]induces a natural map E ,i (SL ( k [ C ])) → RP i ( k ) ⊗ Z Z [1 / , k [ C ] × / ( k [ C ] × ) ]because the differential d : E ,i → E ,i (SL ( k [ C ])) factors through E ,i ( ˜ SN ).Moreover, because the group E ,i is a quotient of E ,i , these maps have the sameimage in RP i ( k ) ⊗ Z Z [1 / , k [ C ] × / ( k [ C ] × ) ]. The elements in E ,i ( ˜ SN ) map-ping to 0 in E ,i ( ˜ SN ) come from E ,i ( ˜ SN ) ∼ = E ,i (SL ( k [ C ])). The elementsin ker( E ,i ( ˜ SN ) → E ,i ( ˜ SN )) can then be identified exactly with the elements ofcoker (cid:16) E ,i ( ˜ SN ) → E ,i (SL ( k [ C ])) (cid:17) , whence we get an exact sequence0 → coker (cid:16) E ,i ( ˜ SN ) → E ,i (SL ( k [ C ])) (cid:17) → E ,i ( ˜ SN ) → E ,i ( ˜ SN ) → . There is a similar exact sequence0 → coker (cid:16) E ,i ( ˜ SN ) → E ,i (SL ( k [ C ])) (cid:17) → ker (cid:16) E ,i ( ˜ SN ) → E ,i (SL ( k [ C ])) (cid:17) → ker (cid:16) E ,i ( ˜ SN ) → E ,i (SL ( k [ C ])) (cid:17) → . The exact sequence of (4) then gives rise to an exact sequence · · · → RP i +1 ( k ) ⊗ Z Z [1 / , k [ C ] × / ( k [ C ] × ) ] → ker (cid:16) E ,i ( ˜ SN ) → E ,i (SL ( k [ C ])) (cid:17) → coker (cid:16) E ,i ( ˜ SN ) → E ,i (SL ( k [ C ])) (cid:17) → RP i ( k ) ⊗ Z Z [1 / , k [ C ] × / ( k [ C ] × ) ] → · · · and the second term is also completely decomposed into stuff coming from thecomparison of E -terms, using the exact sequence just above.Finally, we investigate the map E ,i ( ˜ SN ) → E ,i (SL ( k [ C ])). Let σ ∈ E ,i ( ˜ SN )be such that it becomes trivial in E ,i (SL ( k [ C ])). There is a representative ˜ σ ∈ E ,i ( ˜ SN ) with d ˜ σ = 0. Since the maps E p,i ( ˜ SN ) → E p,i (SL ( k [ C ])) are isomor-phisms for any p ≥ i , if the image of σ in E ,i (SL ( k [ C ])) is trivial, thenalready σ = 0. We see that the map E ,i ( ˜ SN ) → E ,i (SL ( k [ C ])) is injective. ARABOLIC SUBCOMPLEXES AND HOMOLOGY OF SL Recall again that all the comparison morphisms E p,i ( ˜ SN ) → E p,i (SL ( k [ C ]))are isomorphisms for all i and p ≥
1. In particular, the comparison maps on the E terms discussed above (dealing with kernels and cokernels of the maps induced on E ,i and E ,i ) are the only ones which have a possibly non-trivial kernel or cokernel.In particular, the long exact sequence above gives a complete comparison statementfor the E -terms of the two spectral sequences.(6) By Lemma 6.2, the spectral sequence for SL ( k [ C ])-equivariant homology of P ( L ) also has to degenerate at the E -term, because all the differentials (exceptthe d ,i : E ,i → E ,i discussed above) come from the spectral sequence for ˜ SN .In particular, E ∞ p,q (SL ( k [ C ])) = E p,q (SL ( k [ C ])) and E ∞ p,q (SL ( k [ C ])) ∼ = E ∞ p,q ( ˜ SN )for p ≥ E -terms to an exact se-quence of the homology groups proper. Obviously, the equivariant map of themodel complex into the parabolic component induces a morphism of homologygroups H • ( ˜ SN , Z [1 / → H SL ( k [ C ]) • ( P C ( L ) , Z [1 / (cid:16) H i ( ˜ SN , Z [1 / → H SL ( k [ C ]) i ( P C ( L ) , Z [1 / (cid:17) ∼ = ker (cid:16) E ,i ( ˜ SN ) → E ,i (SL ( k [ C ])) (cid:17) . Using this identification, there is a natural map RP i +1 ( k )[ k [ C ] × / ( k [ C ] × ) ] → ker (cid:16) H i ( ˜ SN , Z [1 / → H SL ( k [ C ]) i ( P C ( L ) , Z [1 / (cid:17) coming from the composition RP i +1 ( k )[ k [ C ] × / ( k [ C ] × ) ] → E ,i ( ˜ SN ) → E ,i ( ˜ SN ) , cf. step (5) above.From the analysis of the complex P C ( L ) we see that its equivariant homologyis detected by the maximal torus and the subgroups SL ( k ) sitting at the specialvertices. In particular, there is a natural mapH • (SL ( k ) , Z [1 / k [ C ] × / ( k [ C ] × ) ] → RP • ( k )[ k [ C ] × / ( k [ C ] × ) ]and the analysis of the spectral sequence shows that this map extends to the re-quired natural mapH SL ( k [ C ]) i ( P C ( L ) , Z [1 / → RP i ( k )[ k [ C ] × / ( k [ C ] × ) ]because all the additional relations come from homology of N( k ) and then aresatisfied in the target as well.Using step (5) again, we can describe the cokernel of the comparison map via anextension 0 → coker (cid:16) E ,i ( ˜ SN ) → E ,i (SL ( k [ C ])) (cid:17) →→ coker (cid:16) H i ( ˜ SN , Z [1 / → H SL ( k [ C ]) i ( P C ( L ) , Z [1 / (cid:17) →→ coker (cid:16) E ,i − ( ˜ SN ) → E ,i − (SL ( k [ C ])) (cid:17) → . Also from step (5), we know that there is a natural map RP i ( k )[ k [ C ] × / ( k [ C ] × ) ] → E ,i ( ˜ SN ) whose kernel also sits in an exact sequence with the same outer termsas above. The natural map induces isomorphisms on the graded pieces, by thecomputations in step (5). This implies the exactness claim. (cid:3) Components of normalizer type, case
PGL . Now we state the corre-sponding results on the homology of the normalizer type components for the caseof the group (P)GL . Proposition 6.4.
Let k be an algebraically closed field, let C be a smooth projectivecurve and denote C = C \ { P , . . . , P s } . Let L be a line bundle on C which is -torsion in the Picard group of C , and denote by P C ( L ) the corresponding connectedcomponent of the parabolic subcomplex, cf. Proposition 5.8.Denote ˜ ν : ˜ N →
PGL ( k [ C ]) the analogue of the group homomorphism discussedabove. The ˜ ν -equivariant map φ L : Z s → P C ( L ) induces a homomorphism H • ( ˜ N , Z [1 / ∼ = H ˜ N• ( Z s , Z [1 / → H PGL ( k [ C ]) • ( P C ( L ) , Z [1 / such that there is a long exact sequence of Z [1 / -modules · · · → P i +1 ( k ) → H i ( ˜ N , Z [1 / → H PGL ( k [ C ]) i ( P C ( L ) , Z [1 / → P i ( k ) → · · · The arguments are similar to the ones used for SL above, and we omit a detailedproof: the proof first proceeds through an analogue of Lemma 6.2 for the group ˜ N ,and then mainly uses the PGL -part of Proposition 5.8 in much the same way ashappened in the proof of Proposition 6.3.The groups P i ( k ) appearing above are the generalized scissors congruence groupsof [Dup01], cf. also Section A. Using the long exact sequence connecting homology ofthe normalizer N( k ), homology of SL ( k ) and the scissors congruence groups P i ( k ),we can simplify the above result further and describe the parabolic homology asthe following pushout, all with Z [1 / PGL ( k [ C ]) • ( P C ( L )) ∼ = H • ( ˜ N ) ⊕ H • (N( k )) H • (PGL ( k )) . Functoriality.
We have described the SL ( k [ C ])-equivariant homology of theparabolic subcomplex P C above. We now want to describe the behaviour of para-bolic homology under morphisms of curves. In particular we want to compute thecolimit of these homologies to SL ( k ( C )) and SL ( k ( C )). Proposition 6.5.
Let k be an algebraically closed field. Let f : D → C be a finitemorphism of smooth projective curves over k , let P , . . . , P s be points on C , and let Q , . . . , Q t be points on D not in the preimage of the P i . Set C = C \ { P , . . . , P s } and D = D \ ( { f − ( { P , . . . , P s } ) ∪ { Q , . . . , Q t } ) . Denote by P C and P D theparabolic subcomplexes in X C and X D , respectively. Then we have the followingassertions for the morphism f ∗ : H SL ( k [ C ]) • ( P C ) → H SL ( k [ D ]) • ( P D ) induced from the morphism of Proposition 3.12, where we use Z [1 / -coefficientsthroughout:(1) The composition K ( C ) → π (SL ( k [ C ]) \ P C ) π ( f ∗ ) −→ π (SL ( k [ D ]) \ P D ) → K ( D ) is induced from pullback of line bundles f ∗ : Pic( C ) → Pic( D ) , where thefirst and last map are the bijections of Lemma 5.2.(2) For each line bundle L on C , there is an induced map f ∗L : H SL ( k [ C ]) • ( P C ( L )) → H SL ( k [ D ]) • ( P D ( f ∗ ( L ))) . The map f ∗ is the direct sum of these.(3) If L is not -torsion in Pic( D ) , then f ∗L is identified via the isomorphismsof Proposition 6.1 with the natural map H • ( k [ C ] × ) → H • ( k [ D ] × ) . ARABOLIC SUBCOMPLEXES AND HOMOLOGY OF SL (4) If L is not -torsion in Pic( C ) but becomes -torsion in Pic( D ) , then f ∗L isidentified with the composition H • ( k [ C ] × ) → H • ( ˜ SN ) → H SL ( k [ C ]) • ( P D ( f ∗ L )) , where the first map is the inclusion of diagonal matrices into monomialmatrices, and the second map is the morphism from Proposition 6.3.(5) If L is -torsion in Pic( C ) , then f ∗L sits in a commutative ladder connectingthe respective long exact sequences of Proposition 6.3 for P C and P D . Theother two types of maps of the commutative ladder, H • ( ˜ SN C ) → H • ( ˜ SN D ) and RP • ( k ) ⊗ Z Z [1 / , k [ C ] × / ( k [ C ] × ) ] → RP • ( k ) ⊗ Z Z [1 / , k [ D ] × / ( k [ D ] × ) ] are induced by the natural map k [ C ] × → k [ D ] × .Proof. The result is a direct consequence of Proposition 5.9 and the computationsin Proposition 6.1 and Proposition 6.3. (cid:3)
Obviously, there is a similar functoriality result for the case PGL where againall relevant maps are induced from the natural one f ∗ : k [ C ] × → k [ D ] × . Definition 6.6.
Let k be an algebraically closed field, let C be a smooth projectivecurve over k , and set C = C \ { P , . . . , P s } . We define the parabolic homology of SL ( k ( C )) to be b H • (SL ( k ( C )) , Z /ℓ ) = colim S ⊆ C ( k ) H SL ( k [ C \ S ]) • ( P C \ S , Z /ℓ ) , where the colimit is taken over all finite sets S of closed points of C , ordered byinclusion. As a direct consequence of Proposition 6.5, we can describe the parabolic ho-mology for function fields of curves.
Proposition 6.7.
Let k be an algebraically closed field, let C be a smooth curveover k and let ℓ be an odd prime different from the characteristic of k . Then wehave the following exact sequence · · · → RP i +1 ( k, Z /ℓ )[ k ( C ) × / ( k ( C ) × ) ] → H i (N( k ( C )) , Z /ℓ ) →→ b H i (SL ( k ( C )) , Z /ℓ ) → RP i ( k, Z /ℓ )[ k ( C ) × / ( k ( C ) × ) ] → · · · , where N( k ( C )) denotes the normalizer of a maximal torus in SL ( k ( C )) . Low-dimensional example calculations.
We discuss low-dimensional spe-cial cases of the above computations in Proposition 6.1 and Proposition 6.3. Inlow degrees, we can say more about the precise relation between the homology ofSL ( k ) and the homology of the normalizer of the maximal torus N( k ).We start with the degree 1 case: Corollary 6.8.
Let k be an algebraically closed field, let C be a smooth projectivecurve and denote C = C \ { P , . . . , P s } . There is an isomorphism H SL ( k [ C ])1 ( P C , Z [1 / ∼ = M [ L ] ∈K ( C ) , L6 =0 k [ C ] × ⊗ Z Z [1 / . In the limit, we have b H (SL ( k ( C )) , Z [1 / .Proof. By Proposition 6.1, if L is a line bundle on C which is not 2-torsion inPic( C ), then H SL ( k [ C ])1 ( P C ( L ) , Z [1 / ∼ = k [ C ] × ⊗ Z Z [1 / . Now let L be a line bundle on C which is 2-torsion in Pic( C ). We investigatethe components of the exact sequence computing H SL ( k [ C ])1 ( P C ( L ) , Z [1 / SN is an extension of the free abelian group ˜ ST by a 2-torsion group G .Obviously, H ( G, H ( ˜ ST , Z [1 / ( G, H ( ˜ ST , Z [1 / ∼ = H ( G, ˜ ST ⊗ Z Z [1 / G acts by inversion followed by multiplication with someelement, whence the coinvariants are 2-torsion. The Hochschild-Serre spectral se-quence for the extension then implies H ( ˜ SN , Z [1 / RP ( k ) = 0, cf. Section A, and therefore RP ( k ) ⊗ Z Z [1 / , k ( C ) × / ( k ( C ) × ) ] = 0 . The exact sequence of Proposition 6.3 then implies H SL ( k [ C ])1 ( P C ( L ) , Z [1 / (cid:3) Corollary 6.9.
Let k be an algebraically closed field, let C be a smooth projectivecurve and denote C = C \ { P , . . . , P s } . There is an exact sequence → H ( ˜ SN , Z [1 / → H SL ( k [ C ])3 ( P C ( L ) , Z [1 / →→ RP ( k ) ⊗ Z Z [1 / , k [ C ] × / ( k [ C ] × ) ] →→ H ( ˜ SN , Z [1 / → H SL ( k [ C ])2 ( P C ( L ) , Z [1 / → . The maps are the ones appearing in the Bloch-Wigner exact sequence, cf. [Hut11b] . Appendix A. Refined scissors congruence groups
The following section provides a recollection on the exact sequence relating ho-mology of SL ( k ), homology of the normalizer of a maximal torus N( k ) and suitablegeneralizations of pre-Bloch groups RP • ( k ). The material is well-known from thework of Bloch, Suslin, Dupont, Sah, Hutchinson and others, cf. [Sus90], [Dup01],[Hut11b]. As the statements are somehow scattered over the literature and not usu-ally stated in the generality needed, we provide a detailed review of the definitionsof the refined scissors congruence groups (or point configuration groups) RP • ( k )and the proof of the relevant exact sequences for homology of SL ( k ).A.1. Points on the projective line.
Let F be an infinite field. Consider thestandard action of the general linear group GL ( F ) on the F -points of the projectiveline P ( F ) given by (cid:18) a bc d (cid:19) · z az + bcz + d . As the center obviously acts trivially, the group action factors through PGL ( F ).Restriction to determinant 1 provides the standard group actions of SL ( F ) andPSL ( F ) on P ( F ). Definition A.1.
Let F be an infinite field.(1) Denote by C • ( F ) the complex of points on P , which in degree n has thefree abelian group C n ( F ) generated by ( n + 1) -tuples ( x , . . . , x n ) of distinctpoints on P ( F ) .(2) Denote by C alt • ( F ) the alternating complex of points on P , where C alt n ( F ) is the free abelian group generated by ( n + 1) -tuples ( x , . . . , x n ) of pointson P ( F ) modulo the identifications ( x π (0) , . . . , x π ( n ) ) = sgn π · ( x , . . . , x n ) , where π is any permutation of the set of indices { , . . . , n } . ARABOLIC SUBCOMPLEXES AND HOMOLOGY OF SL In both the above cases, the differential is given by the obvious d ( x , . . . , x n ) = n X i =0 ( − i ( x , . . . , b x i , . . . , x n ) . The complexes are augmented by mapping C ( F ) = C alt0 ( F ) → Z : ( x ) . We recall the well-known acyclicity lemma, cf. [Knu01, Lemma 2.3.2], [Dup01,Lemma 3.6] or any of a huge number of further possible sources.
Lemma A.2.
The augmented complexes C (alt) • ( F ) → Z → of points on P areacyclic.Proof. Let P Nj =1 n j ( x j, , . . . , x j,q ) be a q -cycle in C • ( F ). Since F is infinite, thereexists a point x ∈ P ( F ) different from any of the points x j,q . Then d N X j =1 n j ( x, x j, , . . . , x j,q ) = N X j =1 n j ( x j, , . . . , x j,q ) . For the alternating complexes, one can even choose a global base-point x ∈ P and use the global contraction s q : C alt q ( F ) → C alt q +1 ( F ) : ( x , . . . , x q ) ( x, x , . . . , x q ) , s − (1) = ( x ) . (cid:3) Contractibility implies that the SL ( F )-equivariant homology of the complexesof points on P can indeed be identified with group homology of SL ( F ). Corollary A.3.
Let F be an infinite field, and let Γ be any one of the groups (P)GL ( F ) or (P)SL ( F ) . Then the augmentation induces isomorphisms H • (Γ , C (alt) • ( F )) ∼ = −→ H • (Γ , Z ) . The hyperhomology spectral sequence associated to H • (Γ , C (alt) • ) provides therelation between homology of Γ, the normalizer of a maximal torus in Γ and therefined scissors congruence groups.There is a notion of decomposability of point configurations in projective n -space,cf. [Dup01, p. 126]. Using Z [1 / x, x ) = − ( x, x )implies that ( x, x ) is 2-torsion, we can simplify the notion of decomposability forour purposes: for the projective line, an ( n + 1)-tuple of points ( x , . . . , x n ) isdecomposable if and only if its support contains at most two points. In particu-lar, a non-trivial element in C (alt) q ( F ) is decomposable if and only if q ≤
1. The decomposable subcomplex of the complex of points is defined as follows: F C (alt) q ( F ) = (cid:26) C (alt) q ( F ) q ≤
10 otherwiseIt is obviously stable under the action of (P)GL ( F ). Definition A.4.
Let F be an infinite field. The refined scissors congruence groups (or refined point configuration groups ) are defined as RP • ( F ) := H • (SL ( F ) , C alt • ( F ) /F C alt • ( F )) The scissors congruence groups (or point configuration groups ) are defined as P • ( F ) := H • (PGL ( F ) , C alt • ( F ) /F C alt • ( F )) Remark A.5.
For considerations with finite coefficients, we can define similargroups RP • ( F, Z /ℓ ) := H • (SL ( F ) , C alt • ( F ) /F C alt • ( F ) ⊗ Z Z /ℓ ) Remark A.6.
The notation P q ( F ) is taken from [Dup01] . The notation RP q ( F ) for the refined scissors congruence groups is a combination of the notation from [Dup01] and [Hut11b] . In [Hut11b] , only the groups RP ( F ) are considered (andsimply denoted RP ( F ) ). Proposition A.7. If F is a quadratically closed field, the natural map SL ( F ) → PGL ( F ) induces isomorphisms RP • ( F ) ⊗ Z Z [1 / ∼ = −→ P • ( F ) ⊗ Z Z [1 / . Proof.
The natural map factors as SL ( F ) → PSL ( F ) → PGL ( F ), and we willshow that any of these homomorphisms induces an isomorphism on equivarianthomology with coefficients in C alt • ( F ).If F is quadratically closed, then PSL ( F ) and PGL ( F ) are in fact the samegroups, because any matrix M ∈ PGL ( F ) can be scaled, using the diagonal matrixdiag(det M , det M ), to a matrix in PSL ( F ). Therefore, the second homomor-phism obviously induces an isomorphism on homology.The homomorphism SL ( F ) → PSL ( F ) is surjective and has kernel {± I } ∼ = Z / Z . Since the kernel acts in fact trivially on the complexes C alt • ( F ), the projectionhence induces an isomorphism on homology with Z [1 / (cid:3) A.2.
The hyperhomology spectral sequence.
The hyperhomology spectral se-quence for the action of SL ( F ) on the complexes C (alt) • ( F ) provides some insightsinto the structure of the homology of SL ( F ), which we discuss next.We first recall a suitable resolution of the complex F C alt • ( F ) from [Dup01,Proposition 13.22]. The complex D • ( F ) is the following complex D • ( F ) = M x ∈ P ( F ) (cid:0) Z [ P ( F ) \ { x } ] → Z (cid:1) , with the first term sitting in degree 1 and the differential given by the minus theaugmentation y
7→ −
1. This is a shifted version of the augmented complex, hencethe additional sign. The complex E • ( F ) is defined to be E • ( F ) := M x,y ∈ P ( F ) ,x = y Z concentrated in degree 1. Note that the index set is the set of unordered pairs ofdistinct elements in P ( F ). Lemma A.8.
There is a map of complexes F C alt • ( F ) → D • ( F ) given in degree by ( x ) x and in degree by ( x, y ) ( y ) x − ( x ) y .There is a map of complexes D • ( F ) → E • ( F ) given in degree by the -mapand in degree by ( x ) y ( x,y ) .With these maps, there is an exact sequence of complexes → F C alt • ( F ) ⊗ Z Z [1 / → D • ( F ) ⊗ Z Z [1 / → E • ( F ) ⊗ Z Z [1 / → . Proof.
The first claim is simply that ( x, y ) ( y ) x − ( x ) y
7→ − x + 1 y and ( x, y ) ( y ) − ( x ) y − x are the same map.The second claim is trivially true since E • ( F ) is concentrated in degree 1.We discuss injectivity of the first map: in degree 0, the map is ( x ) x which isobviously injective. In degree 1, the map is ( x, y ) ( y ) x − ( x ) y . If x = y , then theimage is zero, but the element ( x, x ) = − ( x, x ) is 2-torsion, hence also 0. Injectivityin degree 1 follows from this. ARABOLIC SUBCOMPLEXES AND HOMOLOGY OF SL Surjectivity of the second map is clear, since 1 ( x,y ) is in the image of ( x ) y .It remains to see exactness in the middle. In degree 0, exactness in the middleis clear: the last term is 0 and the first map ( x ) x is an isomorphism. In degree1, the composition is obviously 0, since ( x, y ) ( x,y ) − ( y,x ) . For an element c in D • ( F ), the ( x, y )-component of its image in E • ( F ) is trivial if the elements ( x ) y and( y ) x occur with opposite multiplicities. This implies exactness in the middle. (cid:3) We now can identify the homology of the complex F C alt • ( F ) with the homologyof the normalizer, as in [Dup01, Proposition 13.22]. Lemma A.9.
Let F be an infinite field.(1) The complex D • ( F ) is acyclic.(2) There is an induced isomorphism H • (SL ( F ) , F C alt ( F ) ⊗ Z Z [1 / ∼ = H • (N( F ) , Z [1 / , where N( F ) is the normalizer of a maximal torus in SL ( F ) .Proof. (1) The degree 1 part of D • ( F ) is the SL ( F )-module M x ∈ P ( F ) Z [ P ( F ) \ { x } ] ∼ = Ind SL ( F )B ( F ) Z [ P ( F ) \ {∞} ] ∼ = Ind SL ( F )T( F ) Z . In particular, there is an induced isomorphismH • (SL ( F ) , D ( F )) ∼ = H • (T( F ) , Z ) . Similarly, we can identify the degree 0 part of the complex as M x ∈ P ( F ) Z ∼ = Ind SL ( F )B ( F ) Z , which yields an induced isomorphismH • (SL ( F ) , D ( F )) ∼ = H • (B ( F ) , Z ) . Since F is infinite, we can identify the homology of the Borel subgroup B ( F )with the homology of the maximal torus, i.e., the natural projection B ( F ) → T( F ) induces an isomorphism H • (B ( F ) , Z ) ∼ = H • (T( F ) , Z ). The differential d :H • (T , Z ) → H • (T , Z ) is the identity, because the differential of the complex d is induced from the natural inclusion T( F ) → B ( F ) on the stabilizer groups.Therefore, the complex D • ( F ) is acyclic.(2) The result will be proved using the long exact sequence of homology groupsassociated to the exact sequence of complexes from Lemma A.8. From (1), it sufficesto show that the homology of E • ( F ) is identified with a shift of the homology ofthe normalizer.The action of SL ( F ) on E ( F ) = L x = y Z is transitive, we can choose { , ∞} asorbit representative. The stabilizer of the unordered pair { , ∞} is the normalizerN( k ): in addition to the diagonal matrices stabilizing both 0 and ∞ , the Weylgroup generator w = (cid:18) − (cid:19) stabilizes { , ∞} setwise. Therefore, there are induced isomorphismsH i +1 (SL ( F ) , E • ( F )) ∼ = H i (N( F ) , Z ) . (cid:3) From the complex C alt • ( F ), we now obtain the required long exact sequenceconnecting homology of SL ( F ) to the homology of N( F ) and the refined scissorscongruence groups RP • ( F ). Proposition A.10.
Let F be an infinite field.(1) There is a long exact sequence · · · → H • (N( F ) , Z [1 / → H • (SL ( F ) , Z [1 / → RP • ( F ) → · · · where N( F ) denotes the normalizer of a maximal torus in SL ( F ) .(2) There is a long exact sequence · · · → H • (N ′ ( F ) , Z [1 / → H • (PGL ( F ) , Z [1 / → P • ( F ) → · · · where N ′ ( F ) denotes the normalizer of a maximal torus in PGL ( F ) .(3) There are similar long exact sequences with Z /ℓ -coefficients, ℓ an odd prime.Proof. Consider the exact sequence of complexes0 → F C alt • ( F ) → C alt • ( F ) → C alt • ( F ) /F C alt • ( F ) → . The equivariant homology of F C alt • ( F ) is identified with the homology of thenormalizer by Lemma A.9. The equivariant homology of C alt • ( F ) is identified withthe homology of the respective group SL ( F ) or PGL ( F ) by Corollary A.3. Theidentification of equivariant homology of the quotient C alt • ( F ) /F C alt • ( F ) with re-fined scissors congruence groups is built into Definition A.4. (cid:3) Remark A.11.
Note that the groups RP • ( F ; Z /ℓ ) with finite coefficients are notsimply obtained by tensoring RP • ( F ) ⊗ Z Z /ℓ , but instead by tensoring the definingcomplex C alt • ( F ) with Z /ℓ and then computing equivariant homology. We make some remarks on the relation between the spectral sequences associatedto C • ( F ) and C alt • ( F ). Note first that there is a natural map C • ( F ) → C alt • ( F ) givenby the identity on tuples of pairwise distinct points. Both complexes are acyclic,hence their SL ( F )-equivariant homology can be identified with group homology ofSL ( F ) with constant coefficients. The map between the two complexes then pro-vides a map between spectral spectral sequences converging to H • (SL ( F ) , Z [1 / C • ( F ) is the one usually discussed, see for ex-ample [Knu01, Theorem 3.2.2], [Dup01, Theorem 8.19] or [Hut11b]. The spectralsequence for C alt • ( F ) as discussed above seems to appear only in [Dup01, Chapter15].Indexing the E -term as E p,q = H p (SL ( F ) , C q ) and working as always with Z [1 / E -terms of both spectral sequences are both concentratedin the p = 0 column as well as the two lines q = 0 and q = 1. From the computationsin [Hut11b, Section 4], we see that for the complex C • ( F ), we have E p, ∼ = H • (N( F ) , Z [1 / , using that with Z [1 / → T( F ) → N( F ) → Z / → • (N( F ) , Z [1 / ∼ = H ( Z / , H • (T( F ) , Z [1 / . We also saw above that for the complex C alt ( F ), we have E , alt p, ∼ = H • (N( F ) , Z [1 / . But from Lemma A.9, it also follows that the line E , alt p, is in fact trivial. Althoughthe spectral sequence for the complex C alt • ( F ) does not degenerate at the E -term,its differentials are the maps RP q ( F ) → H q − (N( F ) , Z [1 / C • ( F ) → C alt • ( F ), that the differentials d q − : E q − ,q → E q − q − , are surjective (with Z [1 / E ∞ -term, we thenfind E ∞ q, ∼ = E ∞ , alt q, ∼ = 0. Moreover, we can identify the RP q ( F ) groups as the entries ARABOLIC SUBCOMPLEXES AND HOMOLOGY OF SL E q ,q . The connecting map RP q ( F ) → H q − (N( F ) , Z [1 / d q : E q ,q → E qq − , . In particular, the groups RP q ( F ) can be identified (in the spectralsequence for C • ( F )) with the kernel of the two differentials d : E ,q → E ,q − and d q − : E q − ,q → E q − q − , .A.3. Low-dimensional computations.
We note some further well-known state-ments on the (refined) scissors congruence groups, cf. [Dup01] and [Hut11b].
Proposition A.12.
Let F be an infinite field.(1) RP q ( F ) = 0 for q ≤ .(2) P q ( F ) = 0 for q ≤ .(3) RP ( F ) = I( F ) , where I denotes the fundamental ideal in the Witt ring of F .Proof. By definition RP q ( F ) is the SL ( F )-equivariant homology of the quotient C alt • ( F ) /F C alt • ( F ). Since SL ( F ) acts transitively on the set of pairs of distinctpoints in P ( F ), the complex C alt • ( F ) has one copy of Z in degree 0 and one copy of Z in degree 1. But these are both degenerate configurations, so they are containedin the subcomplex F C alt • ( F ). This proves the first claim (and part of the secondclaim).(2) follows similarly, using that the action of PGL ( F ) on 3 distinct points in P ( F ) is also transitive. The result is a single copy of Z in degree 2 of the complexPGL ( F ) \ C alt • ( F ), which is killed by the differential from degree 3.(3) is a reformulation of the computations in [Hut11b, Section 4.5]. (cid:3) We want to note that the low-degree part of the long exact sequence fromProposition A.10 is the Bloch-Wigner sequence proved in [Hut11b]. In the notationof this appendix, it reads (all groups with Z [1 / → H (N( F )) → H (SL ( F )) → RP ( F ) → H (N( F )) → H (SL ( F )) → . There would be a lot more to say about the refined scissors congruence groups.We only make two final remarks. The relation of H (SL ( C )) to actual scissorscongruences in S and H is the subject of the book [Dup01]. The Friedlander-Milnor conjecture is equivalent to unique divisibility of the scissors congruencegroups P • ( F ). This is known so far only for P ( F ) by the work of Suslin, Dupontand Sah. References [AB08] P. Abramenko and K.S. Brown. Buildings. Graduate Texts in Mathematics 248.Springer, 2008.[Ati56] M.F. Atiyah. On the Krull-Schmidt theorem with applications to sheaves. Bull. Soc.Math. France 84 (1956), 307–317.[Ati57] M.F. Atiyah. Complex analytic connections in fibre bundles. Trans. Amer. Math. Soc.85 (1957), no. 1, 181–207.[Bro94] K.S. Brown. Cohomology of groups. Corrected reprint of the 1982 original. GraduateTexts in Mathematics, 87. Springer, 1994.[Dup01] J.L. Dupont. Scissors congruences, group homology and characteristic classes. NankaiTracts in Mathematics 1. World Scientific, 2001.[FM84] E.M. Friedlander and G. Mislin. Cohomology of classifying spaces of complex Lie groupsand related discrete groups. Comment. Math. Helv. 59 (1984), 347–361.[Har77] R. Hartshorne. Algebraic geometry. Graduate Texts in Mathematics 52. Springer, 1977.[Hut11a] K. Hutchinson. A refined Bloch group and the third homology of SL of a field. J. PureAppl. Algebra 217 (2013), 2003–2035.[Hut11b] K. Hutchinson. A Bloch-Wigner complex for SL . J. K-theory 12 (2013), no. 1, 15–68.[Knu01] K.P. Knudson. Homology of linear groups. Progress in Mathematics, 193. BirkhuserVerlag, Basel, 2001. [Mis94] G. Mislin. Tate cohomology for arbitrary groups via satellites. Topology Appl. 56 (1994),no. 3, 293–300.[Rah13] A.D. Rahm. The homological torsion of PSL of the imaginary quadratic integers. Trans.Amer. Math. Soc. 365 (2013), no. 3, 1603–1635.[Rah14] A.D. Rahm. Accessing the cohomology of discrete groups above their virtual cohomo-logical dimension. J. Alg. 404 (2014), no. C, 152–175.[Ros73] M. Rosen. S -units and S -class group in algebraic function fields. J. Algebra 26 (1973),98–108.[Ser80] J.-P. Serre. Trees. Springer, 1980.[Stu76] U. Stuhler. Zur Frage der endlichen Pr¨asentierbarkeit gewisser arithmetischer Gruppenim Funktionenk¨orperfall. Math. Ann. 224 (1976), no. 3, 217–232.[Stu80] U. Stuhler. Homological properties of certain arithmetic groups in the function fieldcase. Invent. Math. 57 (1980), no. 3, 263–281.[Sus90] A.A. Suslin. K of a field and the Bloch group. Trudy Mat. Inst. Steklov., 183:190-199,229, 1990. Translated in Proc. Steklov Inst. Math. 1991, no. 4, 217-239, Galois Theory,rings, algebraic groups and their applications (Russian) Matthias Wendt, Fakult¨at Mathematik, Universit¨at Duisburg-Essen, Thea-Leymann-Strasse 9, 45127, Essen, Germany
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