Homology Stability

for the Special Linear Group of a Field

and Milnor-Witt K-theory

Dedicated to Andrei Suslin

Kevin Hutchinson, Liqun Tao

Received: May 28, 2009 Revised: April 1, 2010

Abstract. Let F be a field of characteristic zero and let f_{t,n} be the stabilization homomorphism from the nth integral homology of SL_{t}(F) to the nth integral homology of SL_{t+1}(F). We prove the following results: For all n, f_{t,n} is an isomorphism if t ≥ n + 1 and is surjective for t = n, confirming a conjecture of C-H. Sah. f_{n,n} is an isomorphism when n is odd and when n is even the kernel is isomorphic to the (n + 1)st power of the fundamental ideal of the Witt Ring of F. When n is even the cokernel of f_{n−1,n} is isomorphic to the nth Milnor-Witt K-theory group of F. When n is odd, the cokernel of f_{n−1,n} is isomorphic to the square of the nth Milnor K-group of F.

2010 Mathematics Subject Classification: 19G99, 20G10 Keywords and Phrases: K-theory, special linear group, group homology

1. Introduction

Given a family of groups {G_{t}}_{t∈ℕ} with canonical homomorphisms G_{t} → G_{t+1}, we say that the family has homology stability if there exist constants K(n) such that the natural maps H_{n}(G_{t}, ℤ) → H_{n}(G_{t+1}, ℤ) are isomorphisms for t ≥ K(n). The question of homology stability for families of linear groups over a ring R - general linear groups, special linear groups, symplectic, orthogonal and unitary groups - has been studied since the 1970s in connection with applications to algebraic K-theory, algebraic topology, the scissors congruence problem, and the homology of Lie groups. These families of linear groups are known to have homology stability at least when the rings satisfy some appropriate finiteness condition, and in particular in the case of fields and local rings ([4],[26],[27],[25], [5],[2], [21],[15],[14]). It seems to be a delicate - but interesting and apparently important - question, however, to decide the minimal possible value of K(n) for a particular class of linear groups (with coefficients in a given class of rings) and the nature of the obstruction to extending the stability range further. The best illustration of this last remark are the results of Suslin on the integral homology of the general linear group of a field in the paper [23]. He proved that, for an infinite field F, the maps H_{n}(GL_{t}(F), ℤ) → H_{n}(GL_{t+1}(F), ℤ) are isomorphisms for t ≥ n (so that K(n) = n in this case), while the cokernel of the map H_{n}(GL_{n−1}(F), ℤ) → H_{n}(GL_{n}(F), ℤ) is naturally isomorphic to the nth Milnor K-group, K_{n}^{M}(F). In fact, if we let

H_{n}(F) := Coker(H_{n}(GL_{n−1}(F), ℤ) → H_{n}(GL_{n}(F), ℤ)),

his arguments show that there is an isomorphism of graded rings H_{•}(F) ≅ K_{•}^{M}(F) (where the multiplication on the first term comes from direct sum of matrices and cross product on homology). In particular, the non-negatively graded ring H_{•}(F) is generated in dimension 1. Recent work of Barge and Morel ([1]) suggested that Milnor-Witt K-theory may play a somewhat analogous role for the homology of the special linear group. The Milnor-Witt K-theory of F is a ℤ-graded ring K_{•}^{MW}(F) surjecting naturally onto Milnor K-theory. It arises as a ring of operations in stable motivic homotopy theory. (For a definition see section 2 below, and for more details see [17, 18, 19].) Let SH_{n}(F) := Coker(H_{n}(SL_{n−1}(F), ℤ) → H_{n}(SL_{n}(F), ℤ)) for n ≥ 1, and let SH_{0}(F) = ℤ [F^{×}] for convenience. Barge and Morel construct a map of graded algebras SH_{•}(F) → K_{•}^{MW}(F) for which the square

\xymatrix{SH_{•}(F) \ar[r] \ar[d] & K_{•}^{MW}(F) \ar[d] \\ H_{•}(F) \ar[r] & K_{•}^{M}(F)}

Homology stability for SL_{n}(F) 269

commutes. A result of Suslin ([24]) implies that the map H_{2}(SL_{2}(F), ℤ) = SH_{2}(F) → K_{2}^{MW}(F) is an isomorphism. Since positive-dimensional Milnor-Witt K-theory is generated by elements of degree 1, it follows that the map of Barge and Morel is surjective in even dimensions greater than or equal to 2. They ask the question whether it is in fact an isomorphism in even dimensions. As to the question of the range of homology stability for the special linear groups of an infinite field, as far as the authors are aware the most general result to date is still that of van der Kallen [25], whose results apply to much more general classes of rings. In the case of a field, he proves homology stability for H_{n}(SL_{t}(F), ℤ) in the range t ≥ 2n+1. On the other hand, known results when n is small suggest a much larger range. For example, the theorems of Matsumoto and Moore imply that the maps H_{2}(SL_{t}(F), ℤ) → H_{2}(SL_{t+1}(F), ℤ) are isomorphisms for t ≥ 3 and are surjective for t = 2. In the paper [22] (Conjecture 2.6), C-H. Sah conjectured that for an infinite field F (and more generally for a division algebra with infinite centre), the homomorphism H_{n}(SL_{t}(F), ℤ) → H_{n}(SL_{t+1}(F), ) is an isomorphism if t ≥ n + 1 and is surjective for t = n. The present paper addresses the above questions of Barge/Morel and Sah in the case of a field of characteristic zero. We prove the following results about the homology stability for special linear groups:

Theorem 1.1. Let F be a field of characteristic 0. For n, t ≥ 1, let f_{t,n} be the stabilization homomorphism H_{n}(SL_{t}(F), ℤ) → H_{n}(SL_{t+1}(F), ℤ)

(1) f_{t,n} is an isomorphism for t ≥ n + 1 and is surjective for t = n.

(2) If n is odd f_{n,n} is an isomorphism

(3) If n is even the kernel of f_{n,n} is isomorphic to I^{n+1}(F).

(4) For even n the cokernel of f_{n−1,n} is naturally isomorphic to K_{n}^{MW}(F).

(5) For odd n ≥ 3 the cokernel of f_{n−1,n} is naturally isomorphic to 2K_{n}^{M}(F).

Proof. The proofs of these statements can be found below as follows: (1) Corollary 5.11.

(2) Corollary 6.12.

(3) Corollary 6.13.

(4) Corollary 6.11.

(5) Corollary 6.13

□

Our strategy is to adapt Suslin’s argument for the general linear group in [23] to the case of the special linear group. Suslin’s argument is an ingenious variation on the method of van der Kallen in [25], in turn based on ideas of Quillen. The broad idea is to find a highly connected simplicial complex on which the group G_{t} acts and for which the stabilizers of simplices are (approximately) the groups G_{r}, with r ≤ t, and then to use this to construct a spectral sequence calculating the homology of the G_{n} in terms of the homology of the G_{r}. Suslin constructs a family E(n) of such spectral sequences, calculating the homology of GL_{n}(F). He constructs partially-defined products E(n) × E(m) → E(n + m) and then proves some periodicity and decomposabilty properties which allow him to conclude by an easy induction. Initially, the attempt to extend these arguments to the case of SL_{n}(F) does not appear very promising. Two obstacles to extending Suslin’s arguments become quickly apparent. The main obstacle is Suslin’s Theorem 1.8 which says that a certain inclusion of a block diagonal linear group in a block triangular group is a homology isomorphism. The corresponding statement for subgroups of the special linear group is emphatically false, as elementary calculations easily show. Much of Suslin’s subsequent results - in particular, the periodicity and decomposability properties of the spectral sequences E(n) and of the graded algebra S_{•}(F) which plays a central role - depend on this theorem. And, indeed, the analogous spectral sequences and graded algebra which arise when we replace the general linear with the special linear group do not have these periodicity and decomposability properties. However, it turns out - at least when the characteristic is zero - that the failure of Suslin’s Theorem 1.8 is not fatal. A crucial additional structure is available to us in the case of the special linear group; almost everything in sight in a ℤ[F^{×}]-module. In the analogue of Theorem 1.8, the map of homology groups is a split inclusion whose cokernel has a completely different character as a ℤ[F^{×}]-module than the homology of the block diagonal group. The former is ‘additive ’, while the latter is ‘multiplicative ’, notions which we define and explore in section 4 below. This leads us to introduce the concept of ‘AM modules’, which decompose in a canonical way into a direct sum of an additive factor and a multiplicative factor. This decomposition is sufficiently canonical that in our graded ring structures the additive and multiplicative parts are each ideals. By working modulo the messy additive factors and projecting onto multiplicative parts, we recover an analogue of Suslin’s Theorem 1.8 (Theorem 4.23 below), which we then use to prove the necessary periodicity (Theorem 5.10) and decomposability (Theorem 6.8) results. A second obstacle to emulating the case of the general linear group is the vanishing of the groups H_{1}(SL_{n}(F), ℤ). The algebra H_{•}(F), according to Suslin’s arguments, is generated by degree 1. On the other hand, SH_{1}(F) = 0 = H_{1}(SL_{1}(F), ℤ) = 0. This means that the best we can hope for in the case of the special linear group is that the algebra SH_{•}(F) is generated by degrees 2 and 3. This indeed turns out to be essentially the case, but it means we have to work harder to get our induction off the ground. The necessary arguments in degree n = 2 amount to the Theorem of Matsumoto and Moore, as well as variations due to Suslin ([24]) and Mazzoleni ([11]). The argument in degree n = 3 was supplied recently in a paper by the present authors ([8]). We make some remarks on the hypothesis of characteristic zero in this paper: This assumption is used in our definition of AM-modules and the derivation of their properties in section 4 below. In fact, a careful reading of the proofs in that section will show that at any given point all that is required is that the prime subfield be sufficiently large; it must contain an element of order not dividing m for some appropriate m. Thus in fact our arguments can easily be adapted to show that our main results on homology stability for the nth homology group of the special linear groups are true provided the prime field is sufficiently large (in a way that depends on n). However, we have not attempted here to make this more explicit. To do so would make the statements of the results unappealingly complicated, and we will leave it instead to a later paper to deal with the case of positive characteristic. We believe that an appropriate extension of the notion of AM-module will unlock the characteristic p > 0 case. As to our restriction to fields rather than more general rings, we note that Daniel Guin [5] has extended Suslin’s results to a larger class of rings with many units. We have not yet investigated a similar extension of the results below to this larger class of rings.

2. Notation and Background Results

2.1. Group Rings and Grothendieck-Witt Rings. For a group G, we let ℤ [G] denote the corresponding integral group ring. It has an additive ℤbasis consisting of the elements g ∈ G, and is made into a ring by linearly extending the multiplication of group elements. In the case that the group G is the multiplicative group, F^{×}, of a field F, we will denote the basis elements by ⟨a⟩, for a ∈ F^{×}. We use this notation in order, for example, to distinguish the elements ⟨1 − a⟩ from 1 − ⟨a⟩, or ⟨−a⟩ from − ⟨a⟩, and also because it coincides, conveniently for our purposes, with the notation for generators of the Grothendieck-Witt ring (see below). There is an augmentation homomorphism ϵ : ℤ [G] → ℤ, ⟨g⟩ ↦ 1, whose kernel is the augmentation ideal I_{G}, generated by the elements g − 1. Again, if G = F^{×}, we denote these generators by ⟨⟨a⟩⟩ := ⟨a⟩ − 1. The Grothendieck-Witt ring of a field F is the Grothendieck group, GW(F), of the set of isometry classes of nondgenerate symmetric bilinear forms under orthogonal sum. Tensor product of forms induces a natural multiplication on the group. As an abstract ring, this can be described as the quotient of the ring ℤ [F^{×}/(F^{×})^{2}] by the ideal generated by the elements ⟨⟨a⟩⟩ · ⟨⟨1 − a⟩⟩, a ≠ 0, 1. (This is just a mild reformulation of the presentation given in Lam, [9], Chapter II, Theorem 4.1.) Here, the induced ring homomorphism ℤ [F^{×}] → ℤ [F^{×}/(F^{×})^{2}] → GW(F), sends ⟨a⟩ to the class of the 1-dimensional form with matrix [a]. This class is (also) denoted ⟨a⟩. GW(F) is again an augmented ring and the augmentation ideal, I(F), - also called the fundamental ideal - is generated by Pfister 1-forms, ⟨⟨a⟩⟩. It follows that the n-th power, I^{n}(F), of this ideal is generated by Pfister n-forms ⟨⟨a_{1}, …, a_{n}⟩⟩ := ⟨⟨a_{1}⟩⟩ ⋯ ⟨⟨a_{n}⟩⟩. Now let h := ⟨1⟩ + ⟨−1⟩ = ⟨⟨−1⟩⟩ + 2 ∈ GW(F). Then h · I(F) = 0, and the Witt ring of F is the ring

W (F) := \frac{GW(F)}{⟨h⟩} = \frac{GW(F)}{h · ℤ}.

Since h ↦ 2 under the augmentation, there is a natural ring homomorphism W (F) → ℤ/2. The fundamental ideal I(F) of GW(F) maps isomorphically to the kernel of this ring homomorphism under the map GW(F) → W (F), and we also let I(F) denote this ideal. For n ≤ 0, we define I^{n}(F) := W (F). The graded additive group I^{•}(F) = {I^{n}(F)}_{n∈ℤ} is given the structure of a commutative graded ring using the natural graded multiplication induced from the multiplication on W (F). In particular, if we let η ∈ I^{−1}(F) be the element corresponding to 1 ∈ W (F), then multiplication by η : I^{n+1}(F) → I^{n}(F) is just the natural inclusion.

2.2. Milnor K-theory and Milnor-Witt K-theory. The Milnor ring of a field F (see [12]) is the graded ring K_{•}^{M}(F) with the following presentation: Generators: {a} , a ∈ F^{×}, in dimension 1. Relations:

(a) {ab} = {a} + {b} for all a, b ∈ F^{×}.

(b) {a} · {1 − a} = 0 for all a ∈ F^{×} \ {1}.

The product {a_{1}} ⋯ {a_{n}} in K_{n}^{M}(F) is also written {a_{1}, …, a_{n}}. So K_{0}^{M}(F) = ℤ and K_{1}^{M}(F) is an additive group isomorphic to F^{×}. We let k_{•}^{M}(F) denote the graded ring K_{•}^{M}(F)/2 and let i^{n}(F) := I^{n}(F)/I^{n+1}(F), so that i^{•}(F) is a non-negatively graded ring. In the 1990s, Voevodsky and his collaborators proved a fundamental and deep theorem - originally conjectured by Milnor ([13]) - relating Milnor K-theory to quadratic form theory:

Theorem 2.1 ([20]). There is a natural isomorphism of graded rings k_{•}^{M}(F) ≅ i^{•}(F) sending {a} to ⟨⟨a⟩⟩. In particular for all n ≥ 1 we have a natural identification of k_{n}^{M}(F) and i^{n}(F) under which the symbol {a_{1}, …, a_{n}} corresponds to the class of the form ⟨⟨a_{1}, …, a_{n}⟩⟩.

The Milnor-Witt K-theory of a field is the graded ring K_{•}^{MW}(F) with the following presentation (due to F. Morel and M. Hopkins, see [17]): Generators: [a], a ∈ F^{×}, in dimension 1 and a further generator η in dimension −1. Relations:

(a) [ab] = [a] + [b] + η · [a] · [b] for all a, b ∈ F^{×}

(b) [a] · [1 − a] = 0 for all a ∈ F^{×} \ {1}
(c) η · [a] = [a] · η for all a ∈ F^{×}
(d) η · h = 0, where h = η · [−1] + 2 ∈ K_{0}^{MW}(F).

Clearly there is a unique surjective homomorphism of graded rings K_{•}^{MW}(F) → K_{•}^{M}(F) sending [a] to {a} and inducing an isomorphism

\frac{K_{•}^{MW}(F)}{⟨η⟩} ≅ K_{•}^{M}(F).

Furthermore, there is a natural surjective homomorphism of graded rings K_{•}^{MW}(F) → I^{•}(F) sending [a] to ⟨⟨a⟩⟩ and η to η. Morel shows that there is an induced isomorphism of graded rings

\frac{K_{•}^{MW}(F)}{⟨h⟩} ≅ I^{•}(F).

The main structure theorem on Milnor-Witt K-theory is the following theorem of Morel:

Theorem 2.2 (Morel, [18]). The commutative square of graded rings

\xymatrix{K_{•}^{MW}(F) \ar[r] \ar[d] & K_{•}^{M}(F) \ar[d] \\ I^{•}(F) \ar[r] & i^{•}(F)}

is cartesian.

Thus for each n ∈ ℤ we have an isomorphism

K_{n}^{MW}(F) ≅ K_{n}^{M}(F) ×_{i^{n}(F)} I^{n}(F).

It follows that for all n there is a natural short exact sequence

0 → I^{n+1}(F) → K_{n}^{MW}(F) → K_{n}^{M}(F) → 0

where the inclusion I^{n+1}(F) → K_{n}^{MW}(F) is given by

⟨⟨a_{1}, …, a_{n+1}⟩⟩ ↦ η[a_{1}] ⋯ [a_{n}].

Similarly, for n ≥ 0, there is a short exact sequence

0 → 2K_{n}^{M}(F) → K_{n}^{MW}(F) → I^{n}(F) → 0

where the inclusion 2K_{n}^{M}(F) → K_{n}^{MW}(F) is given (for n ≥ 1) by

2{a_{1}, …, a_{n}}↦ h[a_{1}] ⋯ [a_{n}].

Observe that, when n ≥ 2,

h[a_{1}][a_{2}] ⋯ [a_{n}] = ([a_{1}][a_{2}] − [a_{2}][a_{1}])[a_{3}] ⋯ [a_{n}] = [a_{1}^{2}][a_{2}] ⋯ [a_{n}].

(The first equality follows from Lemma 2.3 (3) below, the second from the observation that [a_{1}^{2}] ⋯ [a_{n}] ∈ Ker(K_{n}^{MW}(F) → I^{n}(F)) = 2K_{n}^{M}(F) and the fact, which follows from Morel’s theorem, that the composite 2K_{n}^{M}(F) → K_{n}^{MW}(F) → K_{n}^{M}(F) is the natural inclusion map.) When n = 0 we have an isomorphism of rings

GW(F) ≅ W (F) ×_{ℤ/2} ℤ ≅ K_{0}^{MW}(F).

Under this isomorphism ⟨⟨a⟩⟩ corresponds to η[a] and ⟨a⟩ corresponds to η[a]+ 1. (Observe that with this identification, h = η[−1] + 2 = ⟨1⟩ + ⟨−1⟩ ∈ K_{0}^{MW}(F) = GW(F), as expected.) Thus each K_{n}^{MW}(F) has the structure of a GW(F)-module (and hence also of a ℤ [F^{×}]-module), with the action given by ⟨⟨a⟩⟩ · ([a_{1}] ⋯ [a_{n}]) = η[a][a_{1}] ⋯ [a_{n}]. We record here some elementary identities in Milnor-Witt K-theory which we will need below.

Lemma 2.3. Let a, b ∈ F^{×}. The following identities hold in the Milnor-Witt K-theory of F:

(1) [a][−1] = [a][a].

(2) [ab] = [a] + ⟨a⟩[b].
(3) [a][b] = −⟨−1⟩[b][a].

Proof.

(1) See, for example, the proof of Lemma 2.7 in [7].

(2) ⟨a⟩b = (η[a] + 1)[b] = η[a][b] + [b] = [ab] − [a].

(3) See [7], Lemma 2.7.

□

2.3. Homology of Groups. Given a group G and a ℤ [G]-module M, H_{n}(G, M) will denote the nth homology group of G with coefficients in the module M. B_{•}(G) will denote the right bar resolution of G: B_{n}(G) is the free right ℤ [G]-module with basis the elements [g_{1}| ⋯ |g_{n}], g_{i} ∈ G. (B_{0}(G) is isomorphic to ℤ [G] with generator the symbol [ ].) The boundary d = d_{n} : B_{n}(G) → B_{n−1}(G), n ≥ 1, is given by

d([g_{1}| ⋯ |g_{n}]) = ∑_{i=o}^{n−1}(−1)^{i}[g_{1}| ⋯ |ĝ_{i}| ⋯ |g_{n}] + (−1)^{n}[g_{1}| ⋯ |g_{n−1}] ⟨g_{n}⟩ .

The augmentation B_{0}(G) → ℤ makes B_{•}(G) into a free resolution of the trivial ℤ [G]-module ℤ, and thus H_{n}(G, M) = H_{n}(B_{•}(G) ⊗_{ℤ[G]} M). If C_{•} = (C_{q}, d) is a non-negative complex of ℤ [G]-modules, then E_{•,•} := B_{•}(G) ⊗_{ℤ[G]} C_{•} is a double complex of abelian groups. Each of the two filtrations on E_{•,•} gives a spectral sequence converging to the homology of the total complex of E_{•,•}, which is by definition, H_{•}(G, C). (see, for example, Brown, [3], Chapter VII). The first spectral sequence has the form

E_{p,q}^{2} = H_{p}(G, H_{q}(C)) ⟹ H_{p+q}(G, C).

In the special case that there is a weak equivalence C_{•} → ℤ (the complex consisting of the trivial module ℤ concentrated in dimension 0), it follows that H_{•}(G, C) = H_{•}(G, ℤ). The second spectral sequence has the form

E_{p,q}^{1} = H_{p}(G, C_{q}) ⟹ H_{p+q}(G, C).

Thus, if C_{•} is weakly equivalent to ℤ, this gives a spectral sequence converging to H_{•}(G, ℤ). Our analysis of the homology of special linear groups will exploit the action of these groups on certain permutation modules. It is straightforward to compute the map induced on homology groups by a map of permutation modules. We recall the following basic principles (see, for example, [6]): If G is a group and if X is a G-set, then Shapiro’s Lemma says that

H_{p}(G, ℤ[X]) ≅ ⨁_{y∈X/G} H_{p}(G_{y}, ℤ),

the isomorphism being induced by the maps

H_{p}(G_{y}, ℤ) → H_{p}(G, ℤ[X])

described at the level of chains by

B_{p} ⊗_{ℤ[G_{y}]} ℤ → B_{p} ⊗_{ℤ[G]} ℤ[X], z ⊗ 1 ↦ z ⊗ y.

Let X_{i}, i = 1, 2 be transitive G-sets. Let x_{i} ∈ X_{i} and let H_{i} be the stabiliser of x_{i}, i = 1, 2. Let ϕ : ℤ[X_{1}] → ℤ[X_{2}] be a map of ℤ[G]-modules with

ϕ(x_{1}) = ∑_{g∈G/H_{2}} n_{g}gx_{2}, with n_{g} ∈ ℤ.

Then the induced map ϕ_{•} : H_{•}(H_{1}, ℤ) → H_{•}(H_{2}, ℤ) is given by the formula

ϕ_{•}(z) = ∑_{g∈H_{1}\G/H_{2}} n_{g}cor_{g^{−1}H_{1}g∩H_{2}}^{H_{2}}res_{g^{−1}H_{1}g∩H_{2}}^{g^{−1}H_{1}g} (g^{−1} · z) (1)

There is an obvious extension of this formula to non-transitive G-sets.

2.4. Homology of SL_{n}(F) and Milnor-Witt K-theory. Let F be an infinite field. The theorem of Matsumoto and Moore ([10], [16]) gives a presentation of the group H_{2}(SL_{2}(F), ℤ). It has the following form: The generators are symbols ⟨a_{1}, a_{1}⟩, a_{i} ∈ F^{×}, subject to the relations:

(i) ⟨a_{1}, a_{2}⟩ = 0 if a_{i} = 1 for some i

(ii) ⟨a_{1}, a_{2}⟩ = ⟨a_{2}^{−1}, a_{1}⟩

(iii) ⟨a_{1}, a_{2}b_{2}⟩ + ⟨a_{2}, b_{2}⟩ = ⟨a_{1}a_{2}, b_{2}⟩ + ⟨a_{1}, a_{2}⟩

(iv) ⟨a_{1}, a_{2}⟩ = ⟨a_{1}, −a_{1}a_{2}⟩

(v) ⟨a_{1}, a_{2}⟩ = ⟨a_{1}, (1 − a_{1})a_{2}⟩

It can be shown that for all n ≥ 2, K_{n}^{MW}(F) admits a (generalised) Matsumoto-Moore presentation:

Theorem 2.4 ([7], Theorem 2.5). For n ≥ 2, K_{n}^{MW}(F) admits the following presentation as an additive group: Generators: The elements [a_{1}][a_{2}] ⋯ [a_{n}], a_{i} ∈ F^{×}. Relations:

(i) [a_{1}][a_{2}] ⋯ [a_{n}] = 0 if a_{i} = 1 for some i.

(ii) [a_{1}] ⋯ [a_{i−1}][a_{i}] ⋯ [a_{n}] = [a_{1}] ⋯ [a_{i}^{−1}][a_{i−1}] ⋯ [a_{n}]

(iii) [a_{1}] ⋯ [a_{n−1}][a_{n}b_{n}] + [a_{1}] ⋯ { \atop [a_{n−1}]}[a_{n}][b_{n}] = [a_{1}] ⋯ [a_{n−1}a_{n}][b_{n}] +

[a_{1}] ⋯ [a_{n−1}][a_{n}] (iv) [a_{1}] ⋯ [a_{n−1}][a_{n}] = [a_{1}] ⋯ [a_{n−1}][−a_{n−1}a_{n}]

(v) [a_{1}] ⋯ [a_{n−1}][a_{n}] = [a_{1}] ⋯ [a_{n−1}][(1 − a_{n−1})a_{n}]

In particular, it follows when n = 2 that there is a natural isomorphism K_{2}^{MW}(F) ≅ H_{2}(SL_{2}(F), ℤ). This last fact is essentially due to Suslin ([24]). A more recent proof, which we will need to invoke below, has been given by Mazzoleni ([11]). Recall that Suslin ([23]) has constructed a natural surjective homomorphism H_{n}(GL_{n}(F), ℤ) → K_{n}^{M}(F) whose kernel is the image of H_{n}(GL_{n−1}(F), ℤ). In [8], the authors proved that the map H_{3}(SL_{3}(F), ℤ) → H_{3}(GL_{3}(F), ℤ) is injective, that the image of the composite H_{3}(SL_{3}(F), ℤ) → H_{3}(GL_{3}(F), ℤ) → K_{3}^{M}(F) is 2K_{3}^{M}(F) and that the kernel of this composite is precisely the image of H_{3}(SL_{2}(F), ℤ). In the next section we will construct natural homomorphisms T_{n} ∘ ϵ_{n} : H_{n}(SL_{n}(F), ℤ) → K_{n}^{MW}(F), in a manner entirely analogous to Suslin’s construction. In particular, the image of H_{n}(SL_{n−1}(F), ℤ) is contained in the kernel of T_{n} ∘ ϵ_{n} and the diagrams

\xymatrix{H_{n}(SL_{n}(F), ℤ) \ar[r] \ar[d] & K_{n}^{MW}(F) \ar[d] \\ H_{n}(GL_{n}(F), ℤ) \ar[r] & K_{n}^{M}(F)}

commute. It follows that the image of T_{3} ∘ ϵ_{3} is 2K_{3}^{M}(F) ⊂ K_{3}^{MW}(F), and its kernel is the image of H_{3}(SL_{2}(F), ℤ).

3. The algebra S̃(F^{•})

In this section we introduce a graded algebra functorially associated to F which admits a natural homomorphism to Milnor-Witt K-theory and from the homology of SL_{n}(F). It is the analogue of Suslin’s algebra S_{•}(F) in [24], which admits homomorphisms to Milnor K-theory and from the homology of GL_{n}(F). However, we will need to modify this algebra in the later sections below, by projecting onto the ‘multiplicative ’ part, in order to derive our results about the homology of SL_{n}(F). We say that a finite set of vectors v_{1}, …, v_{q} in an n-dimensional vector space V are in general position if every subset of size min(q, n) is linearly independent. If v_{1}, …, v_{q} are elements of the n-dimensional vector space V and if E is an ordered basis of V, we let [v_{1}| ⋯ |v_{q}]_{E} denote the n × q matrix whose i-th column is the components of v_{i} with respect to the basis E.

3.1. Definitions. For a field F and finite-dimensional vector spaces V and W, we let X_{p}(W, V) denote the set of all ordered p-tuples of the form

((w_{1}, v_{1}), …, (w_{p}, v_{p}))

where (w_{i}, v_{i}) ∈ W ⊕ V and the v_{i} are in general position. We also define X_{0}(W, V) := ∅. X_{p}(W, V) is naturally an A(W, V)-module, where

A(W, V) := (\begin{matrix}Id_{W} & Hom(V, W) \\ 0 & GL(V) \end{matrix}) ⊂ GL(W ⊕ V)

Let C_{p}(W, V) = ℤ[X_{p}(W, V)], the free abelian group with basis the elements of X_{p}(W, V). We obtain a complex, C_{•}(W, V), of A(W, V)-modules by introducing the natural simplicial boundary map

d_{p+1} : C_{p+1}(W, V) → C_{p}(W, V)
((w_{1}, v_{1}), …, (w_{p+1}, v_{p+1})) ↦

∑_{i=1}^{p+1}(−1)^{i+1}((w_{1}, v_{1}), …, { \atop (w_{i}, v_{i})}, …, (w_{p+1}, v_{p+1}))

Lemma 3.1. If F is infinite, then H_{p}(C_{•}(W, V)) = 0 for all p.

Proof. If

z = ∑_{i} n_{i}((w_{1}^{i}, v_{1}^{i}), …, (w_{p}^{i}, v_{p}^{i})) ∈ C_{p}(W, V)

is a cycle, then since F is infinite, it is possible to choose v ∈ V such that v, v_{1}^{i}, …, v_{p}^{i} are in general position for all i. Then z = d_{p+1}((−1)^{p}s_{v}(z)) where s_{v} is the ‘partial homotopy operator’ defined by s_{v}((w_{1}, v_{1}), …, (w_{p}, v_{p})) =

\{ \begin{matrix}((w_{1}, v_{1}), …, (w_{p}, v_{p}), (0, v)), & if v, v_{1}, … v_{p} are in general position, \\ 0, & otherwise \end{matrix}

□

We will assume our field F is infinite for the remainder of this section. (In later sections, it will even be assumed to be of characteristic zero.) If n = dim_{F} (V), we let H(W, V) := Ker(d_{n}) = Im(d_{n+1}). This is an A(W, V)-submodule of C_{n}(W, V). Let S̃(W, V) := H_{0}(SA(W, V), H(W, V)) = H(W, V)_{SA(W,V)} where SA(W, V) := A(W, V) ∩ SL(W ⊕ V). If W′ ⊂ W, there are natural inclusions X_{p}(W′, V) → X_{p}(W, V) inducing a map of complexes of A(W′, V)-modules C_{•}(W′, V) → C_{•}(W, V). When W = 0, we will use the notation, X_{p}(V), C_{p}(V), H(V) and S̃(V) instead of X_{p}(0, V), C_{p}(0, V), H(0, V) and S̃(0, V) Since, A(W, V)/SA(W, V) ≅ F^{×}, any homology group of the form

H_{i}(SA(W, V), M), where M is a A(W, V)-module,

is naturally a ℤ[F^{×}]-module: If a ∈ F^{×} and if g ∈ A(W, V) is any element of determinant a, then the action of a is the map on homology induced by conjugation by g on A(W, V) and multiplication by g on M. In particular, the groups S̃(W, V) are ℤ[F^{×}]-modules. Let e_{1}, …, e_{n} denote the standard basis of F^{n}. Given a_{1}, …, a_{n} ∈ F^{×}, we let ⌊a_{1}, …, a_{n}⌉ denote the class of d_{n+1}(e_{1}, …, e_{n}, a_{1}e_{1} + ⋯ + a_{n}e_{n}) in S̃(F^{n}). If b ∈ F^{×}, then ⟨b⟩ · ⌊a_{1}, …, a_{n}⌉ is represented by

d_{n+1}(e_{1}, …, be_{i}, …, e_{n}, a_{1}e_{1} + ⋯ a_{i}be_{i} ⋯ + a_{n}e_{n})

for any i. (As a lifting of b ∈ F^{×}, choose the diagonal matrix with b in the (i, i)-position and 1 in all other diagonal positions.)

Remark 3.2. Given x = (v_{1}, …, v_{v}, v) ∈ X_{n+1}(F^{n}), let A = [v_{1}| ⋯ |v_{n}] ∈ GL_{n}(F) of determinant det A and let A′ = diag(1, …, 1, det A). Then B = A′A^{−1} ∈ SL_{n}(F) and thus x is in the SL_{n}(F)-orbit of

(e_{1}, …, e_{n−1}, det Ae_{n}, A′w) with w = A^{−1}v,

and hence d_{n+1}(x) represents the element ⟨det A⟩ ⌊w⌉ in S̃(F^{n}).

Theorem 3.3. S̃(F^{n}) has the following presentation as a ℤ[F^{×}]-module: Generators: The elements ⌊a_{1}, …, a_{n}⌉, a_{i} ∈ F^{×} Relations: For all a_{1}, …, a_{n} ∈ F^{×} and for all b_{1}, …, b_{n} ∈ F^{×} with b_{i} ≠ b_{j} for i ≠ j

⌊b_{1}a_{1}, …, b_{n}a_{n}⌉ − ⌊a_{1}, …, a_{n}⌉ =

∑_{i=1}^{n}(−1)^{n+i} ⟨(−1)^{n+i}a_{i}⟩ ⌊a_{1}(b_{1} − b_{i}), …, { \atop a_{i}(b_{i} − b_{i})}, …, a_{n}(b_{n} − b_{i}), b_{i}⌉.

Proof. Taking SL_{n}(F)-coinvariants of the exact sequence of ℤ[GL_{n}(F)]- modules

\xymatrix{C_{n+2}(F^{n}) \ar[r]^{d_{n+2}} & C_{n+1}(F^{n}) \ar[r]^{d_{n+1}} & H(F^{n}) \ar[r] & 0}

gives the exact sequence of ℤ[F^{×}]-modules

\xymatrix{C_{n+2}(F^{n})_{SL_{n}(F)} \ar[r]^{d_{n+2}} & C_{n+1}(F^{n})_{SL_{n}(F)} \ar[r]^{d_{n+1}} & S̃(F^{n}) \ar[r] & 0.}

It is straightforward to verify that

X_{n+1}(F^{n}) ≅ ∐_{a=(a_{1},...,a_{n}),a_{i}≠0} GL_{n}(F) · (e_{1}, …, e_{n}, a)

as a GL_{n}(F)-set. It follows that

C_{n+1}(F^{n}) ≅ ⨁_{a} ℤ[GL_{n}(F)] · (e_{1}, …, e_{n}, a)

as a ℤ[GL_{n}(F)]-module, and thus that

C_{n+1}(F^{n})_{SL_{n}(F)} ≅ ⨁_{a} ℤ[F^{×}] · (e_{1}, …, e_{n}, a)

as a ℤ[F^{×}]-module. Similarly, every element of X_{n+2}(F^{n}) is in the GL_{n}(F)-orbit of a unique element of the form (e_{1}, …, e_{n}, a, b · a) where a = (a_{1}, …, a_{n}) with a_{i} ≠ 0 for all i and b = (b_{1}, …, b_{n}) with b_{i} ≠ 0 for all i and b_{i} ≠ b_{j} for all i ≠ j, and b · a := (b_{1}a_{1}, …, b_{n}a_{n}). Thus

X_{n+2}(F^{n}) ≅ ∐_{(a,b)} GL_{n}(F) · (e_{1}, …, e_{n}, a, b · a)

as a GL_{n}(F)-set and

C_{n+2}(F^{n})_{SL_{n}(F)} ≅ ⨁_{(a,b)} ℤ[F^{×}] · (e_{1}, …, e_{n}, a, b · a)

as a ℤ[F^{×}]-module. So d_{n+1} induces an isomorphism

\frac{⊕ℤ[F^{×}] · (e_{1}, …, e_{n}, a)}{⟨d_{n+2}(e_{1}, …, e_{n}, a, b · a)|(a, b)⟩} ≅ S̃(F^{n}).

Now d_{n+2}(e_{1}, …, e_{n}, a, b · a) =

∑_{i=1}^{n}(−1)^{i+1}(e_{1}, …, ê_{i}, …, e_{n}, a, b · a) + (−1)^{i}((e_{1}, …, e_{n}, b · a) − (e_{1}, …, e_{n}, a)).

Documenta Mathematica · Extra Volume Suslin (2010) 267–315

Applying the idea of Remark 3.2 to the terms (e_{1}, …, ê_{i}, …, e_{n}, a, b · a) in the sum above, we let M_{i}(a) := [e_{1}| ⋯ |ê_{i}| ⋯ |e_{n}|a] and δ_{i} = det M_{i}(a) = (−1)^{n−i}a_{i}. Since

M_{i}(a)^{−1} = (\begin{matrix}1 & … & 0 & −a_{1}/a_{i} & 0 & … & 0 \\ 0 & ... & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & … & 1 & −a_{i−1}/a_{i} & 0 & … & 0 \\ 0 & … & 0 & −a_{i+1}a_{i} & 1 & … & 0 \\ 0 & … & 0 & ⋮ & 0 & ... & 0 \\ 0 & … & 0 & −a_{n}/a_{i} & 0 & … & 1 \\ 0 & … & 0 & 1/a_{i} & 0 & … & 0 \end{matrix})

it follows that d_{n+1}(e_{1}, …, ê_{i}, …, e_{n}, a, b·a) represents ⟨δ_{i}⟩ ⌊w_{i}⌉ ∈ S̃(F^{n}) where

w_{i} = M_{i}(a)^{−1}(b · a) = (a_{1}(b_{1} − b_{i}), …, { \atop a_{i}(b_{i} − b_{i})}, …, a_{n}(b_{n} − b_{i}), b_{i}). This

proves the theorem. □

3.2. Products. If W′ ⊂ W, there is a natural bilinear pairing

C_{p}(W′, V) × C_{q}(W) → C_{p+q}(W ⊕ V), (x, y) ↦ x ∗ y

defined on the basis elements by

((w′_{1}, v_{1}), …, (w′_{p}, v_{p})) ∗ (w_{1}, …, w_{q}) := ((w′_{1}, v_{1}), …, (w′_{p}, v_{p}), (w_{1}, 0), …, (w_{q}, 0)) .

This pairing satisfies d_{p+q}(x ∗ y) = d_{p}(x) ∗ y + (−1)^{p}x ∗ d_{q}(y). Furthermore, if α ∈ A(W′, V) ⊂ GL(W ⊕ V) then (αx) ∗ y = α(x ∗ y), and if α ∈ GL(V) ⊂ A(W′, V) ⊂ GL(W ⊕ V) and β ∈ GL(W) ⊂ GL(W ⊕ V), then (αx) ∗ (βy) = (α · β)(x ∗ y). (However, if W′ ≠ 0 then the images of A(W′, V) and GL(W) in GL(W ⊕ V) don’t commute.) In particular, there are induced pairings on homology groups

H(W′, V) ⊗ H(W) → H(W ⊕ V),

which in turn induce well-defined pairings

S̃(W′, V) ⊗ H(W) → S̃(W, V) and S̃(V) ⊗ S̃(W) → S̃(W ⊕ V).

Observe further that this latter pairing is ℤ[F^{×}]-balanced: If a ∈ F^{×}, x ∈ S̃(W) and y ∈ S̃(V), then (⟨a⟩ x) ∗ y = x ∗ (⟨a⟩ y) = ⟨a⟩ (x ∗ y). Thus there is a welldefined map

S̃(V) ⊗_{ℤ[F^{×}]} S̃(W) → S̃(W ⊕ V).

In particular, the groups {H(F^{n})}_{n≥0} form a natural graded (associative) algebra, and the groups {S̃(F^{n})}_{n≥0} = S̃(F^{•}) form a graded associative ℤ[F^{×}]- algebra. The following explicit formula for the product in S̃(F^{•}) will be needed below:

Lemma 3.4. Let a_{1}, …, a_{n} and a′_{1}, …, a′_{m} be elements of F^{×}. Let b_{1}, …, b_{n}, b′_{1}, …, b′_{m} be any elements of F^{×} satisfying b_{i} ≠ b_{j} for i ≠ j and b′_{s} ≠ b′_{t} for s ≠ t.

Then

⌊a_{1}, …, a_{n}⌉ ∗ ⌊a′_{1}, …, a′_{m}⌉ =

= ∑_{i=1}^{n} ∑_{j=1}^{m}(−1)^{m+n+i+j} ⟨(−1)^{i+j}a_{i}a′_{j}⟩ ×

×⌊a_{1}(b_{1} − b_{i}), …, { \atop a_{i}(b_{i} − b_{i})}, …, b_{i}, a′_{1}(b′_{1} − b′_{j}), …, { \atop a′_{j}(b′_{j} − b′_{j})}, …, b′_{j}⌉

+(−1)^{n} ∑_{i=1}^{n}(−1)^{i+1} ⟨(−1)^{i+1}a_{i}⟩ ⌊a_{1}(b_{1} − b_{i}), …, { \atop a_{i}(b_{i} − b_{i})}, …, b_{i}, b′_{1}a′_{1}, …, b′_{m}a′_{m}⌉

+(−1)^{m} ∑_{j=1}^{m}(−1)^{j+1} ⟨(−1)^{j+1}a′_{j}⟩ ⌊b_{1}a_{1}, …, b_{n}a_{n}, a′_{1}(b′_{1} − b′_{j}), …, { \atop a′_{j}(b′_{j} − b′_{j})}, …, b′_{j}⌉

+⌊b_{1}a_{1}, …, b_{n}a_{n}, b′_{1}a′_{1}, …, b′_{m}a′_{m}⌉

Proof. This is an entirely straightforward calculation using the definition of the product, Remark 3.2, the matrices M_{i}(a), M_{j}(a′) as in the proof of Theorem 3.3, and the partial homotopy operators s_{v} with v = (a_{1}b_{1}, …, a_{n}b_{n}, a′_{1}b′_{1}, …, a′_{m}b′_{m}). □

3.3. The maps ϵ_{V}. If dim_{F} (V) = n, then the exact sequence of GL(V)- modules

\xymatrix{0 \ar[r] & H(V) \ar[r] & C_{n}(V) \ar[r]^{d_{n}} & C_{n−1}(V) \ar[r]^{d_{n−1}} & ⋯ \ar[r]^{d_{1}} & C_{0}(V) = ℤ \ar[r] & 0}

gives rise to an iterated connecting homomorphism

ϵ_{V} : H_{n}(SL(V), ℤ) → H_{0}(SL(V), H(V)) = S̃(V).

Note that the collection of groups {H_{n}(SL_{n}(F), ℤ)} form a graded ℤ[F^{×}]- algebra under the graded product induced by exterior product on homology, together with the obvious direct sum homomorphism SL_{n}(F) × SL_{m}(F) → SL_{n+m}(F).

Lemma 3.5. The maps ϵ_{n} : H_{n}(SL_{n}(F), ℤ) → S̃(F^{n}), n ≥ 0, give a well-defined homomorphism of graded ℤ[F^{×}]-algebras; i.e.

(1) If a ∈ F^{×} and z ∈ H_{n}(SL_{n}(F), ℤ), then ϵ_{n}(⟨a⟩ z) = ⟨a⟩ ϵ_{n}(z) in S̃(F^{n}), and

(2) If z ∈ H_{n}(SL_{n}(F), ℤ) and w ∈ H_{m}(SL_{m}(F), ℤ) then

ϵ_{n+m}(z × w) = ϵ_{n}(z) ∗ ϵ_{m}(w) in S̃(F^{n+m}).

Proof.

(1) The exact sequence above is a sequence of GL(V)-modules and hence all of the connecting homomorphisms δ_{i} : H_{n−i+1}(SL(V), Im(d_{i})) → H_{n−i}(SL(V), Ker(d_{i})) are F^{×}-equivariant.

(2) Let C_{•}^{τ} (V) denote the truncated complex.

C_{p}^{τ} (V) = \{ \begin{matrix}C_{p}(V), & p ≤ dim_{F} (V) \\ 0, & p > dim_{F} (V) \end{matrix}

Thus H(V) → C_{•}^{τ} (V) is a weak equivalence of complexes (where we regard H(V) as a complex concentrated in dimension dim (V)). Since the complexes C_{•}^{τ} (V) are complexes of free abelian groups, it follows that for two vector spaces V and W, the map H(V)⊗_{ℤ} H(W) → T_{•}(V, W) is an equivalence of complexes, where T_{•}(V, W) is the total complex of the double complex C_{•}^{τ} (V) ⊗_{ℤ} C_{•}^{τ} (W). Now T_{•}(V, W) is a complex of SL(V) × SL(W)-modules, and the product ∗ induces a commutative diagram of complexes of SL(V) × SL(W)-complexes:

\xymatrix{H(V) ⊗_{ℤ} H(W) \ar[r] \ar[d]^{∗} & C_{•}^{τ} (V) ⊗ C_{•}^{τ} (W) \ar[d]^{∗} \\ H(V ⊕ W) \ar[r] & C_{•}^{τ} (V ⊕ W)}

which, in turn, induces a commutative diagram

\xymatrix{H_{n}(SL(V), ℤ) ⊗ H_{m}(SL(W), ℤ) \ar[r]^{ϵ_{V} ⊗ϵ_{W}} \ar[d]^{×} & H_{0}(SL(V), H(V)) ⊗ H_{0}(SL(W), H(W)) \ar[d]^{×} \\ H_{n+m}(SL(V) × SL(W), ℤ ⊗ ℤ) \ar[r]^{ϵ_{T•}} \ar[d] & H_{0}(SL(V) × SL(W), H(V) ⊗ H(W)) \ar[d] \\ H_{n+m}(SL(V ⊕ W), ℤ) \ar[r]^{ϵ_{V ⊕W}} & H_{0}(SL(V ⊕ W), H(V ⊕ W))}

(where n = dim (V) and m = dim (W)).

□

Lemma 3.6. If V = W ⊕ W′ with W′ ≠ 0, then the composite

\xymatrix{H_{n}(SL(W), ℤ) \ar[r] & H_{n}(SL(V), ℤ) \ar[r]^{ϵ_{V}} & S̃(V)}

is zero.

Proof. The exact sequence of SL(V)-modules

0 → Ker(d_{1}) → C_{1}(V) → ℤ → 0

is split as a sequence of SL(W)-modules via the map ℤ → C_{1}(V), m ↦ m · e where e is any nonzero element of W′. It follows that the connecting homomorphism δ_{1} : H_{n}(SL(W), ℤ) → H_{n−1}(SL(W), Ker(d_{1})) is zero. □

Let SH_{n}(F) denote the cokernel of the map H_{n}(SL_{n−1}(F), ℤ)→H_{n}(SL_{n}(F), ℤ). It follows that the maps ϵ_{n} give well-defined homomorphisms SH_{n}(F) → S̃(F^{n}), which yield a homomorphism of graded ℤ[F^{×}]-algebras ϵ_{•} : SH_{•}(F) → S̃(F^{•}).

3.4. The maps D_{V}. Suppose now that W and V are vector spaces and that dim (V) = n. Fix a basis E of V. The group A(W, V) acts transitively on X_{n}(W, V) (with trivial stabilizers), while the orbits of SA(W, V) are in one-to-one correspondence with the points of F^{×} via the correspondence

X_{n}(W, V) → F^{×}, ((w_{1}, v_{1}), …, (w_{n}, v_{n})) ↦ det ([v_{1}| ⋯ |v_{n}]_{E}) .

Thus we have an induced isomorphism

\xymatrix{H_{0}(SA(W, V), C_{n}(W, V)) \ar[r]^{det}_{≅} & ℤ[F^{×}].}

Taking SA(W, V)-coinvariants of the inclusion H(W, V) → C_{n}(W, V) then yields a homomorphism of ℤ[F^{×}]-modules

D_{W,V} : S̃(W, V) → ℤ[F^{×}].

In particular, for each n ≥ 1 we have a homomorphism of ℤ[F^{×}]-modules D_{n} : S̃(F^{n}) → ℤ[F^{×}]. We will also set D_{0} : S̃(F^{0}) = ℤ → ℤ equal to the identity map. Here ℤ is a trivial F^{×}-module. We set

A_{n} = \{ \begin{matrix}ℤ, & n = 0 \\ I_{F^{×}}, & n odd \\ ℤ[F^{×}], & n > 0 even \end{matrix}

We have A_{n} ⊂ ℤ[F^{×}] for all n and we make A_{•} into a graded algebra by using the multiplication on ℤ[F^{×}].

Lemma 3.7.

(1) The image of D_{n} is A_{n}.

(2) The maps D_{•} : S̃(F^{•}) → A_{•} define a homomorphism of graded ℤ[F^{×}]- algebras.

(3) For each n ≥ 0, the surjective map D_{n} : S̃(F^{n}) → A_{n} has a ℤ[F^{×}]- splitting.

Proof.

(1) Consider a generator ⌊a_{1}, …, a_{n}⌉ of S̃(F^{n}).

Let e_{1}, …, e_{n} be the standard basis of F^{n}. Let a := a_{1}e_{1}+⋯+a_{n}e_{n}. Then

⌊a_{1}, …, a_{n}⌉ = d_{n+1}(e_{1}, …, e_{n}, a)
= ∑_{i=1}^{n}(−1)^{i+1}(e_{1}, …, ê_{i}, …, e_{n}, a) + (−1)^{n}(e_{1}, …, e_{n}).

Thus

D_{n}(⌊a_{1}, …, a_{n}⌉) = ∑_{i=1}^{n}(−1)^{i+1} ⟨det ([e_{1}| ⋯ |ê_{i}| ⋯ |e_{n}|a])⟩ + (−1)^{n} ⟨1⟩

= \{ \begin{matrix}⟨a_{1}⟩ − ⟨−a_{2}⟩ + ⋯ + ⟨a_{n}⟩ − ⟨1⟩ , & n odd \\ ⟨−a_{1}⟩ − ⟨a_{2}⟩ + ⋯ − ⟨a_{n}⟩ + ⟨1⟩ , & n > 0 even \end{matrix}

Thus, when n is even, D_{n}(⌊−1, 1, −1, …, −1, 1⌉) = ⟨1⟩ and D_{n} maps onto ℤ[F^{×}].

When n is odd, clearly, D_{n}(⌊a_{1}, …, a_{n}⌉) ∈ I_{F^{×}}. However, for any a ∈ F^{×}, D_{n}(⌊a, −1, 1, …, −1, 1⌉) = ⟨⟨a⟩⟩ ∈ A_{n} = I_{F^{×}}.

(2) Note that C_{n}(F^{n}) ≅ ℤ[GL_{n}(F)] naturally. Let μ be the homomorphism of additive groups

μ : ℤ[GL_{n}(F)] ⊗ ℤ[GL_{m}(F)] → ℤ[GL_{n+m}(F)],
A ⊗ B ↦ (\begin{matrix}A & 0 \\ 0 & B \end{matrix})

The formula D_{m+n}(x ∗ y) = D_{n}(x) · D_{m}(y) now follows from the commutative diagram

\xymatrix{H(F^{n}) ⊗ H(F^{m}) \ar[r]^{∗} \ar[d] & H(F^{n+m}) \ar[d] \\ C_{n}(F^{n}) ⊗ C_{m}(F^{m}) \ar[r]^{∗} \ar[d]^{≅} & C_{n+m}(F^{n+m}) \ar[d]^{≅} \\ ℤ[GL_{n}(F)] ⊗ ℤ[GL_{m}(F)] \ar[r]^{μ} \ar[d]^{det ⊗ det} & ℤ[GL_{n+m}(F)] \ar[d]^{det} \\ ℤ[F^{×}] ⊗ ℤ[F^{×}] \ar[r]^{·} & ℤ[F^{×}]}

(3) When n is even the maps D_{n} are split surjections, since the image is a

free module of rank 1.

It is easy to verify that the map D_{1} : S̃(F) → A_{1} = I_{F^{×}} is an isomorphism. Now let E ∈ S̃(F^{2}) be any element satisfying D_{2}(E) = ⟨1⟩ (eg. we can take E = ⌊−1, 1⌉). Then for n = 2m + 1 odd, the composite S̃(F) ∗ E^{∗m} → S̃(F^{n}) → I_{F^{×}} = A_{n} is an isomorphism.

□

We will let S̃(W, V)^{+} = Ker(D_{W,V}). Thus S̃(F^{n}) ≅ S̃(F^{n})^{+} ⊕ A_{n} as a ℤ[F^{×}]- module by the results above. Observe that it follows directly from the definitions that the image of ϵ_{V} is contained in S̃(V)^{+} for any vector space V.

3.5. The maps T_{n}.

Lemma 3.8. If n ≥ 2 and b_{1}, …, b_{n} are distinct elements of F^{×} then

[b_{1}][b_{2}] ⋯ [b_{n}] = ∑_{i=1}^{n}[b_{1} − b_{i}] ⋯ [b_{i−1} − b_{i}][b_{i}][b_{i+1} − b_{i}] ⋯ [b_{n} − b_{i}] in K_{n}^{MW}(F).

Proof. We will use induction on n starting with n = 2: Suppose that b_{1} ≠ b_{2} ∈ F^{×}. Then

[b_{1} − b_{2}]([b_{1}] − [b_{2}])

= ([b_{1}] + ⟨b_{1}⟩ [1 − \frac{b_{2}}{b_{1}} ]) (−⟨b_{1}⟩ [\frac{b_{2}}{b_{1}} ]) by Lemma 2.3 (2)
= −⟨b_{1}⟩[b_{1}] [\frac{b_{2}}{b_{1}} ] since [x][1 − x] = 0
= [b_{1}]([b_{1}] − [b_{2}]) by Lemma 2.3(2) again
= [b_{1}]([−1] − [b_{2}]) by Lemma 2.3 (1)
= [b_{1}](−⟨−1⟩[−b_{2}])
= [−b_{2}][b_{1}] by Lemma 2.3 (3).

Thus

[b_{1}][b_{2} − b_{1}] + [b_{1} − b_{2}][b_{2}] = −⟨−1⟩[b_{2} − b_{1}][b_{1}] + [b_{1} − b_{2}][b_{2}]
= −([b_{1} − b_{2}] − [−1])[b_{1}] + [b_{1} − b_{2}][b_{2}]
= −[b_{1} − b_{2}]([b_{1}] − [b_{2}]) + [−1][b_{1}]
= −[−b_{2}][b_{1}] + [−1][b_{1}] = ([−1] − [−b_{2}])[b_{1}]
= −⟨−1⟩[b_{2}][b_{1}] = [b_{1}][b_{2}]

proving the case n = 2. Now suppose that n > 2 and that the result holds for n − 1. Let b_{1}, …, b_{n} be distinct elements of F^{×}.We wish to prove that

( ∑_{i=1}^{n−1}[b_{1} − b_{i}] ⋯ [b_{i}] ⋯ [b_{n−1} − b_{i}])[b_{n}] = ∑_{i=1}^{n}[b_{1} − b_{i}] ⋯ [b_{i}] ⋯ [b_{n} − b_{i}].

We re-write this as:

∑_{i=1}^{n−1}[b_{1} − b_{i}] ⋯ [b_{i}] ⋯ [b_{n−1} − b_{i}]([b_{n}] − [b_{n} − b_{i}]) = [b_{1} − b_{n}] ⋯ [b_{n−1} − b_{n}][b_{n}].

Now

[b_{1} − b_{i}] ⋯ [b_{i}] ⋯ [b_{n−1} − b_{i}]([b_{n}] − [b_{n} − b_{i}])
= (−⟨−1⟩)^{n−i}[b_{1} − b_{i}] ⋯ [b_{n−1} − b_{i}]([b_{i}]([b_{n}] − [b_{n} − b_{i}]))
= (−⟨−1⟩)^{n−i}[b_{1} − b_{i}] ⋯ [b_{n−1} − b_{i}]([b_{i} − b_{n}][b_{n}])
= [b_{1} − b_{i}] ⋯ [b_{i} − b_{n}] ⋯ [b_{n−1} − b_{i}][b_{n}].

So the identity to be proved reduces to

( ∑_{i=1}^{n−1}[b_{1} − b_{i}] ⋯ [b_{i} − b_{n}] ⋯ [b_{n−1} − b_{i}])[b_{n}] = [b_{1} − b_{n}] ⋯ [b_{n−1} − b_{n}][b_{n}].

Letting b′_{i} = b_{i} − b_{n} for 1 ≤ i ≤ n − 1, then b_{j} − b_{i} = b′_{j} − b′_{i} for i, j ≤ n − 1 and this reduces to the case n − 1. □

Theorem 3.9.

(1) For all n ≥ 1, there is a well-defined homomorphism of ℤ[F^{×}]-modules

T_{n} : S̃(F^{n}) → K_{n}^{MW}(F)

sending ⌊a_{1}, …, a_{n}⌉ to [a_{1}] ⋯ [a_{n}].

(2) The maps {T_{n}} define a homomorphism of graded ℤ[F^{×}]-algebras S̃(F^{•}) → K_{•}^{MW}(F): We have

T_{n+m}(x ∗ y) = T_{n}(x) · T_{m}(y), for all x ∈ S̃(F^{n}), y ∈ S̃(F^{m}).

Proof.

(1) By Theorem 3.3, in order to show that T_{n} is well-defined we must prove the identity

[b_{1}a_{1}] ⋯ [b_{n}a_{n}] − [a_{1}] ⋯ [a_{n}] =

∑_{i=1}^{n}(−⟨−1⟩)^{n+i}⟨a_{i}⟩[a_{1}(b_{1} − b_{i})] ⋯ [{ \atop a_{i}(b_{i} − b_{i})}] ⋯ [a_{n}(b_{n} − b_{i}][b_{i}]

in K_{n}^{MW}(F).

Writing [b_{i}a_{i}] = [a_{i}] + ⟨a_{i}⟩[b_{i}] and [a_{j}(b_{j} − b_{i})] = [a_{j}] + ⟨a_{j}⟩[b_{j} − b_{i}] and expanding the products on both sides and using (3) of Lemma 2.3 to permute terms, this identity can be rewritten as

∑_{∅≠I⊂{1,...,n}}(−⟨−1⟩)^{sgn(σ_{I})}⟨a_{i_{1}} ⋯ a_{i_{k}}⟩[a_{j_{1}}] ⋯ [a_{j_{s}}][b_{i_{1}}] ⋯ [b_{i_{k}}] =

∑_{∅≠I⊂{1,...,n}}(−⟨−1⟩)^{sgn(σ_{I})}⟨a_{i_{1}} ⋯ a_{i_{k}}⟩[a_{j_{1}}] ⋯ [a_{j_{s}}]×

×( ∑_{t=1}^{k}[b_{i_{1}} − b_{i_{t}}] ⋯ [b_{i_{t}}] ⋯ [b_{i_{k}} − b_{i_{t}}])

where I = {i_{1} < ⋯ < i_{k}} and the complement of I is {j_{1} < ⋯ < j_{s}} (so that k + s = n) and σ_{I} is the permutation

(\begin{matrix}1 & … & s & s + 1 & … & n \\ j_{1} & … & j_{s} & i_{1} & … & i_{k} \end{matrix}) .

The result now follows from the identity of Lemma 3.8.

(2) We can assume that x = ⌊a_{1}, …, a_{n}⌉ and y = ⌊a′_{1}, …, a′_{m}⌉ with a_{i}, a′_{j} ∈ F^{×}. From the definition of T_{n+m} and the formula of Lemma 3.4,

T_{n+m}(x ∗ y) =

∑_{i=1}^{n} ∑_{j=1}^{m}(−1)^{n+m+i+j} ⟨(−1)^{i+j}a_{i}a′_{j}⟩ ×

× [a_{1}(b_{1} − b_{i})] ⋯ [{ \atop a_{i}(b_{i} − b_{i})}] ⋯ [b_{i}][a′_{1}(b′_{1} − b′_{j})] ⋯ [{ \atop a′_{j}(b′_{j} − b′_{j})}] ⋯ [b′_{j}]

+(−1)^{n} ∑_{i=1}^{n}(−1)^{i+1} ⟨(−1)^{i+1}a_{i}⟩ [a_{1}(b_{1} − b_{i})] ⋯ [{ \atop a_{i}(b_{i} − b_{i})}] ⋯ [b_{i}][b′_{1}a′_{1}] ⋯ [b′_{m}a′_{m}]

+(−1)^{m} ∑_{j=1}^{m}(−1)^{j+1} ⟨(−1)^{j+1}a′_{j}⟩ [b_{1}a_{1}] ⋯ [b_{n}a_{n}][a′_{1}(b′_{1} − b′_{j})] ⋯ [{ \atop a′_{j}(b′_{j} − b′_{j})}] ⋯ [b′_{j}]

+[b_{1}a_{1}] ⋯ [b_{n}a_{n}][b_{i}][b′_{1}a′_{1}] ⋯ [b′_{m}a′_{m}]

which factors as X · Y with X =

∑_{i=1}^{n}(−1)^{n+i+1} ⟨(−1)^{i+1}a_{i}⟩ [a_{1}(b_{1} − b_{i})] ⋯ [{ \atop a_{i}(b_{i} − b_{i})}] ⋯ [b_{i}] + [b_{1}a_{1}] ⋯ [b_{n}a_{n}]

= [a_{1}] ⋯ [a_{n}] = T_{n}(x) by part (1)
and Y =

∑_{j=1}^{m}(−1)^{m+j+1} ⟨(−1)^{j+1}a′_{j}⟩ [a′_{1}(b′_{1} − b′_{j})] ⋯ [{ \atop a′_{j}(b′_{j} − b′_{j})}] ⋯ [b′_{j}] + [b′_{1}a′_{1}] ⋯ [b′_{m}a′_{m}]

= [a′_{1}] ⋯ [a′_{m}] = T_{m}(y) by (1) again.

□

Note that T_{1} is the natural surjective map S̃(F) ≅ I_{F^{×}} → K_{1}^{MW}(F), ⌊a⌉ ↔ ⟨⟨a⟩⟩ ↦ [a]. It has a nontrivial kernel in general. Note furthermore that SH_{2}(F) = H_{2}(SL_{2}(F), ℤ). It is well-known ([24],[11], and [7]) that H_{2}(SL_{2}(F), ℤ) ≅ K_{2}^{M}(F) ×_{k_{2}^{M}(F)} I^{2}(F) ≅ K_{2}^{MW}(F). In fact we have:

Theorem 3.10. The composite T_{2} ∘ ϵ_{2} : H_{2}(SL_{2}(F), ℤ) → K_{2}^{MW}(F) is an isomorphism.

Proof. For p ≥ 1, let X̄_{p}(F) denote the set of all p-tuples (x_{1}, …, x_{p}) of points of ℙ^{1}(F) and let X̄_{0}(F) = ∅. We let C̄_{p}(F) denote the GL_{2}(F) permutation module ℤ[X̄_{p}(F)] and form a complex C̄_{•}(F) using the natural simplicial boundary maps, d̄_{p}. This complex is acyclic and the map F^{2} \ {0} → ℙ^{1}(F), v ↦ \overline{v} induces a map of complexes C_{•}(F^{2}) → C̄_{•}(F). Let H̄_{2}(F) := Ker(d̄_{2} : C̄_{2}(F) → C̄_{1}(F)) and let S̄_{2}(F) = H_{0}(SL_{2}(F), H̄_{2}(F)). We obtain a commutative diagram of SL_{2}(F)-modules with exact rows:

\xymatrix{C_{4}(F^{2}) \ar[d] \ar[r]^{d_{4}} & C_{3}(F^{2}) \ar[d] \ar[r]^{d_{3}} & H(F^{2}) \ar[d] \ar[r] & 0 \\ C̄_{4}(F) \ar[r]^{d̄_{4}} & C̄_{3}(F) \ar[r]^{d̄_{3}} & H̄_{2}(F) \ar[r] & 0}

Taking SL_{2}(F)-coinvariants gives the diagram

\xymatrix{H_{0}(SL_{2}(F), C_{4}(F^{2})) \ar[d] \ar[r]^{d_{4}} & H_{0}(SL_{2}(F), C_{3}(F^{2})) \ar[d] \ar[r]^{d_{3}} & S̃(F^{2}) \ar[d]^{ϕ} \ar[r] & 0 \\ H_{0}(SL_{2}(F), C̄_{4}(F)) \ar[r]^{d̄_{4}} & H_{0}(SL_{2}(F), C̄_{3}(F)) \ar[r]^{d̄_{3}} & S̄_{2}(F) \ar[r] & 0}

Now the calculations of Mazzoleni, [11], show that H_{0}(SL_{2}(F), C̄_{3}(F)) ≅ ℤ[F^{×}/(F^{×})^{2}] via

class of (∞, 0, a) ↦ ⟨a⟩ ∈ ℤ[F^{×}/(F^{×})^{2}],

where a ∈ ℙ^{1}(F) = \overline{e_{1} + ae_{2}} and ∞ := \overline{e_{1}}. Furthermore S̄_{2}(F) ≅ GW(F) in such a way that the induced map ℤ[F^{×}/(F^{×})^{2}] → GW(F) is the natural one. Since ⌊a, b⌉ = d_{3}(e_{1}, e_{2}, ae_{1} + be_{2}), it follows that ϕ(⌊a, b⌉) = ⟨a/b⟩ = ⟨ab⟩ in GW(F). Associated to the complex C̄_{•}(F) we have an iterated connecting homomorphism ω : H_{2}(SL_{2}(F), ℤ) → S̄_{2}(F) = GW(F). Observe that ω = ϕ ∘ ϵ_{2}. In fact, (Mazzoleni, [11], Lemma 5) the image of ω is I^{2}(F) ⊂ GW(F). On the other hand, the module S̃(F^{2})^{+} is generated by the elements [[a, b]] := ⌊a, b⌉−D_{2}(⌊a, b⌉)·E (where E, as above, denotes the element ⌊−1, 1⌉). Note that T_{2}([[a, b]]) = T_{2}(⌊a, b⌉) = [a][b] since T_{2}(E) = [−1][1] = 0 in K_{2}^{MW}(F). Furthermore,

ϕ([[a, b]]) = ϕ(⌊a, b⌉) − D_{2}(⌊a, b⌉)ϕ(E)
= ⟨ab⟩ − (⟨−a⟩ − ⟨b⟩ + ⟨1⟩)⟨−1⟩
= ⟨ab⟩ − ⟨a⟩ + ⟨−b⟩ − ⟨−1⟩
= ⟨ab⟩ − ⟨a⟩ − ⟨b⟩ + ⟨1⟩
= ⟨⟨a, b⟩⟩

(using the identity ⟨b⟩ + ⟨−b⟩ = ⟨1⟩ + ⟨−1⟩ in GW(F)). Using these calculations we thus obtain the commutative diagram

\xymatrix{H_{2}(SL_{2}(F), ℤ) \ar[r]^{ϵ_{2}} \ar[dr]^{ω} & S̃(F^{2})^{+} \ar[d]^{ϕ} \ar[r]^{T_{2}} & K_{2}^{MW}(F) \ar[dl] \\  & I^{2}(F)}

Now, the natural embedding F^{×} → SL_{2}(F), a ↦ diag(a, a^{−1}) := ã induces a homomorphism, μ:

⋀^{2} (F^{×}) ≅ H_{2}(F^{×}, ℤ) → H_{2}(SL_{2}(F), ℤ),
a ∧ b ↦ ([ã|b̃] − [b̃|ã]) ⊗ 1 ∈ B_{2}(SL_{2}(F)) ⊗_{ℤ[SL_{2}(F)]} ℤ.

Mazzoleni’s calculations (see [11], Lemma 6) show that μ(⋀^{2} (F^{×})) = Ker(ω) and that there is an isomorphism μ(⋀^{2} (F^{×})) ≅ 2 · K_{2}^{M}(F) given by μ(a ∧ b) ↦ 2{a, b}. On the other hand, a straightforward calculation shows that ϵ_{2} (μ(a ∧ b)) =

⟨a⟩ ⌊b, \frac{1}{ab}⌉ − ⌊b, \frac{1}{b}⌉ − ⟨a⟩ ⌊1, \frac{1}{a}⌉ + ⟨b⟩ ⌊1, \frac{1}{b}⌉ + ⌊a, \frac{1}{a}⌉ − ⟨b⟩ ⌊a, \frac{1}{ab}⌉ := C_{a,b}

Now by the diagram above,

T_{2}(C_{a,b}) = T_{2}(ϵ_{2} (μ(a ∧ b))) ∈ Ker(K_{2}^{MW}(F) → I^{2}(F)) ≅ 2K_{2}^{M}(F).

Recall that the natural embedding 2K_{2}^{M}(F) → K_{2}^{MW}(F) is given by 2{a, b} ↦ [a^{2}][b] = [a][b] − [b][a] and the composite

\xymatrix{2K_{2}^{M}(F) \ar[r] & K_{2}^{MW}(F) \ar[r]^{κ_{2}} & K_{2}^{M}(F)}

is the natural inclusion map. Since

κ_{2} (T_{2}(C_{a,b})) = \{b, \frac{1}{ab}\} − \{b, \frac{1}{b}\} − \{1, \frac{1}{a}\} + \{1, \frac{1}{b}\} + \{a, \frac{1}{a}\} − \{a, \frac{1}{ab}\}
= {a, b} − {b, a} = 2{a, b},

it follows that we have a commutative diagram with exact rows

\xymatrix{0 \ar[r] & μ(⋀^{2} (F^{×})) \ar[r] \ar[d]^{≅} & H_{2}(SL_{2}(F), ℤ) \ar[r]^{ω} \ar[d]^{T_{2}∘ϵ_{2}} & I^{2}(F) \ar[r] \ar[d]^{=} & 0 \\ 0 \ar[r] & 2K_{2}^{M}(F) \ar[r] & K_{2}^{MW}(F) \ar[r] & I^{2}(F) \ar[r] & 0}

proving the theorem. □

4. AM-modules

From the results of the last section, it follows that there is a ℤ[F^{×}]- decomposition

S̃(F^{2}) ≅ K_{2}^{MW}(F) ⊕ ℤ[F^{×}]⊕?

It is not difficult to determine that the missing factor is isomorphic to the 1-dimensional vector space F (with the tautological F^{×}-action). However, as we will see, this extra term will not play any role in the calculations of H_{n}(SL_{k}(F), ℤ). As ℤ[F^{×}]-modules, our main objects of interest (Milnor-Witt K-theory, the homology of the special linear group, the powers of the fundamental ideal in the Grothendieck-Witt ring) are what we call below ‘multiplicative ’; there exists m ≥ 1 such that, for all a ∈ F^{×}, ⟨a^{m}⟩ acts trivially. This is certainly not true of the vector space F above. In this section we formalise this difference, and use this formalism to prove an analogue of Suslin’s Theorem 1.8 ([23]) (see Theorem 4.23 below). Throughout the remainder of this article, F will denote a field of characteristic 0.

Let S_{F} ⊂ ℤ[F^{×}] denote the multiplicative set generated by the elements {⟨⟨a⟩⟩ = ⟨a⟩ − 1 | a ∈ F^{×} \ {1}}. Note that 0 ∉ S_{F}, since the elements of S_{F} map to units under the natural ring homomorphism ℤ[F^{×}] → F. We will also let S_{ℚ}^{+} ⊂ ℤ[ℚ^{×}] denote the multiplicative set generated by {⟨⟨a⟩⟩ = ⟨a⟩−1 | a ∈ ℚ^{×} \ {±1}}.

Definition 4.1. A ℤ[F^{×}]-module M is said to be multiplicative if there exists s ∈ S_{ℚ}^{+} with sM = 0.

Definition 4.2. We will say that a ℤ[F^{×}]-module is additive if every s ∈ S_{ℚ}^{+} acts as an automorphism on M.

Example 4.3. Any trivial ℤ[F^{×}]-module M is multiplicative , since ⟨⟨a⟩⟩ annihilates M for all a ≠ 1.

Example 4.4. GW(F), and more generally I^{n}(F), is multiplicative since ⟨⟨a^{2}⟩⟩ annihilates these modules for all a ∈ F^{×}.

Example 4.5. Similarly, the groups H_{n}(SL_{n}(F), ℤ) are multiplicative since they are annihilated by the elements ⟨⟨a^{m}⟩⟩.

Example 4.6. Any vector space over F, with the induced action of ℤ[F^{×}], is additive since all elements of S_{F} act as automorphisms.

Example 4.7. More generally, if V is a vector space over F, then for all r ≥ 1, the rth tensor power T_{ℤ}^{r}(V) = T_{ℚ}^{r}(V) is an additive module since, if a ∈ ℚ \ {±1}, ⟨a⟩ acts as multiplication by a^{r} and hence ⟨⟨a⟩⟩ acts as multiplication by a^{r} − 1. For the same reasons, the rth exterior power, ⋀_{ℤ}^{r}(V), is an additive module.

Remark 4.8. Observe that if ⟨⟨a^{m}⟩⟩ acts as an automorphism of the ℤ[F^{×}]- module M for some a ∈ F^{×}, m > 1, then so does ⟨⟨a⟩⟩, since ⟨⟨a^{m}⟩⟩ = ⟨⟨a⟩⟩(⟨a^{m−1}⟩ + ⋯ + ⟨a⟩ + 1) = (⟨a^{m−1}⟩ + ⋯ + ⟨a⟩ + 1)⟨⟨a⟩⟩ in ℤ[F^{×}].

Lemma 4.9. Let

0 → M_{1} → M → M_{2} → 0

be a short exact sequence of ℤ[F^{×}]-modules. Then M is multiplicative if and only if M_{1} and M_{2} are.

Proof. Suppose M is multiplicative . If s ∈ S_{ℚ}^{+} satisfies sM = 0, it follows that sM_{1} = sM_{2} = 0. Conversely, if M_{1} and M_{2} are multiplicative then there exist s_{1}, s_{2} ∈ S_{ℚ}^{+} with s_{i}M_{i} = 0 for i = 1, 2. It follows that sM = 0 for s = s_{1}s_{2} ∈ S_{ℚ}^{+}. □

Lemma 4.10. Let

0 → A_{1} → A → A_{2} → 0

be a short exact sequence of ℤ[F^{×}]-modules. If A_{1} and A_{2} are additive modules, then so is A.

Proof. This is immediate from the definition. □

Lemma 4.11. Let ϕ : M → N be a homomorphism of ℤ[F^{×}]-modules. (1) If M and N are multiplicative , then so are Ker(ϕ) and Coker(ϕ).

(2) If M and N are additive , then so are Ker(ϕ) and Coker(ϕ).

Proof. (1) This follows from Lemma 4.9 above.

(2) If s ∈ S_{ℚ}^{+}, then s acts as an automorphism of M and N, and hence of

Coker(ϕ) and Ker(ϕ).

□

Corollary 4.12. Let C = (C_{•}, d) be a complex of ℤ[F^{×}]-modules. If C_{•} is additive (i.e. if each C_{n} is an additive module), then each H_{n}(C) is an additive module. If each C_{n} is multiplicative then each H_{n}(C) is a multiplicative module.

Lemma 4.13. Let M be a multiplicative ℤ[F^{×}]-module and A an additive ℤ[F^{×}]-module. Then Hom_{ℤ[F^{×}]}(M, A) = 0 and Hom_{ℤ[F^{×}]}(A, M) = 0.

Proof. Let f : M → A be a ℤ[F^{×}]-homomorphism. Every s ∈ S_{ℚ}^{+} acts as an automorphism of A. However, there exists s ∈ S_{ℚ}^{+} with sM = 0. Thus, for m ∈ M, 0 = f(sm) = sf(m) ⟹ f(m) = 0. Let g : A → M be a ℤ[F^{×}]-homomorphism. Again, choose s ∈ S_{ℚ}^{+} acting as an automorphism of A and annihilating M. If a ∈ A, then there exists b ∈ a with a = sb. Hence g(a) = sg(b) = 0 in M. □

Lemma 4.14. If P is a ℤ[F^{×}]-module and if A is an additive submodule and M a multiplicative submodule, then A ∩ M = 0.

Proof. There exists s ∈ ℤ[ℚ^{×}] which annihilates any submodule of M but is injective on any submodule of A. □

Lemma 4.15.

(1) If

\xymatrix{0 \ar[r] & M \ar[r] & H \ar[r]^{π} & A \ar[r] & 0}

is an exact sequence of ℤ[F^{×}]-modules with M multiplicative and A additive then the sequence splits (over ℤ[F^{×}]).

(2) Similarly, if

\xymatrix{0 \ar[r] & A \ar[r] & H \ar[r] & M \ar[r] & 0}

is an exact sequence of ℤ[F^{×}]-modules with M multiplicative and A additive then the sequence splits.

Proof. As above we can find s ∈ ℤ[ℚ^{×}] such that s · M = 0 and s acts as an automorphism of A.

(1) Then sH is a ℤ[F^{×}]-submodule of H and π induces an isomorphism sH ≅ A, since π(sH) = sπ(H) = sA = A and if π(sh) = 0 then sπ(h) = 0 in A, so that π(h) = 0 and h ∈ M.

(2) We have sH = A and multiplication by s gives an automorphism, α, of A. Thus the ℤ[F^{×}]-homomorphism H → A, h ↦ α^{−1}(s · h) splits the sequence.

□

Definition 4.16. We will say that a ℤ[F^{×}]-module H is an AM module if there exists a multiplicative ℤ[F^{×}]-module M and an additive ℤ[F^{×}] module A and an isomorphism of ℤ[F^{×}]-modules H ≅ A ⊕ M.

Lemma 4.17. Let H be an AM module and let ϕ : H → A ⊕ M be an isomorphism of ℤ[F^{×}]-modules, with M multiplicative and A additive . Then

ϕ^{−1}(A) = ⋃_{A′⊂H,A′additive} A′  and  ϕ^{−1}(M) = ⋃_{M′⊂H,M′multiplicative} M′

Proof. Let M′ ⊂ H be multiplicative. Then the composite

\xymatrix{M′ \ar[r] & H \ar[r]^{ϕ} & A ⊕ M \ar[r] & A}

is zero by Lemma 4.13, and thus M′ ⊂ ϕ^{−1}(M). An analogous argument can be applied to ϕ^{−1}(A). □

It follows that the submodules ϕ^{−1}(A) and ϕ^{−1}(M) are independent of the choice of ϕ, A and M. We will denote the first as H_{A} and the second as H_{M}. Thus if H is an AM module then there is a canonical decomposition H = H_{A} ⊕H_{M}, where H_{A} (resp. H_{M}) is the maximal additive (resp. multiplicative ) submodule of H. We have canonical projections

π_{A} : H → H_{A}, π_{M} : H → H_{M}.

Lemma 4.18. Let H be a AM module. Suppose that H is also a module over a ring R and that the action of R commutes with that of ℤ[F^{×}]. Then H_{A} and H_{M} are R-submodules of H.

Proof. Let r ∈ R. Then the composite

\xymatrix{H_{A} \ar[r]^{r·} & H \ar[r]^{π_{M}} & H_{M}}

is a ℤ[F^{×}]-homomorphism and thus is 0 by Lemma 4.13. It follows that r·H_{A} ⊂ Ker(π_{M}) = H_{A}. □

Lemma 4.19. Let f : H → H′ be a ℤ[F^{×}]-homomorphism of AM modules. Then there exist ℤ[F^{×}]-homomorphisms f_{A} : H_{A} → H′_{A} and f_{M} : H_{M} → H′_{M} such that f = f_{A} ⊕ f_{M}. Suppose that H and H′ are modules over a ring R and that the R-action commutes with the ℤ[F^{×}]-action in each case. If f is an R-homomorphism, then so are f_{A} and f_{M}.

Proof. This is immediate from Lemmas 4.13 and 4.18. □

Lemma 4.20. If

\xymatrix{0 \ar[r] & L \ar[r]^{j} & H \ar[r]^{π} & K \ar[r] & 0}

is a short exact sequence of ℤ[F^{×}]-modules and if L and K are AM modules, then so is H.

Proof. Let H̃ = π^{−1}(K_{M}). Then the exact sequence

0 → L → H̃ → K_{M} → 0

gives the exact sequence

0 → \frac{L}{L_{M}} → \frac{H̃}{j(L_{M})} → K_{M} → 0.

Since L/L_{M} ≅ L_{A} is additive , this latter sequence is split, by Lemma 4.15 (2). So H̃/j(L_{M}) is a AM module, and there is a ℤ[k^{×}]-isomorphism

\xymatrix{H̃/j(L_{M}) \ar[r]^{ϕ}_{≅} & L_{A} ⊕ K_{M}.}

Let ϕ̄ be the composite

\xymatrix{H̃ \ar[r] & H̃/j(L_{M}) \ar[r]^{ϕ} & L_{A} ⊕ K_{M}.}

Let H_{m} = ϕ̄^{−1}(K_{M}) ⊂ H̃ ⊂ H. Then, we have an exact sequence

0 → L_{M} → H_{m} → K_{M} → 0

so that H_{m} is multiplicative . On the other hand, since H̃/H_{m} ≅ L_{A} and H/H̃ ≅ K_{A}, we have a short exact sequence

0 → L_{A} → \frac{H}{H_{m}} → K_{A} → 0.

This implies that H/H_{m} is additive , and thus H is AM by Lemma 4.15 (1). □

Lemma 4.21. Let (C_{•}, d) be a complex of ℤ[k^{×}]-modules. If each C_{n} is AM, then H_{•}(C) is AM, and furthermore

H_{•}(C_{A}) = H_{•}(C)_{A}
H_{•}(C_{M}) = H_{•}(C)_{M}

Proof. The differentials d decompose as d = d_{A} ⊕ d_{M} by Lemma 4.19. □

Theorem 4.22. Let (E^{r}, d^{r}) be a first quadrant spectral sequence of ℤ[k^{×}]- modules converging to the ℤ[k^{×}]-module H_{•} = {H_{n}}_{n≥0}. If for some r_{0} ≥ 1 all of the modules E_{p,q}^{r_{0}} are AM, then the same holds for all the modules E_{p,q}^{r} for all r ≥ r_{0} and hence for the modules E_{p,q}^{∞}. Furthermore, H_{•} is AM and the spectral sequence decomposes as a direct sum E^{r} = E^{r}_{A} ⊕ E^{r}_{M} (r ≥ r_{0}) with E^{r}_{A} converging to H_{•}_{A} and E^{r}_{M} converging to H_{•}_{M}.

Proof. Since E^{r+1} = H(E^{r}, d^{r}) for all r, the first statement follows from Lemma 4.21. Since E^{r} is a first quadrant spectral sequence (and, in particular, is bounded), it follows that for any fixed (p, q), E_{p,q}^{∞} = E_{p,q}^{r} for all sufficiently large r. Thus E^{∞} is also AM. Now H_{n} admits a filtration 0 = F_{0}H_{n} ⊂ ⋯ ⊂ F_{n}H_{n} = H_{n} with corresponding quotients gr_{p}H_{n} ≅ E_{p,n−p}^{∞}. Since all the quotients are AM, it follows by Lemma 4.20, together with an induction on the filtration length, that H_{n} is AM. The final two statements follow again from Lemma 4.21. □

If G is a subgroup of GL(V), we let SG denote G ∩ SL(V).

Theorem 4.23. Let V, W be finite-dimensional vector spaces over F and let G_{1} ⊂ GL(W), G_{2} ⊂ GL(V) be subgroups and suppose that G_{2} contains the group F^{×} of scalar matrices. Let M be a subspace of Hom_{F} (V, W) for which G_{1}M = M = MG_{2}. Let

G = (\begin{matrix}G_{1} & M \\ 0 & G_{2} \end{matrix}) ⊂ GL(W ⊕ V).

Then, for i ≥ 1, the groups H_{i}(SG, ℤ) are AM and the natural embedding j : S(G_{1} × G_{2}) → SG induces an isomorphism

H_{i}(S(G_{1} × G_{2}), ℤ) ≅ H_{i}(SG, ℤ)_{M}.

Proof. We begin by noting that the groups H_{i}(SG, ℤ) are ℤ[F^{×}]-modules: The action of F^{×} is derived from the short exact sequence

\xymatrix{1 \ar[r] & SG \ar[r] & G \ar[r]^{det} & F^{×} \ar[r] & 1}

We have a split extension of groups (split by the map j) which is F^{×}-stable:

\xymatrix{0 \ar[r] & M \ar[r] & SG \ar[r]^{π} & S(G_{1} × G_{2}) \ar[r] & 1.}

The resulting Hochschild-Serre spectral sequence has the form

E_{p,q}^{2} = H_{p}(S(G_{1} × G_{2}), H_{q}(M, ℤ)) ⟹ H_{p+q}(SG, ℤ).

This spectral sequence exists in the category of ℤ[F^{×}]-modules and all differentials and edge homomorphisms are ℤ[F^{×}]-maps. Since the map π is split by j it induces a split surjection on integral homology groups. Thus

H_{n}(S(G_{1} × G_{2}), ℤ) = E_{n,0}^{2} = E_{n,0}^{∞} for all n ≥ 0.

Observe furthermore that the ℤ[F^{×}]-module H_{n}(S(G_{1} × G_{2}), ℤ) is multiplicative : Given a ∈ F^{×}, the element

ρ_{a} := (\begin{matrix}Id_{W} & 0 \\ 0 & a · Id_{V} \end{matrix} ) ∈ G

has determinant a^{m} ( m = dim_{F} (V)) and centralizes S(G_{1} × G_{2}). It follows that ⟨a^{m}⟩ acts trivially on H_{n}(S(G_{1} × G_{2}), ℤ) for all n; i.e. ⟨⟨a^{m}⟩⟩ annihilates H_{n}(S(G_{1} × G_{2}), ℤ). Recall (Example 4.7 above) that for q ≥ 1, the modules H_{q}(M, ℤ) = ⋀_{ℤ}^{q}(M), with the ℤ[F^{×}]-action derived from the action of F by scalars on M, are additive modules. Now if a ∈ F^{×}, then conjugation by ρ_{a} is trivial on S(G_{1} × G_{2}) but acts on M as scalar multiplication by a. It follows that for q > 0, ⟨⟨a^{m}⟩⟩ acts as an automorphism on H_{p}(S(G_{1} × G_{2}), H_{q}(M, ℤ)) for all a ∈ ℚ \ {±1}. Thus, for q > 0, the groups H_{p}(S(G_{1} × G_{2}), H_{q}(M, ℤ)) are additive ℤ[F^{×}]-modules; i.e., all E_{p,q}^{2} are additive for q > 0. It follows at once that the groups E_{p,q}^{∞} are additive for all q > 0. Thus, from the convergence of the spectral sequence, we have a short exact sequence

0 → H → H_{n}(SG, ℤ) → E_{n,0}^{∞} = j (H_{n}(S(G_{1} × G_{2}), ℤ)) → 0

and H has a filtration whose graded quotients are all additive . So H_{n}(SG, ℤ) is AM as claimed, and H_{n}(SG, ℤ)_{M} ≅ H_{n}(S(G_{1} × G_{2}), ℤ).

□

Corollary 4.24. Suppose that W′ ⊂ W. Then there is a corresponding inclusion SA(W′, V) → SA(W, V). This inclusion induces an isomorphism

\xymatrix{H_{n}(SA(W′, V), ℤ)_{M} \ar[r]_{≅} & H_{n}(SA(W, V), ℤ)_{M} ≅ H_{n}(SL(V), ℤ)}

for all n ≥ 1.

5. The spectral sequences

Recall that F is a field of characteristic 0 throughout this section. In this section we use the complexes C_{•}(W, V) to construct spectral sequences converging to 0 in dimensions less than n = dim_{F} (V), and to S̃(W, V) in dimension n. By projecting onto the multiplicative part, we obtain spectral sequences with good properties: the terms in the E^{1}-page are just the kernels and cokernels of the stabilization maps f_{t,n} : H_{n}(SL_{t}(F), ℤ) → H_{n}(SL_{t+1}(F), ℤ). We then prove that the higher differentials are all zero. Since the spectral sequences converge to 0 in low degrees, this already implies the main stability result (Corollary 5.11); the maps f_{t,n} are isomorphisms for t ≥ n + 1 and are surjective for t = n. The remainder of the paper is devoted to an analysis of the case t = n − 1, which requires some more delicate calculations. Let C_{•}^{τ} (W, V) denote the truncated complex.

C_{p}^{τ} (W, V) = \{ \begin{matrix}C_{p}(W, V), & p ≤ dim_{F} (V) \\ 0, & p > dim_{F} (V) \end{matrix}

Thus

H_{p}(C_{•}^{τ} (W, V)) = \{ \begin{matrix}0, & p ≠ n \\ H(W, V), & p = n \end{matrix}

where n = dim_{F} (V).

Thus the natural action of SA(W, V) on C_{•}^{τ} (W, V) gives rise to a spectral sequence E(W, V) which has the form

E_{p,q}^{1} = H_{p}(SA(W, V), C_{q}^{τ} (W, V)) ⟹ H_{p+q−n}(SA(W, V), H(W, V)).

The groups C_{q}^{τ} (W, V) are permutation modules for SA(W, V) and thus the E^{1}- terms (and the differentials d^{1}) can be computed in terms of the homology of stabilizers. Fix a basis {e_{1}, …, e_{n}} of V. Let V_{r} be the span of {e_{1}, …, e_{r}} and let V′_{s} be the span of {e_{n−s}, …, e_{n}}, so that V = V_{r} ⊕ V′_{n−r} if 0 ≤ r ≤ n. For any 0 ≤ q ≤ n − 1, the group SA(W, V) acts transitively on the basis of C_{q}^{τ} (W, V) and the stabilizer of

((0, e_{1}), …, (0, e_{q}))

is SA(W ⊕ V_{q}, V′_{n−q}). Thus, for q ≤ n − 1,

E_{p,q}^{1} = H_{p}(SA(W, V), C_{q}^{τ} (W, V)) ≅ H_{p}(SA(W ⊕ V_{q}, V′_{n−q}), ℤ)

by Shapiro’s Lemma. By the results in section 4 we have:

Lemma 5.1. The terms E_{p,q}^{1} in the spectral sequence E(W, V) are AM for q > 0, and

(E_{p,q}^{1})_{M} = H_{p}(SL(V′_{n−q}), ℤ) ≅ H_{p}(SL_{n−q}(F), ℤ).

For q = n, the orbits of SA(W, V) on the basis of C_{n}^{τ}(W, V) are in bijective correspondence with F^{×} via

((w_{1}, v_{1}), …, (w_{n}, v_{n})) ↦ det ([v_{1}| ⋯ |v_{n}]_{E}) .

The stabilizer of any basis element of C_{n}^{τ}(W, V) is trivial. Thus

E_{p,n}^{1} = \{ \begin{matrix}ℤ[F^{×}], & p = 0 \\ 0, & p > 0 \end{matrix}

Of course, E_{p,q}^{1} = 0 for q > n. The first column of the E^{1}-page of the spectral sequence E(W, V) has the form

E_{0,q}^{1} = \{ \begin{matrix}ℤ, & q < n \\ ℤ[F^{×}], & q = n \\ 0, & q > n \end{matrix}

and the differentials are easily computed: For q < n

d_{0,q}^{1} : E_{0,q}^{1} → E_{0,q}^{1} = \{ \begin{matrix}Id_{ℤ}, & q is odd \\ 0, & q is even \end{matrix}

and

d_{0,n}^{1} : ℤ[F^{×}] → ℤ = \{ \begin{matrix}augmentation , & n odd \\ 0, & n even \end{matrix}

It follows that E_{0,q}^{2} = 0 for q ≠ n and

E_{0,n}^{2} = \{ \begin{matrix}I_{F^{×}}, & n odd \\ ℤ[F^{×}], & n even \end{matrix}

Note that the composite

\xymatrix{S̃(W, V) \ar[r]^{edge} & E_{0,n}^{∞} ⊂ E_{0,n}^{2} = A_{n}}

is just the map D_{W,V} of section 3 above.

Lemma 5.2. The map D_{W,V} is a split surjective homomorphism of ℤ[F^{×}]- modules.

Proof. If W = 0, this is Lemma 3.7 (1) and (3), since V ≅ F^{n}. In general the natural map of complexes C_{•}^{τ} (V) → C_{•}^{τ} (W, V) gives rise to a commutative diagram of ℤ[F^{×}]-modules

\xymatrix{S̃(V) \ar[rr] \ar[dr]_{D_{V}} &  & S̃(W, V) \ar[dl]^{D_{W,V}} \\  & A_{n}}

□

We let S̃(W, V)^{+} := Ker(D_{W,V} : S̃(W, V) → A_{n}), so that S̃(W, V) ≅ S̃(W, V)^{+} ⊕ A_{n} for all W, V.

Corollary 5.3. In the spectral sequence E(W, V), we have E_{0,q}^{2} = E_{0,q}^{∞} for all q ≥ 0. All higher differentials d_{0,q}^{r} : E_{0,q}^{r} → E_{r−1,q+r}^{r} are zero.

It follows that the spectral sequences E(W, V) decompose as a direct sum of two spectral sequences

E(W, V) = E^{0}(W, V) ⊕ E^{+}(W, V)

where E^{0}(W, V) is the first column of E(W, V) and E^{+}(W, V) involves only the terms E_{p,q}^{r} with q > 0. The spectral sequence E^{0}(W, V) converges in degree d to

\{ \begin{matrix}0, & d ≠ n \\ A_{n}, & d = n \end{matrix}

The spectral sequence E^{+}(W, V) converges in degree d to

\{ \begin{matrix}0, & d < n \\ S̃(W, V)^{+}, & d = n \\ H_{d−n}(SA(W, V), H(W, V)), & d > n \end{matrix}

By Lemma 5.1 above, all the terms of the spectral sequence E^{+}(W, V) are AM. We thus have

Corollary 5.4.

(1) The ℤ[F^{×}]-modules S̃(W, V)^{+} are AM.
(2) The graded submodule S̃(F^{•})^{+}_{A} ⊂ S̃(F^{•}) is an ideal.

Proof.

(1) This follows from Theorem 4.22.

(2) This follows from Lemma 4.18, since S̃(F^{•})^{+} is an ideal in S̃(F^{•}) by

Lemma 3.7 (2).

□

Corollary 5.5. The natural embedding H(V) → H(W, V) induces an isomorphism

\xymatrix{S̃(V)^{+}_{M} \ar[r]^{≅} & S̃(W, V)^{+}_{M}.}

Proof. The map of complexes of SL(V)-modules C_{•}^{τ} (V) → C_{•}^{τ} (W, V) gives rise to a map of spectral sequences E^{+}(V) → E^{+}(W, V) and hence a map E^{+}(V)_{M} → E^{+}(W, V)_{M}. The induced map on the E^{1}-terms is

\xymatrix{H_{p}(SL_{n−q}(F), ℤ) \ar[r]^{Id} \ar[d]^{≅} & H_{p}(SL_{n−q}(F), ℤ) \ar[d]^{≅} \\ _{H_{p}(SL(V), C_{q}^{τ} (V))_{M}} \ar[r] & _{H_{p}(SA(W, V), C_{q}^{τ} (W, V))_{M}}}

and thus is an isomorphism. It follows that there is an induced isomorphism of abutments

S̃(V)^{+}_{M} ≅ S̃(W, V)^{+}_{M}

and

H_{k}(SL(V), H(V))_{M} ≅ H_{k}(SA(W, V), H(W, V))_{M}.
□

For convenience, we now define

S̃(W, V)_{M} := \frac{S̃(W, V)}{S̃(W, V)^{+}_{A}}

(even though S̃(W, V) is not an AM module). This gives:

Corollary 5.6.

S̃(W, V)_{M} ≅ S̃(W, V)^{+}_{M} ⊕ A_{n} ≅ S̃(V)^{+}_{M} ⊕ A_{n} ≅ S̃(V)_{M}

as ℤ[F^{×}]-modules, and S̃(F^{•})_{M} is a graded ℤ[F^{×}]-algebra.

Lemma 5.7. For any k ≥ 1, the corestriction map

cor : H_{i}(SL_{k}(F), ℤ) → H_{i}(SL_{k+1}(F), ℤ)

is F^{×}-invariant;i.e. if a ∈ F^{×} and z ∈ H_{i}(SL_{k}(F), ℤ), then

cor(⟨a⟩ z) = ⟨a⟩ cor(z) = cor(z).

Proof. Of course, cor is a homomorphism of ℤ[F^{×}]-modules. However, for a ∈ F^{×}, ⟨a^{k}⟩ acts trivially on H_{i}(SL_{k}(F), ℤ) while ⟨a^{k+1}⟩ acts trivially on H_{i}(SL_{k+1}(F), ℤ) so that

cor(⟨a⟩ z) = cor(⟨a^{k+1}⟩ z) = ⟨a^{k+1}⟩ cor(z) = cor(z).
□

Lemma 5.8. For 0 ≤ q < n, the differentials of the spectral sequence

E^{+}(W, V)_{M}
d_{p,q}^{1} : (E_{p,q}^{1})_{M} ≅ H_{p}(SL_{n−q}(F), ℤ) → (E_{p,q−1}^{1})_{M} ≅ H_{p}(SL_{n−q+1}(F), ℤ)

are zero when q is even and are equal to the corestriction map when q is odd.

Proof. d^{1} is derived from the map d_{q} : C_{q}^{τ} (W, V) → C_{q−1}^{τ}(W, V) of permutation modules. Here

d_{q}((0, e_{1}), …, (0, e_{q})) = ∑_{i=1}^{q}(−1)^{i+1}((0, e_{1}), …, { \atop (0, e_{i})}, …, (0, e_{q}))

= ∑_{i=1}^{q}(−1)^{i+1}ϕ_{i}((0, e_{1}), …, (0, e_{q−1}))

where ϕ_{i} ∈ SA(W, V) can be chosen to be of the form

ϕ_{i} = (\begin{matrix}Id_{W} & 0 \\ 0 & ψ_{i} \end{matrix}) , ψ_{i} = (\begin{matrix}σ_{i} & 0 \\ 0 & τ_{i} \end{matrix}) ∈ GL(V)

with σ_{i} ∈ GL(V_{q}) a permutation matrix of determinant ϵ_{i} and τ_{i} ∈ GL(V′_{n−q}) also of determinant ϵ_{i}. ϕ_{i} normalises SA(W ⊕V_{q}, V′_{n−q},) and SL(V′_{n−q}). Thus for z ∈ H_{p}(SL(V′_{n−q}), ℤ),

d^{1}(z) = ∑_{i=1}^{q}(−1)^{i+1}cor(τ_{i}z)

= ∑_{i=1}^{q}(−1)^{i+1}cor(⟨ϵ_{i}⟩ z)

= ∑_{i=1}^{q}(−1)^{i+1}cor(z) = \{ \begin{matrix}cor(z), & q odd \\ 0, & q even \end{matrix}
□

Let E := ⌊−1, 1⌉ ∈ S̃(F^{2})_{M}. E is represented by the element

Ẽ := d_{3}(e_{1}, e_{2}, e_{2}−e_{1}) = (e_{2}, e_{2}−e_{1})−(e_{1}, e_{2}−e_{1})+(e_{1}, e_{2}) ∈ H(F^{2}) ⊂ C_{2}^{τ} (F^{2}).

Multiplication by Ẽ induces a map of complexes of GL_{n−2}(F)-modules

C_{•}^{τ} (F^{n−2})[2] → C_{•}^{τ} (F^{n})

There is an induced map of spectral sequences E(F^{n−2})[2] → E(F^{n}), which in turn induces a map E^{+}(F^{n−2})[2] → E^{+}(F^{n}), and hence a map E^{+}(F^{n−2})_{M}[2] → E^{+}(F^{n})_{M}.

By the work above, the E^{1}-page of E^{+}(F^{n})_{M} has the form

E_{p,q}^{1} = H_{p}(SL_{n−q}(F), ℤ) (p > 0)

while the E^{1}-page of E^{+}(F^{n−2})_{M}[2] has the form

E′_{p,q}^{1} = \{ \begin{matrix}H_{p}(SL_{(n−2)−(q−2)}(F), ℤ) = H_{p}(SL_{n−q}(F), ℤ), & q ≥ 2, p > 0 \\ 0, & q ≤ 1 or p = 0 \end{matrix}

Lemma 5.9. For q ≥ 2 (and p > 0), the map

E′_{p,q}^{1} ≅ H_{p}(SL_{n−q}(F), ℤ) → E_{p,q}^{1} = H_{p}(SL_{n−q}(F), ℤ)

induced by Ẽ ∗ − is the identity map.

Proof. There is a commutative diagram

\xymatrix{E′_{p,q}^{1} = H_{p}(SL_{n−q}(F), ℤ) \ar[r] \ar[d]^{_{(Ẽ∗−)M}} & H_{p}(SA(F^{q−2}, F^{n−q}), ℤ) \ar[r]^{≅} \ar[d]^{Ẽ∗−} & H_{p}(SL_{n−2}(F), C_{q−2}^{τ}(F^{n−2})) \ar[d]^{Ẽ∗−} \\ E_{p,q}^{1} = H_{p}(SL_{n−q}(F), ℤ) \ar[r] & H_{p}(SA(F^{q}, F^{n−q}), ℤ) \ar[r]^{≅} & H_{p}(SL_{n}(F), C_{q}^{τ} (F^{n}))}

We number the standard basis of F^{n−2} e_{3}, …, e_{n} so that the inclusion SL_{n−2}(F) → SL_{n}(F) has the form

A ↦ (\begin{matrix}I_{2} & 0 \\ 0 & A \end{matrix}) .

So we have a commutative diagram of inclusions of groups

\xymatrix{SL_{n−q}(F) \ar[r] \ar[d]^{=} & SA(F^{q−2}, F^{n−q}) \ar[r] \ar[d] & SL_{n−2}(F) \ar[d] \\ SL_{n−q}(F) \ar[r] & SA(F^{q}, F^{n−q}) \ar[r] & SL_{n}(F).}

Let B_{•} = B_{•}(SL_{n}(F)) be the right bar resolution of SL_{n}(F). We can use it to compute the homology of any of the groups occurring in this diagram. Suppose now that q ≥ 2 and we have a class, w, in E′_{p,q}^{1} = H_{p}(SL_{n−q}(F), ℤ) represented by a cycle

z ⊗ 1 ∈ B_{p} ⊗_{ℤ[SL_{n−q}(F)]} ℤ.

Its image in H_{p}(SL_{n−2}(F), C_{q−2}^{τ}(F^{n−2})) is represented by z ⊗ (e_{3}, …, e_{q}). The image of this in H_{p}(SL_{n}(F), C_{q}^{τ} (F^{n})) is

z ⊗ [Ẽ ∗ (e_{3}, …, e_{q})]

= z ⊗ [(e_{2}, e_{2} − e_{1}, e_{3}, …) − (e_{1}, e_{2} − e_{1}, e_{3}, …) + (e_{1}, e_{2}, e_{3}, …)]
= z ⊗ [(g_{1} − g_{2} + 1)(e_{1}, e_{2}, e_{3}, …)] ∈ B_{p} ⊗_{ℤ[SL_{n}(F)]} C_{q}^{τ} (F^{n})

where

g_{1} = (\begin{matrix}0 & −1 & 0 & … & 0 \\ 1 & 1 & 0 & … & 0 \\ 0 & 0 & 1 & 0 & ⋮ \\ ⋮ & ⋮ & 0 & ... & 0 \\ 0 & 0 & 0 & … & 1 \end{matrix}) , g_{2} = (\begin{matrix}1 & −1 & 0 & … & 0 \\ 0 & 1 & 0 & … & 0 \\ 0 & 0 & 1 & 0 & ⋮ \\ ⋮ & ⋮ & 0 & ... & 0 \\ 0 & 0 & 0 & … & 1 \end{matrix}) ∈ SL_{n}(F).

This corresponds to the element in H_{p}(SL_{n−q}(F), ℤ) represented by

z(g_{1} − g_{2} + 1) ⊗ 1 ∈ B_{p} ⊗_{ℤ[SL_{n−q}(F)]} ℤ

Since the elements g_{i} centralize SL_{n−q}(F) it follows that this is (g_{1}−g_{2}+1)·w = w. □

Recall that the spectral sequence E^{+}(F^{n})_{M} converges in degree n to S̃(F^{n})^{+}_{M}. Thus there is a filtration

0 = F_{n,−1} ⊂ F_{n,0} ⊂ F_{n,1} ⊂ ⋯ F_{n,n} = S̃(F^{n})^{+}_{M}

with

\frac{F_{n,i}}{F_{n,i−1}} ≅ E_{n−i,i}^{∞}.

The E^{1}-page of E^{+}(F^{n})_{M} has the form

\xymatrix{0 & 0 & 0 & … & 0 \\ 0 & H_{1}(SL_{2}(F), ℤ) \ar[d] & H_{2}(SL_{2}(F), ℤ) \ar[d] & … & H_{n}(SL_{2}(F), ℤ) \ar[d] \\ ⋮ & ⋮ \ar[d]^{cor} & ⋮ \ar[d]^{cor} & … & ⋮ \ar[d]^{cor} \\ 0 & H_{1}(SL_{n−2}(F), ℤ) \ar[d]^{0} & H_{2}(SL_{n−2}(F), ℤ) \ar[d]^{0} & … & H_{n}(SL_{n−2}(F), ℤ) \ar[d]^{0} \\ 0 & H_{1}(SL_{n−1}(F), ℤ) \ar[d]^{cor} & H_{2}(SL_{n−1}(F), ℤ) \ar[d]^{cor} & … & H_{n}(SL_{n−1}(F), ℤ) \ar[d]^{cor} \\ 0 & H_{1}(SL_{n}(F), ℤ) & H_{2}(SL_{n}(F), ℤ) & … & H_{n}(SL_{n}(F), ℤ)}

Theorem 5.10.

(1) The higher differentials d^{2}, d^{3}, …, in the spectral sequence E^{+}(F^{n})_{M} are all 0.

(2) S̃(F^{n−2})_{M} ≅ E ∗ S̃(F^{n−2})_{M} and this latter is a direct summand of S̃(F^{n})_{M}.

Proof.

(1) We will use induction on n. For n ≤ 2 the statement is true for trivial

reasons.

On the other hand, if n > 2, by Lemma 5.9, the map

Ẽ ∗ − : E^{+}(F^{n−2})_{M}[2] → E^{+}(F^{n})_{M}

induces an isomorphism on E^{1}-terms for q ≥ 2. By induction (and the fact that E′_{p,q}^{1} = 0 for q ≤ 1), the result follows for n.

(2) The map of spectral sequences E^{+}(F^{n−2})_{M}[2] → E^{+}(F^{n})_{M} induces a homomorphism on abutments

\xymatrix{S̃(F^{n−2})^{+}_{M} \ar[r]^{E∗−} & S̃(F^{n})^{+}_{M}}

By Lemma 5.9 again, it follows that the composite

\xymatrix{S̃(F^{n−2})^{+}_{M} \ar[r]^{E∗−} & S̃(F^{n})^{+}_{M} \ar[r] & (S̃(F^{n})^{+}_{M}) /F_{n,1}}

is an isomorphism.

Thus S̃(F^{n−2})^{+}_{M} ≅ E ∗ S̃(F^{n−2})^{+}_{M} and

S̃(F^{n})^{+}_{M} ≅ (E ∗ S̃(F^{n−2})^{+}_{M}) ⊕ F_{n,1}.
□

As a corollary we obtain the following general homology stability result for the homology of special linear groups:

Corollary 5.11. The corestriction maps H_{p}(SL_{n−1}(F), ℤ) → H_{p}(SL_{n}(F), ℤ) are isomorphisms for p < n − 1 and are surjective when p = n − 1.

Proof. Using (1) of Theorem 5.10 and Lemma 5.8, we have (for the spectral sequence E^{+}(F^{n})_{M}) that E_{p,q}^{∞} = E_{p,q}^{2} =

\frac{Ker(d^{1})}{Im(d^{1})} = \{ \begin{matrix}Ker(H_{p}(SL_{n−q}(F), ℤ) → H_{p}(SL_{n−q+1}(F), ℤ)) & q odd \\ Coker(H_{p}(SL_{n−q−1}(F), ℤ) → H_{p}(SL_{n−q}(F), ℤ)) & q even \end{matrix}

But the abutment of the spectral sequence is 0 in dimensions less than n. It follows that E_{p,q}^{∞} = 0 whenever p + q ≤ n − 1. □

Remark 5.12. Note that in the spectral sequence E^{+}(F^{n})_{M},

E_{n,0}^{∞} = Coker(H_{n}(SL_{n−1}(F), ℤ) → H_{n}(SL_{n}(F), ℤ)) = SH_{n}(F).

Clearly, the edge homomorphism H_{n}(SL_{n}(F), ℤ) → E_{n,0}^{∞} → S̃(F^{n})_{M} is just the iterated connecting homomorphism ϵ_{n} of section 3 above. Thus we have:

Corollary 5.13. The maps

ϵ_{•} : SH_{•}(F) → S̃(F^{•})_{M}

define an injective homomorphism of graded ℤ[F^{×}]-algebras.

Corollary 5.14. S̃(F^{2})_{M} = F_{2,1} ⊕ ℤ[F^{×}]E and for all n ≥ 3,

S̃(F^{n})_{M} = (E ∗ S̃(F^{n−2})_{M}) ⊕ F_{n,1} ≅ S̃(F^{n−2})_{M} ⊕ F_{n,1}.

Proof. Clearly S̃(F^{2})^{+}_{M} = F_{1,2}, while for n ≥ 3 we have

S̃(F^{n})_{M} = \{ \begin{matrix}S̃(F^{n})^{+}_{M} ⊕ ℤ[F^{×}]E^{∗ \frac{n}{2}} & n even \\ S̃(F^{n})^{+}_{M} ⊕ (S̃(F) ∗ E^{∗ \frac{n−1}{2}} ) & n odd \end{matrix}

□

Corollary 5.15. For all n ≥ 3,

S̃(F^{n})_{M} ≅ \{ \begin{matrix}F_{n,1} ⊕ F_{n−2,1} ⊕ ⋯ ⊕ F_{2,1} ⊕ ℤ[F^{×}] & n even \\ F_{n,1} ⊕ F_{n−2,1} ⊕ ⋯ ⊕ F_{3,1} ⊕ I_{F^{×}} & n odd \end{matrix}

as a ℤ[F^{×}]-module.

Note that F_{1,1} = S̃(F) = I_{F^{×}}, and for all n ≥ 2, F_{n,1} fits into an exact sequence associated to the spectral sequence E^{+}(F^{n})_{M}:

0 → E_{n,0}^{∞} = F_{n,0} → F_{n,1} → E_{n−1,1}^{∞} → 0.

Corollary 5.16. For all n ≥ 2 we have an exact sequence

H_{n}(SL_{n−1}(F), ℤ) → H_{n}(SL_{n}(F), ℤ) → F_{n,1} →
H_{n−1}(SL_{n−1}(F), ℤ) → H_{n−1}(SL_{n}(F), ℤ) → 0.

Lemma 5.17. For all n ≥ 2, the map T_{n} induces a surjective map F_{n,1} → K_{n}^{MW}(F).

Proof. First observe that since K_{n}^{MW}(F) is generated by the elements of the form [a_{1}] ⋯ [a_{n}] it follows from the definition of T_{n} that T_{n} : S̃(F^{n}) → K_{n}^{MW}(F) is surjective for all n ≥ 1. Next, since K_{•}^{MW}(F) is multiplicative, T_{•} factors through an algebra homomorphism S̃(F^{•})_{M} → K_{•}^{MW}(F). The lemma thus follows from Corollary 5.14 and the fact that T_{2}(E) = 0. □

Lemma 5.18. F_{2,1} = F_{2,0} and T_{2} : F_{2,1} → K_{2}^{MW}(F) is an isomorphism.

Proof. Since H_{1}(SL_{1}(F), ℤ) = 0, F_{2,1} = F_{2,0} = E_{2,0}^{∞} = ϵ_{2}(H_{2}(SL_{2}(F), ℤ)). Now apply Theorem 3.10. □

It is natural to define elements [a, b] ∈ F_{2,0} ⊂ S̃(F^{2})_{M} by [a, b] := T_{2}^{−1}([a][b]).

Lemma 5.19. In S̃(F^{2})_{M} we have the formula

[a, b] = ⌊a⌉ ∗ ⌊b⌉ − ⟨⟨a⟩⟩⟨⟨b⟩⟩E.

Proof. The results above show that the maps T_{2} and D_{2} induce an isomorphism

(T_{2}, D_{2}) : S̃(F^{2})_{M} ≅ K_{2}^{MW}(F) ⊕ ℤ[F^{×}].

Since D_{2}(⌊a⌉ ∗ ⌊b⌉) = ⟨⟨a⟩⟩⟨⟨b⟩⟩, while D_{2}(E) = 1, the result follows. □

Theorem 5.20.

(1) The product ∗ respects the filtrations on S̃(F^{n}); i.e. for all n, m ≥ 1

and i, j ≥ 0

F_{n,i} ∗ F_{m,j} ⊂ F_{n+m,i+j}.

(2) For n ≥ 1, let ϵ_{n+1,1} denote the composite F_{n+1,1} → E_{n,1}^{∞} = E_{n,1}^{2} → H_{n}(SL_{n}(F), ℤ). For all a ∈ F^{×} and for all n ≥ 1 the following diagram commutes:

\xymatrix{F_{n,0} \ar[r]^{⌊a⌉∗} & F_{n+1,1} \ar[d]^{ϵ_{n+1,1}} \\ H_{n}(SL_{n}(F), ℤ) \ar[u]^{ϵ_{n}} \ar[r]^{⟨⟨a⟩⟩·} & H_{n}(SL_{n}(F), ℤ)}

Proof.

(1) The filtration on S̃(F^{n})_{M} is derived from the spectral sequence E(F^{n}). This is the spectral sequence of the double complex B_{•} ⊗_{SL_{n}(F)} C_{•}^{τ} (F^{n}), regarded as a filtered complex by truncating C_{•}^{τ} (F^{n}) at i for i = 0, 1, . . .. Since the product ∗ is derived from a graded bilinear pairing on the complexes C_{•}^{τ} (F^{n}), the result easily follows.

(2) The spectral sequence E(F^{n+1}) calculates

H_{•}(SL_{n+1}(F), C^{τ} (F^{n+1})) ≅ H_{•}(SL_{n+1}(F), H(F^{n+1})[n + 1]

(where [n + 1] denotes a degree shift by n + 1).

Let C[1, n] denote the truncated complex

\xymatrix{C_{1}^{τ} (F^{n+1}) \ar[r]^{d_{1}} & C_{0}^{τ} (F^{n+1})}

and let Z_{1} denote the kernel of d_{1}. Then

H_{•}(SL_{n+1}(F), C[1, n]) ≅ H_{•}(SL_{n+1}(F), Z_{1})[1].

If F_{i} denotes the filtration on H_{•}(SL_{n+1}(F), C^{τ} (F^{n+1})) associated to the spectral sequence E(F^{n+1}), then from the definition of this filtration, F_{1}H_{k}(SL_{n+1}(F), C^{τ} (F^{n+1})) =

Im(H_{k}(SL_{n+1}(F), C[1, n]) → H_{k}(SL_{n+1}(F), C^{τ} (F^{n+1}))).

In particular,

F_{n+1,1} ≅ Im(H_{n+1}(SL_{n+1}(F), C[1, n]) → H_{n+1}(SL_{n+1}(F), C^{τ} (F^{n+1})))

and with this identification the diagram

\xymatrix{H_{n+1}(SL_{n+1}(F), C[1, n]) \ar[r] & F_{n+1,1} \ar[d]^{ϵ_{n+1,1}} \\ H_{n}(SL_{n+1}(F), Z_{1}) \ar[u]^{≅} \ar[r] & H_{n}(SL_{n+1}(F), C_{1}^{τ} (F^{n+1}))}

commutes (and H_{n}(SL_{n+1}(F), C_{1}^{τ} (F^{n+1})) ≅ H_{n}(SA(F, F^{n}), ℤ) by Shapiro’s Lemma, of course).

We consider SL_{n}(F) ⊂ SA(F, F^{n}) ⊂ SL_{n+1}(F) ⊂ GL_{n+1}(F) where the first inclusion is obtained by inserting a 1 in the (1, 1) position. Let B_{•} denote a projective resolution of ℤ over ℤ[GL_{n+1}(F)]. Let z ∈ H_{n}(SL_{n}(F), ℤ) be represented by x ⊗ 1 ∈ B_{n} ⊗_{ℤ[SL_{n}(F)]} ℤ = B_{n} ⊗_{ℤ[SL_{n}(F)]} C_{0}^{τ} (F^{n}). Then ⌊a⌉ ∗ ϵ_{n}(z) is represented by z ⊗ [(ae_{1}) − (e_{1})] ∈ B_{n} ⊗_{SL_{n+1}(F)} Z_{1} which maps to the element of H_{n}(SL_{n+1}(F), C_{1}^{τ} (F^{n+1})) represented by z(g − 1) ⊗ (e_{1}) where g = diag(a, 1, …, 1, a^{−1}). But this is just the image of ⟨⟨a⟩⟩z under the map H_{n}(SL_{n}(F), ℤ) → H_{n}(SA(F, F^{n}), ℤ) ≅ H_{n}(SL_{n+1}(F), C_{1}^{τ} (F^{n+1})).

□

Lemma 5.21. The map T_{3} : F_{3,1} → K_{3}^{MW}(F) is an isomorphism.

Proof. Consider the short exact sequence

0 → E_{3,0}^{∞} → F_{3,1} → E_{2,1}^{∞} → 0.

Here ϵ_{3} induces an isomorphism

E_{3,0}^{∞} ≅ Coker(H_{3}(SL_{2}(F), ℤ) → H_{3}(SL_{3}(F), ℤ)).

By the main result of [8] (Theorem 4.7 - see also section 2.4 of this article), T_{3} thus induces an isomorphism E_{3,0}^{∞} ≅ 2K_{3}^{M}(F) ⊂ K_{3}^{MW}(F). On the other hand,

E_{2,1}^{∞} ≅ Ker(H_{2}(SL_{2}(F), ℤ) → H_{2}(SL_{3}(F), ℤ)) ≅ I^{3}(F)

Thus we have a commutative diagram

\xymatrix{0 \ar[r] & E_{3,0}^{∞} \ar[d]^{T_{3}}_{≅} \ar[r] & F_{3,1} \ar[d]^{T_{3}} \ar[r]^{ρ} & I^{3}(F) \ar[d]^{α} \ar[r] & 0 \\ 0 \ar[r] & 2K_{3}^{M}(F) \ar[r] & K_{3}^{MW}(F) \ar[r]^{p_{3}} & I^{3}(F) \ar[r] & 0}

where the vertical arrows are surjections. Now the inclusion I^{3}(F) → K_{2}^{MW}(F) is given by ⟨⟨a, b, c⟩⟩ ↦ ⟨⟨a⟩⟩[b][c]. Thus the inclusion j : I^{3}(F) → H_{2}(SL_{2}(F), ℤ) is given by ⟨⟨a, b, c⟩⟩ ↦ ⟨⟨a⟩⟩⟨b, c⟩ where ⟨b, c⟩ = ϵ_{2}^{−1}([b, c]). Thus for all a, b, c ∈ F^{×} we have

j ∘ ρ(⌊a⌉ ∗ [b, c]) = ϵ_{3,1}(⌊a⌉ ∗ [b, c]) = ⟨⟨a⟩⟩⟨b, c⟩

using Theorem 5.20 (2), and thus ρ(⌊a⌉ ∗ [b, c]) = ⟨⟨a, b, c⟩⟩ ∈ I^{3}(F). It follows from the diagram that

α(⟨⟨a, b, c⟩⟩) = α ∘ ρ(⌊a⌉ ∗ [b, c]) = p_{3} ∘ T_{3}(⌊a⌉ ∗ [b, c]) = ⟨⟨a, b, c⟩⟩

so that α is the identity map, and the result follows. □

Lemma 5.22. For all a ∈ F^{×}, ⌊a⌉ ∗ E = E ∗ ⌊a⌉ in S̃(F^{3})_{M}.

Proof. By the calculations above, F_{3,1} = S̃(F^{3})^{+}_{M} = Ker(D_{3}). Thus R_{a} := ⌊a⌉ ∗ E − E ∗ ⌊a⌉ ∈ F_{3,1}. But then T_{3}(R_{a}) = 0 since T_{2}(E) = 0 and thus R_{a} = 0 by the previous lemma. □

Lemma 5.23.

(1) For all a, b, c ∈ F^{×}

⌊a⌉ ∗ [b, c] = [a, b] ∗ ⌊c⌉ in S̃(F^{3})_{M}.

(2) For all a, b, c ∈ F^{×}

⌊a⌉ ∗ ⌊b⌉ ∗ ⌊c⌉ = ⌊c⌉ ∗ ⌊a⌉ ∗ ⌊b⌉ in S̃(F^{3})_{M}.

(3) For all a, b, c, d ∈ F^{×}

[a, b] ∗ [c, d] = [a, c^{−1}] ∗ [b, d] in S̃(F^{4})_{M}.

Proof. The calculations above have established that the map

(T_{3}, D_{3}) : S̃(F^{3})_{M} → K_{3}^{MW}(F) ⊕ I_{F^{×}}

is an isomorphism.

(1) This follows from the identities

T_{3}(⌊a⌉ ∗ [b, c]) = [a][b][c] = T_{3}([a, b] ∗ ⌊c⌉)

and

D_{3}(⌊a⌉ ∗ [b, c]) = ⟨⟨a, b, c⟩⟩ = D_{3}([a, b] ∗ ⌊c⌉)

(2) This follows from the fact that [a][b][c] = [c][a][b] in K_{3}^{MW}(F).

(3) We begin by observing that, since S̃(F) ≅ I_{F^{×}} as a ℤ[F^{×}]-module we

have ⟨⟨a⟩⟩⌊b⌉ = ⌊ab⌉ − ⌊a⌉ − ⌊b⌉ = ⟨⟨b⟩⟩⌊a⌉ for all a, b ∈ F^{×}. For x_{1}, …, x_{n} ∈ F^{×} and i, j ≥ 1 with i + j = n we set

L_{i,j}(x_{1}, …, x_{n}) := ⟨⟨x_{1}⟩⟩ ⋯ ⟨⟨x_{i}⟩⟩ (⌊x_{i+1}⌉ ∗ ⋯ ∗ ⌊x_{n}⌉) ∈ S̃(F^{j})_{M}.

By the observation just made, we have

L_{i,j}(x_{1}, …, x_{n}) = L_{i,j}(x_{σ(1)}, …, x_{σ(n)})

for any permutation σ of 1, …, n.

So

[a, b] ∗ [c, d] = (⌊a⌉ ∗ ⌊b⌉ − ⟨⟨a⟩⟩⟨⟨b⟩⟩E) ∗ (⌊c⌉ ∗ ⌊d⌉ − ⟨⟨c⟩⟩⟨⟨d⟩⟩E) = ⌊a⌉ ∗ ⌊b⌉ ∗ ⌊c⌉ ∗ ⌊d⌉ − 2L_{2,2}(a, b, c, d) ∗ E + ⟨⟨a⟩⟩⟨⟨b⟩⟩⟨⟨c⟩⟩⟨⟨d⟩⟩E^{∗2}

Let R = [a, b] ∗ [c, d] − [a, c^{−1}] ∗ [b, d].

So R =

⌊a⌉ ∗ ⌊b⌉ ∗ ⌊c⌉ ∗ ⌊d⌉ − ⌊a⌉ ∗ ⌊c^{−1}⌉ ∗ ⌊b⌉ ∗ ⌊d⌉ − 2(L_{2,2}(a, b, c, d) − L_{2,2}(a, c^{−1}, b, d)) ∗ E +⟨⟨a⟩⟩⟨⟨d⟩⟩ [(⟨⟨b⟩⟩⟨⟨c⟩⟩ − ⟨⟨c^{−1}⟩⟩⟨⟨b⟩⟩)E] ∗ E.

However, since [b, c] = [c^{−1}, b] in S̃(F^{2})_{M} we have (by Lemma 5.19)

(⟨⟨b⟩⟩⟨⟨c⟩⟩ − ⟨⟨c^{−1}⟩⟩⟨⟨b⟩⟩)E = ⌊b⌉ ∗ ⌊c⌉ − ⌊c^{−1}⌉ ∗ ⌊b⌉.

Thus ⟨⟨a⟩⟩⟨⟨d⟩⟩ [(⟨⟨b⟩⟩⟨⟨c⟩⟩ − ⟨⟨c^{−1}⟩⟩⟨⟨b⟩⟩)E] ∗ E =

(L_{2,2}(a, b, c, d) − L_{2,2}(a, c^{−1}, b, d)) ∗ E

and hence R =

⌊a⌉∗⌊b⌉∗⌊c⌉∗⌊d⌉−⌊a⌉∗⌊c^{−1}⌉∗⌊b⌉∗⌊d⌉−(L_{2,2}(a, b, c, d)−L_{2,2}(a, c^{−1}, b, d))∗E.

Now

(L_{2,2}(a, b, c, d) − L_{2,2}(a, c^{−1}, b, d)) ∗ E

= ⌊a⌉ ∗ ⌊d⌉ ∗ [(⟨⟨b⟩⟩⟨⟨c⟩⟩ − ⟨⟨c^{−1}⟩⟩⟨⟨b⟩⟩)E]
= ⌊a⌉ ∗ ⌊d⌉ ∗ [⌊b⌉ ∗ ⌊c⌉ − ⌊c^{−1}⌉ ∗ ⌊b⌉]
= ⌊a⌉ ∗ (⌊d⌉ ∗ ⌊b⌉ ∗ ⌊c⌉) − ⌊a⌉ ∗ (⌊d⌉ ∗ ⌊c^{−1}⌉ ∗ ⌊b⌉)
= ⌊a⌉ ∗ ⌊b⌉ ∗ ⌊c⌉ ∗ ⌊d⌉ − ⌊a⌉ ∗ ⌊c^{−1}⌉ ∗ ⌊b⌉ ∗ ⌊d⌉

using (2) in the last step.

□

Theorem 5.24. For all n ≥ 2 there is a homomorphism μ_{n} : K_{n}^{MW}(F) → F_{n,1} such that the composite T_{n} ∘ μ_{n} is the identity map.

Proof. For n ≥ 2 and a_{1}, …, a_{n} ∈ F^{×}, let {{a_{1}, …, a_{n}}} :=

\{ \begin{matrix}[a_{1}, a_{2}] ∗ ⋯ ∗ [a_{n−1}, a_{n}], & n even \\ ⌊a_{1}⌉ ∗ [a_{2}, a_{3}] ∗ ⋯ ∗ [a_{n−1}, a_{n}], & n odd \end{matrix} \} ∈ F_{n,1} ⊂ S̃(F^{n})_{M}.

By Lemma 5.23 (1) and (3), as well as the definition of [x, y], the elements {{a_{1}, …, a_{n}}} satisfy the ‘Matsumoto-Moore’ relations (see Section 2.4 above), and thus there is a well-defined homomorphism of groups

μ_{n} : K_{n}^{MW}(F) → F_{n,1}, [a_{1}] ⋯ [a_{n}] ↦ {{a_{1}, …, a_{n}}}.

Since T_{n}({{a_{1}, …, a_{n}}}) = [a_{1}] ⋯ [a_{n}], the result follows. □

Corollary 5.25. The subalgebra of SH_{2•}(F) generated by SH_{2}(F) = H_{2}(SL_{2}(F), ℤ) is isomorphic to K_{2•}^{MW}(F) and is a direct summand of SH_{2•}(F).

Proof. This is immediate from Theorems 3.10 and 5.24. □

6. Decomposabilty

Recall that F is a field of characteristic 0 throughout this section. In [24], Suslin proved that H_{n}(GL_{n}(F), ℤ)/H_{n}(GL_{n−1}(F), ℤ) ≅ K_{n}^{M}(F). This is, in particular, a decomposability result. It says that H_{n}(GL_{n}(F), ℤ) is generated, modulo the image of H_{n}(GL_{n−1}(F), ℤ) by products of 1-dimensional cycles. In this section we will prove analogous results for the special linear group, with Milnor-Witt K-theory replacing Milnor K-theory. To do this, we prove the decomposability of the algebra S̃(F^{•})_{M} (for n ≥ 3). Theorem 6.2 is an analogue of Suslin’s Proposition 3.3.1. The proof is essentially identical, and we reproduce it here for the convenience of the reader. From this we deduce our decomposability result (Theorem 6.8), which requires still a little more work than in the case of the general linear group.

Lemma 6.1. For any finite-dimensional vector spaces W and V, the image of the pairing

S̃(W, V) ⊗ H(W) → S̃(W ⊕ V)_{M} (2) coincides with the image of the pairing

S̃(V) ⊗ S̃(W) → S̃(W ⊕ V)_{M} (3)

Proof. The image of the pairing (2) is equal to the image of

S̃(W, V)_{M} ⊗ H(W) → S̃(W ⊕ V)_{M}

which coincides with the image of

S̃(V)_{M} ⊗ S̃(W)_{M} → S̃(W ⊕ V)_{M}

by the isomorphism of Corollary 5.6. □

Let S̃(F^{n})^{dec} ⊂ S̃(F^{n})_{M} be the ℤ[F^{×}]-submodule of decomposable elements; i.e. S̃(F^{n})^{dec} is the image of

\xymatrix{⨁_{p+q=n,p,q>0} (S̃(F^{p})_{M} ⊗ S̃(F^{q})_{M}) \ar[r]^{∗} & S̃(F^{n})_{M}.}

More generally, note that if V = V_{1} ⊕ V_{2} = V′_{1} ⊕ V′_{2} and if dim_{F} (V_{i}) = dim_{F} (V′_{i}) for i = 1, 2, then the image of S̃(V_{1}) ⊗ S̃(V_{2}) → S̃(V) coincides with S̃(V′_{1}) ⊗ S̃(V′_{2}) → S̃(V). This follows from the fact that there exists ϕ ∈ SL(V) with ϕ(V_{i}) = V′_{i} for i = 1, 2. Therefore S̃(F^{n})^{dec} is the image of

\xymatrix{⨁_{F^{n}=V_{1}⊕V_{2},V_{i}≠0} (S̃(V_{1})_{M} ⊗ S̃(V_{2})_{M}) \ar[r]^{∗} & S̃(F^{n})_{M}.}

If x = ∑_{i} n_{i}(x_{1}^{i}, …, x_{p}^{i}) ∈ C_{p}(V) and y = ∑_{j} m_{j}(y_{1}^{j}, …, y_{q}^{j}) ∈ C_{q}(V) and if (x_{1}^{i}, …, x_{p}^{i}, y_{1}^{j}, …, y_{q}^{j}) ∈ X_{p+q}(V) for all i, j, then we let

x ⊛ y := ∑_{i,j} n_{i}m_{j}(x_{1}^{i}, …, x_{p}^{i}, y_{1}^{j}, …, y_{q}^{j}) ∈ C_{p+q}(V).

Of course, if x ∈ C_{p}(V_{1}) and y ∈ C_{q}(V_{2}) with V = V_{1} ⊕ V_{2}, then x ⊛ y = x ∗ y. Furthermore, when x ⊛ y is defined, we have

d(x ⊛ y) = d(x) ⊛ y + (−1)^{p}x ⊛ d(y).

Theorem 6.2. Let n ≥ 1. For any a_{1}, …, a_{n}, b ∈ F^{×} and for any 1 ≤ i ≤ n

⌊a_{1}, …, ba_{i}, …, a_{n}⌉ ≅ ⟨b⟩ ⌊a_{1}, …, a_{n}⌉ (mod S̃(F^{n})^{dec}).

Proof. Let a = a_{1}e_{1} + ⋯ + ba_{i}e_{i} + ⋯ a_{n}e_{n}. We have

⌊a_{1}, …, ba_{i}, …, a_{n}⌉ − ⟨b⟩ ⌊a_{1}, …, a_{n}⌉

= d(e_{1}, …, e_{i}, …, e_{n}, a) − d(e_{1}, …, b_{i}e_{i}, …, e_{n}, a)
= d((e_{1}, …, e_{i−1}) ⊛ ((e_{i}) − (be_{i})) ⊛ (e_{i+1}, …, e_{n}, a))
= d(e_{1}, …, e_{i−1}) ⊛ ((e_{i}) − (be_{i})) ⊛ (e_{i+1}, …, e_{n}, a)
+ (−1)^{i}(e_{1}, …, e_{i−1}) ⊛ ((e_{i}) − (be_{i})) ⊛ d(e_{i+1}, …, e_{n}, a)

Let u = a_{1}e_{1} + ⋯ + a_{i−1}e_{i−1} + ba_{i}e_{i} = a − ∑_{j=i+1}^{n} a_{j}e_{j}.Then

(−1)^{i−1}(e_{1}, …, e_{i−1}) = d((e_{1}, …, e_{i−1}) ⊛ (u)) − d(e_{1}, …, e_{i−1}) ⊛ (u)

and

(e_{i+1}, …, e_{n}, a) = d((u) ⊛ (e_{i+1}, …, e_{n}, a)) + (u) ⊛ d(e_{i+1}, …, e_{n}, a).

Thus ⌊a_{1}, …, ba_{i}, …, a_{n}⌉ − ⟨b⟩ ⌊a_{1}, …, a_{n}⌉ = X_{1} − X_{2} + X_{3} where

X_{1} = d(e_{1}, …, e_{i−1}) ⊛ ((e_{i}) − (be_{i})) ⊛ d(u, e_{i+1}, …, e_{n}, a),
X_{2} = d(e_{1}, …, e_{i−1}, u) ⊛ ((e_{i}) − (be_{i})) ⊛ d(e_{i+1}, …, e_{n}, a), and
X_{3} = d(e_{1}, …, e_{i−1}) ⊛ [((e_{i}) − (be_{i})) ⊛ (u) + (u) ⊛ ((e_{i}) − (be_{i}))] ⊛
⊛ d(e_{i+1}, …, e_{n}, a)

We show that each X_{i} is decomposable: Let V ⊂ F^{n} be the span of u, e_{i+1}, …, e_{n} (which is also equal to the span of a, e_{i+1}, …, e_{n}), and let V′ be the span of e_{1}, …, e_{i−1}. Then F^{n} = V′ ⊕ V and d(u, e_{i+1}, …, e_{n}, a) ∈ H(V) while d(e_{1}, …, e_{i−1}) ⊛ ((e_{i}) − (be_{i})) ∈ H(V, V′). Thus X_{1} lies in the image of

\xymatrix{H(V, V′) ⊗ H(V) \ar[r]^{∗} & S̃(F^{n})_{M}}

and so is decomposable. Similarly, if we let W be the span of e_{1}, …, e_{i} and W′ the span of e_{i+1}, …, e_{n}, then

d(e_{1}, …, e_{i−1}, u) ⊛ ((e_{i}) − (be_{i})), d(e_{1}, …, e_{i−1}) ⊛ [((e_{i}) − (be_{i})) ⊛ (u) + (u) ⊛ ((e_{i}) − (be_{i}))]

belongs to H(W) and d(e_{i+1}, …, e_{n}, a) ∈ H(W, W′).Thus X_{2}, X_{3} lie in the image of

\xymatrix{H(W) ⊗ H(W, W′) \ar[r]^{∗} & S̃(F^{n})_{M}}

and are also decomposable. □

Let S̃(F^{n})^{ind} := S̃(F^{n})_{M}/S̃(F^{n})^{dec}. The main goal of this section is to show that S̃(F^{n})^{ind} = 0 for all n ≥ 3 (Theorem 6.8 below).

Lemma 6.3. For all n ≥ 3, S̃(F^{n})^{ind} is a multiplicative ℤ[F^{×}]-module.

Proof. We have

A_{n} ≅ \{ \begin{matrix}ℤ[F^{×}]E^{∗n/2}, & n even \\ S̃(F) ∗ E^{∗(n−1)/2}, & n odd \end{matrix}

and these modules are decomposable for all n ≥ 3. It follows that the map

S̃(F^{n})^{+}_{M} → S̃(F^{n})^{ind}

is surjective for all n ≥ 3. □

Remark 6.4. Since E ∗ S̃(F^{n−2})_{M} ⊂ S̃(F^{n})^{dec}, in fact we have that F_{n,1} → S̃(F^{n})^{ind} is surjective.

Theorem 6.2 shows that for all a_{1}, …, a_{n} ∈ F^{×}

⌊a_{1}, …, a_{n}⌉ ≅ ⟨∏_{i} a_{i}⟩ ⌊1, …, 1⌉ (mod S̃(F^{n})^{dec}).

In other words the map

ℤ[F^{×}] → S̃(F^{n})^{ind}, α ↦ α⌊1, …, 1⌉

is a surjective homomorphism of ℤ[F^{×}]-modules. Thus, we are required to establish that ⌊1, …, 1⌉ ∈ S̃(F^{n})^{dec} for all n ≥ 3. For convenience below, we will let Σ̃_{n}(F) denote the free ℤ[F^{×}]-module on the symbols [a_{1}, …, a_{n}], a_{1}, …, a_{n} ∈ F^{×}. Let p_{n} : Σ̃_{n}(F) → S̃(F^{n}) be the ℤ[F^{×}]- module homomorphism sending [a_{1}, …, a_{n}] to ⌊a_{1}, …, a_{n}⌉. We will say that σ ∈ S̃(F^{n}) is represented by σ̃ ∈ Σ̃_{n}(F) if p_{n}(σ̃) = σ. Note that Σ̃_{•}(F) can be given the structure of a graded ℤ[F^{×}]-algebra by setting

[a_{1}, …, a_{n}] · [a_{n+1}, …, a_{n+m}] := [a_{1}, …, a_{n+m}];

i.e., we can identify Σ̃_{•}(F) with the tensor algebra over ℤ[F^{×}] on the free module with basis [a], a ∈ F^{×}. Let Π_{•} : Σ̃_{•}(F) → ℤ[F^{×}][x] be the homomorphism of graded ℤ[F^{×}]-algebras sending [a] to ⟨a⟩ x. For all n ≥ 1 we have a commutative square of surjective homomorphisms of ℤ[F^{×}]-modules

\xymatrix{Σ̃_{n}(F) \ar[r]^{Π_{n}} \ar[d]^{p_{n}} & ℤ[F^{×}] · x^{n} \ar[d]^{γ_{n}} \\ S̃(F^{n}) \ar[r] & S̃(F^{n})^{ind}}

where γ_{n}(x^{n}) = ⌊1, …, 1⌉.

Lemma 6.5. If n is odd and n ≥ 3 then S̃(F^{n})^{ind} = 0; i.e.,

S̃(F^{n})_{M} = S̃(F^{n})^{dec}.

Proof. From the fundamental relation in S̃(F^{n}) (Theorem 3.3), if b_{1}, …, b_{n} are distinct elements of F^{×}, then 0 ∈ S̃(F^{n}) is represented by R_{b} :=

[b_{1}, …, b_{n}] − [1, …, 1] − ∑_{j=1}^{n}(−1)^{n+j} ⟨(−1)^{n+j} ⟩ [b_{1} − b_{j} , …, { \atop b_{j} − b_{j}} , …, b_{n} − b_{j} , b_{j} ]

in ∈ Σ̃_{n}(F). Now Π_{n}(R_{b}) =

[⟨∏_{i} b_{i}⟩ − ⟨1⟩ − ∑_{j=1}^{n}(−1)^{n+j} ⟨(b_{j} − b_{1}) ⋯ (b_{j} − b_{j−1}) · (b_{j+1} − b_{j}) ⋯ (b_{n} − b_{j}) · b_{j} ⟩] x^{n}.

We choose b_{i} = i, i = 1, …, n. Then

Π_{n}(R_{b}) = [⟨n!⟩ − ⟨1⟩ − ∑_{j=1}^{n}(−1)^{n+j} ⟨j!(n − j)!⟩] x^{n} = − ⟨1⟩ x^{n} since n is odd.

It follows that −⌊1, …, 1⌉ = 0 in S̃(F^{n})^{ind} as required. □

The case n even requires a little more work. The maps {p_{n}}_{n} do not define a map of graded algebras. However, we do have the following:

Lemma 6.6. For 1 ≠ a ∈ F^{×}, let

L(x) := ⟨−1⟩ [1 − x, 1] − ⟨x⟩ [1 − \frac{1}{x}, \frac{1}{x}] + [1, 1] ∈ Σ̃_{2}(F).

Then for all a_{1}, …, a_{n} ∈ F^{×} \ {1}, the product

∏_{i=1}^{n}⌊1, a_{i}⌉ = ⌊1, a_{1}⌉ ∗ ⋯ ∗ ⌊1, a_{n}⌉ ∈ S̃(F^{2n})

is represented by ∏_{i} L(a_{i}) ∈ Σ̃_{2n}(F).

Proof. For convenience of notation, we will represent standard basis elements of C_{q}(F^{n}) as n × q matrices [v_{1}| ⋯ |v_{q}]. Let e = (1, …, 1) and let σ_{i}(C) denote the sum of the entries in the ith row of the n × n matrix C. By Remark 3.2, if A ∈ GL_{n}(F) and [A|e] ∈ X_{n+1}(F^{n}) then d_{n+1}([A|e]) represents ⟨det A⟩ ⌊σ_{1}(A^{−1}), …, σ_{n}(A^{−1})⌉ ∈ S̃(F^{n}). Now, for a ≠ 1, ⌊1, a⌉ is represented in S̃(F^{2}) by

d_{3} ([\begin{matrix}1 & 0 & 1 \\ 0 & 1 & a \end{matrix}]) = [\begin{matrix}0 & 1 \\ 1 & a \end{matrix}]−[\begin{matrix}1 & 1 \\ 0 & a \end{matrix}]+[\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}] = T_{1}(a)−T_{2}(a)+T_{3}(a) ∈ C_{2}(F^{2}).

From the definition of the product ∗, it follows that ⌊1, a_{1}⌉ ∗ ⋯ ∗ ⌊1, a_{n}⌉ is represented by

Z := ∑_{j=(j_{1},...,j_{n})∈(1,2,3)^{n}}(−1)^{k(j)} [\begin{matrix}T_{j_{1}}(a_{1}) \\ & . & . & . \\ & & & & T_{j_{n}}(a_{n}) \end{matrix}] = ∑_{j}(−1)^{k(j)}T (j, a).

where k(j) := |{i ≤ n|j_{i} = 2}| Since a_{i} ≠ 1 for all i, the vector e = (1, …, 1) is in general position with respect to the columns of all these matrices. Thus we can use the partial homotopy operator s_{e} to write this cycle as a boundary:

Z = ∑_{j}(−1)^{k(j)}d_{2n+1} ([T (j, a)|e]) .

By the remarks above

d_{2n+1} ([T (j, a)|e]) = ⟨∏_{i} det T_{j_{i}}(a_{i})⟩ ×

×⌊σ_{1}(T_{j_{1}}(a_{1})), σ_{2}(T_{j_{1}}(a_{1})), σ_{1}(T_{j_{2}}(a_{2})), …, σ_{1}(T_{j_{n}}(a_{n})), σ_{2}(T_{j_{n}}(a_{n}))⌉.

This is represented by

⟨∏_{i} det T_{j_{i}}(a_{i})⟩ ×

×[σ_{1}(T_{j_{1}}(a_{1})), σ_{2}(T_{j_{1}}(a_{1})), σ_{1}(T_{j_{2}}(a_{2})), …, σ_{1}(T_{j_{n}}(a_{n})), σ_{2}(T_{j_{n}}(a_{n}))]

= ∏_{i=1}^{n} ( ⟨det T_{j_{i}}(a_{i})⟩ [σ_{1}(T_{j_{i}}(a_{i})), σ_{2}(T_{j_{i}}(a_{i}))]) ∈ Σ̃_{2n}(F).

Thus Z is represented by

∑_{j}(−1)^{k(j)} ∏_{i=1}^{n} ( ⟨det T_{j_{i}}(a_{i})⟩ [σ_{1}(T_{j_{i}}(a_{i})), σ_{2}(T_{j_{i}}(a_{i}))]) =

= ∏_{i=1}^{n} ( ∑_{j=1}^{3}(−1)^{j+1} ⟨det T_{j}(a_{i})⟩ [σ_{1}(T_{j}(a_{i})), σ_{2}(T_{j}(a_{i}))])= ∏_{i=1}^{n} L(a_{i}) ∈ Σ̃_{2n}(F).

□

Observe that all of our multiplicative modules (and in particular S̃(F^{n})_{M}) have the following property: they admit a finite filtration 0 = M_{0} ⊂ M_{1} ⊂ ⋯ ⊂ M_{t} = M such that each of the associated quotients M_{r}/M_{r−1} is annihilated by I_{(F^{×})^{kr}} for some k_{r} ≥ 1. From this observation it easily follows that

Lemma 6.7.

S̃(F^{n})^{ind} = 0 ⟺ S̃(F^{n})^{ind}/(I_{(F^{×})^{r}} · S̃(F^{n})^{ind}) = 0 for all r ≥ 1.

Theorem 6.8. S̃(F^{n})^{ind} = 0 for all n ≥ 3.

Proof. The case n odd has already been dealt with in Lemma 6.5 For the even case, by Lemma 6.7 it will be enough to prove that for all r ≥ 1

ℤ[F^{×}/(F^{×})^{r}] ⊗_{ℤ[F^{×}]} S̃(F^{n})^{ind} = 0.

Fix r ≥ 1. If a ∈ (F^{×})^{r} \ {1}, then

Π_{2}(L(a)) = (⟨a − 1⟩ − ⟨1 − \frac{1}{a}⟩ + ⟨1⟩) x^{2} = ⟨1⟩ x^{2} ∈ ℤ[F^{×}/(F^{×})^{r}]x^{2}

since

1 − \frac{1}{a} = \frac{a − 1}{a} ≡ a − 1 (mod (F^{×})^{r}).

Now let n > 1 and choose a_{1}, …, a_{n} ∈ (F^{×})^{r} \ {1}. Let σ = ⌊1, a_{1}⌉ ∗ ⋯ ∗ ⌊1, a_{n}⌉ ∈ S̃(F^{2n}), so that σ ↦ 0 in S̃(F^{2n})^{ind}. By Lemma 6.6, σ is represented by σ̃ = ∏_{i=1}^{n} L(a_{i}) in Σ̃_{2n}(F) and thus

Π_{2n}(σ̃) = ∏_{i=1}^{n} (Π_{2}(L(a_{i}))) = ⟨1⟩ ∈ ℤ[F^{×}/(F^{×})^{r}]x^{2n}

so that the image of σ in ℤ[F^{×}/(F^{×})^{r}] ⊗_{ℤ[F^{×}]} S̃(F^{2n})^{ind} is 1 ⊗ ⌊1, …, 1⌉. This proves the theorem. □

Corollary 6.9. For all n ≥ 2, the map T_{n} induces an isomorphism F_{n,1} ≅ K_{n}^{MW}(F).

Proof. Since, by the computations above, S̃(F^{2})_{M} = S̃(F)^{∗2} + ℤ[F^{×}]E it follows, using Theorem 6.8 and induction on n, that S̃(F^{•})_{M} is generated as a ℤ[F^{×}]-algebra by {⌊a⌉ ∈ S̃(F)|1 ≠ a ∈ F^{×}} and E. Thus E is central in the algebra S̃(F^{•})_{M} and for all n ≥ 2,

\frac{S̃(F^{n})_{M}}{E ∗ S̃(F^{n−2})_{M}}

is generated by the elements of the form ⌊a_{1}⌉ ∗ ⋯ ∗ ⌊a_{n}⌉, and hence also by the elements {{a_{1}, …, a_{n}}} since [a, b] ≡ ⌊a⌉ ∗ ⌊b⌉ (mod ⟨E⟩) for all a, b ∈ F^{×}. Since

F_{n,1} ≅ \frac{S̃(F^{n})_{M}}{E ∗ S̃(F^{n−2})_{M}}

by Corollary 5.14, it follows that F_{n,1} is generated by the elements {{a_{1}, …, a_{n}}}, and thus that the homomorphisms μ_{n} of Theorem 5.24 are surjective. □

Corollary 6.10. For all n ≥ 3,

S̃(F^{n})_{M} ≅ \{ \begin{matrix}K_{n}^{MW}(F) ⊕ K_{n−2}^{MW}(F) ⊕ ⋯ ⊕ K_{2}^{MW}(F) ⊕ ℤ[F^{×}] & n even \\ K_{n}^{MW}(F) ⊕ K_{n−2}^{MW}(F) ⊕ ⋯ ⊕ K_{3}^{MW}(F) ⊕ I_{F^{×}} & n odd \end{matrix}

as a ℤ[F^{×}]-module.

Corollary 6.11. For all even n ≥ 2 the cokernel of the map

H_{n}(SL_{n−1}(F), ℤ) → H_{n}(SL_{n}(F), ℤ)

is isomorphic to K_{n}^{MW}(F).

Proof. Recall that ϵ_{2} induces an isomorphism H_{2}(SL_{2}(F), ℤ) ≅ F_{2,1} = F_{2,0}. Let ⟨a, b⟩ denote the generator ϵ_{2}^{−1}([a, b]) of H_{2}(SL_{2}(F), ℤ). Then for even n

{{a_{1}, …, a_{n}}} = [a_{1}, a_{2}] ∗ ⋯ ∗ [a_{n−1}, a_{n}]
= ϵ_{2}(⟨a_{1}, a_{2}⟩) ∗ ⋯ ∗ ϵ_{2}(⟨a_{n−1}, a_{n}⟩)
= ϵ_{n}(⟨a_{1}, a_{2}⟩ × ⋯ × ⟨a_{n−1}, a_{n}⟩)

by Lemma 3.5 (2).

Since F_{n,1} is generated by the elements {{a_{1}, …, a_{n}}}, it follows that F_{n,1} = ϵ_{n}(H_{n}(SL_{n}(F), ℤ)) = E_{n,0}^{∞} = F_{n,0}, proving the result. □

Corollary 6.12. For all odd n ≥ 1 the maps

H_{n}(SL_{k}(F), ℤ) → H_{n}(SL_{k+1}(F), ℤ)

are isomorphisms for k ≥ n.

Proof. In view of Corollary 5.11, the only point at issue is the injectivity of

H_{n}(SL_{n}(F), ℤ) → H_{n}(SL_{n+1}(F), ℤ).

But the proof of Corollary 6.11 shows that the term

F_{n+1,1}/E_{n+1,0}^{∞} ≅ E_{n,1}^{∞} = Ker(H_{n}(SL_{n}(F), ℤ) → H_{n}(SL_{n+1}(F), ℤ))

in the spectral sequence E^{+}(F^{n+1})_{M} is zero. □

Corollary 6.13. If n ≥ 3 is odd, then

Coker(H_{n}(SL_{n−1}(F), ℤ) → H_{n}(SL_{n}(F), ℤ)) ≅ 2K_{n}^{M}(F)
Ker(H_{n−1}(SL_{n−1}(F), ℤ) → H_{n−1}(SL_{n}(F), ℤ)) ≅ I^{n}(F).

Proof. Since we have already proved this result for n = 3 above, we will assume that n ≥ 5 (n odd). Let a_{1}, …, a_{n} ∈ F^{×} and let z ∈ H_{n−1}(SL_{n−1}(F), ℤ) satisfy ϵ_{n−1}(z) = {{a_{2}, …, a_{n}}} ∈ F_{n−1,0} ≅ K_{n−1}^{MW}(F). Thus {{a_{1}, …, a_{n}}} = ⌊a_{1}⌉ ∗ ϵ_{n−1}(z) and hence ϵ_{n,1}({{a_{1}, …, a_{n}}}) = ⟨⟨a_{1}⟩⟩z by Theorem 5.20 (2). It follows that the diagram

\xymatrix{F_{n,1} \ar[r]^{ϵ_{n,1}} \ar[d]^{T_{n}}_{≅} & H_{n−1}(SL_{n−1}(F), ℤ) \ar[d]^{T_{n−1}∘ϵ_{n−1}} \\ K_{n}^{MW}(F) \ar[r]^{η} & K_{n−1}^{MW}(F)}

commutes. Now Ker(ϵ_{n,1}) = Im(ϵ_{n} : H_{n}(SL_{n}(F), ℤ) → F_{n,1}). Since Im(ϵ_{3}) = T_{3}^{−1}(2K_{3}^{M}(F)) and Im(ϵ_{n−3}) = F_{n−3,1} = T_{n−3}^{−1}(K_{n−3}^{MW}(F)) we have

T_{n}(Im(ϵ_{n})) = Im(T_{n} ∘ ϵ_{n}) ⊃ 2K_{3}^{M}(F) · K_{n−3}^{MW}(F) = 2K_{n}^{M}(F) ⊂ K_{n}^{MW}(F)

(using the fact that T_{•} and ϵ_{•} are algebra homomorphisms). Thus we get a commutative diagram

[figure: \frac{K_{n}^{MW}(F)}{2K_{n}^{M}(F)}; T_{n}^{−1}; η; ≅; \frac{F_{n,1}}{Ker(ϵ_{n,1})}; T_{n−1}∘ϵ_{n−1}∘ϵ_{n,1}; I^{n}(F)]

from which it follows that the map T_{n}^{−1} in this diagram is an isomorphism, and hence Im(ϵ_{n}) = Ker(ϵ_{n,1}) ≅ 2K_{n}^{M}(F) and Im(ϵ_{n,1}) ≅ I^{n}(F). □

7. Acknowledgements

The work in this article was partially funded by the Science Foundation Ireland Research Frontiers Programme grant 05/RFP/MAT0022.

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Kevin Hutchinson School of Mathematical Sciences University College Dublin Dublin 4 Ireland kevin.hutchinson@ucd.ie Liqun Tao School of Mathematical Sciences Luoyang Normal University Luoyang 47 1022, P. R. China lqtao@lynu.edu.cn 316

Documenta Mathematica · Extra Volume Suslin (2010)