MOTIVIC DECOMPOSITION OF CERTAIN SPECIAL LINEAR GROUPS

ALEXANDER S. MERKURJEV

Abstract. We compute the motive of the algebraic group G = SL_{1}(D) for a central simple algebra D of prime degree over a perfect field. As an application we determine certain motivic cohomology groups and differentials in the motivic spectral sequence of G.

1. Introduction

In this paper we study the motive in the triangulated category of geometric mixed effective motives DM_{gm}^{eff}(F) over a perfect field F and the motivic cohomology of the algebraic group SL_{1}(D) of reduced norm 1 elements in a central simple algebra D of prime degree l.

In [18], A. Suslin computed the K-cohomology groups of the (split) special linear group SL_{n} and the symplectic groups Sp_{2n} using higher Chern classes in K-cohomology. O. Pushin in [15] constructed higher Chern classes in motivic cohomology and found decompositions of the motives of the groups SL_{n} and GL_{n} into direct sums of Tate motives. S. Biglari computed in [1] the motives of certain split reductive groups over ℚ. In particular, he showed that

(1.1)  M(SL_{n})_{ℚ} ≃ ∐_{i=0}^{n−1} Sym^{i}(ℚ(2)[3] ⊕ ℚ(3)[5] ⊕ ⋯ ⊕ ℚ(n)[2n − 1]).

A. Huber and B. Kahn determined the motives over ℤ of split reductive groups in [9].

The motives of non-split algebraic groups are more complicated. The slices of the slice filtration of the motive M(GL_{l}(D)) for a division algebra D of prime degree were computed by E. Shinder in [17].

In this paper we study the motive of the group G = SL_{1}(D), where D is a central simple algebra of a prime degree l. As a warm-up, consider the simplest case l = 2. The variety of G is then an open subscheme of a 3-dimensional projective isotropic quadric X given by the homogeneous quadratic equation Nrd = t^{2}, where Nrd is the reduced norm form of D. The surface Y = X \ G, given by Nrd = 0, is isomorphic to S × S, where S is the Severi-Brauer variety of D (a conic curve in the case l = 2). Computing the motives of X and Y as in [17, §4], we get an exact triangle

Key words and phrases. Central simple algebras, Severi-Brauer varieties, special linear groups, motives, motivic cohomology. Mathematical Subject Classification (2010): 20G15; 19E15, 14F42.

The work of the author has been supported by the NSF grant DMS #1160206 and the Guggenheim Fellowship.

M(G) → ℤ ⊕ M(S)(1)[2] ⊕ ℤ(3)[6] → M(S)(1)[2] ⊕ M(S)(2)[4] → M(G)[1].

Canceling out the summands M(S)(1)[2], we obtain an isomorphism

(1.2)  M(G) ≃ ℤ ⊕ N(2)[3],

where the motive N is defined by the exact triangle

ℤ(1)[2] → M(S) → N → ℤ(1)[3]

with the first morphism dual to the canonical one M(S) → ℤ.

In the general case, when l is an arbitrary prime, since the group G and its motive are split over a field extension of degree l, the torsion part of motivic cohomology of G is l-torsion. We work over the coefficient ring ℤ[\frac{1}{(l−1)!}], just inverting insignificant integers.

As in the case l = 2, the motive of G can be computed out of motive of the Severi-Brauer variety S of the algebra D. Let N be the motive defined by the exact triangle

ℤ(l − 1)[2l − 2] → M(S) → N → ℤ(l − 1)[2l − 1].

As (l−1)! is invertible in the coefficient ring, one can define symmetric Sym^{i}(M) and alternating powers Alt^{i}(M) of any motive M for i = 0, 1, …, l − 1. The main result of the paper is the following theorem generalizing (1.1) and (1.2) (see Theorem 11.1).

Theorem. Let D be a central simple algebra of prime degree l over a perfect field F. Then there is an isomorphism

M(SL_{1}(D)) →^{∼} ∐_{i=0}^{l−1} Sym^{i}(N(2)[3]) = ∐_{i=0}^{l−1}(Alt^{i}N)(2i)[3i]

in the category DM_{gm}^{eff}(F) of motives over F with coefficients in ℤ[\frac{1}{(l−1)!}].

The most difficult part of the proof is the construction of a morphism M(G) → N(2)[3] in DM_{gm}^{eff}(F). The main players of the proof are the groups H^{3,2}(G) ≃ ℤ and the Chow group CH^{l+1}(G) = H^{2l+2,l+1}(G) ≃ ℤ/lℤ (when D is not split). These groups are related by a pair of homomorphisms

(1.3)  H^{3,2}(G) ← Hom(M(G), N(2)[3]) → H^{2l+2,l+1}(G).

We prove that there is a morphism M(G) → N(2)[3] with the images in (1.3) generating the two side cyclic groups. This is done in Section 10.

Using Theorem 1 and the exact triangle (Corollary 6.5)

(Alt^{i−1}N)(l − 1)[2l − 2] → Alt^{i}M(S) → (Alt^{i}N) → (Alt^{i−1}N)(l − 1)[2l − 1],

we can compute inductively the motivic cohomology of G. As an application, in Section 12 we compute the motivic cohomology H^{p,q}(G) with 2q − p ≤ 1. We also compute certain differentials in the motivic spectral sequence of G.

Acknowledgements. I thank Shahram Biglari and Bruno Kahn for helpful discussions.

2. Motivic cohomology

The base field F is assumed to be perfect. We fix a prime integer l and work over the coefficient ring Z that is either the ring of integers ℤ or ℤ[\frac{1}{(l−1)!}], or the localization ℤ_{(l)} of ℤ by the prime ideal generated by l. Note that Z /l Z = ℤ/lℤ.

We write DM(F) := DM_{gm}^{eff}(F) for the triangulated category of (geometric mixed effective) motives with coefficients in Z (see [19]). If p and q ≥ 0 are integers, Z(q)[p] denotes the Tate motive and M(X) the motive of a smooth variety X over F. We have M(Spec F) = Z := Z(0)[0]. For a motive M and an integer q ≥ 0, we write M(q) for M ⊗ Z(q).

For a motive M in DM(F) define the motivic cohomology by

H^{p,q}(M) := Hom(M, Z(q)[p]),

where Hom is taken in the category DM(F). If X is a smooth variety, simply write H^{p,q}(X) for H^{p,q}(M(X)). We have

(2.1)  H^{p,q}(X) = 0 if p > 2q or p > q + dim(X).

In particular, H^{p,q}(F) = 0 if p > q. Moreover, H^{p,p}(F) = K_{p}^{M}(F), the Milnor K-groups of F (see [12, Lecture 5]).

The bi-graded group ∐_{p,q} H^{p,q}(X) has a natural structure of a graded commutative ring (with respect to p, [12, Theorem 15.9]).

There is a canonical isomorphism between H^{2p,p}(X) and the Chow groups CH^{p}(X) of (rational equivalence) classes of algebraic cycles on a smooth variety X of codimension p ([12, Lecture 18]). We also write CH^{p}(M) := H^{2p,p}(M) for every motive M.

The cancelation theorem (see [21]) states that the canonical morphism

Hom(M, N) →^{∼} Hom(M(1), N(1))

is an isomorphism for every two motives M and N.

The natural functor from the category of smooth projective varieties over F to DM(F) extends uniquely to a canonical functor from the category Chow(F) of Chow motives over F to DM(F) (see [19, Proposition 2.1.4]). The motives in DM(F) coming from Chow(F) are called pure motives.

Let M be any motive and X a smooth projective variety of pure dimension d over F. The two canonical morphisms (given by the diagonal of X in the category of Chow motives)

Z(d)[2d] → M(X × X) → Z(d)[2d]

together with the cancelation theorem define the two mutually inverse isomorphisms (see [9, Appendix B])

(2.2) Hom(M, M(X)) ⇄ Hom(M ⊗ M(X), Z(d)[2d]) = CH^{d}(M ⊗ M(X)).

In particular, if Y is another smooth projective variety, then

Hom_{DM(F)}(M(Y), M(X)) ≃ CH^{d}(Y × X) = Hom_{Chow(F)}(M(Y), M(X)).

We say that a motive M is of degree d if M is a direct summand of a motive of the form M(X)(q)[p] with 2q−p = d, where X is a smooth projective variety. The pure motives are of degree 0. The following statement is an immediate consequence of (2.1).

Lemma 2.3. Let M and N be motives of degree d and e respectively. If d > e, then Hom(M, N) = 0. □

The coniveau spectral sequence for a smooth variety X over F,

E_{1}^{p,q} = ∐_{x∈X^{(p)}} H^{q−p,n−p}F (x) ⇒ H^{p+q,n}(X),

where X^{(p)} is the set of points in X of codimension p, yields isomorphisms

H^{i+n,n}(X) ≃ A^{i}(X, K_{n}) when n − i ≤ 2

with the K-cohomology groups A^{i}(X, K_{n}) defined in [16].

If X is a variety over F, we write X_{sep} for the variety X ⊗_{F} F_{sep} over a separable closure F_{sep} of F.

3. Severi-Brauer varieties

Let D be a central simple algebra of degree n over F and S the Severi-Brauer variety SB(D) of right ideals in D of rank n. This is a smooth projective variety of dimension n − 1 (see [6]). If D is split, i.e., D = End(V) for an n-dimensional vector space V over F, then S is isomorphic to the projective space ℙ(V). Therefore, in the split case,

M(S) ≃ Z ⊕ Z(1)[2] ⊕ ⋯ ⊕ Z(l − 1)[2l − 2].

Let I → S be the tautological vector bundle of rank n (with the fiber over a right ideal the ideal itself). We have D = End(I)^{op} = End(I^{∨}), where I^{∨} is the vector bundle dual to I. In the split case, when S = ℙ(V),

I = V^{∨} ⊗ L_{t} = Hom(V, L_{t}),

where L_{t} → ℙ(V) is the tautological line bundle. The sheaf of sections of L_{t} is O(−1).

In the split case, when S = ℙ(V), let s ∈ CH^{1}(S) be the class of a hyperplane section. We have

CH^{i}(ℙ(V)) = \{ \begin{matrix}ℤs^{i}, & i = 0, 1, …, n − 1; \\ 0, & otherwise. \end{matrix}

The ring End(M(S)) = CH^{n−1}(S×S) is canonically isomorphic to the product Z^{n} of n copies of Z with the idempotents s^{i} × s^{n−1−i}.

In the non-split case we have the following statement (see [13, Corollary 8.7.2]):

Lemma 3.1. When D is a division algebra of prime degree l, the natural map CH^{∗}(S) → CH^{∗}(S_{sep}) is injective and it identifies the Chow group of S as follows

CH^{i}(S) = \{ \begin{matrix}Z 1, & i = 0; \\ l Z s^{i}, & i = 1, …, l − 1; \\ 0, & otherwise. □ \end{matrix}

Any of the two projections p : S × S → S is the projective bundle of I^{∨}, i.e., S × S = ℙ_{S}(I^{∨}). Let L be the tautological line bundle of this projective bundle. By the projective bundle theorem [19, Proposition 3.5.1], we have:

(3.2) CH^{l−1}(S × S) = CH^{l−1}(S) · 1 ⊕ CH^{l−2}(S) · ξ ⊕ ⋯ ⊕ CH^{0}(S) · ξ^{l−1},

where ξ is the first Chern class of L in CH^{1}(S × S). Consider the composition

type : CH^{l−1}(S × S) = End M(S) → End M(S_{sep}) →^{∼} Z^{l} .

Proposition 3.3. Let D be a division algebra of degree l and S = SB(D). Then the ring homomorphism

type : End M(S) → Z^{l}

is injective. Its image consists of all tuples (a_{1}, a_{2}, …, a_{l}) such that

a_{1} ≡ a_{2} ≡ ⋯ ≡ a_{l} (mod l).

Proof. It follows from Lemma 3.1 and (3.2) that type is injective and [Im(type) : l Z^{l}] = l. Therefore, the identity in Z^{l} and l Z^{l} generate Im(type). □

We will also need the following lemma.

Lemma 3.4. Let M_{1} and M_{2} be direct sum of shifts of M(S) (with arbitrary coefficients) and f : M_{1} → M_{2} a morphism in DM(F). If f is an isomorphism over a field extension, then f is also an isomorphism.

Proof. Write M_{1} and M_{2} as direct sums of the homogeneous (degree k) components M_{1}^{(k)} and M_{2}^{(k)} respectively. By Lemma 2.3, the morphism f is given by a triangular matrix with the diagonal terms f_{k} : M_{1}^{(k)} → M_{2}^{(k)}. By assumption, the matrix is invertible over a splitting field L, hence all f_{k} are isomorphisms over L. Note that f_{k} is a shift of a morphism of pure motives that are direct sums of shifts of M(S). By [4, Corollary 92.7], all f_{k} are isomorphisms. Therefore, the triangular matrix is invertible and hence f is an isomorphism. □

4. The motive N

Let D be a central simple algebra of prime degree l over F and S = SB(D). The motive N is defined by the triangle

(4.1)  Z(l − 1)[2l − 2] → M(S) ⟶^{κ} N ⟶^{ε} Z(l − 1)[2l − 1]

in DM(F) with the first morphism of pure motives given by the identity in CH^{0}(S). We have

(4.2)  N_{sep} ≃ Z ⊕ Z(1)[2] ⊕ ⋯ ⊕ Z(l − 2)[2l − 4],

therefore, Hom(M(S_{sep}), N_{sep}) ≃ Z^{l−1}.

Consider the map

type : Hom(M(S), N) → Hom(M(S_{sep}), N_{sep}) ≃ Z^{l−1} .

For example, type(κ) = (1, 1, …, 1).

Proposition 4.3. Let D be a division algebra of degree l and S = SB(D). Then the homomorphism

type : Hom(M(S), N) → Z^{l−1}

is injective. Its image consists of all tuples (a_{1}, a_{2}, …, a_{l−1}) such that

a_{1} ≡ a_{2} ≡ ⋯ ≡ a_{l−1} (mod l).

Proof. Let φ ∈ Ker(type). The triangle (4.1) yields an exact sequence

CH^{l−1}(S) → End M(S) → Hom(M(S), N) → H^{2l−1,l−1}(S).

The last term is zero as 2l − 1 > 2(l − 1). Therefore, φ = κ ∘ σ for some σ ∈ End M(S). By assumption, type(σ) = (0, …, 0, a), where a ≡ 0 modulo l in view of Proposition 3.3. Then σ comes from CH^{l−1}(S) = l Z by Lemma 3.1 and hence φ = 0. This proves injectivity. The second statement follows from Proposition 3.3. □

Lemma 4.4. There is an isomorphism

N ⊗ M(S) ≃ M(S) ⊕ M(S)(1)[2] ⊕ ⋯ ⊕ M(S)(l − 2)[2l − 4].

In particular, N ⊗ M(S) is a pure motive.

Proof. The triangle (4.1) is split after tensoring with M(S). Indeed, the morphism M(S)(l − 1)[2l − 2] → M(S) ⊗ M(S) has a left inverse given by the class of the diagonal in CH^{2l−2}(S × S × S). □

Lemma 4.5. We have CH^{i}(N) = 0 if i > l.

Proof. In the exact sequence induced by (4.1)

H^{2i−2l+1,i−l+1}(F) → CH^{i}(N) → CH^{i}(S)

the first and the last terms are trivial as 2i−2l+1 > i−l+1 and dim(S) < l. □

Since Hom(Z(q)[p], Z) = 0 if q > 0, the natural morphism M(S) → Z factors uniquely through a morphism ν : N → Z.

5. Higher Chern classes

Let X be a smooth variety. The higher Chern classes with values in motivic cohomology were constructed in [15]:

c_{j,i} : K_{j}(X) → H^{2i−j,i}(X).

We will be using the classes

c_{i} := c_{1,i+1} : K_{1}(X) → H^{2i+1,i+1}(X).

Proposition 5.1 ([17, §4.1]). Let L be a vector bundle over a smooth variety X and α ∈ K_{1}(X). Then

c_{i}(α · [L]) = ∑_{j=0}^{i}(−1)^{j}\binom{i}{j}c_{i−j}(α)h^{j},

where h ∈ CH^{1}(X) = H^{2,1}(X) is the first (classical) Chern class of L.

Let E → X be a vector bundle of rank n. We write SL(E) for the group scheme over X of determinant 1 automorphisms of E.

Let a be the generic element of SL(E) (see [18, §4]). We also write a for the corresponding element in K_{1}(SL(E)). We have c_{0}(a) = 0 since det(a) = 1. For a sequence i = (i_{1}, i_{2}, …, i_{k}) with 1 ≤ i_{1} < i_{2} < ⋯ < i_{k} ≤ n − 1, set d_{i} = i_{1} + i_{2} + ⋯ + i_{k} and e_{i} = k. Let

c_{i}(α) := c_{i_{1}}(α)c_{i_{2}}(α) ⋯ c_{i_{k}}(α) ∈ H^{2d_{i}+e_{i},d_{i}+e_{i}}(SL(E)).

Proposition 5.2. Let E → X be a vector bundle of rank n. Then the H^{∗,∗}(X)- module H^{∗,∗}(SL(E)) is free with basis {c_{i}(α)} over all sequences i.

Proof. This follows from [15, Proposition 3]. □

Write c̃_{i}(α) for the composition

M(SL(E)) ⟶^{diag} M(SL(E)) ⊗ M(SL(E)) ⟶^{j⊗c_{i}(α)} M(X)(d_{i} + e_{i})[2d_{i} + e_{i}],

where j : M(SL(E)) → M(X) is the canonical morphism. The following corollary is deduced from Proposition 5.2 the same way as in [17, Proposition 4.2].

Corollary 5.3. (cf. [17, Proposition 4.4]) The morphisms c̃_{i}(α) yield an isomorphism

M(SL(E)) →^{∼} ∐_{i} M(X)(d_{i} + e_{i})[2d_{i} + e_{i}]. □

Remark 5.4. The natural composition

M(SL(E)) ⟶^{c̃_{i}(α)} M(X)(d_{i} + e_{i})[2d_{i} + e_{i}]→ Z(d_{i} + e_{i})[2d_{i} + e_{i}]

coincides with c_{i}(α).

Corollary 5.5. There is a canonical isomorphism

M(SL_{n}) ≃ ∐_{i} Z(d_{i} + e_{i})[2d_{i} + e_{i}]. □

Let G = SL(E) where E → X is a vector bundle of rank n over a smooth variety X. Consider the grading on M(G) with respect to the value e(i):

M(G)^{(k)} := ∐_{e_{i}=k} M(X)(d_{i} + k)[2d_{i} + k)]

for k = 0, 1, …, n − 1. Thus,

M(G) = ∐_{k=0}^{n−1} M(G)^{(k)}

and each motive M(G)^{(k)} has degree k.

Example 5.6. In the split case, we have a natural isomorphism M(SL_{l})^{(1)} ≃ N(2)[3].

Let D be a central simple algebra of prime degree l over F and G = SL_{1}(D). Let S be the Severi-Brauer variety of D.

Corollary 5.3 yields

Corollary 5.7. There is a canonical isomorphism

M(G × S) ≃ ∐_{i} M(S)(d_{i} + e_{i})[2d_{i} + e_{i}].

In particular, CH^{∗}(G × S) ≃ CH^{∗}(S). □

It follows from Corollary 5.5 that M(G_{sep})^{(1)} ≃ N_{sep}(2)[3] and therefore Hom(M(G_{sep}), M(S_{sep})(2)[3]) is naturally isomorphic to Z^{l−1}. Consider the map

type : Hom(M(G), M(S)(2)[3]) → Hom(M(G_{sep}), M(S_{sep})(2)[3]) ≃ ℤ^{l−1}.

By (2.2) and Corollary 5.7, we have

(5.8) Hom(M(G), M(S)(2)[3]) = H^{2l+1,l+1}(G × S) = ∐_{i=1}^{l−1} CH^{i}(S)c_{l−i}(α).

Lemma 3.1 and (5.8) yield the following proposition.

Proposition 5.9. Let D be a division algebra of degree l and S = SB(D). Then the homomorphism

type : Hom(M(G), M(S)(2)[3]) → Z^{l−1}

is injective and Im(type) = l Z^{l−1}. □

We will need the Chow groups of G that were computed in [10].

Proposition 5.10. Let D be a central division algebra of prime degree l and G = SL_{1}(D). There is an element h ∈ CH^{l+1}(G) such that

CH^{∗}(G) = Z ·1 ⊕ (Z /l Z)h ⊕ (Z /l Z)h^{2} ⊕ ⋯ ⊕ (Z /l Z)h^{l−1}. □

Recall that D ≃ End(I^{∨}) and G × S ≃ SL(I^{∨}), where I is the tautological vector bundle over S of rank n (see Section 3).

Suppose that the algebra D is split. We can compare the generic matrices ᾱ in G = SL_{l} and α in G × S = SL(I^{∨}). The bundle I^{∨} ⊗ L_{t} over G × S is trivial, hence

ᾱ × S = α ⊗ L_{t} in K_{1}(G × S).

We have the Chern classes c_{i}(α) ∈ H^{2i+1,i+1}(G × S) and c_{i} := c_{i}(ᾱ) ∈ H^{2i+1,i+1}(G). We also write c_{i} for its image in H^{2i+1,i+1}(G × S) under the pull-back map given by the projection G × S → G.

By Proposition 5.1, we have

(5.11)  c_{i} = ∑_{j=0}^{i−1} \binom{i}{j}c_{i−j}(α)s^{j} in H^{2i+1,i+1}(G × S)

for all i = 1, 2 …, l − 1, since the first Chern class of L_{t} is equal to −s, where s ∈ CH^{1}(S) is the class of a hyperplane section, and c_{0}(α) = 0 as det(α) = 1. In particular, c_{1} = c_{1}(α).

The group H^{3,2}(G) = A^{1}(G, K_{2}) is infinite cyclic with a canonical generator, and this group does not change under field extensions. (This is true for every absolutely simple simply connected group, see [5, Part II, §9].) Therefore, we can write H^{3,2}(G) = ℤc_{1} viewing c_{1} as a generator of H^{3,2}(G).

6. Symmetric and alternating powers

We consider motives with coefficients in Z = ℤ[\frac{1}{(l−1)!}] in this section. Let i = 0, 1, …, l − 1. The symmetric group Σ_{i} acts naturally on the i-th tensor power M^{⊗i} of a motive M. The elements

τ_{i} = \frac{1}{i!} ∑_{σ∈Σ_{i}} σ and ρ_{i} = \frac{1}{i!} ∑_{σ∈Σ_{i}} sgn(σ)σ

are idempotents in the group ring of Σ_{i}. The motives Sym^{i}(M) := (M, τ_{i}) and Alt^{i}(M) := (M, ρ_{i}), that are split off M by the projectors τ_{i} and ρ_{i}, are called the i-th symmetric power and i-th alternating power of M respectively. We have Sym^{0}(M) = ℤ = Alt^{0}(M) and Sym^{1}(M) = M = Alt^{1}(M).

We will need the following properties of symmetric and alternating powers.

Proposition 6.1 ([1, Proposition 2.3]). Let M and N be two motives. Then (1) Sym^{i}(M[1]) ≃ (Alt^{i}M)[i] and Alt^{i}(M[1]) ≃ (Sym^{i}M)[i],

(2) Sym^{i}(M(q)) ≃ (Sym^{i}M)(iq),

(3) Sym^{i}(M ⊕ N) = ∐_{k+m=i} Sym^{k}(M) ⊗ Sym^{m}(N) and similarly for Alt.

Corollary 6.2. We have

Sym^{i}(Z(q)[p]) ≃ \{ \begin{matrix}Z(iq)[ip], & if p is even; \\ 0, & if p > 1 is odd. □ \end{matrix}

Example 6.3. Let N be the motive Z ⊕ Z(1)[2] ⊕ ⋯ ⊕ Z(l − 2)[2l − 4] (see (4.2)). Then

Sym^{k}(N(2)[3]) = (Alt^{k}N)(2k)[3k] = ∐_{e_{i}=k} Z(d_{i} + k)[2d_{i} + k],

with the notation from Section 5.

Proposition 6.4 ([8, Proposition 15]). Let X → Y → Z → X[1] be an exact triangle. Then there are sequences of morphisms

Alt^{i}X = T_{0} → T_{1} → ⋯ → T_{i} = Alt^{i}Y,
Sym^{i}X = V_{0} → V_{1} → ⋯ → V_{i} = Sym^{i}Y

and exact triangles

T_{j−1} → T_{j} → Alt^{i−j}X ⊗ Alt^{j}Z → T_{j−1}[1],
V_{j−1} → V_{j} → Sym^{i−j}X ⊗ Sym^{j}Z → V_{j−1}[1]

for every j = 1, 2, …, i.

Assuming that Alt^{k}X = 0 for k > 1, we get an exact triangle

X ⊗ Alt^{i−1}Z → Alt^{i}Y → Alt^{i}Z → (X ⊗ Alt^{i−1}Z)[1].

Applying this to the exact triangle (4.1), we have the following proposition.

Corollary 6.5. There is an exact triangle

(Alt^{i−1}N)(l − 1)[2l − 2] → Alt^{i}M(S) → Alt^{i}N → (Alt^{i−1}N)(l − 1)[2l − 1]. □

This proposition will be used in Section 12 to compute inductively the motivic cohomology of Alt^{i}N.

The pure motive Alt^{i}M(S) is a direct summand of M(S^{i}) and the latter is a direct sum of shifts of the motive M(S). If D is a division algebra, the motive M(S) is indecomposable [11, Corollary 2.22]. When the coefficient ring Z is the local ring ℤ_{(l)}, by uniqueness of the decomposition [3, Corollary 35], Alt^{i}M(S) is a pure motive that is a direct sum of pure shifts of M(S). Moreover, since in the split case

Alt^{i}M(S_{sep}) = Z((i(i − 1)/2)[i(i − 1)] ⊕ terms with higher shifts,

we must have

(6.6) Alt^{i}M(S) = M(S)((i(i − 1)/2)[i(i − 1)] ⊕ terms with higher shifts.

7. Split case

We are going to prove the main theorem in the split case. Let G = SL_{l} with prime l. We have

N = Z ⊕ Z(1)[2] ⊕ Z(2)[4] ⊕ ⋯ ⊕ Z(l − 2)[2l − 4].

In fact, N is a direct summand of the motive of S = ℙ^{l−1}:

M(S) = N ⊕ Z(l − 1)[2l − 2].

The Chern classes c_{1}(ā), c_{2}(ā), …, c_{l−1}(ā) with values in the motivic cohomology of G, where ā is the generic matrix in SL_{l}, define a morphism

φ_{1} : M(G) → N(2)[3].

For every i = 0, 1, …, l − 1, consider the composition

φ_{i} : M(G) ⟶^{diag} M(G^{i}) ⟶^{φ^{i}} N(2)[3]^{⊗i} → Sym^{i}(N(2)[3]),

where the first morphism is given by the diagonal embedding.

Proposition 7.1. In the split case G = SL_{l}, the morphism

φ = (φ_{i}) : M(G) → ∐_{i=0}^{l−1} Sym^{i}(N(2)[3])

is an isomorphism.

Proof. By Corollary 5.5, Example 6.3 and Lemma 2.3, for every i = 0, 1, …, l− 1, the morphism φ_{i} is zero on M(G)^{(j)} for j > i and yields an isomorphism

M(G)^{(i)} →^{∼} Sym^{i}(N(2)[3]).

The result follows, as φ is given by an invertible triangular matrix. □

8. Compactifications of G

Let D be a central division algebra over F and G = SL_{1}(D). By [2, §6.2] and [10, §6], G admits a smooth projective G×G-equivariant compactification X. In other words, X is a projective variety equipped with an action of G × G and containing G as an open orbit on which the group G × G acts by the left-right translations. The motive of X is split (i.e., M(X) is a direct sum of Tate motives) over any field extension that splits D by [10, Theorem 6.5].

By Proposition 5.10, the group CH^{l+1}(G) is cyclic of order l. Choose a generator h ∈ CH^{l+1}(G) and let h̄ ∈ CH^{l+1}(X) be any element such that h̄|_{G} = h. Set

(8.1) R = ℤ ⊕ ℤ(l + 1)[2l + 2] ⊕ ℤ(2l + 2)[4l + 4] ⊕ ⋯ ⊕ ℤ(l^{2} − 1)[2l^{2} − 2].

Proposition 8.2 ([10, §8]). When Z = ℤ_{(l)}, the morphism M(X) → R defined by the powers of h̄ has a right inverse. Moreover, we have M(X) ≃ R ⊕ T for a pure motive T that is a direct sum of pure shifts of M(S), where S = SB(D).

Note that the composition π : M(G) → M(X) → R is given by the powers

h^{i} ∈ CH^{i(p+1)}(G) = Hom(M(G), ℤ(i(p + 1)[2i(p + 1)]).

9. The morphism θ

In this section we construct a morphism θ : M(S)(2)[3] → M(G).

As M(G_{sep})^{(1)} ≃ N_{sep}(2)[3] by Example 5.6, there is a natural isomorphism

Hom(M(S_{sep})(2)[3], M(G_{sep})^{(1)}) ≃ Z^{l−1} .

Projecting M(G_{sep}) onto M(G_{sep})^{(1)}, we get a composition

type : Hom(M(S)(2)[3], M(G)) → Hom(M(S_{sep})(2)[3], M(G_{sep})) → Z^{l−1} .

By Corollary 5.7, there is a canonical morphism

θ : M(S)(2)[3] → M(G × S) → M(G),

where the first morphism is the projection to the i-component for i = (1) and the second morphism is given by the projection G × S → G.

Note that by definition of θ, the composition of M(S)(2)[3] → M(G × S) with c̃_{k}(α) : M(G × S) → M(S)(k + 1)[2k + 1] is zero for k > 1. In view of (5.11) and Remark 5.4, the composition

M(S_{sep})(2)[3] ⟶^{θ_{sep}} M(G_{sep}) ⟶^{c_{i}} Z(i + 1)[2i + 1]

for i = 1, 2, …, l − 1 coincides with i = \binom{i}{i−1} times the composition

M(S_{sep})(2)[3] → M(G_{sep} × S_{sep}) ⟶^{c_{1}s^{i−1}} Z(i + 1)[2i + 1].

The latter is equal to the morphism s^{i−1} : M(S_{sep})(2)[3] → Z(i + 1)[2i + 1] that is the identity on the summand Z(i + 1)[2i + 1]. It follows that

type(θ) = (1, 2, …, l − 1).

10. A key lemma

Let D be a central simple algebra of degree l and S = SB(D).

Lemma 10.1. Let Y be a variety over F such that D is split over the residue field F (y) for every y ∈ Y. Then the push-forward homomorphism CH_{j}(Y × S) → CH_{j}(Y) is surjective for every j.

Proof. Let y ∈ Y be a point of dimension j. As S is split over F (y), there is a F (y)-rational point y′ ∈ Y × S in the fiber of the projection q : Y × S → Y over y. We have [y] = q_{∗}([y′]). □

Lemma 10.2. Let U be a smooth variety such that D is split over F (u) for every u ∈ U. Then H^{2j+1,j}(M(U) ⊗ N) = 0 for every j.

Proof. The last term in the exact sequence induced by the triangle (4.1)

H^{2j,j}(U × S) → H^{2k,k}(U) → H^{2j+1,j}(M(U) ⊗ N) → H^{2j+1,j}(U × S),

where k = j − l + 1, is zero as 2j + 1 > 2j. The first map is surjective by Lemma 10.1. □

Let Y be a closed subvariety of a smooth variety X. We define the motive M_{Y} (X) by the triangle

(10.3)  M(X \ Y) → M(X) → M_{Y} (X) → M(X \ Y)[1].

Lemma 10.4. Let X be a smooth irreducible variety and Y ⊂ X an equidimensional closed subvariety such that S is split over F (y) for every y ∈ Y. Then H^{2i+1,i}(M_{Y} (X) ⊗ N) = 0 for every i.

Proof. We proceed by induction on dim(Y). Choose a closed subset Z ⊂ Y of pure codimension 1 such that Y \ Z is smooth. In the exact triangle

M_{Y \Z}(X \ Z) → M_{Y} (X) → M_{Z}(X) → M_{Y \Z}(X \ Z)[1]

the first term is isomorphic to M(Y \ Z)(c)[2c] for c = codim_{X}(Y) since Y \ Z is smooth by [19, Proposition 3.5.4]. In the exact sequence

H^{2i+1,i}(M_{Z}(X) ⊗ N) → H^{2i+1,i}(M_{Y} (X) ⊗ N) → H^{2i+1,i}(M_{Y \Z}(X \ Z) ⊗ N)

the last tern is isomorphic to H^{2i−2c+1,i−c}(M(Y \ Z) ⊗ N) which is zero by Lemma 10.2, and the first term is zero by induction. □

Let X be a smooth G × G-equivariant compactification of G = SL_{1}(D) (see Section 8). Set Y := X \ G. By [10, Lemma 7.1], D is split by F (y) for every y ∈ Y. Applying Lemma 10.4 to the exact cohomology sequence for the exact triangle (10.3) we get the following corollary.

Corollary 10.5. The natural homomorphism

CH^{i}(M(X) ⊗ N) → CH^{i}(M(G) ⊗ N)

is surjective for every i. □

Lemma 10.6. The natural homomorphism

CH^{l+1}(R ⊗ N) → CH^{l+1}(M(G) ⊗ N)

induced by π : M(G) → R is surjective.

Proof. The group CH^{l+1}(M(G) ⊗ N) is l-torsion as it is split over a splitting field. Therefore, we may assume that Z = ℤ_{(l)}. Recall (see Section 8) that M(X) = R ⊕ T, where R is defined in (8.1) and the pure motive T is a direct sum of shifts of M(S). Moreover, π is the composition M(G) → M(X) → R.

Consider the commutative diagram

\xymatrix{CH^{l+1}(R ⊗ N) \ar[r]^{α} \ar@{-->}[dr] & CH^{l+1}(M(G) ⊗ N) & CH^{2l}(M(G) ⊗ N ⊗ S) \ar[l] \\  & CH^{l+1}(M(X) ⊗ N) \ar@{->>}[u]^{β} & CH^{2l}(M(X) ⊗ N ⊗ S) \ar[u] \ar[l] \\  & CH^{l+1}(T ⊗ N) \ar[u] & CH^{2l}(T ⊗ N ⊗ S), \ar[u] \ar[u] \ar@{->>}[l]_{γ}}

where γ (and the two homomorphisms above γ) are induced by the morphism Z(l − 1)[2l − 2] → M(S). By Corollary 10.5, β is surjective. The pure motive T is a direct sum of shifts of M(S), so is T ⊗ N by Lemma 4.4. It follows from Lemma 10.1 that γ is surjective.

Again by Lemma 4.4,

CH^{2l}(M(G) ⊗ N ⊗ S) = ∐_{i=0}^{l−2} CH^{2l−i}(G × S) = 0.

as CH^{j}(G × S) = 0 for j ≥ l in view of Corollary 5.7. Recall that M(X) = R ⊕ T. By diagram chase, α is surjective. □

Consider the following key diagram:

\xymatrix{Hom(M(G), M(S)(2)[3]) \ar[r]^{ξ} \ar[d]^{≀} & Hom(M(G), N(2)[3]) \ar[d]^{ρ} \ar[r] & CH^{l+1}(G) \ar[r] \ar[d]^{σ} & 0 \\ H^{2l+1,l+1}(G × S) \ar[r]^{τ} & H^{3,2}(G) \ar[r] & CH^{l+1}(M(G) ⊗ N) \ar[r] & 0.}

The rows of the diagram are induced by the exact triangle (4.1). The left vertical homomorphism is an isomorphism by (2.2). The rows are exact since CH^{l+1}(G × S) ≃ CH^{l+1}(S) = 0 by Corollary 5.7. The morphisms ρ and σ are induced by the morphism ν : N → Z (see Section 4).

The diagram is commutative. Indeed, both compositions in the right square take a morphism φ : M(G) → N(2)[3] to (ε(2)[3] ∘ φ) ⊗ ν. (The morphisms ε and ν are defined in Section 4.) Both compositions in the left square take a morphism ψ : M(G) → M(S)(2)[3] to ν(2)[3] ∘ ψ.

Now we can prove the following key lemma.

Lemma 10.7. The homomorphism induced by the morphism ν : N → ℤ

σ : CH^{l+1}(G) → CH^{l+1}(M(G) ⊗ N)

is an isomorphism.

Proof. In the commutative diagram

\xymatrix{CH^{l+1}(R) \ar[r] \ar[d] & CH^{l+1}(R ⊗ N) \ar[d] \\ CH^{l+1}(G) \ar[r] & CH^{l+1}(M(G) ⊗ N)}

the right vertical map (induced by π) is surjective by Lemma 10.6. We have CH^{l+1}(R) = Z and by Lemma 4.5, CH^{l+1}(R ⊗ N) = CH^{0}(N) = Z, hence the top map is an isomorphism. It follows that the bottom map in the diagram is surjective. If D is split, the group CH^{l+1}(G) is trivial an we are done.

Suppose D is a division algebra. Since H^{3,2}(G) = Z c_{1} (see Section 5), by Proposition 5.9, the image of τ in the key diagram is equal to l Z c_{1}. It follows that CH^{l+1}(M(G) ⊗ N) is a cyclic group of order l. The group CH^{l+1}(G) is also cyclic of order l by Proposition 5.10. The statement follows from the surjectivity of σ. □

It follows from Corollary 5.5 and Example 5.6 that Hom(M(G_{sep}), N_{sep}(2)[3]) is naturally isomorphic to Z^{l−1}. Consider the map

type : Hom(M(G), N(2)[3]) → Hom(M(G_{sep}), N_{sep}(2)[3]) ≃ Z^{l−1} .

Proposition 10.8. Let D be a division algebra of degree l and S = SB(D). Then the homomorphism

type : Hom(M(G), N(2)[3]) → Z^{l−1}

is injective. Its image consists of all tuples (a_{1}, a_{2}, …, a_{l−1}) such that

a_{1} ≡ 2a_{2} ≡ ⋯ ≡ (l − 1)a_{l−1} (mod l).

Proof. Let β ∈ Hom(M(G), N(2)[3]) have zero type. We have ρ(β) = kc_{1} with k the first component of the type of β. Hence k = 0. It follows from Lemma 10.7 that the image of β in CH^{l+1}(G) is trivial, t.e., β = ξ(γ) for some γ ∈ Hom(M(G), M(S)(2)[3]) with type(γ) = 0. By Proposition 5.9, γ = 0. This proves the injectivity of type.

Take any β ∈ Hom(M(G), N(2)[3]). We have

type(β) = (a_{1}, a_{2}, a_{3}, …, a_{l−1})

for some a_{i} ∈ ℤ. Composing β with θ : M(S)(2)[3] → M(G) (see Section 9) we get a morphism M(S)(2)[3] → N(2)[3] of type (a_{1}, 2a_{2}, 3a_{3}, … (l − 1)a_{l−1}). By Proposition 4.3, we have a_{1} ≡ 2a_{2} ≡ ⋯ ≡ (l − 1)a_{l−1} modulo l.

By Proposition 5.9, the image of the map type contains l Z^{l−1}. To finish the proof it suffices to find a β such that type(β) is not divisible by l. By Lemma 10.7 and diagram chase, the map ρ is surjective. Hence there is a morphism β : M(G) → N(2)[3] such that the composition of β with N(2)[3] → ℤ(2)[3] coincides with c_{1}, i.e., type(β) = (1, …). □

Remark 10.9. If α ∈ Hom(M(G), N(2)[3]) is such that type(α) is not divisible by l, then ρ(α) is not divisible by l in H^{3,2}(G) = Z c_{1}, and hence by Lemma 10.7, the image of α in CH^{l+1}(G) is not zero if D is not split.

11. Main theorem

Now the coefficient ring is Z = ℤ[\frac{1}{(l−1)!}]. By Proposition 10.8, there is a unique morphism β_{1} : M(G) → N(2)[3] with

type(β_{1}) = (1^{−1}, 2^{−1}, …, (l − 1)^{−1}).

For every i = 0, 1, …, l − 1 we have a composition

β_{i} : M(G) ⟶^{diag} M(G^{i}) ⟶^{β^{i}} N(2)[3]^{⊗i} → Sym^{i}(N(2)[3]).

Theorem 11.1. Let D be a central simple algebra of prime degree l over a perfect field F. Then the morphism

β = (β_{i}) : M(SL_{1}(D)) → ∐_{i=0}^{l−1} Sym^{i}(N(2)[3]) = ∐_{i=0}^{l−1}(Alt^{i}N)(2i)[3i]

in the category DM(F) of motives over F with coefficients in ℤ[\frac{1}{(l−1)!}] is an isomorphism.

Proof. We first prove the theorem in the split case. The morphisms β : M(G) → N(2)[3] and φ : M(G) → N(2)[3] of type (1, 1, …, 1) defined in Section 7 differ by an automorphism of N(2)[3] of type (1, 2, …, l − 1). Therefore, the statement follows from Proposition 7.1.

Assume that D is a division algebra. We show next that 1_{M(S)} ⊗ β is an isomorphism. By Corollary 5.7, the motive M(S) ⊗ M(G) = M(S × G) is a direct sum of shifts of M(S). The motive M(S) ⊗ N is a direct sum of shifts of M(S) by Lemma 4.4, hence so is M(S) ⊗ (Alt^{i}N). By the first part of the proof, β is an isomorphisms over a splitting field, hence so is 1_{M(S)} ⊗ β. By Lemma 3.4, 1_{M(S)} ⊗ β is an isomorphism. It follows that

(11.2)  1_{M(S^{i})} ⊗ β is an isomorphism for every i > 0.

We embed the category DM(F) into a larger triangulated category DM_{−}^{eff}(F) of motivic complexes with coefficients in Z as a full subcategory (see [19]).

Let Č(S) be the motive in DM_{−}^{eff}(F) associated with the simplicial scheme given by the powers of S (see [20, Appendix B]). Using the exact triangle in the proof of [20, Proposition 8.1] we see from (11.2) that 1_{Č(S)} ⊗ β is an isomorphism.

It follows from Remark 10.9 that the composition

M(G) ⟶^{β_{1}} N(2)[3] ⟶^{ε(2)[3]} Z(l + 1)[2l + 2]

represents a nontrivial element h ∈ CH^{l+1}(G). Therefore, for every i = 0, 1, …, l − 1, the composition

M(G) ⟶^{β_{i}} Sym^{i}(N(2)[3]) ⟶^{δ_{i}} Sym^{i}(Z(l + 1)[2l + 2]) = Z(i(l + 1))[2i(l + 1)], where δ_{i} = Sym^{i}(ε(2)[3]), is equal to h^{i}. By Section 8, we have a commutative diagram

\xymatrix{M(G) \ar[r]^{β} \ar[d]^{α} & ∐_{i=0}^{l−1} Sym^{i}(N(2)[3]) \ar[d]^{δ} \\ M(X) \ar[r]^{γ} & R,}

where X is a smooth compactification of G and δ = ∐ δ_{i}.

Consider the motive C̃(S) in DM_{−}^{eff}(F) defined by the exact triangle

(11.3)  C̃(S) → Č(S) → Z → C̃(S)[1].

We also have an exact triangle

M(G) → M(X) → M_{Y} (X) → M(G)[1],

where Y = X \ G. The algebra D is split by the residue field F (y) for every y ∈ Y by [10, Lemma 7.1]. Hence, by [17, Lemma 3.4], C̃(S) ⊗ M_{Y} (X) = 0. Therefore, 1_{C̃(S)} ⊗ α is an isomorphism.

By Proposition 8.2, M(X) ≃ R ⊕ T, where T is a direct sum of shifts of M(S) if Z = ℤ_{(l)}. Since C̃(S) ⊗ T = 0, we have 1_{C̃(S)} ⊗ γ is an isomorphism when Z = ℤ_{(l)}. As C̃(S) vanishes over a splitting field of D of degree l, 1_{C̃(S)}⊗γ is an isomorphism when Z = ℤ[\frac{1}{(l−1)!}].

By Proposition 6.4 applied to the exact triangle (4.1), there is a Postnikov tower connecting Sym^{i}(N(2)[3]) and

Sym^{i}(Z(l + 1)[2l + 2]) = Z(i(l + 1))[2i(l + 1)]

with “factors” divisible by M(S). Since C̃(S) ⊗ M(S) = 0, the morphism

1_{C̃(S)} ⊗ Sym^{i}ε : C̃(S) ⊗ Sym^{i}(N(2)[3]) →^{∼} C̃(S)(i(l + 1))[2i(l + 1)]

Is an isomorphism. Therefore, 1_{C̃(S)} ⊗ δ is an isomorphism.

It follows from the commutativity of the diagram that 1_{C̃(S)}⊗β is an isomorphism. Finally, by 5-lemma applied to the exact triangle (11.3), the morphism β is an isomorphism. □

12. Applications

As an application of Theorem 11.1, we compute certain motivic cohomology of G. The Chow groups CH^{i}(G) = H^{2i,i}(G) are given in Proposition 5.10. In Theorem 12.4 below we compute the groups H^{2i+1,i+1}(G).

The following Lemma is an immediate application of the exact triangle in Corollary 6.5.

Lemma 12.1. If p > 2q, then

H^{p,q}(Alt^{i−1}N) ≃ H^{p+2l−1,q+l−1}(Alt^{i}N). □

We compute the Chow groups of N.

Lemma 12.2. We have

CH^{i}(N) = \{ \begin{matrix}Z, & if i = 0; \\ l Z, & if i = 1, 2, …, l − 2; \\ F^{×}/ Nrd(D^{×}), & if i = l; \\ 0, & othervise. \end{matrix}

Proof. We may assume that Z = ℤ. Using (4.1) we get CH^{i}(N) ≃ CH^{i}(S) for i ≤ l − 2 and apply Lemma 3.1. In the exact sequence

0 → CH^{l−1}(N) → CH^{l−1}(S) → CH^{0}(Z)

the last map is injective again by Lemma 3.1, hence CH^{l−1}(N) = 0. In the exact sequence

H^{2l−1,l}(S) → H^{1,1}(F) → CH^{l}(N) → 0

the first map is isomorphic to A^{l−1}(S, K_{l}) → K_{1}^{M}(F) = F^{×} and its image is equal to Nrd(D^{×}) since the image is generated by the norms from finite field extensions that split D. By Lemma 4.5, CH^{i}(N) = 0 if i > l. □

Lemma 12.3. We have

H^{2i+1,i}(Alt^{2}N) = \{ \begin{matrix}Z /l Z, & if i = l − 1; \\ F^{×}/ Nrd(D^{×}), & if i = 2l − 1; \\ 0, & othervise. \end{matrix}

Proof. Using the triangle in Corollary 6.5, we get an exact sequence

CH^{i}(Alt^{2}M(S)) → CH^{i−l+1}(N) → H^{2i+1,i}(Alt^{2}N) → 0.

The middle group is trivial if i < l − 1, l = 2l − 2 and l > 2l − 1 by Lemma 12.2. The first map in the sequence is surjective in the split case since Alt^{2}N is pure and 2i + 1 > 2i. As CH^{i−l+1}(N) = l Z for i = l, l + 1, …, 2l − 3 by Lemma 12.2, the first map is also surjective in general for these values of i. If i = 2l − 1, the first group is trivial as Alt^{2}M(S) is a direct summand of M(S × S) and dim(S × S) = 2l − 2.

Finally consider the case i = l − 1. We may assume that Z = ℤ_{(l)}. As

Alt^{2}M(S) = M(S)(1)[2] ⊕ M(S)(3)[6] ⊕ ⋯ ⊕ M(S)(l − 2)[2l − 4],

we have

CH^{l−1}(Alt^{2}M(S)) = CH^{1}(S) ⊕ CH^{3}(S) ⊕ ⋯ ⊕ CH^{l−2}(S).

This is divisible by l when going to the split case by Lemma 3.1. Whence the case i = l − 1. □

Lemmas 12.1, 12.2 and 12.3 then yield Theorem 12.4. Let D be a central division algebra of degree l over F. Then

H^{2i+1,i+1}(SL_{1}(D)) = \{ \begin{matrix}F^{×}, & if i = 0; \\ ℤc_{1}, & if i = 1; \\ lℤc_{i}, & if i = 2, 3, …, l − 1; \\ ℤ/lℤ, & if i = k(l + 1) + 1 for k = 1, …, l − 2; \\ F^{×}/ Nrd(D^{×}), & if i = k(l + 1) for k = 1, …, l − 1; \\ 0, & otherwise. □ \end{matrix}

Let G = SL_{1}(D). Note that the cup-product maps

F^{×} ⊗ ℤ/lℤ = H^{1,1}(F) ⊗ CH^{k(l+1)}(G) → H^{2k(l+1)+1,k(l+1)+1}(G) = F^{×}/ Nrd(D^{×})

are natural surjections for k = 1, …, l − 1.

Consider the motivic spectral sequence for G when D is not split (see [7]):

E_{2}^{p,q} = H^{p−q,−q}(G) ⇒ K_{−p−q}(G).

The K-groups of G were computed in [18, Theorem 6.1]. In particular, K_{0}(G) = ℤ and K_{1}(G) = K_{1}(F) ⊕ K_{0}(D) ⊕ K_{0}(D^{op}) ≃ F^{×} ⊕ 3ℤ ⊕ 3ℤ. It follows that the zero-diagonal limit terms E_{∞}^{p,−p} are trivial if p ≠ 0. On the other hand, by Proposition 5.10, we have

E_{2}^{p,−p} = CH^{p}(G) = \{ \begin{matrix}ℤ, & if p = 0; \\ ℤ/lℤ, & if p = i(l + 1) for i = 1, …, l − 1; \\ 0, & otherwise. \end{matrix}

Note that in the split case all the differentials coming to the zero diagonal are trivial. It follows that in the general case such differential are l-torsion. By [14, Theorem 3.4], nontrivial differentials coming to the zero diagonal can appear only on pages E_{s}^{∗,∗} with l − 1 dividing s − 1. It follows that the nonzero differentials appear only on page E_{l}^{∗,∗} and they are

(12.5)  d : E_{l}^{1+k(l+1),−2−k(l+1)} → E_{l}^{(k+1)(l+1),−(k+1)(l+1)} = ℤ/lℤ

for k = 0, 1, …, l − 2. These maps are all surjective and are isomorphisms for k > 0, thus “clearing” the zero diagonal and partially the first diagonal. The other differentials on the l-th page coming to the first diagonal are the cup-products with F^{×} of the differentials (12.5). Nontrivial E_{∞}^{∗,∗}-terms on the first diagonal are F^{×} and lℤ (l − 1 times).

Below is a fragment of the third page of the spectral sequence when l = 3.

\xymatrix{ℤ \\ F^{×} & 0 \\ ∗ & ℤ \ar[ddrrr] & 0 \\  & ∗ \ar[ddrrr] & 3ℤ & 0 \\  &  & ∗ & 0 & ℤ/3ℤ \\  &  &  & ∗ & F^{×}/N & 0 \\  &  &  &  & ∗ & ℤ/3ℤ \ar[ddrrr] & 0 \\  &  &  &  &  & ∗ \ar[ddrrr] & 0 & 0 \\  &  &  &  &  &  & ∗ & 0 & ℤ/3ℤ \\  &  &  &  &  &  &  & ∗ & F^{×}/N}

It follows from Theorem 12.4 that the Chern classes c_{2}, c_{3}, …, c_{l−1} (which are defined in the split case) are not defined over F if D is not split. (Recall that c_{1} is always defined over F.) We will show that the product c_{1}c_{2} ⋯ c_{k} is defined over F for all k = 1, 2, …, l − 1.

Lemma 12.6. For every i = 1, 2, …, l − 1, if q < i(i − 1)/2, the group H^{p,q}(Alt^{i}N) is trivial for every p.

Proof. Induction on k. We may assume that Z = ℤ_{(l)}. The basic triangle (6.5) yields an exact sequence

H^{p−2l+1,q−l+1}(Alt^{i−1}N) → H^{p,q}(Alt^{i}N) → H^{p,q}(Alt^{i}M(S)).

The first term is trivial by induction as q − l + 1 < (i − 1)(i − 2)/2. The last group is zero by (6.6). □

Theorem 12.7. Let G = SL_{1}(D) for a central simple algebra D of prime degree l. Then the product of Chern classes c_{1}c_{2} ⋯ c_{k} is defined over F for all k = 1, 2, …, l − 1.

Proof. We may assume that Z = ℤ_{(l)}. The product c_{1}c_{2} ⋯ c_{k} belongs to H^{(k+1)^{2}−1,(k+1)(k+2)/2−1}(G). Consider the following direct summand of this group (see Theorem 11.1):

H^{(k+1)^{2}−1,(k+1)(k+2)/2−1}((Alt^{k}N)(2k)[3k]) = CH^{(k^{2}−k)/2}(Alt^{k}N).

The basic triangle (6.5) yields an exact sequence

H^{k^{2}−k−2l+1,(k^{2}−k)/2−l+1}(Alt^{k−1}N) → CH^{(k^{2}−k)/2}(Alt^{k}N) →

CH^{(k^{2}−k)/2}(Alt^{k}M(S)) → CH^{(k^{2}−k)/2−l+1}(Alt^{k−1}N).

The side terms are trivial by Lemma 12.6. The third term is isomorphic to H^{0,0}(S) = Z by (6.6). Therefore, the group CH^{(k^{2}−k)/2}(Alt^{k}N) contains an element representing c_{1}c_{2} ⋯ c_{k} over a splitting field. □

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Department of Mathematics, University of California, Los Angeles, CA, USA

  E-mail address: merkurev at math.ucla.edu