Cohomological Approaches to SK_{1}

and SK_{2} of Central Simple Algebras

Bruno Kahn

Received: November 12, 2009 Revised: June 11, 2010

Abstract. We discuss several constructions of homomorphisms from SK_{1} and SK_{2} of central simple algebras to subquotients of Galois cohomology groups.

2010 Mathematics Subject Classification: 16Kxx, 19Bxx, 20G15.

Keywords and Phrases: Whitehead group, Galois cohomology, motivic cohomology.

To A. Suslin

Contents

Introduction 318 1. Notation 324 2. Motivic cohomology of smooth geometrically cellular varieties

updated 324

3. Weight 3 and weight 4 étale motivic cohomology 328 4. Some K-cohomology groups 331 5. An approximation of cycle cohomology 335 6. A general K-theoretic construction 336 7. Twisted flag varieties 338 8. Motivic cohomology of some Severi-Brauer varieties 344 9. Étale motivic cohomology of reductive groups 346 10. The generic element 351 11. Some computations 358 Appendix A. A cancellation theorem over imperfect fields 365 References 366

Introduction

Given a simple algebra A with centre F, the group SK_{i}(A) is defined for i = 1, 2 as the kernel of the reduced norm

Nrd_{i} : K_{i}(A) → K_{i}(F).

The definition of Nrd_{1} is classical, and Nrd_{2} was defined by Suslin in [47, Cor. 5.7]. For further reference, let us recall these definitions in a uniform way: let X be the Severi-Brauer variety of A. After Quillen [42, Th. 8.4], there is an isomorphism

⨁_{r=0}^{d−1} K_{i}(A^{⊗r}) ⟶^{∼} K_{i}(X) (d = deg(A))

for any i ≥ 0. The reduced norm is then given by the composition

K_{i}(A) → K_{i}(X) → H^{0}(X, K_{i}) ⟵^{∼} K_{i}(F)

where the right isomorphism is obvious for i = 1 and is due to Suslin [47, Cor. 5.6] for i = 2. Of course, this definition also makes sense for i = 0: in this case, Nrd_{0} is simply multiplication by the index of A:

K_{0}(A) ≃ Z ⟶^{ind(A)} Z ≃ K_{0}(F)

and SK_{0}(A) = 0. [For i > 2, a reduced norm satisfying reasonable properties cannot exist (Rost, Merkurjev [33, p. 81, Prop. 4]): the right generalisation is in the framework of motivic cohomology, see [22].] The groups SK_{1}(A) and SK_{2}(A) remain mysterious and are known only in very special cases. Here are a few elementary properties they enjoy:

(1) SK_{i}(A) is Morita-invariant.

(2) ind(A)SK_{i}(A) = 0 (from Morita invariance, reduce to the case where A is division, and then use a transfer argument thanks to a maximal commutative subfield of A).

(3) The cup-product K_{1}(F) ⊗ K_{1}(A) → K_{2}(A) induces a map

K_{1}(F) ⊗ SK_{1}(A) → SK_{2}(A).

(4) Let v be a discrete valuation of rank 1 on F, with residue field k, and assume that A spreads as an Azumaya algebra A over the discrete valuation ring O_{v}. It can be shown that the map SK_{1}(A) → SK_{1}(A) is surjective and that, if K_{2}(O_{v}) → K_{2}(F) is injective, there is a short exact sequence

SK_{2}(A) → SK_{2}(A) ⟶^{∂} SK_{1}(A_{k})

with

∂({f} · x) = v(f)x̄

for f ∈ F^{∗} and x ∈ SK_{1}(A).

SK_{1} and SK_{2} of Central Simple Algebras 319

(5) Let A(t) = F (t) ⊗_{F} A, and similarly A(x) = F (x) ⊗_{F} A for any closed point x ∈ A_{F}^{1} . Then there is an isomorphism

SK_{1}(A) ⟶^{∼} SK_{1}(A(t))

due to Platonov and an exact sequence

0 → SK_{2}(A) → SK_{2}(A(t)) → ⨁_{x∈A_{F}^{1}} SK_{1}(A(x)).

From (3) and (4), one deduces that SK_{1}(A) is a direct summand of SK_{2}(A(t)) via the map x ↦ {t} · x: in particular, the latter group is nonzero as soon as the former is. More intriguing is the Calmès symbol

cal : Λ^{2} (\frac{K_{1}(A)}{ind(A)K_{1}(A)}) → SK_{2}(A)

a ∧ b ↦ Nrd(a) · b − a · Nrd(b).

The image of this symbol is not detected by residues. Let us now review known results about SK_{1} and SK_{2}. If F is a global field, then SK_{i}(A) = 0 for i = 1, 2: this is classical for i = 1 as a consequence of class field theory, while for i = 2 it is due to Bak and Rehmann using the Merkurjev-Suslin theorem [2]. In the sequel, I concentrate on more general fields F and always assume that the index of A is invertible in F.

0.A. SK_{1}. The first one to give an example where SK_{1}(A) ≠ 0 was Platonov [41]. In his example, F is provided with a discrete valuation of rank 2 and the Brauer group of the second residue field is nontrivial; in particular, cd(F) ≥ 4. Over general fields, a striking and early result for SK_{1} is Wang’s theorem:

Theorem 1 (Wang [58]). If the index of A is square-free, then SK_{1}(A) = 0.

The most successful approach to SK_{1}(A) for other A has been to relate it to Galois cohomology groups. This approach was initiated by Suslin, who (based on Platonov’s results) conjectured the existence of a canonical homomorphism

SK_{1}(A) → H^{4}(F, μ_{n}^{⊗3})/[A] · H^{2}(F, μ_{n}^{⊗2})

where n is the index of A, supposed to be prime to char F [49, Conj. 1.16]. In [49], Suslin was only able to partially carry over this project: he had to assume that μ_{n^{3}} ⊂ F and then could only construct twice the expected map, assuming the Bloch-Kato conjecture in degree 3. The next result in this direction is due to Rost in the case of a biquaternion algebra:

Theorem 2 (Rost [33, th. 4]). If A is a biquaternion algebra, there is an exact sequence

0 → SK_{1}(A) → H^{4}(F, Z/2) → H^{4}(F (Y), Z/2)

where Y is the quadric defined by an ‘Albert form’ associated to A.

The surprise here is that Rost gets in particular a finer map than the one expected by Suslin, as he does not have to mod out by multiples of [A]. Merkurjev generalised Rost’s theorem to the case of a simple algebra of degree 4 but not necessarily of exponent 2:

Theorem 3 (Merkurjev [35, th. 6.6]). If A has degree 4, there is an exact sequence

0 → SK_{1}(A) → H^{4}(F, Z/2)/2[A] · H^{2}(F, Z/2) → H^{4}(F (Y), Z/2)

where Y is the generalised Severi-Brauer variety SB(2, A), a twisted form of the Grassmannian G(2, 4).

Note that the right map makes sense because A_{F (Y)} has exponent 2. Merkurjev’s exact sequence is obtained from Rost’s by descent from F (Z) to F, where Z = SB(A^{⊗2}). The point is that neither SK_{1}(A) nor the kernel of the right map in Theorem 3 changes when one passes from F to F (Z). More recently, Suslin revisited his homomorphism of [49] in [50], where he constructs an (a priori different) homomorphism using motivic cohomology rather than Chern classes in K-theory. He compares it with the one of Rost-Merkurjev and proves the following amazing theorem:

Theorem 4 (Suslin [50, Th. 6]). For any central simple algebra A of degree 4, there exists a commutative diagram of isomorphisms

SK_{1}(A) ⟶_{∼}^{φ} \frac{Ker(H^{4}(F, μ_{4}^{⊗3}) → H^{4}(F (X), μ_{4}^{⊗3}))}{[A] · H^{2}(F, μ_{4}^{⊗2})}
||  _{τ′}↓
SK_{1}(A) ⟶_{∼}^{ψ} \frac{Ker(H^{4}(F, μ_{2}^{⊗3}) → H^{4}(F (Y), μ_{2}^{⊗3}))}{2[A] · H^{2}(F, μ_{2}^{⊗2})}

where X = SB(A), Y = SB(2, A), φ is Suslin’s homomorphism just mentioned and ψ is Merkurjev’s isomorphism from Theorem 3.

0.B. SK_{2}. Concerning SK_{2}(A), the first result (over an arbitrary base field) was the following theorem of Rost and Merkurjev:

Theorem 5 (Rost [43], Merkurjev [31]). For any quaternion algebra A, SK_{2}(A) = 0.

Rost and Merkurjev used this theorem as a step to prove the Milnor conjecture in degree 3; conversely, this conjecture and techniques of motivic cohomology were used in [21, th. 9.3] to give a very short proof of Theorem 5. We revisit this proof in Remark 7.3, in the spirit of the techniques developed here. The following theorem is more recent. In view of the still fluctuant status of the Bloch-Kato conjecture for odd primes, we assume its validity in the statement. (See §2.A for the Bloch-Kato conjecture.)

Theorem 6 (Kahn-Levine [22, Cor. 2], Merkurjev-Suslin [38, Th. 2.4]). Assume the Bloch-Kato conjecture in degree ≤ 3. For any central simple algebra A of square-free index, SK_{2}(A) = 0.

From Theorems 1 and 6, we get by a well-known dévissage argument a refinement of the elementary property (2) given above: for any A and i = 1, 2, \frac{ind(A)}{∏ l_{i}}SK_{i}(A) = 0, where the l_{i} are the distinct primes dividing ind(A). On the other hand, Baptiste Calmès gave a version of Rost’s theorem 2 for SK_{2} of biquaternion algebras:

Theorem 7 (Calmès [5]). Under the assumptions of Theorem 2, assume further that F contains a separably closed field. Then there is an exact sequence

Ker(A_{0}(Z, K_{2}) → K_{2}(F)) → SK_{2}(A) → H^{5}(F, Z/2) → H^{5}(F (Y), Z/2)

where Z is a hyperplane section of Y.

(Note that in the case of SK_{1}, the corresponding group Ker(A_{0}(Z, K_{1}) → K_{1}(F)) is 0 by a difficult theorem of Rost.) Finally, let us mention the construction of homomorphisms à la Suslin

(0.1)  SK_{1}(A) → H^{4}(F, Q/Z(3))/[A] · K_{2}(F)
(0.2)  SK_{2}(A) → H^{5}(F, Q/Z(4))/[A] · K_{3}^{M}(F)

in [22, §6.9], using an étale version of the Bloch-Lichtenbaum spectral sequence for the motive associated to A. The second map depends on the Bloch-Kato conjecture in degree 3 and assumes, as in Theorem 7, that F contains a separably closed field. This construction goes back to 1999 (correspondence with M. Levine), although the targets of (0.1) and (0.2) were only determined in [22, Prop. 6.9.1].

0.C. The results. Calmès’ proof of Theorem 7 is based in part on the methods of [18]. In this paper, I propose to generalise his construction to arbitrary central simple algebras, with the same technique. The methods will also shed some light on the difference between Suslin’s conjecture and the theorems of Rost and Merkurjev. The main new results are the following:

Theorem A. Let F be a field and A a simple algebra with centre F and index e, supposed to be a power of a prime l different from char F. Then, for any divisor r of e, there is a complex

0 → SK_{1}(A) ⟶^{σ_{r}^{1}} H^{4}(F, Q/Z(3))/r[A] · K_{2}(F) → A^{0}(Y^{[r]}, H_{ét}^{4}(Q/Z(3)))

where Y^{[r]} is the generalised Severi-Brauer variety SB(r, A) and the groups A^{0}(Y^{[r]}, −) denote unramified cohomology. If the Bloch-Kato conjecture holds in degree 3 for the prime l, these complexes refine into complexes

0 → SK_{1}(A) → H^{4}(F, μ_{e/r}^{⊗3})/r[A] · H^{2}(F, μ_{e/r}^{⊗2}) → A^{0}(Y^{[r]}, H_{ét}^{4}(μ_{e/r}^{⊗3}))).

They are exact for r = 1, 2 and e = 4.

I don’t know, and don’t conjecture, that these complexes are exact in general. The map of theorem A coincides with those of Rost and Merkurjev, which is the way we get their nontriviality for l = 2 [34].

Theorem B. Let F, A, e and Y^{[r]} be as in Theorem A; assume the Bloch-Kato conjecture in degree ≤ 3 at the prime l and that F contains a separably closed subfield. Then, for any divisor r of e, there is a complex

0 → SK_{2}(A) ⟶^{σ_{r}^{2}} H^{5}(F, Q/Z(4))/r[A] · K_{3}^{M}(F) → A^{0}(Y^{[r]}, H_{ét}^{5}(Q/Z(4))).

If, moreover, the Bloch-Kato conjecture holds in degree 4 for the prime l, these complexes refine into complexes

0 → SK_{2}(A) → H^{5}(F, μ_{e/r}^{⊗4})/r[A] · H^{3}(F, μ_{e/r}^{⊗3}) → A^{0}(Y^{[r]}, H_{ét}^{5}(μ_{e/r}^{⊗4}))).

For l = 2, the maps starting from SK_{2}(A) are nontrivial in general for r = 1, 2 (unless ind(A) ≤ 2).

Theorem C. For any smooth F -variety X, define

SK_{1}(X, A) = \underrightarrow{lim} Hom_{F} (X, SL_{n}(A))^{ab}

where SL_{n}(A) is the reductive group representing the functor R ↦ SL_{n}(A ⊗_{F} R). Then there exists a natural transformation

c_{A}(X) : SK_{1}(X, A) → H_{ét}^{5}(X, Z(3)).

Restricted to fields, c_{A} is the universal invariant with values in H_{ét}^{5}(Z(3)) ≃ H_{ét}^{4}(Q/Z(3)) in the sense of Merkurjev [35].

Loosely speaking, c_{A} is defined out of the “positive” generator of the group H_{ét}^{5}(SL_{1}(A), Z(3))/H_{ét}^{5}(F, Z(3)) which turns out to be infinite cyclic, much like the Rost invariant is defined out of the “positive” generator of the infinite cyclic group H_{ét}^{3}(SL_{1}(A), Z(2)) ≃ H_{ét}^{4}(BSL_{1}(A), Z(2)) (see [8, App. B]). This replies [35, Rk. 5.8] in the same way as what was done for the Arason invariant in [8].

Theorem D. Let K be the function field of SL_{1}(A). If ind(A) = 4, we have

SK_{1}(A_{K})/SK_{1}(A) ≃ Z/2.

In Conjecture 10.16 we conjecture that SK_{1}(A_{K})/SK_{1}(A) is cyclic for any A.

Theorem E. If exp(A) = 2 < ind(A), then

Inv^{4}(SL_{1}(A), H^{∗}(Q/Z(∗ − 1))) ≃ Z/2

where the former group is Merkurjev’s group of invariants of SL_{1}(A) with values in H^{4}(−, Q/Z(3)) [35]. In particular the invariant of Theorem C is nontrivial in this case, and equals the invariant σ_{2}^{1} of Theorem A.

Theorems A, B and C were obtained around 2001/2002, except for the exactness and nontriviality statements for r = 1, which follow from the work of Suslin [50]. They were presented at the 2002 Talca-Pucón conference on quadratic forms [20]. Theorems A and C are used by Tim Wouters in recent work [60]. Theorems D and E were obtained while revising this paper for publication. This paper is organised as follows. We set up notation in Section 1. In Section 2, we recall the slice spectral sequences in the case of geometrically cellular varieties. Sections 3 to 5 are technical. In particular, Section 3 recalls the diagrams of exact sequences from [18, §5], trying to keep track of where the Bloch-Kato conjecture is used; we deduce a simple proof of Suslin’s theorem [50, Th. 1], as indicated by himself in the introduction of [50] (see Remark 3.2). In Section 6 we get our first main result, Theorem 6.1, which constructs functorial injections sending a part of lower K-theory of some projective homogeneous varieties into a certain subquotient of the Galois cohomology of the base field. We apply this result in Section 7 to twisted flag varieties, thus getting Theorems A and B (see Corollaries 7.4 and 7.5); in Remark 7.3, we revisit the proof of Theorem 5 given in [21]. In Section 8, we push the main result of [22] one step further. In Section 9, we do some preliminary computations on the slice spectral sequences associated to a reductive group G: the main result is that, if G is simple simply connected of inner type A_{r} for r ≥ 2, then the complex α^{∗}c_{3}(G) of [14] is quasi-isomorphic to Z[−1] (see Theorem 9.5 for a more complete statement). In section 10, the approach of Merkurjev in [35] plays a central rôle: we prove Theorem C, see Theorem 10.7, Theorem D, see Corollary 10.15 and part of Theorem E, see Proposition 10.11. We conclude with some incomplete computations in Section 11 trying to evaluate the group SK_{1}(A_{K})/SK_{1}(A) in general, where K is the function field of SL_{1}(A): see Theorem 11.9 and Corollary 11.10. At the end of this section we complete the proof of Theorem E, see Corollary 11.12. This paper contains results which are mostly 8 to 9 years old. The main reason why it was delayed so much is that I tried to compare the 3 ways to construct homomorphisms à la Suslin indicated above: in (0.1)–(0.2), Theorems A and B and Theorem C, and to prove their nontriviality in some new cases. In the first version of this work, I wrote that I had been mostly unsuccessful. Since then the situation has changed a bit with Theorems D and E: they were potentially already in the first version, but Wouters’ work [60] was an eye-opener for this. The easy comparisons are, for Theorems A and B, with the Rost and Calmès homomorphisms of Theorems 2 and 7, and with the new Suslin homomorphism of Theorem 4. We can now also compare those of Theorems A and C in certain cases as in Theorem E, see also Corollary 10.10 and [60, §4]. A complete comparison of all invariants still seems challenging^{1}: I give some comments on these comparison issues in Subsection 7.F and Remark 10.12.

Acknowledgements. This project was started in 2001. I would like to thank Marc Levine, Baptiste Calmès, Annette Huber, Sasha Merkurjev, Evgueny Schinder, Philippe Gille, Tim Wouters and Nikita Karpenko for helpful discussions and exchanges. Especially, I thank Gille and Wouters for explaining me my own work. Last, I would like to express my admiration for Suslin’s work: the inspiration I have drawn from it will be obvious throughout this paper.

1Including with the first homomorphism of Suslin in [49], a comparison I had initiated in a preliminary version of this paper. (A vestige remains in §11.C.)

Documenta Mathematica · Extra Volume Suslin (2010) 317–369

1. Notation

If X is a projective homogeneous variety, we denote as in [18] by E_{i} the étale F - algebra corresponding to the canonical Z-basis of CH^{i}(X_{s}) given by Schubert cycles, where X_{s} = X ⊗_{F} F_{s} and F_{s} is a separable closure of F. The motivic cohomology groups used in this paper are (mostly) the Hom groups in Voevodsky’s category DM_{−,ét}^{eff}(F) of [54, §3.3] (étale topology). In particular, the exponential characteristic p of F is inverted in this category by [54, Prop. 3.3.3 2)], so that those groups are Z[1/p]-modules. Very occasionally we shall use Hom groups in the category DM_{−}^{eff}(F) (Nisnevich topology). Let (Q/Z)′ = ⨁_{l≠p} Q_{l}/Z_{l}. We abbreviate the étale cohomology groups H_{ét}^{i}(X, (Q/Z)′(j)) with the notation H^{i}(X, j). Unless otherwise specified, all cohomology groups appearing are étale cohomology groups, with the exception of cycle cohomology groups in the sense of Rost [44]. The latter are denoted by A^{p}(X, M_{q}), where M_{∗} is the relevant cycle module. By Gersten’s conjecture [44, Cor. 6.5], these groups are canonically isomorphic to the Zariski cohomology groups H_{Zar}^{p}(X, M_{q}), where M_{q} is the Zariski sheaf on X associated to M_{q}; we shall occasionally but rarely use this isomorphism, implicitly or explicitly.

2. Motivic cohomology of smooth geometrically cellular varieties updated

2.A. The Bloch-Kato conjecture and the Beilinson-Lichtenbaum conjecture. At the referee’s request, I recall these two conjectures and their equivalence:

2.1. Conjecture (Milnor, Bloch, Kato). Let n ≥ 0, m ≥ 1 be two integers. Then, for any field F of characteristic not dividing m, the “norm residue symbol”

K_{n}^{M}(F)/m → H^{n}(F, μ_{m}^{⊗n})

(first defined by Tate in [52]) is bijective.

2.2. Conjecture (Suslin-Voevodsky). Let n ≥ 0, m ≥ 1, i ∈ Z be three integers. Then, for any field F of characteristic not dividing m and any smooth F -scheme X, the change of topology map

H_{Nis}^{i}(X, Z/m(n)) → H_{ét}^{i}(X, Z/m(n))

is bijective for i ≤ n and injective for i = n + 1, where Z/m(n) is the mod m version of the n-th motivic complex of Suslin-Voevodsky.

Conjecture 2.2 appears in [51] where (among other places like [54]) the complexes Z(n) are introduced. It therefore cannot be literally attributed to Beilinson and Lichtenbaum, although it is indeed a common part of conjectures they made in the eighties on the properties of the still conjectural complexes Z(n). Voevodsky observed in [56] that the special case X = Spec F, i = n of Conjecture 2.2 is a reformulation of Conjecture 2.1. Conversely:

2.3. Theorem ([51, 10], see also [19]). Conjecture 2.1 (for the pair (n, m)) implies Conjecture 2.2 (for the triples (n, m, i)).

We shall actually use in this paper the following variant of Conjecture 2.2 with integral coefficients:

2.4. Proposition. Conjecture 2.2 for m a power of a prime l is equivalent to the following: let n ≥ 0, i ∈ Z be two integers. Then, for any field F of characteristic ≠ l and any smooth F -scheme X, the change of topology map

H_{Nis}^{i}(X, Z(n)) → H_{ét}^{i}(X, Z(n))

is bijective for i ≤ n + 1 and injective for i = n + 2 after localising at l.

The equivalence is an easy consequence of the fact that the map in Proposition 2.4 is an isomorphism after tensoring with Q for any i ∈ Z [53, Prop. 5.28]. The special case X = Spec F, i = n + 1 of Proposition 2.4 enunciates that H_{ét}^{n+1}(F, Z(n))⊗Z_{(l)} = 0: this is called “Hilbert’s theorem 90 in degree n” and is actually equivalent (for all F) to the above conjectures. At the time of writing, the status of Conjecture 2.1 is as follows. For n = 0 it is trivial, for n = 1 it is Kummer theory ( ⟺ Hilbert’s theorem 90), for n = 2 it is the Merkurjev-Suslin theorem [36], for m a power of 2 it is due to Voevodsky [56]. In general it seems now to be fully proven as a combination of works by several authors, merging in [57] (see [59] for an overview). In this paper, we use these conjectures for n = 2 (resp. n = 3) when dealing with SK_{1} (resp. SK_{2}) and Q/Z coefficients, and for n = 3 (resp. n = 4) when dealing with SK_{1} (resp. SK_{2}) and finite coefficients.

2.B. The slice spectral sequences. In [18], we constructed spectral sequences for the étale motivic cohomology of smooth geometrically cellular varieties. These results were limited in two respects:

(1) the ground field F was assumed to be of characteristic 0;

(2) the spectral sequences had a strange abutment, which was nevertheless

sufficient for applications. The results of [14] solved both issues. The first one was due to the fact that [18] worked with motives with compact support in Voevodsky’s triangulated category of motives [54], which are known to be geometric only in characteristic 0: indeed, it was shown that the motive with compact supports of a cellular variety X is a pure Tate motive in the sense of [14], from which it was deduced by duality that the motive of X (without supports) is also pure Tate if X is smooth. In [14, Prop. 4.11], we prove directly that, over any field, the motive of X is pure Tate if X is smooth and cellular. The second issue was more subtle and is discussed in [14, Remark 6.3]. The short answer is that by considering a different filtration than the one used in [18], one gets the “right” spectral sequence. We summarize this discussion by stating the following theorem, which follows from [14, (3.2) and Prop. 4.11] and replaces [18, Th. 4.4]:

2.5. Theorem. Let X be a smooth, equidimensional, geometrically cellular variety over a perfect field F. For all n ≥ 0, there is a spectral sequence E(X, n):

(2.1)  E_{2}^{p,q}(X, n) = H_{ét}^{p−q}(F, CH^{q}(X_{s}) ⊗ Z(n − q)) ⇒ H_{ét}^{p+q}(X, Z(n)).

Note that, by cellularity, each CH^{q}(X_{s}) is a permutation Galois module. These spectral sequences have the following properties:

(i) Naturality. (2.1) is covariant in F and contravariant in X (varying among smooth, equidimensional, geometrically cellular varieties) under any maps (even finite correspondences).

(ii) Products. There are pairings of spectral sequences

E_{r}^{p,q}(X, m) × E_{r}^{p′,q′}(X, n) → E_{r}^{p+p′,q+q′}(X, m + n)

which coincide with the usual cup-product on the E_{2}-terms and the abutments.

(iii) Transfer. For any finite extension E/F and any n ≥ 0, there is a morphism of spectral sequences

E_{r}^{p,q}(X_{E}, n) → E_{r}^{p,q}(X, n)

which coincides with the usual transfer on the E_{2}-terms and the abutment.

(iv) Covariance for closed equidimensional immersions. For any closed immersion i : Y ↪ X of pure codimension c, where X and Y are smooth, geometrically cellular, there is a morphism of spectral sequences

E_{r}^{p−c,q−c}(Y, n − c) ⟶^{i_{∗}} E_{r}^{p,q}(X, n)

“abutting” to the Gysin homomorphisms

H_{ét}^{p+q−2c}(Y, Z(n − c)) →^{i_{∗}} H_{ét}^{p+q}(X, Z(n)).

If X is split, then (2.1) degenerates at E_{2}.

The only nonobvious point in this theorem is (ii) (products). In [14, p. 915], it is claimed that there are pairings of slice spectral sequences for the tensor product of two arbitrary motives M and N. This is not true in general: I thank Evgeny Shinder for pointing out this issue. However, these pairings certainly exist if M or N is a mixed Tate motive: the argument is essentially the same as the one that proves that the Künneth maps of [14, Cor. 1.6] are isomorphisms in this case [14, Lemma 4.8]. For the reader’s convenience, we outline the construction. We take the notation of [14]: Given the way the slice spectral sequence is constructed in [14, §3] (bottom of p. 914), to get a morphism of filtrations, we need to get morphisms

ν_{≤q+q′}(M ⊗ M′) → ν_{≤q}M ⊗ ν_{≤q′}M′

for two motives M, M′ and two integers q, q′. From the canonical maps M → ν_{≤q}M and M′ → ν_{≤q′}M′, we get a morphism

M ⊗ M′ → ν_{≤q}M ⊗ ν_{≤q′}M′

and we would like to prove that its composition with ν^{>q+q′}(M⊗M′) → M⊗M′ is 0. This will be true provided

ν^{>q+q′}(ν_{≤q}M ⊗ ν_{≤q′}M′) =

\underline{Hom}(Z(q + q′ + 1), ν_{≤q}M ⊗ ν_{≤q′}M′)(q + q′ + 1) = 0.

This is false in general (for example M = M′ = h_{1}(C), q = q′ = 0, where C is a curve of genus > 0 over an algebraically closed field), but it is true if M or M′ is a mixed Tate motive. Indeed, we may reduce to M = Z(a) for some integer a. Then

ν_{≤q}M = \begin{cases}0 & if q < a \\ Z(a) & if q ≥ a \end{cases}

hence \underline{Hom}(Z(q + q′ + 1), ν_{≤q}M ⊗ ν_{≤q′}M′) = 0 if q < a, and if q ≥ a we get

\underline{Hom}(Z(q + q′ + 1), ν_{≤q}M ⊗ ν_{≤q′}M′)

= \underline{Hom}(Z(q + q′ + 1), Z(a) ⊗ ν_{≤q′}M′)
= \underline{Hom}(Z(q + q′ + 1 − a), ν_{≤q′}M′) = 0

because q + q′ + 1 − a > q′. Dealing with the spectral sequences for étale motivic cohomology, it will suffice that M or N is geometrically mixed Tate in the sense of [14, §5] to have these products.

2.6. Remark. As stressed in §1, the spectral sequences of Theorem 2.5 are spectral sequences of Z[1/p]-modules, where p is the exponential characteristic of F. Thus all results of this paper are “away from p”. It is nevertheless possible to extend the methods to p-algebras in characteristic p, at some cost: this is briefly discussed in Appendix A. I am grateful to Tim Wouters for a discussion leading to this observation.

2.C. Vanishing of E_{2}-terms. Since this issue may be confusing, we include here an estimate in the case of the spectral sequences (2.1) and of the coniveau spectral sequences, which will be used in the next section (compare [18, p. 161]). It shows that these two spectral sequences live in somewhat complementary regions of the E_{2}-plane.

2.7. Proposition. a) In the spectral sequence (2.1), we have E_{2}^{a,b}(X, n) = 0 in the following cases: (ai) a ≤ b, b ≥ n − 1, except a = b = n. (aii) a = n + 1 under the Bloch-Kato conjecture in degree n − b. Moreover, E_{2}^{a,b}(X, n) is uniquely divisible for a ≤ b and b < n − 1.

b) Let X be a smooth variety. In the coniveau spectral sequence for étale motivic cohomology

E_{1}^{a,b} = ⨁_{x∈X^{(a)}} H^{b−a}(k(x), Z(n − a)) ⇒ H^{a+b}(X, Z(n))

we have E_{1}^{a,b} = 0 in the following cases:

(bi) a ≥ b, a ≥ n − 1, except a = b = n. (bii) b = n + 1 under the Bloch-Kato conjecture in degree n − a. Moreover, E_{2}^{a,b}(X, n) is uniquely divisible for a ≥ b and a < n − 1. Finally, for b = n, the natural map

A^{a}(X, K_{n}^{M})[1/p] → E_{2}^{a,n}

is surjective under the Bloch-Kato conjecture in degrees ≤ n − a, and bijective under the Bloch-Kato conjecture in degrees ≤ n − a + 1.

Proof. For (ai), we use that E_{2}^{a,b}(X, n) = H_{ét}^{a−b}(F, CH^{b}(X_{s}) ⊗ Z(n − b)) ≃ H_{ét}^{a−b−1}(F, CH^{b}(X_{s}) ⊗ Q/Z(n − b)) for n − b < 0 (by definition of Z_{ét}(n − b) for n − b < 0, see [14, Def. 3.1]), and also that Z(0) = Z and Z(1) = 𝔾_{m}[−1]. (aii) follows from Hilbert 90 in degree n − b (see §2.A after Proposition 2.4). The proofs of (bi) and (bii) are similar. The divisibility claims reduce to the unique divisibility of H_{ét}^{i}(K, Z(r)) for i ≤ 0 (r > 0, K/F a function field): this is obvious for i < 0, while for i = 0 we may reduce to finitely generated fields as in [17, proof of Th. 3.1 a)]. Finally, the last claim follows from a diagram chase in the comparison map between the Gersten complexes for Nisnevich and étale cohomology with Z(n) coefficients. □

3. Weight 3 and weight 4 étale motivic cohomology

In this section, we examine in more detail the diagrams obtained in [18] by mixing the slice and coniveau spectral sequences, and expand the results in weight 4. In order to stress the irrelevance of Gersten’s conjecture, we replace the notation H^{p}(X, H^{q}) or H^{p}(X, K_{q}) used in [18] by the notation A^{p}(X, H^{q}) or A^{p}(X, K_{q}) (see §1).

3.A. Weight 3. Let X be a projective homogeneous F -variety. In [18, §5.4], we drew a commutative diagram with some exactness properties, by mixing the coniveau spectral sequence and the spectral sequence of [18, Th. 4.4] for étale motivic cohomology in weight 3. We can now use the spectral sequence (2.1) to get the same diagram over any perfect field. To get the diagram of [18, §5.4], we made the blanket assumption in [18] that all groups were localised at 2, because calculations relied on the Bloch-Kato conjecture in degree 3, which was only proven for l = 2. In this paper, we are also interested in making the dependence on this conjecture explicit. How much exactness remains in this diagram if we don’t wish to use it in degree 3? Using Proposition 2.7, we see that at least the following part of the diagram of [18, §5.4] remains exact by only using the Bloch-Kato conjecture in degree ≤ 2 (= the Merkurjev-Suslin theorem): the exponential characteristic p is implicitly inverted in this diagram as well as in the next one, (3.2).

(3.1) H^{4}(F, Z(3)) ; ↓ ; 0 → A^{1}(X, K_{3}^{M}) ⟶ H^{4}(X, Z(3)) ⟶ A^{0}(X, H^{4}(Z(3))) ; ↓ ; K_{2}(E_{1}) ; ↓^{d_{2}^{3,1}(3)} ; A^{0}(X, H^{4}(Z(3)))  H^{4}(F, 3) ; ↓  ↓  ↘η^{4} ; A^{2}(X, K_{3}^{M}) ⟶ H^{5}(X, Z(3)) →  A^{0}(X, H^{4}(3)) ; ↘ξ^{4}  ↓  ↓ ; E_{2}^{∗}  CH^{3}(X) ; ↓ ; H^{6}(X, Z(3)).

The group A^{0}(X, H^{4}(Z(3))), which appears twice in this diagram, is of course torsion, as well as H^{4}(F, Z(3)), and their l-primary components are 0 under the Bloch-Kato conjecture in degree 3 for the prime l.

3.B. Weight 4. In weight 4, we cannot avoid using the Bloch-Kato conjecture in degree 3. There is a commutative diagram, which was only written down in a special case in [18]:

(3.2) [figure: H^{5}(X, Z(4)) → K_{3}(E_{2})_{ind}  K_{3}^{M}(E_{1}); d_{3}^{3,2}(4); ↓^{d_{2}^{4,1}(4)}; A^{0}(X, H^{5}(Z(4)))  H^{5}(F, 4); ↓; ↓; η^{5}; A^{2}(X, K_{4}^{M})  ⟶ H^{6}(X, Z(4)) → A^{0}(X, H^{5}(4)); ξ^{5}; ↓; ↓; K_{2}(E_{2})  H^{3}(X, K_{4}^{M}); d_{3}^{4,3}(4); ↓^{d_{2}^{4,3}(4)}; ↓; H^{6}(F, 4)  H^{4}(E_{1}, 3)  H^{7}(X, Z(4)).]

In this diagram, the differentials appearing correspond to the spectral sequence (2.1) in weight 4. The path snaking from A^{0}(X, H^{5}(Z(4))) to H^{7}(X, Z(4)) is exact (it comes from the coniveau spectral sequence for weight 4 étale motivic cohomology: see Proposition 2.7). The differential d_{3}^{4,3}(4) is only defined on the kernel of d_{2}^{4,3}(4) and the differential d_{3}^{3,2}(4) takes values in the cokernel of d_{2}^{3,2}(4). The column is a complex, exact at H^{6}(X, Z(4)); its exactness properties at H^{5}(F, 4) and K_{2}(E_{2}) involve the differentials d_{3} in an obvious sense. All these exactness properties depend on the Bloch-Kato conjecture in degree i for any field E and any i ≤ 3, and also on Hilbert’s theorem 90 in degree i under the same conditions (which follows from the Bloch-Kato conjecture, see §2.A). The map η^{5} is the natural map from the Galois cohomology of the ground field to the unramified cohomology of X.

3.C. The groups \overline{Ker} η^{4} and \overline{Ker} η^{5}.

3.1. Definition. For i = 1, 2, we denote by \overline{Ker} η^{i+3} the homology of the complex

K_{i+1}^{M}(E_{1}) ⟶^{d_{2}^{i+2,1}(X,i+2)} H^{i+3}(F, i + 2) ⟶^{η^{i+3}} A^{0}(X, H^{i+3}(i + 2)).

Diagram (3.1) yields an exact sequence

A^{0}(X, H^{4}(Z(3))) → Ker ξ^{4} → \overline{Ker} η^{4} → 0

hence an isomorphism

(3.3)  Ker ξ^{4} ⟶^{∼} \overline{Ker} η^{4}

under the Bloch-Kato conjecture in degree ≤ 3. If F contains an algebraically closed subfield, then K_{3}(E_{2})_{ind} is divisible and the differential d_{3}^{3,2}(4) is 0 since it is a priori torsion [18, Prop. 4.6]. Then diagram (3.2) yields an exact sequence

A^{0}(X, H^{5}(Z(4))) → Ker ξ^{5} → \overline{Ker} η^{5} → 0

under the Bloch-Kato conjecture in degree ≤ 3 and an isomorphism

(3.4)  Ker ξ^{5} ⟶^{∼} \overline{Ker} η^{5}

under the Bloch-Kato conjecture in degree ≤ 4.

3.2. Remark. Let us recover Suslin’s theorem [50, Th. 1] from (3.3). The point is simply that the coniveau spectral sequence for Nisnevich motivic cohomology yields an isomorphism

A^{2}(X, K_{3}^{M}) ⟶^{∼} H_{Nis}^{5}(X, Z(3))

(cf. [50, Lemma 9]). The differential d_{2}^{3,1}(3) was computed in [18, Th. 7.1] for Severi-Brauer varieties.

4. Some K-cohomology groups

4.A. A^{1}(X, K_{3}) and A^{0}(X, K_{3}). Recall from [18, Prop. 4.5] that

(4.1)  A^{i}(X, K_{3}^{M}) ⟶^{∼} A^{i}(X, K_{3}) for i > 0.

For A^{1}(X, K_{3}), we have:

4.1. Proposition. Let X be a projective homogeneous variety over F, and K/F a regular extension. Under the Bloch-Kato conjecture in degree 3, the map

A^{1}(X, K_{3}) → A^{1}(X_{K}, K_{3})

has p-primary torsion kernel, where p is the exponential characteristic of F. More precisely, the kernel of this map is torsion and its l-primary part vanishes for l ≠ p if the Bloch-Kato conjecture holds at the prime l in degree 3.

Proof. Up to passing to its perfect closure, we may assume F perfect. By Diagram (3.1) and (4.1), there is a canonical map

A^{1}(X, K_{3}) → K_{2}(E_{1})

where E_{1} is a certain étale F -algebra associated to X, whose kernel is contained in H_{ét}^{4}(F, Z(3)): hence the l-primary part of this kernel vanishes under the condition in Proposition 4.1. The result now follows from [47, th. 3.6]. □

Let still X be a projective homogeneous F -variety. As in [18, §5.1], for all i ≥ 0 we write E_{i} for the étale F -algebra determined by the Galois-permutation basis of CH^{i}(X_{s}) given by Schubert cycles (see §1).

4.2. Theorem. a) For i ≤ 2, the map K_{i}(F) → A^{0}(X, K_{i}) is bijective.

b) Under the Bloch-Kato conjecture in degree 3, the cokernel of the homomorphism

K_{3}(F) → A^{0}(X, K_{3})

is torsion, and its prime-to-the-characteristic part is

(1) finite if F is finitely generated over its prime subfield;

(2) 0 in the following cases:

(i) F contains a separably closed subfield;

(ii) the map CH^{1}(X_{E_{1}}) → CH^{1}(X_{s}) is surjective.

More precisely, under the Bloch-Kato conjecture in degree 3 for the prime l, the above is true after localisation at l.

Proof. a) is well-known and is quoted for reference purposes: it is obvious for i = 0, 1 (since X is proper geometrically connected), and for i = 2 it is a theorem of Suslin [47, Cor. 5.6].

b) After [17, Th. 3 a)] (see also [27, Th. 16.4]), the homomorphism K_{3}^{M}(K) → K_{3}(K) is injective for any field K. Consider the commutative diagram with exact rows

0 ⟶ K_{3}^{M}(F) ⟶ K_{3}(F) ⟶ K_{3}(F)_{ind} ⟶ 0
↓  ↓  ↓
0 ⟶ A^{0}(X, K_{3}^{M}) ⟶ A^{0}(X, K_{3}) ⟶ A^{0}(X, K_{3}^{ind}).

As X is a rational variety, the right vertical map is bijective [8, lemma 6.2]. It therefore suffices to prove the claims of theorem 4.2 for the left vertical map. Let us first assume F perfect: then we can use Theorem 2.5. Mixing the weight 3 coniveau spectral sequence for étale motivic cohomology with the spectral sequence (2.1) in weight 3, we get modulo the Bloch-Kato conjecture in degree 3 the following commutative diagram with exact rows:

0

↓

K_{3}^{M}(F)

↓  [figure: α]
0→A^{1}(X, H^{2}(Z(3))) ⟶ H^{3}(X, Z(3)) → A^{0}(X, K_{3}^{M})→0
[figure: β]  ↓
K_{3}(E_{1})_{ind}

↓

0.

For the reader’s convenience, let us explain where the Bloch-Kato conjecture in degree 3 is necessary. The weight 3 spectral sequence (2.1) gives a priori an exact sequence

H^{0}(E_{1}, Z(2)) ⟶^{d_{2}^{1,1}(X,3)} H^{3}(F, Z(3)) → H^{3}(X, Z(3))

→ H^{1}(E_{1}, Z(2)) → H^{4}(F, Z(3)).

Recall that all groups are étale cohomology groups here. The group H^{0}(E_{1}, Z(2)) is conjecturally 0; it is uniquely divisible in any case, see proof of Proposition 2.7. Since the differential d_{2}^{1,1}(X, 3) is torsion (proof as in [18, Prop. 4.6]), it must be 0. The identification of H^{1}(E_{1}, Z(2)) with K_{3}(E_{1})_{ind} only depends on the Merkurjev-Suslin theorem. On the other hand, the bijectivity of K_{3}^{M}(F) → H^{3}(F, Z(3)) and the vanishing of H^{4}(F, Z(3)) depend on the Bloch-Kato conjecture in degree 3. This takes care of the vertical exact sequence. Similarly, the Bloch-Kato conjecture in degree 3 is necessary to identify the last term of the horizontal exact sequence (stemming from the coniveau spectral sequence) with A^{0}(X, K_{3}^{M}).

The diagram above gives an isomorphism

Coker α ≃ Coker β.

Let us show that Coker β is m-torsion for some m > 0. The group K_{3}(E_{1})_{ind} appearing in the diagram is really

H^{0}(F, CH^{1}(X_{s}) ⊗ H^{1}(F_{s}, Z(2)))

via Shapiro’s lemma, the isomorphism H^{1}(K, Z(2)) ≃ K_{3}(K)_{ind} for any field and Galois descent for K_{3}(K)_{ind} [37, 23]. A standard computation shows that the corestriction map

H^{0}(E_{1}, CH^{1}(X_{s}) ⊗ H^{1}(F_{s}, Z(2))) ⟶^{Cor} H^{0}(F, CH^{1}(X_{s}) ⊗ H^{1}(F_{s}, Z(2)))

is split surjective. On the other hand, since CH^{1}(X_{s}) is finitely generated, there exists a finite extension E/F such that CH^{1}(X_{E}) → CH^{1}(X_{s}) is surjective. Without loss of generality, we may assume that E contains all the residue fields of the étale algebra E_{1}. A transfer argument then shows that the map CH^{1}(X_{E_{1}}) → CH^{1}(X_{s}) has cokernel killed by some integer m > 0. Hence the composition

CH^{1}(X_{E_{1}}) ⊗ H^{1}(E_{1}, Z(2)) → CH^{1}(X_{s}) ⊗ H^{1}(E_{1}, Z(2))

⟶^{∼} CH^{1}(X_{s}) ⊗ H^{0}(E_{1}, H^{1}(F_{s}, Z(2)))

≃ H^{0}(E_{1}, CH^{1}(X_{s}) ⊗ H^{1}(F_{s}, Z(2)))

has cokernel killed by m, and the same holds for the composition

CH^{1}(X_{E_{1}}) ⊗ H^{1}(E_{1}, Z(2)) → H^{0}(E_{1}, CH^{1}(X_{s}) ⊗ H^{1}(F_{s}, Z(2)))

⟶^{Cor} H^{0}(F, CH^{1}(X_{s}) ⊗ H^{1}(F_{s}, Z(2))).

But this composition factors via cup-product as

CH^{1}(X_{E_{1}}) ⊗ H^{1}(E_{1}, Z(2)) = A^{1}(X_{E_{1}}, H^{2}(Z(1))) ⊗ H^{1}(E_{1}, Z(2))

→ A^{1}(X_{E_{1}}, H^{2}(Z(3))) ⟶^{Cor} A^{1}(X, H^{2}(Z(3)))

⟶^{β} H^{0}(F, CH^{1}(X_{s}) ⊗ H^{1}(F_{s}, Z(2)))

which proves the claim. Coming back the the case where F is not necessarily perfect, let F′ be its perfect (radicial?) closure and α′ the map α “viewed over F′”. Then a transfer argument shows that the natural map Coker α → Coker α′ has p-primary torsion kernel and cokernel, where p is the exponential characteristic of F. In particular, Coker α is torsion, and its prime-to-p part is killed by some m. The integer m equals 1 provided CH^{1}(X_{E_{1}}) → CH^{1}(X_{s}) is surjective, which proves 2) (ii) in Theorem 4.2. In general, the map

K_{3}(F_{0})_{ind}/m → K_{3}(F)_{ind}/m

is bijective, where F_{0} is the field of constants of F [37, 23]. If F_{0} is separably closed, then K_{3}(F_{0})_{ind}/m = 0 (ibid.), which proves 2) (i); if F is finitely generated, then F_{0} is a finite field or a number field with ring of integers A and K_{3}(F_{0})_{ind} is a quotient of K_{3}(A); in both cases it is finitely generated, which proves 1). □

4.3. Example. X is a conic curve. Then Coker β is isomorphic to the cokernel of the map

⨁_{x∈X^{(1)}} K_{3}(F (x))_{ind} ⟶^{(N_{F (x)/F})} K_{3}(F)_{ind}.

Even in the case F = Q, K_{3}(Q)_{ind} ≃ Z/24, I am not able either to produce an example where this map is not onto, or to prove that it is always onto. As a first try, one might restrict to points of degree 2 on X. To have an idea of how complex the situation is, the reader may refer to [15, §8]. In particular, Theorem 8.1 (iv) of loc. cit. shows that the map is onto provided X has a quadratic splitting field of the form Q(\sqrt{−p}), where p is prime and ≡ −1 (mod 8). If X corresponds to the Hilbert symbol (a, b), with a, b two coprime integers, the theorem of the arithmetic progression shows that there

are infinitely many p ≡ −1 (mod 8) such that p ∤ ab and (\frac{−p}{l}) = −1 for all

primes l | ab. Since −p is a square in Q_{2}, this implies that (a, b)_{Q(\sqrt{−p})} = 0 if and only if (a, b)_{Q_{2}} = 0. Thus the above map is surjective if X(Q_{2}) ≠ ∅, but I don’t know the answer in the other case.

4.B. A^{i}(X, K_{4}^{M}) and A^{i}(X, K_{4}).

4.4. Theorem. a) For any smooth variety X, the natural map

φ_{i} : A^{i}(X, K_{4}^{M}) → A^{i}(X, K_{4})

is bijective for i ≥ 3 and surjective for i = 2 with kernel killed by 2.

b) Suppose that F contains a separably closed subfield. Then φ_{2} is bijective.

Proof. a) By definition, both groups are cohomology groups of the respective Gersten complexes

⋯ → ⨁_{x∈X^{(i)}} K_{4−i}^{M}(F (x)) → …

⋯ → ⨁_{x∈X^{(i)}} K_{4−i}(F (x)) → …

Therefore, Theorem 4.4 is obvious for i ≥ 3, and φ_{2} is surjective. Using the Adams operations on algebraic K-theory, we see that, for any field K, the exact sequence

0 → K_{3}^{M}(K) → K_{3}(K) → K_{3}(K)_{ind} → 0

is split up to 2-torsion. It follows that 2 Ker φ_{2} = 0.

b) We have an exact sequence

⨁_{x∈X^{(1)}} K_{3}(F (x))_{ind} ⟶^{ψ} A^{2}(X, K_{4}^{M}) ⟶^{φ_{2}} A^{2}(X, K_{4}) → 0.

By assumption, each group K_{3}(F (x))_{ind} is divisible (compare the proof of Theorem 4.2). Since their images in A^{2}(X, K_{4}^{M}) are killed by 2, they are 0. □

4.5. Remark. I don’t know if the condition on F is necessary for the bijectivity of φ_{2}. Note that ψ factors through the group A^{1}(X, H^{2}(Z(3))) appearing in the proof of Theorem 4.2.

5. An approximation of cycle cohomology

Let M_{∗} be a cycle module in the sense of Rost [44] and let X be projective homogeneous. There are cup-products

(5.1)  CH^{p}(X) ⊗ M_{q−p}(F) → A^{p}(X, M_{q}).

which are isomorphisms when X is split, by [8, Prop. 3.7]. Assume now that X is not necessarily split. Let Y be a splitting variety for X: if X_{s} = G_{s}/P where G is a semi-simple F -algebraic group and P is a parabolic subgroup of G_{s}, we may take Y such that Y_{s} = G_{s}/B for B a Borel subgroup contained in P. Then X_{F (y)} is cellular for any point y ∈ Y. It is possible to define a map

(5.2)  A^{p}(X, M_{q}) \underrightarrow{_{ξ̃}^{p,q}} A^{0}(Y_{E_{p}}, M_{q−p})

which is an isomorphism after tensoring with Q and corresponds to the inverse of (5.1) when X is split. When q − p ≤ 2 and M_{∗} = K_{∗}^{M}, this map refines into a map

(5.3)  A^{p}(X, K_{q}^{M}) ⟶^{ξ^{p,q}} K_{q−p}^{M}(E_{p})

thanks to Suslin’s theorem [47, Cor. 5.6] for q − p = 2 and trivially for q − p = 0, 1. In this paper, we shall only construct such a map in the substantially simpler inner case where all algebras E_{p} are split, which is sufficient for our needs. We note that, if X is split, the functor K ↦ CH^{p}(X_{K}) from field extensions of F to abelian groups is constant, with finitely generated free value. When X is arbitrary, we shall authorise ourselves of this to denote by CH^{p}(X_{s}) the common value of CH^{p}(X_{K}) for all splitting fields K of X. For Y a splitting variety of X as above, consider the Rost spectral sequence [44, §8]

E_{2}^{p,q} = A^{p}(Y, R^{q}π_{∗}M_{∗}) ⇒ A^{p+q}(X × Y, M_{∗})

where π is the projection X × Y → Y and the R^{q}π_{∗}M_{∗} are the higher direct images of M_{∗} in the sense of Rost [44, §7]. Using the fact that (5.1) is an isomorphism in the split case, we get canonical isomorphisms

R^{q}π_{∗}M_{∗} = CH^{q}(X_{s}) ⊗ M_{∗−q}

hence an edge homomorphism

A^{p}(X × Y, M_{q}) → E_{2}^{0,p} = CH^{p}(X_{s}) ⊗ A^{0}(Y, M_{q−p}).

In the inner case, the composition of this map with the obvious map A^{p}(X, M_{q}) → A^{p}(X × Y, M_{q}) is the desired map ξ̃^{p,q} of (5.2). In the special case M_{∗} = K_{∗}^{M}, a functoriality argument shows that the map ξ^{2,3} (resp. ξ^{2,4}) of (5.3) coincides with the map ξ^{4} of Diagram (3.1) (resp. with the map ξ^{5} of Diagram (3.2)).

6. A general K-theoretic construction

Let X be projective homogeneous, and let K be a splitting field for X such that K/F is geometrically rational (for example, take for K the function field of the corresponding full flag variety, see beginning of §5). We assume as in the previous section that the associated algebras E_{p} are split: this is probably not essential. We write K_{∗}(X)^{(i)} for the coniveau filtration on K_{∗}(X), and K_{∗}(X)^{(i/i+1)} for its successive quotients.

6.A. The first steps of the coniveau filtration.

6.1. Theorem. For i ≤ 2,

a) The map

K_{i}(F) ⊕ K_{i}(X)^{(1)} → K_{i}(X)

is an isomorphism.

b) The maps

Ker(K_{i}(X)^{(2)} → K_{i}(X_{K})^{(2)}) → Ker(K_{i}(X)^{(1)} → K_{i}(X_{K})^{(1)})

→ Ker(K_{i}(X) → K_{i}(X_{K}))

are isomorphisms. (For i = 2, we assume the Bloch-Kato conjecture in degree 3 for the torsion primes of X.)

c) There are canonical monomorphisms

Ker(K_{i}(X)^{(2/3)} → K_{i}(X_{K})^{(2/3)}) ⟶ \overline{Ker} η^{i+3}

where \overline{Ker} η^{i+3} was introduced in Definition 3.1. (If i = 2, we assume the Bloch-Kato conjecture in degree 3 for the torsion primes of X, and also that F contains a separably closed field.) These homomorphisms are contravariant in X.

Proof. a) By Theorem 4.2 a), the composition

K_{i}(F) → K_{i}(X) → A^{0}(X, K_{i})

is bijective; hence this composition yields a splitting to the exact sequence

0 → K_{i}(X)^{(1)} → K_{i}(X) → A^{0}(X, K_{i}).

b) It suffices to show that the maps K_{i}(X)^{(j/j+1)} → K_{i}(X_{K})^{(j/j+1)} are injective for j = 0, 1. For j = 0, this is clear from a) (reapplying Theorem 4.2 a)). For j = 1, by the (Brown-Gersten-)Quillen spectral sequence it suffices to show that the map

A^{1}(X, K_{i+1}) → A^{1}(X_{K}, K_{i+1})

is injective. For i = 0, the statement (concerning Pic) is classical; for i = 1, it follows from [32, Theorem] and for i = 2 it follows from Proposition 4.1.

c) The BGQ spectral sequence gives a map

K_{i}(X)^{(2/3)} ⟶^{∼} E_{∞}^{2,−i−2} ↪ Coker(A^{0}(X, K_{i+1}) ⟶^{d_{2}^{0,−i−1}} A^{2}(X, K_{i+2})).

The differential d_{2}^{0,−i−1} is 0 by Theorem 4.2. Therefore, we get an injection

Ker(K_{i}(X)^{(2/3)} → K_{i}(X_{K})^{(2/3)}) ↪ Ker(A^{2}(X, K_{i+2}) → A^{2}(X_{K}, K_{i+2})).

Clearly, the right-hand-side kernel is equal to Ker ξ^{2,i+2}, where ξ^{2,i+2} is the map defined in the previous section. As observed at the end of this section, this map coincides with the map ξ^{i+3} of diagrams (3.1) and (3.2) (for i = 1, 2; similarly for i = 0). The result then follows from (3.3) and (3.4) (and their analogue for i = 0). □

6.B. The reduced norm and projective homogeneous varieties.

6.2. Proposition. Let B be a central simple F -algebra, and let F be a locally free sheaf on X, provided with an action of B. For i ≤ 2, consider the map

u_{F} : K_{i}(B) → K_{i}(X)

induced by the exact functor

(6.1)  P (B) → P (X)
M ↦ F ⊗_{B} M

where P (B) (resp. P (X)) denotes the category of finitely generated [projective] B-modules (resp. of locally free O_{X}-sheaves of finite rank).

a) The composition

K_{i}(B) ⟶^{u_{F}} K_{i}(X) → A^{0}(X, K_{i}) ⟵^{∼} K_{i}(F)

equals rk_{B}(F) Nrd_{B}, where rk_{B}(F) := \frac{rk(F)}{deg(B)}.

b) The map

ũ_{F} : K_{i}(B) → K_{i}(X)

defined by x ↦ u_{F}(X) − rk_{B}(F) Nrd_{B}(x) has image contained in K_{i}(X)^{(1)}. The composition

K_{i}(B) \underrightarrow{_{ũ}_{F}} K_{i}(X)^{(1)} → A^{1}(X, K_{i+1}) ⟶^{ξ^{1,i+1}} K_{i}(E_{1}) = CH^{1}(X) ⊗ K_{i}(F)

where ξ^{1,i+1} is as in Section 5, equals c_{1}(F) ⊗ Nrd_{B}.

Proof. Observe that Nrd_{B} is characterised by the commutation of the diagram

K_{i}(B_{L}) ⟶^{∼} K_{i}(L)
↑  ↑
K_{i}(B) ⟶^{Nrd_{B}} K_{i}(F)

for any extension L/F that splits B and such that L = F (Y), where Y is a smooth projective geometrically rational F -variety and the upper isomorphism is given by Morita theory. Indeed, this diagram then refines to a diagram of the form

A^{0}(Y, K_{i}(B ⊗_{F} O_{Y})) ⟶^{∼} A^{0}(Y, K_{i})
↑  ^{≀}↑
K_{i}(B)  ⟶^{Nrd_{B}} K_{i}(F)

see [47, Cor. 5.6] for the right vertical isomorphism. It is therefore sufficient to check Proposition 6.2 after extending scalars to L = K(Y), where Y is the Severi-Brauer variety of B. Thus, we may assume X and B split. By Morita, u_{F} then corresponds to the map K_{i}(F) → K_{i}(X) given by cupproduct with [F ⊗_{B} S] ∈ K_{0}(X), where S is a simple B-module. a) is now obvious, the first statement of b) follows, and the second one is also obvious since ξ^{i,1} commutes with products in the split case. □

From Proposition 6.2 and Theorem 6.1 a), it follows that the restriction of u_{F} and ũ_{F} to SK_{i}(B) induce the same map: SK_{i}(B) → K_{i}(X)^{(2)}, that we shall still denote by u_{F}. If L/F is chosen as in the proof of Proposition 6.2, then clearly the composition SK_{i}(B) → K_{i}(X)^{(2)} → K_{i}(X_{L})^{(2)} is 0. This yields:

6.3. Definition. Let L/F be a geometrically rational extension splitting both X and B. We denote by σ_{F}^{i} : SK_{i}(B) → \overline{Ker}η^{i+3} the composition

SK_{i}(B) ⟶^{u_{F}} Ker(K_{i}(X)^{(2/3)} → K_{i}(X_{L})^{(2/3)}) ⟶ \overline{Ker} η^{i+3}

where the second map is that of Theorem 6.1 c).

7. Twisted flag varieties

In this section, we define maps from SK_{i}(A) to Galois cohomology as promised in Theorems A and B. We use the results of the previous section. In order to get these maps, it is enough to deal with generalised Severi-Brauer varieties (twisted Grassmannians); however, we start with the apparently greater generality of twisted flag varieties. The reason for doing this is the hope to be able to compare the various maps with each other in the future, see Subsection 7.F.

7.A. K-theory of twisted flag varieties. Let A be a simple algebra of degree d, with centre F. For \underline{r} = (r_{1}, …, r_{k}) with d ≥ r_{1} > ⋯ > r_{k} ≥ 0, let Y^{[\underline{r}]} = SB(\underline{r}; A) be the twist of the flag variety G(r_{1}, …, r_{k}; d) by a 1cocycle defining A: its function field is generic among extensions K/F such that A_{K} acquires a chain I_{1} ⊃ ⋯ ⊃ I_{k} of left ideals of respective K-dimensions dr_{1}, …, dr_{k}. If \underline{s} is a subset of \underline{r}, there is an obvious projection

Y^{[\underline{r}]} → Y^{[\underline{s}]}.

The variety Y^{[\underline{r}]} carries a chain of locally free sheaves

(7.1)  A_{Y^{[\underline{r}]}} ⟶ J_{r_{1}} ⟶ … ⟶ J_{r_{k}}

where A_{Y^{[\underline{r}]}} is the constant sheaf with value A: if A is split, (7.1) corresponds by Morita theory to the tautological flag A_{Y^{[\underline{r}]}}^{d} → → V_{r_{1}} … → → V_{r_{k}} on G(r_{1}, …, r_{k}; d) (J_{r_{j}} is the quotient of End(A^{d})_{Y^{[\underline{r}]}} by the sheaf of ideals consisting of endomorphisms vanishing on Ker(A_{Y^{[\underline{r}]}}^{d} → V_{r_{j}})). There is an action of A on this chain. More generally, for any partition α = (α_{1}, …, α_{m}) of |α| = ∑ α_{i} with α_{1} ≥ ⋯ ≥ α_{m} ≥ 0, with associated Schur functor S^{α}, the sheaf S^{α}(V_{r_{j}}) on G(r_{1}, …, r_{k}; d) defines by faithfully flat descent a sheaf S^{α}(J_{r_{j}}) of A^{⊗|α|}-algebras on Y^{[\underline{r}]} [26, §4]. By Levine-Srinivas-Weyman [26, Th. 4.6], we have an isomorphism

(7.2)  ⨁_{α} K_{∗}(A^{⊗|α|}) ⟶^{(u_{α})} K_{∗}(Y^{[\underline{r}]})

where α = (α^{1}, …, α^{k}) is a family of partitions, with 0 ≤ α_{i}^{j} ≤ r_{i} − r_{i+1}, |α| = ∑ |α^{j}| and u_{α} is induced by the exact functor

P (A^{|α]}) → P (Y^{[\underline{r}]})
M ↦ S^{α}(J) ⊗_{A^{|α|}} M

with S^{α}(J) = S^{α^{1}}(J_{1}) ⊗ ⋯ ⊗ S^{α_{k}}(J_{k}). Actually our choice of generators is not the one of [26], but rather the same as in Panin [40, Th. 7.1], who proves the same results by a different method.

7.B. Maps from SK_{i} to Galois cohomology. We now apply Definition 6.3 with F = J_{r_{j}} for each j: in the above notation, this corresponds to the case α^{j′} = 0 for j′ ≠ j and α^{j} = (1, 0, …). We find maps

(7.3)  σ_{r_{j}}^{i} : SK_{i}(A) → \overline{Ker}η_{Y^{[\underline{r}]}}^{i+1}.

We now proceed to compute the differential d_{2}^{i+2,1}(Y^{[\underline{r}]}, i + 2) involved in Definition 3.1. Using the multiplicativity of (2.1) (Th. 2.5 (ii)), we reduce to computing the differential d_{2}^{1,1}(Y^{[\underline{r}]}, 1) (cf. [18, lemma 6.1]). We have an exact sequence [18, 5.2]

CH^{1}(Y_{s}^{[\underline{r}]})^{G_{F}} ⟶^{d_{2}^{1,1}(Y^{[\underline{r}]},1)} Br(F) → Br(Y^{[\underline{r}]}).

The group CH^{1}(Y_{s}^{[\underline{r}]}) has a basis consisting of the first Chern classes of the bundles V_{r_{j}}: in particular, G_{F} acts trivially on it. For j ∈ [1, k], write Y^{[r_{j}]} for the twisted Grassmannian (generalised Severi-Brauer variety) corresponding to r_{j}. Then we have a commutative diagram

(7.4)  \begin{aligned}& & CH^{1}(Y_{s}^{[\underline{r}]}) & ⟶^{d_{2}^{1,1}(Y^{[\underline{r}]},1)} & Br(F) & ⟶ & Br(Y^{[\underline{r}]}) \\ & & ↑ & & || & & ↑ \\ Z & = & CH^{1}(Y_{s}^{[r_{j}]}) & ⟶^{d_{2}^{1,1}(Y^{[r_{j}]},1)} & Br(F) & ⟶ & Br(Y^{[r_{j}]}). \end{aligned}

This shows that CH^{1}(Y_{s}^{[\underline{r}]}) is generated by the images of the maps CH^{1}(Y_{s}^{[r_{j}]}) → CH^{1}(Y_{s}^{[\underline{r}]}) for j = 1, …, k, and thus there is no loss of generality in assuming k = 1 for the computation of the differential, which we do now. Let us simplify the notation by writing r for r_{j}. We have the following

7.1. Lemma ([39, Cor. 2.7]). Ker(Br(F) → Br(Y^{[r]})) = ⟨r[A]⟩. □

Hence we get d_{2}^{1,1}(Y^{[r]}, 1)(1) = r[A] (up to a unit), and therefore from Diagram (7.4):

d_{2}^{1,1}(Y^{[\underline{r}]}, 1)(V_{r_{j}}) = r_{j}[A] (up to a unit).

We conclude:

7.2. Corollary. a) The maps (7.3) give rise to commutative diagrams of complexes (i = 1, 2):

0→SK_{i}(A)⟶^{σ_{r_{j}}^{i}} \frac{H^{i+4}(F, Z(i + 2))}{gcd(r_{j})[A] · H^{i+1}(F, Z(i + 1))}→ A^{0}(Y^{[\underline{r}]}), H^{i+4}(Z(i + 2)))
||  ↑  ^{p^{∗}}↑
0→SK_{i}(A)⟶^{σ_{r_{j}}^{i}}  \frac{H^{i+4}(F, Z(i + 2))}{r_{j}[A] · H^{i+1}(F, Z(i + 1))} →A^{0}(Y^{[r_{j}]}), H^{i+4}(Z(i + 2)))

where Y^{[r_{j}]} = SB(r_{j}, A) is the generalised Severi-Brauer variety of ideals of rank r_{j}, and the middle vertical map is the natural surjection.

b) If j = k and r_{k} divides the other r_{j}, then both vertical maps are isomorphisms.

Proof. The only thing to remain proven is b). The generic fibre of p : Y^{[\underline{r}]} → Y^{[r_{k}]} is then easily seen to be the split flag variety G(r_{1} − r_{k}, …, r_{k−1} − r_{k}; d); in particular it is rational and the claim follows. □

7.3. Remark. By construction, this homomorphism for i = 2 factors through an injection

SK_{2}(A) ⟶ K_{2}(Y^{[\underline{r}]})^{(2)}.

If A is a quaternion algebra, the only choice for Y^{[\underline{r}]} is the conic corresponding to A and K_{2}(Y^{[\underline{r}]})^{(2)} = 0. This is a variant of the proof of Theorem 5 given in [21].

As seen above, for i = 1, the definition of σ_{r_{j}}^{i} only involves the Merkurjev-Suslin theorem, while for i = 2 it involves the Bloch-Kato conjecture in degree 3 (for the primes dividing d). If we are ready to grant the Bloch-Kato conjecture one degree further, we get a refinement of these maps:

7.4. Corollary. Assume the Bloch-Kato conjecture in degree i + 2 (i = 1, 2). Assume also for simplicity that r_{j} divides d. The the complexes on the bottom row of Corollary 7.2 refine into complexes

(7.5) SK_{1}(A) → H^{4}(F, μ_{d/r_{j}}^{⊗3})/r_{j}[A] · H^{2}(F, μ_{d/r_{j}}^{⊗2}) → A^{0}(Y^{[r_{j}]}, H^{4}(μ_{d/r_{j}}^{⊗3}))

(7.6) SK_{2}(A) → H^{5}(F, μ_{d/r_{j}}^{⊗4})/r_{j}[A] · H^{3}(F, μ_{d/r_{j}}^{⊗3}) → A^{0}(Y^{[r_{j}]}, H^{5}(μ_{d/r_{j}}^{⊗4})).

Proof. Use the fact that d/r Ker η^{i} = 0 (transfer argument), and that the map H^{4}(F, μ_{d/r_{j}}^{⊗3}) → H^{4}(F, Q/Z(3)) = H^{5}(F, Z(3)) (resp. the map H^{5}(F, μ_{d/r_{j}}^{⊗4}) → H^{5}(F, Q/Z(4)) = H^{6}(F, Z(4))) is injective under the Bloch-Kato conjecture in degree 3 (resp. 4). □

7.C. Examples: maps à la Suslin and à la Rost-Merkurjev. The case of Suslin corresponds to r_{j} = 1 for any A. More precisely, the way Suslin constructs his map in [50, §3] shows that it coincides with the one here for r_{j} = 1, compare Remark 3.2. Similarly, the cases of Rost-Merkurjev correspond to d = 4, r_{j} = 2. Using the work of Calmès [5, §2.5], one can check that in the case of a biquaternion algebra we get back Rost’s map for SK_{1} (resp. Calmès’ map for SK_{2}). This implies:

7.5. Corollary. a) For i = 1, the bottom sequence in Corollary 7.2 is exact for r_{j} = 1, 2 and deg(A) = 4.

b) The maps σ_{1}^{1} and σ_{2}^{1} are nonzero in general if 4 | ind(A).

Proof. a) Let us first assume r_{j} = 1. Then, as explained above, the map σ_{1}^{1} coincides with Suslin’s map in [50, §3], and the exactness is loc. cit., Th. 3. Suppose now that r_{j} = 2. If A is a biquaternion algebra, the exactness is Rost’s theorem [33, Th. 4]. If exp(A) = 4, we reduce to the biquaternion case by the same argument as in [35, proof of Th. 6.6].

b) This follows from a) by a standard argument, cf. [34]. □

7.D. Some properties of the maps σ_{r}^{i}. For simplicity, we replace r_{j} by r; we still assume that r divides d.

7.6. Lemma. If r = d, the maps (7.5) and (7.6) are 0.

Proof. In this case the variety Y^{[r]} has a rational point, hence the two kernels are 0. (Alternately, the coefficients of the cohomology groups involved in Corollary 7.4 are 0!) □

7.7. Proposition. Let a ∈ F^{∗}. Then, for all r | d, the diagram

SK_{1}(A) ⟶^{σ_{r}^{1}} H^{4}(F, μ_{d/r}^{⊗3})/r[A] · H^{2}(F, μ_{d/r}^{⊗2})
·{a}↓  ^{·{a}}↓
SK_{2}(A) ⟶^{σ_{r}^{2}} H^{5}(F, μ_{d/r}^{⊗4})/r[A] · H^{3}(F, μ_{d/r}^{⊗3})

commutes, where the vertical maps are cup-product by {a} and the horizontal maps are those of (7.5) and (7.6).

Proof. Since the spectral sequences of [18, Th. 4.4] are multiplicative, it suffices to chek that the diagram

SK_{1}(A) ⟶^{σ_{r}^{1}} Ker ξ_{Y^{[r]}}^{4}
·{a}↓  ^{·{a}}↓
SK_{2}(A) ⟶^{σ_{r}^{2}} Ker ξ_{Y^{[r]}}^{5}

commutes. This in turn reduces to the compatibility of the BGQ spectral sequence and the isomorphisms (7.2) with products. □

Similarly:

7.8. Proposition. Let A be a discrete valuation F -algebra, with quotient field K and residue field E. Then the diagrams

SK_{2}(A_{K}) ⟶^{σ_{r}^{2}} H^{5}(K, μ_{d/r}^{⊗4})/r[A] · H^{3}(K, μ_{d/r}^{⊗3})
∂↓  ^{∂}↓
SK_{1}(A_{E}) ⟶^{σ_{r}^{1}} H^{4}(E, μ_{d/r}^{⊗3})/r[A] · H^{2}(E, μ_{d/r}^{⊗2})

commutes, where the homomorphisms ∂ are induced by the residue maps in K-theory and Galois cohomology respectively.

Proof. Similar. □

Using Corollary 7.5 b), Proposition 7.7 and Proposition 7.8, we find that σ_{1}^{2} and σ_{2}^{2} are nontrivial when 4 | ind(A).

7.E. A refinement. In this subsection, where we keep the previous notation, we assume that A is a division algebra, d is a power of a prime l and r[A] = 0: for r strictly dividing d, this is possible if and only if the exponent ε of A is smaller than d (and then we may choose for r any l-power between ε and d/l). Then we can compute K_{1}(X)^{(1/2)} and extend the map

SK_{i}(A) → K_{i}(X)^{(2)}

of the previous section to a map

K_{i}(A) → K_{i}(X)^{(2)}.

This approach corresponds to that of Rost in the case where A is a biquaternion algebra [33]. Let H be the class of a hyperplane section in K_{0}(Y^{[r]}).

7.9. Proposition. For i ≤ 2,

a) The composition

K_{i}(F) ⟶^{·H} K_{i}(Y^{[r]})^{(1)} → A^{1}(Y^{[r]}, K_{i+1}) ⟶^{ξ^{1,i+1}} K_{i}(F)

is the identity.

b) The induced map

K_{i}(F) → K_{i}(Y^{[r]})^{(1/2)}

is an isomorphism.

c) Let J be the tautological bundle on Y^{[r]}. Then the image of the map

Φ^{[r]} : K_{i}(A) → K_{i}(Y^{[r]})^{(1)}
x ↦ ũ_{J} (x) − Nrd(x) · H

(see Proposition 6.2 b)) sits into K_{i}(X)^{(2)}.

Proof. By Lemma 7.1, the map

CH^{1}(Y^{[r]}) → CH^{1}(Y_{s}^{[r]})

is bijective. In particular, c_{1}(H) = h in CH^{1}(Y_{s}^{[r]}). We then get a) by multiplicativity. b) follows from a) and the fact that the maps

K_{i}(Y^{[r]})^{(1/2)} → H^{1}(Y^{[r]}, K_{i+1}) ⟶^{ξ^{1,i+1}} K_{i}(F)

are injective. c) follows immediately from a). □

7.F. The comparison issue. For s | r | d, let Y^{[r,s]} = SB(r, s, A) be as in 7.A with the two projections

Y^{[r,s]}
[figure: p_{r}; p_{s}]
Y^{[r]}  Y^{[s]}.

We have corresponding diagrams (i = 1, 2)

Ker η_{Y^{[r]}}^{i+3}
[figure: σ_{r}^{i}; p_{r}^{∗}]
SK_{i}(A)  Ker η_{Y^{[r,s]}}^{i+3}
[figure: σ_{s}^{i}; p_{s}^{∗}]
Ker η_{Y^{[s]}}^{i+3}

The comparison issue is to know whether this diagram commutes: if this is the case, then the maps σ_{r}^{i} and σ_{s}^{i} are compatible in an obvious sense thanks to Corollary 7.2 b). In view of Theorem 6.1 c), this commutation is equivalent to the commutation of the diagram

K_{i}(Y^{[r]})^{(2)}
[figure: u_{Jr}; p_{r}^{∗}]
SK_{i}(A)  K_{i}(Y^{[r,s]})^{(2)}
[figure: u_{Js}; p_{s}^{∗}]
K_{i}(Y^{[s]})^{(2)}

or to the vanishing of the map

u_{J_{r}} − u_{J_{s}} : SK_{i}(A) → K_{i}(Y^{[r,s]})^{(2)}.

We may also consider the sheaf I_{r,s} = Ker(I_{r} → I_{s}); then the above amounts to the vanishing of the map

u_{I_{r,s}} : K_{i}(A) → K_{i}(Y^{[r,s]})

on the subgroup SK_{i}(A). In [50, Th. 4], Suslin obtains this commutation (or vanishing) for (s, r, d) = (1, 2, 4) in a very sophisticated and roundabout way. I have no idea how to prove it in general.

8. Motivic cohomology of some Severi-Brauer varieties

In this section, unlike in the rest of the paper, we write H^{∗}(X, Z(n)) (resp. H_{ét}^{∗}(X, Z(n)) for motivic cohomology of some smooth variety X computed in the Nisnevich (resp. étale) topology. We also use Zariski cohomology with coefficients into sheafified étale cohomology groups instead of cycle cohomology, as those are the groups that come naturally.

8.1. Theorem. Let A have prime index l, and let X be its Severi-Brauer variety. Let Z_{A} be the Nisnevich sheaf with transfers defined in [22, 5.3]. Let n ≥ 0, and assume the Bloch-Kato conjecture in degrees ≤ n + 1. Then:

a) There is an exact sequence

0 → H^{n}(F, Z_{A}(n)) ⟶^{Nrd} H^{n}(F, Z(n)) ⟶^{·[A]} H_{ét}^{n+3}(F, Z(n + 1))

→ H^{0}(X, H_{ét}^{n+3}(Z(n + 1))) → 0. b) There is a cross of exact sequences

0

↓

H^{1}(X, H_{ét}^{n+3}(Z(n + 1)))

↓

0→H_{ét}^{n+4}(F, Z(n + 1))→ H^{n+4}(X, Z̄(n + 1)) →H^{0}(X, H_{ét}^{n+2}(Z(n))
↓  ^{·[A]}↓
H^{0}(X, H_{ét}^{n+4}(Z(n + 1))) H_{ét}^{n+5}(F, Z(n + 1))

where Z̄(n) is the cone of the morphism Z(n) → Rα_{∗}α^{∗}Z(n), with α the projection of the big étale site onto the big Nisnevich site.

Proof. This is an extension of [22, Th. 8.1.4 and 8.2.2], and it is proven by the same method. The exact sequence of a) is part 2 of Theorem 8.1.4 of loc. cit. (where the differential is identified with the cup-product with [A] in 8.2), except that in [22, Th. 8.1.4 (2)], the last term is H_{ét}^{n+3}(F (X), Z(n + 3)) and there is no surjectivity claimed. To prove a) and b) we look at the spectral sequence (8.4) of [22]. Let d = dim X(= l − 1). In the proof of Theorem 8.1.4 and in 8.2, the following was established:

• E_{2}^{p,q} = 0 for −q ∉ [0, d], p < d − 1, p = d or (p, q) = (d − 1, −d).

• The differential

d_{2} : Coker(H^{n}(F, Z_{A}(n)) → H_{ét}^{n}(F, Z(n))) ≃ E_{2}^{d−1,1−d}

→ E_{2}^{d+1,−d} ≃ H_{ét}^{n+3}(F, Z(n + 1))

is injective, and induced by the cup-product H_{ét}^{n}(F, Z(n)) ⟶^{·[A]}

H_{ét}^{n+3}(F, Z(n + 1)).

The abutment of this spectral sequence on the diagonal p + q = N is

Hom(Z(d)[2d], M̄(X)(n + 1)[n + 2 + N])

computed in DM^{eff}(F), where

M̄(X) = cone(M(X) → Rα_{∗}α^{∗}M(X)).

Note that M̄(X)(n + 1) ≃ M(X) ⊗ Z̄(n + 1) (by a projection formula). Hence the abutment may be rewritten (by Poincaré duality)

H^{n+2+N}(X, Z̄(n + 1)).

The Bloch-Kato conjecture in degree n + 1 identifies Z̄(n + 1) with τ_{>n+2}(Rα_{∗}α^{∗}Z(n)). The hypercohomology spectral sequence then gives

H^{n+2+N}(X, Z̄(n + 1)) = 0 for N ≤ 0
H^{n+3}(X, Z̄(n + 1)) ≃ H^{0}(X, H_{ét}^{n+3}(Z(n + 1)))

and for N = 2 an exact sequence

0 → H^{1}(X, H_{ét}^{n+3}(Z(n + 1))) → H^{n+4}(X, Z̄(n + 1))

→ H^{0}(X, H_{ét}^{n+4}(Z(n + 1))).

Consider the differentials d_{2}^{d−1,q} : E_{2}^{d−1,q} → E_{2}^{d+1,q−1} for −q ≤ d − 1. We have

E_{2}^{p,q} = Hom(Z, Z̄_{A^{⊗(−q+1)}}(n + 1 − d − q)[n + 2 − 2d + p − q])

where Z̄_{A^{⊗(−q+1)}} = cone(Z_{A^{⊗(−q+1)}} → Rα_{∗}α^{∗}Z_{A^{⊗(−q+1)}}). Therefore

E_{2}^{d−1,q} = Hom(Z, Z̄_{A^{⊗(−q+1)}}(n + 1 − d − q)[n + 1 − d − q])

= Coker(H^{n+1−d−q}(F, Z_{A}(n + 1 − d − q)) → H^{n+1−d−q}(F, Z(n + 1 − d − q)))

and

E_{2}^{d+1,q−1} = Hom(Z, Z̄_{A^{⊗(−q+2)}}(n + 2 − d − q)[n + 4 − d − q])

= H_{ét}^{n+4−d−q}(F, Z(n + 2 − d − q))).

The computation of [22, 8.2] identifies d_{2}^{d−1,q} with the map induced by cupproduct by [A]. By the above, we get that d_{2}^{d−1,q} is injective. The computation of [22, 8.2] also identifies d_{2}^{d+1,q−1} with the cup-product by [A]. This gives both a) and b). □

9. Étale motivic cohomology of reductive groups

9.A. The slice spectral sequence for a reductive group. Let X be a smooth F -variety. There are spectral sequences [14, (3.1), (3.2)], similar to those of Theorem 2.5:

(9.1) E_{2}^{p,q}(X, n)_{Nis} = Hom_{DM_{−}^{eff} (F)}(c_{q}(X), Z(n − q)[p − q]) ⇒ H_{Nis}^{p+q}(X, Z(n))

(9.2)

E_{2}^{p,q}(X, n)_{ét} = Hom_{DM_{−,ét}^{eff}(F)}(α^{∗}c_{q}(X), Z(n − q)[p − q]) ⇒ H_{ét}^{p+q}(X, Z(n))

where c_{q}(X) are complexes of Nisnevich sheaves with transfers associated to X (canonically in the derived category) and α is the projection from the étale site of smooth F -varieties to the Nisnevich site. These spectral sequences have the same formal properties as (2.1): transfers, and products if the motive of X is mixed Tate (resp. geometrically mixed Tate), cf. discussion in the proof of Th. 2.5 (ii). Let X = G be a connected reductive group over F, with maximal torus T defined over F. Set Y = G/T. Assume first G and T split. In [14, Prop. 9.3], it was shown that c_{q}(G) is dual, in the derived category, to the complex of constant Nisnevich sheaves c^{q}(G) (denoted by K(G, q) in loc. cit.) given by

(9.3) 0 → Λ^{q}(T^{∗}) → Λ^{q−1}(T^{∗}) ⊗ CH^{1}(Y) → …

⋯ → T^{∗} ⊗ CH^{q−1}(Y) → CH^{q}(Y) → 0

in which T^{∗} is the group of characters of T, CH^{q}(Y) is in degree 0 and the maps are induced by intersection products and the characteristic map γ : T^{∗} → CH^{1}(X) (compare [8, 3.14]). Thus (9.1) may be rewritten in this case as

E_{2}^{p,q}(G, n)_{Nis} = H_{Nis}^{p−q}(F, c^{q}(G) ⊗ Z(n − q)) ⇒ H_{Nis}^{p+q}(G, Z(n)).

Since c^{q}(G) is concentrated in degrees ≤ 0, c^{q}(G) ⊗ Z(n − q) is concentrated in degrees ≤ n−q and E_{2}^{p,q}(G, n)_{Nis} = 0 for p > n. We also have E_{2}^{p,q}(G, n)_{Nis} = 0 for q > n, since Z(n − q) = 0 in this case. For (p, q) = (n, n) this yields

9.1. Lemma (cf. Grothendieck [13, p. 21, Rem. 2]). If G is split, we have isomorphisms E_{2}^{n,n}(G, n)_{Nis} ≃ E_{∞}^{p,q}(G, n)_{Nis} ≃ H^{2n}(G, Z(n)), hence an exact sequence

T^{∗} ⊗ CH^{n−1}(Y) → CH^{n}(Y) → CH^{n}(G) → 0.

We shall also use:

9.2. Lemma. Suppose G split, simply connected and absolutely simple. Then, for all n > 0, CH^{n}(G) is killed by (n − 1)! and by the torsion index t_{G} of G [7, §5]. In particular, CH^{i}(G) = 0 for i = 1, 2. If G is of type A_{r} or C_{r}, CH^{i}(G) = 0 for all i > 0.

Proof. The first fact follows from K_{0}(G) = Z, cf. [8, Proof of Prop. 3.20 (iii)]. For the second one, Demazure proves in [7, Prop. 5] that the cokernels of the characteristic maps γ^{n} : S^{n}(T^{∗}) → CH^{n}(Y) are killed by t_{G}: the claim then follows from Lemma 9.1 and a small diagram chase. The last fact follows from [7, Lemme 5], which says that t_{G} = 1 for G of type A_{r} or C_{r}. (This also follows from Suslin [48, Th. 2.7 and 2.12].) □

We now relax the assumption that G is split, and would like to study the spectral sequences (9.2). If we knew that

(9.4)  α^{∗}c_{q}(G) ≃ c_{q}(G_{s})

in the derived category of complexes of étale sheaves (or G_{F} -modules), this would allow us to rewrite (9.2) in the form

E_{2}^{p,q}(G, n)_{ét} = H_{ét}^{p−q}(F, c^{q}(G_{s}) ⊗ Z(n − q)) ⇒ H_{ét}^{p+q}(G, Z(n))

as for the split case, in the Nisnevich topology. I don’t know how to prove (9.4), but at least the proof of [14, Prop. 9.3] shows that the two complexes have isomorphic cohomology sheaves. Hence they are quasi-isomorphic at least in the case where the cohomology of c^{p}(G_{s}) is concentrated in at most one degree. We shall therefore make-do with (9.2) and be saved by the fact that, for low values of q and for the groups G we are interested in, the latter fact is true. For simplicity, we shall write

Hom_{DM_{−,ét}^{eff}(F)}(α^{∗}c_{q}(G), Z(n − q)[p − q]) = Ext_{ét}^{p−q}(α^{∗}c_{q}(G), Z(n − q)).

We always have c^{0}(G_{s}) = CH^{0}(Y_{s}) = Z^{π_{0}(G_{s})}. Suppose that G is semi-simple, simply connected. Then c is bijective and one finds [8]

(9.5)  c^{1}(G_{s}) = 0
(9.6)  c^{2}(G_{s}) = S^{2}(T_{s}^{∗})^{W} [1]

where W is the Weyl group of G_{s}. If G is absolutely simple, then rk S^{2}(T_{s}^{∗})^{W} = 1 (with trivial Galois action). We note that the unit section of G splits off from (9.2) spectral sequences

Ẽ_{2}^{p,q}(G, n) ⇒ H̃_{ét}^{p+q}(G, Z(n))

with

Ẽ_{2}^{p,q}(G, n) = \begin{cases}Ext_{ét}^{p−q}(α^{∗}c_{q}(G), Z(n − q)) & for q > 0 \\ 0 & for q = 0 \end{cases}

and H_{ét}^{p+q}(G, Z(n)) = H_{ét}^{p+q}(F, Z(n)) ⊕ H̃_{ét}^{p+q}(G, Z(n)) via the unit section. These spectral sequences are modules over (9.2). From the above spectral sequence in weight 3, the corresponding coniveau spectral sequence, (9.5) and (9.6), we get a commutative diagram analogous to (3.1): (9.7)

0

↓

0→  A^{2}(G, K_{3}^{M})  ⟶  H̃^{5}(G, Z(3)) →Ã^{0}(G, H^{4}(3))
↓  ^{α}↓
Ext_{ét}^{−1}(α^{∗}c_{3}(G), Z)  CH^{3}(G)
_{d̃}_{2}^{2,3}(G,3)↓  ↓
H^{2}(F, 𝔾_{m} ⊗ S^{2}(T_{s}^{∗})^{W}) ⟶^{∼} Ext_{ét}^{2}(α^{∗}c_{2}(G), Z(1)) H̃^{6}(G, Z(3))

↓

H̃^{6}(G, Z(3))

↓

Ext_{ét}^{0}(α^{∗}c_{3}(G), Z)

In this diagram, the column and the row forking downwards are both exact. The groups marked with a are, as above, the direct summands of the corresponding groups without a defined by the unit section of G.

9.B. An invariant computation. In this subsection, we want to compute S^{3}(T_{s}^{∗})^{W} when G is absolutely simple simply connected. We start with the case of type A_{r}. It is then convenient to think of G_{s} as SL_{r+1} embedded into GL_{r+1}. The maximal torus T_{s} of G_{s} is then a subtorus of a maximal torus S of GL_{r+1}, conjugate to its canonical maximal subtorus. The character group S^{∗} is free of rank r + 1, with basis (e_{1}, …, e_{r+1}), and T_{s}^{∗} is the quotient of S^{∗} by Zσ_{1}, with σ_{1} = ∑ e_{i}. The Weyl group W of G_{s} coincides with that of GL_{r+1}; it is isomorphic to 𝔖_{r+1} and permutes the e_{i}. Let σ_{i} be the i-th symmetric function in the e_{i}: by the symmetric functions theorem, we have

S(S^{∗})^{W} = Z[σ_{1}, …, σ_{r+1}].

It is clear that the sequence

(9.8)  0 → σ_{1}S(S^{∗}) → S(S^{∗}) → S(T_{s}^{∗}) → 0

is exact.

9.3. Lemma. If r ≥ 2, the map S^{3}(S^{∗})^{W} → S^{3}(T_{s}^{∗})^{W} is surjective; S^{3}(T_{s}^{∗})^{W} is free of rank 1, with basis the image σ̄_{3} of σ_{3}. If r = 1, S^{3}(T_{s}^{∗})^{W} = 0.

Proof. Suppose first r ≥ 2. In view of (9.8), for the first assertion it suffices to check that H^{1}(W, S^{2}(S^{∗})) = 0. A basis of S^{2}(S^{∗}) is given by (e_{1}^{2}, …, e_{r+1}^{2}, e_{1}e_{2}, …). The group W permutes the squares and the rectangular products transitively; the isotropy group of e_{1}^{2} is 𝔖_{r} while the isotropy group of e_{1}e_{2} is 𝔖_{r−1}. By Shapiro lemma, we get

H^{1}(W, S^{2}(S^{∗})) ≃ H^{1}(𝔖_{r}, Z) ⊕ H^{1}(𝔖_{r−1}, Z) = 0.

For the second assertion, we use (9.8) again and get an exact sequence (thanks to the symmetric functions theorem)

0 → σ_{1}⟨σ_{1}^{2}, σ_{2}⟩ → ⟨σ_{1}^{3}, σ_{1}σ_{2}, σ_{3}⟩ → S(T_{s}^{∗})^{W} → 0.

If r = 1, the same calculation gives the result. □

In the other cases, an application of the theory of exponents [4, V.6.2, Prop. 3 and tables of Ch. VI] gives

9.4. Lemma. If G is not of type A_{r}, S^{3}(T_{s}^{∗})^{W} = 0. □

9.C. Some facts about the c^{q}(G_{s}). Part a) of the following theorem is a version of S. Gille’s theorem [11, th. 1.5]^{2}:

9.5. Theorem. Let G be semi-simple and simply connected. Then:

a) For q ≥ 3, H^{r}(c^{q}(G_{s})) = 0 for r = −q, −q + 1, and H^{−q+2}(c^{q}(G_{s})) is torsion-free.

b) Suppose G simple. For q = 3, H^{−1}(c^{3}(G_{s})) ≃ S^{3}(T_{s}^{∗})^{W} and H^{0}(c^{3}(G_{s})) ≃ CH^{3}(G_{s}).

c) If G is simple of type A_{r}, with r ≥ 2, then c^{3}(G_{s}) ≃ Z(χ)[1], generated by σ̄_{3} (see Lemma 9.3) where χ is the quadratic character of G_{F} corresponding to its (possibly trivial) outer action on the Dynkin diagram of G. If G is of type A_{1}, c^{3}(G_{s}) = 0. If G is not of type A_{r}, c^{3}(G_{s}) = CH^{3}(G_{s})[0].

2For q = 3 and G of type A_{r}, it was obtained in 2001/2002. The general case was inspired by Gille’s work.

Documenta Mathematica · Extra Volume Suslin (2010) 317–369

Proof. a) For two split reductive groups G, H and n ≥ 0, we have the Künneth formula

(9.9)  c^{n}(G × H) ≃ ⨁_{p+q=n} c^{p}(G) ⊗^{L} c^{q}(H)

in the derived category [14, Lemma 4.8], since M(G) and M(H) are mixed Tate motives. Thus we may assume G to be simple. Consider now the commutative diagram

Λ^{q−2}(T_{s}^{∗})⊗S^{2}(T_{s}^{∗})^{W} →^{e}Λ^{q−3}(T_{s}^{∗})⊗S^{2}(T_{s}^{∗})^{W} ⊗T_{s}^{∗}
↓  ^{f}↓
Λ^{q}(T_{s}^{∗})→ Λ^{q−1}(T_{s}^{∗})⊗T_{s}^{∗} → Λ^{q−2}(T_{s}^{∗})⊗S^{2}(T_{s}^{∗}) → Λ^{q−3}(T_{s}^{∗})⊗S^{3}(T_{s}^{∗})
↓^{||}  ^{1⊗γ}↓^{≀}  ^{1⊗γ^{2}}↓  ^{1⊗γ^{3}}↓
Λ^{q}(T_{s}^{∗})→Λ^{q−1}(T_{s}^{∗})⊗CH^{1}(Y_{s})→Λ^{q−2}(T_{s}^{∗})⊗CH^{2}(Y_{s})→ Λ^{q−3}(T_{s}^{∗})⊗CH^{3}(Y_{s})

where the bottom row is the beginning of c^{q}(G_{s}), the middle row is the q-th Koszul complex for T_{s}^{∗}, γ^{i} are induced by the characteristic map, the top row is S^{2}(T_{s}^{∗})^{W} tensored with the beginning of the (q − 2)-nd Koszul complex for T_{s}^{∗}, the middle column is obtained by tensoring the exact sequence of free abelian groups

0 → S^{2}(T_{s}^{∗})^{W} → S^{2}(T_{s}^{∗}) → CH^{2}(Y_{s}) → 0

with Λ^{q−2}(T_{s}^{∗}) and, finally, f is induced by the product S^{2}(T_{s}^{∗})^{W} ⊗ T_{s}^{∗} → S^{3}(T_{s}^{∗}). The middle row is universally exact as the Koszul complex of a free module, and the middle column is (split) short exact. Since G is simple, S^{2}(T_{s}^{∗})^{W} is a rank 1 direct summand of S^{2}(T_{s}^{∗}), which implies that f is injective and remains so after tensoring with Z/m for any m. The same is true for e by the acyclicity of Koszul complexes. A diagram chase then gives the result.

b) For q = 3, let us rewrite part of the above diagram, for clarity:

0→Λ^{3}(T_{s}^{∗})→ Λ^{2}(T_{s}^{∗}) ⊗ T_{s}^{∗} → T_{s}^{∗} ⊗ S^{2}(T_{s}^{∗}) → S^{3}(T_{s}^{∗}) → 0
||↓  ^{1⊗γ}↓  ^{1⊗γ^{2}}↓  ^{γ^{3}}↓
0→Λ^{3}(T_{s}^{∗})→Λ^{2}(T_{s}^{∗}) ⊗ CH^{1}(Y_{s})→T_{s}^{∗} ⊗ CH^{2}(Y_{s})→CH^{3}(Y_{s})→0.

The two left vertical maps are isomorphisms; by (9.6), 1⊗γ^{2} is surjective, with kernel T_{s}^{∗} ⊗ S^{2}(T_{s}^{∗})^{W}; also, by [7, p. 292, Cor. 2] Ker γ^{3} is the Q-span of T_{s}^{∗}S^{2}(T_{s}^{∗})^{W} + S^{3}(T_{s}^{∗})^{W} in S^{3}(T_{s})^{∗}. Using Lemma 9.1, it follows that

H^{i}(c^{3}(G_{s})) = \begin{cases}0 & for i = −3 \\ Ker φ & for i = −2 \\ Coker φ & for i = −1 \\ CH^{3}(G_{s}) & for i = 0 \end{cases}

where φ is the map

T_{s}^{∗} ⊗ S^{2}(T_{s}^{∗})^{W} → ⟨T_{s}^{∗}S^{2}(T_{s}^{∗})^{W} + S^{3}(T_{s}^{∗})^{W} ⟩_{Q},

⟨−⟩_{Q} denoting the Q-span. We have seen in a) that Ker φ = 0 and Coker φ is torsion-free. We may factor φ as a composition

T_{s}^{∗} ⊗ S^{2}(T_{s}^{∗})^{W} \underrightarrow{_{φ̃}} T_{s}^{∗}S^{2}(T_{s}^{∗})^{W} + S^{3}(T_{s}^{∗})^{W} ⟶ ⟨T_{s}^{∗}S^{2}(T_{s}^{∗})^{W} + S^{3}(T_{s}^{∗})^{W} ⟩_{Q}.

Thus Coker φ is an extension of the finite group

\frac{⟨T_{s}^{∗}S^{2}(T_{s}^{∗})^{W} + S^{3}(T_{s}^{∗})^{W} ⟩_{Q}}{T_{s}^{∗}S^{2}(T_{s}^{∗})^{W} + S^{3}(T_{s}^{∗})^{W}}

by a group isomorphic to S^{3}(T_{s}^{∗})^{W} /S^{3}(T_{s}^{∗})^{W} ∩ T_{s}^{∗}S^{2}(T_{s}^{∗})^{W}; but

S^{3}(T_{s}^{∗})^{W} ∩ T_{s}^{∗}S^{2}(T_{s}^{∗})^{W} ⊆ (T_{s}^{∗}S^{2}(T_{s}^{∗})^{W})^{W} = T_{s}^{∗W} S^{2}(T_{s}^{∗})^{W} = 0.

Thus, the map S^{3}(T_{s}^{∗})^{W} → Coker φ̃ is bijective. To conclude, we use the fact that S^{3}(T_{s}^{∗})^{W} is pure in S^{3}(T_{s}^{∗}) (the quotient is torsion-free), which follows from Lemmas 9.3 and 9.4: since Coker φ is torsion-free, this implies that it is isomorphic to S^{3}(T_{s}^{∗})^{W}.

c) now follows from b), Lemmas 9.3, 9.4 and 9.2. For G of type A_{r} with r ≥ 2, the claim on the Galois action follows from the well-known fact that the nontrivial outer automorphism of the Dynkin diagram of G_{s} maps ē_{i} to −ē_{r+1−i}, where ē_{i} is the image of e_{i} in T_{s}^{∗}. □

Here is a complement to Theorem 9.5:

9.6. Lemma. Let r ≥ 2, and consider the embedding ι : SL_{r+1} ↪ SL_{r+2} given by u ↦ ( \begin{matrix}u & 0 \\ 0 & 1 \end{matrix} ). Then the induced morphism ι^{∗} : c^{i}(SL_{r+2}) → c^{i}(SL_{r+1}) is a quasi-isomorphism for i = 2, 3.

Proof. Let T_{r+1}, T_{r+2} be the diagonal tori of SL_{r+1} and SL_{r+2} respectively. It suffices to check that S^{i}(T_{r+2}^{∗})^{𝔖_{r+2}} ⟶^{∼} S^{i}(T_{r+1}^{∗})^{𝔖_{r+1}} for i = 2, 3. This follows from the computations in the proof of Lemma 9.3. □

9.7. Remark. For G of type C_{r}, CH^{i}(G_{s}) = 0 for all i > 0, and for general G, CH^{3}(G_{s}) is a 2-torsion group (see Lemma 9.2). Marlin computed CH^{∗}(G_{s}) for G of type B_{r}, D_{r}, G_{2} or F_{4} in [29]: he finds CH^{3}(G_{s}) = Z/2 in each case. I don’t know the value of CH^{3}(G_{s}) for G of type E_{6}, E_{7}, E_{8}: is it also Z/2?

10. The generic element

In this section we prove Theorem C, see (10.2), (10.3) and Theorem 10.7, Theorem D, see Corollary 10.15, and part of Theorem E, see Proposition 10.11.

10.A. The cohomological generic element. Let G be an absolutely simple simply connected group. From Theorem 9.5 and Diagram (9.7), we first deduce:

10.1. Corollary. If G is not of inner type A_{r} for r ≥ 2, we have A^{2}(G, K_{3}^{M}) = H̃^{5}(G, Z(3)) = 0; the group Ã^{0}(G, H^{4}(3)) is isomorphic to the kernel of the étale motivic cycle map CH^{3}(G) → H^{6}(G, Z(3)) (hence is at most Z/2 except perhaps for types E_{6}, E_{7}, E_{8}, see Remark 9.7).

Proof. All claims follow from the diagram and the fact that we have H^{−1}(F, c^{3}(G_{s})) = 0 in these cases (note that obviously

Ker(CH^{3}(G) → H^{6}(G, Z(3)) = Ker(CH^{3}(G) → H̃^{6}(G, Z(3))).

□

10.2. Proposition. If G is of inner type A_{r} with r ≥ 2, the map α in Diagram (9.7) is 0.

Proof. We have G = SL_{1}(A) for some central simple algebra A. If CH^{3}(G) = 0, there is nothing to prove; by Merkurjev [35, Prop. 4.3], this happens if and only if ind(A) is odd. Suppose now ind(A) even. If A is a quaternion algebra, we have Ã^{0}(G, H^{4}(3)) = 0 by [35, Lemma 5.1]. In general, we proceed as in [35, proof of Prop. 4.3]. Note that α really comes from a map α′ : A^{0}(G, H^{4}(3)) → CH^{3}(G) and that α = 0 if and only if α′ = 0. Let K/F be a field extension such that ind(A_{K}) = 2, so that A_{K} = M_{n}(Q) for some quaternion division algebra Q over K and G_{K} = SL_{n}(Q). Set H = SL_{1}(Q) and X = G_{K}/H. By loc. cit., the generic fibre of the projection G_{K} → X is H_{E}, with E = K(X). We then have a commutative diagram

A^{0}(G, H^{4}(3)) ⟶ A^{0}(G_{K}, H^{4}(3)) ⟶ A^{0}(H_{E}, H^{4}(3))
α′↓  _{α′}↓  _{α′=0}↓
CH^{3}(G) ⟶ CH^{3}(G_{K}) ⟶ CH^{3}(H_{E})

and the bottom horizontal maps are isomorphisms by loc. cit. (see [35, Rk 4.4]). □

From now on, we suppose G of inner type A_{r} with r ≥ 2, i.e. deg(A) > 2 if G = SL_{1}(A). Then H^{−1}(F, c^{3}(G_{s})) is canonically isomorphic to Z, H^{2}(F, 𝔾_{m}⊗ S^{2}(T_{s}^{∗})^{W}) ≃ Br(F) and H^{0}(F, c^{3}(G_{s})) = 0. For the reader’s convenience, let us redraw Diagram (9.7) in this case, taking Proposition 10.2 into account:

(10.1)  0 ; ↓ ; 0→A^{2}(G, K_{3}^{M}) ⟶ H̃^{5}(G, Z(3))→Ã^{0}(G, H^{4}(3))→0 ; ↓ ; Z ; _{_{d̃}_{2}^{2,3}(G,3)}↓ ; Br(F) ; ↓ ; H̃^{6}(G, Z(3)) ; ↓ ; 0

Since A^{0}(G, H^{4}(3)) and Br(F) are torsion, we recover Merkurjev’s result that A^{2}(G, K_{3}^{M}) = A^{2}(G, K_{3}) is infinite cyclic [35, Lemma 5.7]. We also find

10.3. Theorem. The group H̃^{5}(G, Z(3)) is infinite cyclic and the group Ã^{0}(G, H^{4}(3)) is cyclic of order (H̃^{5}(G, Z(3)) : A^{2}(G, K_{3}^{M})).

10.4. Definition. Let G = SL_{1}(A). We denote by c_{A} the “positive” generator of H̃^{5}(G, Z(3)) ⊂ H^{5}(G, Z(3)), that is, the generator that maps to a positive multiple of 1 ∈ Z, and by c̄_{A} its image in Ã^{0}(G, H^{4}(3)) ⊂ A^{0}(G, H^{4}(3)) (c̄_{A} generates Ã^{0}(G, H^{4}(3))).

10.5. Lemma. Let still G = SL_{1}(A), and let p_{1}, p_{2}, μ : G ×_{F} G → G be repectively the first projection, the second projection and the multiplication map. Then

μ^{∗}c_{A} = p_{1}^{∗}c_{A} + p_{2}^{∗}c_{A}.

Proof. Since H̃^{5}(G, Z(3)) → H^{−1}(F, c^{3}(G_{s})) is injective for any group G, it is sufficient to show that the maps μ^{∗} and p_{1}^{∗} + p_{2}^{∗} from c^{3}(G_{s}) to c^{3}(G_{s} ×_{F_{s}} G_{s}) are equal.^{3} The Künneth formula (9.9) gives an isomorphism

c^{3}(G_{s}) ⊕ c^{3}(G_{s}) ⟶^{∼} c^{3}(G_{s} ×_{F_{s}} G_{s})

induced by p_{1}^{∗} ⊕ p_{2}^{∗}, since c^{1}(G_{s}) = 0. Let C = c^{3}(G_{s}). The inclusion ι_{1} : G × {1} → G × G induces a map ι_{1}^{∗} : C ⊕ C → C; since p_{1} ∘ ι_{1} = Id and p_{1} ∘ ι_{1} is the trivial map, ι_{1}^{∗} is the first projection. Similarly, ι_{2} : {1} × G → G × G induces the second projection. We conclude that μ^{∗} : C → C ⊕ C is the diagonal map, using the left and right unit formulas μ ∘ ι_{1} = μ ∘ ι_{2} = Id. □

3Note that morphisms between reductive groups preserving the unit sections act on the spectral sequences (9.2) by preserving the spectral sequences Ẽ_{r}^{p,q}. This applies to μ and to the maps ι_{1} and ι_{2} further below.

Documenta Mathematica · Extra Volume Suslin (2010) 317–369

Let X be a smooth F -variety. To any morphism f : X → SL_{1}(A), we associate the pull-back of c_{A}:

c_{A}(f) = f^{∗}c_{A} ∈ H^{5}(X, Z(3)).

Lemma 10.5 shows that we have

c_{A}(fg) = c_{A}(f) + c_{A}(g)

for two maps f, g, where fg := μ ∘ (f, g). Recall that deg(A) > 2. Consider the embedding ι_{n} : SL_{1}(A) ↪ SL_{n}(A) given by u ↦ ( \begin{matrix}u & 0 \\ 0 & 1 \end{matrix} ). Noting that SL_{n}(A) = SL_{1}(M_{n}(A)), Lemma 9.6 shows that

c_{M_{n}(A)}(ι_{n}) = c_{A}.

In particular, ι_{n}^{∗} : H̃^{5}(SL_{n}(A), Z(3)) → H̃^{5}(SL_{1}(A), Z(3)) is an isomorphism. So, if f is a morphism from X to SL_{n}(A), we may define c_{A}(f) = (ι_{n}^{∗})^{−1}c_{M_{n}(A)}(f), and this definition is “stable”. We record this as:

10.6. Proposition. If deg(A) > 2, the maps

H̃^{5}(SL_{n}(A), Z(3)) → H̃^{5}(SL_{1}(A), Z(3))
Ã^{0}(SL_{n}(A), H^{5}(Z(3))) → Ã^{0}(SL_{1}(A), H^{5}(Z(3)))

induced by the inclusion SL_{1}(A) ↪ SL_{n}(A) are isomorphisms. □

In particular, suppose X = Spec R affine. Then Hom_{F} (X, SL_{n}(A)) = SL_{n}(A ⊗_{F} R). Define SL(A ⊗_{F} R) = \underrightarrow{lim} SL_{n}(A ⊗_{F} R) as usual, and

SK_{1}(X, A) = SL(A ⊗_{F} R)^{ab}.

For X smooth in general, we may similarly define

SL(X, A) = \underrightarrow{lim} Hom_{F} (X, SL_{n}(A)), SK_{1}(X, A) = SL(X, A)^{ab}.

The above discussion then yields a homomorphism

(10.2)  SK_{1}(X, A) → H^{5}(X, Z(3))

which is contravariant in X. In particular, for X = Spec L, L/F a function field, we get a homomorphism

(10.3)  c̄_{A}(L) : SK_{1}(A_{L}) → H^{5}(L, Z(3)) ⟵^{∼} H^{4}(L, 3).

The following theorem was (embarrassingly) pointed out by Philippe Gille, whom I thank here.

10.7. Theorem. In (10.3), L ↦ c̄_{A}(L) defines the universal invariant of SL_{1}(A) of degree 4 with values in H^{4}(3), in the sense of Merkurjev [35, Def. 2.1].

Proof. Let G be an algebraic group and let M be a cycle module of bounded exponent as in [35, p. 133]. By [35, Th. 2.3], we have an isomorphism

Inv^{d}(G, M) ⟶^{∼} A^{0}(G, M_{d})_{mult}, d ∈ Z

induced by evaluation on the generic point of G, where the left group is the group of invariants of G with values in M_{d} as in [35, Def. 2.1] and the right group is the multipicative part of A^{0}(G, M_{d}) as in [35, 1.3]. We cannot apply this directly to M_{d}(K) = H^{d}(K, d − 1), which is not of bounded exponent. However, any cycle module M_{∗} such that M_{d}(K) is torsion prime to char F for any d ∈ Z and any function field K/F may be written as the filtering direct limit of its torsion sub-cycle modules _{m}M_{∗}, m ≥ 1. Then the maps

\underrightarrow{lim} Inv^{d}(G,_{m}M) → Inv^{d}(G, M)_{tors}
\underrightarrow{lim} A^{0}(G,_{m}M_{d})_{mult} → A^{0}(G, M_{d})_{mult}

are bijective, so that [35, Th. 2.3] extends to an isomorphism

(10.4)  Inv^{d}(G, M)_{tors} ⟶^{∼} A^{0}(G, M_{d})_{mult}

for any torsion cycle module M (excluding the characteristic of F) and any d ∈ Z. In the case G = SL_{1}(A), any invariant of G, evaluated at a function field K, factors through G(K)^{ab} = SK_{1}(A_{K}), which is of exponent bounded by ind(A) (see introduction), so any invariant is a torsion invariant. (This argument extends to any simply connected semisimple group G by [35, Cor. 2.6] and a transfer argument. On the other hand, Inv^{1}(𝔾_{m}, K_{∗}^{M}) = Z as the construction of [35, beg. of 2.3] shows.) Thanks to Theorem 10.3, the only thing which remains to be proven is that Ã^{0}(G, H^{4}(3)) = A^{0}(G, H^{4}(3))_{mult} (notation as in [35, 1.3]): this follows from Lemma 10.5. □

10.8. Remark. The above proof yields a little more: if eSK_{1}(A_{K}) = 0 for all K/F, then e Inv^{d}(SL_{1}(A), M) = 0 for any cycle module M and any d ∈ Z. In particular, eÃ^{0}(G, H^{4}(3)) = 0. This will be amplified in Lemma 10.13 below.

A delicate issue is the behaviour of c_{A} and c̄_{A} under extension of scalars: in other words, the universal invariant of Theorem 10.7 might cease to be universal after extending the base field. This is directly related to the differential d̃_{2}^{2,3}(G, 3) in Diagram (10.1). Here is at least one case where this does not happen:

10.9. Lemma. Let L/F be an extension such that exp(A_{L}) = exp(A). Then

H̃^{5}(G, Z(3)) ⟶^{∼} H̃^{5}(G_{L}, Z(3))
Ã^{0}(G, H^{5}(Z(3))) ⟶ Ã^{0}(G_{L}, H^{5}(Z(3))).

In particular, the image of c_{A} (resp. c̄_{A}) under extension of scalars equals c_{A_{L}} (resp. c̄_{A_{L}}).

Proof. We shall show in Corollary 11.3 that d̃_{2}^{2,3}(G, 3)(1) is a multiple of [A] ∈ Br(F). The claim then follows from a diagram chase with (10.1). □

The following corollary to Theorem 10.7 is a special case of [60, Prop. 4.1].

10.10. Corollary. Assume A of degree d = l^{n} (l prime) and of exponent ε < d. Let r be such that ε | r | d/l. Then there is an integer m(A, r) such that

σ_{r}^{1} = m(A, r)c̄_{A}

where σ_{r}^{1} is the invariant in (7.3) (see §7.D). □

As in [60, proof of Prop. 4.3], one might learn more on m(A, r) by considering the generic algebra of degree d and exponent ε. We shall content ourselves with

10.11. Proposition (cf. [34]). For ε = 2 < ind(A), Ã^{0}(SL_{1}(A), H^{4}(3)) ≠ 0 and m(A, 2) is odd.

Proof. 1) If A is a biquaternion algebra, the Rost invariant of Theorem 2 is nontrivial [34, proof of Cor.] and, by Remark 10.8 and the remark after Theorem 6 in the introduction, Ã^{0}(SL_{1}(A), H^{4}(3)) is cyclic of order ≤ 2. Hence this group is cyclic of order 2 and the Rost invariant coincides with c̄_{A} (recovering [35, Th. 5.4]). Thus m(A, 2) = 1 in this case.

2) If ind(A) = 4, let D be the division algebra similar to A, so that A = M_{n}(D) for some n ≥ 1. By Morita invariance of algebraic K-theory, the invariant σ_{2}^{1} is the same for A and D. On the other hand, Proposition 10.6 yields an isomorphism

Ã^{0}(SL_{1}(A), H^{4}(3)) ⟶^{∼} Ã^{0}(SL_{1}(D), H^{4}(3))

so 1) extends to this case.

3) In general, let L = F (SB(4, A)), so that ind(A_{L}) = 4. By 2), c̄_{A_{L}} ≠ 0, hence c̄_{A} ≠ 0 by Lemma 10.9. Since σ_{r}^{1} commutes with any extension of scalars by construction, we have m(A, 2) = m(A_{L}, 2) in Z/2, which shows that m(A, 2) is odd. □

We shall show in Corollary 11.12 that actually Ã^{0}(SL_{1}(A), H^{4}(3)) ≃ Z/2 in Proposition 10.11.

10.12. Remark. Let r be a divisor of d = deg(A). Let us write H^{4}(3)/r[A] for the degree 4 part of the cycle module given by

K ↦ H^{n}(K, n − 1)/r[A]

:= Coker(H^{n−2}(K, μ_{r}^{⊗n−2}) ⟶^{·r[A]} H^{n}(K, Q/Z(n − 1))).

It is tempting to conjecture that the map

A^{0}(SL_{1}(A), H^{4}(3))_{mult} → A^{0}(SL_{1}(A), H^{4}(3)/r[A])_{mult}

is surjective, which would provide a relationship between the invariant c_{A} and the invariant σ_{r}^{1} of Corollary 7.2 in general.^{4} However, since A^{0}(−)_{mult} is left exact rather than right exact, this does not look straightforward at all. A description of the kernel of cup-product with r[A] seems a major issue to solve (cf. (3.3)).

4Since this was written, Wouters has resolved this question in the negative, [60, Prop.

4.2].

Documenta Mathematica · Extra Volume Suslin (2010) 317–369

10.B. The K-theoretic generic element. In the universal case X = SL_{1}(A), we may write SK_{1}(SL_{1}(A), A) = SK_{1}(A) ⊕ S̃K_{1}(SL_{1}(A), A) using the unit section of SL_{1}(A). The induced morphism

S̃K_{1}(SL_{1}(A), A) → H̃^{5}(SL_{1}(A), Z(3))

is surjective, hence split surjective since H̃^{5}(SL_{1}(A), Z(3)) = Z. An explicit splitting sends c_{A} to the class of the inclusions ι_{n} : SL_{1}(A) ↪ SL_{n}(A).

10.13. Lemma. a) For any smooth F -variety Y, the map

H^{0}(Y, SK_{1}(O_{Y} ⊗_{F} A)) → SK_{1}(F (Y) ⊗_{F} A)

is surjective; the image of c_{F (Y)⊗_{F} A} is contained in A^{0}(Y, H^{4}(3)).

b) For Y = SL_{1}(A) and K = F (Y), the map c_{A_{K}} induces a surjection

(10.5)  SK_{1}(A_{K})/SK_{1}(A) ⟶ Ã^{0}(SL_{1}(A), H^{4}(3))

sending the generic element to c̄_{A}.

Proof. The first assertion of a) is classical (Rost, cf. [6, p. 38]), and the second one follows from this and the construction of c_{A}. For b), let η = Spec K be the generic point of SL_{1}(A). It defines an element η̄ ∈ SK_{1}(A_{K}): the generic element. By construction, we have

c_{A_{K}} (η̄) = c̄_{A}

from which b) follows. □

We want to better understand the map (10.5). This is possible if A is biquaternion:

10.14. Theorem. If A is a biquaternion algebra, (10.5) is an isomorphism and both sides are isomorphic to Z/2.

Proof. By Lemma 10.9 and Proposition 10.11, we have a commutative diagram of injections

0 ⟶ SK_{1}(A) ⟶^{c̄_{A}} H^{5}(F, Z(3))
a↓  ^{b}↓
0 ⟶ SK_{1}(A_{K}) ⟶^{c̄_{A_{K}}} H^{5}(K, Z(3)).

Since SL_{1}(A) has a rational point, a and b have compatible retractions and this diagram induces a third injection

(10.6)  0 ⟶ SK_{1}(A_{K})/SK_{1}(A) ⟶ H^{5}(K, Z(3))/H^{5}(F, Z(3))

which is obviously also induced by (10.5). This proves the first claim. The second one follows from [35, Th. 5.4] (or part 1) of the proof of Proposition 10.11). □

10.15. Corollary. If ind(A) = 4, then SK_{1}(A_{K})/SK_{1}(A) ≃ Z/2.

Proof. If A is biquaternion, this follows from Theorem 10.14. In general, let L = F (SB(A^{⊗2})). By [35, Prop. 6.3], the maps SK_{1}(A) → SK_{1}(A_{L}) and SK_{1}(A_{K}) → SK_{1}(A_{KL}) are isomorphisms, so we are reduced to the biquaternion case. □

In an earlier version of this paper, I had conjectured that (10.5) is always an isomorphism. In the light of the proof of Theorem 10.14, this seems a bit optimistic unless all primes factors of ind(A) occur at most with exponent 2. In general, a computation of c^{q}(SL_{1}(A)) for all q > 1 will yield higher cohomological invariants for SK_{1}(A). A still optimistic but more reasonable conjecture is that these future invariants will detect all of SK_{1}(A). Based on this expectation, we propose

10.16. Conjecture. If K = F (SL_{1}(A)), the group SK_{1}(A_{K})/SK_{1}(A) is cyclic, generated by the generic element.

10.17. Remark. The homomorphism

c_{A} : Hom(SL_{1}(A), SL_{1}(A)) → H^{5}(SL_{1}(A), Z(3))

also behaves well with respect to composition: for f ∈ Hom(SL_{1}(A), SL_{1}(A)), we have c_{A}(f) ∈ H̃^{5}(SL_{1}(A), Z(3)) if and only if f(1) = 1. If this is the case, set c_{A}(f) = n(f)c_{A}. Then, clearly, n(g ∘ f) = n(g)n(f). Can one describe this “degree” map in a more naïve fashion?

11. Some computations

We now try and evaluate the groups SK_{1}(A_{K})/SK_{1}(A), where K is the function field of SL_{1}(A), and Ã^{0}(SL_{1}(A), H^{4}(3)): our main results in this direction are Theorem 11.9 and Corollaries 11.10 and 11.12, the latter completing the proof of Theorem E. Unfortunately we are not able to prove the nontriviality of either of these groups when ind(A) is odd (not squarefree) by the present methods. We assume that n = deg(A) is of the form l^{m}, l prime.

11.A. Comparing some quotients. First we have already noted:

11.1. Lemma. |Ã^{0}(SL_{1}(A), H^{4}(3))| ≤ ind(A)/l.

Proof. This follows from Lemma 10.13 b) and the fact that SK_{1}(A_{K}) has exponent ≤ ind(A)/l. □

See Corollary 11.12 for a refinement of this lemma when A is of exponent l. Let G = SL_{1}(A). We note the isomorphisms

A^{2}(G, K_{3}^{M}) ⟶^{∼} A^{2}(G, K_{3})
K_{2}(F) ⟶^{∼} A^{0}(G, K_{2}).

The first one is trivial and the second one is [8, Cor. B.3]. By the second one, the BGQ spectral sequence yields an injection

(11.1)  K_{1}(G)^{(2/3)} ⟶ A^{2}(G, K_{3}).

11.2. Proposition. If G is split, with r ≥ 2, the maps H̃^{5}(G, Z(3)) → Z and A^{2}(G, K_{3}^{M}) → H̃^{5}(G, Z(3)) from (10.1) are both bijective. The same is true of the map (11.1).

Proof. Mixing the coniveau spectral sequence for Nisnevich motivic cohomology with the slice spectral sequence (9.1) (also for Nisnevich motivic cohomology) yields a diagram similar to (10.1) and mapping to it:

(11.2)  \begin{aligned}A^{2}(G, K_{3}^{M}) & ⟶^{∼} & H̃_{Zar}^{5}(G, Z(3)) \\ ≀ & ↓ \\ & & Z \end{aligned}

This proves the first two claims of Proposition 11.2 at once. For the last one, we notice that if G is split then all its Chow groups are 0 by Lemma 9.2, hence all differentials leaving from A^{2}(G, K_{3}) in the BGQ spectral sequence vanish. □

Note that the horizontal map in (11.2) is an isomorphism for any G, whether split or not.

11.3. Corollary. In Diagram (10.1) for G = SL_{1}(A), we have

d̃_{2}^{2,3}(G, 3)(1) = t[A]

for some integer t, where [A] is the class of A in Br(F). In particular, (Z : H̃^{5}(G, Z(3))) divides the exponent of [A].

Proof. Let K be the function field of the Severi-Brauer variety of A. Then A splits over K. The first statement now follows from Proposition 11.2 and Amitsur’s theorem [1] that Ker(Br(F) → Br(K)) = ⟨[A]⟩. The second statement is obvious. □

11.4. Corollary. In general,

(Z : A^{2}(G, K_{3}^{M})) = (A^{2}(G_{s}, K_{3}^{M}) : A^{2}(G, K_{3}^{M}))

| (K_{1}(G_{s})^{(2/3)} : K_{1}(G)^{(2/3)}).

Proof. This follows immediately from Proposition 11.2. □

The following diagram is a little more precise and may be helpful to the reader (G = SL_{1}(A)):

(11.3)

\frac{SK_{1}(A_{K})}{SK_{1}(A)}⟶Ã^{0}(G, H^{4}(3)) \frac{K_{1}(G_{s})^{(2/3)}}{K_{1}(G)^{(2/3)}}
≀↑  ^{onto}↓
0  → \frac{H̃^{5}(G, Z(3))}{A^{2}(G, K_{3}^{M})} →Z/A^{2}(G, K_{3}^{M})→Z/tZ→0

where t is as in Corollary 11.3.

11.B. The map Br(F) → H̃^{6}(G, Z(3)). In order to better understand the differential d̃_{2}^{2,3}(G, 3) in the future, we note:

11.5. Proposition. Let G = SL_{1}(A).

a) We have an exact sequence

0 → A^{1}(G, H^{4}(3)) → H̃^{6}(G, Z(3))/CH^{3}(G) → Ã^{0}(G, H^{5}(3)).

b) The composition

Br(F) → H̃^{6}(G, Z(3)) → H̃^{6}(G, Z(3))/CH^{3}(G) → Ã^{0}(G, H^{5}(3))

from Diagram (10.1) is 0, and so is the map H̃^{6}(G, Z(3))/CH^{3}(G) → Ã^{0}(G, H^{5}(3)). Hence we have in fact an exact sequence

0 → CH^{3}(G) → H̃^{6}(G, Z(3)) → A^{1}(G, H^{4}(3)) → 0.

Proof. a) follows from the coniveau spectral sequence for the étale motivic cohomology of G. b) The second vanishing follows from the first, since Br(F) → H̃^{6}(G, Z(3)) is surjective. For the first vanishing, given the definition of the homomorphism Br(F) → H̃^{6}(G, Z(3)), it suffices to show that the map α^{∗}c_{i}(V) → α^{∗}c_{i}(G) induces 0 on homology sheaves for i = 1, 2, 3 if V is a suitable open subset V of G. Let B be a Borel subgroup containing T_{s} ⊂ G_{s}. Consider the big cell Ū_{0} ⊂ G_{s}/B: it is an affine space, hence all its Chow groups are 0. Observe that U_{0} is defined over a finite extension of F, hence it has only a finite number of Galois conjugates: then their intersection Ū is defined over F, and its geometric Chow groups are still 0. Let U be the inverse image of Ū in Y_{s}: then U is defined over F and all its geometric Chow groups are 0. Hence, for all p > 0, the étale complex α^{∗}c_{p}(U) is concentrated in degrees < 0. We now take for V the inverse image of U (viewed as an open subset of Y) in G. As in [14, Prop. 9.3], we have for all N ≥ 0 a spectral sequence

E_{1}^{p,q}(V_{s}) = H^{q}(c_{N−p}(U_{s})) ⊗ Λ^{p}(T_{s}^{∗}) ⇒ H^{p+q}(c_{N}(V_{s}))

which maps to the corresponding spectral sequence E_{r}^{p,q}(G_{s}) for G_{s} (that yields the complexes (9.3)). For N > 0, we have E_{1}^{p,q}(G_{s}) = 0 for q ≠ 0 and E_{1}^{p,q}(V_{s}) = 0 for q = 0, hence all maps H^{i}(c_{N}(V_{s})) → H^{i}(c_{N}(G_{s})) are 0. This completes the proof of b). □

11.C. A Chern class computation. We use Gillet’s convention for higher Chern classes [12].

11.6. Lemma. For a smooth variety X, consider the higher Chern class

c_{3,1} : K_{1}(X) → A^{2}(X, K_{3}).

Then 2d_{2}^{0,−2} = 0 and the diagram

K_{1}(X)^{(2)} ⟶^{c_{3,1}} A^{2}(X, K_{3}) ⟵^{2} A^{2}(X, K_{3})/d_{2}^{0,−2}A^{0}(X, K_{2})
↓  ↑
K_{1}(X)^{(2/3)}  ⟶^{∼}  E_{∞}^{2,−3}

commutes, where d_{2}^{0,−2} and E_{∞}^{2,−3} are relative to the BGQ spectral sequence for X.

Proof. The BGQ spectral sequence for X may be considered as the coniveau spectral sequence for X relative to algebraic K-theory. For a given i ≥ 0, consider the corresponding coniveau spectral sequence ′E_{r}^{p,q} relative to U ↦ H^{∗}(U, K_{i}) (for U running through open subsets of X). By [12, pp. 239– 240], the i-th Chern class C_{i} defines a morphism of spectral sequences E_{r}^{p,q} → ′E_{r}^{p,q} (r ≥ 1) converging to the higher Chern classes c_{i,−p−q} : K_{−p−q}(X) → H^{p+q+i}(X, K_{i}). The group ′E_{1}^{p,q} is 0 for q ≠ −i and ′E_{1}^{p,−i} = ⨁_{x∈X^{(p)}} K_{i−p}(F (x)). Hence ′E_{2}^{p,q} = 0 for q ≠ −i and ′E_{2}^{p,−i} = H^{p}(X, K_{i}) = ′E_{∞}^{p,−i}. By [12, Th. 3.9], the map from E_{1}^{p,−i} to ′E_{1}^{p,−i} induced by C_{i} equals \frac{(−1)^{p}(i−1)!}{(i−p−1)!}c_{i−p,i−p} on each summand K_{i−p}(F (x)). In particular, for i = 3, c_{1,1} is the identity for fields and we get a commutative diagram

E_{2}^{0,−2} ⟶^{d_{2}^{0,−2}}  E_{2}^{2,−3}
↓  ^{2}↓
0 ⟶ ′E_{2}^{2,−3} = E_{2}^{2,−3}

which proves the first claim of the lemma; the second one follows from the morphism of spectral sequences. □

11.D. Some computations, continued. The group A^{1}(G, H^{4}(3)) of Proposition 11.5 is mysterious and would require a further analysis: we shall refrain from starting it in this paper and will concentrate on computing the index (K_{1}(G_{s})^{(2/3)} : K_{1}(G)^{(2/3)}), which can be done in some interesting cases. For this, we may try and look at the map K_{1}(G) → K_{1}(G_{s}) and use the results of Levine [25] and Suslin [48]. In particular, we have an isomorphism [25, Th. 4.3]

K_{1}(G) ≃ K_{1}(F) ⊕ ⨁_{i=1}^{r} K_{0}(A^{⊗i})

where r = rk G = deg A − 1. If G (equivalently A) is split, the summand K_{0}(A^{⊗i}) ≃ Z is generated by the class of Λ^{i}(ρ_{r}), where ρ_{r} is the standard representation of G = SL_{r+1} into GL_{r+1}. While Levine thinks of ρ_{r} as a representation, Suslin thinks of it as the generic matrix and denotes it by α_{r+1}: the two viewpoints are of course equivalent. If we pass to the separable closure, we get a commutative diagram

K_{1}(G)^{(2/3)} ⟶^{γ_{3}} A^{2}(G, K_{3})
↓  ↓
K_{1}(G_{s})^{(2/3)} ⟶^{γ_{3}} A^{2}(G_{s}, K_{3}) ≃ Z.

11.7. Lemma. Suppose G = SL_{n}, with n = r + 1.

a) All [Λ^{i}(ρ_{r})] belong to K_{1}(G)^{(1)} and the image of [Λ^{i}(ρ_{r})] in A^{1}(G, K_{2}) = Z is \binom{n−2}{i−1}.

b) For all i, [Λ^{i}(ρ_{r})] − \binom{n−2}{i−1}[ρ_{r}] ∈ K_{1}(G)^{(2)} and its image in A^{2}(G, K_{3}) = Z is \binom{n−3}{i−2}.

Proof. (It may not be the most direct, but it works.) For the first assertion of a), we need to show that [Λ^{i}(ρ_{r})]_{|F (SL_{n}))} = 0 or, which amounts to the same, that Λ^{i}(α_{n}) is a product of commutators, where α_{n} is the generic matrix with determinant 1. For this, it suffices to see that det Λ^{i}(α_{n}) = 1. But, for any matrix u, det Λ^{i}(u) is a certain power of det(u), hence the claim. For the second assertion of a) and for b), we first do a Chern class computation. Let γ̄_{j} = γ_{j}([ρ_{r}]) = γ_{j}([α_{n}]), where γ_{j} is the j-th gamma operation in K-theory. Note the formula (cf. [48, p. 65])

∑[Λ^{i}(α_{n})]u^{i} = ∑ γ̄_{i}u^{i}(1 + u)^{n−i}.

Also, from [46, 1.3.4 a) p. 277 and Remark p. 297] (see also [45, IV.6]), we find

c_{2,1}(γ̄_{j}) = \begin{cases}0 & for j > 2 \\ −c_{2,1}(α_{n}) & for j = 2 \\ c_{2,1}(α_{n}) & for j = 1 \end{cases}

and

c_{3,1}(γ̄_{j}) = \begin{cases}0 & for j > 3 \\ 2c_{3,1}(α_{n}) & for j = 3 \\ −3c_{3,1}(α_{n}) & for j = 2 \\ c_{3,1}(α_{n}) & for j = 1 \end{cases}

from which we deduce

(11.4) ∑ c_{2,1}([Λ^{i}(α_{n})])u^{i} = c_{2,1}(α_{n})(u(1 + u)^{n−1} − u^{2}(1 + u)^{n−2})

= c_{2,1}(α_{n})u(1 + u)^{n−2} = c_{2,1}(α_{n}) ∑ \binom{n − 2}{i − 1}u^{i} =: c_{2,1}(α_{n})φ(u)

and

∑ c_{3,1}([Λ^{i}(α_{n})])u^{i}

= c_{3,1}(α_{n})(u(1 + u)^{n−1} − 3u^{2}(1 + u)^{n−2} + 2u^{3}(1 + u)^{n−3})
= c_{3,1}(α_{n})u(1 + u)^{n−3}(1 + u)

hence

(11.5) ∑ c_{3,1}([Λ^{i}(α_{n})])u^{i} − c_{3,1}(α_{n})φ(u) = −2c_{3,1}(α_{n})u^{2}(1 + u)^{n−3}

= −2c_{3,1}(α_{n}) ∑ \binom{n − 3}{i − 2}u^{i}.

We now use the fact that, for i ≥ 1, A^{i}(SL_{n}, K_{i+1}) is generated by c_{i+1,1}([α_{n}]) [48, Th. 2.9]. By an analogue of Lemma 11.6, the edge homomorphism K_{1}(X)^{(1)} → A^{1}(X, K_{2}) of the BGQ spectral sequence coincides with −c_{2,1} for any smooth variety X. With (11.4), this proves the second part of a) and the first part of b). Then the second part of b) follows from Lemma 11.6 and (11.5). □

Let G not be necessarily split anymore. Let e_{i} be the positive generator of the summand K_{0}(A^{⊗i}): e_{i} ↦ ind(A^{⊗i})[Λ^{i}(α_{n})]. Lemma 11.7 shows that \frac{ind(A)}{ind(A^{⊗i})}e_{i} − \binom{n−2}{i−1}e_{1} ∈ K_{1}(G)^{(2)} and that its image in A^{2}(G_{s}, K_{3}) = Z is ind(A)\binom{n−3}{i−2}.

11.8. Lemma. v_{l}(\binom{n−2}{i−1}) = v_{l}(i). (Recall that n = l^{m}.)

Proof. For an integer e, let s_{l}(e) be the sum of the digits of e written in base l. It is well-known that

v_{l}\binom{a}{b} = \frac{s_{l}(b) + s_{l}(a − b) − s_{l}(a)}{l − 1}.

Clearly, we have s_{l}(l^{m}−2) = m(l−1)−1. Let t = v_{l}(i) and write i−1 = ∑ a_{j}l^{j}, with 0 ≤ a_{j} ≤ l−1, a_{j} = l−1 for j < t and a_{t} < l−1. Then l^{m} −i−1 = ∑ b_{j}l^{j} with b_{j} = l − 1 for j < t, b_{t} = l − 2 − a_{t} and b_{j} = l − 1 − a_{j} for t < j ≤ m. Hence

s_{l}(i − 1) + s_{l}(l^{m} − i − 1) − s_{l}(l^{m} − 2) =

2t(l − 1) + (m − t)(l − 1) − 1 − (m(l − 1) − 1) = t(l − 1).

□

11.9. Theorem. We have

(K_{1}(G_{s})^{(2/3)} : K_{1}(G)^{(2/3)}) = \begin{cases}inf(l^{2t}ind(A^{⊗l^{t}})) & if l > 2 \\ inf(l^{2t−1}ind(A^{⊗l^{t}})) & if l = 2. \end{cases}

Proof. Since the index (K_{1}(G_{s})^{(2/3)} : K_{1}(G)^{(2/3)}) a priori divides ind(A) (transfer argument), to evaluate it we may tensor both groups with Z_{l}, as well as A^{2}(G, K_{3}) and A^{2}(G_{s}, K_{3}). Note also that, since K_{1}(G)^{(1/2)} ↪ A^{1}(G, K_{2}) ≃ Z is torsion-free, x ∈ K_{1}(G)^{(1)} ⊗ Z_{l} and mx ∈ K_{1}(G)^{(2)} ⊗ Z_{l} for some m ∈ Z_{l} − {0} implies x ∈ K_{1}(G)^{(2)} ⊗ Z_{l}. This will allow us to divide freely by l-units below. By Lemma 11.7, we have

(11.6)  \frac{n}{ind(A^{⊗i})\binom{n−2}{i−1}}e_{i} − e_{1} ↦ n\begin{matrix}( & {n−3 \atop i−2} & ) \\ ( & {n−2 \atop i−1} & ) \end{matrix} = \frac{n(i − 1)}{n − 2}

under the composite map K_{1}(G)^{(2/3)}⊗Z_{l} → K_{1}(G_{s})^{(2/3)}⊗Z_{l} ⟶^{∼} A^{2}(G_{s}, K_{3})⊗

Z_{l} ⟶^{∼} Z_{l} (note that the coefficient of e_{i} is an l-integer by Lemma 11.8). Let x = ∑ λ_{i}e_{i} ∈ K_{1}(G)^{(2)} ⊗ Z_{l} (with λ_{i} ∈ Z_{l}). In Q_{l}, write

λ_{i} = μ_{i} \frac{n}{ind(A^{⊗i})\binom{n−2}{i−1}}

so that

(11.7)  x = ∑ μ_{i} (\frac{n}{ind(A^{⊗i})\binom{n−2}{i−1}}e_{i} − e_{1}) + ∑ μ_{i}e_{1}

hence x ∈ K_{1}(G)^{(2)} ⊗ Z_{l} if and only if ∑ μ_{i} = 0. Note that

x ↦ ∑ μ_{i} \frac{n(i − 1)}{n − 2} = ∑ μ_{i} \frac{n(i − 1)}{n − 2} + \frac{n}{n − 2} ∑ μ_{i} = ∑ iμ_{i} \frac{n}{n − 2}.

Since v_{l}(μ_{i}) ≥ −v_{l} (\frac{n}{ind(A^{⊗i})\binom{n−2}{i−1}}), we have

v_{l}(iμ_{i} \frac{n}{n − 2}) ≥ \begin{cases}2v_{l}(i) + v_{l}(ind(A^{⊗i})) & if l > 2 \\ 2v_{l}(i) + v_{l}(ind(A^{⊗i})) − 1 & if l = 2 \end{cases}

(see Lemma 11.8). This proves the inequality ≥ in Theorem 11.9. To get equality, let s = inf{t | l^{2t}ind(A^{⊗l^{t}}) is minimum}. Suppose first that l > 2. Choose λ_{l^{s}} = 1, μ_{2l^{s}} = −μ_{l^{s}} and λ_{i} = 0 otherwise, and we are done. Suppose now that l = 2. We can then argue as above by taking μ_{3·2^{s}} = −μ_{2^{s}} provided 3 · 2^{s} < n = 2^{m}, i.e. s ≤ m − 2; s = m is clearly impossible and s = m − 1 may occur only when 2^{2m−3}ind(A^{⊗2^{m−1}}) < 2^{m}, i.e. when 2^{m}ind(A^{⊗2^{m−1}}) ≤ 4. This means m = 1 or m = 2, exp(A) = 2. In the first case we clearly have equality. In the second one we may compute directly

2e_{2} − e_{1} ↦ 2
e_{3} − e_{1} ↦ 4

which shows that (K_{1}(G_{s})^{(2/3)} : K_{1}(G)^{(2/3)}) = 2. So equality still holds in this case. □

11.10. Corollary. a) If ind(A) = exp(A), then (K_{1}(G_{s})^{(2/3)} : K_{1}(G)^{(2/3)}) = ind(A).

b) Suppose exp(A) = l. If l > 2 we have

(K_{1}(G_{s})^{(2/3)} : K_{1}(G)^{(2/3)}) = \begin{cases}l & if ind(A) = l \\ l^{2} & if ind(A) > l \end{cases}

while if l = 2 we always have (K_{1}(G_{s})^{(2/3)} : K_{1}(G)^{(2/3)}) = 2.

Proof. a) is obvious, since in this case necessarily ind(A^{⊗l^{t}}) = l^{q−t} for all t ≤ q, if ind(A) = l^{q}. For b), we have (K_{1}(G_{s})^{(2/3)} : K_{1}(G)^{(2/3)}) = inf(ind(A), l^{2}) (for l = 2) or inf(ind(A), 2) (for l = 2) and the result immediately follows. □

11.11. Remark. An easier computation gives (K_{1}(G_{s})^{(1/2)} : K_{1}(G)^{(1/2)}) = lcm(i · ind(A^{⊗i})) = ind(A). Since A^{1}(G, K_{2}) ⟶^{∼} A^{1}(G_{s}, K_{2}) [8, Cor. B.3], this yields (A^{1}(G, K_{2}) : K_{1}(G)^{(1/2)}) = ind(A).

The first part of the following corollary was (embarrassingly) pointed out by Wouters [60, 2.4 (c)]:

11.12. Corollary. If A is of exponent l, then Ã^{0}(SL_{1}(A), H^{4}(3)) is cyclic of order dividing 2 if l = 2 and dividing l^{2} if l > 2. If moreover l = 2 and ind(A) > 2, then Ã^{0}(SL_{1}(A), H^{4}(3)) ≃ Z/2 and the invariants c_{A} of Theorem 10.7 and σ_{2}^{1} of §7.D coincide. In general

|Ã^{0}(SL_{1}(A), H^{4}(3))| ≤ \begin{cases}exp(A)^{2} & if l is odd \\ exp(A)^{2}/2 & if l = 2. \end{cases}

Proof. The first statement follows from Corollary 11.10, Diagram (11.3) and Theorem 10.3. The second one then follows from Proposition 10.11. The last one follows from taking l^{t} = exp(A) in Theorem 11.9. □

11.13. Question. Let l be odd. Is it true that Ã^{0}(SL_{1}(A), H^{4}(3)) ≃ Z/l if A is of exponent l and index > l?

Appendix A. A cancellation theorem over imperfect fields

A.1. Theorem. Let F be a field and M, N ∈ DM_{−}^{eff}(F) where N is a mixed Tate motive (see [14, Def. 4.1]). Then the map − ⊗ Z(1) induces an isomorphism

Hom_{DM}(M, N) ⟶^{∼} Hom_{DM}(M(1), N(1)).

Proof. It is enough to prove this for M = C_{∗}(X)[i], X a smooth variety and i ∈ Z, and N = Z(n), n ≥ 0. By [54, Prop. 3.2.3] and [55], the left hand side is functorially isomorphic to Bloch’s higher Chow group CH^{n}(X, 2n + i). By [30, Th. 15.12] (projective bundle formula in DM), the right hand side is a direct summand of CH^{n+1}(X × P^{1}, 2n + 2 + i). By the projective bundle formula for higher Chow groups ([3, Th. 7.1], [24, Cor. 5.4]), the latter decomposes as a direct sum

CH^{n+1}(X × P^{1}, 2n + 2 + i) ≃ CH^{n+1}(X, 2n + 2 + i) ⊕ CH^{n}(X, 2n + i).

Moreover, the constructions of the projective bundle isomorphisms in [30] and [3, 24] show that the latter two are compatible via the isomorphism between motivic cohomology and higher Chow groups in [55]. This proves the theorem.

□

Theorem A.1 is sufficient to extend to imperfect fields the construction of the slice spectral sequences in the form of (9.1), i.e. for motivic cohomology computed in the Nisnevich topology (= Bloch’s higher Chow groups). It is not sufficient, however, to obtain a version of the étale spectral sequences of (9.2) which is interesting at p, since p is automatically inverted in DM_{−,ét}^{eff}(F) (see Remark 2.6). In order to achieve this, one may presumably proceed by working directly on Bloch’s cycle complexes, as follows: By the work of Geisser-Levine [9], the étale hypercohomology of Bloch’s cycle complexes provides an interesting theory modulo p. The first thing to do is to find a version of the slice filtration directly on the cycle complexes of a given smooth F -variety X: this can be achieved by using the “homotopy coniveau filtration” (which is at the basis of the construction of the Bloch-Lichtenbaum spectral sequence), see [28] and [22, §4]. This will give spectral sequences comparable to those of Theorem 2.5 and (9.2). The issue is then to identify the E_{2}-terms. This can presumably be done by a slightly tedious imitation of the computations in [14] and §9, where the tediousness comes from the fact that one is limited to work with smooth varieties rather than general motives. In the course of the computation, the following ingredients will certainly appear: étale versions of the localisation theorem for higher Chow groups (see e.g. the proof of [14, Prop. 4.11]) and of Bloch’s projective bundle theorem. They should be obtained much as in [16, Th. 4.2 and Th. 5.1]. Hopefully a partial purity statement similar to [16, Th. 4.2] will be sufficient for the applications. We leave this programme to the interested reader.

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Bruno Kahn

Institut de Mathématiques de Jussieu UPMC UFR 929 Mathématiques 4 place Jussieu 75005 Paris France kahn@math.jussieu.fr 370

Documenta Mathematica · Extra Volume Suslin (2010)