MOTIVIC COHOMOLOGY OF THE SIMPLICIAL MOTIVE OF A ROST VARIETY

ALEXANDER MERKURJEV AND ANDREI SUSLIN

1. Motivic cohomology of X_{θ}

Let l be a prime integer, F a field of characteristic different from l and n ≥ 2 an integer. Let θ be a symbol in H_{et}^{n}(F, μ_{l}^{⊗(n−1)}), i.e., θ is the cup-product of n − 1 elements in H_{et}^{1}(F, μ_{l}) = F^{×}/F^{×l} and an element in H_{et}^{1}(F, ℤ/lℤ) = Hom_{cont}(Gal(F_{sep}/F), ℤ/lℤ). Choose a Rost variety X_{θ} for θ (see [19, Def. 1.1] and [2, Def. 05]). Denote further by X_{θ} the C̆ech simplicial scheme Č(X_{θ}) of X_{θ} (see [16, Appendix B]) and by M(X_{θ}) the motive of X_{θ} in the triangulated category DM(F, ℤ) (see [5]). The motive of X_{θ} in DM(F, ℤ_{(l)}) is independent of the choice of the Rost variety X_{θ} ([15, §5]).

In this paper we compute the motivic cohomology H^{p,q}(X_{θ}, ℤ) of X_{θ} (Theorem 1.14). Some applications are given.

We use the following notation: K_{∗}(F) is the Milnor ring of a field F. If X is a variety over F, we write A_{0}(X, K_{p}) for the cokernel of the residue homomorphism (see [11]):

∐_{x∈X_{(1)}} K_{p+1}F (x) → ∐_{x∈X_{(0)}} K_{p}F (x),

where X_{(i)} is the set of all points of X of dimension i. b = (l^{n−1} − 1)/(l − 1) = 1 + l + ⋯ + l^{n−2}, c = (l^{n} − 1)/(l − 1) = 1 + l + ⋯ + l^{n−1} = bl + 1 = b + l^{n−1}, d = l^{n−1} − 1 = b(l − 1) = c − b − 1.

1.1. The Bloch-Kato Conjecture and the motivic cohomology of X_{θ}. The Bloch-Kato Conjecture asserts that the norm residue homomorphism

h_{n,l} : K_{n}(F)/lK_{n}(F) → H_{et}^{n}(F, μ_{l}^{⊗n})

is an isomorphism. This conjecture was proved in [15] (see also [12], [2], [18] and [19]). In view of [14], the natural maps

H^{p,q}(Y, ℤ) → H_{et}^{p,q}(Y, ℤ)

are isomorphisms for a smooth projective variety Y over F and p ≤ q + 1. Moreover, the natural map

(1)  H^{p,q}(X_{θ}, ℤ) → H_{et}^{p,q}(X_{θ}, ℤ)

is an isomorphism if p ≤ q + 1. Note that X_{θ} = ℤ if θ = 0. It follows that in general,

(2)  H_{et}^{p,q}(X_{θ}, ℤ) ≃ H_{et}^{p,q}(F, ℤ).

For every N ∈ DM(F, ℤ) and every α ∈ H^{p,q}(N , ℤ) the order of α is the integer ord(α) = p − q − 1. The subgroup of H^{∗,∗}(N , ℤ) of elements of nonnegative (respectively, non-positive) order will be denoted by H^{∗,∗}(N , ℤ)^{≥0} (respectively, H^{∗,∗}(N , ℤ)^{≤0}).

1.2. The motive X̃_{θ}. The motive X̃_{θ} is defined by the exact triangle

(3)  X̃_{θ} → X_{θ} → ℤ → X̃_{θ}[1]

in DM(F, ℤ). It follows from (1) and (2) that

(4)  H^{p,q}(X_{θ}, ℤ) ≃ H_{et}^{p,q}(X_{θ}, ℤ) ≃ H_{et}^{p,q}(F, ℤ) ≃ H^{p,q}(F, ℤ)

if p ≤ q + 1. As H^{p,q}(F, ℤ) = 0 when p > q, the exact triangle (3) yields:

Proposition 1.1. There are canonical isomorphisms:

H^{∗,∗}(X̃_{θ}, ℤ)^{≥0} ≃ H^{∗,∗}(X_{θ}, ℤ)^{≥0},
H^{∗,∗}(X_{θ}, ℤ)^{≤0} ≃ H^{∗,∗}(F, ℤ)^{≤0},
H^{∗,∗}(X̃_{θ}, ℤ)^{≤0} = 0.

Note that the motive X̃_{θ} and hence the group H^{p,q}(X̃_{θ}, ℤ) vanish if θ = 0. Since in general θ has a degree l splitting field extension, the group H^{p,q}(X̃_{θ}, ℤ) is l-torsion.

Recall that K_{p}(F) = H^{p,p}(F, ℤ) (see [5, §5]). Hence there is the product

(5)  K_{s}(F) ⊗ H^{p,q}(X̃_{θ}, ℤ) → H^{p+s,q+s}(X̃_{θ}, ℤ), x ⊗ α ↦ x · α.

Let K_{p}^{θ}(F) be the cokernel of the norm homomorphism

∐ K_{p}(E) → K_{p}(F),

where the coproduct is taken over all finite extensions E/F such that θ is split over E. The graded group K_{∗}^{θ}(F) has structure of a graded ring. Clearly, K_{p}^{θ}(F) = 0 if θ = 0. It follows from Proposition 1.1 that in general the product (5) yields structure of a left K_{∗}^{θ}(F)-module on H^{∗,∗}(X̃_{θ}, ℤ) and H^{∗,∗}(X_{θ}, ℤ)^{≥0}.

1.3. Integral elements. We say that an element α ∈ H^{p,q}(X̃_{θ}, ℤ/lℤ) is integral if α belongs to the image of the natural homomorphism

H^{p,q}(X̃_{θ}, ℤ) → H^{p,q}(X̃_{θ}, ℤ/lℤ).

Let

B : H^{∗,∗}(X̃_{θ}, ℤ/lℤ) → H^{∗+1,∗}(X̃_{θ}, ℤ/lℤ)

be the Bockstein homomorphism, i.e., the connecting homomorphism for the exact sequence

0 → ℤ/lℤ → ℤ/l^{2}ℤ → ℤ/lℤ → 0.

The following statement is a consequence of the fact that the group H^{p,q}(X̃_{θ}, ℤ) is l-torsion.

Lemma 1.2. Let α ∈ H^{p,q}(X̃_{θ}, ℤ/lℤ). Then the following conditions are equivalent:

(1) α is integral;

(2) B(α) = 0;

(3) α ∈ Im B. □

1.4. The element δ. As X_{θ} is a splitting variety for θ, the symbol θ belongs to the kernel of the natural homomorphism

res : H_{et}^{n}(F, μ_{l}^{⊗(n−1)}) → H_{et}^{n}(F (X_{θ}), μ_{l}^{⊗(n−1)}).

Proposition 1.3. For any m > 0, there is a canonical isomorphism between H^{m+1,m−1}(X_{θ}, ℤ) and the kernel of the natural homomorphism

res : H_{et}^{m}(F, μ_{l}^{⊗(m−1)}) → H_{et}^{m}(F (X_{θ}), μ_{l}^{⊗(m−1)}).

Proof. As H^{m+1,m−1}(X_{θ}, ℤ) = H^{m+1,m−1}(X̃_{θ}, ℤ) is l-torsion, we have an exact sequence

H^{m,m−1}(X_{θ}, ℤ) → H^{m,m−1}(X_{θ}, ℤ/lℤ) → H^{m+1,m−1}(X_{θ}, ℤ) → 0.

It follows from Proposition 1.1 that the first term of the sequence is trivial. Hence the group H^{m+1,m−1}(X_{θ}, ℤ) is canonically isomorphic to H^{m,m−1}(X_{θ}, ℤ/lℤ). By [15, Lemma 6.5], the latter group is canonically isomorphic to the kernel of the homomorphism res. □

Denote by δ ∈ H^{n+1,n−1}(X_{θ}, ℤ) the element corresponding to the symbol θ ∈ Ker(res) when m = n. Clearly, δ ≠ 0 if θ ≠ 0. We have ord(δ) = 1.

1.5. Cohomological operations. Denote by Q_{i}, i = 0, 1, …, n − 1, the Milnor cohomological operations of bidegree (2l^{i} −1, l^{i} −1) on H^{∗,∗}(X̃_{θ}, ℤ/lℤ) and H^{∗,∗}(X_{θ}, ℤ/lℤ) (see [17, §13]). Note that the Q_{i}’s are K_{∗}(F)-linear, that is Q_{i}(α·x) = Q_{i}(α)·x and Q_{i}(x·α) = (−1)^{deg(x)}x·Q_{i}(α) for all α ∈ H^{∗,∗}(X̃_{θ}, ℤ/lℤ) and x ∈ K_{∗}(F). The operations anti-commute: Q_{k}Q_{j} = −Q_{j}Q_{k} for j ≠ k and Q_{i}^{2} = 0 for all i. Moreover, Q_{0} = B. Note that

ord Q_{i}(α) = ord(α) + l^{i}

for all α.

Proposition 1.4. [16, Th. 3.2], [15, Lemma 4.3] For every i = 1, …, n − 1, the sequence

H^{p−2l^{i}+1,q−l^{i}+1}(X̃_{θ}, ℤ/lℤ) ⟶^{Q_{i}} H^{p,q}(X̃_{θ}, ℤ/lℤ) ⟶^{Q_{i}} H^{p+2l^{i}−1,q+l^{i}−1}(X̃_{θ}, ℤ/lℤ)

is exact for all p and q. □

It follows from the equality Q_{i}B = −BQ_{i} for i ≥ 1 and Lemma 1.2 that Q_{i} takes integral elements to the integral ones. The restriction of Q_{i} on the subgroup of integral elements H^{p,q}(X̃_{θ}, ℤ) is still denoted by Q_{i}.

Proposition 1.5. For every i = 1, …, n − 1, the sequence

H^{p−2l^{i}+1,q−l^{i}+1}(X̃_{θ}, ℤ) ⟶^{Q_{i}} H^{p,q}(X̃_{θ}, ℤ) ⟶^{Q_{i}} H^{p+2l^{i}−1,q+l^{i}−1}(X̃_{θ}, ℤ)

is exact for all p and q.

Proof. Suppose that Q_{i}(α) = 0 for some α ∈ H^{p,q}(X̃_{θ}, ℤ). By induction on ord(α) we prove that α = Q_{i}(β) for some integral β.

By Proposition 1.4, α = Q_{i}(β′) for β′ ∈ H^{p−2l^{i}+1,q−l^{i}+1}(X̃_{θ}, ℤ/lℤ). Since α is integral, we have (Q_{i}B)(β′) = −(BQ_{i})(β′) = −B(α) = 0. Since B(β′) is integral, by the induction hypothesis, B(β′) = Q_{i}(γ) for some integral γ. By Lemma 1.2, we have γ = B(γ′) for some γ′ and hence

B(β′ + Q_{i}(γ′)) = B(β′) + (BQ_{i})(γ′) = B(β′) − Q_{i}(γ) = 0.

Therefore, the element β = β′ +Q_{i}(γ′) is integral and Q_{i}(β) = Q_{i}(β′) = α. □

Propositions 1.1 and 1.5 yield:

Corollary 1.6. Let Q_{i}(α) = 0 for some α ∈ H^{p,q}(X_{θ}, ℤ) and i = 1, …, n − 1. Then

(1) If 0 ≤ ord(α) < l^{i}, then α = 0.

(2) If ord(α) ≥ l^{i}, then α = Q_{i}(β) for some β ∈ H^{p−2l^{i}+1,q−l^{i}+1}(X_{θ}, ℤ).

1.6. The elements γ and μ. Set

μ = (Q_{1}Q_{2} … Q_{n−2})(δ) ∈ H^{2b+1,b}(X_{θ}, ℤ)

and

γ = (Q_{1}Q_{2} … Q_{n−2}Q_{n−1})(δ) = ±Q_{n−1}(μ) ∈ H^{2c,c−1}(X_{θ}, ℤ).

We have ord(μ) = b and ord(γ) = c. By Corollary 1.6, μ ≠ 0 and γ ≠ 0 if θ ≠ 0.

We write ∪ for the product in H^{∗,∗}(X_{θ}, ℤ).

Lemma 1.7. We have Q_{i}(γ) = 0 for any i = 1, …, n − 1 and Q_{i}(x ∪ γ) = Q_{i}(x) ∪ γ for every x ∈ H^{p,q}(X_{θ}, ℤ).

Proof. The first equality follows from Q_{i}^{2} = 0. If l is odd, by the product formula, for every homogeneous x of degree d, we have

Q_{i}(x ∪ γ) = Q_{i}(x) ∪ γ + (−1)^{d}x ∪ Q_{i}(γ) = Q_{i}(x) ∪ γ.

In the case l = 2, the product formula looks as follows [17, Prop. 13.4]:

Q_{i}(x ∪ γ) = x ∪ Q_{i}(γ) + Q_{i}(x) ∪ γ + ∑_{E,E′} u_{E,E′}Q_{E}(x) ∪ Q_{E′}(γ)

for some u_{E,E′} ∈ H^{∗,∗}(F, ℤ), where Q_{E} = Q_{1}^{e_{1}}Q_{2}^{e_{2}} … and the sum is taken over all pairs of nonzero sequences E = (e_{1}, e_{2}, …) and E′ of length less than i. Note that Q_{j}(γ) = 0 for all j = 1, …, n − 1, hence Q_{E′}(γ) = 0. □

The element μ gives rise to a morphism M(X_{θ}) → M(X_{θ})(b)[2b + 1] in DM(F, ℤ), still denoted by μ (see [15, 5.3]). Let M_{θ} be the motive in DM(F, ℤ) defined by the exact triangle

M(X_{θ})(c)[2b] → M_{θ} → M(X_{θ}) ⟶^{μ} M(X_{θ})(b)[2b + 1].

For every i = 0, 1, …, l − 1, let S^{i}M_{θ} be the i-th symmetric power of M_{θ} in DM(F, ℤ_{(l)}) (see [15, §3]). The highest symmetric power R_{θ} := S^{l−1}M_{θ} is called the Rost motive of θ.

There are exact triangles [15, (5.5) and (5.6)] in DM(F, ℤ_{(l)}):

R_{θ} → S^{l−2}M_{θ} → M(X_{θ})(d)[2d + 1] → R_{θ}[1],
S^{l−2}M_{θ}(b)[2b] → R_{θ} → M(X_{θ}) → S^{l−2}M_{θ}(b)[2b + 1].

For every p and q we then have exact sequences

H^{p+2d,q+d}(R_{θ}, ℤ_{(l)}) → H^{p,q}(X_{θ}, ℤ_{(l)}) ⟶^{∂_{1}}

H^{p+2d+1,q+d}(S^{l−2}M_{θ}, ℤ_{(l)}) → H^{p+2d+1,q+d}(R_{θ}, ℤ_{(l)})

and

H^{p+2c−1,q+c−1}(R_{θ}, ℤ_{(l)}) → H^{p+2d+1,q+d}(S^{l−2}M_{θ}, ℤ_{(l)}) ⟶^{∂_{2}}

H^{p+2c,q+c−1}(X_{θ}, ℤ_{(l)}) → H^{p+2c,q+c−1}(R_{θ}, ℤ_{(l)}).

By [15, Lemma 5.15], the motive R_{θ} is a direct summand of M(X_{θ}) in DM(F, ℤ_{(l)}). It follows that H^{p,q}(R_{θ}, ℤ_{(l)}) = 0 if p − q > d = dim X_{θ}. Therefore, both maps in the composition

(6) ∂ : H^{p,q}(X_{θ}, ℤ_{(l)}) ⟶^{∂_{1}} H^{p+2d+1,q+d}(S^{l−2}M_{θ}, ℤ_{(l)}) ⟶^{∂_{2}} H^{p+2c,q+c−1}(X_{θ}, ℤ_{(l)})

are isomorphisms if p > q and the sequence

(7) H^{p+2d,p+d}(R_{θ}, ℤ_{(l)}) → H^{p,p}(X_{θ}, ℤ_{(l)}) ⟶^{∂} H^{p+2c,p+c−1}(X_{θ}, ℤ_{(l)}) → 0

is exact for every p.

Proposition 1.8. The homomorphism ∂ in (7) induces an isomorphism

K_{p}^{θ}(F) →^{∼} H^{p+2c,p+c−1}(X_{θ}, ℤ)

for every p.

Proof. Since both groups are l-torsion, it is sufficient to establish the isomorphism over ℤ_{(l)}. By (4), we have H^{p,p}(X_{θ}, ℤ_{(l)}) = H^{p,p}(F, ℤ_{(l)}) = K_{p}(F)_{(l)}. Let L/F be a finite splitting field extension of θ. The commutativity of the diagram

H^{p,p}(X_{θL}, ℤ_{(l)}) ⟶^{∂_{L}} H^{p+2c,p+c−1}(X_{θL}, ℤ_{(l)})
N_{L/F} ↓  ↓^{N_{L/F}}
H^{p,p}(X_{θ}, ℤ_{(l)}) ⟶^{∂} H^{p+2c,p+c−1}(X_{θ}, ℤ_{(l)})

and the triviality of the top right corner imply that ∂ factors through a surjective homomorphism K_{p}^{θ}(F)_{(l)} → H^{p+2c,p+c−1}(X_{θ}, ℤ_{(l)}). It follows from the proof of [15, Lemma 5.15] that the first homomorphism in (7) factors through the push-forward homomorphism

H^{p+2d,p+d}(X_{θ}, ℤ_{(l)}) → H^{p,p}(F, ℤ_{(l)}) = H^{p,p}(X_{θ}, ℤ_{(l)})

with respect to the structure morphism X_{θ} → Spec F. By [16, Lemma 4.11], the group H^{p+2d,p+d}(X_{θ}, ℤ) is canonically isomorphic to A_{0}(X_{θ}, K_{p}). Hence H^{p+2d,p+d}(X_{θ}, ℤ_{(l)}) is generated by the norms for the field extensions F (x)/F over all closed points x ∈ X. Since the field F (x) splits θ, we see from the exactness of (7) that the homomorphism K_{p}^{θ}(F)_{(l)} → H^{p+2c,p+c−1}(X_{θ}, ℤ_{(l)}) is an isomorphism. □

Suppose that θ ≠ 0. We have K_{0}^{θ}(F) = ℤ/lℤ and therefore by Proposition 1.8, H^{2c,c−1}(X_{θ}, ℤ) ≃ ℤ/lℤ. On the other hand, γ is a nonzero element of this group, hence H^{2c,c−1}(X_{θ}, ℤ) = (ℤ/lℤ)γ.

Note that the motive M_{θ} and its symmetric powers are motives over X_{θ} (see [15]). Moreover, the morphisms in the exact triangles involving these motives are over X_{θ}. In particular, the homomorphism ∂ is H^{∗,∗}(X_{θ}, ℤ)-linear. Therefore, ∂ is given by multiplication by the canonical generator of H^{2c,c−1}(X_{θ}, ℤ) that is a multiple of γ. Note that γ is central in H^{∗,∗}(X_{θ}, ℤ). Then Proposition 1.8 and (6) yield:

Proposition 1.9. (1) The map K_{p}^{θ}(F) → H^{p+2c,p+c−1}(X_{θ}, ℤ) given by multiplication by γ, is an isomorphism for any p.

(2) The map H^{p,q}(X_{θ}, ℤ) → H^{p+2c,q+c−1}(X_{θ}, ℤ), given by multiplication by γ, is an isomorphism if p > q. □

Proposition 1.10. The map K_{p}^{θ}(F) → H^{p+n+1,p+n−1}(X_{θ}, ℤ), x ↦ x · δ, is an isomorphism for all p.

Proof. The composition

K_{p}^{θ}(F) ⟶^{·δ} H^{p+n+1,p+n−1}(X_{θ}, ℤ) ⟶^{Q_{1}...Q_{n−1}} H^{p+2c,p+c−1}(X_{θ}, ℤ)

coincides with the multiplication by γ and therefore is an isomorphism by Proposition 1.9(1). The second map is injective by Corollary 1.6. □

Lemma 1.11. If α ∈ H^{p,q}(X_{θ}, ℤ) such that b < ord(α) ≤ c, then Q_{n−1}(α) = 0.

Proof. Since ord Q_{n−1}(α) > b + l^{n−1} = c, we have Q_{n−1}(α) = β ∪ γ for some β ∈ H^{∗,∗}(X_{θ}, ℤ)^{≥0} by Proposition 1.9. Hence

Q_{n−1}(β) ∪ γ = Q_{n−1}(β ∪ γ) = Q_{n−1}^{2}(α) = 0,

therefore Q_{n−1}(β) = 0 again by Proposition 1.9. Since

ord(β) ≤ c + l^{n−1} − (c + 1) < l^{n−1},

we have β = 0 by Corollary 1.6 and therefore Q_{n−1}(α) = 0. □

Lemma 1.12. If α ∈ H^{p,q}(X_{θ}, ℤ) with 2l^{i} ≤ ord(α) < l^{i+1} for some i = 0, 1, …, n − 2, then α = 0.

Proof. Consider the element α′ = (Q_{n−2} … Q_{i+1})(α). Since

ord(α′) = ord(α) + l^{i+1} + ⋯ + l^{n−2},

we have b < ord(α′) < l^{n−1} < c. By Lemma 1.11, Q_{n−1}(α′) = 0.

Using Corollary 1.6, by descending induction on j = n − 2, …, i, we deduce that (Q_{j} … Q_{i+1})(α) = 0 since ord(Q_{j} … Q_{i+1})(α) < l^{j+1}. Therefore, α = 0. □

Lemma 1.13. If α ∈ H^{p,q}(X_{θ}, ℤ) with ord(α) ≤ c and l^{i} ≤ ord(α) < 2l^{i} for some i = 1, 2, …, n − 1, then α = Q_{i}(β) for some β ∈ H^{∗,∗}(X_{θ}, ℤ)^{≥0}.

Proof. By Corollary 1.6, it is sufficient to show that Q_{i}(α) = 0.

Suppose first that i = n − 1. Since b < l^{n−1} ≤ ord(α) ≤ c, it follows from Lemma 1.11 that Q_{n−1}(α) = 0. In the rest of the proof we assume that i ≤ n − 2.

Case 1: l ≥ 3. We have

2l^{i} ≤ ord Q_{i}(α) < 3l^{i} ≤ l^{i+1}.

By Lemma 1.12, Q_{i}(α) = 0.

Case 2: l = 2. We prove that Q_{i}(α) = 0 by descending induction on i. Since 2^{i+1} ≤ ord Q_{i}(α) < 2^{i} + 2^{i+1}, by the induction hypothesis, (Q_{i+1}Q_{i})(α) = 0. By Proposition 1.5, Q_{i}(α) = Q_{i+1}(ρ) for some ρ ∈ H^{∗,∗}(X_{θ}, ℤ)^{≥0}. Since (Q_{i+1}Q_{i})(ρ) = −(Q_{i}Q_{i+1})(ρ) = −Q_{i}^{2}(α) = 0 and ord(ρ) < 2^{i}, ord Q_{i}(ρ) < 2^{i+1}, it follows from Corollary 1.6 that ρ = 0 and therefore Q_{i}(α) = 0. □

1.7. Main theorem. Consider the exterior algebra

Λ = K_{∗}^{θ}(F)[t][λ_{1}, …, λ_{n−1}]

over the polynomial algebra K_{∗}^{θ}(F)[t], i.e., λ_{i}^{2} = 0 and λ_{i}λ_{j} = −λ_{i}λ_{i} for i ≠ j. We view H^{∗,∗}(X_{θ}, ℤ) as a left Λ-module with t acting by multiplication by γ and λ_{i} acting via the operation Q_{i}. The ring Λ is graded over K_{∗}^{θ}(F) as follows: deg(t) = c + 1, deg(λ_{i}) = l^{i}. For any homogeneous element λ ∈ Λ and any α ∈ H^{p,q}(X_{θ}, ℤ), we have

ord(λα) = deg(λ) + ord(α).

Note that distinct monomials in Λ of the form t^{k}λ_{1}^{a_{1}}λ_{2}^{a_{2}} … λ_{n−1}^{a_{n−1}} with a_{i} = 0 or 1 have different degree.

Theorem 1.14. Let l be a prime integer, F a field of characteristic different from l and θ ∈ H_{et}^{n}(F, μ_{l}^{⊗(n−1)}) a nontrivial symbol. Then

(1) H^{∗,∗}(X_{θ}, ℤ)^{≤0} is canonically isomorphic to H^{∗,∗}(F, ℤ)^{≤0}.

(2) H^{∗,∗}(X_{θ}, ℤ)^{≥0} is a free left K_{∗}^{θ}(F)[t][Q_{1}, … Q_{n−1}]-module with basis {δ}.

Proof. (1) follows from Proposition 1.1.

(2): We shall prove that the map

Λ → H^{∗,∗}(X_{θ}, ℤ)^{≥0}, λ ↦ λδ

is an isomorphism.

Injectivity: In view of the remark preceding the theorem, it suffices to prove that if γδ = 0 for a monomial γ = xt^{k}γ_{1}^{a_{1}}γ_{2}^{a_{2}} … γ_{n−1}^{a_{n−1}} with x ∈ K_{∗}^{θ}(F) and the integers k ≥ 0 and a_{i} such that a_{i} = 0 or 1, i.e.,

γ^{k} ∪ (Q_{1}^{a_{1}}Q_{2}^{a_{2}} … Q_{n−1}^{a_{n−1}})(x · δ) = 0,

then x = 0. By Proposition 1.9,

(Q_{1}^{a_{1}}Q_{2}^{a_{2}} … Q_{n−1}^{a_{n−1}})(x · δ) = 0.

It follows from Corollary 1.6 by descending induction on i ≥ 0 that

(Q_{1}^{a_{1}}Q_{2}^{a_{2}} … Q_{i}^{a_{i}})(x · δ) = 0.

Therefore, x · δ = 0 and hence x = 0 by Proposition 1.10.

Surjectivity: Let α ∈ H^{p,q}(X_{θ}, ℤ) be an element with ord(α) ≥ 0. By induction on ord(α) we show that α = λδ for some λ ∈ Λ. If ord(α) = 0, then α = 0 by Proposition 1.1. The case ord(α) = 1 is covered by Proposition 1.10. If ord(α) ≥ c + 1, then by Proposition 1.9, α = γ ∪ β for β with 0 ≤ ord(β) < ord(α). By induction, β = λδ for some λ ∈ Λ and hence α = (tλ)δ.

Suppose 2 ≤ ord(α) ≤ c. Choose an i = 1, …, n−1 such that l^{i} ≤ ord(α) < l^{i+1}. If ord(α) ≥ 2l^{i} (and hence i ≤ n − 2), then by Lemma 1.12, α = 0. So we may assume that l^{i} ≤ ord(α) < 2l^{i} (and hence i ≥ 1). It follows from Lemma 1.13 that α = Q_{i}(β) for some β with 0 ≤ ord(β) < ord(α). By induction, β = λδ for some λ ∈ Λ and hence α = (λλ_{i})δ. □

Remark 1.15. When l = 2 we have γ = μ^{2}. Moreover if s is the operation of the multiplication by μ, the left K_{∗}^{θ}(F)[s][λ_{1}, … λ_{n−2}]-module H^{∗,∗}(X_{θ}, ℤ)^{≥0} is free with basis {δ}. In this form the statement has appeared in [8].

Remark 1.16. By Theorem 1.14, a nontrivial element α ∈ H^{p,q}(X_{θ}, ℤ) of positive order can be uniquely written in the form

α = γ^{k} ∪ (Q_{1}^{a_{1}}Q_{2}^{a_{2}} … Q_{n−1}^{a_{n−1}})(x · δ)

where x ∈ K_{∗}^{θ}(F) and k, a_{i} are integers such that k ≥ 0 and a_{i} = 0 or 1. Moreover,

ord(α) = 1 + k(c + 1) + ∑_{i=1}^{n−1} l^{a_{i}}.

1.8. The multiplicative structure. The ring structure of H^{∗,∗}(X_{θ}, ℤ)^{≥0} in the case l = 2 has been determined in [8].

Let l be an odd prime. The operation S of the form ±Q_{i_{1}}Q_{i_{2}} … Q_{i_{k}}, where k ≥ 0 and i_{1} < ⋯ < i_{k}, is called a monomial. For a monomial S there exists a unique monomial S′ such that S′S = Q_{1}Q_{2} … Q_{n−1}. We shall compute the cup-product T (δ) ∪ S(δ) for two monomials T and S.

Proposition 1.17. Let S and T be monomials. Then

(1) If S′T′ = 0, then T (δ) ∪ S(δ) = 0.

(2) If S′T′ ≠ 0, i.e., S′T′ is a monomial, let U be (a unique) monomial such that U′ = S′T′. Then T (δ) ∪ S(δ) = U(δ) ∪ γ.

Proof. (1): By assumption, the monomials T and S do not contain Q_{i} for some i. Therefore, the i-th digit of ord T (δ) + ord S(δ) written in base l is equal to 0. By Theorem 1.14, the product T (δ) ∪ S(δ), if not zero, is a K_{∗}^{θ}(F)- multiple of either U(δ) ∪ μ or U(δ) for some monomial U. In the first case ord T (δ) + ord S(δ) = ord U(δ) + b, and this case is impossible since all digits of the right hand side written in base l are nonzero.

In the second case ord T (δ) + ord S(δ) = ord U(δ) − 1 and this case does not occur since ord V (δ) ≡ 1 modulo l for every monomial V. Therefore T (δ) ∪ S(δ) = 0.

(2): If T′ = 1, then U = S, T (δ) = γ and the equality follows.

Assume that T′ ≠ 1. By assumption T′S = 0. Therefore,

S′T′(T (δ) ∪ S(δ)) =S′(T′T (δ) ∪ S(δ) ± T (δ) ∪ T′S(δ))
= S′(γ ∪ S(δ))
= γ ∪ S′S(δ)
= γ^{2},

S′T′(U(δ) ∪ γ) =U′(U(δ) ∪ γ)
= U′U(δ) ∪ γ
= γ^{2}.

By Theorem 1.14, the restriction of every operation Q_{i} on a homogeneous component H^{p,q}(X_{θ}, ℤ) of non-negative order is either injective or zero. Hence T (δ) ∪ S(δ) = U(δ) ∪ γ. □

2. Applications

Let l be a prime integer, F a field of characteristic different from l, θ ∈ H_{et}^{n}(F, μ_{l}^{⊗(n−1)}) a symbol, X_{θ} a Rost variety for θ and R_{θ} the Rost motive of θ.

2.1. An exact sequence.

Theorem 2.1. The sequence

∐ H_{et}^{p}(F (x), μ_{l}^{⊗(p−1)}) \underrightarrow{∑ N_{F (x)/F}} H_{et}^{p}(F, μ_{l}^{⊗(p−1)}) ⟶^{∪θ}

H_{et}^{p+n}(F, μ_{l}^{⊗(p+n−1)}) ⟶^{res_{F (X_{θ})/F}} H_{et}^{p+n}(F (X_{θ}), μ_{l}^{⊗(p+n−1)}),

where the coproduct is taken over all closed point of X_{θ}, is exact.

Proof. By Proposition 1.3, the kernel of the last homomorphism in the sequence is canonically isomorphic to H^{p+n+1,p+n−1}(X_{θ}, ℤ). The statement follows now from Proposition 1.10, the definition of K_{p}^{θ}(F) and the bijectivity of the norm residue homomorphism. □

Remark 2.2. The image of the first homomorphism in the exact sequence in the theorem and the kernel of the last one are independent on the choice of the Rost variety X_{θ}. The image of the first homomorphism is generated by the norms for all finite splitting field extensions of θ. The kernel of res_{F (X_{θ})/F} is the the intersection of Ker res_{L/F} over all splitting fields L of θ.

2.2. Certain motivic cohomology of the Rost motive R_{θ}. Extending to the left the exact sequence (7), we get an exact sequence

H^{p+2c−1,p+c−1}(X_{θ}, ℤ_{(l)}) → H^{p+2d,p+d}(R_{θ}, ℤ_{(l)}) → H^{p,p}(X_{θ}, ℤ_{(l)}) = K_{p}(F)_{(l)}.

By Theorem 1.14 (see also Remark 1.16), the first group in the sequence is trivial. We have proved:

Proposition 2.3. The natural homomorphism

H^{p+2d,p+d}(R_{θ}, ℤ_{(l)}) → K_{p}(F)_{(l)}

is injective.

2.3. Injectivity of the reduced norm map for a central simple algebra. Let D be a central simple algebra of degree m over F. An old theorem of Wang (see [1]) asserts that the classical reduced norm homomorphism

Nrd_{D} : K_{1}(D) → K_{1}(F)

is injective provided m is a square-free integer.

The reduced norm homomorphism Nrd_{D} for the K_{2}-groups has been defined in [13, §26]. It was proven in [10] and [6] that Nrd_{D} is injective in the case m = 2. In the following statement we generalize this result to a K_{2}-analog of Wang’s theorem.

Theorem 2.4. Let D be a central simple algebra of square-free degree over F. Then the reduced norm homomorphism

Nrd_{D} : K_{2}(D) → K_{2}(F)

is injective.

Lemma 2.5. Let B and C be two central simple algebras over F of relatively prime degree. Suppose the reduced norm homomorphisms for B and C are injective over any field extension of F. Then Nrd_{B⊗C} is also injective.

Proof. Let L/F be a splitting field of C of degree r dividing deg C. By assumption, the bottom homomorphism of the diagram

K_{2}(B ⊗ C) ⟶^{Nrd} K_{2}(F)
↓  ↓
K_{2}(B ⊗ L)  K_{2}(B ⊗ C ⊗ L) ⟶^{Nrd} K_{2}(L)

is injective. We deduce that the kernel K of Nrd_{B⊗C} is r-torsion. Similarly we show that K is r-torsion for some integer s relatively prime to r, hence K = 0. □

Since the algebra D is a tensor product of algebras of prime degree, Lemma 2.5 allows us to assume that D is a division algebra of a prime degree l.

Let L/F be a finite extension splitting the algebra D. Then [L : F] = lk for some integer k and there is an embedding of F -algebras L ↪ M_{k}(D) (see [3]). The induced homomorphism

K_{2}(L) → K_{2}(M_{k}(D)) = K_{2}(D)

does not depend on the choice of the embedding.

Let X be the Severi-Brauer variety of right ideals of dimension l in D. As for every closed point x ∈ X the residue field F (x) splits D, we have a canonical homomorphism K_{2}F (x) → K_{2}(D).

Lemma 2.6. There is a homomorphism h : A_{0}(X, K_{2}) → K_{2}(D) satisfying the following properties:

(1) The composition Nrd_{D} ∘h is the norm map A_{0}(X, K_{2}) → K_{2}(F).

(2) For every closed point x ∈ X, the composition of h with the natural homomorphism K_{2}F (x) → A_{0}(X, K_{2}) coincides with the canonical map K_{2}F (x) → K_{2}(D).

Proof. Let J be the canonical vector bundle over X. There is a natural right action of the opposite algebra D^{−1} on J. For every i ≥ 0, the functor M^{⊗i} ↦ J^{i} ⊗_{D^{⊗−i}} M from the category of left finite D^{⊗−i}-modules to the category of vector bundles over X induces a homomorphism

K_{2}(D^{⊗−i}) → K_{2}(X).

By Quillen’s theorem [9, §8, Th. 4.1], the map

∐_{i=0}^{l−1} K_{2}(D^{⊗−i}) → K_{2}(X)

is an isomorphism. We define the map h as the composition

h : A_{0}(X, K_{2}) → K_{2}(X) → K_{2}(D),

where the first map is the edge homomorphism of the Gersten-Quillen spectral sequence [9, §7, Th. 5.4] and the second one is projection on the (l − 1)-th component of the left hand side in the Quillen’s isomorphism. Clearly, h has all reasonable functorial properties, in particular, it commutes with the norm maps for finite field extensions.

Note that the group A_{0}(X, K_{2}) is generated by the norms for finite field extensions that split D. Thus, to prove the first property of h we may assume that D is split. In this case A_{0}(X, K_{2}) ≃ K_{2}(F) canonically and K_{2}(D^{−i}) can be identified with K_{2}(F) via the reduced norm homomorphism. The image of an element α ∈ K_{2}(F) in K_{2}(X) is equal to α · [pt], where [pt] is the class of a rational point in K_{0}(X). Note that the Quillen’s isomorphism takes ∑ a_{i} to ∑ a_{i}η^{i} ∈ K_{∗}(X), where η is the class of the canonical line bundle (with the sheaf of sections O(1)). Since [pt] = (η − 1)^{l−1} = η^{l−1} + …, the element α · [pt] projects to α ∈ K_{2}(F) = K_{2}(D), that proves the first property of h.

To prove the second property let x ∈ X be a closed point of degree l and let L = F (x). Choose a rational point x′ ∈ X_{L} over x. For every α ∈ K_{2}(L) the classes αx and αx′ in the groups A_{0}(X_{L}, K_{2}) and A_{0}(X, K_{2}) respectively satisfy N_{L/F} (αx′) = αx, where N_{L/F} is the left vertical norm homomorphism in the commutative diagram

A_{0}(X ⊗ L, K_{2}) ⟶^{h_{L}} K_{2}(D ⊗ L)
N_{L/F} ↓  ↓^{N_{L/F}}
A_{0}(X, K_{2}) ⟶^{h} K_{2}(D).

Since Nrd_{A_{L}}(h_{L}(αx)) = α, it suffices to prove that the right vertical norm homomorphism coincides with the composition

K_{2}(D ⊗ L) ⟶^{Nrd} K_{2}(L) → K_{2}(D).

This follows from commutativity of the diagram

K_{2}(D ⊗ L) ⟶^{∼} K_{2}M_{l}(L) ⟶ K_{2}M_{l}(D)
Nrd↓  ↓^{≀}  ↓^{≀}
K_{2}(L)  K_{2}(L) ⟶ K_{2}(D). □

Now we can finish the proof of Theorem 2.4. The Rost motive R_{θ} coincides with M(X) in DM(F, ℤ_{(l)}). By Proposition 2.3 (see also the proof of Proposition 1.8), the norm homomorphism

N : A_{0}(X, K_{2}) ⟶^{h} K_{2}(D) ⟶^{Nrd_{D}} K_{2}(F)

is injective after tensoring with Z_{(l)}. It follows from [7, Th. 5.2] that the group K_{2}(D) is generated by the images of natural homomorphisms K_{2}F (x) → K_{2}(D) over all closed points x ∈ X. It follows from Lemma 2.6 that h is surjective. As the kernel of Nrd_{D} is l-torsion, we deduce that Nrd_{D} is injective.

□ Remark 2.7. Theorem 2.4 was proven independently by Kahn and Levine in [4].

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Department of Mathematics, University of California, Los Angeles, CA 90095-1555, USA

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

Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208-2370, USA

E-mail address: suslin@math.northwestern.edu