The Milnor Conjecture.

V. Voevodsky^{1}

December 1996

Contents

1 Introduction. 2

2 Motivic cohomology in the etale topology and splitting varieties. 5 2.1 Beilinson and Lichtenbaum motivic cohomology groups. . . . . 5 2.2 Some computations with Galois cohomology . . . . . . . . . . . 14 2.3 Splitting varieties. . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Motivic cohomology and algebraic cobordisms. 21 3.1 A topological lemma. . . . . . . . . . . . . . . . . . . . . . . . 21 3.2 Homotopy categories of algebraic varieties . . . . . . . . . . . 25 3.3 Eilenberg-MacLane spectra and motivic cohomology. . . . . . . 30 3.4 Topological realization functor. . . . . . . . . . . . . . . . . . . 33 3.5 Algebraic cobordisms. . . . . . . . . . . . . . . . . . . . . . . . 34 3.6 Main theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4 Pfister quadrics and their motives. 39

5 Applications. 46

1Partly supported by NSF

1 Introduction.

The goal of this paper is to prove the following conjecture:

Conjecture 1.1 (Milnor) Let k be a field of characteristic not equal to 2. Then the norm residue homomorphisms K_{n}^{M}(k)/2 → H_{et}^{n}(k, Z/2) are isomorphisms for all n ≥ 0.

As an application we deduce a number of results about motivic cohomology, etale cohomology and algebraic K-theory with Z/2^{m}-coefficients. Conjecture 1.1 was formulated by as a question in [13] and was generalized to the case of Z/l-coefficients for l > 2 in [2, p.118]. It was further generalized and reinterpreted in terms of a comparison conjecture for motivic cohomology in Zariski and etale topologies by A. Beilinson and S. Lichtenbaum ([7]). The following particular cases of this conjecture were known:

1. for n = 0, 1, l-arbitrary it is a corollary of the classical Theorem Hilbert 90

2. for n = 2, l = 2 it was proven by A. Merkurjev in [8]

3. for n = 2 and l arbitrary it was proven by A. Merkurjev and A. Suslin in [9]

4. for n = 3 and l = 2 it was proven by A. Merkurjev and A. Suslin in [10] and, in a different way, by M.Rost in [17]

5. for n = 4 and l = 2 it was announced by M. Rost (unpublished)

The paper is divided into four parts. In the first one we recall the definition of motivic complexes given in [22, p.2] and reduce Conjecture 1.1 for fields of characteristic zero to the existence of algebraic varieties X\underline{a} with certain properties (Theorem 2.25). Except for the extensive use of motivic cohomology and results of [22] the techniques of this part of the paper are just reformulations of the techniques developed by A. Merkurjev and A. Suslin in [10] for the proof of Conjecture 1.1 in dimension 3. Our main innovation here is the use of simplicial schemes Č(X) (see below) in the formulation of Theorem 2.25.

The goal of the second part of the paper is to prove Theorem 3.25. We recall first the construction of the homotopy category of algebraic varieties given in [14] and describe motivic cohomology as the theory represented by an explicitly given Eilenberg-Maclane spectrum. We also define in this section the motivic Steenrod algebra and the algebraic cobordisms. For a variety X over k let Č(X) be the simplicial scheme with terms Č(X)_{n} = X^{n+1} and faces and degeneracy morphisms given by partial projections and diagonals respectively. Assume that our base field k is embedded into the field C of complex numbers. If X is a smooth proper variety over k we denote by s(X) the characteristic number of the complex variety X(C) given by

s(X) = −dim(X)!ch(T_{X})[X]

where ch(T_{X}) is the Chern character of the tangent bundle of X(C) and ch(T_{X})[X] is the value of the highest degree component of ch(T_{X}) on the fundamental class of X. We say that X is a (ν_{n}, l)-variety if dim(X) = l^{n}−1, s(X) ≠ 0(mod l^{2}) and for any integral chracteristic number c of X one has c = 0(mod l). It turns out that if X is a (ν_{n}, l)-variety for some n and l then the motivic cohomology groups of Č(X) with Z/l-coefficients considered as a module over the motivic Steenrod algebra has some very special properties.

Both the first and the second part of the paper deal with motivic cohomology of varieties over fields of characteristic zero with coefficients in Z/l where l is an arbitrary prime.

In the third part we specialize to the case l = 2. It is a well known fact that for any sequence (a_{1}, …, a_{n}) of invertible elements of a field k of characteristic not equal to 2 the quadric Q\underline{a} given by the equation

<< a_{1}, …, a_{n−1} >>= a_{n}t^{2}

satisfies the first condition of Theorem 2.25. One verifies easily that Q\underline{a} is a (ν_{n−1}, 2)-variety which allows us to use Theorem 3.25 to replace the second condition of Theorem 2.25 by the vanishing of the motivic cohomology group H^{2^{n}−1}(Č(Q\underline{a}), Z_{(2)}(2^{n−1})). We prove that this group is zero using two theorems of Markus Rost ([18],[19]) which finishes the proof of Conjecture 1.1 for fields of characteristic zero.

In the fourth part we deduce Conjecture 1.1 for all fields of characteristic not equal to 2 and prove a number of corollaries. Using similar techniques one can also prove the second part of Milnor Conjecture which asserts that the Milnor ring modulo 2 is isomorphic to the graded Witt ring of quadratic forms. The proof of this result together with more detailed computations of motivic cohomology groups of Pfister quadrics will appear in [16].

All through the paper we use the Nisnevich topology [15] instead of the Zariski one. We would like to mention that since all the complexes of sheaves considered in this paper have transfers and homotopy invariant cohomology sheaves [24, Theorem 5.6] implies that one can repalce Nisnevich hypercohomology by Zariski ones everywhere in the paper without changing the answers.

I am glad to be able to use this opportunity to thank all the people who answered a great number of my questions during my work on Milnor Conjecture. First of all I want to thank Andrei Suslin who taught me the techniques used in [9],[10]. Quite a few of the ideas of the first part of the paper are due to numerous conversations with him. Bob Thomason made a lot of comments on the preprint [23] and in particular explained to me why algebraic K-theory with Z/2-coefficients has no multiplicative structure, which helped to eliminate the assumption \sqrt{−1} ∈ k^{∗} in Theorem 4.1. Jack Morava and Mike Hopkins answered a lot of my (mostly meaningless) topological questions and I am in debt to them for not being afraid of things like Steenrod algebra anymore. The same applies to Markus Rost and Alexander Vishik and me not being afraid anymore of the theory of quadratic forms. Dmitry Orlov guessed the form of the distinguished triangle in Theorem 4.4 which was a crucial step to the understanding of the structure of motives of Pfister quadrics. I would also like to thank Fabien Morel, Chuck Weibel, Bruno Kahn and Rick Jardine for a number of discussions which helped me to finish this work.

The mathematics of this paper was invented when I was a Junior Fellow of Harvard Society of Fellows and I wish to express my deep gratitude to the society for providing a unique opportunity to work for three years without having to think of things earthy. In its present form this paper was written during my stay in the Max-Planck Institute in Bonn.

2 Motivic cohomology in the etale topology and splitting varieties.

2.1 Beilinson and Lichtenbaum motivic cohomology groups.

Let us recall first the definition of motivic complexes Z(n) given in [22]. For a smooth variety X over k denote by L(X) the sheaf on the category of smooth schemes over k which sends a connected smooth scheme U to the free abelian group generated by closed irreducible subsets of U × X which are finite and surjective over U. Consider the sheaf L((A^{1} − {0})^{n}) and let F_{n} be its subsheaf which is the sum of images of homomorphisms L((A^{1} − {0})^{n−1}) → L((A^{1} − {0})^{n}) given by the embeddings i_{m} : (A^{1} − {0})^{n−1} → (A^{1} − {0})^{n} of the form

(x_{1}, …, x_{n−1}) ↦ (x_{1}, …, x_{m−1}, 1, x_{m}, …, x_{n−1}).

Let Δ^{•} be the standard cosimplicial object in Sm/k. For any sheaf F of abelian groups we may consider the complex of sheaves \underline{C}_{∗}(F) given by

\underline{C}_{n}(F)(U) = F (U × Δ^{n}).

Denote by Z(n) the complexes of sheaves given by

Z(n) = \{ \begin{matrix}\underline{C}_{∗}(L((A^{1} − \{0\})^{n})/F_{n})[−n] & for n ≥ 0 \\ 0 & for n < 0 \end{matrix}

The complex Z(n) is called the motivic complex of weight n. One defines Beilinson motivic cohomology groups of a smooth variety X as the hypercohomology groups

H_{B}^{p,q}(X, Z) = H_{Nis}^{p}(X, Z(q))

(here “Nis” refers to the Nisnevich topology, see [15]) and the Lichtenbaum motivic cohomology groups as the hypercohomology groups

H_{L}^{p,q}(X, Z) = H_{et}^{p}(X, Z(q))

Similarly, using the complexes Z(n) ⊗ Z/m one defines Beilinson and Lichtenbaum motivic cohomology with finite coefficients. Since motivic cohomology are defined as hypercohomology with coefficients in certain complexes of sheaves on Sm/k we can extend this definition to smooth simplicial schemes in the usual way.

The main technical tools in the study of the basic properties of motivic cohomology are the theory of (pre-)sheaves with transfers developed in [24] and the theory of relative equidimensional cycles and their “cohomology” developed in [5] and [21]. In what follows we will use freely the terminology and notations of [24] and [25]. The statements 2.1-2.4 below summarize the elementary properties of the Beilinson motivic cohomology groups used in this paper and the statements 2.5-2.9 summarize the elementary properties of Lichtenbaum motivic cohomology groups. For any X one has a canonical homomorphism

H_{B}^{∗,∗}(X, Z) → H_{L}^{∗,∗}(X, Z)

As we show below for any field k and any prime l not equal to the characteristic of k there are canonical isomorphisms

H_{L}^{n,n}(K, Z/l) = H_{et}^{n}(k, μ_{l}^{⊗n})
H_{B}^{n,n}(k, Z/l) = K_{n}^{M}(k)/l

which shows that Conjecture 1.1 can be reformulated as a comparison statement between motivic cohomology in Nisnevich and etale topology. The general form of this comparison conjecture is discussed in 2.10-2.15.

Let Z_{Nis}(A^{1}−{0}) be the free sheaf of abelian groups on (Sm/k)_{Nis} generated by the sheaf represented by A^{1}−{0} and let Z̃_{Nis}(A^{1}−{0}) be the kernel of the canonical homomorphism Z_{Nis}(A^{1} − {0}) → Z. For any complex of sheaves K denote by K_{−1} the internal Hom-object \underline{Hom}(Z̃_{Nis}(A^{1} − \{0\}), K) in the derived category of sheaves of abelian groups on (Sm/k)_{Nis}. Observe, that there is a canonical morphism Z(n − 1)[−1] → Z(n)_{−1}. The following result is a form of the Cancellation Theorem ([25, Theorem 4.3.1]).

Theorem 2.1 For any field k of characteristic zero there the canonical morphisms Z(n − 1)[−1] → Z(n)_{−1} are quasi-isomorphisms.

Corollary 2.2 Let X be a (simplicial) smooth scheme. The Beilinson motivic cohomology group H_{B}^{p,q}(X , Z) is zero in the following cases:

1. if q < 0

2. if q = 0 and p < 0

3. if q = 1 and p ≤ 0

Proof: One can easily compute the complexes Z(q) explicitly for q ≤ 1 obtaining the following ([22, Proposition 2.2]):

Z(q) = \{ \begin{matrix}0 & for q < 0 \\ Z & for q = 0 \\ G_{m}[−1] & for q = 1 \end{matrix}

The statement of the corollary is obvious modulo this computation.

Corollary 2.3 Let X be a smooth scheme over k. Then H_{B}^{p,q}(X, Z) is zero in the following cases:

1. if p − q > dim(X)

2. if p > 2q

Proof: By definition of Z(q) the cohomology sheaves \underline{H}^{i}(Z(q)) of this complex are zero for i > q. Thus the first statement is an immediate corollary of the cohomological dimension theorem in Nisnevich topology (see [14] or [15]). To prove the second statement it is sufficient to show that for a smooth X one has H^{n}(X, \underline{H}^{p−n}(Z(q))) = 0 for p > 2q. Using the Gersten resolutions for homotopy invariant sheaves with transfers described in [24, Theorem 4.36] we have only to show that one has \underline{H}^{p−n}(Z(q))_{−n} = 0 for p > 2q. By [24, Proposition 4.33] the functor K ↦ K_{−1} on the category of complexes of sheaves with transfers with homotopy invariant cohomology sheaves commutes with taking cohomology sheaves, thus by Theorem 2.1 we have

\underline{H}^{p−n}(Z(q))_{−n} = \underline{H}^{p−2n}(Z(q − n))

If q < n this sheaf is zero since Z(q − n) = 0 and if q ≥ n and p > 2q we have p − 2n > q − n and the sheaf \underline{H}^{p−2n}(Z(q − n)) is zero since Z(q − n) does not have cohomology sheaves in dimension greater than q − n.

Denote by \underline{K}_{n}^{M} the sheaf on Sm/k such that for a connected smooth scheme X over k the group \underline{K}_{n}^{M}(X) is the subgroup in the n-th Milnor K-group of the function field of X which consists of elements with all residues in points of codimension 1 being zero.

Corollary 2.4 Let X be a smooth scheme over k. Then for any p, q there is a canonical homomorphism H_{B}^{p,q}(X, Z) → H^{p−q}(X, \underline{K}_{q}^{M}) which is an isomorphism in the following cases

1. if p ≥ q + dim(X)

2. if p ≥ 2q − 2

Proof: Consider the cohomology sheaf \underline{H}^{n}(Z(n)). Since \underline{H}^{i}(Z(n)) = 0 for i > n there are canonical homomorphisms H_{B}^{p,q}(X, Z) → H^{p−q}(X, \underline{H}^{q}(Z(q))). Proceeding as in the proof of 2.3 one concludes that these homomorphisms are isomorphisms for values of p and q considred in 2.4. It remains to show that \underline{H}^{n}(Z(n)) = \underline{K}_{n}^{M}. Since \underline{H}^{n}(Z(n)) is a homotopy invariant sheaf with transfers for any smooth connected X the restriction homomorphism

H^{0}(X, \underline{H}^{n}(Z(n))) → H^{0}(Spec(k(X)), \underline{H}^{n}(Z(n))

is a monomorphism ([24, Corollary 4.17]). It was shown in [22, Prop. 4.1] that for any field k one has canonical isomorphisms H_{B}^{n,n}(k, Z) = K_{n}^{M}(k). In particular for any X the group H^{0}(X, \underline{H}^{n}(Z(n))) is a subgroup in K_{n}^{M}(k(X)) and one verifies easily that it coincides with the subgroup \underline{K}_{n}^{M}(X) of elements with all residues being zero.

Theorem 2.5 The canonical homomorphisms

H_{B}^{p,q}(−, Q) → H_{L}^{p,q}(−, Q)

are isomorphisms.

Proof: By the Leray spectral sequence associated to the obvious morphism of sites (Sm/k)_{et} → (Sm/k)_{Nis} it is sufficient to show that for any i, j ∈ Z and any smooth scheme X one has

H_{et}^{i}(X, \underline{H}^{j}(Q(q))_{et}) = H_{Nis}^{i}(X, \underline{H}^{j}(Q(q))).

Since \underline{H}^{j}(Z(q)) are homotopy invariant sheaves with transfers it is a particular case of [24, Prop.5.22 and Prop.5.26].

For n prime to the characteristic of the base field Lichtenbaum motivic cohomology groups with Z/n coefficients are closely related to the “usual” etale cohomology.

Theorem 2.6 Let k be a field and n be an integer prime to characteristic of k. Denote by μ_{n} the etale sheaf of n-th roots of unit on Sm/k and let μ_{n}^{⊗q} be the n-th tensor power of μ_{n} in the category of Z/n-modules. Then there is a canonical isomorphism H_{L}^{p,q}(−, Z/n) = H_{et}^{p}(−, μ_{n}^{⊗q}).

Proof: We have to show that the complex Z/n(q) is canonically quasiisomorphic in the etale topology to the sheaf μ_{n}^{⊗q}. In the category of complexes of sheaves with transfers of Z/n-modules in the etale topology with homotopy invariant cohomology sheaves DM_{−}^{eff}(k, Z/n, et) we have

Z/n(q) = (Z/n(1))^{⊗q}

Since Z(1) = G_{m}[−1] in the etale topology we have Z/n(1) = μ_{n} which proves our claim.

For a smooth variety X over k denote by Č(X) the simplicial smooth scheme with terms Č_{n}(X) = X^{n+1} and faces and degeneracy morphisms given by partial projections and diagonals respectively.

Proposition 2.7 Let X be a nonempty smooth scheme over k. Then the homomorphisms

H_{L}^{p,q}(Spec(k), Z) → H_{L}^{p,q}(Č(X), Z)

induced by the canonical morphism Č(X) → Spec(k) are isomorphisms for all p, q ∈ Z.

Proof: For a smooth scheme X let Z_{et}(X) be the sheaf of abelian groups on (Sm/k)_{et} freely generated by the sheaf of sets represented by X. Similarly, for a simplicial scheme X let Z_{et}(X) be the complex of sheaves with terms Z_{et}(X_{n}) and the differential being given by the alternated sums of morphisms given by the face maps. One can easily see that for any complex of etale sheaves K there are canonical isomorphisms

Hom_{D}(Z_{et}(X), K[n]) = H_{et}^{n}(X , K)

where the group of morphisms on the left hand side is considered in the derived category of etale sheaves on Sm/k. One can easily observe now that for any nonempty X the morphism of complexes

Z_{et}(Č(X)) → Z_{et}(Spec(k))

is a quasi-isomorphism which proves our claim.

Let π : (Sm/k)_{et} → (Sm/k)_{Nis} be the obvious morphism of sites. Consider the complex Rπ_{∗}(π^{∗}(Z(q))) of Nisnevich sheaves with transfers on Sm/k. We have

H_{L}^{p,q}(X, Z) = H_{Nis}^{p}(X, Rπ_{∗}(π^{∗}(Z(q)))).

Lemma 2.8 For a field k of characteristic zero the complex Rπ_{∗}(π^{∗}(Z(q))) has homotopy invariant cohomology sheaves.

Proof: Since the sheaf associated with a homotopy invariant presheaf with transfers is homotopy invariant ([24, Propositions 4.25, 5.5]) it is sufficient to show that the functors U ↦ H_{L}^{p,q}(U, Z) are homotopy invariant i.e. that for any smooth U the homomorphism

H_{L}^{p,q}(U × A^{1}, Z) → H^{p,q}(U, Z)

given by the restriction to U × {0} is an isomorphism. Consider the long exact sequence relating Lichtenbaum motivic cohomology with Z coefficients to Lichtenbaum motivic cohomology with Q-coefficients and Q/Z-coefficients. The cohomology with Q-coefficients are homotopy invariant by Theorem 2.5. The cohomology with Q/Z-coefficients are isomorphic to the etale cohomology by Theorem 2.6 and therefore homotopy invariant as well. Our claim follows now from the five lemma.

Remark: For fields k of positive characteristic the statement Lemma 2.8 is false. Instead one can show that the complex Rπ_{∗}(π^{∗}(Z(q))) ⊗ Z[1/char(k)] has homotopy invariant cohomology sheaves while the complex Rπ_{∗}(π^{∗}(Z(q))) ⊗ (Z[1/char(k)]/Z) is contractible, i.e. represents zero in the motivic category.

Lemma 2.9 Let k be a field of characteristic zero. Then for any q ≥ 0 there is a canonical quasi-isomorphism

Rπ_{∗}(π^{∗}(Z(q)))_{−1} = Rπ_{∗}(π^{∗}(Z(q − 1)))[−1].

Proof: We have to show that the canonical homomorphisms

H_{L}^{p−1,n−1}(X, Z) ⊕ H^{p,n}(X, Z) → H_{L}^{p,n}(X × (A^{1} − {0}), Z)

are isomorphisms. Consider again the cases of rational and Q/Z-coefficients separately. For rational coefficients our result follows from Theorem 2.5 and Theorem 2.1. For Q/Z-coefficients it follows from Theorem 2.6 and the fact that the corresponding homomorphisms for the etale cohomology with μ_{m}^{⊗n}- coefficients are isomorphisms.

The following fundamental conjecture is due to S. Lichtenbaum ([7, p.130]):

Conjecture 2.10 Let k be a field and X be a smooth variety over k. Then the canonical homomorphism H_{B}^{p,q}(X, Z) → H_{L}^{p,q}(X, Z) is an isomorphism for p ≤ q + 1.

Since Z(0) = Z and Z(1) = G_{m}[−1] one can see that this conjecture holds for q = 0 for obvious reasons and for q = 1 because it becomes equivalent to the classical Theorem Hilbert 90.

Let us say that H90(n, l) holds in characteristic zero if for any field k of characteristic zero and any q ≤ n one has H_{L}^{q+1,q}(Spec(k), Z_{(l)}) = 0. Clearly, H90(n,l) is a particular case of Conjecture 2.10 for q = n. In particular it holds in characteristic zero for n = 0, 1 and all l. Our proof of H90(n, 2) in Section 4 goes by induction on n and in order to be able to make the inductive step we will need to know some corollaries of H90(n − 1, l).

Let L(q) be the canonical truncation of complex Rπ_{∗}(π^{∗}(Z(q))) on level q + 1 i.e. L(q) is the subcomplex of sheaves in Rπ_{∗}(π^{∗}(Z(q))) whose cohomology sheaves \underline{H}^{i}(L(q)) are the same as for Rπ_{∗}(π^{∗}(Z(q))) for i ≤ q + 1 and zero for i > q + 1. Since \underline{H}^{i}(Z(q)) = 0 for i > q the canonical morphism Z(q) → Rπ_{∗}(π^{∗}(Z(q))) factors through L(q). Let K(q) be the complex of sheaves with transfers on (Sm/k)_{Nis} defined by the distinguished triangle Z(q) → L(q) → K(q) → Z(q)[1].

One can easily see that Conjecture 2.10 is equivalent to vanishing of the complex K(q). By Lemma 2.8 the cohomology sheaves of K(q) are homotopy invariant sheaves with transfers which are in particular separated ([24, Corollary 4.17]). Thus H90(q,l) is equivalent to the vanishing of the sheaf \underline{H}^{q+1}(K(q)) ⊗ Z_{(l)}. The following result is essentially the main theorem of [22].

Theorem 2.11 Assume that H90(n, l) holds in characteristic zero. Then the complex K(n) ⊗ Z_{(l)} is acyclic.

Proof: By the assumption that H90(n,l) holds we know that \underline{H}^{n+1}(K(n) ⊗ Z_{(l)}) = 0. In order to show that K(n) ⊗ Z_{(l)} is quasi-isomorphic to zero it remains to verify that for any smooth scheme X over k and any p ≤ n the homomorphism H_{B}^{p,n}(X, Z_{(l)}) → H_{L}^{p,n}(X, Z_{(l)}) is an isomorphism. Consider the long exact sequences of motivic cohomology groups associated with the distinguished triangles Z_{(l)}(n) → Q(n) → Q/Z_{(l)}(n) → Z_{l}(n)[1] considered in both Nisnevich and etale topologies. It follows from Theorem 2.5 that in order to prove the theorem it is sufficient to verify that the homomorphisms

H_{B}^{p,n}(X, Q/Z_{(l)}) → H_{L}^{p,n}(X, Q/Z_{(l)})

are isomorphisms for p ≤ n. Since we assume H90(n,l), Theorem 2.6 implies that for any field k of characteristic zero the Bockstein homomorphisms

H_{et}^{n}(k, μ_{l^{m}}^{⊗n}) → H_{et}^{n+1}(k, μ_{l}^{⊗n})

are zero and therefore [22, Proposition 7.1] and [22, Theorem 5.9] imply that homomorphisms in question are isomorphisms.

Corollary 2.12 Assume that H90(n,l) holds in characteristic zero. Then for any field k of characteristic zero and and complex K of Nisnevich sheaves with transfers such that

1. cohomology of K are concentrated in homologically nonnegative degrees

2. K is exact in the etale topology

one has Hom_{DM}(K, Z_{(l)}(q)[p]) = 0 for q ≤ n, p ≤ q + 2.

Proof: Any complex K whose cohomology sheaves are concentrated in nonnegative homological degrees is quasi-isomorphic to a complex K′ such that K′_{n} = 0 for n < 0 and K_{i} is a sum of sheaves of the form L(X) for i ≥ 0. Theorem 2.11 implies that for any K like that and any p, q satisfying the conditions of the corollary the canonical map

Hom_{DM}(K, Z_{(l)}(q)[p]) → Hom_{DM}(K, Rπ_{∗}(π^{∗}(Z_{(l)}(q)[p])))

is a monomorphism. On the other hand the right hand side group is isomorphic to the group of morphisms Hom(K_{et}, π^{∗}(Z_{(l)}(q)[p])) in the derived category of etale sheaves with transfers which is zero since K_{et} is quasi-isomorphic to zero.

Corollary 2.13 Assume that H90(n, l) holds in characteristic zero. Then for any simplicial smooth scheme X one has

1. the homomorphisms

H_{B}^{p,q}(X , Z_{(l)}) → H_{L}^{p,q}(X , Z_{(l)})

are isomorphisms for p − 1 ≤ q ≤ n and monomorphisms for p = q + 2 and q ≤ n

2. the homomorphisms

H_{B}^{p,q}(X , Z/l) → H_{L}^{p,q}(X , Z/l)

are isomorphisms for p ≤ q ≤ n and monomorphisms for p = q +1 and q ≤ n

Corollary 2.14 Assume that H90(n, l) holds in characteristic zero. Then for any field k of characteristic zero any q ≤ n and any cyclic extension E/F of degree l the sequence

K_{q}^{M}(E) →^{1−σ} K_{q}^{M}(E) →^{N_{E/k}} K_{q}^{M}(F)

(where σ is a generator of Gal(E/k)) is exact.

Proof: For any finite separable field extension p : Spec(k′) → Spec(k) the presheaf with transfers L(Spec(k′)) on Sm/k is canonically isomorphic to the direct image p_{∗}(Z). In particular there is a complex of presheaves with transfers of the form

0 → L(k) → L(E) →^{1−σ} L(E) → L(k) → 0

Denote this complex with the last L(k) placed in degree zero by \underline{K}. It is clearly exact in the etale topology and therefore Lemma 2.12 implies that

Hom(\underline{K}, Z_{(l)}(n)[n + 2]) = 0

in the derived category of Nisnevich sheaves with transfers. The standard spectral sequence which converges to this group from morphisms

Hom(L(−), Z_{(l)}(n)[j]) = H_{B}^{j,n}(−, Z_{(l)})

together with the comparison result 2.4 relating motivic cohomology to Milnor K-theory implies the statement of the corollary.

Proposition 2.15 Assume that H90(n,l) holds in characteristic zero. Let X be a smooth scheme over k and U be a dense open subscheme in X. Then the homomorphisms H_{Nis}^{∗}(X, K(n + 1) ⊗ Z_{(l)}) → H_{Nis}^{∗}(U, K(n + 1) ⊗ Z_{(l)}) are isomorphisms.

Proof: By obvious induction it is sufficient to show that the statement of the lemma holds for U = X −Z where Z is a smooth closed subscheme in X. Consider K(n + 1) ⊗ Z_{(l)} as a complex of Nisnevich sheaves with transfers with homotopy invariant cohomology sheaves, i.e. as an object of DM_{−}^{eff}(k). We have a distinguished triangle ([25, Proposition 3.5.4])

M(Z)(c)[2c − 1] → M(U) → M(X) → M(Z)(c)[2c]

where c is the codimension of Z and therefore it is sufficient to show that Hom_{DM^{eff}} (M(Z)(c)[2c], K(n + 1)[∗]) = 0 for any c > 0. For any c > 0 we have

Hom(M(Z)(c)[2c], K(n + 1)[∗]) = Hom(M(Z)(c − 1)[2c − 1], K(n + 1)_{−1}[∗])

By Lemma 2.9 we conclude that K(n + 1)_{−1} = K(n)[−1] and therefore the statement of the proposition follows from Theorem 2.11.

2.2 Some computations with Galois cohomology

In this section we are dealing only with classical objects namely Milnor Ktheory and etale cohomology. More general motivic cohomology do not appear here. The only result of this section which we will directly use below is Theorem 2.23. One may observe that in the case of Z/2-coefficients it can be proven in a much easier way, but we decided to include the case of general l for possible future use. Everywhere below k is a field and l is a fixed prime not equal to the characteristic of k.

Definition 2.16 Let k be a field, l be a prime not equal to the characteristic of k and n ≥ 0 an integer. We say that BK(n, l) holds over k if for any field K over k and any m ≤ n one has:

1. the norm residue homomorphism K_{m}^{M}(K)/l → H_{et}^{m}(K, μ_{l}^{⊗m}) is an isomorphism 2. for any cyclic extension E/K of degree l the sequence

K_{m}^{M}(E) →^{1−σ} K_{m}^{M}(E) →^{N_{E/K}} K_{m}^{M}(K)

where σ is a generator of Gal(E/K) is exact.

Remark: Note that by 2.13 and 2.14 the validity of H90(n, l) in characteristic zero implies the validity of BK(n, l) in characteristic zero.

Proposition 2.17 Let k be a field which has no extensions of degree prime to l. Assume that BK(n, l) holds over k. Then for any cyclic extension E/k of k of degree l there is an exact sequence of the form

H_{et}^{n}(E, Z/l) →^{N_{E/k}} H_{et}^{n}(k, Z/l) →^{−∧[a]} H_{et}^{n+1}(k, Z/l) → H_{et}^{n+1}(E, Z/l)

where [a] ∈ H_{et}^{1}(k, Z/l) is the class which corresponds to E/k.

Proof: In order to prove the proposition we will need to do some preliminary computations. Fix an algebraic closure k̄ of k. Since k has no extensions of degree prime to l there exists a primitive root of unit ξ ∈ k of degree l. Let E ⊂ k̄ be a cyclic extension of k of degree l. We have E = k(b) where b^{l} = a for an element a in k^{∗}. Denote by σ_{ξ} the generator of the Galois group G_{b} = Gal(E/k) which acts on b by multiplication by ξ and by [a]_{ξ} the class in H_{et}^{1}(k, Z/l) which corresponds to the homomorphism Gal(k̄/k) → G → Z/l which takes σ_{ξ} to the canonical generator of Z/l (one can easily see that this class is determined by a and ξ and does notdepend on b).

Let p : Spec(E) → Spec(k) be the projection. Consider the etale sheaf F = p_{∗}(Z/l). The group G acts on F in the natural way. Denote by F_{i} the kernel of the homomorphism (1 − σ)^{i} : F → F. One can verify easily that F_{i} = Im(1 − σ)^{l−i} and that as a Z/l[Gal(k̄/k)]-module F_{i} has dimension i. In particular we have F = F_{l}. Note that the extension 0 → Z/l → F_{2} → Z/l → 0 represents the element [a]_{ξ} in H_{et}^{1}(k, Z/l) = Ext_{Z/l[Gal(k̄/k)]}^{1}(Z/l, Z/l)

Let α_{i} be the element in H^{2}(k, Z/l) = Ext_{Z/l[Gal(k̄/k)]}^{2}(Z/l, Z/l) defined by the exact sequence

0 → Z/l → F_{i} →^{u_{i}} F_{i} → Z/l → 0

where u_{i} = 1 − σ and Im(u_{i}) = F_{i−1}.

Lemma 2.18 One has α_{l} = c([a]_{ξ} ∧ [ξ]_{ξ}) where c is an invertible element of Z/l and α_{i} = 0 for i < l.

Proof: The fact that α_{i} = 0 for i < l follows from the commutativity of the diagram

0 → Z/l → F_{i+1} → F_{i+1} → Z/l → 0
0 ↓ ↓ ↓ ↓ Id
0 → Z/l → F_{i} → F_{i} → Z/l → 0

To compute α_{l} note first that since the action of Gal(k̄/k) on F = F_{l} factors through G = Gal(E/k) it comes from a well defined element in H^{2}(G, Z/l). This element is not zero for trivial reasons. On the other hand the group

H^{2}(G, Z/l) = H^{2}(Z/l, Z/l) = Z/l

is generated by the element β(γ) where γ is the canonical generator of H^{1}(G, Z/l) and β is the Bockstein homomorphism. Thus we conclude that up to multiplication by an invertible element of Z/l our class α_{l} equals β([a]_{ξ}). It remains to show that β([a]_{ξ}) = c[a]_{ξ} ∧ [ξ]_{ξ} which follows by simple explicite computations from the fact that [a]_{ξ} has a lifting to an element of H_{et}^{1}(Spec(k), μ_{l^{2}}).

Lemma 2.19 Assume that BK(n, l) holds over k. Then for all m ≤ n and all i = 1, …, l − 1 one has:

1. The sequence H_{et}^{m}(k, Z/l) ⊕ H_{et}^{m}(k, F_{i+1}) → H_{et}^{m}(k, F_{i+1}) → H_{et}^{m}(k, Z/l) where the first homomorphism is given on the second summand by 1−σ is exact.

2. The homomorphisms ν_{m,i} : H_{et}^{m}(k, Z/l) ⊕ H_{et}^{m}(k, F_{i+1}) → H_{et}^{m}(k, F_{i}) given by the canonical morphisms Z/l → F_{i}, F_{i+1} → F_{i} are surjective.

Proof: We proceed by induction on i. Consider first the case i = l − 1. The first statement follows immediately the assumption that BK(n, l) holds.

Let us prove the second one. The image of H_{et}^{m}(k, F_{l−1}) in H_{et}^{m}(k, F_{l}) = H_{et}^{m}(E, Z/l) coincides with the kernel of the norm homomorphism

H_{et}^{m}(E, Z/l) → H_{et}^{m}(k, Z/l).

The first statement implies then that H_{et}^{m}(k, Z/l) ⊕ H_{et}^{m}(k, F_{l}) maps surjectively to this image. It is sufficient therefore to show that an element γ ∈ H_{et}^{m}(k, F_{l−1}) which goes to zero in H_{et}^{m}(k, F_{l}) belongs to the image of ν_{m,l−1}. Any such element is a composition of a cohomology class in H_{et}^{m−1}(k, Z/l) with the canonical extension

0 → F_{l−1} → F_{l} → Z/l → 0

Thus we may assume that m = 1 and γ is the element which corresponds to this extension. Let δ be the image of c[ξ]_{ξ} (where c is as in Lemma 2.18) under the homomorphism H_{et}^{1}(k, Z/l) → H_{et}^{1}(k, F_{l−1}). The composition

H_{et}^{1}(k, Z/l) → H_{et}^{1}(k, F_{l−1}) → H_{et}^{2}(k, Z/l)

where the later homomorphism corresponds to the extension

0 → Z/l → F_{l} → F_{l−1} → 0

equals to multiplication by [a]_{ξ}. We conclude now by Lemma 2.18 that the image of γ − δ in H^{2}(k, Z/l) is zero. Then it lifts to H_{et}^{1}(k, F_{l}) which proves our Lemma in the case i = l − 1.

Suppose that the lemma is proven for all i > j. Let us show that it holds for i = j. The first statement follows immediately from the inductive assumption and the commutativity of the diagram

F_{j+2} →^{1−σ} F_{j+2}
↓  ↓
F_{j+1} →^{1−σ} F_{j+1}

The proof of the second one is now similar to the case i = l − 1 with a simplification due to the fact that α_{i} = 0 for i < l (Lemma 2.18).

The statement of the proposition follows immediately from Lemma 2.19.

Remark: For l = 2 Proposition 2.17 is a trivial corollary of the exactness of the sequence 0 → Z/2 → F_{2} → Z/2 → 0. In particular it holds without the BK(n, l) assumption and not only in the context of Galois cohomology but for cohomology of any (pro-)finitegroup. For l > 2 this is not true anymore which one can see considering cohomology of Z/l.

Lemma 2.20 Let k be a field which has no extensions of degree prime to l and such that BK(n, l) holds over k. Let further E/k be a cyclic extension of degree l such that the norm homomorphism K_{n}^{M}(E) → K_{n}^{M}(k) is surjective. Then the sequence

K_{n+1}^{M}(E) →^{1−σ} K_{n+1}^{M}(E) →^{N_{E/k}} K_{n+1}^{M}(k)

where σ is a generator for Gal(E/k) is exact.

Proof: It is essentially a version of the proof given in [20] for n = 2 and in [10] for p = 3. Let us define a homomorphism

ϕ : K_{n+1}^{M}(k) → K_{n+1}^{M}(E)/(Im(1 − σ))

as follows. Let a be an element in K_{n+1}^{M}(k) of the form (a_{0}, …, a_{n}) and let b be an element in K_{n}^{M}(E) such that

N_{E/k}(b) = (a_{0}, …, a_{n−1}).

We set ϕ(a) = b ∧ a_{n}. Since BK(n.l) holds over k the element ϕ(a) in K_{n+1}^{M}(E)/(Im(1 − σ)) does not depend on the choice of b and one can easily see that ϕ is a homomorphism from (k^{∗})^{⊗n+1} to K_{n+1}^{M}(E)/(Im(1 − σ)). To show that it is a homomorphism from K_{n}^{M}(k) it is sufficient to verify that ϕ takes an element of the form (a_{0}, …, a_{n}) such that say a_{0} + a_{n} = 1 to zero. Let b be a preimage of (a_{0}, …, a_{n−1}) in K_{n}^{M}(k). We have to show that (b, a_{n}) ∈ (1 − σ)K_{n+1}^{M}(E). Assume first that a_{0} is not in (k^{∗})^{l} and let c be an element in k̄^{∗} such that c^{l} = a_{0}. Let further F = k(c). Then by BK(n.l) one has

b ∧ a_{n} = b ∧ (1 − a_{0}) = N_{EF/E}(b_{EL} ∧ (1 − c)) =
= N_{EF/E}((b − (c, a_{1}, …, a_{n} − 1)) ∧ (1 − c)) ∈ (1 − σ)K_{n+1}^{M}(E)

since N_{EF/F} (b−(c, a_{1}, …, a_{n}−1)) = 0. The proof for the case when a_{0} ∈ (k^{∗})^{l} is similar.

Clearly ϕ is a section for the obvious morphism K_{n+1}^{M}(E)/(Im(1 − σ)) → K_{n+1}^{M}(k). It remains to show that it is surjective. It follows immediately from the fact that under our assumption on k the group K_{n+1}^{M}(E) is generated by symbols of the form (b, a_{1}, …, a_{n}) where b ∈ E^{∗} and a_{1}, …, a_{n} ∈ k^{∗} (see [1]).

Lemma 2.21 Let k be a field which has no extensions of degree prime to l and such that BK(n, l) holds over k. Then the following two conditions are equivalent

1. K_{n+1}^{M}(k) = lK_{n+1}^{M}(k)

2. for any cyclic extension E/k the norm homomorphism K_{n}^{M}(E) → K_{n}^{M}(k) is surjective.

Proof: The 2 ⇒ 1 part is trivial. Since BK(n, l) holds we conclude that there is a commutative square with surjective horizontal arrows of the form

K_{n}^{M}(E) → H^{n}(E, Z/l)
↓  ↓
K_{n}^{M}(k) → H^{n}(k, Z/l)

and thus the cokernel of the left vertical arrow is the same as the cokernel of the right one. By Lemma 2.17 it gives us an exact sequence

K_{n}^{M}(E) → K_{n}^{M}(k) → H^{n+1}(k, Z/l)

and since the last arrow clearly factors through K_{n+1}^{M}(k)/l it is zero. Lemma is proven.

Lemma 2.22 Let k be a field which has no extensions of degree prime to l and such that BK(n, l) holds over k. Assume further that K_{n+1}^{M}(k) = lK_{n+1}^{M}(k). Then for any finite extension E/k one has K_{n+1}^{M}(E) = lK_{n+1}^{M}(E).

Proof: It is a variant of the proof given in [20] for n = 1. It is clearly sufficient to prove the lemma in the case of a cyclic extension E/k of degree l. By Lemma 2.17 and Lemma 2.20 we have an exact sequence

K_{n+1}^{M}(E) →^{1−σ} K_{n+1}^{M}(E) →^{N_{E/k}} K_{n+1}^{M}(k).

Let α be an element in K_{n+1}^{M}(E) and let β ∈ K_{n+1}^{M}(k) be an element such that N_{E/k}(α) = lβ. Then N_{E/k}(α − β_{E}) = 0 and we conclude that the endomorphism

1 − σ : K_{n+1}^{M}(E)/l → K_{n+1}^{M}(E)/l

is surjective. Since (1 − σ)^{l} = 0 it implies that K_{n+1}^{M}(E)/l = 0.

Theorem 2.23 Let k be a field which has no extensions of degree prime to l such that BK(n, l) holds over k and K_{n+1}^{M}(k) = lK_{n+1}^{M}(k). Then

H_{et}^{n+1}(k, Z/l) = 0.

Proof: Let α be an element of H_{et}^{n+1}(k, Z/l). We have to show that α = 0. By Lemma 2.22 and obvious induction we may assume that α vanishes on a cyclic extension of k. Then by Lemma 2.17 it is of the form α_{0} ∧ a where a ∈ H_{et}^{1}(k, Z/l) is the element which represents our cyclic extension. Thus since BK(n, l) holds it belongs to the image of the homomorphism K_{n+1}^{M}(k)/l → H_{et}^{n+1}(k, Z/l) and therefore is zero.

2.3 Splitting varieties.

Definition 2.24 Let \underline{a} = (a_{1}, …, a_{n}) be a sequence of invertible elements of k and l be a prime. A smooth variety X over k is called a (l-)splitting variety for \underline{a} if the symbol \underline{a} is divisible by l in K_{n}^{M}(k(X)) where k(X) is the function field of X.

Theorem 2.25 Assume that H90(n − 1, l) holds in characteristic zero and suppose that for any field k of characteristic zero and any sequence \underline{a} = (a_{1}, …, a_{n}) there exists a splitting variety X\underline{a} with the following properties:

1. X\underline{a} ×_{Spec(k)} Spec(k(X\underline{a})) is rational over k(X\underline{a})

2. H_{B}^{n+1,n}(Č(X\underline{a}), Z_{(l)}) = 0

Then H90(n, l) holds in characteristic zero (see the definition of Č(X) right before Proposition 2.7).

Proof: Let k be a field of characteristic zero which has no extensions of degree prime to l and such that K_{n}^{M}(k) is l-divisible. Then by Theorem 2.23 and Theorem 2.6 we have H_{L}^{n+1,n}(Spec(k), Z/l) = 0 and therefore the group H_{L}^{n+1,n}(Spec(k), Z_{(l)}) must be torsion free. On the other hand by Theorem 2.5 and Corollary 2.3(1) we have H_{L}^{n+1,n}(Spec(k), Q) = 0 which implies that for any k satisfying these conditions we have H_{L}^{n+1,n}(Spec(k), Z_{(l)}) = 0. For a finite extension E/k of degree prime to l the homomorphism H_{L}^{n+1,n}(Spec(k), Z_{(l)}) → H_{L}^{n+1,n}(Spec(E), Z_{(l)}) is a monomorphism by the transfer argument. Thus in order to prove H90(n, l) it is sufficient to show that for any element \underline{a} ∈ K_{n}^{M}(k) there exists an extension K/k such that \underline{a} is divisible by l in K_{n}^{M}(K) and the homomorphism H_{L}^{n+1,n}(Spec(k), Z_{(l)}) → H_{L}^{n+1,n}(Spec(K), Z_{(l)}) is a monomorphism. Since any element in K_{n}^{M}(k) is a sum of symbols of the form \underline{a} = (a_{1}, …, a_{n}) it is sufficient to show that for varieties X\underline{a} satisfying the conditions of the theorem the homomorphisms

H_{L}^{n+1,n}(Spec(k), Z_{(l)}) → H_{L}^{n+1,n}(Spec(k(X\underline{a})), Z_{(l)})

are monomorphisms. Let u be an element in the kernel of this homomorphism. By Proposition 2.7 and our assumption on X\underline{a} it is sufficient to show that the image u′ of u in H_{L}^{n+1,n}(Č(X\underline{a}), Z_{(l)}) belongs to the image of the group H_{B}^{n+1,n}(Č(X\underline{a}), Z_{(l)}).

Let K(n) be the complex of sheaves on (Sm/k)_{Nis} defined at the end of Section 2.1. We have to verify that under our assumptions the image of u′ in the hypercohomology group H_{Nis}^{n+1}(Č(X\underline{a}), K(n)) is zero. Since u becomes zero in the generic point of X\underline{a} there exists a nonempty open subscheme U of X\underline{a} such that the restriction of u′ to U is zero. Then by Proposition 2.15 the restriction of u′ to X\underline{a} is zero. Using the standard spectral sequence which starts with the hypercohomology of Xi ; \underline{a} with coefficients in K(n) and converges to hypercohomology of Č(X\underline{a}) with coefficients in K(n) we see that it is sufficient to show that for p < n + 1 the homomorphisms H^{p}(Xi ; \underline{a}, K(n)) → H^{p}(X\underline{a}, K(n)) given by the diagonal embeddings X\underline{a} → Xi ; \underline{a} are isomorphisms. Since X\underline{a} is rational over k(X\underline{a}) there exist an open neighborhood U of the generic point of X\underline{a} in Xi ; \underline{a} and an open neighborhood W of the generic point of X\underline{a} × \{0\} in X\underline{a} × A^{N} such that the pair (U, U ∩ X\underline{a}) is isomorphic to the pair (W, W ∩ (X\underline{a} × \{0\})). Therefore the required homomorphisms are isomorphisms by Proposition 2.15 and homotopy invariance of hypercohomology with coefficients in K(n) which follow from Lemma 2.8.

3 Motivic cohomology and algebraic cobordisms.

3.1 A topological lemma.

In this section we prove a purely topological result (Lemma 3.6) which will be used below to deduce information on the action of the motivic Steenrod algebra in the motivic cohomology of simplicial schemes of the form Č(X). It should be mentioned that this result is certainly both well known and very basic from the point of view of the modern homotopy theory.

Let A^{∗}(l) be the Steenrod algebra of operations in ordinary cohomology with Z/l-coefficients and A_{∗}(l) its dual. For a CW -complex X and a pair (α_{i}, α^{j}) where α_{i} ∈ H_{i}(X, Z/l) and α^{j} ∈ H^{j}(X, Z/l) one assigns an element (α_{i}, α^{j}) ∈ A_{i−j}(l) which takes an operation ϕ ∈ A^{i−j}(l) to the number given by the value of the pairing H_{i}(X, Z/l) ⊗ H^{i}(X, Z/l) → Z/l on (α_{i}, ϕ(α^{j})). Let β_{i} be the canonical generator of H_{i}(BZ/l, Z/l), α be the canonical generator of H^{1}(BZ/l, Z/l) and β be the canonical generator of H^{2}(BZ/l, Z/l). Following [12, p. 159] one defines elements τ_{i} ∈ A_{2l^{i}−1}(l) and ξ_{i} ∈ A_{2(l^{i}−1)}(l) as the ones represented by the pairs (β_{l^{i}}, α) and (β_{l^{i}}, β) respectively. Considered as a (graded) commutative algebra A_{∗}(l) is isomorphic for odd l to P_{Z/l}[ξ_{1}, …] ⊗_{Z/l} E_{Z/l}[τ_{0}, …] where P_{Z/l} and E_{Z/l} denote the polynomial and the exterior algebras over Z/l with generators ξ_{i} and τ_{i} of degrees 2(p^{i} − 1) and 2p^{i} − 1 respectively and A_{∗}(2) is isomorphic to P_{Z/2}[τ_{0}, …] where deg(τ_{i}) = 2^{i+1} − 1 ([12, Theorem 2]). Observe that for l = 2 one has ξ_{i} = (τ_{i−1}^{2}). One defines the Milnor elements Q_{i} in A^{2l^{i}−1}(l) as the elements dual to τ_{i} with respect to the monomial basis in A_{∗}(l). Let us also define operations q_{i} ∈ A^{2(l^{i}−1)} as the ones dual with respect to the monomial basis in A_{∗}(l) to ξ_{i}.

These operations have the following well known properties which we list here mostly to provide a topological “background” for the similar result in motivic cohomology given Section 3.3.

Proposition 3.1 For any ptime l one has:

1. Operations Q_{i} generate an exterior subalgebra in A^{∗}(l), i.e. one has

(a) Q_{i}Q_{j} + Q_{j}Q_{i} = 0

(b) Q_{i}^{2} = 0

2. For any u ∈ H^{∗}(X, Z/l) and v ∈ H^{∗}(Y, Z/l) one has

Q_{i}(u × v) = Q_{i}(u) × v ± u × Q_{i}(v)

3. Q_{0} = β is the Bockstein homomorphism and for any i > 0 one has Q_{i} = [q_{i}, β]

4. for any i > 0 one has Q_{i} = ±[Q_{i−1}, P^{l^{i−1}}]

Note that the fourth property of Q_{i}’s may be used to define them inductively in terms of the Bockstein and the reduced power operations.

The following expression for the coproduct of q_{i} which will be used in the proof of Lemma 3.6.

Corollary 3.2 For any i ≥ 1 one has

Δ(q_{i}) = \{ \begin{matrix}q_{i} ⊗ 1 ± 1 ⊗ q_{i} & for l ≠ 2 \\ q_{i} ⊗ 1 + 1 ⊗ q_{i} + Q_{i−1} ⊗ Q_{i−1} & for l = 2 \end{matrix}

i.e. for any x ∈ H^{∗}(X, Z/l), y ∈ H^{∗}(Y, Z/l) one has

q_{i}(x × y) = \{ \begin{matrix}q_{i}(x) × y ± x × q_{i}(y) & for l ≠ 2 \\ q_{i}(x) × y + x × q_{i}(y) + Q_{i−1}(x) × Q_{i−1}(y) & for l = 2 \end{matrix}

Finally let us prove the following result which describes the action of operations q_{i} in the cohomology ring of CP^{∞}.

Lemma 3.3 Let x ∈ H^{2}(CP^{∞}, Z/2) be the canonical generator. Then q_{i}(x) = x^{l^{i}−1}.

Proof: The canonical map CP^{∞} → BZ/l gives a monomorphism on even cohomology groups and therefore it is sufficient to verify that q_{i}(β) = β^{l^{i}−1}. That follows immediately from our definition of q_{i} as a dual to ξ_{i} and ξ_{i} as the element represented by the pair (β_{l^{i}}, β).

Let MU be the spectrum representing the complex cobordisms theory and τ : MU → H_{Z/l} be the Thom class with values in ordinary cohomology with Z/l-coefficients. For any cohomological operation P ∈ A^{2n}(l) we may consider the cohomological class P (τ) ∈ H^{∗}(MU, Z/l) = H^{∗}(BGL, Z/l). Let s_{i} ∈ H^{2i}(BGL, Z/l) be the element which corresponds to the characteristic class defined by the properties:

1. s_{i}(E ⊕ F) = s_{i}(E) + s_{i}(F)

2. s_{i}(L) = c_{1}(L)^{i} for a line bundle L.

Lemma 3.4 One has q_{i}(τ) = s_{l^{i}−1} in H^{∗}(MU, Z/l).

Proof: Let E be a complex vector bundle on a CW -complex X, T h(E) be its Thom space and τ_{E} ∈ H^{2n}(T h(E), Z/l) be its Thom class. It is sufficient to verify that q_{i}(τ_{E}) = s_{i}(E)τ. Using the standard splitting principle we may assume that E is a sum of line bundles L_{j}, j = 1, …, m. By Lemma 3.3 and Corollary 3.2 one can easily see that in this case one has q_{i}(τ_{E}) = (∑_{j=1}^{m} c_{1}(L_{j})^{l^{i}−1})τ_{E} which proves our claim in view of the definition of s_{i}.

Remark: One can also show by the same method that P^{i}(τ) corresponds to the characteristic class which takes a sum of line bundles L_{1} ⊕…⊕L_{m} to the i-th elementary symmetric function of the classes c_{1}^{l−1}(L_{j}) ∈ H^{2(l−1)}(−, Z/l). In particular for l = 2 the element P^{i}(τ) = Sq^{2i}(τ) corresponds to the i-th Chern class.

For an almost complex manifold X of complex dimension n denote by s(X) the value of the characteristic class s_{n} of the normal bundle of X on the fundamental class of X. If [X] is the class of X in MU_{2n} one can verify easily that s_{2n}([X]) = s(X).

Definition 3.5 An element a in MU_{∗} is called a (ν_{n}, l)-element if it belongs to MU_{2(l^{n}−1)} and satisfies the following two conditions

1. the image of a in H_{∗}(MU, Z) is divisible by l

2. s_{l^{n}−1}(a) is not zero modulo l^{2}.

A complex algebraic variety X is called a (ν_{n}, l)-variety if the class of X in MU_{∗} is a (ν_{n}, l)-element, i.e. if

1. dim(X) = l^{n} − 1

2. all characteristic numbers of X are divisible by l

3. s(X) ≠ 0(mod l^{2})

Consider Q_{i} as a morphism in the stable homotopy category of the form

Q_{i} : H_{Z/l} → S^{2l^{n}−1} ∧ H_{Z/l}

where H_{Z/l} is the Eilenberg-MacLane spectrum representing ordinary cohomology with Z/l-coefficients and denote by Φ_{i}^{top} the homotopy fiber of this map such that we have a distinguished triangle

S^{2(l^{i}−1)} ∧ H_{Z/l} →^{u} Φ_{i}^{top} →^{v} H_{Z/l} →^{Q_{i}} S^{2l^{i}−1} ∧ H_{Z/l}.

We are ready now to formulate our topological lemma.

Lemma 3.6 Let ψ : MU → Φ_{n}^{top} be a morphism such that the diagram

MU →^{ψ} Φ_{n}^{top}
τ ↓  ↓ v
H_{Z/l} →^{Id} H_{Z/l}

commutes. Then for any (ν_{n}, l)-element a ∈ MU_{2(l^{n}−1)} one has ψ(a) ≠ 0.

Proof: Let us fix a and let E be the cofiber of the map a : S^{2(l^{n}−1)} → MU. The Thom class on MU defines a cohomology class in H^{0}(E, Z/l) which we denote by τ_{a}. One obseves easily that all we have to show is that Q_{n}(τ_{a}) is not zero in H^{2l^{n}−1}(E, Z/l). Let us assume that n > 0. Then Q_{i}(τ_{a}) = βq_{i}(τ_{a}) since β(τ_{a}) = 0 for n > 0. Therefore all we have to show is that the cohomology class q_{i}(τ_{a}) in H^{2(l^{n}−1)}(E, Z/l) can not be lifted to a class in H^{2(l^{n}−1)}(E, Z/l^{2}). Since the restriction homomorphism H^{2(l^{n}−1)}(E, Z/l) → H^{2(l^{n}−1)}(MU, Z/l) is a monomorphism it is sufficient to show that for any lifting ψ̃ of the class q_{i}(τ) in H^{2(l^{n}−1)}(MU, Z/l) to H^{2(l^{n}−1)}(MU, Z/l^{2}) one has ψ̃(a) ≠ 0. By Lemma 3.4 we have ψ̃ = s_{l^{n}−1} + lc where c is an integral characteristic class. By Definition 3.5 the number c(a) is divisible by l. Therefore ψ̃(a) = s_{l^{n}−1}(a) in Z/l^{2} which is nonzero by the definition of (ν_{n}, l)-classes.

3.2 Homotopy categories of algebraic varieties

Let Δ^{op}Shv_{Nis}(Sm/k) be the category of simplicial sheaves of sets in the Nisnevich topology on the category of smooth schemes over k. This category replaces in our theory the category of topological spaces. Note that the categories of smooth schemes, sheaves of sets, simplicial smooth schemes and simplicial sets naturally embed into Δ^{op}Shv_{Nis}(Sm/k).

For a smooth variety U over k and a point u of U let O_{U,u}^{h} be the henselisation of the local ring of u in U. Then Spec(O_{U,u}^{h}) is a limit of smooth schemes U_{α} over k and for any object X of Δ^{op}Shv_{Nis}(Sm/k) one defines a simplicial set X (Spec(O_{U,u}^{h})) as the corresponding colimit of simplicial sets X (U_{α}).

Definition 3.7 A morphism of simplicial sheaves f : X → Y is called a simplicial weak equivalence if for any smooth scheme U over k and any point u of U the corresponding map of simplicial sets

X (Spec(O_{U,u}^{h})) → Y(Spec(O_{U,u}^{h}))

is a weak equivalence.

A class of important examples of a simplicial weak equivalences can be obtained as follows. Recall, that for a smooth variety X over we denote by Č(X) the simplicial scheme such that Č(X)_{n} = X^{n+1} and faces and degeneracy morphisms are given by partial projections and diagonals respectively. We have a canonical morphism Č(X) → Spec(k).

Lemma 3.8 Let X, Y be smooth schemes over k such that

Hom(Y, X) ≠ ∅.

Then the projection Č(X) × Y → Y is a simplicial weak equivalence.

Proof: Let U be a smooth scheme over k. Then the simplicial set Č(X)(U) is the simplex generated by the set Hom(U, X) which is contractible if and only if Hom(U, X) ≠ ∅. It implies immediately that for any U the map of simplicial sets

(Č(X) × Y)(U) → Y (U) = Hom(Y, U)

is a weak equivalence.

Remark: One can strengthen the statement of Lemma 3.8 by showing that the map Č(X) × Y → Y is a simplicial weak equivalence if and only if X has a rational point over the residue field of any point of Y.

Let H_{s}(Δ^{op}Shv_{Nis}(Sm/k)) be the localization of Δ^{op}Shv_{Nis}(Sm/k) with respect to simplicial weak equivalences.

Definition 3.9 An object X of Δ^{op}Shv_{Nis}(Sm/k) is called A^{1}-local if for any other object Y the map

Hom_{H_{s}}(Y, X) → Hom_{H_{s}}(Y × A^{1}, X)

induced by the projection Y × A^{1} → Y is a bijection.

A morphism f : Y → Y′ is called an A^{1}-weak equivalence if for any A^{1}-local object X the corresponding map

Hom_{H_{s}}(Y′, X) → Hom_{H_{s}}(Y, X)

is a bijection.

As was shown in [14] there is a proper closed model structure on the category Δ^{op}Shv_{Nis}(Sm/k) where all monomorphisms are cofibrations and A^{1}-weak equivalences are weak equivalences. This closed model structure is called the A^{1}-closed model structure (as opposed to the simplicial closed model structure where weak equivalences are given by Definition 3.7) and the corresponding homotopy category H(k) is called the homotopy category of schemes over Spec(k).

Let Δ_{•}^{op}Shv_{Nis}(Sm/k) be the category of pointed simplicial sheaves. For a simplicial sheaf X we denote by X_{+} the simplicial sheaf X ∐ Spec(k) pointed by the canonical embedding Spec(k) → X ∐ Spec(k). Define the smash product X ∧ Y as the sheaf associated with the presheaf of the form U ↦ X (U) ∧ Y(U). One observes easily that this provides us with a symmetric monoidal structure on Δ_{•}^{op}Shv_{Nis}(Sm/k) and on H_{•}(k) with the unit object being S^{0} = Spec(k)_{+}.

Denote by T the pointed simplicial sheaf given by the cocartesian square

A^{1} − {0} → A^{1}
↓  ↓
Spec(k) → T

This object plays the role of S^{2} in our theory (note that there is a canonical isomorphism in H_{•}(k) of the form T ≅ (P^{1}, 0)). We also have two different “circles”

S_{s}^{1} - the simplicial circle considered as an object of Δ_{•}^{op}Shv_{Nis}(Sm/k)

S_{t}^{1} - the scheme A^{1} − {0} pointed by 1 considered as an object of the same category.

A T-spectrum over k is a sequence E = (E_{i}, e_{i} : T ∧E_{i} → E_{i+1}). T-spectra and the obvious morphisms between them form a category which we denote Spect_{T} (k). Using the A^{1}-closed model structure on Δ_{•}^{op}Shv_{Nis}(Sm/k) one defines in the usual way ([4]) the strict and stable closed model structures on Spect_{T} (k). The homotopy category SH(k) corresponding to the stable closed model structure is called the stable homotopy category of schemes over Spec(k). For any pointed simplicial sheaf X the sequence

Σ_{T}^{∞}X = (T^{∧i} ∧ X , Id)

is a T-spectrum called the suspension spectrum of X. This construction gives us a functor H_{•}(k) → SH(k). By abuse of notations we will usually write S_{s}^{1}, S_{t}^{1} ant T instead of Σ_{T}^{∞}S_{s}^{1}, Σ_{T}^{∞}S_{t}^{1} and Σ_{T}^{∞}T.

For any morphism f : X → Y of pointed simplicial sheaves define cone(f) to be the sheaf associated with the presheaf of the form U ↦ cone(X (U) → Y(U)). We have a canonical morphism Y → cone(f) which extends in the usual way to a sequence

X →^{f} Y → cone(f) → S_{s}^{1} ∧ X

called the cofibration sequence associated to f.

Theorem 3.10 There exists a structure of a tensor triangulated category on HS(k) with the following properties

1. the shift functor E → E[1] is given by E[1] = S_{s}^{1} ∧ E

2. the functor Σ_{T}^{∞} takes cofibration sequences to distinguished triangles

3. the functor Σ_{T}^{∞} is a symmetric monoidal functor from (H_{•}(k), ∧) to (SH_{•}(k), ∧)

4. the object T of SH(k) is invertible

Let E = (E_{i}, e_{i}) be a spectrum. The morphisms e_{i} : Σ_{T} E_{i} → E_{i+1} define morphisms in the stable category of the form

ẽ_{i} : Σ_{T}^{∞}E_{i}[−i] → Σ_{T}^{∞}E_{i+1}[−(i + 1)]

and one can describe the spectrum E in terms of the spectra Σ_{T}^{∞}E_{i} and morphisms ẽ_{i} as follows.

Lemma 3.11 For any spectrum E = (E_{i}, e_{i}) there is a canonical distinguished triangle in SH of the form

⊕_{i≥0}Σ_{T}^{∞}E_{i}[−i] →^{⊕(ẽ_{i}−Id)} ⊕_{i≥0}Σ_{T}^{∞}E_{i}[−i] → E → ⊕_{i≥0}Σ_{T}^{∞}E_{i}[−i + 1]

There is a canonical isomorphism in H_{•} of the form S_{s}^{1} ∧ S_{t}^{1} = T. Using this relation and Theorem 3.10 we conclude that both S_{s}^{1} and S_{t}^{1} are invertible in SH(k). Let us fix objects S_{s}^{−1} and S_{t}^{−1} and isomorphisms

\begin{matrix}S_{s}^{1} ∧ S_{s}^{−1} ≅ S^{0} \\ S_{t}^{1} ∧ S_{t}^{−1} ≅ S^{0} \end{matrix}

Let further

S_{s}^{n} = \{ \begin{matrix}(S_{s}^{1})^{∧n} & for n ≥ 0 \\ (S_{s}^{−1})^{∧−n} & for n ≤ 0 \end{matrix} S_{t}^{n} = \{ \begin{matrix}(S_{t}^{1})^{∧n} & for n ≥ 0 \\ (S_{t}^{−1})^{∧−n} & for n ≤ 0 \end{matrix}

and

S^{p,q} = S_{t}^{q} ∧ S_{s}^{p−q}

For any object E in SH(k) denote by E(q)[p] the object S^{p,q} ∧ E. Define the cohomology theory Ẽ^{∗,∗} associated to E by

Ẽ^{p,q}(X) = Hom_{SH}(X, E(p)[q])

and the homology theory E_{∗,∗} associated to E by

Ẽ_{p,q}(X) = Hom_{SH}(S^{p,q}, E ∧ X).

These theories take distinguished triangles to long exact sequences and have suspension isomorphisms with respect to smashing with S_{s}^{1}, S_{t}^{1} and T. For a simplicial smooth scheme X we set

E^{∗,∗}(X) = {♡ \atop E^{∗,∗}}(±_{T}^{∞}X_{+})

E_{∗,∗}(X) = {♡ \atop E_{∗,∗}}(±_{T}^{∞}X_{+}).

Remark: The obvious functor from the category of pointed simplicial sets to Δ_{•}^{op}(Shv_{Nis}(Sm/k)) extends to a tesor triangulated functor from the topological stable homotopy category to SH(k) which takes S^{1} to S_{s}^{1}. This implies in particular that the permutation isomorphism on S_{s}^{1} ∧ S_{s}^{1} equals −1. It is not true however that the permutation isomorphism on S_{t}^{1} ∧ S_{t}^{1} is −1. Instead it equals −ϵ where ϵ is a certain element in End(S^{0}). This element equals 1 if the base field k contains a square root of −1 but in general it is nontrivial.

3.3 Eilenberg-MacLane spectra and motivic cohomology.

Consider the quotient sheaf of abelian groups L(A^{n})/L(A^{n} − {0}) as a sheaf of sets pointed by zero and denote it by K(Z(n), 2n).

One has a canonical morphism of pointed sheaves

e_{n} : T ∧ K(Z(n), 2n) → K(Z(n + 1), 2n + 2)

and we define the Eilenberg-MacLane spectrum H_{Z} as the sequence (K(Z(n), 2n), e_{n}). Similarly starting with the sheaves (L(A^{n})/L(A^{n}−{0}))⊗ Z/n one defines Eilenberg-MacLane spectra H_{Z/n}. The external product of cycles give us morphisms of pointed sheaves K(Z(n), 2n) ∧ K(Z(m), 2m) → K(Z(n + m), 2(n + m)) which fit together and produce the multiplication morphism

m : H_{Z} ∧ H_{Z} → H_{Z}

in SH(k) which satisfies the usual associativity and commutativity condition.

One defines motivic cohomology as the cohomology theory H̃^{p,q}(−, Z) represented by the spectrum H_{Z}.

Remark: One can easily see that if k is a field of characteristic zero and X is a smooth variety over k the sheaf L(X) considered as an object of H_{•}(k) is the group completion of the abelian monoid ∐_{i≥0} SP^{i}X where SP^{i}X is the symmetric product of X. One can also observe that A^{n}/(A^{n} − {0}) is canonically isomorphic in H_{•}(k) to the sphere S^{2n,n} and therefore our definition of K(Z(n), 2n) is a version of the Dold-Thom equivalence

(∐_{i≥0} SP^{i}(S^{2n}))^{+} ≅ K(Z, 2n)

in topology.

Theorem 3.12 Let k be a field which admits resolution of singularities and X be a simplicial smooth scheme over k. Then for all p, q ∈ Z there are canonical isomorphisms

H^{p,q}(X, Z) = H_{B}^{p,q}(X, Z)

and the same holds for finite coefficients.

Proof: See [14, ].

Starting from this point we will denote the Beilinson motivic cohomology groups simply by H^{∗,∗}(−, −).

Define the motivic Steenrod algebra as the algebra of endomorphisms of H_{Z/l} in SH(k)

A^{p,q}(k, Z/l) = Hom_{SH}(H_{Z/l}, H_{Z/l}(q)[p]) = H^{p}(H_{Z/l}, Z/l(q))

The algebra A^{p,q}(k, Z/l) is analized in details in [26]. For our present purposes we will only need a few results of this paper explained below. Denote by H^{∗,∗} the motivic cohomology ring of Spec(k) with Z/l-coefficients. Using Corollaries 2.2-2.4 one immediately gets the following result.

Proposition 3.13 For any field k with resolution of singularities one has:

1. H^{p,q} = 0 in the following cases

(a) if p > q

(b) if q < 0

(c) if q = 0 and p ≠ 0

(d) if q = 1 and p ≠ 0, 1

2. H^{p,p} = K_{p}^{M}(k)/l

3. H^{0,1} = μ_{l}(k) where μ_{l} is the sheaf of l-th roots of unit.

We denote by τ the generator of H^{0,1} = μ_{2} = Z/2 for l = 2 and we assume τ = 0 for l ≠ 2. We also denote by ρ the class of −1 in H^{1,1} = k^{∗}/(k^{∗})^{l} (which is also zero for l ≠ 2).

Theorem 3.14 For a field k of characteristic zero the group A^{p,q}(k, l) is zero for q < 0 and the group A^{0,0}(k, l) is the cyclic group of order Z/l generated by the identity morphism Id_{H_{Z/l}}.

Theorem 3.15 The Kunnet homomorphism

H̃^{∗,∗}(H_{Z/l}, Z/l) ⊗_{H^{∗,∗}} H̃^{∗,∗}(H_{Z/l}, Z/l) → H̃^{∗,∗}(H_{Z/l} ∧ H_{Z/l}, Z/l)

is an isomorphism.

Composing the inverse to Kunnet homomorphism with the map

H̃^{∗,∗}(H_{Z/l}, Z/l) → H̃^{∗,∗}(H_{Z/l} ∧ H_{Z/l}, Z/l)

given by the multiplication morphism H_{Z/l} ∧ H_{Z/l} → H_{Z/l} we get a homomorphism

Δ : H̃^{∗,∗}(H_{Z/l}, Z/l) → H̃^{∗,∗}(H_{Z/l}, Z/l(∗)) ⊗_{H^{∗,∗}} H^{∗}(H_{Z/l}, Z/l).

In the following theorems β ∈ A^{1,0}(k, l) denotes the Bockstein homomorphism given by the distinguished triangle

H_{Z/l} → H_{Z/l^{2}} → H_{Z/l} →^{β} S_{s}^{1} ∧ H_{Z/l}.

Theorem 3.16 Let k be a field of characteristic zero. Then there exists a unique sequence of cohomological operations P^{i} ∈ A^{2i(l−1),l−1}, i ≥ 0 with the following properties

1. P^{0} = Id

2. for any simplicial smooth scheme X and any u ∈ H^{n}(X, Z/l(i)) one has P^{i}(u) = 0 for n < 2i and P^{i}(u) = u^{l} for n = 2i

3. Δ(P^{i}) = ∑_{a+b=i} P^{a} ⊗ P^{b} + τ ∑_{a+b=i−2} βP^{a} ⊗ βP^{b}

Define operations Q_{i} ∈ A^{2l^{i}−1,l^{i}−1} inductively setting

Q_{0} = β
Q_{i+1} = [Q_{i}, P^{l^{i}}].

Theorem 3.17 Let k be a field of characteristic zero. Then operations Q_{i} have the following properties:

1. Q_{i}’s generate an exterior subalgebra in A^{∗,∗}(l, k) i.e. Q_{i}Q_{j} + Q_{j}Q_{i} = 0 and Q_{i}^{2} = 0

2. Δ(Q_{i}) = 1 ⊗ Q_{i} + Q_{i} ⊗ 1 + ∑ ρ^{n_{j}}ϕ_{j} ⊗ ψ_{j} where n_{j} > 0 and ϕ_{i}, ψ_{i} are some operations of bidegrees (p, q) with p > 2q.

3. for any i > 0 there exist operations q_{i} such that Q_{i} = [β, q_{i}]

Just as in the topological situation we can use operations Q_{i} to define Margolis motivic cohomology H̃M_{i}^{∗,∗}(X) as cohomology groups of the complex

H̃^{∗−2l^{i}+1,∗−l^{i}+1}(X , Z/l) →^{Q_{i}} H̃^{∗,∗}(X , Z/l) →^{Q_{i}} H̃^{∗+2l^{i}−1,∗+l^{i}−1}(X , Z/l)

3.4 Topological realization functor.

In this section we assume that our base field can be embedded into the fields C of complex numbers. Any such embedding i : k ⊂ C gives us a functor from Sm/k to the category of simplicial sets given by X ↦ Sing(X(C)) where Sing is the singular simplicial set functor and X(C) is the complex manifold of C-points of X with respect to the embedding i. As was shown in [14, ] there is an endofunctor rr on Δ^{op}Shv_{Nis}(Sm/k) and a natural transformation rr → Id such that for any simplicial sheaf X the terma rr(X)_{n} are direct sums of representable sheaves and the morphism rr(X) → Id is a (simplicial) weak equivalence. Define a functor t_{C} from Δ^{op}Shv_{Nis}(Sm/k) to the category of simplicial sets setting t_{C}(X) to be the diagonal simplicial set of the bisimplicial set Sing(rr(X)(C)). One observes easily that this functor takes A^{1}-weak equivalences to weak equivalences and therefore defines the topological realization functor t_{C} : H(k) → H (here H is the usual homotopy category) such that for any smooth scheme X the object t_{C}(X) is canonically isomorphisc in H to the object X(C).

Similarly, one can define the topological realization functor t_{C} from SH(k) to the usual stable homotopy category SH. This is a tensor triangulated functor and one has canonical isomorphisms t_{C}(S_{s}^{1}) = S^{1} and t_{C}(S_{t}^{1}) = S^{1}.

For an object E in SH(k) denote by E_{top}^{p} the cohomology theory given by

E_{top}^{p}(X) = Hom_{SH}(t_{C}(X), S^{p} ∧ t_{C}(E))

and by E_{p}^{top} the homology theory given by

E_{p}^{top}(X) = Hom_{SH}(S^{p}, t_{C}(X) ∧ t_{C}(E))

such that for a smooth variety X we have

E_{top}^{p}(X) = (t_{C}(E)^{p}(X(C))
E_{p}^{top}(X) = (t_{C}(E)_{p}(X(C)).

Then we have canonical morphisms E^{p,q}(X) → E_{top}^{p}(X), E_{p,q}(X) → E_{p}^{top}(X).

Remark: One can easily see that the theory E_{top}^{p} is represented by an object E_{top} of SH(k) which is (0, 1)-periodic i.e. such that there is a canonical isomorphism E_{top} ≅ E_{top}(1). Form the point of view of the homotopy theory of algebraic varieties spectra of the form E_{top} are usually completely misterious.

The following result is a corollary of our definition of motivic cohomology and the Dold-Thom theorem.

Proposition 3.18 For any embedding i : k ⊂ C there is a canonical isomorphism

t_{C}(H_{Z}) = H_{Z}

where H_{Z} is the usual topological Eilenberg-Maclane spectrum. The same holds for finite coefficients.

For an embedding k ⊂ C let t_{C} : A^{∗,∗}(l, k) → A(l) be the homomorphism induced by Proposition 3.18 by the topological realization functor t_{C}. One can choose an algebraic model B_{et}Z/l for the classifying space BZ/l such that the generator a ∈ H^{1}(BZ/l, Z/l) comes from a motivic cohomology class a ∈ H_{B}^{1,1}(B_{et}Z/l, Z/l). Thus properties of operations P^{i} given in Theorem 3.16 together with the standard topological results on the uniqueness of reduced power operations imply immediately that for any embedding k ⊂ C one has t_{C}(P^{i}) = P^{i} and in particular one has:

Lemma 3.19 For any embedding k ⊂ C one has t_{C}(Q_{i}) = Q_{i}.

3.5 Algebraic cobordisms.

In this section we define a spectrum MGL called the algebraic cobordisms spectrum and prove some of its very basic properties. To do it we need to introduce Thom spaces of vector bundles on smooth schemes over k. Let E → X be such a bundle and let E − s(X) be the complement to the zero section. Then the Thom space T h(E) of E is the pointed sheaf given by the cocartesian square

E − s(X) → E
↓  ↓
Spec(k) → T h(E)

One observes easily that there is a canonical isomorphism of pointed sheaves of the form T h(E ⊕ O) = T ∧ T h(E) and therefore considering universal bundles on the Grassmannians G(m, n) one gets in the usual way a spectrum which we denote MGL and call the algebraic cobordisms spectrum.

Lemma 3.20 Let X be a connected smooth scheme over a field k which admits resolution of singularities and E be a vector bundle over X of dimension d. Then there are canonical isomorphisms

H̃^{p,q}(T h(E), Z) = H^{p−2d,q−d}(X, Z)

and the same holds for finite coefficients. In particular there is a canonical isomorphism H̃^{2d,d}(T h(E), Z) = Z.

Proof: By [14, ] we have a canonical isomorphism in the homotopy category H_{•}(k) of the form

T h(E) ≅ P(E ⊕ O)/P(E).

Together with the projective bundle theorem in DM ([25, Prop. 3.5.1]) this implies the statement of the lemma.

The generator of the group H̃^{2d,d}(T h(E), Z) is called the Thom class of E. Combining Lemma 3.20 with 3.11 and 2.3(2) we get the following result.

Theorem 3.21 Let k be a field which admits resolution of singularities. Then one has H̃^{p,q}(MGL, Z) = 0 for p > 2q and H̃^{0,0}(MGL, Z) = Z. The same holds for finite coefficients.

Denote by τ : MGL → H_{Z/l} the canonical generator of H̃^{0,0}(MGL, Z/l). Note that the first part of Theorem 3.21 implies immediately that if α is an element of the motivic Steenrod algebra of bidegree (p, q) with p > 2q then α(τ) = 0. Let k ⊂ C be an embedding of our base field into the field of complex numbers. Then there is a canonical isomorphism in the stable homotopy category of the form t_{C}(MGL) = MU. In particular for any smooth scheme X over k we get natural homomorphisms MGL_{p,q}(X) → MU_{p}(X(C)). The following theorem is the only nontrivial result on algebraic cobordisms which we will use in this paper.

Theorem 3.22 Let X be a smooth projective variety of dimension d over a subfield k of C. Then there exists an element ϕ_{X} ∈ MGL_{2d,d}(X) such that the image t_{C}(ϕ_{X}) of ϕ_{X} in MU_{2d}(X(C)) is the fundamental class of X(C) in MU-homology.

3.6 Main theorem.

We are ready now to prove the main theorem of this section (Theorem 3.25), which is essentially the only result of the general theory described here which we use in the second part of the paper. We start with some preliminary results. Consider the operation Q_{i} as a morphism in the category SH of the form H_{Z/l} → S_{s}^{1} ∧ T^{l^{i}−1} ∧ H_{Z/l} and let Φ_{i} be its homotopy fiber such that we have a distinguished triangle of the form

T^{l^{i}−1} ∧ H_{Z/l} →^{u} Φ_{i} →^{v} H_{Z/l} →^{Q_{i}} S_{s}^{1} ∧ T^{l^{i}−1} ∧ H_{Z/l}.

Denote by τ̃ the composition

MGL ∧ H_{Z/l} →^{τ∧Id} H_{Z/l} ∧ H_{Z/l} →^{m} H_{Z/l}

where τ is the Thom class described in Section 3.5 and m is the multiplication homomorphism.

Lemma 3.23 There exists a morphism ϕ̃_{n} : MGL ∧ Φ_{n} → Φ_{n} such that the diagram

MGL ∧ T^{l^{n}−1} ∧ H_{Z/l} →^{τ̃} T^{l^{n}−1} ∧ H_{Z/l}
↓  ↓ u
MGL ∧ Φ_{n}  →^{ϕ̃_{n}}  Φ_{n}
↓  ↓ v
MGL ∧ H_{Z/l}  →^{τ̃}  H_{Z/l}

commutes.

Proof: Since both columns are parts of distinguished triangles it is sufficient to show that the diagram

MGL ∧ H_{Z/l}  →^{τ̃}  H_{Z/l}
Id∧Q_{i}  ↓  ↓ Q_{n}
MGL ∧ S_{s}^{1} ∧ T^{l^{i}−1} ∧ H_{Z/l} →^{τ̃} S_{s}^{1} ∧ T^{l^{i}−1} ∧ H_{Z/l}

commutes. This follows from the definition of τ̃, the formula 3.17(2) for Δ(Q_{n}) and the fact that H̃^{p,q}(MGL, Z/l) = 0 for p > 2q (Theorem 3.21).

Let Y be an object of SH and Y → S^{0} be a morphism. Let further ϕ̃_{n} be a morphism satisfying the condition of Lemma 3.23 and ϕ_{Y} be an element of MGL_{2d,d}(Y). For any object X of SH consider the homomorphism

p_{ϕ_{Y},ϕ_{n}} : Φ̃_{n}^{∗,∗}(Y ∧ X) → Φ̃_{n}^{∗−2d,∗−d}(X)

which takes an element α in Φ̃_{n}^{∗,∗}(Y ∧ X) given by a morphism

α : Y ∧ X → Φ_{n}(∗)[∗]

to the element given by the composition

T^{d} ∧ X →^{ϕ_{Y}∧Id} MGL ∧ Y ∧ X →^{Id∧α} MGL ∧ Φ_{n}(∗)[∗] →^{ϕ̃_{n}(∗)[∗]} Φ_{n}(∗)[∗]

Proposition 3.24 Let Y, X be objects of SH, p : Y → S^{0} be a morphism and ϕ_{Y} be an element in M̃GL_{2(l^{n}−1),l^{n}−1}(Y) such that the image t_{C}(p(ϕ_{Y})) of ϕ_{Y} in MU_{∗}(pt) is a (ν_{n}, l)-element. Then for any choice of a morphism ϕ_{n} satisfying the conditions of Lemma 3.23 the following diagram commutes up to multiplication by a nonzero element in Z/l

Φ̃_{n}^{∗,∗}(X) →^{p^{∗}}  Φ̃_{n}^{∗,∗}(Y ∧ X)
v ↓  ↓ p_{ϕ_{Y} ,ϕn}
H̃^{∗,∗}(X , Z/l) →^{u} Φ̃_{n}^{∗−2(l^{n}−1),∗−(l^{n}−1)}(X)

Proof: It follows from the construction of p_{ϕ_{Y},ϕ_{n}} that the composition p_{ϕ_{Y},ϕ_{n}}∘ p^{∗} is given by the morphism

f : T^{l^{n}−1} ∧ Φ_{n} ⟶^{p_{∗}(ϕ_{Y})∧Id} MGL ∧ Φ_{n} →^{ϕ_{n}} Φ_{n}

By Theorem 3.14 and the definition of Φ_{n} we have

Hom_{SH}(T^{l^{n}−1} ∧ Φ_{n}, Φ_{n}) = Z/l

The generator of this group corresponds to the composition of the left vertical and the lower horizontal arrows of our diagram. Thus to prove the proposition it is sufficient to show that the morphism f is not zero. Denote the (ν_{n}, l)-element t_{C}(p_{∗}(ϕ_{Y})) by a. By Proposition 3.18 and Lemma 3.19 we have t_{C}(Φ_{n}) = Φ_{n}^{top}. Thus applying the topological realization functor we reduce the problem to showing that the composition

t_{C}(f) : S^{2(l^{n}−1)} ∧ Φ_{n}^{top} →^{a∧Id} MU ∧ Φ_{n}^{top} →^{t_{C}(ϕ_{n})} Φ_{n}^{top}

is not zero. This follows immediately from Lemma 3.6.

Let k be a subfield in C and X be a variety over k. Denote by I_{X} the image of the homomorphism

⊕_{i≥0}MGL_{2i,i}(X) → ⊕_{i≥0}MGL_{2i,i}(Spec(k)) →^{t_{C}} MU_{∗}(pt).

Note that Theorem 3.22 implies that I_{X} contains classes of complex varieties Y (C) for all smooth projective varieties Y over k such that Hom(Y, X) ≠ ∅.

Theorem 3.25 Let k be a subfield in C and X be a smooth variety over k such that I_{X} contains a (ν_{n}, l)-element. Denote by X̃ the object of SH given by the distinguished triangle

X̃ → Σ_{T}^{∞}Č(X)_{+} → S^{0} → X̃[1]

Then H̃M_{n}^{p,q}(X̃) = 0 for all p, q ∈ Z.

Proof: Since Hom(X, X) ≠ ∅, Lemma 3.8 implies that the morphism

Σ_{T}^{∞}Č(X)_{+} ∧ Σ_{T}^{∞}X_{+} → Σ_{T}^{∞}X_{+}

is an isomorphism in SH and therefore Σ_{T}^{∞}X_{+} ∧ X̃ = 0. Proposition 3.24 implies now that the homomorphisms

Φ̃_{n}^{p,q}(X̃) → H̃^{p,q}(X̃, Z/l) → Φ̃_{n}^{p−2(l^{n}−1),q−(l^{n}−1)}(X̃)

are zero for all p, q ∈ Z which is tautologically equivalent to the statement of the theorem.

Remark: One can easily see that for any X the subset I_{X} is an ideal in MU_{∗}. Using a little more of the general theory one can show that this ideal is in fact invariant under the action of the Landweber-Novikov operations on MU_{∗}. In particular it is true that if I_{X} contains a (ν_{n}, l)-element for some n and l then it also contains (ν_{m}, l)-elements for all m ≤ n. It is also true that if I_{X} contains an element satisfying the second condition of Definition 3.5 for siome n and l then it contains a (ν_{n}, l)-element.

Remark: There is another way to prove Theorem 3.25 which does not use algebraic cobordisms and the topological realization functor but it is less enlightening than the one we give here. In order to get an alternative proof replace the condition that I_{X} contains a (ν_{n}, l)-element by the condition that there exists a (ν_{n}, l)-variety Y such that Č(X)×Y → Y is a weak equivalence. Consider the Thom spectrum T h(N_{X}) of the normal bundle to X which comes together with a canonical stable map T^{∧(l^{n}−1)} → T h(N_{X}). Let cone be the cone of this map in SH. Then we have T h(N_{X})∧X̃ ≅ 0 and therefore the morphism cone[−1]∧X̃ → T^{∧(l^{n}−1)}∧X̃ is a weak equivalence. The Thom class T h(N_{X}) → H_{Z} gives a canonical morphism cone[−1] → H_{Z}[−1] which, after smashing with X̃, gives a map S_{s}^{1} ∧ T^{∧(l^{n}−1)} ∧ X̃ → X̃ ∧ H_{Z}. This map gives a homomorphism of graded abelian groups

ϕ : H^{∗,∗}(X̃, Z/l) → H^{∗−2l^{n}+1,∗−l^{n}+1}(X̃, Z/l).

Using the realations between the action of the Steenrod operations on the Thom class and characteristic classes (which can be proven in the algebrogeometrical context using the splitting principle in exactly the same way as in topology) one can show that for any x in H^{∗,∗}(X̃, Z/l) one has

x = Q_{n}(ϕ(x)) ± ϕ(Q_{n}(x)).

Therefore, ϕ is a contracting homotopy for the complex (H^{∗,∗}(X̃, Z/l), Q_{n}) which shows that H̃M_{n}^{∗,∗} = 0.

4 Pfister quadrics and their motives.

In this section we prove the following theorem by constructing varieties satisfying the conditions of Theorem 2.25 for l = 2 and any n.

Theorem 4.1 Let k be a field of characteristic zero. Then for any q ∈ Z one has H_{L}^{q+1,q}(Spec(k), Z_{(2)}) = 0.

We will deduce Conjecture 1.1 from Theorem 4.1 in Section 5.

Let k be a field of characteristic not equal to 2. For elements a_{1}, …, a_{n} in k^{∗} let < a_{1}, …, a_{n} > be the quadratic form ∑ a_{i}x_{i}^{2}. Let further

<< a_{1}, …, a_{n} >>=< 1, −a_{1} > ⊗ … ⊗ < 1, −a_{n} > .

Denote by Q\underline{a} = Q_{a_{1},...,a_{n}} the projective quadric of dimension 2^{n−1} − 1 given by the equation << a_{1}, …, a_{n−1} >>= a_{n}t^{2}. This quadric is called the norm quadric associated with the sequence (a_{1}, …, a_{n}). The following result is well known but we decided to include the proof since it is crucial for our main theorem.

Proposition 4.2 The quadric Q\underline{a} is a 2-splitting variety for \underline{a}.

Proof: We are going to show now that if Q\underline{a} has a rational point over k then \underline{a} is divisible by 2 in K_{n}^{M}(k). In particular it implies that \underline{a} is divisible by 2 in the generic point of Q\underline{a}. We need first the following result.

Lemma 4.3 For any \underline{a} = (a_{1}, …, a_{n}) the following two conditions are equivalent

1. Q\underline{a} has a k-rational point (i.e. the form

<< a_{1}, …, a_{n−1} >> ⊕ < −a_{n} >

represents zero over k).

2. The form << a_{1}, …, a_{n} >> represents zero over k.

Proof: Since << a_{1}, …, a_{n−1} >> ⊕ < −a_{n} > is a subform in << a_{1}, …, a_{n} >> the first condition implies the second one for obvious reasons. Let us show that if << a_{1}, …, a_{n} >> represents zero over k then so does << a_{1}, …, a_{n−1} >> ⊕ < −a_{n} >. Denote the quadric given by the equation << a_{1}, …, a_{n} >>= 0 by P\underline{a}. By [6, Corollary 1.6] we may assume that << a_{1}, …, a_{n} >> is hyperbolic. The quadric Q\underline{a} is a section of P\underline{a} by a linear subspace L of codimension 2^{n−1} − 1 in P^{2^{n}−1}. On the other hand since P\underline{a} is hyperbolic for any rational point p of P\underline{a} there exists a linear subspace H of dimension dim(P\underline{a})/2 = 2^{n−1} − 1 which lies on P\underline{a} and passes through p. The intersection of H and L is a point on Q\underline{a}.

To rpove Proposition 4.2 we proceed by induction on n. Consider first the case n = 2. Then Q\underline{a} is given by the equation x^{2} − a_{1}y^{2} = a_{2}z^{2}. We may assume that it has a point of the form (x_{0}, y_{0}, 1) (otherwise a_{1} is a square root in k and the statement is obvious). Then a_{2} is the norm of the element x_{0} + a_{1}^{1/2}y_{0} from k(a_{1}^{1/2}) and thus the symbol (a_{1}, a_{2}) is divisible by 2.

Suppose that the lemma is proven for weights smaller than n. The quadric Q\underline{a} is given by the equation << a_{1}, …, a_{n−1} >>= a_{n}t^{2}. The form << a_{1}, …, a_{n−1} >> is of the form < 1 > ⊕μ\underline{a}. By induction we may assume that our point q ∈ Q\underline{a}(k) belongs to the affine part t ≠ 0. Consider the plane L generated by points (1, 0, …, 0) and q. The restriction of the quadratic form << a_{1}, …, a_{n−1} >> to L is of the form << b >> for some b (the idea is that L is a “subfield” in the vector space where << a_{1}, …, a_{n−1} >> lives). Consider the field extension k(b^{1/2}). The form << b >> and therefore the form << a_{1}, …, a_{n−1} >> represents zero over k(b^{1/2}) and thus by the inductive assumption (a_{1}, …, a_{n−1}) = 0 in K_{n−1}^{M}(k(b^{1/2}))/2. On the other hand by the construction << b >> represents a_{n} and therefore we have a_{n} ∈ ImN_{k(b^{1/2})/k} ⊂ k^{∗} which proves the proposition.

The fact that Q\underline{a} is rational over its function field follows from the observation that any quadric with a rational point is rational and that any variety has a rational point over its function field. Therefore to finish the proof of Theorem 4.1 it remains to show that H^{n+1,n}(Č(Q\underline{a}), Z_{(2)}) = 0.

Denote the simplicial scheme Č(Q\underline{a}) by X\underline{a}. Our proof goes in two steps. First we use two results due to Markus Rost to show that

H^{2^{n}−1,2^{n−1}}(X\underline{a}, Z_{(2)}) = 0.

Then we use Theorem 3.25 to show that the group H^{n+1,n}(X\underline{a}, Z_{(2)}) embeds into H^{2^{n}−1,2^{n−1}}(X\underline{a}, Z_{(2)}) and therefore is zero as well.

Theorem 4.4 For any n > 1 there exists a distinguished triangle in DM_{−}^{eff} of the form

M(X\underline{a})(2^{n−1} − 1)[2^{n} − 2] → M\underline{a} → M(X\underline{a})→M(X\underline{a})(2^{n−1} − 1)[2^{n} − 1]

such that M\underline{a} is a direct summand of M(Q\underline{a}).

The proof of Theorem 4.4 is based on the following important result which is due to M. Rost (see [19])^{2}.

2There is another approach to the proof of Theorem 4.4 which does not require Theorem 4.5 but uses more of the theory developed in Section 3.

Theorem 4.5 (M. Rost) There exists a direct summand M\underline{a} of the motive M(Q\underline{a}) together with two morphisms

ψ^{∗} : Z(2^{n−1} − 1)[2^{n} − 2] → M\underline{a}
ψ_{∗} : M\underline{a} → Z

such that for any field F over k where Q\underline{a} has a point the sequence

Z(2^{n−1} − 1)[2^{n} − 2] → (M\underline{a})_{F} → Z

is a splitting distinguished triangle in DM^{eff}(F).

The compositions

Z(2^{n−1} − 1)[2^{n} − 2] → M\underline{a} → M(Q\underline{a})
M(Q\underline{a}) → M\underline{a} → Z

are equal to the fundamental class and the canonical morphism M(Q\underline{a}) → Z = M(Spec(k) respectively.

Remark: Theorem 4.5 is formulated in [19] in the context of Chow motives. It implies the same result in the category DM^{eff}(k) by [25].

Theorem 4.4 is, in fact, a rather formal corollary of Theorem 4.5. We start with the following lemmas.

Lemma 4.6 The sequence of morphisms

M(Q\underline{a})(2^{n−1} − 1)[2^{n} − 2] →^{Id⊗ψ^{∗}} M(Q\underline{a}) ⊗ M\underline{a} →^{Id⊗ψ_{∗}} M(Q\underline{a})

is a splitting distinguished triangle.

Proof: Let cone be a cone of the morphism ψ^{∗} : Z(2^{n−1} − 1)[2^{n} − 2] → M\underline{a}. The morphism ψ_{∗} : M\underline{a} → Z factors through a well defined morphism ϕ : cone → Z. By Theorem 4.5 ϕ is an isomorphism in the generic point of Q\underline{a}. It implies immediately by general properties of category DM^{eff}(k) that ϕ ⊗ Id_{M(Q\underline{a})} is an isomorphism. Thus the sequence

M(Q\underline{a})(2^{n−1} − 1)[2^{n} − 2] → M(Q\underline{a}) ⊗ M\underline{a} → M(Q\underline{a})

can be extended to a distinguished triangle. This triangle splits since by duality ([25, Theorem 4.3.2]) we have

Hom(M(Q\underline{a}), M(Q\underline{a})(2^{n−1} − 1)[2^{n} − 1]) = H^{2^{n+1}−3,2^{n}−2}(Q\underline{a} × Q\underline{a}, Z)

and the right hand side group is zero by Corollary 2.3(2).

Lemma 4.7 Consider the canonical morphism M(X\underline{a}) → Z. Let further N be an object of the thick^{3} (resp. localizing) subcategory in DM^{eff}(k) generated by objects of the form M(Q\underline{a})⊗L. Then this morphism induces isomorphisms

Hom(Z, N) → Hom(M(X\underline{a}), N)

(resp. isomorphisms Hom(N, M(X\underline{a})) → Hom(N, Z)).

Proof: It is clearly sufficient to consider N = M(Q\underline{a}) ⊗ L for some L in DM_{−}^{eff}(k). In this case our result follows from duality since the morphism M(X\underline{a}) ⊗ M(Q\underline{a}) → M(Q\underline{a}) is an isomorphism by Lemma 3.8.

By Lemma 4.7 the morphism ψ_{∗} : M\underline{a} → Z has a canonical lifting to a morphism ψ̃_{∗} : M\underline{a} → M(X\underline{a}). Together with the composition

ψ̃^{∗} : M(X\underline{a})(2^{n−1} − 1)[2^{n} − 2] → Z(2^{n−1} − 1)[2^{n} − 2] →^{ψ^{∗}} M\underline{a}

it gives us a sequence of morphisms

M(X\underline{a})(2^{n−1} − 1)[2^{n} − 2] →^{ψ̃^{∗}} M\underline{a} →^{ψ̃_{∗}} M(X\underline{a})

such that ψ̃_{∗} ∘ ψ̃^{∗} = 0. Let cone be a cone of ψ̃^{∗}. Then ψ̃_{∗} factors through a unique morphism ϕ : cone → M(X\underline{a}) and we have to show that ϕ is an isomorphism. The category of objects N such that ϕ⊗Id_{N} is an isomorphism is a localizing subcategory. Since M(X\underline{a}) ⊗ M(Q\underline{a}) = M(Q\underline{a}) we conclude from Lemma 4.6 that it contains M(Q\underline{a}) and therefore it contains M(X\underline{a}). On the other hand we have a commutative diagram

cone ⊗ M(X\underline{a}) → M(X\underline{a}) ⊗ M(X\underline{a})
↓  ↓
cone  →  M(X\underline{a})

3The thick (resp. localizing) subcategory in a triangulated category D generated by a class of objects A is the minimal triangulated subcategory in D which is closed under direct summands (resp. direct summands and arbitrary direct sums) and contains A.

with both vertical arrows are clearly isomorphisms. This finishes the proof of Theorem 4.4.

Theorem 4.8 (M. Rost) For \underline{a} = (a_{1}, …, a_{n}) the natural homomorphism

H^{2^{n}−1,2^{n−1}}(Q\underline{a}, Z) → k^{∗}

is a monomorphism.

Proof: Since the dimension of Q\underline{a} equals 2^{n−1} − 1 the left hand side group is isomorphic to the group H^{2^{n−1}−1}(Q\underline{a}, \underline{K}_{2^{n−1}}^{M}) by Corollary 2.4(1) and our result follows from [18].

Corollary 4.9 For \underline{a} = (a_{1}, …, a_{n}) we have

H^{2^{n}−1,2^{n−1}}(X\underline{a}, Z) = 0.

Proof: By Theorem 4.4 we have an exact sequence

H^{0,1}(X\underline{a}, Z_{(2)}) → H^{2^{n}−1,2^{n−1}}(X\underline{a}, Z) → H^{2^{n}−1,2^{n−1}}(M\underline{a}, Z) → H^{1,1}(X\underline{a}, Z)

Since M\underline{a} is a direct summand of M(Q\underline{a}) Theorem 4.8 implies that the motivic cohomology group H^{2^{n}−1,2^{n−1}}(M\underline{a}, Z) embeds to the group H^{2^{n}−1,2^{n−1}}(M̄\underline{a}, Z) where M̄\underline{a} is a lifting of M\underline{a} to the algebraic closure of k. Since

H^{2^{n}−1,2^{n−1}}(X\underline{a} × Spec(k̄), Z) = H^{2^{n}−1,2^{n−1}}(Spec(k̄), Z) = 0

we conclude that the second arrow in this exact sequence is zero. On the other hand by H90(1, 2), Corollary 2.13(1) and Proposition 2.7 we have

H^{0,1}(X\underline{a}, Z_{(2)}) = H^{0,1}(Spec(k), Z) = 0

which finishes the proof of the corollary.

Proposition 4.10 For any sequence (a_{1}, …, a_{n}) one has

H^{n+1,n}(X\underline{a}, Z_{(2)}) = 0.

Proof: Consider the object X̃\underline{a} in SH(k) given by the distinguished triangle

X̃\underline{a} → Σ_{T}^{∞}(X\underline{a})_{+} → S^{0} → X̃\underline{a}[1]

Since H^{p,q}(Speck(k), Z) = 0 for p > q (2.3(1)) the canonical homomorphisms H̃^{p,q}(X\underline{a}, Z) → H^{p,q}(X̃\underline{a}, Z) are isomorphisms for p > q. Thus it is sufficient to verify that H̃^{n+1,n}(X̃\underline{a}, Z_{(2)}) = 0. Since there exists an extension of degree two E/k such that Q\underline{a}(Spec(E)) ≠ ∅ Lemma 3.8 together with the standard transfer argument implies that all the motivic cohomology groups of X̃\underline{a} have exponent at most 2. Thus it is sufficient to show that the image of H̃^{n+1,n}(X̃\underline{a}, Z_{(2)}) in H̃^{n+1,n}(X̃\underline{a}, Z/2) is zero. Let u be an element of this image. Consider the composition of cohomological operations Q_{n−2}Q_{n−3} … Q_{1}. It maps u to an element of H̃^{2^{n}−1,2^{n−1}}(X̃\underline{a}, Z/2) and Theorem 3.17(3) implies that this element belongs to the image of the corresponding integral motivic cohomology group. Thus by Corollary 4.9 it is zero. It remains to verify that the composition Q_{n−2}Q_{n−3} … Q_{1} is a monomorphism.

Lemma 4.11 The ideal I_{Q\underline{a}} contains (ν_{i}, 2)-elements for all i ≤ n − 1.

Proof: Fix an embedding k ⊂ C. By Theorem 3.22 the ideal I_{Q\underline{a}} contains the cobordisms classes of all smooth subvarieties inQ\underline{a} which includes in particular all the plane sections. Thus it is sufficient to show that a quadric Q in CP^{2^{i}} satisfies the conditions of Definition 3.5 for l = 2 and n = i. Let j : Q → CP^{2^{i}} be the embedding. Since Q is the set of zeros of a generic section of the line bundle O(2) on CP^{2^{i}} we have a short exact sequence

0 → T_{Q} → j^{∗}(T_{CP}_{2^{i}}) → j ∗ (O(2)) → 0

and therefore the class of the normal bundle to Q equals to j^{∗}(T_{CP}_{2^{i}} − O(2)) in K_{0}(Q). In particular any characteristic number of Q can be obtained by evaluating a cohomology class from H^{2(2^{i}−1)}(CP^{2^{i}}, Z) on the fundamental class of the quadric [Q] in H_{2(2^{i}−1)}(CP^{2^{i}}, Z). The calss [Q] is divisible by 2 which implies that all the chracteristic numbers of Q are divisible by 2. To compute s(Q) we may use the fact that the characteristic classes s_{i}(−) are additive. In particular we have:

s(Q) = −s_{2^{i}−1}(T_{Q})[Q] = (s_{2^{i}−1}(T_{CP}_{2^{i}}) − s_{2^{i}−1}(O(2)))[Q]

where on the right hand side [Q] denotes the fundamental class of Q in H_{2(2^{i}−1)}(CP^{2^{i}}, Z). We have T_{CP}_{2^{i}} = (2^{i} + 1)O(1) in K_{0}(CP^{2^{i}}) and therefore

−s(Q) = (s_{2^{i}−1}(T_{CP}_{2^{i}}) − s_{2^{i}−1}(O(2)))[Q] = 2((2^{i} + 1) − 2^{2^{i}−1}) = 2(mod 4).

By Theorem 3.25 and Lemma 4.11 we have H̃M_{i}^{∗,∗}(X̃\underline{a}) = 0 for all i ≤ n − 1. To show that the composition Q_{n−2} … Q_{1} is a monomorphism it is sufficient to show that the operation Q_{i} acts monomorphically on the group H̃^{n−i+2(2^{i−1}−1),n−i+2^{i−1}−1}(X̃\underline{a}, Z/2) for i = 1, …, n − 2. Since H̃M_{i}^{∗,∗}(X̃\underline{a}) = 0 the kernel of Q_{i} on this group is covered by the image of H̃^{n−i−1,n−i}(X̃\underline{a}, Z/2). The later group is zero by the inductive assumption that H90(n-1,2) holds, Proposition 2.7 and Corollary 2.13(2).

5 Applications.

In this section we prove some corollaries of our main theorem 4.1. We start with the following result.

Theorem 5.1 (Milnor Conjecture) For any field k of characteristic not equal to 2 and any n ≥ 0, m > 0 the norm residue homomorphisms

K_{n}^{M}(k)/2^{m} → H_{et}^{n}(k, μ_{2^{m}}^{⊗n})

are isomorphisms.

Proof: For fields k of characteristic zero this follows immediately from Theorem 4.1 and Corollary 2.13. To prove the theorem for fields of positive characteristic we need the following lemma.

Lemma 5.2 Let n be an integer and l be a prime such that for any field k of characteristic zero the norm-residue homomorphisms

K_{n}^{M}(k)/l → H_{et}^{n}(k, μ_{l}^{⊗n})

are isomorphisms. Then the same holds for all fields k of characteristic different from l.

Proof: The standard transfer argument shows that it is sufficient to prove that the norm-residue homomorphism is an isomorphism for fields k which have no extensions of degree prime to l. In particular we may assume that k is a perfect field. Then the standard Witt ring construction gives us a complete discrete valuation ring R with the function field K of characteristic zero and the residue field k. Define the Milnor K-ring K_{∗}^{M}(R) as the quotient of the tensor algebra of R^{∗} with respect to the ideal generated by elements of the form x ⊗ (1 − x). The natural homomorphism K_{n}^{M}(R)/l → K_{n}^{M}(k)/l is an isomorphism since R is henselian. Choosing a generator π ∈ K^{∗} of K^{∗} over R^{∗} we may construct a commutative diagram of the form

K_{n}^{M}(k)/l →^{i_{K}}  K_{n+1}^{M}(K)/l  →^{p_{K}} K_{n}^{M}(k)/l
↓  ↓  ↓
H_{et}^{n}(k, μ_{l}^{⊗n}) →^{i_{H}} H_{et}^{n+1}(K, μ_{l}^{⊗(n+1)}) →^{p_{H}} H_{et}^{n}(k, μ_{l}^{⊗n})

as follows. Take the vertical arrows to be the norm-residue homomorphisms. To get the left horizontal arrows observe that there are canonical homomorphisms

K_{n}^{M}(k)/l ≅ K_{M}^{n}(R)/l → K_{n}^{M}(K)/l
H_{et}^{n}(k, μ_{l}^{⊗n}) ≅ H_{et}^{n}(Spec(R), μ_{l}^{⊗n}) → H_{et}^{n}(K, μ_{l}^{⊗n})

and we define i_{K} and i_{H} as compositions of these homomorphisms with the multiplication by the class of π. We define p_{K} as the standard residue map which is determined by the property that its value on a symbol of the form (π^{m}a_{1}, …, a_{n+1}) where a_{i} ∈ R^{∗} equals m(a_{2}, …, a_{n+1}). To define p_{H} proceed as follows. Let j : Spec(K) → Spec(R) and x : Spec(k) → Spec(R) be the canonical open and closed embeddings. By [11, Lemma 1.4 p.183] we have Rj_{∗}(G_{m}) = j_{∗}(G_{m}). We have a natural homomorphism j_{∗}(G_{m}) → i_{∗}(Z) and therefore we get a morphism in the derived category

Rj_{∗}(μ_{l}^{⊗n+1}) = (Rj_{∗}(G_{m}) ⊗^{L}_{Z} Z/l) ⊗_{Z/l} μ_{l}^{⊗n} → i_{∗}(Z/l) ⊗_{Z/l} μ_{l}^{⊗n} = i_{∗}(μ_{l}^{⊗n})

which gives the morphism p_{H} on the etale cohomolgy groups. One verifies easily that this diagram is indeed commutative and that the compositions of the horizontal arrows are identity homomorphisms. That finishes the proof of Lemma 5.2.

Corollary 5.3 For any field k of characteristic not equal to 2 and any n ≥ 0 the etale cohomology group H_{et}^{n+1}(k, Z_{2}(n)) is torsion free.

The following nice corollary of Theorem 5.1 is due to S. Bloch.

Corollary 5.4 Let α ∈ H^{i}(X, Z) be a 2-torsion element in the integral cohomology of a complex algebraic variety X. Then there exists a divisor Z on X such that the restriction of α to X − Z is zero.

Proof: Since 2α = 0, α is the image of an element α′ in H^{i−1}(X, Z/2) with respect to the Bockstein homomorphism β : H^{i−1}(X, Z/2) → H^{i}(X, Z). By Theorem 5.1 there exists a dense open subset U = X −Z of X such that α′ on U is in the image of the canonical map (O^{∗}(U))^{⊗(i−1)} → H^{i−1}(U, Z/2). Since this map factors through the integral cohomology group we have β(α′) = 0 on U.

Denote by B/m(n) the complex truncation at the cohomological level n of the complex of sheaves Rπ_{∗}(π^{∗}(μ_{m}^{⊗n})) where π : (Sm/k)_{et} → (Sm/k)_{Nis} is the canonical morphism of sites. Combining Theorem 4.1 with [22, Theorem 5.9] we get.

Theorem 5.5 For a smooth variety X over a field of characteristic zero and any m > 0 there are canonical isomorphisms

H^{p,q}(X, Z/2^{m}) = H_{Nis}^{p}(X, B/2^{m}(q))

In particular, for any X as above H^{p,q}(X, Z/2^{m}) = 0 for p < 0.

In [3] S. Bloch and S. Lichtenbaum constructed a spectral sequence which starts from Higher Chow groups of a field and converges to its algebraic Ktheory. Combining their result with Theorem 5.5 and the comparison result relating Higher Chow groups and motivic cohomology ([25, Prop. 4.2.9 and Th. 4.3.2]) we obtain the following.

Theorem 5.6 Let k be a field of characteristic zero. Then there exists a spectral sequence, natural with respect to extensions of k, with the E_{2}-term of the form

E_{2}^{p,q} = \{ \begin{matrix}H_{et}^{p−q}(k, Z/2) & for p, q ≤ 0 \\ 0 & otherwise \end{matrix}

which converges to K_{−p−q}/2(k) (K-theory with Z/2-coefficients).

Corollary 5.7 Let X be a complex algebraic variety of dimension d. Then the canonical homomorphisms K_{i}^{alg}/2(X) → K_{i}^{top}/2(X) are isomorphisms for i ≥ d − 1 and monomorphisms for i = d − 2.

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