On 2-torsion in motivic cohomology

Vladimir Voevodsky^{1}^{,}^{2}

June 2001

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 The degree map . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 The motivic analog of Margolis homology . . . . . . . . . . . . 13 4 Norm quadrics and their motives . . . . . . . . . . . . . . . . 18 5 Computations with Galois cohomology . . . . . . . . . . . . . 25 6 Beilinson-Lichtenbaum conjectures . . . . . . . . . . . . . . . 31 7 Main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 8 Appendix A. Hypercohomology of pointed simplicial sheaves . 42 9 Appendix B. Čech simplicial schemes . . . . . . . . . . . . . . 44

1 Introduction

In the early 1980-ies Alexander Beilinson and Stephen Lichtenbaum (see [13], [2, 5.10.D]) introduced the idea of motivic complexes Z(n) and formulated a set of conjectures describing their properties. Among these conjectures was the following one which they called the generalized Hilbert 90 property.

Conjecture 1.1 For any field k and any n ≥ 0 one has

H_{et}^{n+1}(Spec(k), Z(n)) = 0

1The work on this paper was supported by NSF grants DMS-96-25658, DMS-97-29992 and DMS-99-01219, Abrose Monell Foundation, Clay Mathematics Institute, Sloan Research Fellowship and Veblen Fund

2School of Mathematics, Institute for Advanced Study, Princeton NJ, USA. e-mail:

vladimir@ias.edu

As we know today it implies an amazing number of other conjectures about Galois cohomology, motivic cohomology and algebraic K-theory. The goal of this paper is to prove the 2-local version of Conjecture 1.1.

Theorem 1.2 For any field k and any n ≥ 0 one has

H_{et}^{n+1}(Spec(k), Z_{(2)}(n)) = 0

As a corollary of Theorem 1.2 we get the following result.

Theorem 1.3 Let k be a field of characteristic not equal to 2. Then the norm residue homomorphism

K_{∗}^{M}(k)/2 → H^{∗}(k, Z/2)

is an isomorphism.

The statement of Theorem 1.3 is known as the Milnor Conjecture or the Z/2-coefficients case of the Bloch-Kato conjecture. The general “Bloch-Kato Conjecture” can be described as follows. Let k be a field and l a prime number different from the characteristic of k. Fix a separable closure k_{sep} of k and let μ_{l} denote the group of l-th roots of unity in k_{sep}. One may consider μ_{l} as a Gal(k_{sep}/k)-module. By definition of μ_{l} one has a short exact sequence

1 ⟶ μ_{l} ⟶ k_{sep}^{∗} ⟶^{z^{l}} k_{sep}^{∗} ⟶ 1

which is called the Kummer sequence. The boundary map in the associated long exact sequence of Galois cohomology is a homomorphism

k^{∗} → H^{1}(k, μ_{l})  (1)

In [1], Bass and Tate proved that for a ∈ k^{∗} − {1} the cohomology class (a) ∧ (1 − a) lying in H^{2}(k, μ_{l}^{⊗2}) is zero i.e. that the homomorphism (1) extends to a homomorphism of rings

T (k^{∗})/I → H^{∗}(k, μ_{l}^{⊗∗})  (2)

where T (k^{∗}) is the tensor algebra of the abelian group k^{∗} and I the ideal generated by elements of the form a ⊗ b for a, b ∈ k^{∗} such that a + b = 1. The graded components of the quotient T (k^{∗})/I are known as the Milnor K-groups of k and the homomorphism (2) is usually written as

K_{∗}^{M}(k) → H^{∗}(k, μ_{l}^{⊗∗})  (3)

Let K_{∗}^{M}(k)/l be the quotient of K_{∗}^{M}(k) by the ideal of elements divisible by l. It is clear that (3) factors through a map

K_{∗}^{M}(k)/l → H^{∗}(k, μ_{l}^{⊗∗})  (4)

which is called the norm residue homomorphism. Theorem 1.3 proves the following conjecture for l = 2.

Conjecture 1.4 The map (4) is an isomorphism for any field k of characteristic ≠ l.

This conjecture has a long and convoluted history. The map (4) is clearly an isomorphism in degree zero. In degree one, (4) is a monomorphism and its cokernel is the group of l-torsion elements in H^{1}(k, k_{sep}^{∗}) which is known to be zero as a corollary of the classical Hilbert 90 Theorem.

In degree 2 the homomorphism (4) has an interpretation in terms of central simple algebras. The question about central simple algebras which is equaivalent to surjectivity of (4) in degree 2 seems to be very old. The question of injectivity of (4) in degree 2 was explicitly stated by John Milnor in [20, Remark on p. 147].

In 1981 Alexander Merkurjev published a paper [16] where he proved that (4) is an isomorphism in degree 2 for l = 2 and any field k (such that char(k) ≠ 2). This paper is the starting point of all the further work on bijectivity of (4). In 1982 Merkurjev together with Andrei Suslin proved that (4) is an isomorphism in degree 2 for all l (see [17]).

In degree 3 and l = 2 the bijectivity of (4) was proved by Merkurjev and Suslin in [18] and independently by Markus Rost in [25].

In [19] Milnor considered the homomorphism (4) in all degrees for l = 2 as a part of his investigation of the relations between K_{∗}^{M}(k)/2 and quadratic forms over k. He mentioned that he does not know of any fields k for which (4) fails to be an isomorphism and gave several examples of classes of fields k for which (4) is an isomorphism in all degrees. His examples extend a computation made by Bass and Tate in the same paper where they introduce (1). This paper of Milnor is the reason why Conjecture 1.4 for l = 2 is called the Milnor Conjecture.

The name “Bloch-Kato conjecture” comes from a 1986 paper by Spencer Bloch and Kazuya Kato [3] where they mention (on p. 118) that “one conjectures [the homomorphism (4)] to be an isomorphism quite generally”. This name is certainly incorrect because Conjecture 1.4 appeared quite explicitly in the work of Beilinson [2] and especially Lichtenbaum [13] which was published much earlier.

The list of hypothetical properties of motivic complexes of Beilinson and Lichtenbaum implied that

H_{Zar}^{n}(Spec(k), Z/l(n)) = K_{M}^{n}(k)/l  (5)

and if l ≠ char(k) then

H_{et}^{n}(Spec(k), Z/l(n)) = H^{n}(k, μ_{l}^{⊗n})  (6)

The norm residue homomorphism in this language is just the canonical map from the motivic cohomology in the Zariski topology to the motivic cohomology in the etale topology. The conjectures made by Beilinson and Lichtenbaum implied the following:

Conjecture 1.5 Let X be a smooth variety over a field k. Then the map

H_{Zar}^{p}(X, Z(q)) → H_{et}^{p}(X, Z(q))

from the motivic cohomology of X in the Zariski topology to the motivic cohomology of X in the etale topology is an isomorphism for p ≤ q + 1.

In view of (5) and (6) Conjecture 1.5 implies Conjecture 1.4. Note also that since for any field k one has H_{Zar}^{n+1}(Spec(k), Z(n)) = 0, Conjecture 1.1 is a particular case of Conjecture 1.5.

The relation between Conjecture 1.5 and Conjecture 1.4 was further clarified in [31] and [8] where it was shown that the l-local version of Conjecture 1.5 for fields k such that char(k) ≠ l is in fact equivalent to Conjecture 1.4 and moreover that it is sufficient to show only the surjectivity of (4), because the injectivity formally follows. In Theorem 6.4 we show using results of [31] and [8] that the l-local version of Conjecture 1.1 implies the l-local version of Conjecture 1.5 for char(k) ≠ l.

In [9] Thomas Geisser and Marc Levine proved the p-local version of Conjecture 1.5 for schemes over fields of characterstic p. They use the version of motivic cohomology based on the higher Chow groups which is now known to be equivalent to the version used here by [30], [7] and [35]. Together with our Theorem 6.4 this result implies in particular that Conjecture 1.5 is a corollary of Conjecture 1.1 for all k.

The first version of the proof of Theorem 1.2 appeared in [32]. It was based on the idea that there should exist algebraic analogs of the higher Morava K-theories and that the m-th algebraic Morava K-theory can be used for the proof of Conjecture 1.4 for l = 2 and n = m + 2 in the same way as the usual algebraic K-theory is used in Merkurjev-Suslin proof of Theorem 1.3 in degree 3 in [18]. This approach was recently validated by Simone Borghesi [5] who showed how to construct algebraic Morava K-theories (at least in characteristic zero).

The second version of the proof appeared in [33]. Instead of algebraic Morava K-theories it used small pieces of these theories which are easy to construct as soon as one knows some facts about the cohomological operations in motivic cohomology and their interpretation in terms of the motivic stable homotopy category.

The main difference between the present paper and [33] is in the proof of Theorem 3.2 ([33, Theorem 3.25]). The approach used now was outlined in [33, Remark on p.39]. It is based on the connection between cohomological operations and characteristic classes and circumvents several technical ingredients of the older proof. The most important simplification is due to the fact that we can now completely avoid the motivic stable homotopy category and the topological realization functor.

Another difference between this paper and [33] is that we can now prove all the intermediate results for fields of any characteristic. Several developments made this possible. The new proof of the suspension theorem for the motivic cohomology [38, Theorem 2.4] based on the comparison between the motivic cohomology and the higher Chow groups established in [30], [7] and [35] does not use resolution of singularities. The same comparison together with the new proof of the main result of [31] by Thomas Geisser and Marc Levine in [8] allows one to drop the resolution of singularities assumption in the proof of Theorem 6.4. Finally, the new approach to the proof of Theorem 3.2 does not require the topological realization functor which only exists in characteristic zero.

The paper is organized as follows. In Sections 2-5 we prove several results about motivic cohomology which are used in the proof of our main theorem but which are not directly related to the Belinson-Lichtenbaum conjecturs. In Section 3 this is Corollary 3.5, in Section 4 Theorems 4.3 and 4.9 and in Section 5 Theorem 5.9. The proof of Corollary 3.5 uses Theorem 2.6 of Section 2. There are no other connections of these four sections to each other or to the remaining sections of the paper.

In Section 6 we show that Conjecture 1.5 is a corollary of Conjecture 1.1 and that Conjecture 1.4 is a corollary of Conjecture 1.5. This section is independent of the previous four sections.

In Section 7 we prove Theorem 1.2. This section also contains some corollaries of the main theorem. Using similar techniques one can also prove the Milnor Conjecture which asserts that the Milnor ring modulo 2 is isomorphic to the graded Witt ring of quadratic forms. For the proof of this result together with a more detailed computations of motivic cohomology groups of norm quadrics see [23].

Two appendixes contain the material which is used throughout the paper and which I could not find good references for.

All through the paper we use the Nisnevich topology [22] instead of the Zariski one. Since all the complexes of sheaves considered in this paper have transfers and homotopy invariant cohomology sheaves [36, Theorem 5.7, p.128] implies that one can replace 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 the Beilinson-Lichtenbaum conjectures. First of all I want to thank Andrei Suslin who taught me the techniques used in [17],[18]. 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 [32] 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 7.1. 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 the Steenrod algebra anymore. The same applies to Markus Rost and Alexander Vishik for not being afraid anymore of the theory of quadratic forms. Dmitri Orlov guessed the form of the distinguished triangle in Theorem 4.3 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. Finally, I would like to thank Eric Friedlander who introduced me to Anderei Suslin and helped me in many ways during the years when I was working on the Theorem 1.3.

Most of the mathematics of this paper was invented when I was a Junior Fellow of the 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 earthly. The first complete version was written during my stay in the Max-Planck Institute in Bonn. Further work was done when I was at the Northwestern University and in its final form the paper was written when I was a member of the Institute for Advanced Study in Princeton.

2 The degree map

In this section and Section 3 we use the motivic homotopy theory of algebraic varieties developed in [21]. For an introduction to this theory see also [34] and [6]. For a smooth projective variety X of pure dimension d we consider the degree map H^{2d,d}(X, Z) → Z (“evaluation on the fundamental class”). Most of the section is occupied by the proof of Theorem 2.6 where we show that the degree map can be described as the composition of the Thom isomorphism for an appropriate vector bundle and a map defined by a morphism in the pointed motivic homotopy category H_{•}. This theorem is a formal corollary of Spanier-Whitehead duality in the motivic stable homotopy theory but since the details of this duality are not worked out yet we are forced to give a very direct but not a very conceptual proof here.

Let X be a smooth projective variety of pure dimension d. As we know from [30], [7] and [35] there is an isomorphism H^{2q,q}(X, Z) = CH^{q}(X) where CH^{q}(X) is the group of rational equivalence classes of cycles of codimension q on X. In particular, H^{2d,d}(X, Z) = CH^{0}(X) and there is the degree map deg : H^{2d,d}(X, Z) → Z.

For a closed embedding i : Z → X of smooth varieties over k with the normal bundle N denote by γ_{i} the composition of the projection X_{+} → X/(X − Z) with the weak equivalence X/(X − Z) → T h_{Z}(N) constructed in [21, Theorem 2.23]. We will use the following properties of the degree map.

Lemma 2.1 Let i : X → Y be a closed embedding of smooth projective schemes over k of pure dimensions d_{X} and d_{Y} respectively. Let N be the normal bundle to i and t_{N} the Thom class of N. Then for any a ∈ H^{2d_{X},d_{X}}(X, Z) one has:

deg(γ_{i}^{∗}(a ∧ t_{N})) = deg(a)

Lemma 2.2 Let i : X → P^{d} be a smooth hypersurface of degree n. Then deg(i^{∗}(e(O(1))^{d−1})) = n.

Lemma 2.3 Let X be a smooth projective variety of pure dimension d. Then the map

H^{2d,d}(X, Z) →^{deg} Z → Z/n

is surjective if and only if X has a rational point over an extension of k of degree prime to n.

Remark 2.4 For a smooth projective variety X of dimension d and elements a ∈ H^{∗,>0}(Spec(k), Z) and x ∈ H^{∗,∗}(X) such that ax ∈ H^{2d,d}(X) one clearly has deg(ax) = 0. Lemma 2.1 together with this property and the property that deg(1) = 1 uniquely characterize the degree maps H^{2d,d}(X, Z) → Z for projective X. This can be easily seen using [38, Lemma 4.2].

For the proof of the Theorem 2.6 we will need the following construction. Let i : Z → X be a closed embedding of smooth varieties over k with the normal bundle N. Let further V be a vector bundle on X and X → V the zero section. Consider the embedding Z → X → V. Since the normal bundle to Z in V is N⊕i^{∗}(V) we have a weak equivalence V/(V −Z) → T h_{Z}(N⊕i^{∗}(V)). This gives us a map

γ_{i,V} : T h_{X}(V) ≅ V/(V − X) → V/(V − Z) ≅ T h_{Z}(N ⊕ i^{∗}(V)).

Lemma 2.5 Let x be a class in H^{∗,∗}(Z). Then one has:

γ_{i,V}^{∗} (xt_{N⊕i^{∗}(V)}) = γ_{i}^{∗}(xt_{N})t_{V}

where t_{N⊕i^{∗}(V)}, t_{V} and t_{N} are the Thom classes of the corresponding vector bundles.

Theorem 2.6 Let X be a smooth projective variety of pure dimension d over a field k. Then there exists an integer n and a vector bundle V on X of dimension n such that:

1. V + T_{X} = O^{n+d} in K_{0}(X)

2. There exists a morphism in H_{•} of the form f_{V} : T^{n+d} → T h_{X}(V) such that the map H^{2d,d}(X) → Z defined by f_{V} and the Thom isomorphisms coincides with the degree map.

Proof: In Proposition 2.7 below we show that the statement holds for X = P^{d} and a vector bundle V_{d} on P^{d} of dimension n_{d}. Let i : X → P^{m} be a closed embedding and N the normal bundle to i. Consider the composition

T^{n_{m}+m} →^{f_{Vm}} T h_{P^{m}}(V_{m}) →^{γ_{i,Vm}} T h_{X}(i^{∗}(V_{m}) ⊕ N)

If γ_{i} : P^{m} → T h_{X}(N) is the canonical map and x is a class in H^{∗,∗}(X, Z) then by Lemma 2.5 one has

γ_{i,V_{m}}^{∗}(xt_{i^{∗}(V_{m})⊕N}) = γ_{i}^{∗}(xt_{N})t_{V_{m}}

Let t be the tautological class in H̃^{2(n_{m}+m),n_{m}+m}(T^{n_{m}+m}, Z), then

f_{V_{m}}^{∗}γ_{i,V_{m}}^{∗}(xt_{i^{∗}(V_{m})⊕N}) = f_{V_{m}}^{∗}(γ_{i}^{∗}(xt_{N})t_{V_{m}}) = deg(γ_{i}^{∗}(xt_{N}))t =
= deg(x)t

where the second equality holds by Proposition 2.7 and the third one by Lemma 2.1. Finally, in K_{0}(X) we have T_{X} = i^{∗}(T_{P^{m}}) − N and therefore,

T_{X} + N + i^{∗}(V_{m}) = O^{n_{m}+m}.

Proposition 2.7 There exists a vector bundle V_{d} of dimension n_{d} on P^{d} and a morphism f_{V} : T^{d+n} → T h_{P^{d}}(V) in H_{•} such that the homomorphism

H^{2d,d}(P^{d}, Z) → Z

defined by the Thom isomorphisms and f_{V} coincides with the degree map and V + T_{P^{d}} = O^{n_{d}+d} in K_{0}(P^{d}).

Proof: Let W = Ω ⊕ (Ω ⊗ T) where T is the tangent bundle on P^{d} and Ω is its dual. The dimension of W is n = d^{2} + d.

Lemma 2.8 W + T = O^{d^{2}+2d} Proof: We have to show that T +Ω+Ω⊗T = O^{d^{2}+2d}. Consider the standard exact sequence

0 → Ω → O(−1)^{d+1} → O → 0

(see [10, Th. 8.13]) and its dual

0 → O → O(1)^{d+1} → T → 0.

The first sequence implies that Ω + O = (d + 1)O(−1), hence

T + Ω ⊗ T = T ⊗ (O + Ω) = (d + 1)T (−1).

Form the second sequence T (−1) = (d + 1)O − O(−1), hence

Ω+T +Ω⊗T = Ω+(d+1)T (−1) = Ω+(d+1)^{2}O−(d+1)O(−1) = (d^{2}+2d)O.

Consider the incidence hyperplane H in P^{d} × P^{d} where the first projective space is thought of as the projective space of a vector space O^{d+1} and the second one as the projective space of (O^{d+1})^{∗}. The complement

P̃^{d} = P^{d} × P^{d} − H

considered as a scheme over P^{d} by means of a projection p : P̃^{d} → P^{d} is an affine bundle over P^{d}. On the other hand, the Segre embedding

i_{d,d} : P^{d} × P^{d} → P^{d^{2}+2d}

gives H as the divisor at infinity for an appropriate choice of the intersecting hyperplane H_{∞}. Therefore, P̃^{d} is an affine variety. This construction is known as the Jouanolou trick (see [11]).

Consider the fiber product:

\begin{matrix}P̃^{d} × P^{d} & ⟶^{h} & P^{d} × P^{d} \\ ^{v}↓ & & ↓^{p} \\ P̃^{d} & ⟶^{p} & P^{d} \end{matrix}  (7)

The open embedding P̃^{d} → P^{d} × P^{d} defines a section s of v. Let N be the normal bundle to the Segre embedding i_{d,d} and let E be the normal bundle to s.

Lemma 2.9 There is an integer m and an isomorphism of vector bundles

E ⊕ N ⊕ O^{m} ≅ p^{∗}(W) ⊕ O^{m}  (8)

on P̃^{n}.

Proof: Since P̃^{d} is affine, two vector bundles give the same class in K_{0} if and only if they become isomorphic after the addition of O^{m} for some m ≥ 0. Therefore, to prove the lemma it is sufficient to show that one has E + N + p^{∗}(T_{P^{d}}) = O^{d^{2}+2d} in K_{0}(P̃^{d}). Let p′ : P̃^{d} → P^{d} be the second of the two projections. One can easily see from the definition of E that E = (p′)^{∗}(T_{P^{d}}). Hence E + p^{∗}(T_{P^{d}}) is the restriction to P̃^{d} of the tangent bundle on P^{d}×P^{d}. On the other hand the short exact sequence which defines N shows that over P̃^{d} we have

T_{P^{d}×P^{d}} + N = O^{d^{2}+d}

which finishes the proof of the lemma.

Lemma 2.10 There exists a map f : T^{d^{2}+2d} → T hP̃^{d}(E ⊕ N) such that the homomorphism H^{2d,d}(P^{d}) → Z defined by p^{∗}, f^{∗} and the Thom isomorphisms coincides with the degree map.

Proof: Consider the pointed sheaf

F = P^{d^{2}+2d}/(H_{∞} ∪ (P^{d^{2}+2d} − i_{d,d}(P^{d} × P^{d})))

There is an obvious map η from T^{d^{2}+2d} ≅ P^{d^{2}+2d}/H_{∞} to F. We claim that F is isomorphic in H_{•} to T hP̃^{n}(E ⊕ N) and that the composition of η with this isomorphism satisfies the condition of the lemma.

The same reasoning as in the proof of the homotopy purity theorem in [21, Theorem 2.23], shows that in H_{•} one has

F ≅ T h_{P^{d}×P^{d}}(N)/T h_{H}(N).

Consider again the pull-back square (7). The map

T h_{P̃^{d}×P^{d}}(h^{∗}(N))/T h_{h^{−1}(H)}(h^{∗}(N)) → T h_{P^{d}×P^{d}}(N)/T h_{H}(N)

is clearly an A^{1}-weak equivalence. On the other hand one verifies easily that h^{−1}(H) is contained in P̃^{d} × P^{d} − s(P̃^{d}) and that this embedding is an A^{1}-weak equivalence. Hence we have a weak equivalence from

T h_{P̃^{d}×P^{d}}(h^{∗}(N))/T h_{h^{−1}(H)}(h^{∗}(N))

to

T h_{P̃^{d}×P^{d}}(h^{∗}(N))/T h_{P̃^{d}×P^{d}−s(P̃^{d})}(h^{∗}(N))

The later quotient is isomorphic to h^{∗}(N)/(h^{∗}(N) − s(P̃^{d})) and since the normal bundle to s in h^{∗}N is N ⊕ E we conclude by [21, Theorem 2.23] that it is weakly equivalent to T hP̃^{d}(E ⊕ N). This finishes the construction of the map f : T^{d^{2}+2d} → T hP̃^{d}(E ⊕ N). It remains to check that for a ∈ H^{2d,d}(P^{d}) we have f^{∗}(at_{E⊕N}) = deg(a)t.

Consider the following diagram

P_{+}^{d^{2}+2d}  ⟶  T^{d^{2}+2d}
γ_{i_{d,d}} ↓  ↓
T h_{P^{d}×P^{d}}(N) ⟶ T h_{P^{d}×P^{d}}(N)/T h_{H}(N)
h̃↑  ↑
T h_{P̃^{d}×P^{d}}(N) ⟶ T h_{P̃^{d}×P^{d}}(N)/T h_{h^{−1}(H)}(N)
γ_{s,N} ↓  ↓
T hP̃^{d}(E ⊕ N) ⟶^{=}  T hP̃^{d}(E ⊕ N)

where the arrows going up are weak equivalences. One verifies immediately that it commutes. Denote the upper horizontal map of the diagram by ϕ. Since this map coincides with γ_{i_{0}} where i_{0} : Spec(k) → P^{d^{2}+2d} is the embedding of a point (see [38, Lemma 4.2]), Lemma 2.1 shows that deg(ϕ^{∗}(t)) = 1 where t is the tautological class in H̃^{2(d^{2}+2d),d^{2}+2d}(T^{d^{2}+2d}). Therefore, it it sufficient to show that for a class a ∈ H^{∗,∗}(P^{d}) we have

deg(ϕ^{∗}(f^{∗}(p^{∗}(a)t_{E⊕N}))) = deg(a)  (9)

Let h̃ denote the map

T h_{P̃^{d}×P^{d}}(N) → T h_{P^{d}×P^{d}}(N)

defined by h. Then

ϕ^{∗}(f^{∗}(p^{∗}(a)t_{E⊕N})) = γ_{i_{d,d}}^{∗}(h̃^{−1})^{∗}γ_{s,N}^{∗}(p^{∗}(a)t_{E⊕N}) =
= γ_{i_{d,d}}^{∗}(h̃^{−1})^{∗}(γ_{s}^{∗}(p^{∗}(a)t_{E})t_{N}) = γ_{i_{d,d}}^{∗}((h^{−1})^{∗}(γ_{s}^{∗}(p^{∗}(a)t_{E}))t_{N})

The commutative diagram

P̃^{d} × P^{d} ⟶^{γ_{s}} T hP̃^{d}(E)
h↓  ↓
P^{d} × P^{d} ⟶^{γ_{Δ}} T h_{P^{d}}(T_{P^{d}})

implies further that

γ_{s}^{∗}(p^{∗}(a)t_{E}) = h^{∗}γ_{Δ}^{∗}(at_{T_{P}^{d}})

and therefore

ϕ^{∗}(f^{∗}(p^{∗}(a)t_{E⊕N})) = γ_{i_{d,d}}^{∗}(γ_{Δ}^{∗}(at_{T_{P}^{d}})t_{N})

The equality (9) follows now from Lemma 2.1.

We are ready now to finish the proof of Proposition 2.7. We take V = W ⊕O^{m} and n = d^{2} + d + m. The map p : P̃^{d} → P^{d} defines a map of Thom spaces T hP̃^{n}(p^{∗}(V)) → T h_{P^{n}}(V). The isomorphism (8) defines an isomorphism

Σ_{T}^{m}T hP̃^{n}(E ⊕ N) ≅ T hP̃^{n}(p^{∗}(V))

Composing these maps with the m-fold suspension of f we get a map

f_{V} : T^{d^{2}+2d+m} → T h_{P^{d}}(V)

For a ∈ H^{2d,d}(P^{d}) we have

f_{V}^{∗} (at_{V}) = (Σ_{T}^{m}f)^{∗}(p^{∗}(a)t_{E⊕N⊕O^{m}}) =
= f^{∗}(p^{∗}(a)t_{E⊕N}) = deg(a)t

which shows that f_{V} satisfies the condition of the proposition.

3 The motivic analog of Margolis homology

In this section we introduce the motivic version of Margolis homology. The definition of topological Margolis homology^{3} is based on the fact that the Steenrod algebra contains some very special elements Q_{i} called Milnor’s primitives. These elements generate an exterior subalgebra in the Steenrod algebra and in particular Q_{i}^{2} = 0. Hence, one may consider the cohomology of a space or a spectrum as a complex with the differential given by Q_{i}. The homology of this complex are known in topology as Margolis homology M̃H_{i}^{∗}. Spaces or spectra whose Margolis homology vanish for i ≤ n play an important role in the proofs of the amazing recent results on the structure of the stable homotopy category.

3which appeared in [14] and which I learned about from [24].

Since we only know how to construct reduced power operations in the motivic cohomology with Z/l coefficients for l ≠ char(k)

everywhere in this section l is a prime not equal to the characteristic of the base field.

Recall that we defined in [38, §13] operations Q_{i} in the motivic cohomology with Z/l-coefficients of the form

Q_{i} : H̃^{p,q}(−, Z/l) → H̃^{p+2l^{i}−1,q+l^{i}−1}(−, Z/l)

We proved in [38, Proposition 13.4] that operations Q_{i} have the property Q_{i}^{2} = 0. For any X the cohomology of the complex

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

at the term H̃^{p,q} is called the i-th motivic Margolis cohomology of X of bidegree (p, q) denoted by M̃H_{i}^{p,q}(X). By [38, Lemma 13.5] Q_{0} is the Bockstein homomorphism and we get the following important sufficient condition for the vanishing of M̃H_{0}^{∗,∗}(X).

Lemma 3.1 Let X be a pointed simplicial sheaf such that lH̃^{∗,∗}(X , Z_{(l)}) = 0. Then M̃H_{0}^{∗,∗}(X) = 0.

Proof: Follows easily from the long exact sequences in the motivic cohomology defined by the short exact sequence 0 → Z →^{l} Z → Z/l → 0.

For a smooth variety X over k we denote by s_{d}(T_{X}) the d-th Milnor class of X i.e. the characteristic class of the tangent bundle T_{X} which corresponds to the Newton polynomial in the Chern classes (see [38, Corollary 14.3] for a more careful definition).

Theorem 3.2 Let Y be a smooth projective variety over k such that ther exists a map X → Y where X is a smooth projective variety of dimension l^{m} − 1 satisfying

deg(s_{l^{m}−1}(X)) ≠ 0(mod l^{2})

Then M̃H_{m}^{p,q}(C̃(Y), Z/l) = 0 for all p and q (see Appendix B. for the definition of C̃(Y)).

Proof: For m = 0 our condition means that Y has a rational point over a separable extension of k of degree n where n ≠ 0(mod l^{2}). By Lemma 9.3 this implies that lH̃^{∗,∗}(C̃(Y), Z_{(l)}) = 0 and by Lemma 3.1 we conclude that M̃H_{0}^{∗,∗}(C̃(Y)) = 0.

Assume that m > 0. Set d = l^{m} − 1. Let V be a vector bundle on X and f_{V} : T^{d+n} → T h_{X}(V) a map satisfying the conclusion of Theorem 2.6. Consider the cofibration sequence

T^{d+n} →^{f_{V}} T h_{X}(V) → cone →^{ψ} Σ_{s}^{1}T^{n+d}

corresponding to f_{V}. The long exact sequence of motivic cohomology corresponding to this cofibration sequence shows that ther exists a unique class α in H̃^{2n,n}(cone) whose restriction to T h_{X}(V) is the Thom class t_{V}. Multiplication with this class gives us a map

H̃^{p,q}(C̃(Y)) → H̃^{p+2n,q+n}(cone ∧ C̃(Y))  (10)

Consider the map cone ∧ C̃(Y) → (Σ_{s}^{1}T^{n+d}) ∧ C̃(Y). We claim that this map is a weak equivalence. Indeed, it is a part of a cofibration sequence and to verify that it is a weak equivalence it is enough to check that T h_{X}(V)∧C̃(Y) is contractible. This follows from the cofibration sequence

(V − {0})_{+} ∧ C̃(Y) → V_{+} ∧ C̃(Y) → T h_{X}(V) ∧ C̃(Y)

and Lemma 9.2. Since cone∧C̃(Y) → (Σ_{s}^{1}T^{n+d})∧C̃(Y) is a weak equivalence,

(10) defines a map

ϕ : H̃^{p,q}(C̃(Y)) → H̃^{p−2d−1,q−d}(C̃(Y))

We claim that ϕ is something like a contracting homotopy for the complex (H̃_{n}^{∗,∗}(C̃(Y)), Q_{m}). More precisely we have the following lemma which clearly imply the statement of the theorem.

Lemma 3.3 There exists c ∈ (Z/l)^{∗} such that for any x ∈ H̃^{p,q}(C̃(Y), Z/l) one has

cx = ϕQ_{m}(x) − Q_{m}ϕ(x)

Proof: Let γ ∈ H̃^{2n+2d+1,n+d}(cone) be the pull-back of the tautological motivic cohomology class on Σ_{s}^{1}T^{n+d} with respect to ψ. Since the map

cone ∧ C̃(Y) → (Σ_{s}^{1}T^{n+d}) ∧ C̃(Y)

is a weak equivalence it is sufficient to verify that there exists c ∈ (Z/l)^{∗} such that

cγ ∧ x = α ∧ Q_{m}(x) − Q_{m}(α ∧ x)

Let us show that for i < m we have Q_{i}(α) = 0. Indeed, the restriction of Q_{i}(α) to T h_{X}(V) is zero by [38, Theorem 14.2(1)] for any i. On the other hand for i < m the restriction map is a mono since there are no motivic cohomology of negative weight. This fact together with [38, Proposition 13.4] implies that we have

Q_{m}(α ∧ x) = α ∧ Q_{m}(x) + Q_{m}(α) ∧ x

and therefore

Q_{m}(α) ∧ x = α ∧ Q_{m}(x) − Q_{m}(α ∧ x)

We now have two possibilities. If Y has a point over an extension of degree prime to l then H̃^{p,q}(C̃(X)) = 0 and the statement of our lemma obviously holds. Assume that Y has no points over extensions of degree prime to l. Let us show that under this assumption Q_{m}(α) = cγ for c ∈ (Z/l)^{∗}. Note first that since Y has no points over extensions of degree prime to l, X does not have points over such extensions either. Since Q_{m}(α) restricts to zero on T h_{X}(V) by [38, Theorem 14.2(1)] it is sufficient to show that Q_{m}(α) ≠ 0. By [38, Proposition 13.6] we have Q_{m} = βq_{m} ± q_{m}β. Since α is a reduction of an integral class we have Q_{m}(α) = βq_{m}(α) and to show that it is non-zero we have to check that q_{m}(α) can not be lifted to cohomology with Z/l^{2}-coefficients. If it could there would be a lifting y of q_{m}(t_{V}) to the cohomology with Z/l^{2}-coefficients such that f_{V}^{∗} (y) = 0. Our condition that X has no points over extensions of degree prime to l implies, by Lemma 2.3, that the value of f_{V}^{∗} (y) does not depend on the choice of y. On the other hand by [38, Corollary 14.3] we know that q_{m}(t_{V}) is the reduction of the integral class s_{d}(X) which we assumed is non-zero mod l^{2}.

The following result provides us with a class of varieties X satisfying the condition of Theorem 3.2.

Proposition 3.4 Let X be a smooth hypersurface of degree d in P^{n}. Then

deg(s_{n−1}(X)) := deg(s_{n−1}(T_{X})) = d(n + 1 − d^{n−1})

In particular, if X is a smooth hypersurface of degree l in P^{l^{n}} then

deg(s_{l^{n}−1}(X)) = l(mod l^{2})

Proof: For the hypersurface given by a generic section of a vector bundle L the normal bundle is canonically isomorphic to L. In particular we have a short exact sequence of the form

0 → T_{X} → i^{∗}(T_{P^{n}}) → i^{∗}(O(d)) → 0

The tangent bundle on P^{n} fits into an exact sequence of the form

0 → O → O(1)^{n+1} → T_{P^{n}} → 0

Therefore, in K_{0}(X) we have

T_{X} = i^{∗}(T_{P^{n}}) − i^{∗}(O(d)) = (n + 1)i^{∗}(O(1)) − O − i^{∗}(O(d))

Since s_{n−1} is an additive characteristic class which on line bundles is given by s_{n−1}(L) = e(L)^{n−1} we get for n > 1

s_{n−1}(T_{X}) = (n + 1 − d^{n−1})i^{∗}((−σ)^{n−1})

where σ = e(O(−1)). By Lemma 2.2 we conclude that

deg(s_{n−1}(X)) := deg(s_{n−1}(T_{X})) = d(n + 1 − d^{n−1})

Combining Theorem 3.2 and Proposition 3.4 we get the following result which is the only result of this section used for the proof of Theorem 7.1.

Corollary 3.5 Let Q be a smooth quadric in P^{2^{n}}. Then

M̃H_{i}^{∗,∗}(C̃(Q), Z/2) = 0

for all i ≤ n.

4 Norm quadrics and their motives

The goal of this section is to prove Theorems 4.3 and 4.9. Everywhere in this section (except for Lemmas 4.5, 4.7 and 4.11) k is a field of characteristic ≠ 2.

For elements a_{1}, …, a_{n} in k^{∗} let ⟨a_{1}, …, a_{n}⟩ be the quadratic form ∑ a_{i}x_{i}^{2}. One defines the Pfister form ⟨⟨a_{1}, …, a_{n}⟩⟩ as the tensor product

⟨⟨a_{1}, …, a_{n}⟩⟩ := ⟨1, −a_{1}⟩ ⊗ … ⊗ ⟨1, −a_{n}⟩  (11)

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}. For n = 2 the rational points of the affine part of this quadric (t ≠ 0) can be identified with non-zero elements x of E = k(\sqrt{a_{1}}) such that N_{E/k}(x) = a_{2}. Because of this interpretation the quadric given by (11) 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 4.1 The symbol \underline{a} is divisible by 2 in K_{n}^{M}(k(Q\underline{a})).

Proof: We are going to show that if Q\underline{a} has a rational point over k then \underline{a} is divisible by 2 in K_{n}^{M}(k). Since any variety has a pojnt over its function field this implies that \underline{a} is divisible by 2 in the generic point of Q\underline{a}. Let P\underline{a} denote the quadric given by the equation ⟨⟨a_{1}, …, a_{n}⟩⟩ = 0.

Lemma 4.2 For any \underline{a} = (a_{1}, …, a_{n}) the following two conditions are equivalent

1. Q\underline{a} has a k-rational point

2. P\underline{a} has a k-rational point

Proof: The first condition implies the second one because the form

⟨⟨a_{1}, …, a_{n−1}⟩⟩ ⊕ ⟨ − a_{n}⟩

is a subform in ⟨⟨a_{1}, …, a_{n}⟩⟩ and therefore Q\underline{a} is a subvariety in P\underline{a}. Assume that the second condition holds. By [12, Corollary 1.6] it implies that the form ⟨⟨a_{1}, …, a_{n}⟩⟩ is hyperbolic. Hence, 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 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}. The intersection of H and L is a rational point on Q\underline{a}.

To prove Proposition 4.1 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} + \sqrt{a_{1}}y_{0} from k(\sqrt{a_{1}}) and thus the symbol (a_{1}, a_{2}) is divisible by 2.

Suppose that the proposition is proved for sequences (a_{1}, …, a_{i}) of length 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(\sqrt{b}). The form ⟨⟨b⟩⟩ and therefore the form ⟨⟨a_{1}, …, a_{n−1}⟩⟩ represents zero over k(\sqrt{b}) and thus by the inductive assumption (a_{1}, …, a_{n−1}) = 0 in K_{n−1}^{M}(k(\sqrt{b}))/2. On the other hand by the construction ⟨⟨b⟩⟩ represents a_{n} and therefore we have a_{n} ∈ ImN_{k(\sqrt{b})/k} ⊂ k^{∗} which proves the proposition.

Let X\underline{a} denote the simplicial scheme Č(Q\underline{a}) (see Appendix B). The following result is formulated in terms of the triangulated category of mixed motives DM_{−}^{eff} = DM_{−}^{eff}(k) introduced in [37].

Theorem 4.3 Let k be a perfect field. Then, 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] (12)

Proof: Denote by C the category of Chow motives over k and let Z{n} be the n-th Tate motive in this category. The proof of the theorem is based on the following important result.

Theorem 4.4 (Markus Rost) There exists a direct summand M\underline{a} of Q\underline{a} in C together with two morphisms

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

such that

1. the composition ψ_{∗} : Q\underline{a} → M\underline{a} → Z is the morphism defined by the projection Q\underline{a} → Spec(k)

2. for any field F over k where Q\underline{a} has a point the pull-back of the sequence

Z\{2^{n−1} − 1\} → M\underline{a} → Z

to F is split-exact.

Proof: See [27], [28].

The Friedlander-Lawson moving lemma for families of cycles shows that for any k there is a functor from the category of Chow motives over k to DM_{−}^{eff}(k) (it is shown in [37] that this functor is a full embedding if k is perfect). Therefore, the Rost motive M\underline{a} can be also considered in DM_{−}^{eff} where it is a direct summand of the motive M(Q\underline{a}) of the norm quadric. To show that it fits into a distinguished triangle of the form (12) we need the following lemmas.

Lemma 4.5 Let k be a perfect field, M an object of DM_{−}^{eff}(k) and X a smooth variety over k such that for any generic point η of X the pull-back of M to the residue field k_{η} is zero. Then one has

Hom_{DM}(M(X), M) = 0
M ⊗ M(X) = 0.

Proof: Since k is perfect, the group Hom(M(X), M) is the hypercohomology group of X with coefficients in the complex of sheaves with transfers with homotopy invariant cohomology sheaves K which represents M. Our condition on X and M implies easily that the cohomology sheaves h_{i}(K) of K vanish on the generic points of schemes etale over X. Since h_{i}(K) are homotopy invariant sheaves with transfers we conclude by [36, Corollary 4.19, p.116] that they vanish on all schemes etale over X. Therefore, Hom(M(X), M) = 0.

To prove that M ⊗ M(X) = 0 it is sufficient to show that the class of objects N such that

Hom(N, M ⊗ M(X)[m]) = 0

for all m, contains M ⊗ M(X). Since this class is a localizing subcategory^{4}, it is sufficient to show that for any smooth Y over k and any m one has

Hom(M(Y × X), M ⊗ M(X)[m]) = 0  (13)

If X and M satisfy the conditions of the lemma then for any Y and any N, Y × X and M ⊗ N satify these conditions. Therefore, (13) follows from the first assertion of the lemma.

Lemma 4.6 Let k be a perfect field. Then the sequence of

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})  (14)

is split-exact.

Proof: Observe first that Id ⊗ ψ_{∗} is a split epimorphism. Indeed, Theorem 4.4(1) implies that the morphism

M(Q\underline{a}) → M(Q\underline{a}) ⊗ M(Q\underline{a}) → M(Q\underline{a}) ⊗ M\underline{a}

where the first arrow is defined by the diagonal and the second by the projection M(Q\underline{a}) → M\underline{a} is a section of Id ⊗ ψ_{∗}. It remains to show that (14) extends to a distinguished triangle. Let cone be a cone of the morphism ψ^{∗} : Z(2^{n−1} − 1)[2^{n} − 2] → M\underline{a}. Since there are no motivic cohomology of negative weight, the morphism ψ_{∗} : M\underline{a} → Z factors through a morphism ϕ : cone → Z. Let cone′ be the a cone of ϕ. Standard properties of triangles in triangulated categories imply that the sequence (14) extends to a distinguished triangle if and only if cone′ ⊗ M(Q\underline{a}) = 0. It follows from Theorem 4.4(b) and Lemma 4.5.

4A subcategory in a triangulated category is called localizing if it is closed under triangles, direct sums and direct summands.

Lemma 4.7 Let X be a smooth scheme over k and M an object of the localizing subcategory generated by M(X). Then one has

1. the morphism M ⊗ M(Č(X)) → M is an isomorphism

2. the homomorphism Hom(M, Č(X)) → Hom(M, Z) is an isomorphism.

Proof: It is clearly sufficient to prove the lemma for M = M(X). In this case the first statement follows immediately from Lemma 9.2 and the fact that M takes simplicial weak equivalences to isomorphisms. Let M(C̃(X)) be the cone of the morphism M(Č(X)) → Z. To prove the second statement we have to show that any morphism f : M(X) → M(C̃(X)) is zero. The morphism f can be written as the composition

M(X) →^{M(Δ)} M(X) ⊗ M(X) →^{f⊗Id} M(C̃(X)) ⊗ M(X) → M(C̃(X))

which is zero because the first part of the lemma implies that

M(C̃(X)) ⊗ M(X) = 0.

(Proof of Theorem 4.3 continues) The morphism ψ_{∗} : M\underline{a} → Z has a canonical lifting to a morphism ψ̃_{∗} : M\underline{a} → M(X\underline{a}) by Lemma 4.7(2). 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}

this lifting 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})

The composition ψ̃_{∗} ∘ ψ̃^{∗} is zero by Lemma 4.7(2) and the fact that

Hom(Z(2^{n−1} − 1)[2^{n} − 2], Z) = 0

Let cone be a cone of ψ̃^{∗}. The morphism ψ̃_{∗} factors through a morphism ϕ : cone → M(X\underline{a}) and we have to show that ϕ is an isomorphism. The category C of objects N such that ϕ ⊗ Id_{N} is an isomorphism is a localizing subcategory. By Lemma 4.7(1), the morphism M(X\underline{a}) ⊗ M(Q\underline{a}) → M(Q\underline{a}) is an isomorphism and we conclude by Lemma 4.6 that C 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})

with both vertical arrows are isomorphisms by Lemma 4.7(1). This finishes the proof of Theorem 4.3.

Remark 4.8 It is essential to use the category DM_{−}^{eff} in Theorem 4.3 because the triangle (12) can not be lifted to the motivic homotopy category (stable or unstable). One can see this for n = 2 using the fact that the map X\underline{a} → Spec(k) defines an isomorphism on algebraic K-theory but K^{∗,∗}(Q\underline{a}) ≠ K^{∗,∗}(k) ⊕ K^{∗,∗}(k).

Theorem 4.9 H^{2^{n}−1,2^{n−1}}(X\underline{a}, Z_{(2)}) = 0

Proof: Since char(k) ≠ 2 any purely inseparable extension of k is of odd degree and the transfer argument shows that it is sufficient to consider the case of a perfect k. 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 u such that all residues of u in points of codimension 1 are zero. The proof is based on the following result.

Theorem 4.10 (Markus Rost) The natural homomorphism

H^{2^{n−1}−1}(Q\underline{a}, \underline{K}_{2^{n−1}}^{M}) → k^{∗}  (15)

is a monomorphism.

Proof: See [26].

The following lemma shows that the cohomology group on the left hand side of (15) can be replaced by a motivic cohomology group.

Lemma 4.11 Let X be a smooth scheme over a field k. Then for any p, q there is a canonical homomorphism H^{p,q}(X, Z) → H^{p−q}(X, \underline{K}_{q}^{M}) which is an isomorphism if p ≥ q + dim(X).

Proof: Considering X as a limit of smooth schemes over the subfield of constants of k we may assume that k is perfect. Let h^{i} = \underline{H}^{i}(Z(q)) denote the cohomology sheaves of the complex Z(q). Since h^{i} = 0 for i > q the standard spectral sequence which goes from cohomology with coefficients in h^{i} and converges to the hypercohomology with coefficients in Z(q) shows that there is a canonical homomorphism

H^{p,q}(X, Z) → H^{p−q}(X, h^{q})  (16)

The same spectral sequence implies that the kernel and cokernel of this homomorphism are bulit out of groups H^{p−i}(X, h^{i}) and H^{p−i+1}(X, h^{i}) respectively, where i < q. Since p ≥ q + dim(X) we get p − i > p − q ≥ dim(X) and the cohomological dimension theorem for the Nisnevich topology implies that these groups are zero.

It remains to show that h^{q} = \underline{K}_{q}^{M}. Since h^{q} is a homotopy invariant sheaf with transfers for any smooth connected X the restriction homomorphism

H^{0}(X, h^{q}) → H^{0}(Spec(k(X)), h^{q})

is a monomorphism ([36, Corollary 4.18, p.116]). It was shown in [31, Prop. 4.1] that for any field k one has canonical isomorphisms H^{q,q}(k, Z) = K_{q}^{M}(k). In particular for any X the group H^{0}(X, h^{q}) is a subgroup in K_{q}^{M}(k(X)) and one verifies easily that it coincides with the subgroup \underline{K}_{q}^{M}(X) of elements with zero residues.

By Theorem 4.3 and the suspension isomorphism (see [38, Theorem 2.4]) we have an exact sequence

H^{0,1}(X\underline{a}, Z) → 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) (17)

Since the dimension of Q\underline{a} equals 2^{n−1} − 1, the left hand side group in (15) is isomorphic to the group H^{2^{n}−1,2^{n−1}}(Q\underline{a}, Z) by Lemma 4.11 and we obtain a natural monomorphism

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

Let M̄\underline{a} be the lifting of M\underline{a} to the algebraic closure of k. Since M\underline{a} is a direct summand of M(Q\underline{a}) we conclude that the map

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

is injective. 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 (17) is zero. On the other hand since Z(1) = G_{m}[−1] we have

H^{0,1}(X\underline{a}, Z) = H^{−1}(X\underline{a}, G_{m}) = 0

5 Computations with Galois cohomology

In this section we are only concerned with classical objects - Milnor K-theory and etale cohomology. More general motivic cohomology does not appear here. The only result of this section which we will directly use below is Theorem 5.9. One may observe that in the case of Z/2-coefficients it can be proved in a much easier way, but we decided to include the case of general l for possible future use.

Definition 5.1 We say that BK(w, l) holds if for any field k of characteristic ≠ l and any q ≤ w one has:

1. the norm residue homomorphism K_{q}^{M}(k)/l → H_{et}^{q}(k, μ_{l}^{⊗q}) is an isomorphism

2. for any cyclic extension E/k of degree l the sequence

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

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

Proposition 5.2 Let k be a field of characteristic ≠ l which has no extensions of degree prime to l. Assume that BK(w, l) holds. Then for any cyclic extension E/k of k of degree l there is an exact sequence of the form

H_{et}^{w}(E, Z/l) →^{N_{E/k}} H_{et}^{w}(k, Z/l) →^{−∧[a]} H_{et}^{w+1}(k, Z/l) → H_{et}^{w+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 5.3 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 5.4 Assume that BK(w, l) holds. Then for all fields k of characteristic ≠ l, all q ≤ w and all i = 1, …, l − 1 one has:

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

2. The homomorphisms ν_{q,i} : H_{et}^{q}(k, Z/l)⊕H_{et}^{m}(k, F_{i+1}) → H_{et}^{q}(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(w, l) holds.

Let us prove the second one. The image of H_{et}^{q}(k, F_{l−1}) in

H_{et}^{q}(k, F_{l}) = H_{et}^{q}(E, Z/l)

coincides with the kernel of the norm homomorphism

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

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

0 → F_{l−1} → F_{l} → Z/l → 0

Thus we may assume that q = 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 5.3) 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 5.3 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 proved 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 5.3).

The statement of the proposition follows immediately from Lemma 5.4.

Remark 5.5 For l = 2 Proposition 5.2 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(w, l) assumption and not only in the context of Galois cohomology but for cohomology of any (pro-)finite group. For l > 2 this is not true anymore which one can see considering cohomology of Z/l.

Lemma 5.6 Assume that BK(w, l) holds and let k be a field of characteristic ≠ l which has no extensions of degree prime to l. Let further E/k be a cyclic extension of degree l such that the norm homomorphism K_{w}^{M}(E) → K_{w}^{M}(k) is surjective. Then the sequence

K_{w+1}^{M}(E) →^{1−σ} K_{w+1}^{M}(E) →^{N_{E/k}} K_{w+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 [29] for w = 2 and in [18] for w = 3. Let us define a homomorphism

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

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

N_{E/k}(b) = (a_{0}, …, a_{w−1})

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

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

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

Clearly ϕ is a section for the obvious morphism

K_{w+1}^{M}(E)/(Im(1 − σ)) → K_{w+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_{w+1}^{M}(E) is generated by symbols of the form (b, a_{1}, …, a_{w}) where b ∈ E^{∗} and a_{1}, …, a_{w} ∈ k^{∗} (see [1]).

Lemma 5.7 Assume that BK(w, l) holds and let k be a field of characteristic ≠ l which has no extensions of degree prime to l. Then the following two conditions are equivalent

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

2. for any cyclic extension E/k the norm homomorphism

K_{w}^{M}(E) → K_{w}^{M}(k)

is surjective.

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

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

and thus the cokernel of the left vertical arrow is the same as the cokernel of the right one. By Proposition 5.2 it gives us an exact sequence

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

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

Lemma 5.8 Assume that BK(w, l) holds and let k be a field of characteristic not equal to l which has no extensions of degree prime to l. Assume further that K_{w+1}^{M}(k) = lK_{w+1}^{M}(k). Then for any finite extension E/k one has K_{w+1}^{M}(E) = lK_{w+1}^{M}(E).

Proof: This proof is a variant of the proof given in [29] for w = 1. Since k has no extensions of degree prime to l it is separable and its Galois group is an l-group. Therefore it is sufficient to prove the lemma in the case of a cyclic extension E/k of degree l. By Proposition 5.2 and Lemma 5.6 we have an exact sequence

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

Let α be an element in K_{w+1}^{M}(E) and let β ∈ K_{w+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_{w+1}^{M}(E)/l → K_{w+1}^{M}(E)/l

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

Theorem 5.9 Assume that BK(w, l) holds and let k be a field of characteristic not equal to l which has no extensions of degree prime to l such that K_{w+1}^{M}(k) = lK_{w+1}^{M}(k). Then H_{et}^{w+1}(k, Z/l) = 0.

Proof: Let α be an element of H_{et}^{w+1}(k, Z/l). We have to show that α = 0. By Lemma 5.8 and obvious induction we may assume that α vanishes on a cyclic extension of k. Then by Proposition 5.2 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(w, l) holds it belongs to the image of the homomorphism K_{w+1}^{M}(k)/l → H_{et}^{w+1}(k, Z/l) and therefore is zero.

6 Beilinson-Lichtenbaum conjectures

For a smooth variety X and an abelian group A define the Lichtenbaum motivic cohomology groups of X with coefficients in A as the hypercohomology groups

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

The following fundamental conjecture is due to Alexander Beilinson and Stephen Lichtenbaum (see [2],[13]).

Conjecture 6.1 Let k be a field. Then for any n ≥ 0 one has

H_{L}^{n+1,n}(Spec(k), Z) = 0

For n = 0 we have Z(0) = Z and this conjecture follows from the simple fact that H^{1}(k, Z) = 0. For n = 1 we have Z(1) = G_{m}[−1] and the conjecture is equivalent to the cohomological form of the Hilbert 90 Theorem. Because of this fact Conjecture 6.1 is called the generalized Hilbert 90 Conjecture.

The standard proofs of Conjecture 6.1 for n = 0 and n = 1 work integrally. In the proofs of all the known cases of Conjecture 6.1 for n > 1 one considers the vanishing of the groups H_{L}^{n+1,n}(Spec(k), Z_{(l)}) = 0 for different primes l separately.

Definition 6.2 We say that H90(n, l) holds if for any k one has

H_{L}^{n+1,n}(Spec(k), Z_{(l)}) = 0.

The following result is proved in [9, Theorem 8.6].^{5}

Theorem 6.3 (Geisser-Levine) Let S be the local ring of a smooth scheme over a field of characteristic p > 0. Then for any n ≥ 0 one has

H_{L}^{n+1,n}(Spec(k), Z_{(p)}) = 0.

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)))).

5The authors of [9] define motivic cohomology through the higher Chow groups. Their definition is equivalent to the one used here by the comparison theorem of [30], [7] and [35].

Let L(q) be the canonical truncation of the complex Rπ_{∗}(π^{∗}(Z(q))) at 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].

Theorem 6.4 Assume that H90(w, l) holds. Then for any k such that char(k) ≠ l the complex K(w) ⊗ Z_{(l)} is quasi-isomorphic to zero.

Proof: We will use the following two lemmas.

Lemma 6.5 Let k be a field of characteristic not equal to l. Then for any q ≥ 0 and any Z_{(l)}-module A, the complex Rπ_{∗}(π^{∗}(A ⊗ Z(q))) has homotopy invariant cohomology sheaves.

Proof: Since the sheaf associated with a homotopy invariant presheaf with transfers is homotopy invariant ([36, Propositions 4.26, p. 118] and [36, Propositions 5.5, p. 128]) it is sufficient to show that the functors H_{L}^{p,q}(−, A) are homotopy invariant i.e. that for any smooth U the homomorphism

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

given by the restriction to U ×{0} is an isomorphism. Consider the universal coefficients long exact sequence relating Lichtenbaum motivic cohomology with coefficients in A to Lichtenbaum motivic cohomology with coefficients in A⊗Q and A⊗Q/Z_{(l)}. The cohomology with A⊗Q-coefficients is homotopy invariant by Lemma 6.6. The cohomology with A ⊗ Q/Z_{(l)}-coefficients is isomorphic to the etale cohomology by Theorem 6.8 and therefore homotopy invariant as well. Our claim follows now from the five lemma.

Lemma 6.6 The canonical homomorphisms

H^{p,q}(−, Q) → H_{L}^{p,q}(−, Q)

are isomorphisms.

Proof: 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 the \underline{H}^{j}(Z(q)) are sheaves with transfers it is a particular case of [36, Propositions 5.24, 5.27, p. 135].

Lemma 6.5 implies that \underline{H}^{n+1}(K(w) ⊗ Z_{(l)}) is a homotopy invariant sheaf with transfers and by the assumption that H90(w, l) holds we know that it vanishes over fields. Therefore, by [36, Corollary 4.18, p. 116] we conclude that \underline{H}^{n+1}(K(w) ⊗ Z_{(l)}) = 0.

In order to show that K(w) ⊗ Z_{(l)} is quasi-isomorphic to zero it remains to verify that for any smooth scheme X over k and any p ≤ w the homomorphism H^{p,w}(X, Z_{(l)}) → H_{L}^{p,w}(X, Z_{(l)}) is an isomorphism. Lemma 6.6 and the universal coefficients long exact sequence imply that it is sufficient to verify that the homomorphisms

H^{p,w}(X, Q/Z_{(l)}) → H_{L}^{p,w}(X, Q/Z_{(l)})  (18)

are isomorphisms for p ≤ w. Since we assume H90(w, l), we conclude that the group H_{L}^{p,w}(X, Q/Z_{(l)}) is infinitely divisible and using the trick of [31, Theorem 11.4] we conclude that the map (18) is surjective for p = w and X = Spec(F) where F is a field. By the analog of [8, Theorem 1.1] for Q/Z_{(l)}-coefficients and the comparison between the higher Chow groups and the etale cohomology we conclude that (18) is an isomorphism.

Corollary 6.7 Assume that H90(w, l) holds. Then for any field k and any smooth simplicial scheme X over k one has

1. the homomorphisms

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

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

2. the homomorphisms

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

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

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 6.8 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.

Remark 6.9 A very detailed proof of this theorem which uses only the most basic facts about motivic cohomology can be found in [15].

Corollary 6.10 Assume that H90(w, l) holds. Then for any k of characteristic not equal to l and any q ≤ w the norm residue map

K_{q}^{M}(k)/l → H_{et}^{n}(Spec(k), μ_{l}^{⊗q})  (19)

is an isomorphism.

Proof: By Corollary 6.7(2) the homomorphisms

H^{p,q}(X, Z/l) → H_{L}^{p,q}(X, Z/l)  (20)

are isomorphisms for p ≤ q ≤ w. By Theorem 6.8, for p = q and X = Spec(k) the homomorphism (20) is isomorphic to the homomorphism (19).

Lemma 6.11 Assume that H90(w, l) holds. Then for any field k, any q ≤ w and any cyclic extension E/k of degree l the sequence

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

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

Proof: One verifies easily that this complex becomes exact after tensoring with Z[1/l]. It remains to show that it becomes eact after tensoring with Z_{(l)}. Recall that for a smooth scheme X over k we let Z_{tr}(X) denote the free sheaf with transfers generated by X. Consider the complex of presheaves with transfers of the form

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

where the second arrow is the transfer map. Denote this complex with the last Z_{tr}(k) placed in degree zero by \underline{K}. One can easily see that it is exact in the etale topology and therefore

Hom_{D}(\underline{K}, Rπ_{∗}(π^{∗}(Z(q)))[∗]) = Hom_{D_{et}}(\underline{K}, Z(q)[∗]) = 0

where D_{et} is the category of etale sheaves with transfers. Since

\underline{H}^{q+1}(Rπ_{∗}(π^{∗}(Z(q)))) = 0,

the map

Hom_{D}(\underline{K}, L(q)[q + 2]) → Hom_{D}(\underline{K}, Rπ_{∗}(π^{∗}(Z(q)))[q + 2])

is injective and we conclude by Theorem 6.4 that

Hom_{D}(\underline{K}, Z_{(l)}(q)[q + 2]) = 0  (21)

We have

H^{p,q}(X, Z) = Hom_{D}(Z_{tr}(X), Z(q)[p])

and in particular for a separable extension F of k we have

Hom_{DM}(Z_{tr}(Spec(F)), Z(q)[q]) = \{ \begin{matrix}K_{q}^{M}(F) & for p = q \\ 0 & for p > q \end{matrix}

Our result follows now from (21) and the standard spectral sequence which computes morphisms in a triangulated category from a complex in terms of morphisms from its terms.

7 Main theorem

In this section we prove the following theorem.

Theorem 7.1 For any field k and any w ≥ 0 one has

H_{L}^{w+1,w}(Spec(k), Z_{(2)}) = 0.

Proof: For char(k) = 2 the theorem is proved in [8, Theorem 8.6]. Assume that char(k) ≠ 2. By induction on w we may assume that H_{L}^{q+1,q}(k, Z_{(2)}) = 0 for all q < w.

Let k be a field which has no extensions of degree prime to 2 and such that K_{w}^{M}(k) is 2-divisible. By Lemma 6.11 our inductive assumption implies that BK(w − 1, 2) holds. By Theorem 5.9 we conclude that

H_{et}^{w}(Spec(k), Z/2) = 0.

Together with Theorem 6.8 this shows that the group H_{L}^{w+1,w}(k, Z_{(2)}) is torsion free. On the other hand H_{L}^{w+1,w}(k, Q) = 0 by Lemma 6.6 and we conclude that H_{L}^{w+1,w}(Spec(k), Z_{(2)}) = 0.

For a finite extension E/k of degree prime to 2 the homomorphism

H_{L}^{w+1,w}(Spec(k), Z_{(2)}) → H_{L}^{w+1,w}(Spec(E), Z_{(2)})

is a monomorphism by the transfer argument. Thus in order to prove H90(w, 2) it is sufficient to show that for any element \underline{a} ∈ K_{w}^{M}(k) there exists an extension K\underline{a}/k such that \underline{a} is divisible by 2 in K_{w}^{M}(K\underline{a}) and the homomorphism

H_{L}^{w+1,w}(Spec(k), Z_{(2)}) → H_{L}^{w+1,w}(Spec(K\underline{a}), Z_{(2)})  (22)

is a monomorphism. Since any element in K_{w}^{M}(k) is a sum of symbols it is sufficient to construct K\underline{a} for \underline{a} of the form (a_{1}, …, a_{w}).

Let us show that the function field K = k(Q\underline{a}) of the norm quadric has the required properties. The fact that \underline{a} becomes divisible by 2 in the Milnor K-theory of K proved in Proposition 4.1. It remains to show that the map (22) is injective. We do it in two steps - first we prove that the kernel of (22) is covered by the group H^{w+1,w}(X\underline{a}, Z_{(2)}) and then that the later group is zero. For the first step we will need the following three lemmas.

Lemma 7.2 Let X be a non empty smooth scheme over k. Then the homomorphisms

H_{L}^{p,q}(Spec(k), Z) → H_{L}^{p,q}(Č(X), Z)  (23)

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

Proof: By definition, (see Appendix A) we can rewrite the homomorphism (23) as the homomorphism

Hom_{D_{et}}(Z, Z(q)[p]) → Hom_{D_{et}}(Z(Č(X)), Z(q)[p])

where the morphisms are in the derived category of sheaves of abelian groups in the etale topology on Sm/k. We claim that the morphism of complexes Z(Č(X)) → Z is a quasi-isomorphism. To check this statement we have to verify that for any strictly henselian local scheme S the map of complexes of abelian groups Z(Č(X))(S) → Z is a quasi-isomorphism. Since X is non-empty there exists a morphism S → X and therefore, the simplicial set Č(X)(S) is contractible (see the proof of Lemma 9.2). Therefore we have

Z(Č(X))(S) = Z(Č(X)(S)) ≅ Z

Let K(w) be the complex of sheaves on (Sm/k)_{Nis} defined in Section 6.

Lemma 7.3 Let k be a field of characteristic not equal to l and assume that H90(w − 1, l) holds. Then for any smooth scheme X the map

H^{∗}(X, K(w) ⊗ Z_{(l)}) → H^{∗}(X × (A^{1} − {0}), K(w) ⊗ Z_{(l)})

defined by the projection A^{1} − {0} → Spec(k), is an isomorphism.

Proof: Recall from [36] that for a functor F from schemes to abelian groups we denote by F_{−1} the functor X ↦ coker(F (X) → F (X × A^{1} − {0})) where the map is defined by the projection. To prove the lemma we have to show that H^{∗}(X, K(w) ⊗ Z_{(l)})_{−1} = 0. This is clearly equivalent to checking that the standard map

H^{∗,w}(X, Z_{(l)})_{−1} = H^{∗}(X, Z_{(l)}(w))_{−1} → H_{L}^{∗}(X, Z_{(l)}(w))_{−1}

is an isomorphism. For a complex of sheaves with transfers K there is a complex K_{−1} (defined up to a canonical quasi-isomorphism) such that H^{∗}(−, K)_{−1} = H^{∗}(−, K_{−1}). By [36, Proposition 4.34, p.124], if K is a complex of sheaves with transfers with homotopy invariant cohomology sheaves \underline{H}^{i} then

\underline{H}^{i}(K_{−1}) = (\underline{H}^{i}(K))_{−1}

Therefore, it is sufficient to check that the maps

\underline{H}^{i}(Z_{(l)}(w))_{−1} → \underline{H}^{i}(L(w) ⊗ Z_{(l)})_{−1}

are isomorphisms. Since both sides are zero for i > w + 1 and since \underline{H}^{i}(K) are the sheaves associated with the presheaves X ↦ H^{i}(X, K) it remains to check that the maps

H^{i,w}(X, Z_{(l)})_{−1} → H^{i}(X, L(w) ⊗ Z_{(l)})_{−1}

are isomorphisms for i ≤ w + 1. In this range the map

H^{i}(X, L(w) ⊗ Z_{(l)}) → H^{i}(X, Rπ_{∗}(π^{∗}(Z_{l}(w)))) = H_{L}^{i,w}(X, Z_{(l)})

is an isomorphism and therefore it remains to check that the map

H^{i,w}(X, Z_{(l)})_{−1} → H_{L}^{i,w}(X, Z_{(l)})_{−1}

is an isomorphism for i ≤ w + 1. Consider the commutative diagram

H^{i−1,w−1}(X, Z_{(l)}) ⟶ H_{L}^{i−1,w−1}(X, Z_{(l)})
↓  ↓
H^{i,w}(X, Z_{(l)})_{−1} ⟶ H_{L}^{i,w}(X, Z_{(l)})_{−1}

where the vertical arrows are defined by the multiplication with the canonical class η ∈ H^{1,1}(A^{1} − {0}, Z). The upper horizontal arrow is an isomorphism by our assumption that H90(w − 1, l) holds and Corollary 6.7(1). The left hand side vertical arrow is an isomorphism by the supension theorem [38, Theorem 2.4]. It remains to check that the right hand side vertical arrow is an isomorphism. This we can verify separately for rational coefficients and Q/Z_{(l)}-coefficients. In the former case the result follows from Lemma 6.6 and again [38, Theorem 2.4]. In the later case it follows from Theorem 6.8 and the corresponding result for the etale cohomology.

Lemma 7.4 Assume that H90(w − 1, l) holds and let k be a field of characteristic not equal to l, X a smooth scheme over k and U a dense open subscheme in X. Then the map

H^{∗}(X, K(w)) → H^{∗}(U, K(w))  (24)

is an isomorphism.

Proof: Considering X to be a limit of smooth schemes (possibly of greater dimension) over the subfield of constants in k we may assume that k is perfect. 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. Moreover, one can easily see that it is sufficient to prove that for any point z of Z there exists a neighborhood V of z in X such that

H^{∗}(V, K(w)) → H^{∗}(V − V ∩ Z, K(w))

is an isomorphism. This follows from the easiest case of the Gysin distinguished triangle [37, Proposition 3.5.4, p.221] or from the homotopy purity theorem [21, Theorem 2.23, p.115] and Lemma 7.3.

(Proof of Theorem 7.1 continues) Let u be an element in the kernel of (22). By Lemma 7.2 it is sufficient to show that the image u′ of u in H_{L}^{w+1,w}(X\underline{a}, Z_{(2)}) belongs to the image of the group H^{w+1,w}(X\underline{a}, Z_{(2)}) i.e. that under our assumptions the image of u′ in the hypercohomology group H_{Nis}^{w+1}(X\underline{a}, K(w)) is zero. Since u becomes zero in the generic point of Q\underline{a} there exists a nonempty open subscheme U of Q\underline{a} such that the restriction of u′ to U is zero. By Lemma 7.4 we conclude that the restriction of u′ to Q\underline{a} is zero.

The canonical morphism M(Q\underline{a}) → M(X\underline{a}) factors through the morphism M\underline{a} → M(X\underline{a}) which is a part of the distinguished triangle of Theorem 4.3. Since M(Q\underline{a}) → M\underline{a} has a section, our class u′ becomes zero on M\underline{a} and by (12) we conclude that it belongs to the image of the group Hom(M(X\underline{a})(2^{w−1} − 1)[2^{w} − 1], K(w)[w + 1]). Since w > 1 this group is zero by Lemma 7.3.

It remains to show that H^{w+1,w}(X\underline{a}, Z_{(2)}) = 0. Consider the pointed simplicial scheme X̃\underline{a} = C̃(Q\underline{a}) defined by the cofibration sequence

(X\underline{a})_{+} → S^{0} → X̃\underline{a} → Σ_{s}^{1}((X\underline{a})_{+})

Since H^{p,q}(Spec(k), Z) = 0 for p > q the homomorphisms

H^{p−1,q}(X\underline{a}, Z) → H̃^{p,q}(X̃\underline{a}, Z)

defined by the third arrow of this sequence are isomorphisms for p − 1 > q. Thus it is sufficient to verify that H̃^{w+2,w}(X̃\underline{a}, Z_{(2)}) = 0. Since there exists an extension of degree two E/k such that Q\underline{a}(Spec(E)) ≠ ∅ Lemma 9.3 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^{w+2,w}(X̃\underline{a}, Z_{(2)}) in H̃^{w+2,w}(X̃\underline{a}, Z/2) is zero. Let u be an element of this image. Consider the composition of cohomological operations Q_{w−2}Q_{w−3} … Q_{1}. It maps u to an element of

H̃^{2^{w},2^{w−1}}(X̃\underline{a}, Z/2) = H^{2^{w}−1,2^{w−1}}(X\underline{a}, Z/2)

Lemma 7.5 Let a be a class in H^{∗,∗}(X , Z/l) which the image of an integral class. Then Q_{i}(a) is the image of an integral class for any i.

Proof: By [38, Proposition 13.6] we have Q_{i} = βq_{i}±q_{i}β. Since a is the image of an integral class we have β(a) = 0 and Q_{i}(a) = βq_{i}(a). On the other hand, the Bockstein homomorphism β can be written as the composition

H̃^{∗,∗}(−, Z/l) →^{B} H̃^{∗+1,∗}(−, Z) → H̃^{∗+1,∗}(−, Z/l)

where the first map is the connecting homomorphism for the exact sequence 0 → Z → Z → Z/l → 0 and the second map is the reduction modulo l. Therefore, any element of the form β(x) is the image of an integral class.

Lemma 7.5 implies that this element belongs to the image of the corresponding integral motivic cohomology group. Therefore by Theorem 4.9 it is zero. It remains to verify that the composition Q_{w−2}Q_{w−3} … Q_{1} is a monomorphism i.e that the operation Q_{i} acts monomorphically on the group H̃^{w−i+2^{i}−1,w−i+2^{i−1}−1}(X̃\underline{a}, Z/2) for i = 1, …, w − 2. By Corollary 3.5, the motivic Margolis homology of X̃\underline{a} are zero for Q_{i} with i ≤ w − 1. Therefore, the kernel of Q_{i} on this group is covered by the image of H̃^{w−i,w−i}(X̃\underline{a}, Z/2). The later group is zero by the inductive assumption that H90(w−1, 2) holds, Lemma 7.2 and Corollary 6.4(2).

Theorem 7.1 is proved.

Corollary 7.6 Let k be a field of characteristic not equal to 2. Then the norm residue homomorphisms K_{w}^{M}(k)/2 → H_{et}^{w}(k, Z/2) are isomorphisms for all w ≥ 0.

Corollary 7.7 For any field k of characteristic not equal to 2 and any q ≥ 0 the etale cohomology group H_{et}^{q+1}(k, Z_{2}(q)) is torsion free.

The following nice corollary of Theorem 7.1 is due to S. Bloch.

Corollary 7.8 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 Corollary 7.6 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/n(q) the canonical truncation at the cohomological level q of the complex of sheaves Rπ_{∗}(π^{∗}(μ_{n}^{⊗q})) where π : (Sm/k)_{et} → (Sm/k)_{Nis} is the usual morphism of sites.

Theorem 7.9 For a smooth variety X and any n > 0 there are canonical isomorphisms

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

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

In [4] Spencer Bloch and Stephen Lichtenbaum constructed the motivic spectral sequence which starts from Higher Chow groups of a field and converges to its algebraic K-theory. Their construction was reformulated in much more natural terms and extended to all smooth varieties over fields in [7]. Combining the motivic spectral sequence with Theorem 7.9 and using [35] to identify motivic cohomology of [7] with the motivic cohomology of this paper we obtain the following result.

Theorem 7.10 Let k be a field of characteristic ≠ 2. Then for all smooth X over k and all n > 0 there exists a natural spectral sequence with the E_{2}-term of the form

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

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

Corollary 7.11 Let X be a complex algebraic variety of dimension d. Then the canonical homomorphisms K_{i}^{alg}(X, Z/2^{n}) → K_{i}(X(C), Z/2^{n}) are isomorphisms for i ≥ d − 1 and monomorphisms for i = d − 2.

8 Appendix A. Hypercohomology of pointed simplicial sheaves

We recall here some basic notions related to the hypercohomology of simplicial sheaves. Let T be a site with final object pt. For the purpose of the present paper one may assume that T is the category of smooth schemes over a field k with the Nisnevich or the etale topology and pt = Spec(k). Denote by AbShv(T) the category of sheaves of abelian groups on T. For an object X of T let Z(X) be the sheaf characterized by the property that

Hom(Z(X), F) = F (X)  (25)

for any sheaf of abelian groups F. Note that Z(Spec(k)) is the constant sheaf Z.

For a pointed simplicial object X, Z(X) is a simplical sheaf of abelian groups and we may form a complex Z(X)_{∗} by taking the alternating sums of boundary maps. For a complex of sheaves of abelian groups K we define the hypercohomology of X with coefficients in K by the formula

H^{n}(X , K) := Hom_{D}(Z(X), K)

where D = D(AbShv(T)) is the derived category of complexes of sheaves of abelian groups on T. The formula (25) together with the standard method of computing cohomology by means of injective resolutions implies that for X = X an object of T our definition agrees with the usual one.

Let now Y be a sheaf of sets on T. Then we can define Z(Y) as the free sheaf of abelian groups generated by Y such that for every sheaf of abelian groups F one has Hom(Z(Y), F) = Hom(Y, F) where on the left hand side we have morphisms of sheaves of abelian groups and on the right hand side morphisms of sheaves of sets. The Yoneda Lemma shows that for an object X, Z(X) = Z(h_{X}) where h_{X} is the sheaf of sets represented by X. Therefore, we may immediately extend our definition of hypercohomology groups to simplicial sheaves in a way which agrees with the definition for simplicial objects of T on representable sheaves. Starting from this point we consider objects of T to be a particular type of sheaves of sets.

If X is a pointed simplicial sheaf the distinguished point defines a homomorphism H^{∗}(X , K) → H^{∗}(pt, K) which has a canonical section defined by the projection X → pt. We define reduced hypercohomology of X by the formula:

H̃^{∗}(X , K) = ker(H^{∗}(X , K) → H^{∗}(pt, K))

Alternatively, we can define Z̃(X) setting

Z̃(X) := coker(Z → Z(X))

where the morphism is defined by the distinguished point. Then

H̃^{n}(X , K) = Hom_{D}(Z̃(X), K[n])

If X_{+} denotes the simplicial sheaf X ∐ pt pointed by the canonical embedding pt → X ∐ Spec(k), then

H^{∗,∗}(X , −) = H̃^{∗,∗}(X_{+}, −)

There is a standard spectral sequence with the E_{1}-term consisting of the hypercohomology of the terms X_{i} of X with coefficients in K which tries to converge to hypercohomology of X. However, this spectral sequence is of limited use since Z(X)_{∗} is unbounded on the left and the convergence properties of this spectral sequence are uncertain. The proof of the following proposition contains the trick which allows one get around this problem.

Proposition 8.1 Let n ≥ 0 be an integer and X and K be such that

H^{m}(X_{j}, K) = 0

for all j ≥ 0 and m ≤ n. Then H^{m}(X , K) = 0 for m ≤ n.

Proof: Let s_{≤i}Z(X)_{∗} be the subcomplex of Z(X)_{∗} such that

s_{≤i}Z(X)_{j} = \{ \begin{matrix}Z(X)_{j} & for j ≤ i \\ 0 & for j > i \end{matrix}

Since the complexes s_{≤i}Z(X)_{∗} are bounded the usual argument shows that Hom_{D}(s_{≤i}Z(X)_{∗}, K[m]) = 0 for m ≤ n and all i. On the other hand one has an exact sequence of complexes

0 → ⊕_{i}s_{≤i}Z(X)_{∗} → ⊕_{i}s_{≤i}Z(X)_{∗} → Z(X)_{∗} → 0

which expresses the fact that Z(X)_{∗} = colim_{i}s_{≤i}Z(X)_{∗}. This short exact sequence defines a long exact sequence of groups of morphisms in the derived category which shows that H^{m}(X , K) = 0 for m ≤ n.

For a morphism of simplicial sheaves f : X → X′ let cone(f) denote the simplicial sheaf such that for a smooth scheme U one has

cone(f) : U ↦ cone(X (U) → X′(U))

Lemma 8.2 The sequence

X →^{f} X′ → cone(f)

defines a long exact sequence of hypercohomology groups of the form

… → H̃^{n}(cone(f), K) → H̃^{n}(X , K) → H̃^{n}(X′, K) →
→ H̃^{n+1}(cone(f), K) → …

9 Appendix B. Čech simplicial schemes

We define here for any smooth scheme X a simplicial scheme Č(X) such that the map π : Č(X) → Spec(k) is a weak equivalence if and only if X has a rational point. This map defines an isomorphism in motivic cohomology with Z_{(n)}-coefficients if and only if X has a 0-cycle of degree prime to n. In particular, the motivic cohomology of C̃(X) := cone(π) with Z/n-coefficients provide obstructions to the existence of 0-cycles on X of degree prime to n.

Definition 9.1 For a variety X over k, Č(X) is the simplicial scheme with terms Č(X)_{n} = X^{n+1} and face and degeneracy morphisms given by partial projections and diagonals respectively.

Lemma 9.2 Let X, Y be smooth schemes over k such that

Hom(X, Y) ≠ ∅

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

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

(Č(Y) × X)(U) → X(U) = Hom(X, U)

is a weak equivalence.

Let C̃(X) denote the unreduced suspension of Č(X) i.e. the cone of of the morphism Č(X)_{+} → Spec(k)_{+}.

Lemma 9.3 Let Y be a smooth scheme which has a rational point over an extension of k of degree n. Then nH̃^{∗,∗}(C̃(Y), Z) = 0.

Proof: Let E be an extension of k and Y_{E} = Y ×_{Spec(k)} Spec(E) considered as a smooth scheme over E. Then there are homomorphisms

H̃^{∗,∗}(C̃(Y), Z) → H̃^{∗,∗}(C̃(Y_{E}), Z)

and

H̃^{∗,∗}(C̃(Y_{E}), Z) → H̃^{∗,∗}(C̃(Y), Z)

and the composition of the first one with the second is multiplication by deg(E/k). Let E be an extension of degree n such that Y (E) ≠ ∅. Since Y (E) ≠ ∅ the pointed sheaf C̃(Y_{E}) is contractible and therefore,

H̃^{∗,∗}(C̃(Y_{E}), Z) = 0.

We conclude that nH̃^{∗,∗}(C̃(Y), Z).

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