VOEVODSKY’S SEATTLE LECTURES:

K-THEORY AND MOTIVIC COHOMOLOGY

V. Voevodsky (Notes by C. Weibel)

These notes are based on a series of talks given by V. Voevodsky at the AMS Summer Research Conference on algebraic K-theory, held in Seattle during July 1997. The purpose of these talks was to outline the proof of “Milnor’s conjecture” [V] that the Milnor K-theory of a field F, reduced modulo two, is isomorphic to the étale cohomology of F with coefficients ℤ/2: K_{∗}^{M}(F)/2 ≅ H_{et}^{∗}(F, ℤ/2).

The first two parts develop a motivic homotopy theory, and are joint work with Fabien Morel. In the third part we define the motivic cohomology of a variety in this setting, a setting where it carries cohomology operations. The final part uses a series of reductions to reduce Milnor’s conjecture to a vanishing assertion in motivic cohomology, which can be checked using these operations and two results of Rost.

§1. Unstable Motivic Homotopy Theory (joint with Fabien Morel)

Fix a field k. In these notes, we will set up the foundations of a homotopy theory for algebraic varieties over k. Morally, we will replace the category Top of topological spaces by the category of algebraic varieties over k. The actual details are, of course, more involved.

First we recall how topologists construct stable homotopy theory. Using a subscript ‘∗’ to denote pointed categories, the topologists’ homotopy category Hot_{∗} is obtained from Top_{∗} by inverting weak homotopy equivalences. Then the stable homotopy category SHot_{∗} is obtained from Hot_{∗} by inverting the suspension, i.e., the smash product with the circle S^{1}. If we view the singular chain complex C_{∗}(X) of a topological space X as a functor to the derived category D(Ab) of abelian groups, then this functor factors through SHot_{∗}:

Top_{∗} ⟶^{invert w.e.} Hot_{∗}
C_{∗}↓  ↓_{invert S^{1}∧}
D(Ab) ⟵ SHot_{∗}

Remark. Thinking of SHot_{∗} as the category of S^{0}-modules, the bottom arrow is the base-change functor induced by the morphism S^{0} → K(ℤ, 0) of ring spectra. This is only a special case of a more general construction. For example, the morphisms from S^{0} to the cobordism spectrum MU and the topological complex K-theory spectrum KU give rise to functors from SHot_{∗} to the derived categories of MUmodules and KU-modules.

Typeset by AMS-TEX

Now let us begin our construction of a homotopy theory for algebraic varieties. Our approach is motivated by an analysis of what operations we would like to perform.

Let SmAff = SmAff/k denote the category of smooth affine algebraic varieties of finite type over k. As nice as it is, this category has some undesirable properties, such as the fact that quotients do not always exist. As a first step in correcting this defect, we replace varieties by presheaves. This has the advantage that we can construct arbitrary colimits of presheaves, and in particular quotients of presheaves.

Definition 1.1. Let PreShv = PreShv(SmAff/k) denote the category of contravariant functors from SmAff/k to the category of sets; we call such functors presheaves. Any scheme Y over k gives a representable presheaf h_{Y}, defined by h_{Y} (X) = Hom(X, Y). Thus we have Yoneda embeddings:

SmAff ⊂ Schemes ⊂ PreShv .

We will abuse notation and write Y for the presheaf h_{Y}.

Colimits. Any presheaf F is a colimit of representable functors. This well-known observation is due to Kan: F is the colimit of the h_{Y} over an indexing category of pairs (Y, α), where α ∈ F (Y). The maps α_{∗}: h_{Y} → F are determined from (Y, α) by the Yoneda lemma; a morphism from (Y′, α′) to (Y, α) is a morphism f: Y′ → Y sending α′ to α, and f induces a factorization h_{Y′} ⟶^{f} h_{Y} ⟶^{α_{∗}} F of α′_{∗}.

If Z is a subscheme of Y then we shall write Y/Z for the quotient presheaf h_{Y} /h_{Z}. This presheaf is not usually representable by a scheme. However, Y/Z is always a pushout of representable presheaves, in the sense that the following diagram is a cocartesian square of presheaves:

Z ⟶ Y
↓  ↓
Spec(k) ⟶ Y/Z.

Examples 1.2. (1) The 0-sphere S^{0} is the disjoint union of two copies of Spec(k).

(2) Let S_{s}^{1} denote the simplicial circle 𝔸^{1}/{0, 1}, which is the quotient presheaf obtained from the affine line 𝔸^{1} by identifying the two points 0 and 1. The presheaf S_{s}^{1} is represented by the affine node, which by coincidence is the corresponding pushout in the category of schemes.

(3) The Tate object T is obtained as the quotient of the affine line 𝔸^{1} by 𝔸^{1}−{0}. This makes sense as a cocartesian square as above.

(4) The Thom space of a vector bundle E → X is the quotient presheaf T h(E) = E/(E − s(X)). Here s(X) is the zero-section of E. For example, the Tate object T is the Thom space of the vector bundle 𝔸^{1} → Spec(k).

For comparison, the Thom space of a topological vector bundle E is usually described as the one-point compactification of E. But it can also be described as the quotient of the unit disk bundle by the unit sphere bundle. The complement of the zero-section is clearly the algebraic analogue of the sphere bundle.

(5) If (X, x) and (Y, y) are pointed presheaves, their smash product is the pointed presheaf

X ∧ Y = X × Y/((X × {y}) ∪ ({x} × Y)).

Unfortunately, the presheaf T h(E_{1} × E_{2}) need not equal T h(E_{1}) ∧ T h(E_{2}), because the Yoneda embedding from schemes to presheaves does not commute with colimits.

Exercise 1.3. Show that if O_{X} is the trivial line bundle on X, and X_{+} denotes the union of X and a disjoint basepoint, then T h(O_{X}) ≅ X_{+} ∧ T.

The limitations of the presheaf approach become apparent when we consider the Mayer-Vietoris construction, starting with a covering of X by two open subschemes U and V. In order to study this we consider the pushout (in PreShv) of U and V along U ∩ V; it is not represented by the pushout scheme X = U ∪ V. Indeed, the canonical element 1_{X} ∈ h_{X}(X) cannot come from an element of h_{U}(X) unless U = X.

To fix this problem, we pass to sheaves. Although the Zariski topology suffices for the pushout example above, we will eventually want to use the Nisnevich topology for our sheaves. Here is a quick way to define Nisnevich sheaves, which suffices for our purposes.

Definition 1.4. An elementary distinguished square is a cartesian square

U ×_{X} V ⟶ V
↓  ↓^{p}
U  ⟶^{i} X

in which i is an open embedding, p is an étale morphism, and p induces an isomorphism of closed subschemes: p^{−1}(X − U)_{red} ⟶^{≅} (X − U)_{red}.

A presheaf is called a Nisnevich sheaf if it takes elementary distinguished squares into cartesian squares of sets. We write Spc for the category of Nisnevich sheaves on SmAff/k. We will often refer to objects of Spc as “spaces.”^{†}

Every representable scheme h_{X} is a Nisnevich sheaf. This is equivalent to the fact that the family {U → X, V → X} is a universal effective epimorphism for every elementary distinguished square.

The Nisnevich topology. A Nisnevich sheaf is the same as a sheaf for the Nisnevich topology, which is a Grothendieck topology defined as follows.

A family {U_{α} → X} of étale morphisms is called a Nisnevich cover if, for every point x of X, there is an α and a point u ∈ U_{α} such that U_{α} → X sends u to x and induces an isomorphism of residue fields: k(x) ≅ k(u). This is equivalent to the requirement that for every field K the function ∐ U_{α}(K) → X(K) is surjective.

†There are alternative approaches to the construction of Hot which take a different basic category of spaces. One such construction uses the category of cdh sheaves on all varieties, including singular varieties. In the presence of resolution of singularities, the theorem below on Blow-up squares shows that both constructions yield the same stable homotopy category SHot.

Any sheaf for the Nisnevich topology satisfies the cartesian condition above, because {U, V } is a Nisnevich cover in any elementary distinguished square. A proof that Nisnevich sheaves are sheaves for the Nisnevich topology is given in [MV, 3.1], and is based upon the following observation. If {U_{α} → X} is a Nisnevich cover, then for each generic point x of X there is an α such that U_{α} → X is a birational isomorphism.

Example 1.5. The Nisnevich topology reflects many of the arithmetic properties of k. For example, for each a ≠ 0 in k, the square

𝔸^{1} − \{0, ±\sqrt{a}\} ↪ 𝔸^{1} − \{0, \sqrt{a}\}
↓  ↓^{square}
𝔸^{1} − {a}  ↪  𝔸^{1}.

is an elementary distinguished square if and only if the equation y^{2} = a has a solution in k.

Note that colimits of sheaves (e.g., pushouts, quotients and smash products) are the sheafification of the corresponding colimits of presheaves. From now on, when we refer to the Tate object T, the Thom space T h(E) or the smash product X ∧ Y, we shall mean the corresponding Nisnevich sheaf, rather than the presheaf.

Thom Spaces 1.6. Let E and E′ be vector bundles over X and X′, respectively. Then E × E′ is a vector bundle over X × X′. On the level of Nisnevich sheaves we have

T h(E × E′) ≅ T h(E) ∧ T h(E′).

In particular, if O_{X} is the trivial line bundle on X then T h(E ⊕ O_{X}) ≅ T h(E) ∧ T. If Σ_{T} denotes smashing with T, then this implies that T h(E ⊕ O_{X}^{n}) ≅ Σ_{T}^{n} T h(E).

Since T = T h(𝔸^{1}), this also implies that 𝔸^{n}/𝔸^{n} − {0} = T h(𝔸^{n}) ≅ T^{∧n}.

Exercise 1.7. Suppose that U, V and X form an elementary distinguished square, and let U ∪_{U∩V} V denote the pushout of U and V along U ∩ V = U ×_{X} V in the category of Nisnevich sheaves. Show that a) U ∪_{U∩V} V → X is an isomorphism of Nisnevich sheaves, and hence that b) X/U is isomorphic to V/(U ∩ V). (See [MV, 3.1.6].)

Remark. This exercise shows that we could have started with the category Sm/k of all smooth schemes over k, instead of the category SmAff of smooth affine schemes. The resulting category Spc of spaces would have been the same.

The homotopy category Hot.

In order to form the homotopy category of spaces, we shall define a model structure on the category Spc, in the sense of Quillen. The role of cofibrations will be played by the class C of all monomorphisms.

Since we want the affine line 𝔸^{1} to be the analogue of the unit interval, we require X × 𝔸^{1} → X to be a weak equivalence. This leads to the following definition.

Definition 1.8. The class W = W_{𝔸^{1}} of (𝔸^{1}–)weak equivalences is the smallest class of morphisms in Spc containing all isomorphisms and satisfying the following axioms. A trivial cofibration is defined to be a cofibration which is also a weak equivalence.

(1) (Homotopy) each projection X × 𝔸^{1} → X is in W;

(2) (Saturation) If two of f, g and fg are in W, so is the third;

(3) (Continuity) If X_{α} ⟶^{f_{αβ}} X_{β} is a filtered system of trivial cofibrations, then each X_{α} ⟶ colim_{β} X_{β} is in W.

(4) The pushout of a weak equivalence along a cofibration is also a weak equivalence. That is, if c is a cofibration and w a weak equivalence in the left pushout diagram below then w′ is also a weak equivalence.

\begin{matrix}X & ⟶^{w} & Y \\ ^{c}↓ & & ↓ \\ Z & ⟶^{w′} & Z ∪_{X} Y \end{matrix}  \begin{matrix}X & ⟶^{w} & Y \\ ↓ & & ↓ \\ Z & ⟶^{w′} & Z ∪_{X} Y \end{matrix}

(5) Similarly, the pushout of a trivial cofibration along any map is a weak equivalence. That is, if w is both a cofibration and a weak equivalence in the right pushout diagram above then w′ is also a weak equivalence.

We define the 𝔸^{1}-fibrations in Spc to be the class F_{𝔸^{1}} of all morphisms with the right lifting property relative to trivial cofibrations.

The following result is based upon results Joyal [Jo] and Jardine [J]. Its proof requires the use of simplicial sheaves, and will be sketched shortly.

Theorem 1.9. The classes C, W_{𝔸^{1}} and F_{𝔸^{1}} form a proper (left and right) closed model structure on Spc.

Definition 1.10. The homotopy category Hot = Hot_{𝔸^{1}}(k) of schemes over k is the category obtained from Spc by the standard process of inverting the weak equivalences in a model category. In particular, morphisms in Hot are of the form fw^{−1} and composition is described by a calculus of fractions.

There is a variant of these constructions, starting from the category Spc_{∗} of pointed spaces. There is a proper closed model structure on Spc_{∗}; a map in Spc_{∗} is a cofibration, weak equivalence or fibration if the underlying map in Spc is one. This allows us to form the corresponding homotopy category Hot_{∗} of pointed spaces over k. It is not hard to show that the smash product makes Hot_{∗} into a symmetric monoidal category with unit S^{0}.

Example of a fibration 1.11. A scheme X is called rigid if Hom(Y × 𝔸^{1}, X) = Hom(Y, X) for every Y. If X is rigid, then every projection Y × X → Y is an 𝔸^{1}-fibration, and Hom_{Hot}(Y, X) = Hom_{Spc}(Y, X).

Any curve X of genus > 0 is rigid, because any map 𝔸^{1} → X is trivial. Thus there is a full embedding of the category of such curves into Hot.

Simplicial Structure.

Let us write Δ^{op} Spc for the category of simplicial “spaces,” i.e., simplicial objects in Spc. Since we may regard any set as a scheme (a disjoint union of copies of Spec(k)), and hence as an object of Spc, we may regard every simplicial set as a simplicial space. It should not be surprising that the standard n-simplex Δ[n] is a very useful simplicial space.

Here is another way to obtain simplicial spaces. Let Δ^{•} denote the standard cosimplicial object in SmAff which in degree n is the scheme

Δ^{n} = Spec(k[t_{0}, …, t_{n}]/(∑ t_{i} = 1)).

There is a “chains” functor C_{∗}: Spc → Δ^{op} Spc sending the sheaf X to the simplicial sheaf X(– × Δ^{•}), which in degree n is Y ↦ C_{n}(X)(Y) = X(Y × Δ^{n}).

There is also a geometric realization functor X ↦ |X|_{𝔸^{1}}, from Δ^{op} Spc to Spc. It is the left adjoint functor to C_{∗}, just as the geometric realization of topological spaces is left adjoint to the singular chain complex. By adjointness, the realization of a constant simplicial space n ↦ X is just X. Hence the geometric realization functor is characterized by the fact that it sends the n-simplex Δ[n] to the scheme Δ^{n}. For example, the realization of the boundary of the n-simplex is the union of the “faces” of the scheme Δ^{n}, and the realization of the simplicial circle S^{1} = Δ^{1}/{0, 1} is the node S_{s}^{1}.

Joyal proved in [Jo] that there is a “simplicial” model structure on Δ^{op} Spc, which is closed and proper in the sense of [BF]. The “simplicial cofibrations” are the monomorphisms, and the “simplicial weak equivalences” are the maps which induce isomorphisms on all sheaves of homotopy groups. The “simplicial mapping spaces” in this structure are the simplicial sets Hom_{•}(X, Y) = Hom(X × Δ^{•}, Y).

We shall write Hot(Δ^{op} Spc) for the corresponding “simplicial” homotopy category, obtained from Δ^{op} Spc by inverting the simplicial weak equivalences.

With this simplicial structure available, it is possible to prove that the monomorphisms, 𝔸^{1}-weak equivalences (defined using C_{∗}𝔸^{1} in place of 𝔸^{1}) and 𝔸^{1}-fibrations provide the category Δ^{op} Spc with the structure of a proper closed model category; see [MV, 2.2.5]. The corresponding homotopy category Hot_{𝔸^{1}}(Δ^{op} Spc) may be obtained from the simplicial homotopy category Hot(Δ^{op} Spc) by classical f-localization, i.e., by inverting all the projections f: C_{∗}𝔸^{1} × X → X. (See [B].)

The usual argument in topology shows that the adjoint functors C_{∗} and |–|_{𝔸^{1}} take 𝔸^{1}-weak equivalences to 𝔸^{1}-weak equivalences, and 𝔸^{1}-fibrations to 𝔸^{1}-fibrations. It follows that the 𝔸^{1}-structure also makes Spc into a proper closed model category, and we can identify the homotopy categories Hot of Spc and Hot_{𝔸^{1}}(Δ^{op} Spc).

Up to this point, everything we have said has been equally true for Zariski sheaves. Here are two results whose Zariski analogues are false.

Theorem 1.12 (“Homotopy Purity”). Let Z ⊂ X be a smooth pair, with normal bundle N_{X}Z. Then T h(N_{X}Z) is 𝔸^{1}-weak equivalent to X/(X − Z), i.e., there is a homotopy cocartesian square

X − Z ⟶  X
↓  ↓
point ⟶ T h(N_{X}Z)

Theorem 1.13 (Blow-ups). Let Z ⊂ X be a smooth pair, and X̃ the blowup of X along Z. Then the first S_{s}^{1}-suspension of the blowup square

p^{−1}Z ⟶ X̃
↓  ↓^{p}
Z ⟶ X

is homotopy cocartesian.

We do not know if the suspension is necessary in this theorem. The first step in the proof is to show that the homotopy cofiber of the left column is the same as the homotopy cofiber of X̃/U → X/U.

Classifying spaces of algebraic groups.

We will write B_{gm}(GL_{n}) for the infinite Grassmannian G(n, ∞). This is a geometric substitute for the classifying space of GL_{n} in our setting.

Here is a geometric construction which works more generally for any linear algebraic group G defined over k. The geometric classifying space B_{gm}(G) of G is an object of Spc which is well-defined up to 𝔸^{1}-weak equivalence. To construct it, we fix a faithful representation ρ: G → GL_{n}. In the diagonal representation ρ^{i} of G on 𝔸^{ni} = (𝔸^{n})^{i}, let U_{i} ⊂ 𝔸^{ni} be the maximal open subscheme where G acts freely. The geometric quotient V_{i} = U_{i}/G is smooth over k. We define B_{gm}(G) to be the colimit V_{∞} of the spaces V_{i} along the system of closed embeddings V_{i} ↪ V_{i+1} induced by the 𝔸^{ni} ⊂ 𝔸^{n(i+1)}.

For example, if G = GL_{n} and we consider 𝔸^{ni} as parametrizing n × i matrices, then V_{i} is the classical Grassmann variety of n-dimensional linear subspaces of 𝔸^{i}, and V_{∞} is the usual infinite Grassmannian G(n, ∞) classifying n-dimensional vector bundles. Thus our two definitions of B_{gm}(GL_{n}) agree.

Lemma 1.14. Up to isomorphism in Hot(k), the space B_{gm}(G) doesn’t depend upon ρ.

In topology, the classifying space BG of a group G is a simplicial set. This provides another way to construct a classifying space for a group scheme G. Since each G(U) is a group, the simplicial sets B(G(U)) assemble to form the simplicial (nisnevich) sheaf B_{nis}G: U ↦ B(G(U)). By abuse of notation, we shall write BG for the geometric realization |B_{nis}G|_{𝔸^{1}}. The following is a restatement of [MV, 4.1.18 and 4.2.7].

Lemma 1.15. There is a natural map BG → B_{gm}G. This map is an 𝔸^{1}-weak equivalence if and only if “Hilbert’s Theorem 90” holds for G, i.e., H_{et}^{1}(K, G) = 0 for every finitely generated field extension K of k.

Example 1.16. Consider the algebraic group GL_{n}. The usual proof of Hilbert’s Theorem 90 [Milne, III(4.10)] shows that H_{nis}^{1}(K, GL_{n}) = H_{et}^{1}(K, GL_{n}) = 0. Thus BGL_{n} is weak equivalent to B_{gm}(GL_{n}) = G(n, ∞).

Example 1.17. Consider the algebraic group μ_{2} with \frac{1}{2} ∈ k. The spaces Bμ_{2} and B_{gm}(μ_{2}) are not weak equivalent, because H_{et}^{1}(K, μ_{2}) = K^{×}/K^{×2} is nonzero for K = k(t).

§2. The stable motivic homotopy category

The pointed category Spc_{∗} has two circles, the simplicial circle S_{s}^{1} (or node) and the “Tate circle” S_{t}^{1} = 𝔸^{1} − {0}, with 1 as basepoint. The smash product with these two circles gives two different types of suspension (Σ_{s} and Σ_{t}) on the category Spc_{∗}, as well as on the homotopy category Hot_{∗}.

The simplicial suspension Σ_{s}F of a pointed simplicial sheaf F is the smash product of S_{s}^{1} with F. This is the sheaf associated to the presheaf U ↦ ΣF (U), where Σ denotes the usual suspension of a pointed simplicial set. If X is a space then |Σ_{s}C_{∗}X|_{𝔸^{1}} ≃ |S_{s}^{1}| ∧ X. If we only invert the simplicial suspension on Hot_{∗} we obtain the stable simplicial homotopy category, which we write as Hot_{∗}^{s}.

Definition 2.1. The stable homotopy category SHot(k) of schemes over k is defined to be the homotopy category Hot_{∗} with T -suspension Σ_{T} (X) = T ∧ X inverted.

The next exercise shows that T -suspension is Σ_{t} ∘ Σ_{s}. Hence SHot(k) may also be obtained from Hot_{∗}^{s} by inverting only the “Tate” suspension Σ_{t}.

Exercise 2.2. Show that the Tate object T is weak equivalent to both S_{s}^{1}∧S_{t}^{1} and the pointed projective line ℙ_{∗}^{1}. Hint: Construct weak equivalences S_{s}^{1} ∧ S_{t}^{1} ⟵ P ⟶ T, where P is the pushout of 𝔸^{1} and 𝔸^{1} ∧ S_{t}^{1} along S_{t}^{1}. For the second assertion, show that T ≃ ℙ^{1}/𝔸^{1} ≃ ℙ^{1}/point = ℙ_{∗}^{1}.

Theorem 2.3. SHot(k) is a triangulated category, where the translation functor is simplicial suspension: E[1] = Σ_{s}E. The smash product ∧ provides SHot(k) with a tensor structure compatible with the triangulated structure.

The proof of this result is long but straightforward; see [V, 3.10]. It uses the fact that the cyclic permutation on S^{1} ∧ S^{1} ∧ S^{1} is equivalent to the identity. However the reader should beware that, although the transpose map on S^{2}∧S^{2} is (homotopic to) the identity in topology, the transpose map τ_{T} on T ∧ T is not the identity in general. We remark that τ_{T} is the identity whenever k contains \sqrt{−1}.

Here is another approach to constructing SHot(k), the stable homotopy category: instead of inverting 𝔸^{1}-weak equivalences first and T -suspensions second, we could first invert T -suspensions. This leads to the notion of T -spectra.

Definition 2.4. A T -spectrum E is a sequence of pointed spaces E_{i}, together with bonding maps T ∧E_{i} → E_{i+1}. We write T -Spectra for the category of T -spectra; a morphism E → F of T -spectra is just a sequence of maps E_{n} → F_{n} which commute with the bonding maps.

The category of T -spectra has an evident notion of stable 𝔸^{1}-weak equivalence. If we localize the category of T -spectra by inverting stable 𝔸^{1}-weak equivalences á la Bousfield-Friedlander [BF], we get a homotopy category of T -spectra, SHot^{T} (k).

As in topology, there is a suspension T -spectrum functor Σ_{T}^{∞} from Spc_{∗} to T -Spectra; the ith space of Σ_{T}^{∞}X is T^{∧i} ∧ X, and the bonding maps are the associativity maps. The functor Σ_{T}^{∞} preserves 𝔸^{1}-weak equivalences, and smash products: Σ_{T}^{∞}(X ∧ Y) is equivalent to Σ_{T}^{∞}(X) ∧ Σ_{T}^{∞}(Y). Since the smash product with T is a self-equivalence on the category of T -spectra, Σ_{T}^{∞} induces a functor from SHot to SHot^{T} (k). This functor is an equivalence of homotopy categories.

Theorem 2.5. ( [V, 3.10]) The functor Σ_{T}^{∞}: Spc_{∗} → T -Spectra takes cofibration sequences to distinguished triangles.

We shall write S_{s}^{m} and S_{t}^{m} for the suspension T -spectra Σ_{T}^{∞}S_{s}^{m} and Σ_{T}^{∞}S_{t}^{m}, respectively. Note that if m < 0 the object S_{s}^{m} belongs to Hot_{∗}^{s} but not Hot_{∗}, while the object S_{t}^{m} of SHot does not even belong to Hot_{∗}^{s}; it exists as the formal object Σ_{t}^{−m}(S^{0}).

Definition 2.6. For integers p, q the bigraded sphere T -spectrum is

S^{p,q} = S_{s}^{p−q} ∧ S_{t}^{q}.

If E is a T -spectrum, we write E(q)[p] for S^{p,q} ∧ E.

For example, applying Σ_{s}^{−1} to S^{2n,n} ≅ T^{∧n} ≅ 𝔸^{n}/𝔸^{n} − {0} yields a canonical isomorphism in Hot_{∗}^{s}:

𝔸^{n} − {0} ≅ S^{2n−1,n} = (S_{s}^{1})^{n−1} ∧ (S_{t}^{1})^{n}.

E-Cohomology 2.7. If E is a T -spectrum, we can define a bigraded cohomology theory on Hot(k):

E^{p,q}(X) = Hom_{SHot}(Σ_{T}^{∞}(X_{+}), E(q)[p]).

This definition is parallel to the usual definition in topology, except for the fact that we need to allow for two kinds of suspension. The same sort of bigrading occurs in Atiyah’s KR-theory, for example. The (p, q) indexing convention we use is chosen to enhance the comparison with étale cohomology (which is also bigraded).

In the next part, we will introduce the Eilenberg-Mac Lane T -spectra H_{R}. Here are some other important examples of T -spectra, and their associated cohomology theories.

Example 2.8. Let BGL denote the infinite Grassmannian, the union over N of the Grassmannians G(N, ∞). Consider the T -spectrum K = (BGL, T ∧BGL → BGL). This is the analogue of the BU-spectrum in topology, where the bonding maps come from Bott periodicity. If we take T = ℙ_{∗}^{1}, as in the exercise above, the bonding map ℙ_{∗}^{1} ∧ BGL → BGL is the classifying map for the canonical “Bott” element of K_{0}(ℙ_{∗}^{1} ∧ BGL).

Theorem 2.9. If X is smooth over k, then K^{p,q}(X) is isomorphic to K_{2q−p}^{alg}(X), the algebraic K-theory of X. In particular, K^{p,q}(X) = 0 if p > 2q.

Note that this bigraded cohomology theory has a (2, 1)-periodicity: K^{p+2,q+1} ≅ K^{p,q}. The periodicity is induced by the identifications K^{2p,p}(k) ≅ KU^{2p}(point).

Example 2.10. Consider the T -spectrum MGL = {T h(E_{n} → BGL_{n})}_{n}, with bonding maps coming from E_{n} ⊕ O → E_{n+1} via T h(E ⊕ O) ≅ Σ_{T} T h(E).

The T -spectrum MGL represents algebraic cobordism, once we impose étale descent and pass to finite coefficients. It is not known what MGL^{pq}(X) is, but it can be proven that MGL^{2p,p}(k) is the topological cobordism group MU^{2p}(point).

§3. Motivic Cohomology and the Motivic Steenrod Algebra

If X is a scheme of finite type over k, there is a sheaf L(X), the “free sheaf with transfers generated by X,” defined as follows.

Definition 3.1. L(X): SmAff/k^{op} → Ab is the presheaf sending a connected U to the free abelian group on the set of all closed irreducible W ⊂ U × X such that the projection W → U is finite and surjective.

If R is a ring, we define L(X, R) to be the presheaf sending U to the free Rmodule on the above set. It is a theorem that L(X) and L(X, R) are Nisnevich sheaves, i.e., objects of Spc. (See [SV2, 6.6 and 5.18] or [SV, 4.2.9].)

The graph of a function gives a map Hom(U, X) → L(X)(U), inducing a canonical map X → L(X) in Spc. This is the analogue of the Dold-Thom construction:

Let S^{n}X denote the nth symmetric product of X, i.e., S^{n}X = X^{n}/Σ_{n}, and let S^{·}X denote ∐_{n≥0} S^{n}X. In topology S^{·}X is the free abelian monoid on X, and it represents homology by the Dold-Thom theorem [DT]: H_{n}(X, ℤ) = π_{n}(S^{·}X) for all n. In algebraic geometry, we need to group complete the Hom monoid:

Lemma 3.2. ( [SV2, 6.8]) If U is smooth over k, then L(X)(U) is the group completion of the abelian monoid Hom_{Varieties}(U, S^{·}X).

Definition 3.3 (H_{R}). Given a ring R, we define K(R(n), 2n) to be the sheaf of abelian groups (considered as a sheaf of sets, i.e., as an object of Spc):

K(R(n), 2n) = L(𝔸^{n}, R)/L(𝔸^{n} − {0}, R).

There are product maps K(R(m), 2m) ∧ K(R(n), 2n) → K(R(m + n), 2m + 2n), induced from the external product of cycles. Bonding maps T ∧ K(R(n), 2n) → K(R(n + 1), 2n + 2) are obtained by composing with the natural map from T = 𝔸^{1}/𝔸^{1} − {0} to L(𝔸^{1})/L(𝔸^{1} − {0}) = K(R(1), 2).

We write H_{R} for the resulting T -spectrum, formed from the K(R(n), 2n), and call it the “Eilenberg-Mac Lane T -spectrum associated to R.”

Definition 3.4. The motivic cohomology of X over k is defined to be its H_{R}- cohomology:

H^{pq}(X, R) = H_{R}^{pq}(X) = Hom_{SHot}(Σ_{T}^{∞}(X_{+}), H_{R}(q)[p]).

Here are some of the main properties of motivic cohomology. We assume that k admits resolution of singularities, and that X is smooth.

(1) H^{2p,p}(X, ℤ) = CH^{p}(X), the classical Chow group of p-cycles on X;

(2) H^{pp}(Spec k, ℤ) ≅ K_{p}^{M}(k), the Milnor K-groups of the field k;

(3) H^{pq}(X, R) = 0 when p > q + dim(X);

(4) H^{pq}(X, R) = 0 when p > 2q;

(5) H^{pq}(X, R) = 0 when q < 0;

(6) H^{pq}(X, ℤ) = CH^{q}(X, 2q − p), the higher Chow groups of Bloch;

(7) H^{∗∗}(X, R) is a bigraded ring, natural in X.

Beilinson-Soulé Vanishing Conjecture. For smooth X, is H^{pq}(X, ℤ) = 0 for p < 0?

Example 3.5. For each integer ℓ > 0 we have H^{pp}(k, ℤ/ℓ) = K_{p}^{M}(k)/ℓ. This follows from the above properties and the exact sequence

H^{pp}(k, ℤ) ⟶^{ℓ} H^{pp}(k, ℤ) ⟶ H^{pp}(k, ℤ/ℓ) ⟶ H^{p+1,p}(k, ℤ).

Exercise 3.6. Let ℓ be a prime. (1) Show that H^{pq}(X, ℤ_{(ℓ)}) is the localization of the abelian group H^{pq}(X, ℤ) at the prime ℓ.

(2) Show that K_{p}^{M}(k) ⊗ ℚ/ℤ_{(ℓ)} ≅ H^{pp}(Spec k, ℚ/ℤ_{(ℓ)}).

Here is the only theorem in this part that cannot presently be proven without using resolution of singularities. Let Ω_{T} be the right adjoint of Σ_{T}, so that the bonding maps Σ_{T} E_{i} → E_{i+1} of an T -spectrum E induce maps E_{i} → Ω_{T} E_{i+1}.

Theorem 3.7. Assume that k admits resolution of singularities. Then H_{R} is an “Ω_{T} -spectrum,” i.e., the spaces K(R(n), 2n) satisfy:

K(R(n), 2n) ≃ Ω_{T} K(R(n + 1), 2n + 2) in Hot_{∗}.

In particular, the unstable group Hom_{Hot}(X, K(R(n), 2n)) equals the stable group H^{2n,n}(X, R) = Hom_{SHot}(Σ_{T}^{∞}(X_{+}), K(R(n), 2n)).

As in algebraic topology, we can interpret a natural motivic cohomology operation ϕ_{X}: E^{p,q}(X) → E^{p+i,q+j}(X) as a (stable) map ϕ: E → E(j)[i] = S^{i,j} ∧ E in SHot. Indeed, ϕ_{E} sends the identity element in E^{00}(E) = Hom(E, E) to ϕ, considered as an element of E^{ij}(E) = Hom(E, E(j)[i]).

Conversely, the map ϕ determines the operations ϕ_{X}. For each x in E^{pq}(X), the identification E^{pq}(X) = Hom_{SHot}(Σ_{T}^{∞}(X_{+}), E(p)[q]) allows us to compose x with

E(q)[p] = S^{pq} ∧ E ⟶^{1∧ϕ} S^{pq} ∧ S^{ij} ∧ E = S^{p+i,q+j} ∧ E = E(q + j)[p + i]

to get the element ϕ_{X}(x) of E^{p+i,q+j}(X). As in topology, the fact that ϕ is stable implies that each ϕ_{X} is additive: ϕ_{X}(x + y) = ϕ_{X}(x) + ϕ_{X}(y).

Restricting our attention to the T -spectrum H_{ℤ/ℓ} which yields the motivic cohomology groups H_{ℤ/ℓ}^{pq}(X) = H^{pq}(X, ℤ/ℓ), we see that natural motivic cohomology operations ϕ_{X}: H^{p,q}(X, ℤ/ℓ) → H^{p+i,q+j}(X, ℤ/ℓ) correspond to elements of H^{ij}(H_{ℤ/ℓ}, ℤ/ℓ).

Definition 3.8. The (mod ℓ) motivic Steenrod algebra A^{pq} = A^{pq}(k, ℤ/ℓ) is the algebra of endomorphisms of the T -spectrum H_{ℤ/ℓ} in the stable category SHot(k). That is,

A^{pq} = Hom_{SHot}(H_{ℤ/ℓ}, H_{ℤ/ℓ}(q)[p]) = H^{pq}(H_{ℤ/ℓ}, ℤ/ℓ).

Examples 3.9. (1) The bigraded ring H^{∗∗} = H^{∗∗}(k, ℤ/ℓ) is a subring of the bigraded A^{∗∗}. Indeed, if a ∈ H^{ij} then left multiplication by a induces a natural map from H^{pq}(X, ℤ/ℓ) to H^{p+i,q+j}(X, ℤ/ℓ), and is nonzero on H^{00}(k, ℤ/ℓ) if a ≠ 0.

When p = q, we see by example 3.5 that the group K_{p}^{M}(k)/ℓ is a subgroup of A^{pp}. In particular, we will be interested in the cohomology operation ρ ∈ H^{11}(k, ℤ/ℓ) corresponding to the class of −1 ∈ k^{×}/k^{×ℓ}. When ℓ is odd, or when ℓ = 2 and \sqrt{−1} ∈ k, then clearly ρ = 0.

(2) ([V, 3.14]) If char(k) = 0, then A^{pq} = A^{pq}(k, ℤ/ℓ) is zero for q < 0, and A^{00} ≅ ℤ/ℓ is generated by the identity, 1.

(3) The Bockstein β ∈ A^{10} is the connecting homomorphism in the distinguished triangle

H_{ℤ/ℓ} ⟶ H_{ℤ/ℓ^{2}} ⟶ H_{ℤ/ℓ} ⟶^{β} H_{ℤ/ℓ}.

Theorem 3.10. Fix a field k of characteristic zero, and a prime ℓ. Then there are motivic cohomology operations Q_{i} ∈ A^{2ℓ^{i}−1,ℓ^{i}−1} satisfying the following properties:

(i) Q_{i}Q_{j} = −Q_{j}Q_{i} and Q_{i}^{2} = 0; (ii) Q_{0} is the Bockstein β, and there are operations q_{i} so that Q_{i} = [β, q_{i}], i > 0. (iii) Δ: H^{∗∗}(H_{ℤ/ℓ}, ℤ/ℓ) → H^{∗∗}(H_{ℤ/ℓ}, ℤ/ℓ) ⊗_{H^{∗∗}} H^{∗∗}(H_{ℤ/ℓ}, ℤ/ℓ) satisfies

Δ(Q_{i}) = 1 ⊗ Q_{i} + Q_{i} ⊗ 1 + ∑(ρ^{n_{j}}ϕ_{j}) ⊗ ψ_{j}

where the operations ϕ_{j} and ψ_{j} have bidegrees (p, q) with p > 2q. In particular, Q_{i} is primitive if ℓ is odd, or if ℓ = 2 and \sqrt{−1} ∈ k.

The case i = 1 of formula (iii) is Δ(Q_{1}) = 1 ⊗ Q_{1} + Q_{1} ⊗ 1 + ρQ_{0} ⊗ Q_{0}.

Realization 3.11. Any embedding of k in ℂ gives a functor from Sm/k to complex manifolds, and hence to topological spaces. Using the fact that any sheaf is a colimit of representable sheaves, this functor can be extended to a topological realization functor t_{ℂ} from Spc to topological spaces. This sends H_{ℤ/ℓ} to the usual Eilenberg-MacLane spectrum with coefficients ℤ/ℓ, and maps A^{∗∗} to the usual Steenrod algebra. Under this realization, t_{ℂ}(Q_{i}) is the usual cohomology operation Q_{i} (see [M58] or [Mar, ch. 15]). For example, when ℓ = 2 we have Q_{1} = [Sq^{2}, Sq^{1}] = Sq^{3} + Sq^{2}Sq^{1}.

Remark 3.12. There should be a spectral sequence converging to K_{∗}(X; ℤ/2):

E_{2}^{p,q} = H^{p−q}(X; ℤ/2(−q)) ⟹ K_{−p−q}(X; ℤ/2).

Bloch and Lichtenbaum have constructed such a spectral sequence when X = Spec(k). The differential d_{2} goes from H^{p,q}(X, ℤ/2) to H^{p+3,q+1}(X, ℤ/2), and it is reasonable to ask if d_{2} equals the motivic operation Q_{1}.

For motivation, consider the topological analogue. If X is a topological space then there is an Atiyah-Hirzebruch spectral sequence converging to KU^{∗}(X; ℤ/2); the first nontrivial differential d_{3} : H^{p}(X, ℤ/2) → H^{p+3}(X, ℤ/2) is a cohomology operation, so d_{3} is a degree 3 element of the usual Steenrod algebra A. Now A^{3} is spanned by Sq^{2}Sq^{1} and Sq^{3}, and simple computations with X = ℝℙ^{4} and X = ℝℙ^{6} show that d_{3} must equal Q_{1} = Sq^{2}Sq^{1} + Sq^{3}. (This folklore result dates back to the calculations in [St, p. 542].)

§4. Motivic Cohomology of Quadrics and the Milnor Conjecture

Recall (from §3) that the motivic cohomology of X was defined as

H^{pq}(X, R) = Hom_{SHot}(Σ_{T}^{∞}X_{+}, Σ^{pq}H_{R}),

where the bigraded suspension Σ^{pq} is the smash product with S^{p,q} and H_{R} is the T -spectrum with nth space K(R(n), 2n) = L(𝔸^{n}, R)/L(𝔸^{n} − {0}, R).

Also recall from §1 that if F is a presheaf then C_{∗}(F) is the simplicial presheaf F (– × Δ^{•}). If F is a presheaf of abelian groups then we shall also write C_{∗}(F) for the chain complex of presheaves (of abelian groups) corresponding to this simplicial presheaf under the Dold-Kan correspondence.

Definition 4.1. ([V2, 4.1.8]) If R is a ring, the motivic complex R(n) over k is the chain complex of presheaves

R(n) = C_{∗}(L(𝔸^{n}, R)/L(𝔸^{n} − {0}, R))[−2n]

Indexing convention. Here we use the shifting convention that if F_{∗} is a chain complex then the degree i part of the chain complex F [n] is F_{i−n}. In order to take the hypercohomology of F_{∗}, we reindex as a cochain complex (F^{n} = F_{−n}), so that ℍ^{i}(X, F_{∗}[n]) = ℍ^{i+n}(X, F_{∗}).

Theorem 4.2. H^{pq}(X, R) ≅ ℍ_{zar}^{p}(X, R(q)).

In particular, this shows that the groups H^{pq}(X, R) do not depend upon the choice of the ground field k, since the presheaves R(q) do not.

This theorem follows by combining the following lemma, partly due to K. Brown [Br, p. 426], with the (hard) theorem 3.6 which states that the stable groups H^{pq}(X, R) coincide with the unstable Hom groups in the lemma.

Main Lemma 4.3. ℍ_{zar}^{2n}(X, R(n)) ≅ ℍ_{Nis}^{2n}(X, R(n)) ≅ Hom_{Hot}(X, K(R(n), 2n)).

Definition 4.4 (H_{L}^{∗∗}). The Lichtenbaum motivic cohomology of X is defined to be the étale hypercohomology of R(q): H_{L}^{pq}(X, R) = ℍ_{et}^{p}(X, R(q)).

Warning. There is a subtleness in this definition. There is a stable étale homotopy category SHot_{et}, obtained by replacing Nisnevich sheaves by étale sheaves in the definitions, and “étale motivic cohomology” groups Hom_{SHot_{et}}(Σ_{T}^{∞}X_{+}, Σ^{pq}H_{R}) of X. In characteristic zero, they agree with H_{L}^{pq}(X, R). However, they are not the same as H_{L}^{pq}(X, R) in characteristic p, unless we replace the coefficients R by R[\frac{1}{p}].

To see why, consider the Frobenius morphism F in characteristic p. Let 𝔾_{a} denote the sheaf of abelian groups X ↦ H^{0}(X, O_{X}). Since 𝔾_{a} ≃ 0 in the derived category DM_{k} of [V2], the Artin-Schreier sequence 0 → ℤ/p → 𝔾_{a} ⟶^{1−F} 𝔾_{a} → 0 implies that the constant sheaf ℤ/p is zero in DM_{k}, and that DM_{k} is ℤ[1/p]-linear in characteristic p. (See [V1, 4.1.7].) Applying X ↦ L(X) to ℤ/p yields H_{ℤ/p}. Hence H_{ℤ/p} is 𝔸^{1}-weak equivalent to zero in SHot_{et}.

We can identify Lichtenbaum motivic cohomology with finite coefficients.

Theorem 4.5. (Suslin-Voevodsky) If 1/ℓ ∈ k then for all X

H_{L}^{p,q}(X, ℤ/ℓ) ≅ H_{et}^{p}(X, μ_{ℓ}^{⊗q}).

The proof doesn’t require resolution of singularities, provided that we use the above definition of H_{L}^{pq}. The case q = 1 follows from the fact that ℤ(1)_{et} = 𝔾_{m}[−1] in the étale topology, which implies that ℤ/ℓ(1)_{et} = μ_{ℓ}.

We are interested in the following fundamental conjecture, made independently by Beilinson ([Bei, p. 22]) and Lichtenbaum ([L, p. 130]). It effectively connects motivic cohomology to étale cohomology.

Beilinson-Lichtenbaum Conjecture 4.6. The map H^{pq}(X, ℤ) → H_{L}^{pq}(X, ℤ) is an isomorphism for all p ≤ q + 1.

If true, this would imply as a corollary that H^{pq}(X, ℤ/ℓ) ≅ H_{et}^{p}(X, μ_{ℓ}^{⊗q}) for p ≤ q. Since H^{pp}(k, ℤ/ℓ) = K_{p}^{M}(k)/ℓ, the case p = q of Lichtenbaum’s Conjecture includes the:

Norm Residue Homomorphism Conjecture 4.7. ( [BK, p. 118]) If \frac{1}{ℓ} ∈ k, the Norm Residue homomorphism

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

is an isomorphism for all p.

This in turn is a generalization to all primes ℓ of Milnor’s conjecture, that K_{p}^{M}(k)/2 ≅ H_{et}^{p}(k, ℤ/2). (This was originally asked as a question in [M70].)

It is helpful to localize the Lichtenbaum Conjecture at a prime ℓ. Since the sheaf ℤ_{(ℓ)}(q) is the localization of ℤ(q) at ℓ, we see that the groups H^{p,q}(X, ℤ_{(ℓ)}) and H_{L}^{p,q}(X, ℤ_{(ℓ)}) are the localizations of H^{pq}(X, ℤ) and H_{L}^{pq}(X, ℤ) at the prime ℓ, respectively.

Example 4.8 (R = ℚ). We have H^{pq}(X, ℚ) ≅ H_{L}^{pq}(X, ℚ) for all p and q. This follows from the fact that étale and Nisnevich hypercohomology agree for any complex of étale sheaves of ℚ-vector spaces, such as ℚ(q), which is true in turn because all higher Galois cohomology groups vanish for uniquely divisible coefficients.

Hence the kernel and cokernel of H^{pq}(X, ℤ) → H_{L}^{pq}(X, ℤ) are torsion groups.

This leads to the following definition, based on the Beilinson-Lichtenbaum Conjecture for q; a modified version of this was discussed in the Suslin-Voevodsky paper [SV1, 5.6].

Definition 4.9 (BL condition). We say that BL(q, ℓ) holds for k if for every p ≤ q + 1 and every X of finite type over k we have

H^{pq}(X, ℤ_{(ℓ)}) ≅ H_{L}^{pq}(X, ℤ_{(ℓ)}).

The reduction above shows that if BL(q, ℓ) holds for k then the Norm Residue Homomorphism Conjecture holds for q and ℓ.

The special case p = q + 1, X = Spec(K) of the BL(q, ℓ) condition turns out to be critical; since H^{pq}(K, ℤ_{(ℓ)}) = 0 for p > q, it amounts to asking whether H_{L}^{q+1,q}(K, ℤ_{(ℓ)}) = 0 for every field K over k. Lichtenbaum called this vanishing condition the (generalized) Hilbert’s Theorem 90 in [L], because when q = 1 it reduces to the classical version of Hilbert’s Theorem 90 (see below). In fact, this vanishing condition implies the rest of the BL condition:

Theorem 4.10. (Suslin-Voevodsky [SV1, 5.9]) Assume that k admits resolution of singularities. Then BL(q, ℓ) holds for k if and only

H_{L}^{q+1,q}(K, ℤ_{(ℓ)}) = 0 for every field extension K of k.

If k has characteristic p > 0, and ℓ ≠ p, Geisser and Levine proved in [GL] that the analogue of this theorem for BL(q, ℓ) holds for all q, provided that the groups H^{pq}(K, ℤ_{(ℓ)}) are replaced by Bloch’s higher Chow groups.

Low degree cases 4.11. We know that BL(q, ℓ) holds for q ≤ 1, because we know ℤ(q) in this range. BL(q, ℓ) is trivial for q < 0 because then ℤ(q) = 0. It holds for q = 0 because ℤ(0) = ℤ and H_{et}^{1}(K, ℤ) = Hom(Gal(K̄/K), ℤ) = 0. Finally, it holds for q = 1 because ℤ(1)_{et} ≅ 𝔾_{m}[−1] and H_{L}^{2}(K, ℤ(1)) = H_{et}^{1}(K, 𝔾_{m}) = 0.

Regarding the case q = 2, Lichtenbaum has shown in [L] that the vanishing of H_{L}^{3,2}(K, ℤ) amounts to the “Hilbert Theorem 90 for K_{2}” of [MS, 14.1].

Main Theorem 4.12 for ℓ = 2. Let k be a field of characteristic zero. Then:

H_{L}^{q+1,q}(k, ℤ_{(2)}) = 0 for all q.

This implies Milnor’s Conjecture for fields of characteristic zero. We can use this case to verify Milnor’s Conjecture for all fields of characteristic ≠ 2.

Corollary 4.13 (Milnor’s Conjecture). If \frac{1}{2} ∈ k, the mod 2 Norm Residue homomorphism is an isomorphism for all p:

K_{p}^{M}(k)/2 ⟶^{≅} H_{et}^{p}(k, ℤ/2).

Our proof of the Main Theorem will proceed by induction on q, following the outline of the proof of the Merkurjev-Suslin theorem [MS]. Here is the first step, which works for any prime ℓ, using an inductive knowledge that BL(q − 1, ℓ) holds. (The induction is not needed when ℓ = 2.)

Lemma 4.14. If K_{q}^{M}(K)/ℓ = 0 then H_{L}^{q+1,q}(K, ℤ_{(ℓ)}) = 0.

Proof. The exact sequence of coefficients 0 → ℤ_{(ℓ)} → ℚ → ℚ/ℤ_{(ℓ)} → 0 induces an exact sequence

H_{et}^{q}(K, ℚ/ℤ_{(ℓ)}(q)) → H_{L}^{q+1,q}(K, ℤ_{(ℓ)}) → H_{L}^{q+1,q}(K, ℚ).

We have seen that the last term is zero. So it suffices to show that H_{et}^{q}(K, ℤ/ℓ(q)) is zero, since this will imply that the (ℓ-torsion) first term is also zero.

To prove this, we may reduce to the case in which K has no extensions of degree prime to ℓ. Using the arguments of [Su] for q = 2, 3, a Galois cohomology calculation shows that K_{q}^{M}(E)/ℓ = 0 for all finite extensions E of K.

An element β of H^{q}(K, ℤ/ℓ(q)) vanishes on some finite extension E, which we may assume has the form E = K(\sqrt[ℓ]{b}). A calculation shows that this implies that β = [b]∪α for some α in H^{q−1}(K, ℤ/ℓ(q−1)), where [b] is the class of b in K^{×}/K^{×ℓ} = H^{1}(K, ℤ/ℓ(1)). Since BL(q − 1, ℓ) holds, H^{q−1}(K, ℤ/ℓ(q − 1)) = K_{q−1}^{M}(K)/ℓ = 0. Hence α = 0, and we have β = 0.

Exercise 4.15. Let K be a field extension of k. Compare the motivic and étale sequences for 0 → ℤ_{(ℓ)}(q) → ℚ(q) → ℚ/ℤ_{(ℓ)}(q) → 0, using 3.6, 4.5 and 4.8 to show that there is an exact sequence:

H_{L}^{q,q}(K, ℤ_{(ℓ)}) → K_{q}^{M}(K) ⊗ ℚ/ℤ_{(ℓ)} → H_{et}^{q}(K, ℚ/ℤ_{(ℓ)}(q)) → H_{L}^{q+1,q}(K, ℤ_{(ℓ)}) → 0.

The second step in the proof of the Main Theorem 4.12, outlined below, uses a geometric construction. For every a = (a_{1}, …, a_{q}) ∈ (k^{×})^{q}, we must find an extension K_{a}/k such that: 1) {a_{1}, …, a_{q}} is in ℓK_{q}^{M}(K_{a}), and 2) the map H_{L}^{q+1,q}(k, ℤ_{(ℓ)}) → H_{L}^{q+1,q}(K_{a}, ℤ_{(ℓ)}) is injective.

The Main Theorem follows from these two steps. Indeed, by repeatedly taking maximal prime-to-ℓ extensions and then the composite of the fields K_{a} in step two, we form an extension K of k so that 1) K_{q}^{M}(K)/ℓ = 0 and 2) H_{L}^{q+1,q}(k, ℤ_{(ℓ)}) embeds into the group H_{L}^{q+1,q}(K, ℤ_{(ℓ)}), which is zero by step one. Thus H_{L}^{q+1,q}(k, ℤ_{(ℓ)}) = 0.

Definition 4.16. Let a = (a_{1}, …, a_{q}) be a sequence of nonzero elements of k, and let ℓ be a prime. We say that a variety X_{a} over k is a generic splitting variety for (a, ℓ), and that its function field K_{a} = k(X_{a}) is a generic splitting field, if the following condition holds for every extension field K of k:

X_{a} has a rational point over K ⇔ {a_{1}, …, a_{q}} ∈ ℓK_{q}^{M}(K).

We don’t know how to construct generic splitting fields unless ℓ = 2 or q is small. For example, when q = 1 it is easy to see that the generic splitting field is the finite field extension K_{a} = k(\sqrt[ℓ]{a}_{1}) of k. The use of Brauer-Severi varieties as generic splitting varieties for q = 2 was central to the approach in [MS].

Brauer-Severi Varieties. When q = 2, the generic splitting variety for (a, b) is the Severi-Brauer Variety X_{(a,b)} associated to the division algebra A(a, b) = k{x, y}/(x^{ℓ} − a, y^{ℓ} − b, yx − ζxy), ζ^{ℓ} = 1. It is classical that X_{(a,b)} has a rational point over k if and only if A(a, b) is a matrix algebra. By the Merkurjev-Suslin theorem [MS], this holds if and only if {a, b} vanishes in K_{2}(k)/ℓ.

Pfister Quadrics. When ℓ = 2, we use the following construction. For a_{i} ∈ k, let ⟨1, −a_{i}⟩ denote the quadratic form x^{2} − a_{i}y^{2}. Now fix a sequence a = (a_{1}, …, a_{q}) of units in k. The Pfister form of dimension 2^{q} is the quadratic form

≪ a_{1}, …, a_{q} ≫= ⟨1, −a_{1}⟩ ⊗ ⋯ ⟨1, −a_{q}⟩.

The Pfister quadric X_{a} is the smooth projective quadric of dimension 2^{q−1} −1 given by the quadratic equation ≪ a_{1}, …, a_{q−1} ≫= a_{q}t^{2}. It is a classical fact that X_{a} has a K-rational point if and only if ≪ a_{1}, …, a_{q} ≫ represents zero over a field K.

For example, if q = 2 then X_{a} is the plane curve x^{2} = a_{1}y^{2} + a_{2}t^{2}, which is the Brauer-Severi Variety for ℓ = 2.

Theorem 4.17. The Pfister quadric X_{a} is a generic splitting variety for (a, 2), q ≥ 2.

Idea of proof. If X_{a} has a K-rational point with t ≠ 0 then the quadratic form β =≪ a_{1}, …, a_{q−1} ≫ represents a_{q} over K. This means that there is a quadratic extension E = K(\sqrt{b}) so that β represents zero over E, and a_{q} = N(e) is the norm of some e ∈ E. Since β represents zero, the Pfister quadric for (a_{1}, …, a_{q−1}) has an E-rational point; by induction on q, this implies that {a_{1}, …, a_{q−1}} = 0 in K_{q−1}^{M}(E)/2. But {a_{1}, …, a_{q}} is the norm of {a_{1}, …, a_{q−1}, e}, so it vanishes in K_{q}^{M}(k)/2.

To complete step two, we need to prove that H_{L}^{q+1,q}(k, ℤ_{(2)}) → H_{L}^{q+1,q}(K_{a}, ℤ_{(2)}) is a monomorphism, where the generic splitting field K_{a} is the function field of X_{a}.

For this, we use a reduction borrowed from descent theory. Given a scheme X over k, consider the simplicial scheme Č(X) defined by

Č(X): X\overleftarrow{←}X × X← ; ← ; ←X^{3} · · · .

If X(k) ≠ ∅, it is easy to see that the augmented simplicial scheme Č(X) → Spec(k) is aspherical. If X(k) = ∅ and we consider Č(X) as a simplicial Nisnevich sheaf then Č(X)(k) is empty. This proves most of the following lemma; the rest follows by “étale descent” from an E/k with X(E) ≠ ∅, which shows that Č(X) → Spec(k) induces an isomorphism in étale cohomology.

Lemma 4.18. (i) The augmentation Č(X) → Spec(k) is a simplicial weak equivalence in Spc if and only if X(k) ≠ ∅.

(ii) In any case, it induces an isomorphism H_{L}^{pq}(Č(X), −) ≅ H_{L}^{pq}(Spec(k), −).

Corollary 4.19. Assume that BL(q − 1, ℓ) holds for k. Then for all p ≤ q,

H^{p,q−1}(k, ℤ/ℓ) ≅ H^{p,q−1}(Č(X), ℤ/ℓ).

Indeed, the BL condition allows us to replace both sides by étale cohomology, where we can invoke (ii).

We now specialize X to the Pfister quadric X_{a}.

Proposition 4.20. The map H_{L}^{q+1,q}(k, ℤ_{(2)}) → H_{L}^{q+1,q}(K_{a}, ℤ_{(2)}) is injective if and only if

H^{q+1,q}(Č(X_{a}), ℤ_{(2)}) = 0.

We have reduced step two to a vanishing assertion in motivic cohomology. Note that the L subscript has disappeared from our assertion!

Sketch of the proof. Pick u ∈ H_{L}^{q+1,q}(k, ℤ_{(ℓ)}) vanishing in H_{L}^{q+1,q}(K_{a}, ℤ_{(ℓ)}). Then u must vanish on some dense open U in X_{a}. A Gysin sequence argument and induction on q shows that the image of u in H_{L}^{q+1,q}(X_{a}, ℤ_{(ℓ)}) comes from some u_{0} in H^{q+1,q}(X_{a}, ℤ_{(ℓ)}). Because X_{a} is a quadric with a K_{a}-rational point, it is rational over K_{a}. Using the homotopy invariance of hypercohomology, we can show that the image of u in H_{L}^{q+1,q}(Č(X_{a}), ℤ_{(ℓ)}) comes from H^{q+1,q}(Č(X_{a}), ℤ_{(ℓ)}), a group which is zero by assumption.

We shall deduce the vanishing assertion from the following vanishing theorem. It is a combination of two theorems proven by Markus Rost in the late 1980’s, using the language of Chow motives.

Theorem 4.21. (Rost) If s = 2^{q} − 1 and t = 2^{q−1} then: H^{st}(Č(X_{a}), ℤ_{(2)}) = 0.

We shall connect Rost’s result to our situation by reducing modulo two and using a mod 2 motivic cohomology operation ϕ. For legibility, we write Č for Č(X_{a}) and set s = 2^{q} − 1, t = 2^{q−1}.

Since there is a degree two extension E of k so that Č(E) ≠ ∅, the usual transfer argument shows that H^{pq}(Č, ℤ_{(2)}) has exponent 2 for p > q. In particular, if ρ denotes reduction ℤ_{(2)} → ℤ/2, then ρ and β fit into an exact sequence

0 → H^{p,q}(Č, ℤ_{(2)}) ⟶^{ρ} H^{p,q}(Č, ℤ/2) ⟶^{β} H^{p+1,q}(Č, ℤ/2).

We define the mod 2 cohomology operation ϕ to be the composite of the successive cohomology operations Q_{1}, Q_{2}, Q_{3}, …, Q_{q−2}. Each operation Q_{i} maps H^{pq} to H^{p+2^{i+1}−1,q+2^{i}−1}, so their composite ϕ maps H^{q+1,q} to H^{s,t}.

Because Q_{0} = β commutes with ϕ (mod 2), there is a homomorphism Φ making the following diagram commute:

0 ⟶ H^{q+1,q}(Č, ℤ_{(2)}) ⟶^{ρ} H^{q+1,q}(Č, ℤ/2) ⟶^{β} H^{q+2,q}(Č, ℤ/2)
Φ↓  ^{ϕ}↓  ^{ϕ}↓
0 ⟶ H^{st}(Č, ℤ_{(2)}) ⟶^{ρ} H^{st}(Č, ℤ/2) ⟶^{β} H^{s+1,t}(Č, ℤ/2).

By Rost’s theorem, the lower left group H^{st}(Č, ℤ_{(2)}) is zero. The following result, that the middle map ϕ is a monomorphism implies that the upper left group H^{q+1,q}(Č, ℤ_{(2)}) is zero. This proves the vanishing assertion, and completes the proof of the Main Theorem 4.12.

Theorem 4.22. For s = 2^{q} − 1 and t = 2^{q−1}, the cohomology operation

ϕ: H^{q+1,q}(Č, ℤ/2) ↣ H^{s,t}(Č, ℤ/2)

is a monomorphism.

To prove this, we need some machinery which works for any prime ℓ.

Just as in topology [Mar], we can use the fact that Q_{i}^{2} = 0 to define the motivic Margolis cohomology HM_{i}^{∗∗} of X with coefficients ℤ/ℓ. Set s = 2ℓ^{i}−1 and t = ℓ^{i}−1. Then HM_{i}^{p,q}(X, ℤ/ℓ) is defined to be the cohomology of the complex

H^{∗−s,∗−t}(X, ℤ/ℓ) ⟶^{Q_{i}} H^{∗,∗}(X, ℤ/ℓ) ⟶^{Q_{i}} H^{∗+s,∗+t}(X, ℤ/ℓ).

Example 4.23. If i = 0 then Q_{0} is the Bockstein. A routine diagram chase shows that HM_{0}^{pq}(X, ℤ/ℓ) is the subgroup of elements divisible by ℓ in H^{pq}(X, ℤ/ℓ^{2}).

One way to study Margulis cohomology is to introduce Φ_{i}-cohomology, where Φ_{i} is the homotopy fiber of Q_{i}: H_{ℤ/ℓ} → H_{ℤ/ℓ}(t)[s]. From the long exact cohomology sequence

⋯ Φ_{i}^{pq}(X) ⟶^{u} H^{pq}(X, ℤ/ℓ) ⟶^{Q_{i}} H^{p+s,q+t}(X, ℤ/ℓ) ⟶^{w} Φ_{i}^{p+1,q}(X) ⋯

and a diagram chase, we see that HM_{i}^{p,q}(X, ℤ/ℓ) = 0 if and only if the “obstruction map” wu: Φ_{i}^{pq}(X) → Φ_{i}^{p+1−s,q−t}(X) is zero.

Define the simplicial space X̃ = X̃(X) by the distinguished triangle

X̃(X) → Č(X) → Spec(k)

Since Hom(X, X) ≠ ∅, it is not hard to see that Σ_{T}^{∞}(X_{+}) ∧ X̃(X) = 0.

Theorem 4.24. Let X be a smooth hypersurface of degree ℓ in ℙ^{ℓ^{n}}. Then

HM_{i}^{∗∗}(X̃(X), ℤ/ℓ) = 0 for all i ≤ n.

The proof of this theorem is given in [V, 3.22, 3.25 and 4.11]. It uses the motivic Thom class τ: MGL → H_{ℤ/ℓ} and the classes in MGL_{2p,p}(X) of the plane sections of X, which correspond to important bordism classes in MU_{2p}(point). With these tools, and the formula 3.10(iii) above, we can show that the obstruction map wu for X̃ factors through the group Φ_{i}^{∗,∗}(Σ_{T}^{∞}(X_{+}) ∧ X̃(X), ℤ/ℓ) = 0.

Applying Theorem 4.24 to the Pfister quadric hypersurface in ℙ^{2^{q−1}}, we see that for i < q there is an exact sequence

H^{q−i−1,q−i}(X̃) ⟶^{Q_{i}} H^{q−i+2^{i+1}−2,q−i+2^{i}−1}(X̃) ⟶^{Q_{i}} H^{q−i+2^{i+2}−3,q−i+2^{i+1}−2}(X̃).

Because we know that BL(q −i, 2) holds by induction, and H_{L}^{∗,∗} vanishes on X̃(X_{a}) by (ii) of the previous lemma, the first group vanishes:

H^{q−i−1,q−i}(X̃(X_{a}), ℤ/2) ≅ H_{L}^{q−i−1,q−i}(X̃(X_{a}), ℤ/2) = 0.

This proves that Q_{i} is a monomorphism on the middle group. For i = 1 this Q_{1} is the beginning H^{q+1,q} → H^{q+4,q+1} of the operation ϕ. By induction on i, each composite Q_{i} ⋯ Q_{2}Q_{1} (and in particular ϕ) is a monomorphism on H^{q+1,q}(X̃, ℤ/2). So we are done with the proof of the Main Theorem 4.12.

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