Lecture X K-theory and Motivic Cohomology

§1. Basic definitions

Let’s fix a perfect field k, and consider the category Sm/k of smooth schemes over k. This lecture concerns two important families of contravariant functors on Sm/k, which we shall now define: Cor(, X), also called c_{equi}(X, 0), and z_{equi}(X, r).

Definition 1.1 (Cor(U, X)). Fix a scheme X of finite type over k.

(a) For a smooth connected scheme U, let Cor(U, X) denote the free abelian group on the set of closed subvarieties Z ⊂ U × X so whose projection Z → U is finite and surjective. We call this group the finite correspondences from U to X.

(b) Similarly, for each r ≥ 0 we let z_{equi}(X, r)(U) denote the free abelian group on the set of closed subvarieties Z ⊂ U ×X whose projection Z → U is equidimensional of relative dimension r. (This condition implies that Z dominates U, but the projection need not be surjective.)

If U is not connected, but has several connected components U_{i}, we write Cor(U, X) for ⊕ Cor(U_{i}, X), and write z_{equi}(X, r)(U) for ⊕z_{equi}(X, r)(U_{i}).

Examples 1.1.1. (1) If r > dim(X) then z_{equi}(X, r) = 0.

(2) If r = dim(X) and U is connected then z_{equi}(X, r)(U) = ℤ on Z = U × X.

(3) In general Cor(U, X) ⊆ z_{equi}(X, 0). When X is proper, Cor(U, X) = z_{equi}(X, 0).

(4) If U = Spec(k), z_{equi}(X, r)(k) = z_{equi}(X, r)(U) is the divisor group of dimension r cycles on X, and Cor(k, X) = z_{equi}(X, 0) is the divisor group of zero-cycles on X.

(5) There is an inclusion Hom_{Sm/k}(U, X) ⊂ Cor(U, X), sending a morphism U → X to the graph Γ_{f} which is in Cor(U, X).

The effect of Cor(, X) on a morphism is clear: take the pullback. The effect of z_{equi}(X, r) on a morphism is less clear, but is characterized by the following condition.

Lemma 1.2. The map f^{∗}: z_{equi}(X, r)(U) → z_{equi}(X, r)(V) associated to a map f: V → U of smooth schemes is characterized by the following condition. Fix Z ⊂ U × X equidimensional over U. If f is dominant, or if Z is flat over U, then:

(1.2.1)

f^{∗}(Z) = cycle(Z ×_{U} V)

= sum of components with multiplicity, as in Lecture IX(1.0).

Example 1.2.2. If R is a discrete valuation domain with residue field k, the map i^{∗}: z_{equi}(X, r)(R) → z_{equi}(X, r)(k) induced by the inclusion i of Spec(k) in U = Spec(R) is given by the formula (1.2.1), because any Z ⊂ U × X which is equidimensional over U is also flat over U.

Exercise 1.2.3. (a) If V is open in U, show that z_{equi}(X, r)(U) ⊆ z_{equi}(X, r)(V).

(b) If i: Y ↪ U is a closed immersion, show that i^{∗} is given by the intersection formula (1.2.1).

Typeset by AMS-TEX

Warning: If U is not smooth, the map f^{∗} may still be defined, but the target becomes z_{equi}(X, r)(V) ⊗ ℚ. For details, see [SV-1].

Applying the functor Cor(, X) to Δ^{·} = Spec(k[Δ^{·}]) gives a simplicial abelian group

Cor(k, X) \overleftarrow{←} Cor(k[t_{1}], X) ← ; ← ; ← Cor(k[t_{1}, t_{2}], X) · · · .

As in Lecture 7(4.5), we write \underline{C}_{∗}X or \underline{C}_{∗} Cor(k, X) for the chain complex associated to Cor(Δ^{·}, X). We write \underline{C}_{∗}X(U) or \underline{C}_{∗} Cor(U, X) for the chain complex associated to Cor(U ×Δ^{·}, X). As U varies, it is easy to see that this is actually a chain complex of sheaves, at least for the Zariski topology.

Definition 1.3. The chain complexes of Zariski sheaves ℤ(i) are defined using iterates of the scheme 𝔾_{m} = Spec(k[t, t^{−1}]):

ℤ(1) = (\underline{C}_{∗}𝔾_{m}/\underline{C}_{∗}k)[−1] = Tot(\underline{C}_{∗}k → \underline{C}_{∗}𝔾_{m}),
ℤ(i) = (\underline{C}_{∗}(𝔾_{m})^{i}/F_{i})[−i],

where F_{i} denotes the subsheaf given by the i standard embeddings of (𝔾_{m})^{i−1} into (𝔾_{m})^{i}.

For each m > 0 we set ℤ/m(i) = ℤ(i) ⊗ ℤ/m. Note that each ℤ(i)(U) is a chain complex of free abelian groups, so the chain complex ℤ/m(i) is the total derived tensor product ℤ(i) ⊗^{L} (ℤ/m).

Exercise 1.3.1. Show that ℤ(0) = \underline{C}_{∗}k is the constant sheaf ℤ, regarded as a chain complex concentrated in degree zero.

We define the motivic cohomology of X to be the Zariski hypercohomology of ℤ(i). Hypercohomology requires that we reindex a chain complex C_{∗} by C^{−n} = C_{n} so we view ℤ(i) as a (bounded above!) cochain complex of sheaves. For such complexes, hypercohomology is well-behaved because X is noetherian and finitedimensional.

Definition 1.4. The motivic cohomology groups of a smooth scheme X, with coefficients in ℤ(i) and ℤ/m(i), are:

H^{∗}(X, ℤ(i)) = ℍ_{zar}^{∗}(X, ℤ(i)),
H^{∗}(X, ℤ/m(i)) = ℍ_{zar}^{∗}(X, ℤ/m(i)).

Warning 1.4.1: The usual definition of motivic cohomology is in terms of the cdh topology on X. If char(k) = 0, this not only agrees with our more naive definition by [FV, 8.1], but may be extended to non-smooth X using resolution of singularities.

The long exact hypercohomology sequence associated to the short exact sequence

0 → ℤ(i) ⟶^{m} ℤ(i) → ℤ/m(i) → 0

yields the Universal Coefficient sequence:

Lemma 1.4.2 (Universal Coefficients). For each m, n and i there is an exact sequence:

0 → H^{n}(X, ℤ(i))/m → H^{n}(X, ℤ/m(i)) → _{m}H^{n+1}(X, ℤ(i)) → 0.

§2. Connection to Bloch’s higher Chow groups

Applying the functor z_{equi}(X, r) to Δ^{·} = Spec(k[Δ^{·}]) gives a simplicial abelian group

z_{equi}(X, r)(Δ^{·}): z_{equi}(X, r)(k) \overleftarrow{←} z_{equi}(Z, r)(k[t_{1}]) ← ; ← ; ← z_{equi}(X, r)(k[t_{1}, t_{2}]) · · · .

Exercise 2.0. If Z ⊂ X × Δ^{n} is equidimensional over Δ^{n}, show that Z meets all faces of Δ^{n} properly. This shows that z_{equi}(X, r)(Δ^{·}) is a simplicial subgroup of Bloch’s cycle group z^{i}(X, ·), where i + r = dim(X).

Theorem 2.1. (Suslin [S2]) If 0 ≤ i ≤ dim(X) and r = dim(X) − i, then the inclusion z_{equi}(X, r)(Δ^{·}) ⊂ z^{i}(X, ·) is a homotopy equivalence. Thus

CH^{i}(X, n) ≅ π_{n}z_{equi}(X, r)(Δ^{·}).

Of course, CH^{i}(X, n) is also defined for i > d = dim(X). But we saw in Lecture 9 that CH^{i}(X, n) ≅ CH^{i}(X × Δ^{i}, n), and of course i ≤ d + i = dim(X × Δ^{i}). So we always have the interpretation:

(2.1.1)  CH^{i}(X, n) ≅ π_{n}z_{equi}(X × Δ^{i}, d)(Δ^{·}).

Friedlander and Voevodsky [FV] studied the chain complex \underline{C}_{∗}z_{equi}(X, r)(U) associated to the simplicial group U ↦ z_{equi}(X, r)(U × Δ^{·}), discussed in Lecture 7(4.5). They defined bivariant Chow groups A_{rn} in terms of hypercohomology; the negative superscript in their definition comes from the reindexing of chain complexes as cochain complexes.

Definition 2.2 (Bivariant cycle homology). For each r and n, the bivariant Chow groups are defined as:

A_{rn}(U, X) = ℍ_{zar}^{−n}(U, \underline{C}_{∗}z_{equi}(X, r)).

For U = Spec(k) this yields A_{rn}(k, X) = H_{zar}^{−n}(k, \underline{C}_{∗}z_{equi}(X, r)) = CH^{i}(X, n).

The following proposition lists the crucial properties we need. The hypothesis that char(k) = 0 may be replaced by the possibly more general hypothesis that k admits resolutions of singularities.

Proposition 2.3. ([FV]) Assume that char(k) = 0, or that U is smooth. Then:

(a) A_{rn}(U, X) ≅ π_{n}z_{equi}(X, r)(U × Δ^{·}) if r ≥ 0;

(b) A_{rn}(Y × U, X) ≅ A_{r+d,n}(Y, U × X) where d = dim(Y);

(c) A_{0n}(X, 𝔸^{i}) ≅ H^{2i−n}(X, ℤ(i)).

We will describe this result in more detail later, in §4–5. Combining (2.1.1) with all three parts of Proposition 2.3 yields the following result, proven in [FV].

Theorem 2.4. Assume that char(k) = 0, and that X is smooth. Then

CH^{i}(X, n) ≅ H^{2i−n}(X, ℤ(i)).

§3. Presheaves with transfer

By adding the extra “correspondence” morphisms in Cor(U, X) to the category Sm/k, we get a larger category SmCor/k.

Definition 3.1 (SmCor). The objects of SmCor/k are the smooth schemes over k, and the morphisms are

Hom_{SmCor}(U, X) = Cor(U, X).

To define the composition Cor(U, X) × Cor(X, Y) ⟶^{∘} Cor(U, Y), suppose given subvarieties W_{1} ⊂ U × X and W_{2} ⊂ X × Y which are finite over U and X, respectively. Let W_{1} ∗ W_{2} denote the intersection of W_{1} × Y and U × W_{2} inside U × X × Y. We define W_{2} ∘ W_{1} to be the pushforward p_{∗}(W_{1} ∗ W_{2}) along the projection p: U × X × Y → U × Y. Note that the pushforward p_{∗} is defined because of our finiteness assumptions.

An ordinary morphism of smooth schemes f: U → X can be thought of as the class of its graph Γ_{f} ⊂ U ×X in Cor(U, X). It is not hard to see that Γ_{f} ∘Γ_{g} = Γ_{gf}, so that we may identify Sm/k with a subcategory of SmCor/k.

Definition 3.2. (Voevodsky [V-2]) A presheaf with transfers over k is a contravariant, additive functor from SmCor/k to abelian groups.

We may think of a presheaf with transfers as a contravariant functor on Sm/k, equipped with group homomorphisms

F: Cor(U, X) → Hom(F (X), F (U))

satisfying the usual conditions for ∘ and Hom_{Sm/k}.

Example 3.2.1. If f: Y → X is a finite flat map of smooth schemes, we may view its graph Γ_{f} as an element of both Cor(Y, X) and Cor(X, Y). Hence every presheaf with transfers determines maps

f^{∗}: F (X) → F (Y)  and  f_{∗}: F (Y) → F (X).

Exercise 3.2.2. Show that f_{∗}f^{∗}: F (X) → F (X) is multiplication by d, the degree of f. Hint: As codimension zero cycles, f_{∗}([Y]) = d[X].

Examples 3.3. (a) Cor(, X) was born as a presheaf with transfers.

(b) Each z_{equi}(X, r) is a presheaf with transfers; if Z ⊂ V × X is equidimensional over V and W ⊂ U × X is such that the map W → U is finite surjective, then W ×_{V} Z is equidimensional over U.

(c) If F is a presheaf with transfers then the chain complex \underline{C}_{∗}F associated to U ↦ F (U × Δ^{·}) is a simplicial presheaf with transfers, and its homology groups

h^{−n}(F)(U) = H_{n}\underline{C}_{∗}F (U)

are homotopy invariant presheaves with transfer. When F = Cor(, X), for example, we saw in Friedlander’s Lecture VI that \underline{C}_{∗}F (k) = Cor(Δ^{·}, X) is the Suslin complex Sus_{∗}(X), and that if char(k) = 0 then [SV-2, 7.8 and 8.3] state that the étale homology of X is

H_{∗}^{et}(X; ℤ/m) = H_{∗}(Sus_{∗}(X) ⊗ ℤ/m),

and the étale cohomology of X is

H_{et}^{∗}(X; ℤ/m) = H^{∗}(Hom(Sus_{∗}(X), ℤ/m)).

Warning: Some early papers of Voevodsky, as well as Friedlander-Voevodsky [FV] and Suslin-Voevodsky [SV-2] use a condition that a contravariant functor F from Sm/k to abelian groups be a “pre-theory of homological type.” Every presheaf with transfers satisfies this condition by [V-2, 3.1.10].

Theorem 3.4. (Voevodsky [V-2, 3.2.22]) Let F be a homotopy invariant presheaf with transfers, and write F̃ for the associated Zariski sheaf. Then F̃ is also a sheaf for the Nisnevich topology, and

H_{zar}^{∗}(U, F̃) ≅ H_{nis}^{∗}(U, F̃).

Recall that the Nisnevich topology on Sm/k is a Grothendieck topology intermediate between the Zariski and étale topologies. A Nisnevich covering {U_{i} → U}_{i∈I} is an étale covering such that for every point u ∈ U there is some point u_{i} ∈ U_{i} over u such the induced residue field map k(u) → k(u_{i}) is an isomorphism.

Definition 3.5. A presheaf with transfers F is called a Nisnevich sheaf with transfers if it is also a sheaf for the Nisnevich topology on Sm/k. We shall write S for the category (or topos) Shv_{nis}(SmCor/k) of all Nisnevich sheaves with transfers.

Exercise 3.5.1. Show that Cor(, X) is a Nisnevich presheaf with transfers.

§4. The triangulated category of Motives

We now make the blanket assumption that k admits resolutions of singularities, which means char(k) = 0 in practice. Voevodsky’s approach to motives for schemes of finite type over k uses the category S of all Nisnevich sheaves with transfers, defined in 3.5 above.

Definition 4.1. If X is a scheme of finite type over k, its motive is

M(X) = \underline{C}_{∗} Cor(, X)

and its motive with compact supports is

M^{c}(X) = \underline{C}_{∗}z_{equi}(X, 0).

These are bounded below chain complexes of Nisnevich sheaves with transfer (by Ex. 3.5.1). If we reindex them as cochain complexes they are bounded above, and we shall view them as objects in the derived category D_{−}(S).

Definition 4.2. The derived category of motives DM = DM_{k} is defined to be the full subcategory of D_{−}(S) consisting of all cochain complexes of sheaves C^{∗} whose cohomology sheaves are homotopy invariant in the sense that for all (smooth) U:

H^{∗}C^{∗}(U) ≅ H^{∗}C^{∗}(U[t]).

Note that M(X) and M^{c}(X) were designed to belong to DM, and that DM is a triangulated subcategory of the derived category D(S). Here is a fairly formal consequence of the above definitions, proven in [V2, 3.1. and 3.2.6].

Proposition 4.3. If X is smooth and K is a complex in DM then

Hom_{DM}(M(X), K[n]) = ℍ_{zar}^{n}(X, K).

For K = ℤ(i) this yields the motivic cohomology defined in 1.4:

H^{n}(X, ℤ(i)) = Hom_{DM}(M(X), ℤ(i)[n]).

For K = M(Y) with Y smooth this yields the formula:

Hom_{DM}(M(X), M(Y)[n]) = ℍ^{n}(X, \underline{C}_{∗}Y).

Exercise 4.3.1. Show that the Suslin homology of X is H_{n}Sus_{∗}(Y) = Hom_{DM}(ℤ(n), M(Y)).

Mayer-Vietoris Theorem 4.4. Assume that char(k) = 0, and that X is the union of two open sets X = U ∪ V. Then there are distinguished triangles

M(U ∩ V) → M(U) ⊕ M(V) → M(X) → M(U ∩ V)[1],

M^{c}(U ∩ V) → M^{c}(U) ⊕ M^{c}(V) → M^{c}(X) → M^{c}(U ∩ V)[1].

If Y = X − U, there is also a distinguished triangle:

M^{c}(Y) → M^{c}(X) → M^{c}(U) → M^{c}(Y)[1].

Definition 4.5. We define the Tate twists M(X)(1) and M^{c}(X)(1) as two-fold shifts of the mapping cone complexes:

M(X)(1)[2] = cone of M(X) → M(ℙ_{X}^{1}),

M^{c}(X)(1)[2] = cone of M^{c}(X) → M^{c}(ℙ_{X}^{1}).

Exercise 4.5.1. Show that ℤ(1) = M(k)(1).

§5. Duality and bivariant cycle homology

In any triangulated category the Hom-sets Hom(A, B) are cohomological functors in A, and homological functors in B. In the derived category D(S), and hence in its full subcategory DM, these Hom-sets are the cohomology of an internal Hom — the cochain complex \underline{Hom}.

We saw in 4.3 and 4.3.1 that the cohomology theory Hom_{DM}(M(X), ℤ(i)) is motivic cohomology of X, while the homology theory Hom_{DM}(ℤ(n), M(Y)) is the Suslin homology of Y. If we use motives with compact supports, we get a homology theory with locally compact supports.

Definition 5.1. The homology of X with locally compact supports is

H_{n}^{lc}(X, ℤ(i)) = Hom_{DM}(ℤ(i)[n], M^{c}(X)).

The next result shows that the higher Chow groups are an example of homology with locally compact supports. The case r = 0 is just definition 4.1: M^{c}(X) = \underline{C}_{∗}z_{equi}(X, 0).

Proposition 5.2. ([V-2, 4.2.8]) If X has finite type over k and char(k) = 0, then

\underline{C}_{∗}z_{equi}(X, r) ≅ \underline{Hom}(ℤ(r)[2r], M^{c}(X))

for each r ≥ 0. Hence if i + r = dim(X) we have

CH^{i}(X, n) ≅ H_{2r+n}^{lc}(X, ℤ(r)) = Hom_{DM}(ℤ(r)[2r + n], M^{c}(X)).

In order to prove this result, we use 4.3 to reformulate the Friedlander-Voevodsky definition 2.2 of bivariant cycle homology.

Lemma 5.3. ([V-2, 4.2.3]) Let X and Y be schemes of finite type over k. Then for every r ≥ 0:

A_{rn}(X, Y) ≅ Hom_{DM}(M(X)(r)[2r + n], M^{c}(Y)).

The twist-and-shift contortions are designed to make the following duality theorem look better. It is proven in [FV, 8.2], and requires resolution of singularities, i.e., characteristic zero.

Duality Theorem 5.4. Assume that char(k) = 0, and that U is a smooth scheme of dimension d. Then for any schemes X, Y of finite type over k we have a natural duality isomorphism for U:

A_{rn}(X × U, Y) ≅ A_{r+d,n}(X, U × Y).

This is the final step in proving Proposition 2.3 above, identifying Bloch’s higher Chow groups with motivic cohomology. We end this lecture by repeating the following important consequence, due to Friedlander-Voevodsky [FV] and already listed above as Corollary 2.4: if char(k) = 0 and X is smooth, then

CH^{i}(X, n) ≅ H^{2i−n}(X, ℤ(i)).