Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic.

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

In this short paper we show that the motivic cohomology groups defined in [3] are isomorphic to the motivic cohomology groups defined in [1] for smooth schemes over any field. In view of [1, Proposition 12.1] this implies that motivic cohomology groups of [3] are isomorphic to higher Chow groups. This fact was previously known only under the resolution of singularities assumption. The new element in the proof is Proposition 4.

This paper was written when I was a member of the Institute for Advanced Study in Princeton and, part of the time, an employee of the Clay Mathematics Institute. I am grateful to both institutions for their support. I would also like to thank Charles Weibel for his useful comments on the previous version of the paper.

Recall that the motivic complex Z(q) of weight q was defined in [3] as C_{∗}(Z_{tr}(G_{m}^{∧q}))[−q]. In [1, §8] Friedlander and Suslin defined complexes which we will denote Z_{tr}^{SF} (q) as C_{∗}(z_{equi}(A^{q}, 0))[−2q] where z_{equi}(X, 0) is the sheaf of equidimensional cycles on X of relative dimension zero. In this paper we prove the following result.

Theorem 1 For any field k the complexes of sheaves with transfers Z(q) and Z^{SF} (q) on Sm/k are quasi-isomorphic in the Zariski topology.

Corollary 2 For any field k, any smooth scheme X over k and any p, q ∈ Z one has a natural isomorphism

H^{p,q}(X, Z) = CH^{q}(X, 2q − p)

and the same holds for the motivic cohomology and higher Chow groups with coefficients.

1Supported by the NSF grants DMS-97-29992 and DMS-9901219, Sloan Research Fellowship and Veblen Fund

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

vladimir@ias.edu

Proof: The hypercohomology groups with coefficients in Z_{tr}^{SF} (q) are shown in [1, Propisition 12.1] to coincide with the higher Chow groups for smooth varieties over all fields. Motivic cohomology groups are defined in [3, Definition 3.1] as hypercohomology groups with coefficients in Z(q). Therefore, Theorem 1 implies Corollary 2.

To prove the theorem we have to show that for any smooth scheme X over k and any point x of X the complexes of abelian groups Z(q)(Spec(O_{X,x})) and Z^{F S}(q)(Spec(O_{X,x})) are quasi-isomorphic. Let k_{0} be the subfield of constants in k. Then there exists a smooth variety X_{0} over k_{0} (possibly of dimension greater than the dimension of X) and a point x_{0} on X_{0} such that the local rings O_{X,x} and O_{X_{0},x_{0}} are isomorphic. Therefore, we may always assume that k = k_{0} and in particular that k is perfect.

Since both complexes are complexes of presheaves with transfers with homotopy invariant cohomology presheaves it is sufficient to show that they are quasi-isomorphic in the Nisnevich topology. Consider the triangulated category of motives DM = DM_{−}^{eff}(k) defined in [5]. For our purposes it will be convenient to think of it as of the localization of the derived category of complexes of sheaves with transfers by A^{1}-contractible objects. Since the complexes we consider are A^{1}-local it is sufficient to show that they are isomorphic in DM. Since for any sheaf with transfers F the natural morphism F → C_{∗}(F) is an isomorphism in DM it is sufficient to show that Z_{tr}(G_{m}^{∧n})[n] ≅ z_{equi}(A^{n}, 0) in DM. We construct two isomorphisms. The first one is of the form

Z_{tr}(P^{n})/Z_{tr}(P^{n−1}) → Z_{tr}(G_{m}^{∧n})[n]  (1)

and the second one of the form

Z_{tr}(P^{n})/Z_{tr}(P^{n−1}) → z_{equi}(A^{n}, 0)  (2)

The following construction of (1) is well known. For a family of open embeddings U = {U_{i} → X}_{i=1,...,n} let S(U) denote the complex of sheaves with transfers

0 → Z_{tr}(∩_{i=1}^{i=n}U_{i}) → … → ⊕_{i=1}^{i=n}Z_{tr}(U_{i}) → 0

with the sum of Z_{tr}(U_{i}) placed in degree zero. We have an obvious map S(U) → Z_{tr}(X). The following lemma is a version of [5, Prop. 3.1.3].

Lemma 3 If U = {U_{i} → X} is a Zariski covering of X then the morphism S(U) → Z_{tr}(X) is a quasi-isomorphism in the Nisnevich topology.

Let U_{n} be the standard covering of P^{n} by n + 1 copies of A^{n}. Let further V_{n} be the family of maps {U_{i} → A^{n}} where

U_{i} = {(x_{1}, …, x_{i}) : x_{i} ≠ 0}.

The embedding P^{n−1} → P^{n} defines a morphism S(U_{n−1}) → S(U_{n}) and the complimentary embedding A^{n} → P^{n} defines a morphism S(V_{n}) → S(U_{n}). By Lemma 3 the cokernel of the first morphism represents in DM the object Z_{tr}(P^{n})/Z_{tr}(P^{n−1}). The complex S(V_{n}) is the n-th tensor power of the complex Z_{tr}(A^{1} − {0}) → Z_{tr}(A^{1}) and therefore it represents in DM the object

Z_{tr}(G_{m}^{∧n})[n] = ((Z̃_{tr}(A^{1} − {0}; 1))[1])^{⊗n}

It remains to show that the map

S(V_{n}) → S(U_{n})/S(U_{n−1})

is an isomorphism in DM. This can be easilly seen term-by-term. The first isomorphism is constructed.

We define (2) as the morphism given by the obvious homomorphisms of presheaves with transfers

Z_{tr}(P^{n})/Z_{tr}(P^{n−1}) → z_{equi}(A^{n}, 0)  (3)

Proposition 4 The morphism (3) is an isomorphism in DM.

Proof: Define F_{n} as the subpresheaf in z_{equi}(A^{n}, 0) such that for a smooth connected X the group F_{n}(X) is generated by closed irreducible subschemes Z of X × A^{n} which are equidimensional over X of relative dimenison 0 and which do not intersect X × {0}. Consider Z_{tr}(P^{n} − {0}) as a subpresheaf in Z_{tr}(P^{n}). If U is a smooth scheme and Z is a closed irreducible subset in U × P^{n} which represents an element of Z_{tr}(P^{n} − {0})(U) then it does not intersect U ×{0} and therefore the image of Z under the map (4) is contained in F_{n}(U). We get the following diagram of morphisms of presheaves

Z_{tr}(P^{n−1}) = Z_{tr}(P^{n−1})
↓  ↓
Z_{tr}(P^{n} − {0}) → Z_{tr}(P^{n})
↓  ↓
F_{n}  → z_{equi}(A^{n}, 0)

(4)

The statement of the proposition follows from the diagram (4) and Lemmas 5,6,7.

Lemma 5 The morphism of Nisnevich sheaves

Z_{tr}(P^{n})/Z_{tr}(P^{n} − {0}) → z_{equi}(A^{n}, 0)/F_{n}

is an isomorphism.

Proof: Let S be a henselian local scheme and Z be an irreducible closed subset of S × A^{n} which is equidimensional of relative dimension zero over S. If Z does not belong to F_{n}(S) then the intersection of Z with S × {0} is nonempty. Since it is closed it must contain the closed point of S and thus the image of Z in S contains the closed point. But then Z is finite over S by [2, Ch. 1, Th.4.2(c)] and thus it is closed in S × P^{n}. This proves surjectivity. The proof of injectivity is similar.

Lemma 6 One has F_{n} ≅ 0 in DM.

Proof: Let us show that F_{n} is contactible i.e. that there exist a collection homomorphisms

ϕ_{X} : F_{n}(X) → F_{n}(X × A^{1})

natural in X and such that the composition of ϕ_{X} with the restriction to X × {0} is zero and to X × {1} is identity.

Consider the morphism π : X × A^{n} × A^{1} → X × A^{n} given by the formula π(x, r, t) = (x, rt). An element of F_{n}(X) is a cycle Z on X × A^{n} which does not intersect X × {0}. Since π is flat over X × (A^{n} − {0}) the cycle π^{∗}(Z) is well defined and one checks immediately that it belongs to F_{n}(X × A^{1}). One further verifies that the homomorphisms we constructed are compatible with the functoriality in X and satisfy the required conditions for the restrictions to X × {0} and X × {1}.

The homomorphisms ϕ_{X} define for any X a homomorphism

C_{∗}(F_{n})(X) → C_{∗}(F_{n})(X × A^{1}).

By [4, Prop.3.6] the restriction maps C_{∗}(F_{n})(X × A^{1}) → C_{∗}(F_{n})(X) corresponding to X ×{0} and X ×{1} are a quasi-isomorphisms. This implies that our homomorphism is a quasi-isomorphism which equals zero on cohomology. Therefore C_{∗}(F_{n}) is exact and we conclude that F_{n} ≅ 0 in DM.

Lemma 7 The morphism

Z_{tr}(P^{n−1}) → Z_{tr}(P^{n} − {0})

is an isomorphism in DM.

Proof: This morphism is a section of the morphism Z_{tr}(p) where

p : P^{n} − {0} → P^{n−1}

is a locally trivial in the Zariski topology bundle whose fibers are affine spaces. Any such p gives an isomorphism of motives because of the homotopy invariance and the Mayer-Vietoris properties.

References

[1] Eric M. Friedlander and Andrei Suslin. The spectral sequence relating algebraic K-theory to motivic cohomology. www.math.uiuc.edu/Ktheory/432, 2000.

[2] J.S. Milne. Etale Cohomology. Princeton Univ. Press, Princeton, NJ, 1980.

[3] Andrei Suslin and Vladimir Voevodsky. Bloch-Kato conjecture and motivic cohomology with finite coefficients. In The arithmetic and geometry of algebraic cycles, pages 117–189. Kluwer, 2000.

[4] Vladimir Voevodsky. Cohomological theory of presheaves with transfers. In Cycles, transfers and motivic homology theories, Annals of Math Studies, pages 87–137. Princeton Univ. Press, 2000.

[5] Vladimir Voevodsky. Triangulated categories of motives over a field. In Cycles, transfers and motivic homology theories, Annals of Math Studies, pages 188–238. Princeton Univ. Press, 2000.