Motivic Complexes and Bloch's Higher Chow Groups

Motivic cohomology provides the "universal" cohomology theory for algebraic varieties, serving as the source of comparison maps to all other theories (etale, de Rham, Hodge). Bloch's higher Chow groups give a concrete cycle-theoretic definition that is equivalent to Voevodsky's more abstract approach via motivic complexes.

Bloch's higher Chow groups

Definition6.1Algebraic simplex

The algebraic $n$ -simplex is

$\Delta^n = \operatorname{Spec} \mathbb{Z}[t_0, \ldots, t_n] / (t_0 + \cdots + t_n - 1) \cong \mathbb{A}^n_\mathbb{Z}$

with faces $\partial_i: \Delta^{n-1} \hookrightarrow \Delta^n$ defined by $t_i = 0$ and degeneracies by doubling coordinates. These form a cosimplicial scheme $\Delta^\bullet$ .

A face of $\Delta^n$ is a closed subscheme defined by setting some subset of the $t_i$ to zero. A codimension- $k$ face is isomorphic to $\Delta^{n-k}$ .

Definition6.2Bloch's higher Chow groups

For a smooth variety $X$ over a field $k$ , the higher Chow complex $z^p(X, \bullet)$ is the chain complex:

$\cdots \to z^p(X, n) \xrightarrow{\partial} z^p(X, n-1) \to \cdots \to z^p(X, 0)$

where $z^p(X, n)$ is the free abelian group generated by irreducible codimension- $p$ subvarieties $Z \subset X \times \Delta^n$ meeting all faces $X \times F$ properly (i.e., in the expected codimension), and $\partial = \sum_{i=0}^n (-1)^i \partial_i^*$ is the alternating sum of pullbacks to faces.

The higher Chow groups are the homology:

$CH^p(X, n) = H_n(z^p(X, \bullet)).$

For $n = 0$ : $CH^p(X, 0) = CH^p(X)$ , the classical Chow group.

ExampleCH¹(X, 1) and units

For $n = 1$ , $p = 1$ : $z^1(X, 1)$ consists of codimension-1 cycles on $X \times \mathbb{A}^1$ meeting $X \times \{0\}$ and $X \times \{1\}$ properly. The boundary map sends such a cycle $Z$ to $Z|_{t=0} - Z|_{t=1} \in z^1(X, 0) = Z^1(X)$ .

For $X = \operatorname{Spec} k$ : $CH^1(\operatorname{Spec} k, 1) = k^{\times}$ . A cycle $Z \in z^1(\operatorname{Spec} k, 1)$ is a 0-cycle on $\mathbb{A}^1_k$ meeting $\{0\}$ and $\{1\}$ properly, i.e., a finite set of closed points $\{a_1, \ldots, a_r\} \subset \mathbb{A}^1 \setminus \{0, 1\}$ with multiplicities. The identification with $k^{\times}$ sends such a cycle to $\prod a_i^{n_i} \in k^{\times}$ .

More generally, $CH^1(X, 1) = \mathcal{O}(X)^{\times}$ for smooth affine $X$ .

Connection to K-theory

Theorem6.1Bloch--Lichtenbaum / Voevodsky

For a smooth variety $X$ over a field $k$ , there are natural isomorphisms:

$CH^p(X, n) \cong H^{2p-n}_{\text{mot}}(X, \mathbb{Z}(p))$

where the right side is Voevodsky's motivic cohomology. Moreover, there is a motivic spectral sequence (Bloch--Lichtenbaum, Friedlander--Suslin, Levine):

$E_2^{p,q} = H^{p-q}_{\text{mot}}(X, \mathbb{Z}(-q)) = CH^{-q}(X, -p-q) \implies K_{-p-q}(X)$

This spectral sequence degenerates rationally (by weight arguments), giving:

$K_n(X) \otimes \mathbb{Q} \cong \bigoplus_{p \geq 0} CH^p(X, n+p) \otimes \mathbb{Q} = \bigoplus_p H^{2p-n}_{\text{mot}}(X, \mathbb{Q}(p)).$

ExampleMotivic spectral sequence for a field

For $X = \operatorname{Spec} F$ (a field):

$E_2^{p,q} = H^{p-q}_{\text{mot}}(F, \mathbb{Z}(-q)) = \begin{cases} K_{-q}^M(F) & p = q \\ 0 & p > q \text{ (Beilinson--Soule vanishing)} \end{cases}$

The spectral sequence has $E_2^{p,q}$ concentrated on the diagonal $p = q$ (at least conjecturally for $p < q$ ). On the diagonal: $E_2^{n,n} = K_n^M(F)$ . The edge map gives the natural map

$K_n^M(F) = E_2^{n,n} \twoheadrightarrow E_\infty^{n,n} \hookrightarrow K_n(F)$

which is the canonical comparison from Milnor to Quillen K-theory.

Voevodsky's motivic cohomology

Definition6.3Motivic complexes (Voevodsky)

Voevodsky defines motivic cohomology using the derived category of motives $\mathbf{DM}^{\text{eff}}(k)$ . The motivic complex $\mathbb{Z}(p)$ is an object in $D^-(\text{Sh}_{\text{Nis}}(\text{Sm}/k))$ (the derived category of Nisnevich sheaves on smooth $k$ -schemes) defined as:

$\mathbb{Z}(p) = C_*(\mathbb{Z}_{\text{tr}}(\mathbb{G}_m^{\wedge p}))[-p]$

where:

$\mathbb{Z}_{\text{tr}}(\mathbb{G}_m^{\wedge p})$ is the sheaf with transfers represented by $\mathbb{G}_m^{\wedge p} = (\mathbb{A}^1 \setminus \{0\})^p / (\text{faces})$ .
$C_*$ is the singular complex functor (using $\Delta^\bullet$ -homotopy).
The shift $[-p]$ is a cohomological convention.

Motivic cohomology groups are:

$H^{n}_{\text{mot}}(X, \mathbb{Z}(p)) = H^n_{\text{Nis}}(X, \mathbb{Z}(p)) = \operatorname{Hom}_{\mathbf{DM}(k)}(M(X), \mathbb{Z}(p)[n]).$

RemarkKey identifications

The following identifications connect motivic cohomology to classical invariants:

$H^{0}_{\text{mot}}(X, \mathbb{Z}(0)) = H^0_{\text{Zar}}(X, \mathbb{Z})$ (connected components).
$H^{1}_{\text{mot}}(X, \mathbb{Z}(1)) = \mathcal{O}(X)^{\times}$ (units).
$H^{2}_{\text{mot}}(X, \mathbb{Z}(1)) = \operatorname{Pic}(X)$ (Picard group).
$H^{2p}_{\text{mot}}(X, \mathbb{Z}(p)) = CH^p(X)$ (Chow groups).
$H^{n}_{\text{mot}}(\operatorname{Spec} F, \mathbb{Z}(n)) = K_n^M(F)$ (Milnor K-theory).

More generally, $H^{n}_{\text{mot}}(X, \mathbb{Z}(p)) = CH^p(X, 2p - n)$ (Bloch's higher Chow groups), providing the cycle-theoretic interpretation.

The motivic-to-etale comparison

RemarkComparison with etale cohomology

For $\ell$ invertible in $k$ , there is a natural comparison map:

$H^n_{\text{mot}}(X, \mathbb{Z}/\ell(p)) \to H^n_{\text{et}}(X, \mu_\ell^{\otimes p})$

The Beilinson--Lichtenbaum conjecture (now a theorem, following from Bloch--Kato) states this is an isomorphism for $n \leq p$ and an injection for $n = p + 1$ .

For $X = \operatorname{Spec} F$ and $n = p$ , this gives the Bloch--Kato isomorphism $K_n^M(F)/\ell \cong H^n_{\text{et}}(F, \mu_\ell^{\otimes n})$ .

The failure for $n > p + 1$ is measured by the higher etale cohomology, which is sensitive to the absolute Galois group but not to algebraic cycles.