The Motivic Spectral Sequence

The motivic spectral sequence (also called the Atiyah--Hirzebruch spectral sequence for algebraic K-theory) relates motivic cohomology to algebraic K-theory. It is the algebraic analogue of the classical Atiyah--Hirzebruch spectral sequence from singular cohomology to topological K-theory.

Statement

Theorem6.2Motivic spectral sequence

For a smooth variety $X$ over a field $k$ , there is a convergent spectral sequence:

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

where $H^n_{\text{mot}}(X, \mathbb{Z}(m))$ denotes motivic cohomology. The $d_r$ differentials have bidegree $(r, r-1)$ .

Equivalently, with the reindexing $a = p - q$ , $b = -q$ (so $a =$ cohomological degree, $b =$ weight):

$E_2^{a,b} = H^a_{\text{mot}}(X, \mathbb{Z}(b)) \implies K_{2b-a}(X).$

RemarkThe E₂ page

The $E_2$ -terms $H^a_{\text{mot}}(X, \mathbb{Z}(b)) = CH^b(X, 2b-a)$ include:

On the row $a = 2b$ : $CH^b(X, 0) = CH^b(X)$ (classical Chow groups).

On the row $a = 2b - 1$ : $CH^b(X, 1)$ (related to algebraic cycles modulo algebraic equivalence).

On the diagonal $a = b$ : $H^b_{\text{mot}}(\operatorname{Spec} F, \mathbb{Z}(b)) = K_b^M(F)$ for $X = \operatorname{Spec} F$ .

The spectral sequence converges to $K_n(X) = \bigoplus_{a+b=n}$ (associated graded of the filtration).

Rational degeneration

Definition6.4Weight filtration on K-theory

The motivic spectral sequence defines a weight filtration (or gamma filtration) on $K_n(X)$ :

$\cdots \subseteq F^{p+1} K_n(X) \subseteq F^p K_n(X) \subseteq \cdots \subseteq K_n(X)$

with associated graded pieces $\operatorname{gr}^p K_n(X) = E_\infty^{2p-n, -p}$ , a subquotient of $H^{2p-n}_{\text{mot}}(X, \mathbb{Z}(p)) = CH^p(X, n)$ .

After tensoring with $\mathbb{Q}$ , the Adams operations $\psi^k$ act on $\operatorname{gr}^p K_n(X) \otimes \mathbb{Q}$ by $k^p$ , and the spectral sequence degenerates at $E_2$ :

$E_2^{p,q} \otimes \mathbb{Q} = E_\infty^{p,q} \otimes \mathbb{Q}$

giving the rational decomposition:

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

ExampleThe spectral sequence for Spec k

For $X = \operatorname{Spec} k$ with $k$ a field:

The $E_2$ page has $E_2^{p,q} = H^{p-q}_{\text{mot}}(k, \mathbb{Z}(-q))$ . Non-zero terms include:

$E_2^{0,0} = \mathbb{Z}$ (weight 0, giving $K_0(k) \supseteq \mathbb{Z}$ ).
$E_2^{1,1} = k^{\times}$ (weight 1, contributing to $K_1(k)$ ).
$E_2^{n,n} = K_n^M(k)$ (Milnor K-theory on the diagonal).

All other terms with $p \neq q$ should vanish by the Beilinson--Soule conjecture. Assuming this, $E_2 = E_\infty$ and we get exact sequences:

$0 \to K_n^M(k) \to K_n(k) \to (\text{lower weight}) \to 0$

The map $K_n^M(k) \to K_n(k)$ is the canonical one from Milnor to Quillen K-theory. For $n \leq 2$ , this is an isomorphism (by Matsumoto). For $n = 3$ , the kernel measures the difference between Milnor and Quillen K-theory.

Integral differentials

ExampleNon-trivial differentials

The motivic spectral sequence can have non-trivial differentials integrally. The first potential differential is:

$d_2: E_2^{p,q} \to E_2^{p+2, q+1}$

In motivic terms: $d_2: H^a_{\text{mot}}(X, \mathbb{Z}(b)) \to H^{a+2}_{\text{mot}}(X, \mathbb{Z}(b+1))$ .

For $X = \operatorname{Spec} \mathbb{F}_q$ : The $E_2$ page has $E_2^{n,n} = K_n^M(\mathbb{F}_q)$ . Since $K_n^M(\mathbb{F}_q) = 0$ for $n \geq 2$ and $K_1^M = \mathbb{F}_q^{\times}$ , the spectral sequence collapses. But $K_{2i-1}(\mathbb{F}_q) \cong \mathbb{Z}/(q^i - 1)$ is nonzero, which is captured by a differential or extension in the spectral sequence.

In fact, the groups $K_{2i-1}(\mathbb{F}_q)$ arise from extensions in the spectral sequence between the weight- $i$ and weight-0 terms: the element $q^i - 1 \in \mathbb{Z}$ from the Adams eigenvalue determines the extension class.

With finite coefficients

Definition6.5Motivic spectral sequence mod ℓ

With $\mathbb{Z}/\ell$ coefficients, the motivic spectral sequence becomes:

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

By the Bloch--Kato conjecture (Voevodsky's theorem), for $p - q \leq -q$ (i.e., $p \leq 0$ ):

$H^{p-q}_{\text{mot}}(X, \mathbb{Z}/\ell(-q)) \cong H^{p-q}_{\text{et}}(X, \mu_\ell^{\otimes (-q)}).$

This identifies the $E_2$ terms with etale cohomology in a range, and the spectral sequence becomes the etale descent spectral sequence:

$E_2^{p,q} = H^{p-q}_{\text{et}}(X, \mu_\ell^{\otimes(-q)}) \implies K_{-p-q}(X; \mathbb{Z}/\ell)$

in the appropriate range. This is the Quillen--Lichtenbaum conjecture: algebraic K-theory with finite coefficients agrees with etale K-theory in sufficiently high degrees.

RemarkLichtenbaum's conjectures and zeta values

For $X = \operatorname{Spec} \mathcal{O}_F$ (ring of integers of a number field $F$ ), the motivic spectral sequence with finite coefficients, combined with the Wiles/Huber--Kings proof of the Bloch--Kato conjecture for number fields, gives:

$|K_{2n-2}(\mathcal{O}_F)| \cdot R_n(F) \sim_{\mathbb{Q}^{\times}} |\zeta_F(1-n)| \cdot w_n(F)$

where $R_n$ is a regulator, $\zeta_F$ is the Dedekind zeta function, and $w_n$ is the number of roots of unity in a related sense. This is the Lichtenbaum conjecture, connecting:

Algebraic K-theory (left side)
Special values of L-functions (right side)
Motivic/etale cohomology (mediating via the spectral sequence)