Motivic spectral sequence

Remarks

The spectral sequence is the motivic analogue of the Atiyah–Hirzebruch spectral sequence from ordinary cohomology to topological K-theory. For a field it was constructed by Bloch and Lichtenbaum, as the Bloch–Lichtenbaum spectral sequence; Friedlander and Suslin extended it to smooth varieties, and Levine gave another construction. Rationally the spectral sequence degenerates, and $K_n(X) \otimes \Q \cong \bigoplus_q H^{2q - n}(X, \Q(q))$, where the summands are eigenspaces of Adams operations. Example: the term $E_2^{q, -q} = H^{2q}(X, \Z(q)) = CH^q(X)$ contributes to $K_0(X)$, which links K-theory to the Chow groups. For $X = \operatorname{Spec} F$, the term $E_2^{0, -n} = H^n(F, \Z(n)) \cong K^M_n(F)$ contributes to $K_n(F)$, which gives the map from Milnor K-theory to Quillen K-theory.

Going deeper

For a student

The spectral sequence is the main bridge between algebraic K-theory and arithmetic. With finite coefficients and the norm residue isomorphism theorem, it gives the Quillen–Lichtenbaum conjecture, and so a computation of K-groups of number rings in terms of étale cohomology. Its rational degeneration gives $K_0(X) \otimes \Q \cong \bigoplus_q CH^q(X) \otimes \Q$, a result going back to Grothendieck's Riemann–Roch theory.