Bloch–Lichtenbaum spectral sequence

Remarks

Bloch and Lichtenbaum built it from a tower of spectra made of K-theory of the algebraic simplices with supports on cycles in good position. For a field, $CH^j(F, m) = 0$ when $j > m$, so the nonzero terms satisfy $p \le 0$ and $q \le 0$. The term $E_2^{0, -n} = CH^n(F, n) \cong K^M_n(F)$ gives the edge map from Milnor K-theory to Quillen K-theory. Example: in total degree $1$, the only nonzero term is $E_2^{0,-1} = CH^1(F, 1) \cong F^\times$, and the spectral sequence recovers $K_1(F) \cong F^\times$. Rationally the spectral sequence degenerates at $E_2$.

Going deeper

For a student

This was the first motivic spectral sequence, and it was later extended from fields to all smooth varieties. It is the analogue for fields of the Atiyah–Hirzebruch spectral sequence in topology. Together with the comparison of motivic and étale cohomology, it is a route from the Bloch–Kato conjecture to the Quillen–Lichtenbaum conjecture.