Quillen–Lichtenbaum conjecture

In mathematics, the Quillen–Lichtenbaum conjecture is a conjecture relating étale cohomology to algebraic K-theory introduced by Quillen (1975, p. 175), who was inspired by earlier conjectures of Lichtenbaum (1973). Kahn (1997) and Rognes & Weibel (2000) proved the Quillen–Lichtenbaum conjecture at the prime 2 for some number fields. Voevodsky, using some important results of Markus Rost, have proved the Bloch–Kato conjecture, which implies the Quillen–Lichtenbaum conjecture for all primes.

The conjecture in Quillen's original form states that if A is a finitely-generated algebra over the integers and l is prime, then there is a spectral sequence analogous to the Atiyah–Hirzebruch spectral sequence, starting at

and abutting to

for −p − q > 1 + dim A.

Assuming the Quillen–Lichtenbaum conjecture and the Vandiver conjecture, the K-groups of the integers, K_n(Z), are given by:

where c_k/d_k is the Bernoulli number B_2k/k in lowest terms and n is 4k − 1 or 4k − 2 (Weibel 2005).

...
Wikipedia