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.
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Statement
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.
K-theory of the integers
Assuming the Quillen–Lichtenbaum conjecture and the Vandiver conjecture, the K-groups of the integers, Kn(Z), are given by:
where ck/dk is the Bernoulli number B2k/k in lowest terms and n is 4k − 1 or 4k − 2 (Weibel 2005).