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. Rost and Voevodsky have announced proofs of the Bloch–Kato conjecture, which implies the Quillen–Lichtenbaum conjecture for all primes.

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

E_2^{pq}=H^p_\text{étale}(\text{Spec }A[\ell^{-1}], Z_\ell(-q/2)), (which is understood to be 0 if q is odd)

and abutting to

K_{-p-q}A\otimes Z_\ell

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).

References