# Quillen–Lichtenbaum conjecture

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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.

## 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

${\displaystyle E_{2}^{pq}=H_{\text{etale}}^{p}({\text{Spec }}A[\ell ^{-1}],Z_{\ell }(-q/2)),}$ (which is understood to be 0 if q is odd)

and abutting to

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

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:

• 0 if n = 0 mod 8 and n > 0, Z if n = 0
• Z ⊕ Z/2 if n = 1 mod 8 and n > 1, Z/2 if n = 1.
• Z/ckZ/2 if n = 2 mod 8
• Z/8dk if n = 3 mod 8
• 0 if n = 4 mod 8
• Z if n = 5 mod 8
• Z/ck if n = 6 mod 8
• Z/4dk if n = 7 mod 8

where ck/dk is the Bernoulli number B2k/k in lowest terms and n is 4k − 1 or 4k − 2 (Weibel 2005).