In mathematics, the Quillen–Lichtenbaum conjecture is a conjecture relating étale cohomology

Étale cohomology

In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures...

 to algebraic K-theory

Algebraic K-theory

In mathematics, algebraic K-theory is an important part of homological algebra concerned with defining and applying a sequenceof functors from rings to abelian groups, for all integers n....

 introduced by , who was inspired by earlier conjectures of . and 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

Atiyah-Hirzebruch spectral sequence

In mathematics, the Atiyah–Hirzebruch spectral sequence is a spectral sequence for calculating generalized cohomology, introduced by in the special case of K-theory. For a CW complex X, it relates the generalized cohomology groups...

, starting at
(which is understood to be 0 if q is odd)

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 are given by
K_n(Z) =

• 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/c_k ⊕ Z/2 if n = 2 mod 8
• Z/8d_k if n = 3 mod 8
• 0 if n = 4 mod 8
• Z if n = 5 mod 8
• Z/c_k if n = 6 mod 8
• Z/4d_k if n = 7 mod 8

where c_k/d_k is the Bernoulli number

Bernoulli number

In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers with deep connections to number theory. They are closely related to the values of the Riemann zeta function at negative integers....

B_2k/k in lowest terms and n is 4k − 1 or 4k − 2 .