mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Quillen–Lichtenbaum conjecture is a conjecture relating

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 conjecture ...

algebraic K-theory Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense o ...

introduced by , who was inspired by earlier conjectures of . and proved the Quillen–Lichtenbaum conjecture at the prime 2 for some number fields. Voevodsky, using some important results of

Markus Rost Markus Rost is a German mathematician who works at the intersection of topology and algebra. He was an invited speaker at the International Congress of Mathematicians in 2002 in Beijing, China. He is a professor at the University of Bielefeld. He ...

, has 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 In mathematics, the Atiyah–Hirzebruch spectral sequence is a spectral sequence for calculating generalized cohomology, introduced by in the special case of topological K-theory. For a CW complex X and a generalized cohomology theory E^\bullet, i ...

, starting at :

Z_\ell(-q/2)),

(which is understood to be 0 if ''q'' is odd) and abutting to :

K_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, ''K''_''n''(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/''c''_''k'' ⊕ Z/2 if ''n'' = 2 mod 8 *Z/8''d''_''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/4''d''_''k'' if ''n'' = 7 mod 8 where ''c''_''k''/''d''_''k'' is the

Bernoulli number In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...

''B''_2''k''/''k'' in lowest terms and ''n'' is 4''k'' − 1 or 4''k'' − 2 .

References

* * * * * * {{DEFAULTSORT:Quillen-Lichtenbaum conjecture Algebraic K-theory Conjectures that have been proved