In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Quillen–Lichtenbaum conjecture is a
conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...
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 conjectu ...
to
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 sens ...
introduced by , who was inspired by earlier conjectures of . and
proved the Quillen–Lichtenbaum conjecture at the
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
2 for some
number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a ...
s.
Voevodsky, using some important results of
Markus Rost, 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
:
(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, ''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 function ...
''B''
2''k''/''k'' in lowest terms and ''n'' is 4''k'' − 1 or 4''k'' − 2 .
References
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{{DEFAULTSORT:Quillen-Lichtenbaum conjecture
Algebraic K-theory
Conjectures that have been proved