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, 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
E 2 p q = H etale p ( Spec A [ ℓ − 1 ] , Z ℓ ( − q / 2 ) ) , {\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
K − p − q A ⊗ Z ℓ {\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/ck ⊕ Z/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).
References Grayson, Daniel R. (1994), "Weight filtrations in algebraic K-theory", in Jannsen, Uwe; Kleiman, Steven; Serre, Jean-Pierre (eds.), Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., vol. 55, Providence, R.I.: American Mathematical Society, pp. 207–237, ISBN 978-0-8218-1636-3, MR 1265531 Kahn, Bruno (1997), The Quillen-Lichtenbaum conjecture at the prime 2 (PDF) Lichtenbaum, Stephen (1973), "Values of zeta-functions, étale cohomology, and algebraic K-theory", in Bass, H. (ed.), Algebraic K-theory, II: Classical algebraic K-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Lecture Notes in Mathematics, vol. 342, Berlin, New York: Springer-Verlag, pp. 489–501, doi:10.1007/BFb0073737, ISBN 978-3-540-06435-0, MR 0406981 Quillen, Daniel (1975), "Higher algebraic K-theory", Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1, Canad. Math. Congress, Montreal, Que., pp. 171–176, MR 0422392, archived from the original on 2011-09-27, retrieved 2011-04-28 Rognes, J.; Weibel, Charles (2000), "Two-primary algebraic K-theory of rings of integers in number fields", Journal of the American Mathematical Society, 13 (1): 1–54, doi:10.1090/S0894-0347-99-00317-3, hdl:10852/39337, ISSN 0894-0347, MR 1697095 Weibel, Charles (2005), "Algebraic K-theory of rings of integers in local and global fields", in Friedlander, Eric M.; Grayson, Daniel R. (eds.), Handbook of K-theory. Vol. 1, Berlin, New York: Springer-Verlag, pp. 139–190, doi:10.1007/3-540-27855-9_5, ISBN 978-3-540-23019-9, MR 2181823