Quillen–Lichtenbaum conjecture

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, 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, 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, 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/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 B_2k/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
• 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, 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