Let F be a global field and let l≥ 2. If l=2, assume that mu_4⊂ F. For n>1 and l≠operatornamechar F, let K_n^et denote étale K-theory. Quillen–Lichtenbaum conjecture. The natural map K_n( O_F)otimes Z_l longrightarrow K_n^et( O_F[frac1l]) is an isomorphism. The conjecture is a comparison between algebraic and étale K-theory for global fields. The surrounding text states that it resulted from the Voevodsky results and related motivic-cohomology work, so it is now solved.

Quillen–Lichtenbaum conjecture for global fields

Let $F$ be a global field and let $l\ge 2$ . If $l=2$ , assume that $\mu_4\subset F$ . For $n>1$ and $l\ne\operatorname{char} F$ , let $K_n^{\mathrm{et}}$ denote étale $K$ -theory. Quillen–Lichtenbaum conjecture. The natural map

K_n(\mathcal O_F)\otimes\mathbb Z_l\longrightarrow K_n^{\mathrm{et}}\left(\mathcal O_F\left[\frac{1}{l}\right]\right)

is an isomorphism. The conjecture is a comparison between algebraic and étale $K$ -theory for global fields. The surrounding text states that it resulted from the Voevodsky results and related motivic-cohomology work, so it is now solved.

Sources & referencesView supporting material

Primary source

Grzegorz Banaszak, “Wild Kernels and divisibility in K-groups of global fields”, arXiv:1208.2137 (2012).

Additional references

3 papers in this index state this conjecture (2010–2012). The statement above is taken from the most recent of them; the others are arXiv:1106.0513, arXiv:1008.1000.

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Progress summary

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Solved

The comparison conjecture is settled: work on motivic cohomology established that algebraic and étale K-theory agree in the stated global-field setting.

Quillen introduced the conjecture in 1975, building on Lichtenbaum’s 1973 conjectures. The Bloch–Kato theorem of Rost and Voevodsky implies the required comparison for all primes.

Known results

The prime- $2$ case was already established by Rognes and Weibel by 2003; later work treated remaining hypotheses and refinements.

Voevodsky–Rost resolution, recorded 2012

A 2012 paper states the Quillen–Lichtenbaum comparison as a theorem, attributing it to Voevodsky’s proof of Bloch–Kato, building on Rost’s work. Later literature uses the result as settled mathematics, including applications to higher $K$ -groups of rings of integers.

Current status (as of August 2026): The Quillen–Lichtenbaum conjecture is resolved for the stated global-field comparison; no unresolved objection or counterexample was found.

Sources

No solutions have been posted yet.