Let k be a field and let w≥ 0. The group H_operatornameet^w+1(operatornameSpec k,Z_(p)(w)) denotes étale hypercohomology with p-local motivic coefficients. Beilinson–Lichtenbaum conjecture. One has H_operatornameet^w+1(operatornameSpec k,Z_(p)(w))=0. This is presented as a reduction step toward the general Bloch–Kato conjecture before motivic cohomology operations are used. Its resolution status is not specified in the supplied source context.

Beilinson–Lichtenbaum conjecture, p-local version

The conjecture asserts vanishing of $\mathbb{H}_{\operatorname{et}}^{w+1}(\operatorname{Spec} k,\mathbb{Z}_{(p)}(w))$ . Retrieved expositions identify the field-case Beilinson–Lichtenbaum theorem with the established Bloch–Kato theory, although they do not display this exact $p$ -local formulation.

Known results

Geisser (2004) and Voevodsky (2011): the classical Beilinson–Lichtenbaum comparison was proved for the relevant smooth schemes.
Rost and Voevodsky: the Bloch–Kato conjecture was proved in general, with the field case essentially equivalent to Beilinson–Lichtenbaum.
Elmanto–Morrow: the comparison was extended to smooth schemes over fields and to ind-smooth schemes over Prüfer rings.

2025 comparison extensions

Recent work gives broader mod- $p^k$ and syntomic Beilinson–Lichtenbaum comparisons, but reports no new gap, counterexample, or competing claim concerning the field-level $p$ -local vanishing.

Current status (as of August 2026): The standard field-case theorem is settled through Bloch–Kato and the Beilinson–Lichtenbaum comparison; the displayed $p$ -local reformulation has no separately reported unresolved issue.

Beilinson–Lichtenbaum conjecture, $p$ -local version

Primary source

Progress summary

Solutions 0