Parshin's conjecture, also called the Beilinson-Parshin conjecture, is a claim in algebraic geometry stating that for any smooth projective variety defined over a finite field, the higher algebraic K-groups of that variety vanish up to torsion. It is named for the mathematicians Aleksei Parshin and Alexander Beilinson.
Facts
Partially Attested
Progress Toward ResolutionThe conjecture holds when the variety has dimension 0, by Quillen's computation of the K-groups of finite fields, and when it has dimension 1, by a result of Harder. 1 Only the dimension 0 and dimension 1 cases are known; the general case remains open. StatementFor any smooth projective variety X defined over a finite field, the higher algebraic K-groups vanish up to torsion. 1 Classification
Resolution Status Prize Status
Prize Status (category) Connections
In Branch
Source Parshin's Conjecture (Wikipedia)
Sources
1. Parshin's Conjecture (Wikipedia)
Lead section
Parshin's conjecture (also referred to as the Beilinson-Parshin conjecture) states that for any smooth projective variety X defined over a finite field, the higher algebraic K-groups vanish up to torsion
Known cases section
The conjecture holds if dim X = 0 by Quillen's computation of the K-groups of finite fields. The conjecture holds if dim X = 1 by the proof of Corollary 3.2.3 of Harder.
In Branch: Algebraic Geometry, Lead sentence
In mathematics, more specifically in algebraic geometry, Parshin's conjecture (also referred to as the Beilinson-Parshin conjectur
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