Parshin's conjecture and motivic cohomology with compact support (Q2809886)

Parshin's conjecture states that the higher algebraic \(K\)-groups of a smooth projective variety over a finite field are torsion. The article under review explores consequences of this conjecture for motivic cohomology.NEWLINENEWLINESince Parshin's conjecture requires certain groups to be torsion, it is equivalent to asking that their rationalizations vanish: \(K_i(X) \otimes \mathbb{Q} = 0\) for \(i > 0\) and \(X\) smooth projective over a finite field. Rational algebraic \(K\)-theory coincides with rational motivic cohomology, so this statement is equivalent to \(H^i(X, \mathbb{Q}(n)) = 0\) in a certain range, and for all \textit{smooth projective} \(X\) (over a finite field).NEWLINENEWLINEOne of the aims of the article is to find equivalent statements which are expressed in terms of more varieties. For example, the author shows that Parshin's conjecture is equivalent to \(H^i_c(X, \mathbb{Q}(n)) = 0\) for \(i < 2n\) and \textit{all} varieties \(X\) over finite fields. Here \(H^i_c\) denotes motivic cohomology with compact support.NEWLINENEWLINEThe proofs involve careful investigation of coniveau spectral sequences and comparison of weight cohomology groups and motivic cohomology groups. The author also relies on resolution of singularities, so many results are conditional.