Cycles on curves over global fields of positive characteristic

Akhtar, Reza

doi:10.1090/S0002-9947-05-03777-3

Cycles on curves over global fields of positive characteristic
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Trans. Amer. Math. Soc. 357 (2005), 2557-2569 Request permission

Abstract:

Let $k$ be a global field of positive characteristic, and let $\sigma : X \longrightarrow \operatorname {Spec} k$ be a smooth projective curve. We study the zero-dimensional cycle group $V(X) =\operatorname {Ker}(\sigma _*: SK_1(X) \rightarrow K_1(k))$ and the one-dimensional cycle group $W(X) =\operatorname {coker}(\sigma ^*: K_2(k) \rightarrow H^0_{Zar}(X, \mathcal {K}_2))$, addressing the conjecture that $V(X)$ is torsion and $W(X)$ is finitely generated. The main idea is to use Abhyankar’s Theorem on resolution of singularities to relate the study of these cycle groups to that of the $K$-groups of a certain smooth projective surface over a finite field.

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