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Cycles on curves over global fields of positive characteristic


Author: Reza Akhtar
Journal: Trans. Amer. Math. Soc. 357 (2005), 2557-2569
MSC (2000): Primary 14C15, 14C25
DOI: https://doi.org/10.1090/S0002-9947-05-03777-3
Published electronically: March 1, 2005
MathSciNet review: 2139518
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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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Additional Information

Reza Akhtar
Affiliation: Department of Mathematics and Statistics, Miami University, Oxford, Ohio 45056
Email: reza@calico.mth.muohio.edu

DOI: https://doi.org/10.1090/S0002-9947-05-03777-3
Received by editor(s): January 20, 2003
Published electronically: March 1, 2005
Article copyright: © Copyright 2005 American Mathematical Society

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