Zero-cycles on varieties over finite fields. (English) Zbl 1062.14013
Summary: We prove that for a smooth projective variety \(X\) of dimension \(d\) defined over a finite field \(k\), the structure map \(\sigma: X \to\text{Spec}\;k\) induces an isomorphism \(\sigma_* : \text{CH}^{d+1}(X,1) \cong \text{CH}^1(k, 1) = k^*\). We also prove that the higher Chow groups \(\text{CH}^{d+s-1}(X,s)\) of one-dimensional cycles are torsion for \(s \geq 2\).
MSC:
| 14C25 | Algebraic cycles |
| 11G25 | Varieties over finite and local fields |
| 14C15 | (Equivariant) Chow groups and rings; motives |
| 14G15 | Finite ground fields in algebraic geometry |
| 19E08 | \(K\)-theory of schemes |
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