Finite generation conjectures for motivic cohomology theories over finite fields (Q2906504)

Let \(X\) be a regular scheme of finite type over a finite field and let \(\mathrm{CH}_n(X,i)\) the higher Chow groups. These groups are conjecturally finitely generated and this conjecture is equivalent to the finite generation of the motivic cohomology groups \(H^I_{{\mathcal M}}(X\mathbb{Z}(n))\) for smooth \(X\). Similarly the Lichtenbaum's Weil-étale cohomology groups \(H^i_W(X,\mathbb{Z}(n))\) should be finitely generated. In this paper the author defines an intermediate cohomology theory \(H^i_F(X,\mathbb{Z}(n))\) and natural maps NEWLINE\[NEWLINEH^i_{{\mathcal M}}(X, \mathbb{Z}(n))\to H^i_F(X, \mathbb{Z}(n)\to H^i(X, \mathbb{Z}(n))NEWLINE\]NEWLINE which fit into a diagram NEWLINE\[NEWLINE\begin{tikzcd} NEWLINE\rar & H^i_{{\mathcal M}}(X, \mathbb{Z}(n)) \rar["\alpha"]\dar["f" '] & H^i_F(X, \mathbb{Z}(n)) \rar["\beta"]\dar["g" '] & H^{i-1}_K(X, \mathbb{Z}(n)) \dar["h"]\rar &{} \\ NEWLINE\rar & H^i_{\text{ét}}(X, \mathbb{Z}(n)) \rar["\alpha" '] & (X, \mathbb{Z}(n)) \rar["\beta" '] & H^{i-1}_{{\mathcal M}}K(X, \mathbb{Q}(n))\rar &{}NEWLINE\end{tikzcd}NEWLINE\]NEWLINEwhere the groups \(H^*_K(X,\mathbb{Z}(n))\) are Kato cohomology. Conjecturally all groups in the upper row should be finitely generated. In the paper several particular cases of this conjecture are considered.NEWLINENEWLINEFor the entire collection see [Zbl 1242.14001].