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<p>Building on our previous work (Hiranouchi in Ann. K-Theory 9(3):499–518, 2024), we investigate an analogue of the differential symbol map used in the Bloch–Gabber–Kato theorem. Within this framework, for an appropriate variety over a field, the higher Chow group corresponds to the 0-th Kato homology group. Inspired by Akhtar’s theorem (Commun Algebra 32(1):279–294, 2004) on higher Chow groups, we investigate the structure of the 0-th Kato homology groups for varieties over arithmetic fields, including finite fields, local fields, and global fields of positive characteristic. We also express our results in terms of reciprocity sheaves.</p>

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Extended differential symbol and the Kato homology groups

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  • 发表日期：

    2025年7月31日

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  • Toshiro Hiranouchi,
  • Rin Sugiyama

摘要

Building on our previous work (Hiranouchi in Ann. K-Theory 9(3):499–518, 2024), we investigate an analogue of the differential symbol map used in the Bloch–Gabber–Kato theorem. Within this framework, for an appropriate variety over a field, the higher Chow group corresponds to the 0-th Kato homology group. Inspired by Akhtar’s theorem (Commun Algebra 32(1):279–294, 2004) on higher Chow groups, we investigate the structure of the 0-th Kato homology groups for varieties over arithmetic fields, including finite fields, local fields, and global fields of positive characteristic. We also express our results in terms of reciprocity sheaves.