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PreprintCC BY-NC-ND 4.0arXiv

Representation Theory and the Algebraic K-Theory of Geometric Fields

  • Gunnar Carlsson
  • Roy Joshua
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Abstract

We apply a derived completion construction for commutative ring spectra to give a description of the K-theory spectrum of a geometric field F (i.e. containing an algebraically closed subfield k) of characteristic zero purely in terms of the representation theory of its absolute Galois group GF. A result that follows is that the l-adically completed Milnor K-theory of F is isomorphic to the derived completion of the representation ring of GF. The main result, from which this follows, is that K(F) completed at l is equivalent to the derived completion of the K-theory of the category of finite dimensional representations of GF in k.

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Cite

BibTeXRIS
BibTeX
@misc{carlsson2026representation,
  title = {{Representation Theory and the Algebraic K-Theory of Geometric Fields}},
  author = {Gunnar Carlsson and Roy Joshua},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.15605},
  eprint = {2611.15605},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.15605},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Representation Theory and the Algebraic K-Theory of Geometric Fields
AU  - Gunnar Carlsson
AU  - Roy Joshua
PY  - 2026
DA  - 2026/11/11
PB  - arXiv
DO  - 10.48550/arxiv.2611.15605
UR  - https://arxiv.org/abs/2611.15605
AB  - We apply a derived completion construction for commutative ring spectra to give a description of the K-theory spectrum of a geometric field F (i.e. containing an algebraically closed subfield k) of characteristic zero purely in terms of the representation theory of its absolute Galois group G_F. A result that follows is that the l-adically completed Milnor K-theory of F is isomorphic to the derived completion of the representation ring of G_F. The main result, from which this follows, is that K(F) completed at l is equivalent to the derived completion of the K-theory of the category of finite dimensional representations of G_F in k.
ER  -