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PreprintCC BY 4.0arXiv

Discretisation in algebraic K-theory of independent ample groupoids and their Steinberg algebras

  • Guillermo Cortiñas
  • Xin Li
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Abstract

An independent ample groupoid G comes equipped with a discrete groupoid Gd and a proper étale correspondence Ω:Gd→ G. Discretisation with respect to a homological invariant ℋ holds when ℋ(Ω) is an equivalence. We show for example that discretisation with respect to the K-theory spectrum 𝕂 BG of the permutative category of bisections always holds, establish excision and continuity for this invariant, and use all this and explicit resolutions by independent groupoids to compute it for several families of ample groupoids. We also prove discretisation with respect to invariants of the Steinberg algebra ℓ[G] such as (homotopy) algebraic K-theory in several particular cases, and carry out computations.

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Cite

BibTeXRIS
BibTeX
@misc{cortinas2026discretisation,
  title = {{Discretisation in algebraic $K$-theory of independent ample groupoids and their Steinberg algebras}},
  author = {Guillermo Cortiñas and Xin Li},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.38791},
  eprint = {2610.38791},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.38791},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Discretisation in algebraic K-theory of independent ample groupoids and their Steinberg algebras
AU  - Guillermo Cortiñas
AU  - Xin Li
PY  - 2026
DA  - 2026/10/28
PB  - arXiv
DO  - 10.48550/arxiv.2610.38791
UR  - https://arxiv.org/abs/2610.38791
AB  - An independent ample groupoid G comes equipped with a discrete groupoid G_d and a proper étale correspondence Ω:G_d→ G. Discretisation with respect to a homological invariant ℋ holds when ℋ(Ω) is an equivalence. We show for example that discretisation with respect to the K-theory spectrum 𝕂 B_G of the permutative category of bisections always holds, establish excision and continuity for this invariant, and use all this and explicit resolutions by independent groupoids to compute it for several families of ample groupoids. We also prove discretisation with respect to invariants of the Steinberg algebra ℓ[G] such as (homotopy) algebraic K-theory in several particular cases, and carry out computations.
ER  -