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PreprintCC BY 4.0arXiv

Remarks on refined scissors congruence group and the third homology of SL2

  • Elvis Torres Pérez
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Abstract

This work revisits the key sufficient conditions on a ring A that guarantee the existence of the exact sequence H3(SM2(A), ℤ)→ H3(SL2(A), ℤ) → ℛℬ(A) → 0 and of the isomorphism H3(SL2(A), SM2(A); ℤ) ≃ ℛ𝒫1(A). We show that there are rings which satisfy the relations above but do not satisfy the condition that -1 is an square, for example some rings ℤ[1/m] and finite local rings. In addition, we study the special case of non-dyadic local fields with characteristic not 2, obtaining a refined Bloch-Wigner exact sequence when -1 is not an square.

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Cite

BibTeXRIS
BibTeX
@misc{perez2026remarks,
  title = {{Remarks on refined scissors congruence group and the third homology of $\mathrmSL_2$}},
  author = {Elvis Torres Pérez},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.08890},
  eprint = {2608.08890},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.08890},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Remarks on refined scissors congruence group and the third homology of SL_2
AU  - Elvis Torres Pérez
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2608.08890
UR  - https://arxiv.org/abs/2608.08890
AB  - This work revisits the key sufficient conditions on a ring A that guarantee the existence of the exact sequence H_3(SM_2(A), ℤ)→ H_3(SL_2(A), ℤ) → ℛℬ(A) → 0 and of the isomorphism H_3(SL_2(A), SM_2(A); ℤ) ≃ ℛ𝒫_1(A). We show that there are rings which satisfy the relations above but do not satisfy the condition that -1 is an square, for example some rings ℤ[1/m] and finite local rings. In addition, we study the special case of non-dyadic local fields with characteristic not 2, obtaining a refined Bloch-Wigner exact sequence when -1 is not an square.
ER  -