关于精细剪刀同余群与$\boldsymbol{\textrm{SL}_2}$的三阶同调的注记
Remarks on refined scissors congruence group and the third homology of $\mathrm{SL}_2$
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中文总结 AI 辅助
该研究探讨环$A$上与$\textrm{SL}_2$三阶同调相关的正合序列和同构的充分条件，指出两类环满足相关关系但不满足-1为平方数的条件，并在非阿基米德局部域情形下得到精细Bloch-Wigner正合序列。
中文摘要 AI 辅助
本研究重新探讨环$A$上保证正合序列$H_3(\textrm{SM}_2(A), \boldsymbol{\textrm{Z}})\to H_3(\textrm{SL}_2(A), \boldsymbol{\textrm{Z}})\to \boldsymbol{\textrm{RB}}(A)\to 0$存在，以及同构$H_3(\textrm{SL}_2(A), \textrm{SM}_2(A); \boldsymbol{\textrm{Z}})\backsimeq \boldsymbol{\textrm{RP}}_1(A)$存在的关键充分条件。研究表明，存在满足上述关系但不满足-1是平方数的环，例如$\boldsymbol{\textrm{Z}}[\frac{1}{2}]$和$\boldsymbol{\textrm{Z}}[\frac{1}{10}]$。此外，研究非阿基米德局部域的特殊情形，得到当-1不是平方数时的精细Bloch-Wigner正合序列。
英文摘要
This work revisits the key sufficient conditions on a ring $A$ that guarantee the existence of the exact sequence \[ H_3(\mathrm{SM}_2(A), \mathbb{Z})\to H_3(\mathrm{SL}_2(A), \mathbb{Z}) \to \mathcal{RB}(A) \to 0 \] and of the isomorphism \[ H_3(\mathrm{SL}_2(A), \mathrm{SM}_2(A); \mathbb{Z}) \simeq \mathcal{RP}_1(A). \] 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 $\mathbb{Z}[\frac{1}{m}]$ 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.
发表机构
- Pontifical Catholic University of Peru (PUCP)（秘鲁天主教 Pontificia 大学）
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