Remarks on refined scissors congruence group and the third homology of $\mathrm{SL}_2$

Abstract

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.