Characterize when the localization sequence is short exact

Determine for which rings of S-integers \mathcal{O}_{K,S} the sequence relating H_2(SL_2(\mathbb{Z}[1/2n]),\mathbb{Z}), H_2(SL_2(\mathbb{Q}),\mathbb{Z}), and the residue-field unit groups generalizes to a short exact sequence, and more generally characterize the cases in which the analogous localization sequence is short exact.

Background

For \mathcal{O}_{K,S}=\mathbb{Z}[1/2n], the authors compare the exact sequence obtained from their main theorem with a known description of H_2(SL_2(\mathbb{Z}[1/2n]),\mathbb{Z}) and observe that the sequence is actually short exact.

They then explicitly ask in what other cases this phenomenon occurs, connecting the question to the study of the group \widetilde{K}2(2,\mathcal{O}{K,S}).

References

In this case, \ref{seq-Q} is actually a short exact sequence, motivating us to ask in what other cases this is true.

— The Schur multiplier of $\rm{SL}_2$, $K_2$, and Dedekind zeta-functions over $S$-integers  (2609.28677 - Amorim et al., 23 Sep 2026) in Example 3.1, subsection “The group K_2(2, \mathcal{O}_{K,S})”