The third homology of $$\mathrm {SL}_2$$ of local rings

Abstract

We describe the third homology of \(SL_2\) of local rings, over \(\mathbb {Z}\left[ \tfrac{1}{2}\right] \), in terms of a refined Bloch group. We use this result to elucidate the relationship of this homology group to the indecomposable part of \(K_3\) of the ring, extending and generalizing recent results in the case of fields. In particular, we prove that if A is a local domain with sufficiently large (possibly finite) residue field then the natural map \(\mathrm {H}_{3}\left( {\text {SL}}_{2}\left( A\right) ,{\mathbb {Z}}\left[ \tfrac{1}{2}\right] \right) \rightarrow {K^{\mathrm { ind}}_3(A)}\left[ \tfrac{1}{2}\right] \) induces an isomorphism \(\mathrm {H}_{3}\left( {\text {SL}}_{2}\left( A\right) ,{\mathbb {Z}}\left[ \tfrac{1}{2}\right] \right) _{A^\times }\cong {K^{\mathrm { ind}}_3(A)}\left[ \tfrac{1}{2}\right] \) on coinvariants for the natural action of units \(A^\times \). We prove that the action of \(A^\times \) on \(\mathrm {H}_{3}\left( {\text {SL}}_{2}\left( A\right) ,{\mathbb {Z}}\left[ \tfrac{1}{2}\right] \right) \) is trivial when A is a complete discrete valuation ring with finite residue field of odd characteristic, and we show by example that this action is non-trivial for certain complete discrete valuation rings with infinite residue field.