The third homology of SL2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {SL}_2$$\end{document} of local rings

We describe the third homology of SL2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$SL_2$$\end{document} of local rings, over Z12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}\left[ \tfrac{1}{2}\right] $$\end{document}, in terms of a refined Bloch group. We use this result to elucidate the relationship of this homology group to the indecomposable part of K3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_3$$\end{document} 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 H3SL2A,Z12→K3ind(A)12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {H}_{3}\left( {\text {SL}}_{2}\left( A\right) ,{\mathbb {Z}}\left[ \tfrac{1}{2}\right] \right) \rightarrow {K^{\mathrm { ind}}_3(A)}\left[ \tfrac{1}{2}\right] $$\end{document} induces an isomorphism H3SL2A,Z12A×≅K3ind(A)12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {H}_{3}\left( {\text {SL}}_{2}\left( A\right) ,{\mathbb {Z}}\left[ \tfrac{1}{2}\right] \right) _{A^\times }\cong {K^{\mathrm { ind}}_3(A)}\left[ \tfrac{1}{2}\right] $$\end{document} on coinvariants for the natural action of units A×\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A^\times $$\end{document}. We prove that the action of A×\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A^\times $$\end{document} on H3SL2A,Z12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {H}_{3}\left( {\text {SL}}_{2}\left( A\right) ,{\mathbb {Z}}\left[ \tfrac{1}{2}\right] \right) $$\end{document} 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.