The third homology of SL2 of local rings

We describe the third homology of S L 2 of local rings, over Z 1 2 , 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 H 3 SL 2 A , Z 1 2 → K 3 ind ( A ) 1 2 induces an isomorphism H 3 SL 2 A , Z 1 2 A × ≅ K 3 ind ( A ) 1 2 on coinvariants for the natural action of units A × . We prove that the action of A × on H 3 SL 2 A , Z 1 2 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.