Bloch–Wigner exact sequence for H3(SL2(A), Z) over local domains

Establish, for every local domain A, the exact sequence 0 -> Tor_1(μ(A), μ(A)~) -> H3(SL2(A), Z) -> RB(A) -> 0, where μ(A) is the group of all roots of unity in A, μ(A)~ is the canonical 2-fold extension of μ(A), and RB(A) is the refined Bloch group of A.

Background

The paper develops refined Bloch–Wigner type exact sequences for E2(A) and PE2(A) over semilocal rings under mild hypotheses, generalizing prior results. Coronado and Hutchinson posed a question asking whether a Bloch–Wigner exact sequence holds directly for SL2(A) over local domains.

The authors frame their main results (Theorems 6.2 and 6.5) as steps towards this broader goal and explicitly restate the question for SL2(A). The desired sequence relates the third homology of SL2(A) with the refined Bloch group RB(A), with a Tor term built from the roots of unity in A.