The homology of SL2 of discrete valuation rings

Let A be a discrete valuation ring with field of fractions F and residue field k such that |k| =&NOTEQUexpressionL;2, 3, 4, 5, 7, 8, 9, 16, 27, 32, 64. We prove that there is a natural exact sequence & nbsp;H-3(SL2(A), Z[1/2]) -> H-3(SL2(F), Z[1/2]) ->& nbsp; RP1(k)[1/2] -> 0,& nbsp;& nbsp;where RP1(k) is the refined scissors congruence group of k. Let gamma(0)(m(A)) denote the congruence subgroup consisting of matrices in SL2(A) whose lower off-diagonal entry lies in the maximal ideal m(A). We also prove that there is an exact sequence & nbsp;0 & nbsp; -> (P) over bar (k)[1/2] -> H-2(gamma(0)(m(A)), Z[1/2]) ->& nbsp; H-2(SL2(A), Z[1/2]) -> I-2(k)[1/2]& nbsp;-> 0,& nbsp;& nbsp;where I-2(k) is the second power of the fundamental ideal of the Grothendieck-Witt ring GW(k) and (P) over bar(k) is a certain quotient of the scissors congruence group (in the sense of Dupont-Sah) P(k) of k. (C) 2022 Elsevier Inc. All rights reserved.