The homology of SL2 of discrete valuation rings

Abstract

Let A be a discrete valuation ring with field of fractions F and residue field k such that |k|≠2,3,4,5,7,8,9,16,27,32,64. We prove that there is a natural exact sequenceH3(SL2(A),Z[12])→H3(SL2(F),Z[12])→RP1(k)[12]→0, where RP1(k) is the refined scissors congruence group of k. Let Γ0(mA) denote the congruence subgroup consisting of matrices in SL2(A) whose lower off-diagonal entry lies in the maximal ideal mA. We also prove that there is an exact sequence0→P‾(k)[12]→H2(Γ0(mA),Z[12])→H2(SL2(A),Z[12])→I2(k)[12]→0, where I2(k) is the second power of the fundamental ideal of the Grothendieck-Witt ring GW(k) and P‾(k) is a certain quotient of the scissors congruence group (in the sense of Dupont-Sah) P(k) of k.