The homology of SL₂ of discrete valuation rings

2020/07/22 by Hutchinson, Kevin, Mirzaii, Behrooz, Mokari, Fatemeh Yeganeh
#19D99 #20J05 #FOS: Mathematics #K-Theory and Homology (math.KT)

paper · doi:10.48550/arxiv.2007.11159

Abstract

Let A be a discrete valuation ring with field of fractions F and (sufficiently large) residue field k. We prove that there is a natural exact sequence H₃(SL₂(A),ℤ[(1)/(2)]) → H₃(SL₂(F),ℤ[(1)/(2)])→ RP₁(k)[(1)/(2)]→ 0, where RP₁(k) is the refined scissors congruence group of k. Let Γ₀(\mathfrakm_A) denote the congruence subgroup consisting of matrices in SL₂(A) whose lower off-diagonal entry lies in the maximal ideal \mathfrakm_A. We also prove that there is an exact sequence 0→ P(k)[(1)/(2)]→ H₂(Γ₀(\mathfrakm_A),ℤ[(1)/(2)])→ H₂(SL₂(A),ℤ[(1)/(2)])→ I²(k)[(1)/(2)]→ 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.

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