The homology of $\mathrm{SL}_2$ of discrete valuation rings

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 $H_3(\mathrm{SL}_2(A),\mathbb{Z}[\frac{1}{2}]) \to H_3(\mathrm{SL}_2(F),\mathbb{Z}[\frac{1}{2}])\to \mathcal{RP}_1(k)[\frac{1}{2}]\to 0$, where $\mathcal{RP}_1(k)$ is the refined scissors congruence group of $k$. Let $\Gamma_0(\mathfrak{m}_A)$ denote the congruence subgroup consisting of matrices in $\mathrm{SL}_2(A)$ whose lower off-diagonal entry lies in the maximal ideal $\mathfrak{m}_A$. We also prove that there is an exact sequence $0\to \overline{\mathcal{P}}(k)[\frac{1}{2}]\to H_2(\Gamma_0(\mathfrak{m}_A),\mathbb{Z}[\frac{1}{2}])\to H_2(\mathrm{SL}_2(A),\mathbb{Z}[\frac{1}{2}])\to I^2(k)[\frac{1}{2}]\to 0$, where $I^2(k)$ is the second power of the fundamental ideal of the Grothendieck-Witt ring $\mathrm{GW}(k)$ and $\overline{\mathcal{P}}(k)$ is a certain quotient of the scissors congruence group (in the sense of Dupont-Sah) $\mathcal{P}(k)$ of $k$.