论文标题
$ \ mathrm {sl} _2 $离散估值戒指的同源性
The homology of $\mathrm{SL}_2$ of discrete valuation rings
论文作者
论文摘要
让$ a $是一个离散的估值戒指，带有分数$ f $和（足够大的）残留场$ k $。我们证明，有一个自然的精确序列$ h_3（\ mathrm {sl} _2（a），\ mathbb {z} [\ frac {\ frac {1} {2} {2}]） \ Mathcal {rp} _1（k）[\ frac {1} {2}] \至0 $，其中$ \ Mathcal {rp} _1（k）$是$ k $的精制剪刀一致性组。令$γ_0（\ Mathfrak {M} _a）$表示由$ \ Mathrm {sl} _2 _2（a）$组成的一致性子组，其较低的偏外条目在于最大的理想$ \ mathfrak {m} _a $。我们还证明，有一个精确的序列$ 0 \ to \ editline {\ mathcal {p}}}（k）[\ frac {1} {2} {2} {2}] \ to H_2（γ_0（\ m atrafrak {\ Mathfrak {m} _a），\ Mathbb {Z} h_2（\ mathrm {sl} _2（a），\ mathbb {z} [\ frac {\ frac {1} {2}]） $ \ mathrm {gw}（k）$和$ \ overline {\ mathcal {p}}（k）$是剪刀一致组的某些商（从dupont-sah）$ \ nathcal {p}（p}（p}（k）$ k $ k $）。
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 $Γ_0(\mathfrak{m}_A)$ 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 $0\to \overline{\mathcal{P}}(k)[\frac{1}{2}]\to H_2(Γ_0(\mathfrak{m}_A),\mathbb{Z}[\frac{1}{2}])\to H_2(\mathrm{SL}_2(A),\mathbb{Z}[\frac{1}{2}])\to I^2(k)[\frac{1}{2}]\to 0$, 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 $k$.
