关于$S$-整数上$\mathrm{SL}_2$的同余子群的阿贝尔化
On the abelianization of congruence subgroups of $\mathrm{SL}_2$ over $S$-integers
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中文总结 AI 辅助
本研究计算了局部环及算术型戴德金环上$\mathrm{SL}_2$同余子群的一阶整同调（即阿贝尔化），明确其结构并推导二阶上同挠子群，为后续相关研究提供关键支撑。
中文摘要 AI 辅助
在本研究中，我们计算了同余子群$Γ(A, \mathfrak{m}_A), Γ_1(A, \mathfrak{m}_A)$与$Γ_0(A, \mathfrak{m}_A)$的一阶整同调（即阿贝尔化），其中$A$是含极大理想$\mathfrak{m}_A$的局部环，证明了$H_1(Γ(A, \mathfrak{m}_A), \mathbb{Z})$同构于$\mathfrak{sl}_2(\mathfrak{m}_A/\mathfrak{m}_A^2)$的加法群。随后我们利用这些结果确定了群$H_1(Γ(\mathcal{O}_{K, S}, \mathfrak{p}), \mathbb{Z})$、$H_1(Γ_1(\mathcal{O}_{K, S}, \mathfrak{p}), \mathbb{Z})$与$H_1(Γ_0(\mathcal{O}_{K, S}, \mathfrak{p}), \mathbb{Z})$的结构，其中$\mathcal{O}_{K,S}$是算术型戴德金环且非全虚，$|S| \geq 2$，$\mathfrak{p}$是非零素理想。相关计算通过剩余域$κ(\mathfrak{p})$与已知的$H_1(\mathrm{SL}_2(\mathcal{O}_{K, S}), \mathbb{Z})$给出。作为推论，我们还得到了它们二阶整上同调的挠子群。这些结果对一项关于$H_2(\mathrm{SL}_2(\mathcal{O}_{K,S}), \mathbb{Z})$的即将开展的研究至关重要。
英文摘要
In this work, we compute the first integral homology, or abelianization, of the congruence subgroups $Γ(A, \mathfrak{m}_A), Γ_1(A, \mathfrak{m}_A)$, and $Γ_0(A, \mathfrak{m}_A)$ for a local ring $A$ with maximal ideal $\mathfrak{m}_A$, showing that $H_1(Γ(A, \mathfrak{m}_A), \mathbb{Z})$ is isomorphic to the additive group of $\mathfrak{sl}_2(\mathfrak{m}_A/\mathfrak{m}_A^2)$. We then use these results to determine the structure of the groups $H_1(Γ(\mathcal{O}_{K, S}, \mathfrak{p}), \mathbb{Z})$, $H_1(Γ_1(\mathcal{O}_{K, S}, \mathfrak{p}), \mathbb{Z})$ and $H_1(Γ_0(\mathcal{O}_{K, S}, \mathfrak{p}), \mathbb{Z})$, where $\mathcal{O}_{K,S}$ is a Dedekind domain of arithmetic type, not totally imaginary, $|S| \geq 2$, and $\mathfrak{p}$ is a nonzero prime ideal. The computations are given in terms of the residue field $κ(\mathfrak{p})$ and the known $H_1(\mathrm{SL}_2(\mathcal{O}_{K, S}), \mathbb{Z})$. As a consequence, we also obtain the torsion subgroup of their second integral cohomology. These results will be of paramount importance for a forthcoming work concerning $H_2(\mathrm{SL}_2(\mathcal{O}_{K,S}), \mathbb{Z})$.
