One Decomposition of $K_2$-Group for Certain Quotients over $\mathbb{Z}[G]$ with $G$ a Finite Abelian $p$-Group
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| Format: | Preprint |
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2026
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| _version_ | 1866917293165903872 |
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| author | Zhang, Yakun |
| author_facet | Zhang, Yakun |
| contents | This paper investigates the structure of $K_2$-groups for certain quotient rings of the integral group ring $\mathbb{Z}[G]$. Let $G$ be a finite abelian $p$-group with $p$-rank $r$, let $Γ$ be the maximal $\mathbb{Z}$-order of $\mathbb{Q}[G]$, and let $\widetilde{G}$ denote the sum of all elements of $G$ in the group ring. By employing the framework of Kähler differentials, we first determine that the relative $K$-group $K_2(\mathbb{F}_p[G], (\widetilde{G}))$ is an elementary abelian $p$-group of rank $r$ when $|G|>2$. Building upon this result, we establish an explicit isomorphism for $r > 1$: $$ K_2(\mathbb{Z}[G]/(|G|Γ\cap p\mathbb{Z}[G])) \cong K_2(\mathbb{Z}[G]/|G|Γ) \oplus K_2(\mathbb{F}_p[G], (\widetilde{G})). $$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_14112 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | One Decomposition of $K_2$-Group for Certain Quotients over $\mathbb{Z}[G]$ with $G$ a Finite Abelian $p$-Group Zhang, Yakun K-Theory and Homology 16S34, 19C20, 19C99 This paper investigates the structure of $K_2$-groups for certain quotient rings of the integral group ring $\mathbb{Z}[G]$. Let $G$ be a finite abelian $p$-group with $p$-rank $r$, let $Γ$ be the maximal $\mathbb{Z}$-order of $\mathbb{Q}[G]$, and let $\widetilde{G}$ denote the sum of all elements of $G$ in the group ring. By employing the framework of Kähler differentials, we first determine that the relative $K$-group $K_2(\mathbb{F}_p[G], (\widetilde{G}))$ is an elementary abelian $p$-group of rank $r$ when $|G|>2$. Building upon this result, we establish an explicit isomorphism for $r > 1$: $$ K_2(\mathbb{Z}[G]/(|G|Γ\cap p\mathbb{Z}[G])) \cong K_2(\mathbb{Z}[G]/|G|Γ) \oplus K_2(\mathbb{F}_p[G], (\widetilde{G})). $$ |
| title | One Decomposition of $K_2$-Group for Certain Quotients over $\mathbb{Z}[G]$ with $G$ a Finite Abelian $p$-Group |
| topic | K-Theory and Homology 16S34, 19C20, 19C99 |
| url | https://arxiv.org/abs/2602.14112 |