Computations of $K_2$ for certain $\mathbb{Z}/p^s\mathbb{Z}$-algebras and the extension of Oliver's logarithm
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908832675921920 |
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| author | Zhang, Yakun |
| author_facet | Zhang, Yakun |
| contents | This paper describes the $K$-theory structure for three algebra classes. For cyclic $p$-group rings and truncated polynomial rings over $\mathbb{Z}/p^s\mathbb{Z}$, we determine reduced $K_2$-structures via a common algebraic framework. For abelian $p$-group rings over $\widehat{\mathbb{Z}}_p$, we extend the isomorphism between reduced continuous $K_2$ and the first cyclic homology group to all finite abelian $p$-groups. A constructive proof using a generalized Artin-Hasse map yields an explicit splitting. This isomorphism is realized by extending Oliver's $p$-adic logarithm. We also characterize the map from reduced continuous $K_2$ to reduced linearized $K_2$, clarifying the links between $K_2$, cyclic homology, and Kähler differentials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_00773 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Computations of $K_2$ for certain $\mathbb{Z}/p^s\mathbb{Z}$-algebras and the extension of Oliver's logarithm Zhang, Yakun K-Theory and Homology 19C20, 16S34, 19D55, 13N05 This paper describes the $K$-theory structure for three algebra classes. For cyclic $p$-group rings and truncated polynomial rings over $\mathbb{Z}/p^s\mathbb{Z}$, we determine reduced $K_2$-structures via a common algebraic framework. For abelian $p$-group rings over $\widehat{\mathbb{Z}}_p$, we extend the isomorphism between reduced continuous $K_2$ and the first cyclic homology group to all finite abelian $p$-groups. A constructive proof using a generalized Artin-Hasse map yields an explicit splitting. This isomorphism is realized by extending Oliver's $p$-adic logarithm. We also characterize the map from reduced continuous $K_2$ to reduced linearized $K_2$, clarifying the links between $K_2$, cyclic homology, and Kähler differentials. |
| title | Computations of $K_2$ for certain $\mathbb{Z}/p^s\mathbb{Z}$-algebras and the extension of Oliver's logarithm |
| topic | K-Theory and Homology 19C20, 16S34, 19D55, 13N05 |
| url | https://arxiv.org/abs/2503.00773 |