The algebraic $K$-theory of the projective line associated with a strongly $\mathbb{Z}$-graded ring
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2018
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| _version_ | 1866910240097697792 |
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| author | Huettemann, Thomas Montgomery, Tasha |
| author_facet | Huettemann, Thomas Montgomery, Tasha |
| contents | A Laurent polynomial ring $A[t,1/t]$ with coefficients in a unital ring $A$ determines a category of quasi-coherent sheaves on the projective line over $A$; its $K$-theory is known to split into a direct sum of two copies of the $K$-theory of $A$. In this paper, the result is generalised to the case of an arbitrary strongly $\mathbb{Z}$-graded ring $R$ in place of the Laurent polynomial ring. The projective line associated with $R$ is indirectly defined by specifying the corresponding category of quasi-coherent sheaves. Notions from algebraic geometry like sheaf cohomology and twisting sheaves are transferred to the new setting, and the $K$-theoretical splitting is established. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1810_06272 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | The algebraic $K$-theory of the projective line associated with a strongly $\mathbb{Z}$-graded ring Huettemann, Thomas Montgomery, Tasha K-Theory and Homology Rings and Algebras 19D55, 16W50, 19E99 A Laurent polynomial ring $A[t,1/t]$ with coefficients in a unital ring $A$ determines a category of quasi-coherent sheaves on the projective line over $A$; its $K$-theory is known to split into a direct sum of two copies of the $K$-theory of $A$. In this paper, the result is generalised to the case of an arbitrary strongly $\mathbb{Z}$-graded ring $R$ in place of the Laurent polynomial ring. The projective line associated with $R$ is indirectly defined by specifying the corresponding category of quasi-coherent sheaves. Notions from algebraic geometry like sheaf cohomology and twisting sheaves are transferred to the new setting, and the $K$-theoretical splitting is established. |
| title | The algebraic $K$-theory of the projective line associated with a strongly $\mathbb{Z}$-graded ring |
| topic | K-Theory and Homology Rings and Algebras 19D55, 16W50, 19E99 |
| url | https://arxiv.org/abs/1810.06272 |