The "fundamental theorem" for the algebraic $K$-theory of strongly $\mathbb{Z}$-graded rings
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arXiv
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| Formato: | Preprint |
| Publicado: |
2020
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| _version_ | 1866911701516943360 |
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| author | Huettemann, Thomas |
| author_facet | Huettemann, Thomas |
| contents | The "fundamental theorem" for algebraic $K$-theory expresses the $K$-groups of a Laurent polynomial ring $L[t,t^{-1}]$ as a direct sum of two copies of the $K$-groups of $L$ (with a degree shift in one copy), and certain "nil" groups of $L$. It is shown here that a modified version of this result generalises to strongly $\mathbb{Z}$-graded rings; rather than the algebraic $K$-groups of $L$, the splitting involves groups related to the shift actions on the category of $L$-modules coming from the graded structure. (These action are trivial in the classical case). The nil groups are identified with the reduced $K$-theory of homotopy nilpotent twisted endomorphisms, and analogues of Mayer-Vietoris and localisation sequences are established. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2003_01506 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | The "fundamental theorem" for the algebraic $K$-theory of strongly $\mathbb{Z}$-graded rings Huettemann, Thomas K-Theory and Homology Rings and Algebras Primary 19D50, Secondary 19D35 16E20 18G35 The "fundamental theorem" for algebraic $K$-theory expresses the $K$-groups of a Laurent polynomial ring $L[t,t^{-1}]$ as a direct sum of two copies of the $K$-groups of $L$ (with a degree shift in one copy), and certain "nil" groups of $L$. It is shown here that a modified version of this result generalises to strongly $\mathbb{Z}$-graded rings; rather than the algebraic $K$-groups of $L$, the splitting involves groups related to the shift actions on the category of $L$-modules coming from the graded structure. (These action are trivial in the classical case). The nil groups are identified with the reduced $K$-theory of homotopy nilpotent twisted endomorphisms, and analogues of Mayer-Vietoris and localisation sequences are established. |
| title | The "fundamental theorem" for the algebraic $K$-theory of strongly $\mathbb{Z}$-graded rings |
| topic | K-Theory and Homology Rings and Algebras Primary 19D50, Secondary 19D35 16E20 18G35 |
| url | https://arxiv.org/abs/2003.01506 |