The 'fundamental theorem' for the algebraic $K$-theory of strongly $\mathbb{Z}$-graded rings
| Autor: | Huettemann, Thomas |
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| Rok vydání: | 2020 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 35 pages; v2: 36 pages, minor changes |
| Databáze: | arXiv |
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