• Corpus ID: 211818072

The "fundamental theorem" for the algebraic $K$-theory of strongly $\mathbb{Z}$-graded rings

@article{Huettemann2020TheT,
  title={The "fundamental theorem" for the algebraic \$K\$-theory of strongly \$\mathbb\{Z\}\$-graded rings},
  author={Thomas Huettemann},
  journal={arXiv: K-Theory and Homology},
  year={2020}
}

• T. Huettemann
• Published 3 March 2020
• Mathematics
• arXiv: K-Theory and Homology

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.