# 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} }

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…

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We systematically develop a theory of graded semigroups, that is, semigroups S partitioned by groups Γ, in a manner compatible with the multiplication on S . We define a smash product S #Γ, and show…

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