The "fundamental theorem" for the algebraic $K$theory of strongly $\mathbb{Z}$graded rings
Abstract
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 MayerVietoris and localisation sequences are established.
 Publication:

arXiv eprints
 Pub Date:
 March 2020
 DOI:
 10.48550/arXiv.2003.01506
 arXiv:
 arXiv:2003.01506
 Bibcode:
 2020arXiv200301506H
 Keywords:

 Mathematics  KTheory and Homology;
 Mathematics  Rings and Algebras;
 Primary 19D50;
 Secondary 19D35 16E20 18G35
 EPrint:
 35 pages