The "fundamental theorem" for the higher 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 groups ${\text{NK}}_q^{\pm }$. 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 actions are trivial in the classical case). The analogues of the groups ${\text{NK}}_q^{\pm }$ are identified with the reduced $K$-theory of homotopy nilpotent twisted endomorphisms, and appropriate versions of Mayer-Vietoris and localisation sequences are established.