The "fundamental theorem" for the higher algebraic K-theory of strongly Z-graded rings

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 NK^±_q​. It is shown here that a modified version of this result generalises to strongly 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 NK^±_q​​ are identified with the reduced K-theory of homotopy nilpotent twisted endomorphisms, and appropriate versions of Mayer-Vietoris and localisation sequences are established.