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 Mayer-Vietoris and localisation sequences are established.

35 pages; v2: 36 pages, minor changes

References in corpus (3)

• Non-commutative localisation and finite domination over strongly Z-graded rings
• A Fundamental Theorem for the $K$-theory of connective $\mathbb{S}$-algebras
• The algebraic $K$-theory of the projective line associated with a strongly $\mathbb{Z}$-graded ring