The ``fundamental theorem'' for the higher algebraic \(K\)-theory of strongly \(\mathbb{Z}\)-graded rings (Q2238033)

The ``classical'' fundamental theorem for the higher algebraic \(K\)-theory of rings has been proved by \textit{D. Grayson} [Lect. Notes Math. 551, 217--240 (1976; Zbl 0362.18015)]. It has been extended from the \(K\)-theory of rings to the \(K\)-theory of schemes by \textit{R. W. Thomason} and \textit{T. Trobaugh} [Prog. Math. 88, 247--435 (1990; Zbl 0731.14001)], and to the algebraic \(K\)-theory of spaces by \textit{T. Hüttemann} et al. [J. Pure Appl. Algebra 160, No. 1, 21--52 (2001; Zbl 0982.19001)]. More recently, the result has been established by \textit{W. Lück} and \textit{W. Steimle} [Forum Math. 28, No. 1, 129--174 (2016; Zbl 1338.19003)] for skew Laurent extensions of additive categories. \par The ``fundamental theorem'' for algebraic \(K\)-theory, also know as the Bass-Heller-Swan formula, 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 \(L\) and certain groups \(NK^\pm_q\). The present paper goes beyond previous generalisations and the author shows 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. The analogues of the groups \(NK^\pm_q\) are identified with the reduced \(K\)-theory of homotopy nilpotent twisted endomorphisms, and appropriate versions of Mayer-Vietoris and localisation sequences are established.