The fundamental theorem via derived Morita invariance, localization, and \(\mathbb A^{1}\)-homotopy invariance (Q2905331)

Let \(E\) be a functor from the category of dg categories to a triangulated category. The author says that \(E\) ``satisfies the fundamental theorem'' if, for any dg category \(A\), NEWLINE\[NEWLINEE(A[t,t^{-1}]) \cong E(A) \oplus \Sigma (E(A)).NEWLINE\]NEWLINE Here \(A[t,t^{-1}]\) is the tensor product of \(A\) with the algebra of Laurent polynomials.NEWLINENEWLINEThe author shows that if \(E\) is derived Morita invariant, localizing, and \(\mathbb A ^{1}\)-homotopy invariant, \(E\) satisfies the fundamental theorem. This result unifies and simplifies the proof of the fundamental theorem for homotopy algebraic \(K\)-theory in \textit{C. A. Weibel} [Contemp. Math. 83, 461--488 (1989; Zbl 0669.18007)] and the proof of the fundamental theorem for periodic cyclic homology in \textit{C. Kassel} [J. Algebra 107, 195--216 (1987; Zbl 0617.16015)]. The author notes applications to quasi-compact, quasi-separated schemes. The author's proof of the fundamental theorem is based on ideas introduced by him in [Int. Math. Res. Not. 2005, No. 53, 3309--3339 (2005; Zbl 1094.18006)].