A^1-homotopy invariants of dg orbit categories

Let A be a dg category, F : A -> A be a dg functor inducing an equivalence of categories in degree-zero cohomology, and A/F be the associated dg orbit category. For every A(1)-homotopy invariant E (e.g. homotopy K-theory, K-theory with coefficients, etale K-theory, and periodic cyclic homology), we construct a distinguished triangle expressing E(A/F) as the cone of the endomorphism E(F) - Id of E(A). In the particular case where F is the identity dg functor, this triangle splits and gives rise to the fundamental theorem. As a first application, we compute the A(1)-homotopy invariants of cluster (dg) categories, and consequently of Kleinian singularities, using solely the Coxeter matrix. As a second application, we compute the A(1)-homotopy invariants of the dg orbit categories associated with Fourier-Mukai autoequivalences. (C) 2015 Elsevier Inc. All rights reserved.