A1A1-homotopy invariants of dg orbit categories

Let AA be a dg category, F:A→AF:A→A be a dg functor inducing an equivalence of categories in degree-zero cohomology, and A/FA/F be the associated dg orbit category. For every A1A1-homotopy invariant E (e.g. homotopy K-theory, K-theory with coefficients, étale K  -theory, and periodic cyclic homology), we construct a distinguished triangle expressing E(A/F)E(A/F) as the cone of the endomorphism E(F)−IdE(F)−Id of E(A)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 A1A1-homotopy invariants of cluster (dg) categories, and consequently of Kleinian singularities, using solely the Coxeter matrix. As a second application, we compute the A1A1-homotopy invariants of the dg orbit categories associated with Fourier–Mukai autoequivalences.