A¹-homotopy invariants of dg orbit categories

Abstract

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 A¹-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 )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 A¹-homotopy invariants of cluster (dg) categories, and consequently of Kleinian singularities, using solely the Coxeter matrix. As a second application, we compute the A¹-homotopy invariants of the dg orbit categories associated with Fourier–Mukai autoequivalences.