A1-homotopy invariants of dg orbit categories

TL;DR

This paper develops a framework to compute A1-homotopy invariants of dg orbit categories, providing explicit formulas and applications to cluster categories, Kleinian singularities, and Fourier-Mukai autoequivalences.

Contribution

It introduces a method to express A1-homotopy invariants of dg orbit categories as cones of endomorphisms, extending fundamental theorems and enabling new computations.

Findings

Computed A1-homotopy invariants of cluster dg categories using Coxeter matrices

Determined homotopy K-theory and cyclic homology of dg orbit categories from Fourier-Mukai autoequivalences

Established a distinguished triangle relating invariants of A and A/F

Abstract

Let A be a dg category, F:A->A a dg functor inducing an equivalence of categories in degree-zero cohomology, and A/F the associated dg orbit category. For every A1-homotopy invariant (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 A1-homotopy invariants of cluster (dg) categories, and consequently of Kleinian singularities, using solely the Coxeter matrix. As a second application, we compute the homotopy K-theory and periodic cyclic homology of the dg orbit categories associated to Fourier-Mukai autoequivalences.