A^1-homotopy invariants of dg orbit categories

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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.
Original languageEnglish
Pages (from-to)169-192
JournalJournal of Algebra
Publication statusPublished - 2015


  • Dg orbit category
  • A1A1-homotopy
  • Algebraic K-theory
  • Cluster category
  • Kleinian singularities
  • Fourier–Mukai transform
  • Noncommutative algebraic geometry


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