A¹-homotopy invariants of dg orbit categories
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Tabuada_A1-homotopy.pdf
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Author(s)
Trigo Neri Tabuada, Goncalo Jorge
Date Issued
April 2015
Journal
Journal of Algebra
Publisher
Elsevier
Citation
Tabuada, Gonçalo. “A¹-homotopy invariants of dg orbit categories.” Journal of Algebra 434 (July 2015): 169–192.
Version
Original manuscript
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution-NonCommercial-NoDerivs License
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DOI of Published Version
https://doi.org/10.1016/j.jalgebra.2015.03.028
