A1-homotopy invariants of dg orbit categories

JOURNAL OF ALGEBRA

卷:434

Article

Tabuada, Goncalo^1,2,3 

[1] MIT, Dept Math, Cambridge, MA 02139 USA

[2] Univ Nova Lisboa, FCT, Dept Matemat, Lisbon, Portugal

[3] Univ Nova Lisboa, FCT, CMA, Lisbon, Portugal

关键词: Dg orbit category;    A(1)-homotopy;    Algebraic K-theory;    Cluster category;    Kleinian singularities;    Fourier-Mukai transform;    Noncommutative algebraic geometry;   

DOI  :  10.1016/j.jalgebra.2015.03.028

来源: Elsevier

【 摘 要 】

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.

【 授权许可】

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