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On the construction of triangle equivalences

Lectures given at the Workshop on Derived Equivalences (Pappenheim 1994)

Part of the Lecture Notes in Mathematics book series (LNM,volume 1685)

Abstract

We give a self-contained account of an alternative proof of J. Rickard's Morita-theorem for derived categories [135] and his theorem on the realization of derived equivalences as derived functors [139]. To this end, we first review the basic facts on unbounded derived categories (complexes unbounded to the right and to the left) and on derived functors between such categories (cf. [159], [19]). We then extend the formalism of derived categories to differential graded algebras (cf. [77]). This allows us to write down a formula for a bimodule complex given a tilting complex. We then deduce J. Rickard's results.

As a second application of the differential graded algebra techniques, we prove a structure theorem for stable categories admitting infinite sums and a small generator. This yields a natural construction of D. Happel's equivalence [63] between the derived category of a finite-dimensional algebra and the stable category of the associated repetitive algebra.

Finally, we use differential graded algebras to show that cyclic homology is preserved by derived equivalences (following [93]).

Keywords

  • Simple Complex
  • Direct Summand
  • Triangulate Category
  • Homotopy Category
  • Cyclic Homology

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 1998 Springer-Verlag

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Keller, B. (1998). On the construction of triangle equivalences. In: Derived Equivalences for Group Rings. Lecture Notes in Mathematics, vol 1685. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0096374

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  • DOI: https://doi.org/10.1007/BFb0096374

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-64311-1

  • Online ISBN: 978-3-540-69748-0

  • eBook Packages: Springer Book Archive