manuscripta mathematica

, Volume 67, Issue 1, pp 379–417 | Cite as

Chain complexes and stable categories

  • Bernhard Keller


Under suitable assumptions, we extend the inclusion of an additive subcategory\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mathcal{X}} \subset \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mathcal{A}}\) (=stable category of an exact category with enough injectives) to anS-functor [15]\(\mathcal{H}_{0]} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mathcal{X}} \to \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mathcal{A}}\), where\(\mathcal{H}_{0]} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mathcal{X}}\) is the homotopy category of chain complexes concentrated in positive degrees. We thereby obtain a new proof for the key result of J. Rickard’s ‘Morita theory for Derived categories’ [17] and a sharpening of a theorem of Happel [12,10.10] on the ‘module-theoretic description’ of the derived category of a finite-dimensional algebra.


Exact Sequence Short Exact Sequence Direct Limit Full Subcategory Abelian Category 
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Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Bernhard Keller
    • 1
  1. 1.ETH-Zentrum, G 28.2ZurichSwitzerland

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