Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. A. Beilinson,Coherent sheaves on P n and problems of linear algebra, Funct. anal. and appl., Vol.12, 1979, 214–216
N. Bourbaki,Algèbre Commutative, Hermann, Paris, 1961
H. Cartan, S. Eilenberg,Homological algebra, Princeton University Press, 1956
P. Freyd,Abelian Categories, Harper & Row, New York, 1964
P. Gabriel,Sur les catégories abéliennes, Bull. Soc. Math. France,90, 1962, 323–448
P. Gabriel,The universal cover of a representation-finite algebra, Representations of algebras, Springer LNM903, 1981, 68–105
P. Gabriel, A. V. Roiter,Representation theory, to appear
A. Grothendieck,Sur quelques points d’algèbre homologique, Tôhoku Math. Journal,9, 1957, 119–221
A. Grothendieck,Eléments de Géométrie Algébrique III, Etude cohomologique des faisceaux cohérents, Publ. Math. IHES,11, 1961
A. Grothendieck, J.L. Verdier,Préfaisceaux=Exposé I in SGA 4: Théorie des Topos et Cohomologie Etale des Schémas, Springer LNM269, 1974
D. Happel,Triangulated Categories in the Representation Theory of Finite Dimensional Algebras, London Math. Soc. Lecture Note Series,119, 1988
D. Happel,On the derived Category of a finite-dimensional Algebra, Comment. Math. Helv.,62, 1987, 339–389
R. Hartshorne,Residues and Duality, Springer LNM20, 1966
A. Heller,The loop-space functor in homological algebra, Trans. Amer. Math. Soc.,96, 1960, 382–394
B. Keller, D. Vossieck,Sous les catégories dérivées, C. R. Acad. Sci. Paris,305, Série I, 1987, 225–228
D. Quillen,Higher Algebraic K-theory I, Springer LNM341, 1973, 85–147
J. Rickard,Morita theory for Derived Categories, Journal of the London Math. Soc.,39, 1989, 436–456
J.-E. Roos,Sur les foncteurs dérivés de \(\underleftarrow {\lim }\).Applications, C. R. Acad. Sci. Paris,252, Série I, 1961, 3702–3704
J.-L. Verdier,Catégories dérivées, état O, SGA 4 1/2, Springer LNM569, 1977, 262–311
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Keller, B. Chain complexes and stable categories. Manuscripta Math 67, 379–417 (1990). https://doi.org/10.1007/BF02568439
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02568439