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Chain complexes and stable categories. (English) Zbl 0753.18005
Supposing \(\mathcal A\) is an exact category with enough injectives, for any full additive subcategory \(\mathcal X\) of the stable category \(\underline{\mathcal A}\) one constructs an \(S\)-functor \({\mathcal H}_{0]}{\mathcal X}\to\underline{\mathcal A}\) extending the inclusion. (Here \({\mathcal H}_{0]}{\mathcal X}\) is the full subcategory of the homotopy category \(\mathcal H\mathcal X\) consisting of the positive chain complexes.) Applications to the case of some specific categories whose objects are obtained from finitely (countably) generated modules over a finite-dimensional \(k\)-algebra are considered.

MSC:
18E30 Derived categories, triangulated categories (MSC2010)
18G35 Chain complexes (category-theoretic aspects), dg categories
18G10 Resolutions; derived functors (category-theoretic aspects)
18E10 Abelian categories, Grothendieck categories
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References:
[1] A. A. Beilinson,Coherent sheaves on P n and problems of linear algebra, Funct. anal. and appl., Vol.12, 1979, 214–216 · Zbl 0424.14003
[2] N. Bourbaki,Algèbre Commutative, Hermann, Paris, 1961 · Zbl 0108.04002
[3] H. Cartan, S. Eilenberg,Homological algebra, Princeton University Press, 1956
[4] P. Freyd,Abelian Categories, Harper & Row, New York, 1964 · Zbl 0121.02103
[5] P. Gabriel,Sur les catégories abéliennes, Bull. Soc. Math. France,90, 1962, 323–448
[6] P. Gabriel,The universal cover of a representation-finite algebra, Representations of algebras, Springer LNM903, 1981, 68–105
[7] P. Gabriel, A. V. Roiter,Representation theory, to appear
[8] A. Grothendieck,Sur quelques points d’algèbre homologique, Tôhoku Math. Journal,9, 1957, 119–221 · Zbl 0118.26104
[9] A. Grothendieck,Eléments de Géométrie Algébrique III, Etude cohomologique des faisceaux cohérents, Publ. Math. IHES,11, 1961
[10] A. Grothendieck, J.L. Verdier,Préfaisceaux=Exposé I in SGA 4: Théorie des Topos et Cohomologie Etale des Schémas, Springer LNM269, 1974
[11] D. Happel,Triangulated Categories in the Representation Theory of Finite Dimensional Algebras, London Math. Soc. Lecture Note Series,119, 1988 · Zbl 0635.16017
[12] D. Happel,On the derived Category of a finite-dimensional Algebra, Comment. Math. Helv.,62, 1987, 339–389 · Zbl 0626.16008
[13] R. Hartshorne,Residues and Duality, Springer LNM20, 1966
[14] A. Heller,The loop-space functor in homological algebra, Trans. Amer. Math. Soc.,96, 1960, 382–394 · Zbl 0096.25502
[15] B. Keller, D. Vossieck,Sous les catégories dérivées, C. R. Acad. Sci. Paris,305, Série I, 1987, 225–228 · Zbl 0628.18003
[16] D. Quillen,Higher Algebraic K-theory I, Springer LNM341, 1973, 85–147 · Zbl 0292.18004
[17] J. Rickard,Morita theory for Derived Categories, Journal of the London Math. Soc.,39, 1989, 436–456 · Zbl 0672.16034
[18] J.-E. Roos,Sur les foncteurs dérivés de \(\underleftarrow {\lim }\) .Applications, C. R. Acad. Sci. Paris,252, Série I, 1961, 3702–3704 · Zbl 0102.02501
[19] J.-L. Verdier,Catégories dérivées, état O, SGA 4 1/2, Springer LNM569, 1977, 262–311
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