Deriving DG categories.(English)Zbl 0799.18007
The author investigates the derived category of a differential \(\mathbb{Z}\)- graded category.
As applications:
1. a “triangulated analogue” of a theorem ofP. Freyd [“Abelian Categories”, Harper & Row (1964; Zbl 0121.021)] andP. Gabriel [Bull. Soc. Math. France 90, 323-448 (1962; Zbl 0201.356)] characterizing module categories among abelian categories is deduced;
2. a “Morita theorem” is proved and a formalism for Koszul duality is developed.
As applications:
1. a “triangulated analogue” of a theorem ofP. Freyd [“Abelian Categories”, Harper & Row (1964; Zbl 0121.021)] andP. Gabriel [Bull. Soc. Math. France 90, 323-448 (1962; Zbl 0201.356)] characterizing module categories among abelian categories is deduced;
2. a “Morita theorem” is proved and a formalism for Koszul duality is developed.
Reviewer: M.Golasiński (Toruń)
MSC:
| 18E30 | Derived categories, triangulated categories (MSC2010) |
| 16D90 | Module categories in associative algebras |
| 13D03 | (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) |
Cite
References:
| [1] | A. A. BEILINSON , V. GINSBURG and V. A. SCHECHTMAN , Koszul duality (Journal of Geometry and Physics, Vol. 5, 1988 , pp. 317-350). MR 91c:18011 | Zbl 0695.14009 ·Zbl 0695.14009 ·doi:10.1016/0393-0440(88)90028-9 |
| [2] | A. A. BEILINSON V. GINSBURG and W. SOERGEL , Koszul Duality Patterns in Representation Theory , preprint, 1991 . |
| [3] | E. H. BROWN , Cohomology Theories (Ann. of Math., Vol. 75, 1962 , pp. 467-484). MR 25 #1551 | Zbl 0101.40603 ·Zbl 0101.40603 ·doi:10.2307/1970209 |
| [4] | H. CARTAN and S. EILENBERG , Homological Algebra , Princeton University Press, 1956 . MR 17,1040e | Zbl 0075.24305 ·Zbl 0075.24305 |
| [5] | P. FREYD , Abelian Categories , Harper, 1966 . MR 35 #231 | Zbl 0121.02103 ·Zbl 0121.02103 |
| [6] | P. GABRIEL , Des catégories abéliennes (Bull. Soc. Math. France, Vol. 90, 1962 , pp. 323-448). Numdam | MR 38 #1144 | Zbl 0201.35602 ·Zbl 0201.35602 |
| [7] | P. GABRIEL and A. V. ROITER , Representations of Finite-dimensional Algebras, Encyclopaedia of Mathematical Sciences , Vol. 73, Springer, 1992 . MR 94h:16001b | Zbl 0839.16001 ·Zbl 0839.16001 |
| [8] | A. GROTHENDIECK , Éléments de Géométrie algébrique, III, Étude cohomologique des faisceaux cohérents (Publ. Math. IHES, Vol. 11, 1961 ). Numdam | Zbl 0118.36206 ·Zbl 0118.36206 ·doi:10.1007/BF02684778 |
| [9] | D. HAPPEL , On the derived Category of a Finite-dimensional Algebra (Comment. Math. Helv., Vol. 62, 1987 , pp. 339-389). MR 89c:16029 | Zbl 0626.16008 ·Zbl 0626.16008 ·doi:10.1007/BF02564452 |
| [10] | G. HOCHSCHILD , B. KOSTANT and A. ROSENBERG , Differential Forms on regular affine Algebras (Trans. Amer. Math. Soc., Vol. 102, 1962 , pp. 383-408). MR 26 #167 | Zbl 0102.27701 ·Zbl 0102.27701 ·doi:10.2307/1993614 |
| [11] | L. ILLUSIE , Complexe cotangent et déformations II (Springer LNM, Vol. 283, 1972 ). MR 58 #10886b | Zbl 0238.13017 ·Zbl 0238.13017 ·doi:10.1007/BFb0059573 |
| [12] | B. KELLER , Chain Complexes and Stable Categories (Manus. Math., Vol. 67, 1990 , pp. 379-417). Article | MR 91h:18006 | Zbl 0753.18005 ·Zbl 0753.18005 ·doi:10.1007/BF02568439 |
| [13] | B. KELLER , A Remark on Tilting Theory and DG algebras (Manus. Math., Vol. 79, 1993 , pp. 247-252). Article | MR 94d:18016 | Zbl 0810.16006 ·Zbl 0810.16006 ·doi:10.1007/BF02568343 |
| [14] | B. KELLER and D. VOSSIECK , Sous les catégories dérivées (C. R. Acad. Sci. Paris, Vol. 305, Série I, 1987 , pp. 225-228). MR 88m:18014 | Zbl 0628.18003 ·Zbl 0628.18003 |
| [15] | S. MACLANE , Homology , Springer-Verlag, 1963 . MR 28 #122 | Zbl 0133.26502 ·Zbl 0133.26502 |
| [16] | D. QUILLEN , Higher Algebraic K-theory I (Springer LNM, Vol. 341, 1973 , pp. 85-147). MR 49 #2895 | Zbl 0292.18004 ·Zbl 0292.18004 |
| [17] | A. NEEMAN , The Connection between the K-theory Localization Theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel , Preprint. Numdam | Zbl 0868.19001 ·Zbl 0868.19001 |
| [18] | D. C. RAVENEL , Localization with Respect to Certain Periodic Homology Theories (Amer J. of Math., Vol. 106, 1984 , pp. 351-414). MR 85k:55009 | Zbl 0586.55003 ·Zbl 0586.55003 ·doi:10.2307/2374308 |
| [19] | J. RICKARD , Morita Theory for Derived Categories (Journal of the London Math. Soc., 39, 1989 , 436-456). MR 91b:18012 | Zbl 0642.16034 ·Zbl 0642.16034 ·doi:10.1112/jlms/s2-39.3.436 |
| [20] | J. RICKARD Derived Equivalences as Derived Functors (J. London Math. Soc., Vol. 43, 1991 , pp. 37-48). MR 92b:16043 | Zbl 0683.16030 ·Zbl 0683.16030 ·doi:10.1112/jlms/s2-43.1.37 |
| [21] | G. RINEHART , Differential Forms on General Commutative Algebras (Trans. Amer. Math. Soc., Vol. 108, 1965 , pp. 195-222). MR 27 #4850 | Zbl 0113.26204 ·Zbl 0113.26204 ·doi:10.2307/1993603 |
| [22] | H. TODA , Composition Methods in Homotopy Groups of Spheres , Princeton University Press, 1962 . MR 26 #777 | Zbl 0101.40703 ·Zbl 0101.40703 |
| [23] | J.-L. VERDIER , Catégories dérivées, état 0, SGA 4 1/2 (Springer LNM, Vol. 569, 1977 , pp. 262-311). MR 57 #3132 | Zbl 0407.18008 ·Zbl 0407.18008 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
