Deformation theory of abelian categories
HTML articles powered by AMS MathViewer
- by Wendy Lowen and Michel Van den Bergh PDF
- Trans. Amer. Math. Soc. 358 (2006), 5441-5483 Request permission
Abstract:
In this paper we develop the basic infinitesimal deformation theory of abelian categories. This theory yields a natural generalization of the well-known deformation theory of algebras developed by Gerstenhaber. As part of our deformation theory we define a notion of flatness for abelian categories. We show that various basic properties are preserved under flat deformations, and we construct several equivalences between deformation problems.References
- Jiří Adámek, Horst Herrlich, and George E. Strecker, Abstract and concrete categories, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1990. The joy of cats; A Wiley-Interscience Publication. MR 1051419
- Théorie des topos et cohomologie étale des schémas. Tome 3, Lecture Notes in Mathematics, Vol. 305, Springer-Verlag, Berlin-New York, 1973 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4); Dirigé par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de P. Deligne et B. Saint-Donat. MR 0354654
- M. Artin and J. J. Zhang, Abstract Hilbert schemes, Algebr. Represent. Theory 4 (2001), no. 4, 305–394. MR 1863391, DOI 10.1023/A:1012006112261
- Francis Borceux, Handbook of categorical algebra. 1, Encyclopedia of Mathematics and its Applications, vol. 50, Cambridge University Press, Cambridge, 1994. Basic category theory. MR 1291599
- N. Bourbaki, Éléments de mathématique. 22. Première partie: Les structures fondamentales de l’analyse. Livre 1: Théorie des ensembles. Chapitre 4: Structures, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1258, Hermann, Paris, 1957 (French). MR 0097335
- M. Gerstenhaber and S. D. Schack, On the deformation of algebra morphisms and diagrams, Trans. Amer. Math. Soc. 279 (1983), no. 1, 1–50. MR 704600, DOI 10.1090/S0002-9947-1983-0704600-5
- Murray Gerstenhaber and Samuel D. Schack, Algebraic cohomology and deformation theory, Deformation theory of algebras and structures and applications (Il Ciocco, 1986) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 247, Kluwer Acad. Publ., Dordrecht, 1988, pp. 11–264. MR 981619, DOI 10.1007/978-94-009-3057-5_{2}
- Murray Gerstenhaber and Samuel D. Schack, The cohomology of presheaves of algebras. I. Presheaves over a partially ordered set, Trans. Amer. Math. Soc. 310 (1988), no. 1, 135–165. MR 965749, DOI 10.1090/S0002-9947-1988-0965749-X
- Murray Gerstenhaber, On the deformation of rings and algebras, Ann. of Math. (2) 79 (1964), 59–103. MR 171807, DOI 10.2307/1970484
- Murray Gerstenhaber, On the deformation of rings and algebras. II, Ann. of Math. 84 (1966), 1–19. MR 0207793, DOI 10.2307/1970528
- Alexander Grothendieck, Sur quelques points d’algèbre homologique, Tohoku Math. J. (2) 9 (1957), 119–221 (French). MR 102537, DOI 10.2748/tmj/1178244839
- Théorie des intersections et théorème de Riemann-Roch, Lecture Notes in Mathematics, Vol. 225, Springer-Verlag, Berlin-New York, 1971 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6); Dirigé par P. Berthelot, A. Grothendieck et L. Illusie. Avec la collaboration de D. Ferrand, J. P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud et J. P. Serre. MR 0354655
- Henning Krause, The spectrum of a module category, Mem. Amer. Math. Soc. 149 (2001), no. 707, x+125. MR 1803703, DOI 10.1090/memo/0707
- Henning Krause, A Brown representability theorem via coherent functors, Topology 41 (2002), no. 4, 853–861. MR 1905842, DOI 10.1016/S0040-9383(01)00010-6
- W. Lowen, A generalization of the Gabriel-Popescu theorem, J. Pure Appl. Algebra 190 (2004), no. 1-3, 197–211. MR 2043328, DOI 10.1016/j.jpaa.2003.11.016
- Wendy Lowen, Obstruction theory for objects in abelian and derived categories, Comm. Algebra 33 (2005), no. 9, 3195–3223. MR 2175388, DOI 10.1081/AGB-200066155
- Wendy Lowen and Michel Van den Bergh, Hochschild cohomology of abelian categories and ringed spaces, Adv. Math. 198 (2005), no. 1, 172–221. MR 2183254, DOI 10.1016/j.aim.2004.11.010
- Saunders MacLane, Categories for the working mathematician, Graduate Texts in Mathematics, Vol. 5, Springer-Verlag, New York-Berlin, 1971. MR 0354798
- Barry Mitchell, Rings with several objects, Advances in Math. 8 (1972), 1–161. MR 294454, DOI 10.1016/0001-8708(72)90002-3
- Amnon Neeman, Triangulated categories, Annals of Mathematics Studies, vol. 148, Princeton University Press, Princeton, NJ, 2001. MR 1812507, DOI 10.1515/9781400837212
- Nicolae Popesco and Pierre Gabriel, Caractérisation des catégories abéliennes avec générateurs et limites inductives exactes, C. R. Acad. Sci. Paris 258 (1964), 4188–4190 (French). MR 166241
- N. Popescu, Abelian categories with applications to rings and modules, London Mathematical Society Monographs, No. 3, Academic Press, London-New York, 1973. MR 0340375
- L. A. Takhtadzhyan, The noncommutative homology of quantum tori, Funktsional. Anal. i Prilozhen. 23 (1989), no. 2, 75–76 (Russian); English transl., Funct. Anal. Appl. 23 (1989), no. 2, 147–149. MR 1011367, DOI 10.1007/BF01078791
Additional Information
- Wendy Lowen
- Affiliation: Departement DWIS, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium
- Email: wlowen@vub.ac.be
- Michel Van den Bergh
- Affiliation: Departement WNI, Limburgs Universitair Centrum, Universitaire Campus, Building D, 3590 Diepenbeek, Belgium
- MR Author ID: 176980
- Email: vdbergh@luc.ac.be
- Received by editor(s): September 24, 2004
- Received by editor(s) in revised form: October 7, 2004
- Published electronically: July 21, 2006
- Additional Notes: The first author is a Postdoctoral Fellow of the Research Foundation–Flanders (FWO)
The second author is a senior researcher at the FWO - © Copyright 2006 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 358 (2006), 5441-5483
- MSC (2000): Primary 13D10, 14A22, 18E15
- DOI: https://doi.org/10.1090/S0002-9947-06-03871-2
- MathSciNet review: 2238922