Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Deformation theory of abelian categories


Authors: Wendy Lowen and Michel Van den Bergh
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
Published electronically: July 21, 2006
MathSciNet review: 2238922
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. J. Adámek, H. Herrlich, and G. E. Strecker, Abstract and concrete categories, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1990, The joy of cats, A Wiley-Interscience Publication. MR 1051419 (91h:18001)
  • 2. M. Artin, A. Grothendieck, and J. L. Verdier, Theorie des topos et cohomologie étale des schémas, SGA4, Tome 3, Lecture Notes in Mathematics, vol. 305, Springer-Verlag, 1973. MR 0354654 (50:7132)
  • 3. M. Artin and J. J. Zhang, Abstract Hilbert schemes, Algebr. Represent. Theory 4 (2001), no. 4, 305-394. MR 1863391 (2002h:16046)
  • 4. F. Borceux, Handbook of categorical algebra. 2, Encyclopedia of Mathematics and its Applications, vol. 51, Cambridge University Press, Cambridge, 1994, Categories and structures. MR 1313497 (96g:18001b)
  • 5. N. Bourbaki, Eléments de mathématique. 22. Première partie: Les structures fondamentales de l'analyse. Livre 1: Théorie des ensembles. Chapitre 4: Structures, Actualités Sci. Ind. no. 1258, Hermann, Paris, 1957. MR 0097335 (20:3804)
  • 6. 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 0704600 (85d:16021)
  • 7. -, Algebraic cohomology and deformation theory, Deformation theory of algebras and structures and applications (Il Ciocco, 1986) (Dordrecht), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 247, Kluwer Acad. Publ., Dordrecht, 1988, pp. 11-264. MR 0981619 (90c:16016)
  • 8. -, The cohomology of presheaves of algebras. I. Presheaves over a partially ordered set, Trans. Amer. Math. Soc. 310 (1988), no. 1, 135-165.MR 0965749 (89k:16052)
  • 9. M. Gerstenhaber, On the deformation of rings and algebras, Ann. of Math. (2) 79 (1964), 59-103. MR 0171807 (30:2034)
  • 10. -, On the deformation of rings and algebras. II, Ann. of Math. 84 (1966), 1-19. MR 0207793 (34:7608)
  • 11. A. Grothendieck, Sur quelques points d'algèbre homologiques, Tôhoku Math. J. (2) 9 (1957), 119-221. MR 0102537 (21:1328)
  • 12. L. Illusie, Existence de résolutions globales, SGA6, Lecture Notes in Math., vol. 225, Springer-Verlag, 1971. MR 0354655 (50:7133)
  • 13. H. Krause, The spectrum of a module category, Mem. Amer. Math. Soc. 149 (2001), no. 707, x+125. MR 1803703 (2001k:16010)
  • 14. -, A Brown representability theorem via coherent functors, Topology 41 (2002), no. 4, 853-861. MR 1905842 (2003c:18011)
  • 15. W. Lowen, A generalization of the Gabriel-Popescu theorem, Journal Pure and Appl. Algebra 190 (2004), 197-211. MR 2043328
  • 16. -, Obstruction theory for objects in abelian and derived categories, Communications in Alg. 33 (2005), no. 9, 3195-3223. MR 2175388
  • 17. W. Lowen and M. Van den Bergh, Hochschild cohomology of abelian categories and ringed spaces, Advances in Math. 198 (2005), 172-221. MR 2183254
  • 18. S. MacLane, Categories for the working mathematician, Springer-Verlag, Berlin, 1971. MR 0354798 (50:7275)
  • 19. B. Mitchell, Rings with several objects, Advances in Math. 8 (1972), 1-161. MR 0294454 (45:3524)
  • 20. A. Neeman, Triangulated categories, Annals of Mathematics Studies, vol. 148, Princeton University Press, Princeton, NJ, 2001. MR 1812507 (2001k:18010)
  • 21. N. Popesco and P. Gabriel, Caractérisation des catégories abéliennes avec générateurs et limites inductives exactes, C. R. Acad. Sci. Paris 258 (1964), 4188-4190. MR 0166241 (29:3518)
  • 22. N. Popescu, Abelian categories with applications to rings and modules, Academic Press, London, 1973, London Mathematical Society Monographs, no. 3. MR 0340375 (49:5130)
  • 23. L. A. Takhtadjian, Noncommutive homology of quantum tori, Functional Anal. Appl. 23 (1989), 147-149. MR 1011367 (90m:18015)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 13D10, 14A22, 18E15

Retrieve articles in all journals with MSC (2000): 13D10, 14A22, 18E15


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
Email: vdbergh@luc.ac.be

DOI: https://doi.org/10.1090/S0002-9947-06-03871-2
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
Article copyright: © Copyright 2006 American Mathematical Society

American Mathematical Society