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A Hochschild cohomology comparison theorem for prestacks


Authors: Wendy Lowen and Michel Van den Bergh
Journal: Trans. Amer. Math. Soc. 363 (2011), 969-986
MSC (2000): Primary 16E40, 18D30
DOI: https://doi.org/10.1090/S0002-9947-2010-05288-2
Published electronically: September 21, 2010
MathSciNet review: 2728592
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Abstract: We generalize and clarify Gerstenhaber and Schack's ``Special Cohomology Comparison Theorem''. More specifically we obtain a fully faithful functor between the derived categories of bimodules over a prestack over a small category $ \mathcal{U}$ and the derived category of bimodules over its corresponding fibered category. In contrast to Gerstenhaber and Schack we do not have to assume that $ \mathcal{U}$ is a poset.


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

  • 1. H. J. Baues and G. Wirsching, Cohomology of small categories, J. Pure Appl. Algebra 38 (1985), no. 2-3, 187-211. MR 814176 (87g:18013)
  • 2. 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 (85d:16021)
  • 3. -, 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 (90c:16016)
  • 4. -, 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 (89k:16052)
  • 5. W. Lowen, Hochschild cohomology of presheaves as map-graded categories, Internat. Math. Res. Notices 118 (2008), 32pp. MR 2449052 (2009i:18015)
  • 6. W. Lowen and M. Van den Bergh, A local to global spectral sequence for Hochschild cohomology, in preparation.
  • 7. A. Neeman, Noncommutative localisation in algebraic $ K$-theory. II, Adv. Math. 213 (2007), no. 2, 785-819. MR 2332610 (2008d:19001)
  • 8. A. Neeman and A. Ranicki, Noncommutative localisation in algebraic $ K$-theory. I, Geom. Topol. 8 (2004), 1385-1425 (electronic). MR 2119300 (2005k:19006)
  • 9. A. Neeman, A. Ranicki, and A. Schofield, Representations of algebras as universal localizations, Math. Proc. Cambridge Philos. Soc. 136 (2004), no. 1, 105-117. MR 2034017 (2005b:16031)
  • 10. A. H. Schofield, Representation of rings over skew fields, London Mathematical Society Lecture Note Series, vol. 92, Cambridge University Press, Cambridge, 1985. MR 800853 (87c:16001)
  • 11. A. Vistoli, Grothendieck topologies, fibered categories and descent theory, Fundamental algebraic geometry, Math. Surveys Monogr., vol. 123, Amer. Math. Soc., Providence, RI, 2005, pp. 1-104. MR 2223406
  • 12. C. Weibel, Cyclic homology for schemes, Proc. Amer. Math. Soc. 124 (1996), no. 6, 1655-1662. MR 1277141 (96h:19003)

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Additional Information

Wendy Lowen
Affiliation: Departement Wiskunde-Informatica, University of Antwerpen, Middelheimcampus, Middelheimlaan 1, 2020 Antwerp, Belgium
Email: wendy.lowen@ua.ac.be

Michel Van den Bergh
Affiliation: Department WNI, Hasselt University, Agoralaan, 3590 Diepenbeek, Belgium
Email: michel.vandenbergh@uhasselt.be

DOI: https://doi.org/10.1090/S0002-9947-2010-05288-2
Keywords: Hochschild cohomology, fibered categories, special cohomology comparison theorem
Received by editor(s): May 31, 2009
Published electronically: September 21, 2010
Additional Notes: The first author is a postdoctoral fellow with the Fund of Scientific Research Flanders (FWO)
The second author is a director of research at the FWO
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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