A Hochschild cohomology comparison theorem for prestacks
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- by Wendy Lowen and Michel Van den Bergh PDF
- Trans. Amer. Math. Soc. 363 (2011), 969-986 Request permission
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
- Hans Joachim Baues and Günther Wirsching, Cohomology of small categories, J. Pure Appl. Algebra 38 (1985), no. 2-3, 187–211. MR 814176, DOI 10.1016/0022-4049(85)90008-8
- 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
- Wendy Lowen, Hochschild cohomology of presheaves as map-graded categories, Int. Math. Res. Not. IMRN , posted on (2008), Art. ID rnn118, 32. MR 2449052, DOI 10.1093/imrn/rnn118
- W. Lowen and M. Van den Bergh, A local to global spectral sequence for Hochschild cohomology, in preparation.
- Amnon Neeman, Noncommutative localisation in algebraic $K$-theory. II, Adv. Math. 213 (2007), no. 2, 785–819. MR 2332610, DOI 10.1016/j.aim.2007.01.010
- Amnon Neeman and Andrew Ranicki, Noncommutative localisation in algebraic $K$-theory. I, Geom. Topol. 8 (2004), 1385–1425. MR 2119300, DOI 10.2140/gt.2004.8.1385
- Amnon Neeman, Andrew Ranicki, and Aidan Schofield, Representations of algebras as universal localizations, Math. Proc. Cambridge Philos. Soc. 136 (2004), no. 1, 105–117. MR 2034017, DOI 10.1017/S030500410300700X
- A. H. Schofield, Representation of rings over skew fields, London Mathematical Society Lecture Note Series, vol. 92, Cambridge University Press, Cambridge, 1985. MR 800853, DOI 10.1017/CBO9780511661914
- Angelo 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
- Charles Weibel, Cyclic homology for schemes, Proc. Amer. Math. Soc. 124 (1996), no. 6, 1655–1662. MR 1277141, DOI 10.1090/S0002-9939-96-02913-9
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
- MR Author ID: 176980
- Email: michel.vandenbergh@uhasselt.be
- 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 - © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2728592