A sheaf of Hochschild complexes on quasi-compact opens
HTML articles powered by AMS MathViewer
- by Wendy Lowen PDF
- Proc. Amer. Math. Soc. 136 (2008), 3045-3050 Request permission
Abstract:
For a scheme $X$, we construct a sheaf $\mathbf {C}$ of complexes on $X$ such that for every quasi-compact open $U \subset X$, $\mathbf {C}(U)$ is quasi-isomorphic to the Hochschild complex of $U$ (Lowen and Van den Bergh, 2005). Since $\mathbf {C}$ is moreover acyclic for taking sections on quasi-compact opens, we obtain a local to global spectral sequence for Hochschild cohomology if $X$ is quasi-compact.References
- Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos, Lecture Notes in Mathematics, Vol. 269, Springer-Verlag, Berlin-New York, 1972 (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 N. Bourbaki, P. Deligne et B. Saint-Donat. MR 0354652
- 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
- E. Getzler and J. D. S. Jones, Operads, homotopy algebra and iterated integrals for double loop spaces, preprint hep-th/9403055.
- Alexander Grothendieck, Sur quelques points d’algèbre homologique, Tohoku Math. J. (2) 9 (1957), 119–221 (French). MR 102537, DOI 10.2748/tmj/1178244839
- Vladimir Hinich, Deformations of sheaves of algebras, Adv. Math. 195 (2005), no. 1, 102–164. MR 2145794, DOI 10.1016/j.aim.2004.07.007
- B. Keller, Derived invariance of higher structures on the Hochschild complex, preprint, http://www.math.jussieu.fr/~keller/publ/dih.pdf.
- Maxim Kontsevich, Deformation quantization of algebraic varieties, Lett. Math. Phys. 56 (2001), no. 3, 271–294. EuroConférence Moshé Flato 2000, Part III (Dijon). MR 1855264, DOI 10.1023/A:1017957408559
- Maxim Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003), no. 3, 157–216. MR 2062626, DOI 10.1023/B:MATH.0000027508.00421.bf
- W. Lowen, Algebroid prestacks and deformations of ringed spaces, Trans. Amer. Math. Soc. 360 (2008), 1631–1660.
- W. Lowen and M. Van den Bergh, A local to global spectral sequence for Hochschild cohomology, in preparation.
- 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
- Wendy Lowen and Michel Van den Bergh, Deformation theory of abelian categories, Trans. Amer. Math. Soc. 358 (2006), no. 12, 5441–5483. MR 2238922, DOI 10.1090/S0002-9947-06-03871-2
- Barry Mitchell, Rings with several objects, Advances in Math. 8 (1972), 1–161. MR 294454, DOI 10.1016/0001-8708(72)90002-3
- Richard G. Swan, Hochschild cohomology of quasiprojective schemes, J. Pure Appl. Algebra 110 (1996), no. 1, 57–80. MR 1390671, DOI 10.1016/0022-4049(95)00091-7
- Michel Van den Bergh, On global deformation quantization in the algebraic case, J. Algebra 315 (2007), no. 1, 326–395. MR 2344349, DOI 10.1016/j.jalgebra.2007.02.012
- Amnon Yekutieli, Deformation quantization in algebraic geometry, Adv. Math. 198 (2005), no. 1, 383–432. MR 2183259, DOI 10.1016/j.aim.2005.06.009
Additional Information
- Wendy Lowen
- Affiliation: Departement DWIS, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium
- Email: wlowen@vub.ac.be
- Received by editor(s): September 18, 2006
- Received by editor(s) in revised form: June 25, 2007, and July 10, 2007
- Published electronically: April 17, 2008
- Additional Notes: The author is a Postdoctoral fellow FWO/CNRS. She acknowledges the hospitality of the Institut de Mathématiques de Jussieu (IMJ) and of the Institut des Hautes Études Scientifiques (IHES) during her postdoctoral fellowship with CNRS
- Communicated by: Ted Chinburg
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 3045-3050
- MSC (2000): Primary 18E15, 18F20
- DOI: https://doi.org/10.1090/S0002-9939-08-09471-9
- MathSciNet review: 2407066