Gerstenhaber algebra and Deligne’s conjecture on the Tate–Hochschild cohomology
HTML articles powered by AMS MathViewer
- by Zhengfang Wang PDF
- Trans. Amer. Math. Soc. 374 (2021), 4537-4577 Request permission
Abstract:
Using noncommutative differential forms, we construct a complex called the singular Hochschild cochain complex for any associative algebra over a field. The cohomology of this complex is isomorphic to the Tate–Hochschild cohomology in the sense of Buchweitz. By a natural action of the cellular chain operad of the spineless cacti operad, introduced by R. Kaufmann, on the singular Hochschild cochain complex, we provide a proof of the Deligne conjecture for this complex. More concretely, the complex is an algebra over the (dg) operad of singular chains of the little $2$-discs operad. By this action, we also obtain that the singular Hochschild cochain complex has a $B_{\infty }$-algebra structure and its cohomology ring is a Gerstenhaber algebra.
Inspired by the original definition of Tate cohomology for finite groups, we define a generalized Tate–Hochschild complex with the Hochschild chains in negative degrees and the Hochschild cochains in nonnegative degrees. There is a natural embedding of this complex into the singular Hochschild cochain complex. In the case of a self-injective algebra, this embedding becomes a quasi-isomorphism. In particular, for a symmetric algebra, this allows us to show that the Tate–Hochschild cohomology ring, equipped with the Gerstenhaber algebra structure, is a Batalin–Vilkovisky algebra.
References
- Hossein Abbaspour, On the Hochschild homology of open Frobenius algebras, J. Noncommut. Geom. 10 (2016), no. 2, 709–743. MR 3519050, DOI 10.4171/JNCG/246
- Luchezar L. Avramov and Oana Veliche, Stable cohomology over local rings, Adv. Math. 213 (2007), no. 1, 93–139. MR 2331239, DOI 10.1016/j.aim.2006.11.012
- H. J. Baues, The double bar and cobar constructions, Compositio Math. 43 (1981), no. 3, 331–341. MR 632433
- Petter Andreas Bergh and David A. Jorgensen, Tate-Hochschild homology and cohomology of Frobenius algebras, J. Noncommut. Geom. 7 (2013), no. 4, 907–937. MR 3148613, DOI 10.4171/JNCG/139
- Apostolos Beligiannis, The homological theory of contravariantly finite subcategories: Auslander-Buchweitz contexts, Gorenstein categories and (co-)stabilization, Comm. Algebra 28 (2000), no. 10, 4547–4596. MR 1780017, DOI 10.1080/00927870008827105
- Michel Broué, Higman’s criterion revisited, Michigan Math. J. 58 (2009), no. 1, 125–179. MR 2526081, DOI 10.1307/mmj/1242071686
- R. O. Buchweitz, Maximal Cohen-Macaulay modules and Tate-cohomology over Gorenstein rings, preprint, 1986.
- Buenos Aires Cyclic Homology Group, Hochschild and cyclic homology of hypersurfaces, Adv. Math. 95 (1992), no. 1, 18–60. MR 1176152, DOI 10.1016/0001-8708(92)90043-K
- Alberto Canonaco and Paolo Stellari, A tour about existence and uniqueness of dg enhancements and lifts, J. Geom. Phys. 122 (2017), 28–52. MR 3713870, DOI 10.1016/j.geomphys.2016.11.030
- X. W. Chen, H. H. Li, and Z. F. Wang, The Hochschild cohomology of Leavitt path algebras and Tate–Hochschild cohomology, to appear.
- M. Chas and D. Sullivan, String topology, arXiv:9911159.
- Kai Cieliebak, Urs Frauenfelder, and Alexandru Oancea, Rabinowitz Floer homology and symplectic homology, Ann. Sci. Éc. Norm. Supér. (4) 43 (2010), no. 6, 957–1015 (English, with English and French summaries). MR 2778453, DOI 10.24033/asens.2137
- Frederick R. Cohen, Thomas J. Lada, and J. Peter May, The homology of iterated loop spaces, Lecture Notes in Mathematics, Vol. 533, Springer-Verlag, Berlin-New York, 1976. MR 0436146
- Joachim Cuntz and Daniel Quillen, Algebra extensions and nonsingularity, J. Amer. Math. Soc. 8 (1995), no. 2, 251–289. MR 1303029, DOI 10.1090/S0894-0347-1995-1303029-0
- Tobias Dyckerhoff, Compact generators in categories of matrix factorizations, Duke Math. J. 159 (2011), no. 2, 223–274. MR 2824483, DOI 10.1215/00127094-1415869
- Ching-Hwa Eu and Travis Schedler, Calabi-Yau Frobenius algebras, J. Algebra 321 (2009), no. 3, 774–815. MR 2488552, DOI 10.1016/j.jalgebra.2008.11.003
- Murray Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math. (2) 78 (1963), 267–288. MR 161898, DOI 10.2307/1970343
- Ezra Getzler, Cartan homotopy formulas and the Gauss-Manin connection in cyclic homology, Quantum deformations of algebras and their representations (Ramat-Gan, 1991/1992; Rehovot, 1991/1992) Israel Math. Conf. Proc., vol. 7, Bar-Ilan Univ., Ramat Gan, 1993, pp. 65–78. MR 1261901
- E. Getzler, Batalin-Vilkovisky algebras and two-dimensional topological field theories, Comm. Math. Phys. 159 (1994), no. 2, 265–285. MR 1256989
- A. A. Voronov and M. Gerstenkhaber, Higher-order operations on the Hochschild complex, Funktsional. Anal. i Prilozhen. 29 (1995), no. 1, 1–6, 96 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 29 (1995), no. 1, 1–5. MR 1328534, DOI 10.1007/BF01077036
- V. Ginzburg, Lectures on noncommutative geometry, arXiv:0506603.
- François Goichot, Homologie de Tate-Vogel équivariante, J. Pure Appl. Algebra 82 (1992), no. 1, 39–64 (French, with English summary). MR 1181092, DOI 10.1016/0022-4049(92)90009-5
- G. Hochschild, On the cohomology groups of an associative algebra, Ann. of Math. (2) 46 (1945), 58–67. MR 11076, DOI 10.2307/1969145
- André Joyal and Ross Street, The geometry of tensor calculus. I, Adv. Math. 88 (1991), no. 1, 55–112. MR 1113284, DOI 10.1016/0001-8708(91)90003-P
- T. V. Kadeishvili, The structure of the $A(\infty )$-algebra, and the Hochschild and Harrison cohomologies, Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 91 (1988), 19–27 (Russian, with English summary). MR 1029003
- Ralph M. Kaufmann, On several varieties of cacti and their relations, Algebr. Geom. Topol. 5 (2005), 237–300. MR 2135554, DOI 10.2140/agt.2005.5.237
- Ralph M. Kaufmann, On spineless cacti, Deligne’s conjecture and Connes-Kreimer’s Hopf algebra, Topology 46 (2007), no. 1, 39–88. MR 2288726, DOI 10.1016/j.top.2006.10.002
- Ralph M. Kaufmann, Moduli space actions on the Hochschild co-chains of a Frobenius algebra. I. Cell operads, J. Noncommut. Geom. 1 (2007), no. 3, 333–384. MR 2314100, DOI 10.4171/JNCG/10
- Ralph M. Kaufmann, A proof of a cyclic version of Deligne’s conjecture via cacti, Math. Res. Lett. 15 (2008), no. 5, 901–921. MR 2443991, DOI 10.4310/MRL.2008.v15.n5.a7
- B. Keller, Derived invariance of higher structures on the Hochschild complex, 2003, https://webusers.imj-prg.fr/~bernhard.keller/publ/dih.pdf
- Bernhard Keller and Wendy Lowen, On Hochschild cohomology and Morita deformations, Int. Math. Res. Not. IMRN 17 (2009), 3221–3235. MR 2534996, DOI 10.1093/imrp/rnp050
- Maxim Kontsevich, Operads and motives in deformation quantization, Lett. Math. Phys. 48 (1999), no. 1, 35–72. Moshé Flato (1937–1998). MR 1718044, DOI 10.1023/A:1007555725247
- Maxim Kontsevich and Yan Soibelman, Deformations of algebras over operads and the Deligne conjecture, Conférence Moshé Flato 1999, Vol. I (Dijon), Math. Phys. Stud., vol. 21, Kluwer Acad. Publ., Dordrecht, 2000, pp. 255–307. MR 1805894
- Henning Krause, The stable derived category of a Noetherian scheme, Compos. Math. 141 (2005), no. 5, 1128–1162. MR 2157133, DOI 10.1112/S0010437X05001375
- Jean-Louis Loday, Cyclic homology, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 301, Springer-Verlag, Berlin, 1992. Appendix E by María O. Ronco. MR 1217970, DOI 10.1007/978-3-662-21739-9
- Jean-Louis Loday and Bruno Vallette, Algebraic operads, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 346, Springer, Heidelberg, 2012. MR 2954392, DOI 10.1007/978-3-642-30362-3
- 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
- J. Lurie, Derived algebraic geometry X: formal moduli problems, http://www.math.harvard.edu/~lurie/papers/DAG-X.pdf
- Martin Markl, Steve Shnider, and Jim Stasheff, Operads in algebra, topology and physics, Mathematical Surveys and Monographs, vol. 96, American Mathematical Society, Providence, RI, 2002. MR 1898414, DOI 10.1090/surv/096
- James E. McClure and Jeffrey H. Smith, A solution of Deligne’s Hochschild cohomology conjecture, Recent progress in homotopy theory (Baltimore, MD, 2000) Contemp. Math., vol. 293, Amer. Math. Soc., Providence, RI, 2002, pp. 153–193. MR 1890736, DOI 10.1090/conm/293/04948
- Luc Menichi, Batalin-Vilkovisky algebras and cyclic cohomology of Hopf algebras, $K$-Theory 32 (2004), no. 3, 231–251. MR 2114167, DOI 10.1007/s10977-004-0480-4
- Van C. Nguyen, Tate and Tate-Hochschild cohomology for finite dimensional Hopf algebras, J. Pure Appl. Algebra 217 (2013), no. 10, 1967–1979. MR 3053527, DOI 10.1016/j.jpaa.2013.01.008
- D. O. Orlov, Triangulated categories of singularities and D-branes in Landau-Ginzburg models, Tr. Mat. Inst. Steklova 246 (2004), no. Algebr. Geom. Metody, Svyazi i Prilozh., 240–262 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 3(246) (2004), 227–248. MR 2101296
- Jeremy Rickard, Derived categories and stable equivalence, J. Pure Appl. Algebra 61 (1989), no. 3, 303–317. MR 1027750, DOI 10.1016/0022-4049(89)90081-9
- Manuel Rivera and Zhengfang Wang, Singular Hochschild cohomology and algebraic string operations, J. Noncommut. Geom. 13 (2019), no. 1, 297–361. MR 3941481, DOI 10.4171/JNCG/325
- Paul Seidel, Fukaya categories and deformations, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 351–360. MR 1957046
- D. Tamarkin, Formality of chain operad of small squares, arXiv:9809164.
- Bertrand Toën, The homotopy theory of $dg$-categories and derived Morita theory, Invent. Math. 167 (2007), no. 3, 615–667. MR 2276263, DOI 10.1007/s00222-006-0025-y
- Thomas Tradler, The Batalin-Vilkovisky algebra on Hochschild cohomology induced by infinity inner products, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 7, 2351–2379 (English, with English and French summaries). MR 2498354
- Thomas Tradler and Mahmoud Zeinalian, On the cyclic Deligne conjecture, J. Pure Appl. Algebra 204 (2006), no. 2, 280–299. MR 2184812, DOI 10.1016/j.jpaa.2005.04.009
- Alexander A. Voronov, Homotopy Gerstenhaber algebras, Conférence Moshé Flato 1999, Vol. II (Dijon), Math. Phys. Stud., vol. 22, Kluwer Acad. Publ., Dordrecht, 2000, pp. 307–331. MR 1805923
- Z. F. Wang, Singular Hochschild cohomology of radical square zero algebra, arXiv:1511.08348.
- Alexander Zimmermann, Representation theory, Algebra and Applications, vol. 19, Springer, Cham, 2014. A homological algebra point of view. MR 3289041, DOI 10.1007/978-3-319-07968-4
Additional Information
- Zhengfang Wang
- Affiliation: Beijing International Center for Mathematical Research, Peking University, No. 5 Yiheyuan Road Haidian District, Beijing 100871, People’s Republic of China; and Université Paris Diderot-Paris 7, IMJ-PRG CNRS UMR 7586, Bâtiment Sophie Germain, Case 7012, 75205 Paris Cedex 13, France
- MR Author ID: 1118527
- Email: zhengfang.wang@imj-prg.fr, zhengfangw@gmail.com
- Received by editor(s): January 18, 2018
- Published electronically: April 20, 2021
- Additional Notes: This work was partially supported by Jun Yu’s grants.
Part of the results in this work was presented at the workshop on Hochschild Cohomology in Algebra, Geometry, and Topology at Oberwolfach (MFO) in 2016. - © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 374 (2021), 4537-4577
- MSC (2020): Primary 16E40, 13D03; Secondary 14B07
- DOI: https://doi.org/10.1090/tran/7886
- MathSciNet review: 4273171