The homotopy Gerstenhaber algebra of Hochschild cochains of a regular algebra is formal. (English) Zbl 1144.18007
Deligne’s Hochschild conjecture, having stimulated much interest in string theoriests, claims that the Hochschild complex of an
associative algebra admits a canonical structure of a differential graded
algebra over the chain operad of the little discs operads. Several authors
(M. Kontsevich and Y. Soibelman [Volume I. Dordrecht: Kluwer Academic Publishers. Math. Phys. Stud. 21, 255–307 (2000; Zbl 0972.18005)], J. E. McClure and J. H. Smith [Contemp. Math. 293, 153–193 (2002; Zbl 1009.18009)], V. Hinich [Forum Math. 15, No. 4, 591–614 (2003; Zbl 1081.16014)] and A. A. Voronov [Volume II. Dordrecht: Kluwer Academic Publishers. Math. Phys. Stud. 22, 307–331 (2000; Zbl 0974.16005)])
have settled the conjecture affirmatively, which has demonstrated a remarkable fact that the
complex \(C^{\cdot}(A)\) of Hochschild cochains of an associative
algebra \(A\)is endowed with the distinguishable structure of a homotopy
M. Gerstenhaber algebra [Ann. Math. (2) 78, 267–288 (1963; Zbl 0131.27302)]. This paper constructs a natural chain
of quasi-isomorphisms of homotopy Gerstenhaber algebras between the Hochschild
cochain complex of a regular commutative algebra \(A\) and the Gerstenhaber
algebras of multiderivations of \(A\), which enables us to have an explicit
construction of a chain of quasi-isomorphisms and to dispense with the really bulky
Gelfand-Fuchs machinery in establishing the formality theorem.
Reviewer: Hirokazu Nishimura (Tsukuba)
MSC:
| 18D50 | Operads (MSC2010) |
| 16E40 | (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) |
| 16E45 | Differential graded algebras and applications (associative algebraic aspects) |
| 58B99 | Infinite-dimensional manifolds |
| 53D55 | Deformation quantization, star products |
References:
| [1] | J. M. Boardman and R. M. Vogt, Homotopy invariant algebraic structures on topologi- cal spaces . Lecture Notes in Math. 347, Springer-Verlag, Berlin 1973. · Zbl 0285.55012 |
| [2] | A. C\?ald\?araru, The Mukai pairing, II: the Hochschild-Kostant-Rosenberg isomorphism. Adv. Math. 194 (2005), 34-66. · Zbl 1098.14011 |
| [3] | A. S. Cattaneo and G. Felder, Relative formality theorem and quantisation of coisotropic submanifolds. · Zbl 1106.53060 · doi:10.1016/j.aim.2006.03.010 |
| [4] | A. V. Dolgushev, Covariant and equivariant formality theorems. Adv. Math. 191 (2005), 147-177. · Zbl 1116.53065 · doi:10.1016/j.aim.2004.02.001 |
| [5] | A. V. Dolgushev, A proof of Tsygan’s formality conjecture for an arbitrary smooth mani- fold. PhD thesis, M.I.T. |
| [6] | B. V. Fedosov, A simple geometrical construction of deformation quantization. J. Differ- ential Geom. 40 (1994), 213-238. · Zbl 0812.53034 |
| [7] | B. Fresse, Koszul duality of operads and homology of partition posets. In Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic K -theory (Evanston, 2002), Contemp. Math. 346, Amer. Math. Soc. Providence, RI, 2004, 115-215. · Zbl 1077.18007 |
| [8] | I. M. Gel’fand and D. B. Fuks, Cohomology of the Lie algebra of formal vector fields. Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), 322-337; English transl. Math. USSR-Izv. 4 (1970), 327-342. · Zbl 0216.20302 · doi:10.1070/IM1970v004n02ABEH000908 |
| [9] | I. M. Gel’fand and D. A. Každan, Some problems of differential geometry and the calculation of cohomologies of Lie algebras of vector fields. Dokl. Akad. Nauk SSSR 200 (1971), 269-272; English transl. Soviet Math. Dokl. 12 (1971), 1367-1370. · Zbl 0238.58001 |
| [10] | M. Gerstenhaber, The cohomology structure of an associative ring. Ann. of Math. (2) 78 (1963), 267-288. · Zbl 0131.27302 · doi:10.2307/1970343 |
| [11] | E. Getzler, Cartan homotopy formulas and the Gauss-Manin connection in cyclic homol- ogy. In Quantum deformations of algebras and their representations , Israel Math. Conf. Proc. 7, Bar-Ilan Univ., Ramat Gan, 1993, 65-78. · Zbl 0844.18007 |
| [12] | E. Getzler and J. D. S. Jones, Operads, homotopy algebra and iterated integrals for double loop spaces. |
| [13] | V. Ginzburg and M. Kapranov, Koszul duality for operads. Duke Math. J. 76 (1994), 203-272. · Zbl 0855.18006 · doi:10.1215/S0012-7094-94-07608-4 |
| [14] | G. Halbout, Globalization of Tamarkin’s formality theorem. Lett. Math. Phys. 71 (2005), 39-48. · Zbl 1081.53079 · doi:10.1007/s11005-004-5926-3 |
| [15] | V. Hinich, Homological algebra of homotopy algebras. Comm. Algebra 25 (1997), 3291-3323. · Zbl 0894.18008 · doi:10.1080/00927879708826055 |
| [16] | V. Hinich, Tamarkin’s proof of Kontsevich formality theorem. Forum Math. 15 (2003), 591-614. · Zbl 1081.16014 · doi:10.1515/form.2003.032 |
| [17] | G. Hochschild, B. Kostant, and A. Rosenberg, Differential forms on regular affine alge- bras. Trans. Amer. Math. Soc. 102 (1962), 383-408. · Zbl 0102.27701 · doi:10.2307/1993614 |
| [18] | T. V. Kadeishvili, The structure of the A.1/-algebra, and the Hochschild and Harrison cohomologies. Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 91 (1988), 19-27 (in Russian). · Zbl 0717.55011 |
| [19] | M. Kontsevich, Operads and motives in deformation quantization. Lett. Math. Phys. 48 (1999), 35-72. · Zbl 0945.18008 · doi:10.1023/A:1007555725247 |
| [20] | M. Kontsevich, Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66 (2003), 157-216. · Zbl 1058.53065 · doi:10.1023/B:MATH.0000027508.00421.bf |
| [21] | M. Kontsevich and Y. Soibelman, Deformations of algebras over operads and Deligne’s conjecture. In Conférence Moshé Flato 1999 , Vol. I (Dijon), Math. Phys. Stud. 21, Kluwer Acad. Publ., Dordrecht 2000, 255-307. · Zbl 0972.18005 |
| [22] | S. L. Lyakhovich and A. A. Sharapov, BRST theory without Hamiltonian and Lagrangian. J. High Energy Phys. 03 (2005), 011. |
| [23] | J. E. McClure and J. H. Smith, A solution of Deligne’s Hochschild cohomology conjecture. In Recent progress in homotopy theory (Baltimore, MD, 2000), Contemp. Math. 293, Amer. Math. Soc., Providence, RI, 2002, 153-193. · Zbl 1009.18009 |
| [24] | A. Nijenhuis, A Lie product for the cohomology of subalgebras with coefficients in the quotient. Bull. Amer. Math. Soc. 73 (1967), 962-967. · Zbl 0153.36105 · doi:10.1090/S0002-9904-1967-11865-2 |
| [25] | S. B. Priddy, Koszul resolutions. Trans. Amer. Math. Soc. 152 (1970), 39-60. · Zbl 0261.18016 · doi:10.2307/1995637 |
| [26] | D. Tamarkin, Another proof of M. Kontsevich formality theorem. |
| [27] | D. Tamarkin, Formality of chain operad of little discs. Lett. Math. Phys. 66 (2003), 65-72. · Zbl 1048.18007 · doi:10.1023/B:MATH.0000017651.12703.a1 |
| [28] | M. Van den Bergh, On global deformation quantization in the algebraic case. · Zbl 1133.14021 · doi:10.1016/j.jalgebra.2007.02.012 |
| [29] | J. Vey, Déformation du crochet de Poisson sur une variété symplectique. Comment. Math. Helv. 50 (1975), 421-454. · Zbl 0351.53029 · doi:10.1007/BF02565761 |
| [30] | A. Voronov, Homotopy Gerstenhaber algebras. In Conférence Moshé Flato 1999 , Vol. II (Dijon), Math. Phys. Stud. 22, Kluwer Acad. Publ., Dordrecht 2000, 307-331. · Zbl 0974.16005 |
| [31] | A. Yekutieli, The continuous Hochschild cochain complex of a scheme. Canad. J. Math. 54 (2002), 1319-1337. · Zbl 1047.16004 · doi:10.4153/CJM-2002-051-8 |
| [32] | A. Yekutieli, Mixed resolutions, simplicial sections and unipotent group actions. · Zbl 1143.14002 · doi:10.1007/s11856-007-0085-8 |
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