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Uniqueness of \(A_\infty\)-structures and Hochschild cohomology. (English) Zbl 1246.16011

Summary: Working over a commutative ground ring, we establish a Hochschild cohomology criterion for uniqueness of derived \(A_\infty\)-algebra structures in the sense of S. Sagave [J. Reine Angew. Math. 639, 73-105 (2010; Zbl 1209.18011)]. We deduce a Hochschild cohomology criterion for intrinsic formality of a differential graded algebra. This generalizes a classical result of T. V. Kadeishvili [Russ. Math. Surv. 35, No. 3, 231-238 (1980); translation from Usp. Mat. Nauk 35, No. 3(213), 183-188 (1980; Zbl 0521.55015)] for the case of a graded algebra over a field.

MSC:

16E45 Differential graded algebras and applications (associative algebraic aspects)
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
55S30 Massey products
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[1] D Benson, H Krause, S Schwede, Introduction to realizability of modules over Tate cohomology (editors R O Buchweitz, H Lenzing), Fields Inst. Commun. 45, Amer. Math. Soc. (2005) 81 · Zbl 1107.20043
[2] H Cartan, S Eilenberg, Homological algebra, Princeton Univ. Press (1956) · Zbl 0075.24305
[3] D Dugger, B Shipley, Topological equivalences for differential graded algebras, Adv. Math. 212 (2007) 37 · Zbl 1118.55008
[4] D Dugger, B Shipley, A curious example of triangulated-equivalent model categories which are not Quillen equivalent, Algebr. Geom. Topol. 9 (2009) 135 · Zbl 1159.18305
[5] A Fialowski, M Penkava, Deformation theory of infinity algebras, J. Algebra 255 (2002) 59 · Zbl 1038.17012
[6] M Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math. \((2)\) 78 (1963) 267 · Zbl 0131.27302
[7] E Getzler, Batalin-Vilkovisky algebras and two-dimensional topological field theories, Comm. Math. Phys. 159 (1994) 265 · Zbl 0807.17026
[8] T V Kadeishvili, On the theory of homology of fiber spaces, Uspekhi Mat. Nauk 35 (1980) 183 · Zbl 0521.55015
[9] 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 · Zbl 0717.55011
[10] B Keller, Introduction to \(A\)-infinity algebras and modules, Homology Homotopy Appl. 3 (2001) 1 · Zbl 0989.18009
[11] K Lefèvre-Hasegawa, Sur les \(A_\infty\)-catégories, PhD thesis, Université de Paris 7 - Denis Diderot (2003)
[12] J McCleary, A user’s guide to spectral sequences, Cambridge Studies in Advanced Math. 58, Cambridge Univ. Press (2001) · Zbl 0959.55001
[13] M Penkava, A Schwarz, \(A_\infty\) algebras and the cohomology of moduli spaces (editors S G Gindikin, E B Vinberg), Amer. Math. Soc. Transl. Ser. 2 169, Amer. Math. Soc. (1995) 91 · Zbl 0863.17017
[14] D C Ravenel, Complex cobordism and stable homotopy groups of spheres, Pure and Applied Math. 121, Academic Press (1986) · Zbl 0608.55001
[15] S Sagave, A derived \(A_\infty\)-algebra model for a dga studied by Dugger and Shipley, Unpublished note (2009)
[16] S Sagave, DG-algebras and derived \(A_\infty\)-algebras, J. Reine Angew. Math. 639 (2010) 73 · Zbl 1209.18011
[17] J D Stasheff, Homotopy associativity of \(H\)-spaces. I, II, Trans. Amer. Math. Soc. 108 \((1963)\), 275-292; ibid. 108 (1963) 293 · Zbl 0114.39402
[18] C A Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Math. 38, Cambridge Univ. Press (1994) · Zbl 0797.18001
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