The cup product on Hochschild cohomology via twisting cochains and applications to Koszul rings. (English) Zbl 1376.16010
Summary: Given an acyclic twisting cochain \(\pi:C\to A\), from a curved dg coalgebra \(C\) to a dg algebra \(A\), we show that the associated twisted hom complex \(\mathrm{Hom}_k^\pi(C,A)\) has cohomology equal to the Hochschild cohomology of \(A\), as a graded ring. As a corollary we find that the Hochschild cohomology of a Koszul algebra \(A\), along with its cup product, is a subquotient of the tensor product algebra \(A^!\otimes A\) of \(A\) with its Koszul dual.
MSC:
| 16E40 | (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) |
| 16S37 | Quadratic and Koszul algebras |
References:
| [1] | Barthel, T.; May, J.; Riehl, E., Six model structures for dg-modules over dgas: model category theory in homological action, N.Y. J. Math., 20, 1077-1159 (2014) · Zbl 1342.16006 |
| [2] | Brown, E. H., Twisted tensor products, I, Ann. of Math., 223-246 (1959) · Zbl 0199.58201 |
| [3] | Buchweitz, R. O.; Green, E. L.; Snashall, N.; Solberg, Ø., Multiplicative structures for Koszul algebras, Q. J. Math., 59, 4, 441-454 (2008) · Zbl 1191.16027 |
| [4] | Chen, K.-T., Iterated path integrals, Bull. Am. Math. Soc., 83, 5, 831-879 (1977) · Zbl 0389.58001 |
| [5] | Felix, Y.; Thomas, J.-C.; Vigué-Poirrier, M., The Hochschild cohomology of a closed manifold, Publ. Math. IHÉS, 99, 235-252 (2004) · Zbl 1060.57019 |
| [6] | Gerstenhaber, M., The cohomology structure of an associative ring, Ann. of Math., 267-288 (1963) · Zbl 0131.27302 |
| [7] | Gerstenhaber, M., On the deformation of rings and algebras, Ann. of Math., 59-103 (1964) · Zbl 0123.03101 |
| [8] | Grimley, L.; Nguyen, V. C.; Witherspoon, S., Gerstenhaber brackets on Hochschild cohomology of twisted tensor products, preprint · Zbl 1382.16005 |
| [9] | Gugenheim, V., On a perturbation theory for the homology of the loop-space, J. Pure Appl. Algebra, 25, 2, 197-205 (1982) · Zbl 0487.55003 |
| [10] | Gugenheim, V.; Lambe, L.; Stasheff, J., Algebraic aspects of Chen’s twisting cochain, Ill. J. Math., 34, 2, 485-502 (1990) · Zbl 0684.55006 |
| [11] | Husemoller, D.; Moore, J. C.; Stasheff, J., Differential homological algebra and homogeneous spaces, J. Pure Appl. Algebra, 5, 2, 113-185 (1974) · Zbl 0364.18008 |
| [12] | Keller, B., A-infinity algebras in representation theory, online · Zbl 1086.18007 |
| [13] | Keller, B., Derived invariance of higher structures on the Hochschild complex, preprint at |
| [14] | Keller, B., Introduction to \(A\)-infinity algebras and modules, Homol. Homotopy Appl., 3, 1, 1-35 (2001) · Zbl 0989.18009 |
| [15] | Keller, B., Koszul duality and coderived categories (after K. Lefevre) (2003) |
| [16] | Keller, B., \(A\)-infinity algebras, modules and functor categories, (Trends in Representation Theory of Algebras and Related Topics. Trends in Representation Theory of Algebras and Related Topics, Contemp. Math., vol. 406 (2006), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 67-93 · Zbl 1121.18008 |
| [18] | Lefévre-Hasegawa, K., Sur les \(A_\infty \)-catégories (November 2003), Université Denis Diderot - Paris 7, PhD thesis, Université Denis Diderot Paris 7, November 2003 |
| [19] | Loday, J.-L.; Vallette, B., Algebraic Operads, Grundlehren der Mathematischen Wissenschaften, vol. 346 (2012), Springer: Springer Heidelberg · Zbl 1260.18001 |
| [20] | Lu, D.-M.; Palmieri, J. H.; Wu, Q.-S.; Zhang, J. J., \(A_\infty \)-algebras for ring theorists, (Proceedings of the International Conference on Algebra, vol. 11 (2004)), 91-128 · Zbl 1066.16049 |
| [21] | Manetti, M., Differential graded Lie algebras and formal deformation theory, (Algebraic Geometry—Seattle 2005, Part 2 (2009)), 785-810 · Zbl 1190.14007 |
| [22] | Menichi, L., The cohomology ring of free loop spaces, Homol. Homotopy Appl., 3, 1, 193-224 (2001) · Zbl 0974.55005 |
| [23] | Montgomery, S., Hopf Algebras and Their Actions on Rings, CBMS Regional Conference Series in Mathematics, vol. 82 (1993), published for the Conference Board of the Mathematical Sciences, Washington, DC · Zbl 0804.16041 |
| [24] | Negron, C., Alternate Approaches to the Cup Product and Gerstenhaber Bracket on Hochschild Cohomology (2015), University of Washington, PhD thesis |
| [25] | Negron, C.; Witherspoon, S., An alternate approach to the Lie bracket on Hochschild cohomology, Homol. Homotopy Appl., 18, 265-285 (2016) · Zbl 1353.16008 |
| [26] | Nicolás, P., The bar derived category of a curved dg algebra, J. Pure Appl. Algebra, 212, 12, 2633-2659 (2008) · Zbl 1152.18008 |
| [27] | Positselski, L., Two Kinds of Derived Categories, Koszul Duality, and Comodule-Contramodule Correspondence (2011), American Mathematical Society · Zbl 1275.18002 |
| [28] | Positsel’skiĭ, L. E., Nonhomogeneous quadratic duality and curvature, Funkc. Anal. Prilozh., 27, 3, 57-66 (1993), 96 · Zbl 0826.16041 |
| [29] | Priddy, S., Koszul resolutions, Trans. Am. Math. Soc., 152, 39-60 (1970) · Zbl 0261.18016 |
| [30] | Snashall, N.; Solberg, Ø., Support varieties and Hochschild cohomology rings, Proc. Lond. Math. Soc., 88, 3, 705-732 (2004) · Zbl 1067.16010 |
| [31] | Van den Bergh, M., Noncommutative homology of some three-dimensional quantum spaces, (Proceedings of Conference on Algebraic Geometry and Ring Theory in Honor of Michael Artin, Part III. Proceedings of Conference on Algebraic Geometry and Ring Theory in Honor of Michael Artin, Part III, Antwerp, 1992, vol. 8 (1994)), 213-230 · Zbl 0814.16006 |
| [32] | Van den Bergh, M., Existence theorems for dualizing complexes over non-commutative graded and filtered rings, J. Algebra, 195, 2, 662-679 (1997) · Zbl 0894.16020 |
| [33] | Yekutieli, A., The rigid dualizing complex of a universal enveloping algebra, J. Pure Appl. Algebra, 150, 1, 85-93 (2000) · Zbl 1004.16007 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
