Abstract
In this paper we study classical deformations of diagrams of commutative algebras over a field of characteristic 0. In particular we determine several homotopy classes of DG-Lie algebras, each one of them controlling this above deformation problem: the first homotopy type is described in terms of the projective model structure on the category of diagrams of differential graded algebras, the others in terms of the Reedy model structure on truncated Bousfield-Kan approximations.
The first half of the paper contains an elementary introduction to the projective model structure on the category of commutative differential graded algebras, while the second half is devoted to the main results.
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References
Bousfield, A., Gugenheim, V.: On PL De Rham Theory and Rational Homotopy Type. Memoirs of the American Mathematical Society, vol. 8, no. 179. American Mathematical Society, Providence (1976)
Bousfield, A.K., Kan, D.M.: Homotopy Limits, Completions and Localizations. Springer Lecture Notes in Mathematics, vol. 304. Springer, New York (1972)
Chachólski, W., Scherer, J.: Homotopy Theory of Diagrams. Memoirs of the American Mathematical Society, no. 736. American Mathematical Society, Providence (2002)
Gelfand, S.I., Manin, Y.I.: Methods of Homological Algebra. Springer Monographs in Mathematics. Springer, Berlin (2003)
Hinich, V.: Deformations of homotopy algebras. Commun. Algebra 32(2), 473–494 (2004)
Hirschhorn, P.S.: Model Categories and their Localizations. Mathematical Surveys and Monographs, vol. 99. American Mathematical Society, Providence (2003)
Hovey, M.: Model Categories. Mathematical Surveys and Monographs, vol. 63. American Mathematical Society, Providence (1999)
Jardine, J.F.: A closed model structure for differential graded algebras. In: Cyclic Cohomology and Noncommutative Geometry (Waterloo, ON, 1995). Fields Institute Communications, vol. 17, pp. 55–58. American Mathematical Society, Providence (1997)
Leinster, T.: Basic Category Theory. Cambridge Studies in Advanced Mathematics, vol. 143. Cambridge University Press, Cambridge (2014)
Lurie, J.: Higher Topos Theory. Annals of Mathematical Studies, vol. 170. Princeton University Press, Princeton (2009)
Lurie, J.: Higher Algebra. Available on J. Lurie’s web page (2014). Preprint
Manetti, M.: Deformation theory via differential graded Lie algebras. In: Seminari di Geometria Algebrica 1998–1999. Scuola Normale Superiore, Pisa (1999). arXiv: math.AG/0507284
Manetti, M.: Deformation of singularities via differential graded Lie algebras. In: Seminar Notes, Roma (2001). www1.mat.uniroma1.it/people/manetti/dispense/defosing.pdf
Manetti, M.: Differential graded Lie algebras and formal deformation theory. In: Algebraic Geometry: Seattle 2005. Proceedings of Symposia in Pure Mathematics, vol. 80, pp. 785–810 (2009)
Manetti, M., Meazzini, F.: Formal deformation theory in left-proper model categories. New York J. Math. 25, 1259–1311 (2019)
Manetti, M., Meazzini, F.: Deformations of algebraic schemes via Reedy-Palamodov cofibrant resolutions. Indag. Math. (N.S.) (2019). https://doi.org/10.1016/j.indag.2019.08.007
Palamodov, V.P.: Deformations of complex spaces. Usp. Mat. Nauk 31(3), 129–194 (1976). English translation: Russian Math. Surveys 31, 129–197 (1976)
Quillen, D.: Homotopical Algebra. Lecture Notes in Mathematics, vol. 43. Springer, New York (1967)
Sernesi, E.: Deformations of Algebraic Schemes. Grundlehren der mathematischen Wissenschaften, vol. 334. Springer, New York (2006)
Tate, J.: Homology of Noetherian rings and local rings. Illinois J. Math. 1, 14–27 (1957)
Acknowledgements
Both authors thank Francesco Meazzini for useful discussions about the topics of this paper, and the anonymous referee for useful comments and remarks. M.M. is partially supported by Italian MIUR under PRIN project 2017YRA3LK “Moduli and Lie theory”.
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Lepri, E., Manetti, M. (2020). On Deformations of Diagrams of Commutative Algebras. In: Colombo, E., Fantechi, B., Frediani, P., Iacono, D., Pardini, R. (eds) Birational Geometry and Moduli Spaces. Springer INdAM Series, vol 39. Springer, Cham. https://doi.org/10.1007/978-3-030-37114-2_6
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