Strictly unital \(\mathrm{A}_{\infty}\)-algebras. (English) Zbl 1418.16024
Summary: Given a graded module over a commutative ring, we define a dg-Lie algebra whose Maurer-Cartan elements are the strictly unital \(\mathrm{A}_\infty\)-algebra structures on that module. We use this to generalize L. E. Positsel’skiĭ’s result [Funct. Anal. Appl. 27, No. 3, 197–204 (1993; Zbl 0826.16041); translation from Funkts. Anal. Prilozh. 27, No. 3, 57–66 (1993)] that a curvature term on the bar construction compensates for a lack of augmentation, from a field to arbitrary commutative base ring. We also use this to show that the reduced Hochschild cochains control the strictly unital deformation functor. We motivate these results by giving a full development of the deformation theory of a nonunital \(\mathrm{A}_\infty\)-algebra.
MSC:
| 16W50 | Graded rings and modules (associative rings and algebras) |
| 16E40 | (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) |
| 17B60 | Lie (super)algebras associated with other structures (associative, Jordan, etc.) |
| 18D50 | Operads (MSC2010) |
Citations:
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