DG-algebras and derived A-infinity algebras

Details DG-algebras and derived A-infinity algebras DG-algebras and derived A-infinity algebras

2007-11-28

arxiv.org/abs/0711.4499v3

J. Reine Angew. Math. (Crelles Journal), 639 (2010), 73-105.

Mathematics

K-Theory and Homology

v3: 27 pages. Minor corrections, to appear in Crelle's Journal

Scientific paper

10.1515/CRELLE.2010.011

A differential graded algebra can be viewed as an A-infinity algebra. By a theorem of Kadeishvili, a dga over a field admits a quasi-isomorphism from a minimal A-infinity algebra. We introduce the notion of a derived A-infinity algebra and show that any dga A over an arbitrary commutative ground ring k is equivalent to a minimal derived A-infinity algebra. Such a minimal derived A-infinity algebra model for A is a k-projective resolution of the homology algebra of A together with a family of maps satisfying appropriate relations. As in the case of A-infinity algebras, it is possible to recover the dga up to quasi-isomorphism from a minimal derived A-infinity algebra model. Hence the structure we are describing provides a complete description of the quasi-isomorphism type of the dga.

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