DG-algebras and derived A-infinity algebras

Abstract

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.

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

References in corpus (2)

• Sur les A-infini catégories
• Universal Toda brackets of ring spectra

Cited by in corpus (7)

• Uniqueness of $A_{\infty }$-structures and Hochschild cohomology
• Derived A-infinity algebras in an operadic context
• Presentations of non-commutative deformation rings via $A_{\infty }$-algebras and applications to deformations of Galois representations and pseudorepresentations
• Derived homotopy algebras
• The Homotopy Theory of $A_{\infty }$Categories
• Fixed points of $p$-toral groups acting on partition complexes
• Homotopy Theory of Bicomplexes