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On the construction of \(A_\infty\)-structures. (English) Zbl 1202.55007
Let \(R\) be a commutative ring and \(A\) be a differential graded \(R\)-algebra. Suppose the homology \(H(A)\) of \(A\) is free as a graded \(R\)-module. Then \(H(A)\) acquires an \(A_{\infty}\)-algebra structure that is equivalent to \(A\). Over a field, this fact is the “minimality theorem” for differential graded algebras [see T. V. Kadeishvili, Russ. Math. Surv. 35, No. 3, 231–238 (1980); translation from Usp. Mat. Nauk 35, No. 3(213), 183–188 (1980; Zbl 0521.55015)].
Let \(C\) be a simply connected coaugmented differential graded \(R\)-coalgebra such that its homology \(H(C)\) is free as an \(R\)-module. Then \(H(C)\) acquires a coaugmented graded coalgebra structure and an \(A_{\infty}\)-coalgebra structure equivalent to the original coalgebra \(C\). This is a version of the “minimality theorem” for coalgebras [see V. K. A. M. Gugenheim, J. Pure Appl. Algebra 25, 197–205 (1982; Zbl 0487.55003)].
The aim of this paper is to relate the previous two constructions and to establish links between them.

55P62 Rational homotopy theory
13N05 Modules of differentials
57T30 Bar and cobar constructions
12H05 Differential algebra
16E45 Differential graded algebras and applications (associative algebraic aspects)
17B55 Homological methods in Lie (super)algebras
17B56 Cohomology of Lie (super)algebras
17B65 Infinite-dimensional Lie (super)algebras
17B70 Graded Lie (super)algebras
18G55 Nonabelian homotopical algebra (MSC2010)
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