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This paper builds a general framework in which to study cohomology theories of strongly homotopy algebras, namely A_∞, C_∞ and L_∞-algebras. This framework is based on noncommutative geometry as expounded by Connes and Kontsevich. The developed machinery is then used to establish a general form of Hodge decomposition of Hochschild and cyclic cohomology of C_∞-algebras. This generalizes and puts in a conceptual framework previous work by Loday and Gerstenhaber-Schack.doi:10.2140/agt.2009.9.1503 fatcat:24nbgtlwcja37hgcyayl422j6e