Applications of one-point extensions to compute the A∞-(co)module structure of several Ext (resp., Tor) groups

Let A be a nonnegatively graded connected algebra over a noncommutative separable k-algebra K, and let M be a bounded below graded right A-module. If we denote by T the -coalgebra , we know that there exists an -comodule structure on over T. The structure of the -algebra and the corresponding -module on are just obtained by taking the bigraded dual. In this article we prove that there is partial description of the -comodule over T and of the structure of -module over E, similar to and also generalizing the partial description of the -algebra structure on E given by Keller's higher-multiplication theorem in [19]. We also provide a criterion to check if a given -comodule structure on is a model by regarding if the associated twisted tensor product is a minimal projective resolution of M, analogous to a theorem of B. Keller explained by the author of this article in [9]. Finally, we give an application of this result by computing the -module structure on for any generalized Koszul algebra A and any generalized Koszul module M.