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

Abstract

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 A∞-coalgebra Tor•A(K,K), we know that there exists an A∞-comodule structure on T′=Tor•A(M,K) over T. The structure of the A∞-algebra E=ExtA•(K,K) and the corresponding A∞-module on E′=ExtA•(M,K) are just obtained by taking the bigraded dual. In this article we prove that there is partial description of the A∞-comodule T′ over T and of the structure of A∞-module E′ over E, similar to and also generalizing the partial description of the A∞-algebra structure on E given by Keller's higher-multiplication theorem in [19]. We also provide a criterion to check if a given A∞-comodule structure on T′ 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 A∞-module structure on E′ for any generalized Koszul algebra A and any generalized Koszul module M.