On long exact (π¯, ExtΛ)-sequences in module theory

In (2003), we proved the injective homotopy exact sequence of modules by a method that does not refer to any elements of the sets in the argument, so that the duality applies automatically in the projective homotopy theory (of modules) without further derivation. We inherit this fashion in this paper during our process of expanding the homotopy exact sequence. We name the resulting doubly infinite sequence the long exact (π̄,ExtΛ)-sequence in the second variable—it links the (injective) homotopy exact sequence with the long exact ExtΛ-sequence in the second variable through a connecting term which has a structure containing traces of both a π̄ -homotopy group and an ExtΛ-group. We then demonstrate the nontriviality of the injective/projective relative homotopy groups (of modules) based on the results of Su (2001). Finally, by inserting three (π̄,ExtΛ)-sequences into a one-of-a-kind diagram, we establish the long exact (π̄,ExtΛ)-sequence of a triple, which is an extension of the homotopy sequence of a triple in module theory.