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(π¯,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
π¯-homotopy group and an ExtΛ-group. We then
demonstrate the nontriviality of the injective/projective
relative homotopy groups (of modules) based on the results ofs
Su (2001). Finally, by inserting three
(π¯,ExtΛ)-sequences into a one-of-a-kind diagram,
we establish the long exact (π¯,ExtΛ)-sequence of
a triple, which is an extension of the homotopy sequence of a
triple in module theory.
