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International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 26, Pages 1347-1361

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

Department of Mathematics and Computer Science, Providence College, Providence 02918, RI, USA

Received 11 June 2003

Copyright © 2004 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.