Syzygy problems by method of stable categories

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We mainly studied homotopy categories of chain complexes of projective modules because our standpoint is to use a triangle structure of homotopy categories to investigate stable categories. The results of this project are divided into the following three groups. The first one is on the symmetry caused by torsion pairs. We found a structure of torsion pairs with strong symmetry which leads us to a new triangle equivalence. The second one is on representability by monomorphisms (rbm) which measures the obstruction for the category to be triangulated.The third one is on syzygies of modules with positive codimension. We obtained that every module is equivalent to some module with positive codimension up to first syzygy if and only if the ring is an integral domain.