Applications Of Triangulated Categories In Relative Homological Theory

The investigation of objects admitting some projective (resp. injective) properties in abelian categories is a main subject of relative homological algebra. The Bousfield’s localization theory is one of its foundations after the methods of derived categories were extensively used in the study of the classical homological algebra. In fact, the Bousfield’s (co)localization gives rise to recollements for triangulated categories [45]. From another point, the Bousfield’s (co)localization (or recollement) induced by pro-jective and injective objects in fact gives rise to resolutions for complexes [54], based on this one can introduce and study homological dimensions for complexes.In Chapter 2, we investigate how a recollement can be induced by a balanced pair. In Chapter 3, we consider in homotopy category, the representations of some balanced pair induced by purity. In particular, we define and study pure homological dimensions for complexes.The Auslander-Reiten theory is a main tool in the study of the representation the-ory of artin algebra. This theory has been used extensively for studies of triangulated categories. In Chapter 4, we prove that the bounded homotopy category of finitely presented modules over an artin algebra has Auslander-Reiten triangles and give some applications.Singularity equivalence based on finitely generated Gorenstein projective modules over a Gorenstein ring has attracted many authors. In Chapter 5, we try to reduce the conditions for this equivalence as weak as possible. We prove that this equivalence holds for arbitrary Gorenstein projective modules over any ring.