Constructions of Tilting Complexes and Relative Projective Covers in Representation Theory

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In the research of this project, we studied theory of the relative projective covers of modules over group algebras to apply it for construction of tilting complexes over group algebras. And we obtained some good results as we shall describe in the following."Relative projective cover" is a generalization of usual projective cover. For the group SL(2,q), investigating structures of projective indecomposable modules in detail, we completed to check abelian defect conjecture by Broue for this group over the field of defining characteritic. We already decided unknown parameters in decomposition numbers of the group SU(3,q^2) over the field of odd characteritic deviding q^<+1>. Kunugi-Waki used this result to check the conjecture by Broue for this group. A similar situation as SU(3,q^2) occurs in the group Sp(4,q). Improving their method we could checked the conjecture for Sp(4,q). In the investigation, relative projective covers with respect to its parabolic subgroups were useful. We continued the investigation to the group G_2(q) which is closely related to SU(3,q^2) and obtained some interesting complexes, but could not check the conjecture.Using relative projective covers with respect to a family of modules, we obtained some results which can be used to investigate cohomology algebras of finite groups of small rank. And we could give a partial answer to the problem by Carlson on the existence of quasi regular sequences in cohomology algebras of finite groups.