Study of Invariant on Derived Categories over Algebras

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First, we study Picard groups and derived Picard groups of finite dimensional hereditary algebras. We obtain general results on the structure of Picard groups and derived Picard groups, as well as explicit calculations for the Dynkin and affine quivers, and for some wild quivers with multiple arrows. In addition we prove that when A is hereditary, the derived Picard group of A coincides with the full group of Minear triangle auto-equivalences of the derived category of A. Hence we can calculate the group of non-commutative projective spaces introduced by Kontsevich-Roseriberg.Second, we show that a compact object C in a triangulated category, which satisfies suitable conditions, induces a t-structure. Third, in an abelian category we show that a complex P of small projective objects of term length two, which satisfies suitable conditions, induces a torsion theory. In the case of module categories, using a torsion theory, we give equivalent conditions for P' to be a tilting complex. Finally, in the case of artin algebras, we give a one to one correspondence between tilting complexes of term length two and torsion theories with certain conditions.Moreover, We have the following related results :1) First, we define test modules to calculate Dutta multiplicities and study the relation. Second, we characterize Roberts rings by some Galois extensions. Third, it is shown that a Chow group A.(A) of A is determined by cycles and a rational equivalence with respect to certain G-graded ideals of a Noetherian ring that graded by a finitely generated Abelian group G.Finally,we prove that the induced map G_0(A) → G_0(A) by completion is injective if A is an excellent Noetherian local ring that satisfies one of the three conditions (K. Kurano).2) It is shown that for a quasi-Frobenius extension A of a right non-singular ring B if A is a right self-injective ring, then so is B (Y. Kitamura).