Study of Invariant on Derived Categories over Algebras

首先，我们研究了有限维遗传代数的Picard群及其导子Picard群。我们得到了关于Picard群和派生Picard群的结构的一般结果，以及对动态箭图、仿射箭图和一些带多箭头的野生箭图的显式计算。此外，我们还证明了当A是遗传的时，A的派生Picard群与A的派生范畴的M-线性三角自等价的满群重合，从而可以计算Kontsevich-Roseriberg引入的非交换射影空间群。其次，我们证明了三角范畴中的紧对象C满足适当的条件，诱导出t-结构。第三，在阿贝尔范畴中，我们证明了满足适当条件的项长为2的小射影对象的复数P导出了一个挠率理论。在模范畴的情形下，利用挠理论，给出了P‘为倾斜复形的等价条件。最后，在Artin代数的情况下，我们给出了项长为2的倾斜复形与满足一定条件的挠理论之间的一一对应关系，并得到了以下相关结果：1)首先，我们定义了用于计算duta重数的测试模，并研究了它们之间的关系。其次，我们用一些Galois扩张刻画了Roberts环。证明了A的Chow群A(A)是由圈和关于由有限生成Abelian群G分次的Notherian环的某些G-分次理想的有理等价所决定的。最后，我们证明了如果A是满足三个条件(K.Kurano)之一的优秀Notherian局部环，则完备诱导映射G_0(A)→G_0(A)是内射的。2)证明了对于右非奇异环B的拟Frobenius扩张，如果A是右自内射环，那么B(Y.Kitamura)也是。

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).