Study on Representations of Algebras and Derived Categories

我们定义了余倾双模复形，并发展了导出对偶理论来处理非交换Noether代数的情况。证明了余倾双模复形包含所有可积不可分解模。这一性质类似于对偶化复形的残余性。此外，我们还给出了派生范畴的一个“Morita对偶定理”。将上述结果应用于Gorenstein环和Auslander-Gorenstein环，它们是交换Gorenstein环的推广，我们证明了如果$M$是一个左$R$-模，且它等于$R$的对合维度，则$M$的最小对合归结中的最后一项$E^n(M)$出现在$R$的最小对合归结的最后一项中.特别地，我们得到了如果$R$是Auslander-Gorenstein，则$E^n(M)$有本质解，并且得到了以下相关结果：1)给出了中心为等重理想的爆破是斑化的条件。和…给出了关于交重数的Serre猜想的一个等价条件，并研究了素数理想的符号幂。找到了亚当斯算子与局部陈特征标之间的关系，并证明了特征0(K.Kurano)中duta重数的正性。2)利用由Lambek挠理论导出的商范畴，给出了环的自感染性的一个刻画。并且，利用Morita对偶理论，我们得到了投射不可分解模是内射的条件(M.Hoshino)。3)e处理与具有无限多个自由度的某些类玻色子的任何系统处于热平衡状态的量子谐振子。利用自旋玻色子哈密顿量的基态能量E{SB}$的表达式，我们证明了[-1，\，0]$中参数$G的一个充要条件，使得一个含有$G的公式达到$E{SB}$(M.Hirokawa，他一直是这项研究的研究者，直到1998年9月)。较少

We define cotilting bimodule complexes, and develop the derived duality theory to deal with case of non-commutative Noetherian algebras. We show that cotilting bimodule complexes contain all invective indecomposable modules. This property is similar to residuality of dualizing complexes. Furthermore, we give a "Morita duality theorem" for derived categories. Applying the above to the cases of Gorenstein and Auslander-Gorenstein rings, that are generalizations of commutative Gorenstein rings, we prove that if $M$ is a left $R$-module of invective dimension $n$ which is equal to the invective dimension of $R$, then the last term $E^n(M)$ in a minimal invective resolution of $M$ appears in the Last term of a minimal invective resolution of $R$. In particular, we obtain that if $R$ is Auslander-Gorenstein, then $E^n(M)$ has essential socle.Moreover, We have the following related results :1) e give a condition that a blow-up whose center is an equi-multiple ideal is a macaulayfication. And … More we give a equivalent condition for a Serre conjecture concerning intersection multiplicities, and study symbolic powers of prime ideals with respect to the above. We find a relation between Adams operation and localized Chern character, and prove the positivity of Dutta multiplicity in characteristic 0 (K.Kurano).2) We give a characterization for self-infectivity of rings by using quotient categories which are induced from Lambek torsion theory. And, using Morita duality theory, we find a condition that a projective indecomposable module is injective (M.Hoshino).3) e treat a quantum harmonic oscillator in thermal equilibrium with any systems in certain classes of bosons with infinitely many degrees of freedom. By using the expression of the ground state energy $E_{SB}$ of the spin-boson Hamiltonian, we show a necessary and sufficient condition with respect to a parameter $G\in [- 1, \, 0]$ such that a formula with $G$ attains to $E_{SB}$ (M.Hirokawa who was an investigator of this research until September 1998). Less