Study on Representations of Algebras and Derived Categories

我们定义了可变换双模复形，并发展了非交换noether代数的推导对偶理论。我们证明了可倾双模复合物包含所有的逆射不可分解模。这种性质类似于二元复合体的残馀性。进一步给出了派生范畴的“森田对偶定理”。将上述结论应用于Gorenstein环和Auslander-Gorenstein环的推广，证明了如果$M$是左$R$-模的射维$n$等于$R$的射维，则$M$的最小射维分解中的最后一项$E^n(M)$出现在$R$的最小射维分解的最后一项中。特别地，我们得到了如果$R$是Auslander-Gorenstein，则$E^n(M)$具有本质式。此外，我们还得到了以下相关结果：1)我们给出了中心为等多重理想的爆破是一个macaulayation的条件。给出了一个关于交多重的Serre猜想的等价条件，并在此基础上研究了素理想的符号幂。我们发现了Adams运算与局部陈特征之间的关系，并证明了特征0 （K.Kurano）上Dutta多重性的正性。2)利用Lambek扭转理论导出的商范畴，给出了环的自侵染性的表征。并且，利用Morita对偶理论，我们发现了一个射光不可分解模是内射的条件。（M.Hoshino） 3)我们处理一个量子谐振子与任意具有无限多自由度的玻色子类系统的热平衡态。利用自旋玻色子哈密顿量的基态能量$E_{SB}$的表达式，我们给出了关于参数$G\in[- 1, \, 0]$的一个充分必要条件，使得含有$G$的公式达到$E_{SB}$ （M.Hirokawa， 1998年9月以前是本研究的研究者）。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