Study on Representations of Algebras and Derived Categories

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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