Project/Area Number  09640014 
Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Algebra

Research Institution  Tokyo Gakugei University 
Principal Investigator 
MIYACHI Junichi Tokyo Gakugei University, Department of Mathematics, Associate Professor, 教育学部, 助教授 (50209920)

CoInvestigator(Kenkyūbuntansha) 
TOKUHIRO Yoshimi Tokyo Gakugei University, Department of Mathematics, Professor, 教育学部, 教授 (00014811)
HOSHINO Mitsuo Tsukuba University, Department of Mahtematics, Lecturer, 数学系, 講師 (90181495)
KURANO Kazuhiko Tokyo Metropolitan University, Department of Mathematics, Associate Professor, 理学部, 助教授 (90205188)
廣川 真男 東京学芸大学, 教育学部, 講師 (70282788)

Project Period (FY) 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥3,000,000 (Direct Cost : ¥3,000,000)
Fiscal Year 1998 : ¥1,000,000 (Direct Cost : ¥1,000,000)
Fiscal Year 1997 : ¥2,000,000 (Direct Cost : ¥2,000,000)

Keywords  Derived category / Chain complex / Morita duality theory / AuslanderGorenstein ring / Dutta multiplicity / Injective module / Perfect ring / Hamiltonian / Chern character / 双対鎖複体 / CohenMacaulay環 / blowup / 量子 
Research Abstract 
We define cotilting bimodule complexes, and develop the derived duality theory to deal with case of noncommutative 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 AuslanderGorenstein 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 AuslanderGorenstein, then $E^n(M)$ has essential socle. Moreover, We have the following related results : 1) e give a condition that a blowup whose center is an equimultiple ideal is a macaulayfication. And
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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 selfinfectivity 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 spinboson 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
