1998 Fiscal Year Final Research Report Summary
Study on Representations of Algebras and Derived Categories
| Project/Area Number |
09640014
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Tokyo Gakugei University |
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Principal Investigator |
MIYACHI Jun-ichi Tokyo Gakugei University, Department of Mathematics, Associate Professor, 教育学部, 助教授 (50209920)
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| Co-Investigator(Kenkyū-buntansha) |
TOKUHIRO Yoshimi (KITAMUR) 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)
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| Project Period (FY) |
1997 – 1998
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| Keywords | Derived category / Chain complex / Morita duality theory / Auslander-Gorenstein ring / Dutta multiplicity / Injective module / Perfect ring / Hamiltonian |
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| Research Abstract |
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
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