2003 Fiscal Year Final Research Report Summary
Study of Derived Categories and Chain complexes
| Project/Area Number |
14540011
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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 |
MIYAUCHI Jun-ichi Tokyo Gakugei University, Department of Mathematics, Associate, 教育学部, 助教授 (50209920)
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| Co-Investigator(Kenkyū-buntansha) |
TOKUHIRO Yoshimi (KITAMURA Yoshimi) Tokyo Gakugei University, Department of Mathematics, Professor, 教育学部, 教授 (00014811)
KURANO Kazuhiko Meiji University, Department of Mathematics, Professor, 理工学部, 教授 (90205188)
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| Project Period (FY) |
2002 – 2003
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| Keywords | Derived category / Chain complex / Tilting complex / Recollement / Symmetric algebra / Grothendieck group / Grassmanian / Quasi-Frobenius extension |
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| Research Abstract |
For an idempotent e of an algebra A, we study the structure of a recollement {D_<A/AeA>(A), D(A), D(eAe)} and the distinguished triangle ξ(e) which induduce the recollement. We give equivalent conditions that a tilting complex induces an equivalence between recollements {D_<A/AeA>(A), D(A), D(eAe)} and{D_<B/BfB>(B),D(fBf)}.We call this tilting complex a recollement tilting complex. We study constructions of tilting complexes for a symmetric algebra A over a field. First, we construct a family of recollement tilting complexes {T_n}_<n≧0> which induce an equivalence between recollements {D_<A/AeA>(A), D(A), D(eAe)} and{D_<B/BfB>(B),D(fBf)} from a partial tilting complex. As applications, we show that a partial tilting complex P of length 2 is a tilting complex if and only if the number of indecomposable types of P is equal to the number of generators of the Grothendieck group of A. This is a complex version over symmetric algebras of Bongartz's result on classical tilting modules. The recollement tilting complex which is constructed in the above induces an Morita equivalence between A/AeA and B/BfB. Moreover, We have the following related results : We study when the induced map between Grothendieck groups of finitely generated modules of a noetherian local ring and its completion is injective. We determine when the affine cone of the Grassmanian variety is a Roberts ring. We generalize the relation between divisor class groups of normal projective variety and its total coordinate ring. Let A/B be a quasi-Frobenius extension. In the case that B is non-singular, if A is self-injective, then so is B. If B is self-injective and has a right ideal of which socle is non-zero, then A sarisfies the same condition.
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