2006 Fiscal Year Final Research Report Summary
Study of perfect complexes over algebras and their properties
| Project/Area Number |
16540012
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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, Professor, 教育学部, 教授 (50209920)
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| Co-Investigator(Kenkyū-buntansha) |
KURANO Kazuhiko Meiji University, Department of Mathematics, Professor, 理工学部, 教授 (90205188)
TOKUHIRO Yoshimi (KITAMURA Yoshimi) Tokyo Gakugei University, Department of Mathematics, Professor, 教育学部, 教授 (00014811)
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| Project Period (FY) |
2004 – 2006
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| Keywords | Unbounded derived category / Grothendieck group / Gorenstein / Auslander condition / Tilting complex / Derived Picard group / Chow group / Hilbert-Kunz function |
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| Research Abstract |
The Grothendieck group K_0(D^b(modA)) of a bounded derived category of finitely generated modules over an Artinian ring A is a free abelian group of rank n, where n is the number of non-isomorphic simple modules. But it is not known what is the Grothendieck group of a unbounded derived category. By using the notions of compact objects and generators which are generalizations of perfect complexes, we introduce the notion of an additive T-set of classical generators, and show that The Grothendieck group K_0(D^-(modA)) (resp., K_0(D^-(modA))) of bounded above (resp., bounded below) complexes of finitely generated A-modules is isomorphic to (Z{T, T^<-1>}/(1+T))^n. As a consequence, the Grothendieck groups of D^-(modA), D^+(modA), D(modA) are trivial.
Let R be a coherent ring, and T a tilting R-module. For an R-module M which has a T-resolutibn 0->T_<n^->>...->T_<1->>T_<0->>M_->0, we show that the Auslander condition for T is equivalent to the condition that the flat dimension of an End_R(T)-module Hom_R(T, E^i (M)) is less than or equal to i+ n, where 0- >M_->E^0(M)->E^1(M)->...is an minimal injective resolution of M. We study the case of algebraic varieties which have tilting sheaves, and applying them investigate the case that derived Picard group of a finite dimensional algebra is isomorphic to the automorphism group of the derived category of modules.
We show that the Rees algebra of the second syzygy module of the residue field of a regular local ring is a Gorenstein factorial domain. We define a notion of numerical equivalence on Chow groups or Grothendieck groups of Noetherian local rings. Under a mild condition, it is proved that the Chow group modulo numerical equivalence is a finite dimensional Q-vector space. We prove that the coefficient of the second term of the Hilbert-Kunz function of e vanishes over a Q-Gorenstein ring.
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