| Co-Investigator(Kenkyū-buntansha) |
KAWASAKI Takesi Tokyo Metropolitan University, Department of Mathematics, Assistant, 大学院・理学研究科, 助手 (40301410)
NISHIDA Koji Chiba University, Graduate school of Science and Technology, Associate Professor, 大学院・自然科学研究科, 助教授 (60228187)
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| Research Abstract |
Let us make a survey of the main part of our results given in the term of this project 12640051.
(1):Our definition of an unconditioned strong d-sequence(abbrev.USD) apparently seems very "strong", because "any" powers must form an unconditioned d-sequence. Clarifying the relation between the decomposition law by S.Goto and another sequence property, say a d^*-sequence, it is shown that a_1,a_2,..., a_s forms a USD-sequence if and only if a_1^{m_1}, a_2^{m_2},..., a_s^{m_s) forms a d-sequence in any order for all m_i=1,2(1≦i≦s). This lead us good effects about the criterion whether a given sequence forms a USD-sequence, especially this guarantees to do it in finitely many steps.
(2):We deal with the Rees modules instead of Rees algebras. Then we shown certain sufficient condition for Rees modules to obtain the Buchsbaumness. If a given primary ideal is of minimal multiplicity in the equi-I-invariant case, then the positively graded submodule of Rees module must be Buchsbaum, moreover tha
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t the Rees module itself is also Buchsbaum if the dimension of a given module is greater than one. In particular, in the case where the dimension of a given module is equal to one, it is completely determined the necessary and sufficient condition whether Rees module is Buchsbaum, but it is still open in general.
(3):Under the same situation as above, there are several equivalent conditions for the positively graded submodule of the fiber cone to be Buchsbaum. In particular, some conditions are described by looking at the appearence of homogeneous components of local cohomology modules of Rees modules. Applying this result, it is also clarified the condition for the fiber cone itself to be Buchsbaum.
(4):we prove that the I-invariant of the associated graded module defining by any n-th power of a given ideal has the stability in some sense when n is increasing. It is also shown that the I-invariant of such the associated graded module becomes a constat if n is sufficiently large.
(5):Finally, the following result is quite surprising. Namely, it is shown that the extended Rees module has the Buchsbaumness over the Rees module(in extended sense), under the same situation as above. Less
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