The basic theory of USD-sequences and the ring theoretical structure of Rees algebras

让我们对我们在这个项目12640051中给出的结果的主要部分进行一个综述。(1)我们对无条件强d-序列的定义（焦v.USD）显然看起来非常“强”，因为“任何”幂必须形成无条件d-序列。阐明了S.后藤的分解律与另一个序列性质（如d^*-序列）之间的关系，证明了a_1，a_2，. a_s形成USD序列当且仅当a_1^{m_1}，a_2^{m_2}，.，a_s^（m_s）对于所有m_i= 1，2（1 i s）形成任意顺序的d序列。这对判断一个序列是否构成USD序列有很好的效果，特别是保证了在一定的步骤内就能构成USD序列。(2)：我们处理的是Rees模而不是Rees代数。然后给出了Rees模获得Buchsbaumness的某个充分条件。若给定准素理想在等I-不变情形下具有极小重数，则Rees模的正分次子模必为Buchsbaum，且 ...更多信息 如果一个给定的模的维数大于1，则Rees模本身也是Buchsbaum。特别地，在给定模的维数等于1的情况下，完全确定了Rees模是Buchsbaum模的充要条件，但它在一般情况下仍然是开的。(3)在上述相同的情况下，纤维锥的正分次子模为Buchsbaum的几个等价条件。特别地，通过观察Rees模的局部上同调模的齐次分量的出现来描述某些条件。应用这一结果，还阐明了纤维锥本身是布赫斯鲍姆的条件。(4)证明了当n增加时，由理想的任意n次幂所定义的分次模的I-不变量在某种意义下具有稳定性.当n充分大时，这种分次模的I-不变量成为常数. (5)最后，下面的结果是相当令人惊讶的。也就是说，在与上述相同的情形下，证明了扩展的Rees模在扩展意义上比Rees模具有Buchsbaumness性.少

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 … More 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