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

让我们对12640051项目期间给出的主要结果进行调查。(1)：给出了无条件强d序列的定义。美元)显然看起来非常“强大”，因为“任何”力量都必须形成无条件的d序列。澄清了S.Goto分解定律与另一个序列性质（如d^*-序列）之间的关系，得到了a_1，a_2，…，当且仅当a_1^{m_1}， a_2^{m_2}，…， a_s^{m_s)对所有m_i=1,2（1≦i≦s）形成任意顺序的d序列。这使得我们对一个给定序列是否构成usd序列的判定效果很好，特别是它保证了在有限多步中完成usd序列。我们处理的是Rees模而不是Rees代数。然后给出了Rees模获得Buchsbaumness的若干充分条件。如果给定的初等理想在等i不变情况下具有最小多重性，则Rees模的正梯度子模必然是Buchsbaum，而且，如果给定模的维数大于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，Rees模是否为Buchsbaum的充要条件已经完全确定，但一般来说仍然是开放的。(3)：在上述相同情况下，光纤锥的正梯度子模为Buchsbaum存在几个等价条件。特别地，通过观察Rees模的局部上同模的齐次分量的出现，描述了一些条件。应用这一结果，也阐明了光纤锥体本身为布赫斯鲍姆的条件。(4)：证明了当n增加时，由给定理想的任意n次幂定义的相关梯度模的i不变量具有一定意义上的稳定性。还证明了当n足够大时，这种相关的梯度模的i不变量成为一个常数。最后，下面的结果是相当令人惊讶的。即证明了在与上述相同的情况下，扩展Rees模对扩展意义上的Rees模具有Buchsbaumness。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