Research on Dutta multiplicity and Roberts ring

1. 证明了交多重性的正性问题与Dutta多重性的正性有很深的关系。复数的Dutta多重是一个有理数，是由Dutta在正特征的情况下定义的。我们在没有特征假设的情况下定义了Dutta多重性。利用复合体的Adams运算（由于Gillet-Soule），我们成功地描述了Dutta多重性，并在一些特殊情况下证明了其正性。我们还给出了不使用k理论的Dutta多重性的另一种描述。由Roberts引起的交多重性消失的证明的关键是局部环Todd类的消失。局部环的Todd类非常有趣，但是很难计算出具体的例子。在一些特殊情况下，我们成功地给出了局部环Todd类的一个公式。用它计算了一些例子。我们发现有很多例子满足Todd类的消失，我们称这种环为Roberts环。我们研究了罗伯茨环的基本性质。环的平面态射引出了Grothendieck群的图。对于一个northern局部环，我们研究了由补全引起的Grothendieck群之间的映射。更准确地说，我们研究了它什么时候是注入的。我们找到了注入性的几个充分条件。例如，如果给定的局部环是孤立奇点，则它是内射。

1. We proved that the positivity problem of intersection multiplicity had a deep relation with the positivity of Dutta multiplicity. The Dutta multiplicity of a complex is a rational number that was defined by Dutta in the case of positive characteristic. We defined the Dutta multiplicity without the assumption on the characteristic. Using Adams operation for complexes (due to Gillet-Soule), we succeeded to describe the Dutta multiplicity and prove the positivity in some special cases. We also gave another description of Dutta multiplicity without using K-theory.2. The key point of the proof of the vanishing of intersection multiplicity due to Roberts is the vanishing of Todd classes of local rings. The Todd classes of local rings are very interesting, but it is very difficult to calculate concrete examples. We succeeded to give an formula on Todd classes of local rings in some special cases. Using it, we calculate some examples. We found that there are many examples that satisfy the vanishing of Todd classes, we call such rings Roberts rings. We studied basic properties on Roberts rings.3. Flat morphisms of rings induces a map of Grothendieck groups. For a Northerian local ring, we studied a map between Grothendieck groups induced by the completion. More precisely, we studied when it is injective. We found some sufficient conditions for the injectivity. For example, it is injective if the given local ring is isolated singularity.