Research on Dutta multiplicity and Roberts ring

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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.