Numerical equivalence on Chow groups of local rings and its applications

点击翻译按钮获取中文摘要

Serre defined the intersection number for two subvarieties of a smooth algebraic variety in an algebraic method. Many researchers tried to extend this definition for two subvarieties of an arbitrary scheme, but in 1985 Dutta, Hochster and MacLaughlin gave an example that implies that it is impossible to do so. For many years, this example has been regarded as a very special bad one. Recently, the development of algebraic K-theory has well explained why such examples exist by the research due to Levin, Roberts and Srinivas. By the project, we proved that the existence of such examples is deeply related to standard conjectures, that is the most important conjecture in the theory of algebraic cycles. Precisely speaking, we defined numerical equivalence on the Chow group of Noetherian local ring, as that on the Chow ring of smooth projective variety. We obtained a lattice if we divided the Chow group by numerical equivalence, and studied its fundamental properties.We proved a formula, that is a natural extension of a relation between the divisor class group of a normal projective variety and the divisor class group of its (normal) homogeneous coordinate ring. Furthermore, we proved that the total coordinate ring of a normal projective variety whose divisor class group is finitely generated free abelian group is factorial.We found a formula that involves the canonical class and the Frobenius direct image for a Noetherian normal local ring of positive characteristic. We proved the vanishing of the second coefficient of the Hilbert-Kunz function of Q-Gorenstein rings.