Numerical equivalence on Chow groups of local rings and its applications

Serre用代数方法定义了光滑代数簇的两个子簇的交数。许多研究人员试图将这个定义扩展到任意方案的两个亚种，但在1985年，Duta，Hochster和MacLaughlin给出了一个例子，暗示这样做是不可能的。多年来，这个例子一直被认为是一个非常特别的坏例子。最近，代数K-理论的发展很好地解释了为什么Levin，Roberts和Srinivas等人的研究中存在这样的例子。通过这个项目，我们证明了这些例子的存在与标准猜想密切相关，标准猜想是代数圈理论中最重要的猜想。确切地说，我们定义了Notherian局部环的Chow群与光滑射影簇的Chow环上的数值等价。研究了Chow群的基本性质，证明了正规射影簇的因子类群与其(正规)齐次坐标环的因子类群之间关系的一个自然推广公式.进一步证明了因子类群为有限生成自由阿贝尔群的正规射影簇的全坐标环是阶乘的，给出了具有正特征的Noether正规局部环的典范类和Frobenius有向映象的公式.证明了q-Gorenstein环的Hilbert-kunz函数的第二系数为零。

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.