On Several Homological Problems in Commutative Algebra

点击翻译按钮获取中文摘要

The main goal of this project is to study the following problems incommutative algebra: 1) The Canonical Element Conjecture of Hochster,2) Serr's Conjecture on Intersection Multiplicity and the Chow group problem,3) the study of asymptotic behavior of homologies in positive characteristic and4) the Strong Intersection Conjecture due to Peskine and Szpiro. Each of these isa longstanding open problem and occupies a central position in the areas ofcommutative algebra and algebraic geometry.Commutative algebra and algebraic geometry may be thought of as studying solutions of many equations in many unknowns when, typically, the solution is not unique. The set of solutions can then be viewed geometrically, but one can use instead encode all the pertinent information about the equations in abstract algebraic objects called commutative rings. The two points of view interact in beautiful and productive ways. A breakthrough method in commutative algebra that has had explosive development in this decade is the theory of tight closure. Part of the underlying idea is to consider the system of equations, after sutiable modification, or the ring, modulo many different prime integers. This is a powerful technique that has solved many problems, while opening up vast new areas for research.