Mathematical Sciences: Topics in Commutative Algebra

该项目涉及三个不同的领域。第一部分涉及与局部诺瑟环的构造有关的各种主题。主要研究者将考虑形式纤维和双伸展，关于低维形式纤维环的问题，以及局部优秀环的局部态射的闭合运算。第二部分研究环上具有近似性质的除数类群问题，第三部分研究局部上同的局部-整体湮灭问题。本研究涉及交换代数和代数几何中的若干问题。代数几何研究多项式方程族的解。人们既可以研究解集的几何，也可以通过研究解集上的某些函数来代数地解决问题，这些函数形成了所谓的交换环。这种对偶视角在交换代数和代数几何之间建立了紧密的联系。

This project addresses three different areas. The first is concerned with various topics in connection with the construction of local noetherian rings. The principal investigator will consider formal fibers and birational extensions, questions concerning rings with low dimensional formal fibers, and a closure operation for local morphisms of local excellent rings. The second part deals with a problem on divisor class groups in rings with the approximation property while the third part is concerned with a question on local-global annihilation of local cohomology. This research is concerned with a number of questions in commutative algebra and algebraic geometry. Algebraic geometry studies solutions of families of polynomial equations. One can either study the geometry of the solution set or approach problems algebraically by investigating certain functions on the solution set that form what is called a commutative ring. This dual perspective creates a close connection between commutative algebra and algebraic geometry.