Topics in Local Algebra

ROTTHAUS 98 - 01222提案由两部分组成。第一节涉及域上有限超越度局部Noether环的构造技巧。 这种技术在过去已经非常强大;交换代数中的许多（著名的）反例都涉及到这种技术。 提议者将调查是否该构造提供了一种描述所有包含系数域k并且在k上具有有限超越度的局部诺特域的方法。 该提议者也感兴趣的应用程序，这一建设的地方一致化和阿丁猜想的正规态射。 该建议的第二部分是关于局部上同调的问题。在域上具有有限超越度的局部诺特域是交换代数和代数几何中的主要组成部分;它们最终涉及代数方程的解集的描述。局部上同调是在代数几何中作为描述代数方程解集的工具而发展起来的。

ROTTHAUS98-01222The proposal consists of two parts. The first section involves a construction technique for local noetherian rings of finite transcendence degree over a field. This technique has been very powerful in the past; many(famous) counterexamples in commutative algebrahave been produced involving this technique. The proposer will investigate if the construction provides a way of describing all local Noetherian domains which contain a coefficient field k and are of finite transcendence degree over k.. The proposer is also interested in applications of this construction to local uniformization and to Artin's conjecture on regular morphism. The second part of the proposal is concerned with questions about local cohomology.Local Noetherian domains that are of finite transcendence degree over a field occur as major building blocks in commutative algebra and algebraic geometry; they are ultimately involved in the description of the solution sets of algebraic equations. Local cohomology was developed in algebraic geometry as a tool for the description of solution sets of algebraic equations.