Topics in Local Algebra

提案由两部分组成。第一部分涉及域上有限超越度局部诺瑟环的构造技术。这种技术在过去非常强大；在交换代数中，许多（著名的）反例都涉及到这一技术。本文将探讨该构造是否提供了一种描述包含系数域k且在k上具有有限超越度的所有局部noether域的方法。作者还对这种构造在局部均匀化和Artin正则态射猜想中的应用感兴趣。第二部分是关于局部上同的问题。域上有限超越度的局部诺瑟域是交换代数和代数几何中的重要组成部分；它们最终涉及到代数方程解集的描述。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.