Topics in Local Algebra

该提案由两部分组成。第一节讨论了域上有限超越度的局部Noether环的构造方法。这一技巧在过去一直非常强大；在交换代数中已经产生了许多涉及这一技巧的(著名的)反例。作者还对这种结构在局部一致化和Artin关于正则态射的猜想中的应用感兴趣。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Notherian域是交换代数和代数几何中的主要构件，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发展起来的，作为描述代数方程解集的工具。

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.