The 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.
