Homological Questions in Commutative Algebra

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The main part of the research in this proposal is concerned with a set of conjectures in Commutative Algebra, often referred to as the"`Homological Conjectures". The Principal Investigator has studied these conjectures intensively in previous years, and interest in them has recently been revived by the proof of the Direct Summand Conjecture in dimension three by Ray Heitmann. This part of the proposal will investigate properties of local cohomology modules of Noetherian rings, and in particular whether it is possible to find elements of small valuation that annihilate elements of the local cohomology modules. The proposal deals primarily with the development of new methods for rings of mixed characteristic using ideas from Arithmetic Geometry. In addition, research will be carried out work on several related questions on intersection multiplicities.Applications of Algebra to Geometry go back to the introduction of coordinate systems several centuries ago. In recent years many fundamental problems in the field of Commutative Algebra have come from the question of defining the order of tangency for geometricspaces defined by algebraic equations. This proposal investigatesseveral questions in this area, concentrating primarily on the so-called arithmetic case, which is analogous to the situation where equations are defined over the integers instead of a field such as the complex numbers.The main focus of the research in this proposal is to develop new tools to apply in this situation.