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 geometric spaces defined by algebraic equations. This proposal investigates several 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.
One of the central problems in Algebra is finding solutions to equations. A system of elements which can be added and multiplied a in ordinary arithmetic and algebra is called a ring; one example of a ring that is not just the ordinary numbers is the set of polynomials, say polynomials in two variables $x$ and $y$. A very useful property of a ring is that the only solutions to certain equations are the obvious ones. For example in the ring of polynomials in $x$ and $y$, the only solutions for $a$ and $b$ to the equation $ax=by$ are the obvious solutions $a=cy$ and $b=cx$ for some polynomial $c$. In higher dimension and for more complicated rings the equations whose solutions can be predicted in this way are more varied and can involve expressions such as determinants, and the situation is much more interesting. A ring with this property is called a Cohen-Macaulay ring after two mathematicians who studied this condition. Not every ring is Cohen-Macaulay, but a major question in Commutative Algebra is whether any ring can be embedded into a Cohen-Macaulay ring. In fact, there are numerous conjectures that can be settled if this is known to be possible. The research supported by this grant is concerned with the question of embedding rings into Cohen-Macaulay rings. One approach to this question is to try to embed the ring into finite extensions, or into an a succession of finite extensions, hoping that the result is Cohen-Macaulay. While this process is sometimes successful, in general it is not. The new methods applied here involve recent developments in Arithmetic Geometry, and they attempt to find a succession of finite extensions to make the ring almost Cohen-Macaulay, a concept which we now describe. As we have seen, a Cohen-Macaulay ring is one in which certain equations have only the obvious solutions. We say that a ring is almost Cohen-Macaulay if these equations may have other solutions, but the solutions can be multiplied by arbitrarily small elements to make them expressible in terms of the solutions we want. The definition of ``small element" is rather technical, but the idea can be seen by noting that if $x$ is an element, its cube root is small and its hundredth root is very small. If it is possible to embed rings in almost Cohen-Macaulay rings, then known methods can be applied to show that they can be embedded into rings that are actually Cohen-Macaulay. The results of this research give a method for carrying out the program and showing that it works and can be completed for large classes of rings.
