Intersection Theory and Commutative Algebra

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Around forty years ago Serre proposed an algebraic definition of intersection multiplicities which satisfied many of the properties required of such a definition but left several unanswered questions. These problems concern, among other things, whether the intersection multiplicities are always greater than or equal to zero (nonnegativity) and precise conditions for them to vanish (vanishing) or to be greater than zero (positivity). These problems have become known as Serre's conjectures for intersection multiplicities. This proposal has two parts. The first part is a continuation of research of the principal investigator on the positivity conjecture that uses recent advances on the resolution of singularities which have led to a solution of the nonnegativity conjecture. This investigation will also study relations between these ideas and other questions on multiplicities in commutative algebra. The second part concerns modules of finite projective dimension over nonregular rings. It will use a recent result of the principal investigator and V. Srinivas to study intersection properties of modules of finite length and finite projective dimension and extend these results to study the vanishing conjecture for two modules of finite projective dimension.In this proposal the principal investigator studies some fundamental questions in the relations between Algebra and Geometry. Geometric sets are often defined as sets of solutions to polynomial equations. In studying the behaviour of these solutions, one has to define multiplicities, which give the number of times a solution should be counted. This concept generalizes the multiplicity of a root of a polynomial, which is crucial to most applications of polynomials. The investigation of these ideas has led to several fundamental questions in Algebra. One of these questions is the problem of determining when these multiplicities are positive. Another group of questions concerns modules of finite projective dimension, which are modules which can be described by a finite resolution; the finiteness of this description is used, for example, in computer algebra for computing invariants. The principal investigator will study properties of these modules and their relation to questions on multiplicities as well as to other branches of Algebra.