2005/09/13 by Terence Gaffney, Gaffney, Terence
Mathematics · #13H15 #14B05 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #math.AG #msc:13H15 #msc:14B05
paper · pdf · doi:10.48550/arxiv.math/0509285
22 pages
arxiv created 2005/09/13 · openalex publication_date 2005/09/13 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Using invariants from commutative algebra to count geometric objects is a basic idea in singularities. For example, the multiplicity of an ideal is used to count points of intersection of two analytic sets at points of non-transverse intersection. A problem with the multiplicity of an ideal or module is that it is only defined for modules and ideals of finite colength. In this paper we use pairs of modules and their multiplicities as a way around this difficulty. A key tool is the multiplicity-polar theorem which applies to families of pairs of modules, linking the multiplicity at a general pair of the family to the multiplicity of the pair at the special fiber. The specific problem we work on is: Suppose X,0 is an equidimensional complex analytic set in Cn, and T^*X(Cn) is the closure of the conormal vectors to the smooth part of X, s, a section of T^*(Cn), the cotangent bundle of Cn restricted to X. Suppose the intersection of the image of s, denoted im(s), with T^*X(Cn) is isolated. Then we calculate the intersection multiplicity im(s)⋅ T^*X(Cn) using the multiplicity of pairs. It is easy to see that if X=Cn and s(x)=df(x), where f has an isolated singularity at the origin, then this number is the Milnor number of f. Variations of this number appear in several different situations, such as, the theory of differential forms on singular spaces, the theory of D-modules, the Milnor fiber of a function with an isolated singularity, the Af stratification condition of Thom, and the theory of the Euler invariant. We discuss the application of our result to all of these situations.