Holomorphic Mappings and Local Algebra
This chapter aims to understand the properties of those ideals of holomorphic functions that define isolated points. By treating mappings rather than complex-valued functions as the basic object, one can recover many of the basic properties of holomorphic functions of one variable. Any injective mapping defines the origin only once. For other equidimensional finite analytic mappings, the analogue of “order of the zero” will be the generic number of inverse images of nearby points. The index or winding number of a mapping will depend only on the ideal generated by its component functions in the ring of germs of holomorphic functions. The chapter offers two other corollaries, each of which arises in the study of germs of holomorphic mappings in several variables. The idea is to use enough test functions in the last slot to obtain a system of linear equations for the rrij.