Topological and Cohomological Structure of Zero-Dimensional Mappings

A modern interpretation of the author's results on the theory of zero-dimensional mappings is given, special attention being paid to its relations to commutative algebra, algebraic geometry, function algebras, and sheaf theory. These results are classified under three fields: representation of zero-dimensional mappings as a limit of maximally simple finite-to-point mappings, algebraic characterization of zero-dimensional mappings and its applications, and resolutions of sheaves related to a zero-dimensional mapping and their applications. The first part of the paper is based on the properties of the limit of local systems of spaces, which is a parametric generalization of the inverse limit of spaces. The second part rests on the fact that, from the point of view of the rings of continuous functions, zero-dimensional mappings are topological analogues of the integral closure of rings. The third part is devoted to the cohomological structure of zero-dimensional mappings. Here, the main idea is establishing a relation between the limit of a local system of simple finite-to-point mappings and two classical functors in sheaf theory, the direct and inverse images of sheaves. This relation leads to new resolutions of sheaves and new spectral sequences for the case of zero-dimensional mappings. Applications concern a characterization of the rings of continuous functions of the Menger universal compacta and the dimension-raising zero-dimensional mappings, which is one of the favorite areas of investigation of L. V. Keldysh.