2004/02/18 by Tristram de Piro, de Piro, Tristram
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AG
paper · pdf · doi:10.48550/arxiv.math/0402301
34 pages
arxiv created 2004/02/18 · openalex publication_date 2004/02/18 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The purpose of this paper is to provide a new account of multiplicity for finite morphisms between smooth projective varieties. Traditionally, this has been defined using commutative algebra in terms of the length of integral ring extensions. In model theory, a different approch to multiplicity was developed by Zilber using the techniques of non-standard analysis. Here, we first reformulate Zilber's method in the language of algebraic geometry and secondly show that, in classical projective situations, the two notions essentially coincide. As a consequence, we can recover intersection theory in all characteristics from the non-standard method and sketch further developments in connection with etale cohomology and deformation theory. The usefulness of this approach can be seen from the increasing interplay between Zariski structures and objects of non-commutative geometry.