## On the intersection multiplicity of images under an etale morphism

### Volume 75 / 1998

Colloquium Mathematicum 75 (1998), 167-174
DOI: 10.4064/cm-75-2-167-174

#### Abstract

We present a formula for the intersection multiplicity of the images of two subvarieties under an etale morphism between smooth varieties over a field k. It is a generalization of Fulton's Example 8.2.5 from [3], where a strong additional assumption has been imposed. In a special case where the base field k is algebraically closed and a proper component of the intersection is a closed point, intersection multiplicity is an invariant of etale morphisms. This corresponds with analytic geometry where intersection multiplicity is an invariant of local biholomorphisms.