Multiplicity theory

In abstract algebra, multiplicity theory concerns the multiplicity of a module M at an ideal I (often a maximal ideal)

${\displaystyle \mathbf {e} _{I}(M).}$

The notion of the multiplicity of a module is a generalization of the degree of a projective variety. By Serre's intersection formula, it is linked to an intersection multiplicity in the intersection theory.

The main focus of the theory is to detect and measure a singular point of an algebraic variety (cf. resolution of singularities). Because of this aspect, valuation theory, Rees algebras and integral closure are intimately connected to multiplicity theory.