In abstract algebra, multiplicity theory concerns the multiplicity of a module "M" at an ideal "I" (often a maximal ideal)
formula_1
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.
Multiplicity of a module.
Let "R" be a positively graded ring such that "R" is finitely generated as an "R"0-algebra and "R"0 is Artinian. Note that "R" has finite Krull dimension "d". Let "M" be a finitely generated "R"-module and "F""M"("t") its Hilbert–Poincaré series. This series is a rational function of the form
formula_2
where formula_3 is a polynomial. By definition, the multiplicity of "M" is 
formula_4
The series may be rewritten 
formula_5
where "r"("t") is a polynomial. Note that formula_6 are the coefficients of the Hilbert polynomial of "M" expanded in binomial coefficients. We have
formula_7
As Hilbert–Poincaré series are additive on exact sequences, the multiplicity is additive on exact sequences of modules of the same dimension.
The following theorem, due to Christer Lech, gives a priori bounds for multiplicity.
References.
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