1. Multiplicity of a module

2. See also

3. References

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

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 generated as an R₀-algebra and R₀ 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

where is a polynomial. By definition, the multiplicity of M is

The series may be rewritten

where r(t) is a polynomial. Note that are the coefficients of the Hilbert polynomial of M expanded in binomial coefficients. We have

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.^[1][2]

| name = Lech

See also

• Dimension theory (algebra)
• j-multiplicity
• Hilbert–Samuel multiplicity
• Hilbert–Kunz function
• Normally flat ring

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