Multiplicity theory explained

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

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.

Multiplicity of a module

Let R be a positively graded ring such that R is finitely 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

(P(t))/(( 1- t) ^{d}) ,

where

P(t) is a polynomial. By definition, the multiplicity of M is e (M) = P( 1 ).

The series may be rewritten

F(t) = \sum ^{d} _{1} {a _{d-i} \over ( 1 - t) ^{d}} + r(t).

where r(t) is a polynomial. Note that

a _{d-i} are the coefficients of the Hilbert polynomial of M expanded in binomial coefficients. We havee (M) = a _{0.}

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]}

See also

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

Notes and References

1. Book: Vasconcelos, Wolmer. Integral Closure: Rees Algebras, Multiplicities, Algorithms. 2006-03-30. Springer Science & Business Media. 9783540265030. 129. en.
2. Lech. C.. 1960. Note on multiplicity of ideals. Arkiv för Matematik. 4. 1 . 63 - 86. 10.1007/BF02591323. 1960ArM.....4...63L . free.