Multiplicity theory

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

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

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.

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

${\frac {P(t)}{(1-t)^{d}}},$

where $P(t)$ is a polynomial. By definition, the multiplicity of M is

$\mathbf {e} (M)=P(1).$

The series may be rewritten

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

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 have

$\mathbf {e} (M)=a_{0}.$

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.

Lech—Suppose R is local with maximal ideal ${\mathfrak {m}}$. If an I is ${\mathfrak {m}}$-primary ideal, then

$e(I)\leq d!\deg(R)\lambda (R/{\overline {I}}).$

See also

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