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.

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

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

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

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

The series may be rewritten

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

where r(t) is a polynomial. Note that ${\displaystyle a_{d-i}}$ are the coefficients of the Hilbert polynomial of M expanded in binomial coefficients. We have

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

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.[1][2]

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

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

See also

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

References

1. Vasconcelos, Wolmer (2006-03-30). Integral Closure: Rees Algebras, Multiplicities, Algorithms. Springer Science & Business Media. p. 129. ISBN 9783540265030.
2. Lech, C. (1960). "Note on multiplicity of ideals". Arkiv för Matematik. 4: 63–86. doi:10.1007/BF02591323.