Multiplicity theory

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]

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