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 FM(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.