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

\mathbf_I(M).

The notion of the multiplicity of a module is a generalization of the

degree of a projective variety In mathematics, the degree of an affine or projective variety of dimension is the number of intersection points of the variety with hyperplanes in general position.In the affine case, the general-position hypothesis implies that there is no int ...

. By Serre's intersection formula, it is linked to an intersection multiplicity in the

intersection theory In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem o ...

. The main focus of the theory is to detect and measure a

singular point of an algebraic variety In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In c ...

(cf.

resolution of singularities In algebraic geometry, the problem of resolution of singularities asks whether every algebraic variety ''V'' has a resolution, a non-singular variety ''W'' with a proper birational map ''W''→''V''. For varieties over fields of characteri ...

). Because of this aspect,

valuation theory In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of size or multiplicity of elements of the field. It generalizes to commutative algebra the notion of size ...

Rees algebra In commutative algebra, the Rees algebra of an ideal ''I'' in a commutative ring ''R'' is defined to be R t\bigoplus_^ I^n t^n\subseteq R The extended Rees algebra of ''I'' (which some authors refer to as the Rees algebra of ''I'') is defined asR ...

s and

integral closure In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over ''A'', a subring of ''B'', if there are ''n'' ≥ 1 and ''a'j'' in ''A'' such that :b^n + a_ b^ + \cdots + a_1 b + a_0 = 0. That is to say, ''b'' ...

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 ''R''₀-algebra and ''R''₀ is Artinian. Note that ''R'' has finite

Krull dimension In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generall ...

''d''. Let ''M'' be a finitely generated ''R''-module and ''F''_''M''(''t'') its

Hilbert–Poincaré series In mathematics, and in particular in the field of algebra, a Hilbert–Poincaré series (also known under the name Hilbert series), named after David Hilbert and Henri Poincaré, is an adaptation of the notion of dimension to the context of gra ...

. This series is a rational function of the form :

\frac,

where

P(t)

is a polynomial. By definition, the multiplicity of ''M'' is :

\mathbf(M) = P(1).

The series may be rewritten :

F(t) = \sum_1^d  + r(t).

where ''r''(''t'') is a polynomial. Note that

a_

are the coefficients of the Hilbert polynomial of ''M'' expanded in binomial coefficients. We have :

\mathbf(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.

References

{{reflist Theorems in ring theory

Multiplicity of a module

See also

References