In abstract algebra, multiplicity theory concerns the multiplicity of a module ''M'' at an
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
''I'' (often a maximal ideal)
:
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 mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple (more than 2) curves, and accounting properly for ta ...
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 cas ...
(cf.
resolution of singularities). 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 i ...
,
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 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 generally th ...
''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 grade ...
. This series is a rational function of the form
:
where
is a polynomial. By definition, the multiplicity of ''M'' is
:
The series may be rewritten
:
where ''r''(''t'') is a polynomial. Note that
are the coefficients of the Hilbert polynomial of ''M'' expanded in binomial coefficients. We have
:
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.
See also
*
Dimension theory (algebra)
In mathematics, dimension theory is the study in terms of commutative algebra of the notion dimension of an algebraic variety (and by extension that of a scheme). The need of a ''theory'' for such an apparently simple notion results from the existe ...
*
j-multiplicity
In algebra, a j-multiplicity is a generalization of a Hilbert–Samuel multiplicity. For ''m''-primary ideals, the two notions coincide.
Definition
Let (R, \mathfrak) be a local Noetherian ring of Krull dimension d > 0. Then the j-multiplicity ...
*
Hilbert–Samuel multiplicity
*
Hilbert–Kunz function
*
Normally flat ring
References
{{reflist
Theorems in ring theory