Module multiplicity is a number associated with every nonzero finitely generated graded module 
over a graded ring 
for which the Hilbert series
is defined. If 
,
the Hilbert series of 
can be written in the form
 |
and the multiplicity of 
is the integer
 |
If 
is the polynomial ring ![K[X_1,...,X_n]](/images/equations/ModuleMultiplicity/Inline7.svg)
over the field 
, the multiplicity of the quotient
ring 
,
where 
is a polynomial of degree 
, is equal to 
. This example shows the geometric origin of the notion.
The number 
is in fact the so-called intersection multiplicity of the algebraic
variety 
of 
defined by the equation 
, of which 
is the coordinate ring (i.e., a line of 
chosen in a sufficiently general way intersects 
in 
distinct points).
The definition of multiplicity can be extended to nonzero finitely generated modules over a Noetherian local
ring 
.
If 
is the maximal ideal of 
, one can define the multiplicity of 
as the multiplicity of the associated
graded module of 
with respect to 
.
