Module multiplicity is a number associated with a nonzero finitely generated nonnegatively graded module 
over a standard graded
ring 
. Here 
is a finitely generated algebra over a field 
, with 
, generated by elements of
degree 1. 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/Inline10.svg)
over the field 
, the multiplicity of the quotient
ring 
, where 
is a homogeneous polynomial
of total degree 
, is equal to 
. This example shows the geometric origin of the
notion. The number 
is the degree of the hypersurface scheme 
defined by 
, whose coordinate ring
is 
:
a line of 
chosen in a sufficiently general way intersects 
in 
points over the algebraic
closure of 
, counted with multiplicity. If
is not squarefree, 
has nilpotent elements and is not the coordinate ring
of the corresponding reduced algebraic variety.
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 
.
