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 
.
