The Hilbert-Samuel function of a nonzero finitely generated module 
over a commutative Noetherian local ring 
with maximal ideal 
, with respect to an 
-primary ideal 
, is the map 
such that, for all 
,
 |
where 
denotes module length. It is related to the Hilbert
function of the associated graded module 
by the identity
 |
For sufficiently large 
, it coincides with a polynomial
in 
of degree equal to the Krull
dimension of 
.
