The Hilbert-Samuel multiplicity 
of a nonzero finitely
generated module 
over a Noetherian local
ring 
is the leading coefficient of its eventual Hilbert-Samuel
function for a 
-primary ideal 
, multiplied by 
, where 
is the Krull dimension
of 
.
Equivalently,
 |
where 
denotes module length (Bruns and Herzog 1998). For
and 
,
write 
.
For the formal power series ring ![R=k[[x_1,...,x_d]]](/images/equations/Hilbert-SamuelMultiplicity/Inline13.svg)
over a field 
,
 |
so 
.
In Krull dimension 0, the eventual Hilbert-Samuel
function is the module length, and 
. This is the local form of module
multiplicity. The Lech multiplicity
conjecture compares these numbers under flat local homomorphisms.
