
C.1 Standard bases
Definition
Let
and let be a submodule of .
Note that for r=1 this means that is an ideal in .
Denote by the submodule of generated by the leading terms
of elements of , i.e. by
.
Then
is called a standard basis of
if
generate .
A standard basis is minimal if
.
A minimal standard basis is completely reduced if
Properties
 normal form:

A function
, is called a normal
form if for any and any standard basis the following
holds: if
then does not divide
for all .
(Note that such a function is not unique).
is called a normal form of with
respect to  ideal membership:

For a standard basis of the following holds:
if and only if
.
 Hilbert function:
 Let
be a homogeneous module, then the Hilbert function
of (see below)
and the Hilbert function of the leading module
coincide, i.e.,
.
