
C.2 Hilbert function
Let M
be a graded module over with
respect to weights .
The Hilbert function of , , is defined (on the integers) by
The HilbertPoincare series of is the power series
It turns out that
can be written in two useful ways
for weights :
where and are polynomials in .
is called the first Hilbert series,
and the second Hilbert series.
If
, and ,
then
(the Hilbert polynomial) for .
Generalizing this to quasihomogeneous modules we get
where is a polynomial in .
is called the first (weighted) Hilbert series of M.
