Betti numbers of modules over Noetherian rings with ... - IPM

Chapter 1: Preliminaries 6
i < 0, then QM(t) is an ordinary polynomial with integer coefficients in the variable
t. If moreover d = 0, then HM(t) = QM(t), i.e., the Hilbert series is a polynomial.
If M has a finite graded free resolution
· · · −→
R(−j) βs
j −→ · · · −→ R(−j) β1
j −→
then
j∈Z
j∈Z
j∈Z
R(−j) β0 j −→ M −→ 0,
HM(t) = HR(t)
(−1) i β R i j(M)t j . (1.2)
i,j
Moreover, if x1, · · · , xr is a regular sequence over M of homogeneous elements of
degree 1, then the Hilbert function of the n − r-dimensional quotient module ¯ M =
M/(x1, · · · , xr)M is
and in particular, Q ¯ M(t) = QM(t).
HM(t) ¯ = QM(t)
, (1.3)
(1 − t) n−r
These properties suggest effective methods for computing the Hilbert series of a
finitely generated graded module over the polynomial ring R = K[X1, · · · , Xn], where
K is a field.
The Hilbert series of R, which has dimension n, can be obtained by considering
the maximal regular sequence X1, · · · , Xn of R, and the Hilbert function of the
0-dimensional quotient ring ¯ R = R/(X1, · · · , Xr), which is the same as K. Now
H(K, 0) = 1, and H(K, i) = 0 for all i = 0. Hence HK(t) = 1. It follows that QR(t) is
the constant polynomial 1, so that
HR(t) =
1
.
(1 − t) n