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

Chapter 1: Preliminaries 7
This approach can be applied to all Cohen-Macaulay quotient rings S = R/I, where I
is an ideal generated by homogeneous polynomials. The first step is to find a maximal
regular sequence f1, · · · , fm of S composed of homogeneous polynomials of degree 1;
here, by virtue of the Cohen-Macaulay property, m = dim S. This will produce a
0-dimensional ring ¯ S = S/(f1, · · · , fm) whose Hilbert series is the polynomial Q ¯ S. By
(1.2) and (1.3) we get that
HS(t) = Q ¯ S(t)
.
(1 − t) m
Example 1.2.1. Let f be a homogenous polynomial of R = K[x1, · · · , xn] of degree
d > 0 and I = (f). The graded free resolution of R/I is
0 −→ R(−d) 1↦→f
−−→ R −→ R/I −→ 0,
which yields β0 0, β1 d = 1, whereas the remaining βi j are zero. Hence (1.2) implies
that,
so the Hilbert series of R/I is
HR/I(t) = HR(t)(1 − t d )
= (1 − td )
(1 − t) n
= (1 + t + t2 + · · · + td−1 )
(1 − t) n−1 ,
1 + t + t 2 + · · · + t d−1
(1 − t) n−1
For more complicated ideals I, the computation requires the use of Gröbner bases;
see [45, 58] for a fruitful resource of techniques.
In [29, Chapter 6] some techniques for computing the multiplicity is provided.
Thanks to Francisco, we include them in this section.
.