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

Chapter 1: Preliminaries 3
of the same degree, is called the Graded free resolution. It is usually written in the
form
· · · −→
R(−j) βs
j −→ · · · −→ R(−j) β1
j −→
j∈Z
j∈Z
j∈Z
R(−j) β0 j −→ M −→ 0,
(1.1)
where R(−j) indicates the ring R with the shifted graduation such that, for all a ∈ Z,
(R(−j))a = Ra−j.
For all nonnegative integers i and all integers j, βi j is the number of copies of R(−j)
appearing in the ith module of the resolution, and is called a graded Betti number,
i.e., Fi requires βi j minimal generators of degree j. The ordinary ith Betti number is
βi =
βi j. Furthermore, if R is a polynomial ring over a field K we have βR i j(M) =
j∈Z
dimK Tor R i (M, K)j.
Example 1.1.1. Let R be the polynomial ring K[x1, x2, x3] over a field K, with the
usual graduation. Then the graded free resolution of M = R/(x 2 1, x 3 2) is
where
0 −→ R(−5) d2
d1 d0
−→ R(−2) ⊕ R(−3) −→ R −→ M −→ 0,
d0 : 1 ↦→ 1,
d1 : (1, 0) ↦→ (−x 2 1), (0, 1) ↦→ (−x 3 2),
d2 : 1 ↦→ (−x 3 2, x 2 1).
In R(−2), the constant polynomials have degree 2. It follows that −x 3 2 has degree
5. Similarly, x 2 1 has degree 5 in R(−3).