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

Chapter 2: Linear resolution of powers of an ideal 26
follows. Given the first order syzygies of M,
0 −→ L −→ R n −→ M −→ 0,
S(M) = S(R n )/(L) = R[t1, · · · , tn]/(L),
where L = (a1t1 +· · ·+amtm : a1f1 +· · ·+amfm = 0). Now take a = (a1, · · · , am) ∈ T
such that a1f1 + · · · + amfm = 0. In other words, a1t1 + · · · + amtm ∈ Kerϕ = P. So
looking only at t-degrees we get
P1
as R-module
∼ = L = First syzygy of (f1, · · · , fm). (2.6)
In addition for an ideal I of R we have the following short exact sequence:
S(R n ) = R[x1, · · · , xn] ϕ1
−→ S(I) −→ R(I),
where ϕ 1 is the natural surjective map induced from a free presentation R m ρ −→ I. So
ϕ 1 ( aiti) = aifi. Thus by (2.6) we actually have Kerϕ 1 = P1.
The difficulty in finding the equations is partly governed by the notion of relation
type.
Definition 2.2.2. The ideal I is said to be of linear type if P = P1. More generally,
I is said to be of relation type r if P can be generated by forms of degree ≤ r.
Hence; I is linear type ⇐⇒ Kerϕ 1 = Kerϕ ⇐⇒ S(I) ∼ = R(I).
We remark that any ideal which is generated by a regular sequence is of linear
type. The same is also true for ideals generated by a d-sequence; see [38, 66] for the
source of information.
Definition 2.2.3. Let x = {x1. · · · , xn} be a sequence of elements in a ring R generat-
ing the ideal I. Then x is called a d-sequence if (x1, · · · , xi) : xi+1xk = (x1, · · · , xi) : xk
for i = 0, · · · , n − 1 and k ≥ i + 1.