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

Chapter 4: Graded minimal free resolution of ideals 62
4.3 Analysis of a special class of Stanley-Reisner
ideals
Let ∆ be the following simplicial complex which corresponds to the n-gon with
vertices at the points 1, 2, . . . , n. Clearly ∆ is a pure simplical complex (of dimension
1).
∆ = {∅, {1}, {2}, . . . , {n}, {1, 2}, {2, 3}, . . . , {n, 1}}. (4.9)
Let S = K[x1, x2, . . . , xn], and let J1 be the Stanley-Reisner ideal associated to ∆
in (4.9), i.e., J1= the ideal in S generated by all monomials of the form xi1xi2 . . . xir,
where 1 ≤ i1 < i2 < . . . < ir ≤ n and {i1, . . . , ir} /∈ ∆. Then it easily follows that for
each n ≥ 3 we get:
⎧
⎪⎨ (x1x2x3), n=3;
J1 =
⎪⎩ (x1x3, x1x4, · · · , x1xn−1, x2x4, · · · , x2xn, · · · , xn−2xn), otherwise.
(4.10)
In [2] the author showed that the ith Betti number of the S-module S/J1, denoted
by β S i (S/J1) or simply β S i , which is the ith Betti number of the n-gon, for n ≥ 3 is
given by
As well we have
β S i =
⎧
⎪⎨
⎪⎩
1, i=0,
n i(n−i−2)
i+1
n−1 , i=1,2,. . . ,n-3,
1, i=n-2,
0, otherwise.
0 → S βS n−2 fn−2
−−→ S βS n−3 → . . . → S βS 1
f1
−→ S βS 0
f0
−→ S
J1
(4.11)
→ 0 (4.12)