FREE MINIMAL RESOLUTIONS AND THE BETTI NUMBERS OF THE SUSPENSION OF AN n-GON

Consider the general n-gon with vertices at the points 1,2, . . . ,n. Then its suspension involves two more vertices, say at n+1 and n+2. Let R be the polynomial ring k[x1,x2, . . . ,xn], where k is any field. Then we can associate an ideal I to our suspension in the Stanley-Reisner sense. In this paper, we find a free minimal resolution and the Betti numbers of the R-module R/I.

By definition, a free-minimal resolution of the R-module R/I is an exact sequence of the form where each M i is a free R-module with the smallest possible rank.For material on free-minimal resolutions, the reader can refer to [5] or [7].The Betti numbers B i (n) of the R-module R/I are just the ranks of those free modules M i , i.e., B i (n) = rank R (M i ) for i = 0, 1,....
In this paper, we find a free-minimal resolution and the Betti numbers of the Rmodule R/I.Sometimes we simply refer to them as a free-minimal resolution and the Betti numbers of the suspension of the n-gon.
2. Some useful results.In this section, we recall some results on free-minimal resolutions and the Betti numbers of the n-gon.These results are needed to obtain the theorems on the suspension of the n-gon.The proofs of most of these theorems can be found in [1] or [2].
(1) Let ∆ be the finite abstract simplicial complex corresponding to the n-gon with vertices at the points 1, 2,...,n.Let S = k[x 1 ,...,x n ] and J 1 be the Stanley-Reisner ideal associated to ∆.Then, it easily follows that J 1 = (x 1 x 3 ,x 1 x 4 ,...,x 2 x 4 ,...,x 2 x n ,..., x n−2 x n ) for n > 3, and (2) Let β i (n) denote the ith Betti number of the S-module S/J 1 .In other words, it is the ith Betti number of the n-gon.Then, for n ≥ 3, (3) We can show that, is a free-minimal resolution of the S-module S/J 1 .Even though we do not need the specific definitions of the maps f j for what follows, the inquisitive reader can find them in [1].

Main results.
Let J 1 be the ideal in the polynomial ring S = k[x 1 ,...,x n ] as in Section 2. Let J be the ideal in the polynomial ring R = k[x 1 ,...,x n ,x n+1 ,x n+2 ] generated by the same generators as that of J 1 .
Tensor the exact sequence (2.2) with the k-module k[x n+1 ,x n+2 ], which is a free module.Hence we obtain the following exact sequence of R-modules.
where d i are the same as the maps f i ⊗ id.This means that the following complex is exact at all places except at degree 0: Consider the following diagram where the two rows are the same as the complex (3.2) and the vertical maps are multiplication by the element y = x n+1 x n+2 of R.
Hence, even though J is not a prime ideal of R, by considering the primary decomposition of J, one can easily obtain that q ∈ J. Therefore, the exact sequence (3.1) gives us q = d 1 (q 1 ) for some Hence, p = d 2 (p 1 ) + yq 1 .Now we have two equations d 2 (p 1 ) + yq 1 = p, and d 1 (q 1 ) = q where (p 1 ,q 1 ) ∈ D 2 ⊕ D 1 .This yields ∂ 2 (p 1 ,q 1 ) = (p, q) and hence Ker Finally, for i = 0, we have However, the exact sequence (3.1) implies that d 1 (D 1 ) = J and hence im ∂ 1 = {j −yq | j ∈ J, q ∈ R} = J +(y) = I.Therefore the homology of the complex (3.4) at the zeroth spot is equal to R/I.Theorem 3.2 says more about the free resolution (3.5).

Theorem 3.2. The sequence (3.5) is a free-minimal resolution of the R-module R/I.
Proof.To show the minimality, it is enough to show that the maps [4, p. 136]).However, this is an easy consequence of commutativity of the following diagram and the minimality of (2.2): ( Proof.Let n ≥ 3 be a positive integer.Since (3.5) is a free-minimal resolution, the Betti numbers of R/I are just the respective ranks of the free modules appearing in (3.5).Hence, we obtain, for n ≥ 3, Let us denote B i (n) by B i .The theorem is clear for n = 3. Therefore assume that n > 3.So and by using formula (2.1).A similar calculation shows that B n−2 = n−1

( 3 . 6 )Theorem 3 . 3 .
Theorem 3.3 enables us to calculate the Betti numbers B i (n) of the suspension of the n-gon.Let n ≥ 3 be a positive integer.Then the ith Betti number B i (n) of the suspension of the n-gon is given by