On the Betti Numbers of Certain Local Rings of Embedding Dimension 3

Let R be a commutative Noetherian local ring with maximal ideal m and residue field k and let K be the Koszul complex associated with R. The relationship between the homology algebra H(K) and the homological invariants of R such as the Betti numbers B_p=dim_k Tor^R_p(k, k) and the Betti series B(R)=ΣB_pZ^p of R has been investigated. Especially, for the local ring of embedding dimension n≦2, the Betti series of R is completely determined by the multiplicative property of the homology algebra H(K) [5, 7]. In the case when n=3 Wiebe [9] proved the rationality of B(R) under the assumption that R is a Gorenstein ring and is not complete intersection by calculating the syzygy modules of k. In this paper, mostly we assume that n=3 and we calculate the Betti numbers by using the spectral sequence associated with the Koszul complex introduced by T. H. Gulliksen and G. Levin [3]. Then, as an application of this, we give the recurrence relation between the Betti numbers and give the explicit form of Betti series under the additional assumption that H_1(K)^2=0 and H_1(K)H_2(K)=H_3(K). This gives an alternating proof of a theorem due to Wiebe above in some extended form. As a second application, we will calculate the fourth deflection ε_4 which is also an invariant of R by means of H(K) in a similar restricted case. The author wishes to express his hearty thanks to professor M. Sakuma for his kind encouragement and for his helpful suggestions. Unless otherwise specified, we shall use the similar notations and the similar terminology which appeared in [6].

公開日:2010年1月24日で登録したコンテンツは、国立情報学研究所において電子化したものです。