Extremal Betti Numbers of t-Spread Strongly Stable Ideals

Abstract

Let K be a field and let S = K[x1, . . . , xn] be a polynomial ring over K. We analyze the extremal Betti numbers of special squarefree monomial ideals of S known as the t-spread stronglystable ideals, where t is an integer ≥ 1. A characterization of the extremal Betti numbers of such a class of ideals is given. Moreover, we determine the structure of the t-spread strongly stable idealswith the maximal number of extremal Betti numbers when t = 2.

A numerical characterization of the extremal Betti numbers of t-spread strongly stable ideals

“…We quote the next result from [5]. Such a result gives the maximal number of corners allowed for a t-spread strongly stable ideal.…”

“…Remark 2. 5 If we put t = 0 in (1), then we obtain the well-known formula of Eliahou and Kervaire [14] for the (strongly) stable ideals; whereas, if we put t = 1 in (1), then we obtain the Aramova et al [8] formula for squarefree (strongly) stable ideals.…”

“…The next example illustrates Theorem 3.10. 6,4). The greatest monomial of A 3 (6, 4), with respect to > slex , is v = x 1 x 4 x 7 x 16 .…”

“…We have to find 10 monomials of A 2 (3, 4) not belonging to Shad 2 − 1 2 (G(I ) 1 ). In more detail, such monomials of A 2 (3,4) have to be smaller than all monomials in Shad 2 − 1 2 (G(I ) 1 ) with respect to > slex . In particular, they have to be smaller than min Shad 2 − 1 2 (G(I ) 1 ).…”

“…. , (k r , r ), under which conditions there exists a t-spread strongly stable ideal I of S = For t = 1, a positive answer can be found in [3]; whereas in [5] the maximal number r of extremal Betti numbers allowed for a t-spread strongly stable ideals, for t ≥ 2, has been computed. In this paper, we are able to give a positive answer to question ( * ) for t ≥ 2.…”