Powers of t-spread principal Borel ideals

Abstract

We prove that t-spread principal Borel ideals are sequentially Cohen-Macaulay and study their powers. We show that these ideals possess the strong persistence property and compute their limit depth.2010 Mathematics Subject Classification. 13D02,13H10, 05E40, 13C14. 1 particular, it follows that every t-spread principal Borel ideal has linear quotients and such an ideal is Cohen-Macaulay if and only if it is t-spread Veronese.Since B t (u) is a squarefree monomial ideal, we may interpret it as the Stanley-Reisner ideal of a simplicial complex ∆. In the proof of Theorem 1.1, we characterize the facets of ∆. This gives explicitly all the generators of the ideal of the Alexander dual ∆ ∨ . We then derive that the Stanley-Reisner ideal of ∆ ∨ has linear quotients, which yields the sequential Cohen-Macaulay property of B t (u) by Alexander duality; see Theorem 1.2.In Section 2, we consider the Rees algebra R(B t (u)) of a t-spread principal Borel ideal B t (u). In Proposition 2.2, we show that B t (u) satisfies the ℓ-exchange property with respect to the sorting order < sort . This allows us to describe in Theorem 2.3 the reduced Gröbner basis of the defining ideal of R(B t (u)) with respect to a suitable monomial order. The form of the binomials in this Gröbner basis shows that B t (u) satisfies an x-condition which guarantees that all the powers of B t (u) have linear quotients. Moreover, the Rees algebra R(B t (u)) is a normal Cohen-Macaulay domain which implies in turn, by a result of [6], that B t (u) possesses the strong persistence property as considered in [6]. Consequently, B t (u) satisfies the persistence property, which means that Ass (B t Note that the associated primes of B t (u) can be read from Theorem 1.1.In Section 3, we study the limit behavior of the depth for the powers of t-spread principal Borel ideals. In Theorem 3.1 we show that if the generator u = x i 1 · · · x i d satisfies the condition i 1 ≥ t + 1, then lim k→∞ depth(S/B t (u) k ) = 0. In particular, we determine the analytic spread of B t (u), that is, the Krull dimension of the fiber ring R(B t (u))/mR(B t (u)) where m = (x 1 , . . . , x n ).

Extremal Betti Numbers of t-Spread Strongly Stable Ideals

“…If n = 13, then w 3 = x 2 x 5 x 8 x 10 x 13 is the greatest two-spread monomial of degree five not belonging to Shad 2 (B 2 (x 2 x 5 x 7 x 13 )). On the other hand, the greatest two-spread monomial of degree 6 = 13−1 2 not belonging to Shad 2…”

“…During the last years, many authors have focused their attention toward problems and questions involving such a class of ideals. Recently, Ene, Herzog, and Qureshi have introduced the notion of t-spread monomial ideal [1] (see also [2,3]), where t is a non-negative integer. More precisely, if t ≥ 0 is an integer, a monomial x i 1 x i 2 · · · x i d with 1 ≤ i 1 ≤ i 2 ≤ · · · ≤ i d ≤ n is called t-spread, if i j − i j−1 ≥ t for 2 ≤ j ≤ d. A monomial ideal in S is called a t-spread monomial ideal, if it is generated by t-spread monomials.…”

On the Associated Prime Ideals and the Depth of Powers of Squarefree Principal Borel Ideals

“…Proof. It is enough to show that m ∈ Ass(I d ), since I satisfies the persistence property, see [1,Proposition 2.4].…”

“…has a linear resolution, see[1, Proposition 2.4].Therefore, S/I is Cohen-Macaulay, see for example[14, Theorem 8.1.9]. The following results [3, Theorem 2.1 and Corollary 2.3] is also important for this paper.…”

“…This class of ideals was introduced by Hibi and the first author of this paper in [2]. In the sequel, algebraic properties of squarefree strongly stable ideals and their powers have been studied by many authors, see for example [1], [3], [11], [12] and [18].…”