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 ).

We study algebraic properties of powers of squarefree principal Borel ideals I, and show that astab(I) = dstab(I). Furthermore, the behaviour of the depth function depth S/I k is considered.

In this paper, we have studied the stability of $t$-spread principal Borel ideals in degree two. We have proved that $\Ass^\infty(I) =\Min(I)\cup \{\mathfrak{m}\}$ , where $I=B_t(u)\subset S$ is a $t$-spread Borel ideal generated in degree $2$ with $u=x_ix_n, t+1\leq i\leq n-t.$ Indeed, $I$ has the property that $\Ass(I^m)=\Ass(I)$ for all $m\geq 1$ and $i\leq t,$ in other words, $I$ is normally torsion free. Moreover, we have shown that $I$ is a set theoretic complete intersection if and only if $u=x_{n-t}x_n$. Also, we have derived some results on the vanishing of Lyubeznik numbers of these ideals.

No abstract

No abstract