Properties of $T$-Spread Principal Borel Ideals Generated in Degree Two

Abstract

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.

Almost normally torsionfree ideals

“…Moreover, we prove that in this case, the properties normally torsionfree and almost normally torsionfree coincide. We notice that for degree 2, the behaviour of the set of associated prime ideals of the powers of a t-spread principal Borel ideal was given in [13,Theorem 1.1]. We recover here this result by a different proof; see Proposition 3.5.…”

“…Let u = x i x n be a t spread monomial in S, where t is a positive integer. In [13,Theorem 1.1], it was proved that Ass ∞ (B t (u)) = Min(B t (u)) ∪ (x 1 , x 2 , . .…”

Vector-spread monomial ideals and Eliahou-Kervaire type resolutions

“…The t-spread monomial ideals have been introduced in 2019 by Ene, Herzog and Qureshi [18]. The homological and combinatorial properties of these and some related classes of ideals are the subject of a large body of research [1,2,3,4,5,6,7,8,13,14,15,16,18,20,21,24,28,29,30].…”