Geometric regularity of powers of two-dimensional squarefree monomial ideals

The $a_0$-invariants of powers of a two-dimensional squarefree monomial ideal

“…We claim that order I (x i u) = n for every i ∈ [s], and then a is a 0-th critical vector of I n . In fact, if i ∈ [6], say i = 1, then…”

“…Here an ideal is pure if all its minimal ideals have the same height. The algebraic properties of (symbolic) powers of the Stanley-Reisner ideal I ∆ of ∆ were studied in [5,6,7]. It was in [5] that the regularity of S/I (n) ∆ is computed and in [6] the geometric regularity of S/I n ∆ was obtained and the following formula was given: g-reg(S/I n ∆ ) = reg(S/I…”

“…Regarding a 0 -invariants, we know well that a 0 (S/I (n) ∆ ) always vanishes, but the computing of a 0 (S/I n ∆ ) seems very difficult. This is why the geometric regularity instead of the regularity of S/I n ∆ was obtained in [6]. In [7], although the authors presented the regularity of S/I n ∆ , they didn't give a 0 (S/I n ∆ ) explicitly.…”

Regularity of powers of Stanley–Reisner ideals of one‐dimensional simplicial complexes

“…Nonetheless, for a given simplicial complex Δ, studying the sequence $$\lbrace \operatorname{reg}I_\Delta ^s \mid s \ge 1\rbrace$$ is a challenging problem. When $$\dim \Delta = 1$$, Hoa and Trung [7] computed $$\operatorname{reg}I_\Delta ^{(s)}$$ for all s , while Lu [9] computed the geometric regularity of $$I_\Delta ^s$$. Note that $$a_0(I_\Delta ^{(s)}) = -\infty$$, while Lu did not compute $$a_0(I_\Delta ^s)$$ (see Section 2.3 for the definition of the $$a_i$$‐invariants).…”

“…Note that $$a_0(I_\Delta ^{(s)}) = -\infty$$, while Lu did not compute $$a_0(I_\Delta ^s)$$ (see Section 2.3 for the definition of the $$a_i$$‐invariants). In this paper, we compute the regularity of all intermediate ideals lying between $$I_\Delta ^s$$ and $$I_\Delta ^{(s)}$$ extending work of Hoa and Trung [7] and Lu [9]. More precisely, for monomial ideals $$J \subseteq K$$, we define $$\operatorname{Inter}(J,K)$$ the set of monomial ideals L such that $$L = J + (f_1, \ldots , f_t)$$ where $$f_i$$ are among minimal monomial generators of K .…”