Comparison between regularity of small symbolic powers and ordinary powers of an edge ideal

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

“…Proof This is [11, Lemma 2.3], we include an argument here for completeness. Let $$P_1, \ldots , P_r$$ be the minimal prime ideals of I .…”

“…Proof This is [11, Lemma 2.8], we include an argument here for completeness. By [3, Proposition 2.5], $$$$\begin{equation*} \operatorname{reg}(S/I)=\max {\left\lbrace |\mathbf {a}|+|G_\mathbf {a}|+i\nobreakspace |\nobreakspace \mathbf {a}\in \operatorname{\mathbb {Z}}^n,i\ge 0,\widetilde{H}_{i-1}(\Delta _\mathbf {a}(I);K)\ne 0 \right\rbrace} .$$…”

“…Theorem 1.1 settles Conjecture A for edge ideals of graphs with independence number 2. This paper is a continuation of [11], where we analyze the degree complex Δ 𝐚 (𝐽) via the radical ideal √ 𝐽 ∶ 𝑥 𝐚 . We now describe the idea of proof of Theorem 1.1.…”

“…We noticed the rigidity of regularity for intermediate ideals $$J \in \operatorname{Inter}(I^s,I^{(s)})$$ for $$s = 2,3$$ (see [10] for more detail) from the proofs in our previous work with Nam, Phong, and Thuy [11]. Prior to this, we were not aware of any rigidity result for the regularity of ideals in non‐trivial sequences.…”

Differences in regularities of a power and its integral closure and symbolic power

“…Most prior work in this area has been on establishing asymptotic properties and upper and lower bounds on the regularities of I s , I s and I (s) , as well as on finding cases of ideals for which the regularities of these ideals agree. See Kumar and Kumar [4] and Minh, Nam, Phong, Thuy and Vu [5] for a summary of known results of cases of edge ideals for which the regularity of a power equals the regularity of the integral closure of the same power, and for the very few known classes of edge ideals for which the regularity of a power equals the regularity of that symbolic power.…”