Integral closure of small powers of edge ideals and their regularity

Abstract

Let $I(G)$ be the edge ideal of a simple graph $G$ over a field k. We prove that $$\mathrm{reg}(\overline{I(G{)}^s})=\mathrm{reg}(I(G{)}^s),$$ for all $s\le 4$. Furthermore, we provide an example of a graph $G$ such that $$\mathrm{reg}I(G{)}^s=\mathrm{reg}\overline{I(G{)}^s}=\mathrm{reg}I(G{)}^{(s)}=\{\begin{array}{ll}5+2s& \text{ if char k}=2\\ 4+2s& \text{ if char k}\ne 2,\end{array}$$ for all $s\ge 1.$

22 pages. arXiv admin note: text overlap with arXiv:2109.06396, arXiv:2109.05242. Add a section about integral closure of powers of Stanley-Reisner ideals of one-dimensional simplicial complexes

References in corpus (1)

• Comparision between regularity of small symbolic powers and ordinary powers of an edge ideal